src/HOL/Real/RealVector.thy
author huffman
Sun Sep 02 23:36:21 2007 +0200 (2007-09-02)
changeset 24520 40b220403257
parent 24513 f39e79334052
child 24588 ed9a1254d674
permissions -rw-r--r--
fix sgn_div_norm class
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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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lemma (in additive) setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
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apply (cases "finite A")
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apply (induct set: finite)
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apply (simp add: zero)
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apply (simp add: add)
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apply (simp add: zero)
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done
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subsection {* Real vector spaces *}
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class scaleR = type +
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  fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"
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notation
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  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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  real_scaleR_def [simp]: "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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interpretation scaleR_left: additive ["(\<lambda>a. scaleR a x::'a::real_vector)"]
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by unfold_locales (rule scaleR_left_distrib)
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interpretation scaleR_right: additive ["(\<lambda>x. scaleR a x::'a::real_vector)"]
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by unfold_locales (rule scaleR_right_distrib)
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lemmas scaleR_zero_left [simp] = scaleR_left.zero
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lemmas scaleR_zero_right [simp] = scaleR_right.zero
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lemmas scaleR_minus_left [simp] = scaleR_left.minus
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lemmas scaleR_minus_right [simp] = scaleR_right.minus
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lemmas scaleR_left_diff_distrib = scaleR_left.diff
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lemmas scaleR_right_diff_distrib = scaleR_right.diff
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lemma scaleR_eq_0_iff [simp]:
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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" by (simp add: 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" by (simp add: 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)
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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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text{*Every real algebra has characteristic zero*}
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instance real_algebra_1 < ring_char_0
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proof
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  fix m n :: nat
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  have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
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    by (simp only: of_real_eq_iff of_nat_eq_iff)
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  thus "(of_nat m = (of_nat n::'a)) = (m = n)"
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    by (simp only: of_real_of_nat_eq)
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qed
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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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   327
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   328
apply (auto simp add: Reals_def)
huffman@20584
   329
apply (rule range_eqI)
huffman@20584
   330
apply (rule of_real_inverse [symmetric])
huffman@20584
   331
done
huffman@20584
   332
huffman@20584
   333
lemma nonzero_Reals_divide:
huffman@20584
   334
  fixes a b :: "'a::real_field"
huffman@20584
   335
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   336
apply (auto simp add: Reals_def)
huffman@20584
   337
apply (rule range_eqI)
huffman@20584
   338
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   339
done
huffman@20584
   340
huffman@20584
   341
lemma Reals_divide [simp]:
huffman@20584
   342
  fixes a b :: "'a::{real_field,division_by_zero}"
huffman@20584
   343
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   344
apply (auto simp add: Reals_def)
huffman@20584
   345
apply (rule range_eqI)
huffman@20584
   346
apply (rule of_real_divide [symmetric])
huffman@20584
   347
done
huffman@20584
   348
huffman@20722
   349
lemma Reals_power [simp]:
huffman@20722
   350
  fixes a :: "'a::{real_algebra_1,recpower}"
huffman@20722
   351
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   352
apply (auto simp add: Reals_def)
huffman@20722
   353
apply (rule range_eqI)
huffman@20722
   354
apply (rule of_real_power [symmetric])
huffman@20722
   355
done
huffman@20722
   356
huffman@20554
   357
lemma Reals_cases [cases set: Reals]:
huffman@20554
   358
  assumes "q \<in> \<real>"
huffman@20554
   359
  obtains (of_real) r where "q = of_real r"
huffman@20554
   360
  unfolding Reals_def
huffman@20554
   361
proof -
huffman@20554
   362
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   363
  then obtain r where "q = of_real r" ..
huffman@20554
   364
  then show thesis ..
huffman@20554
   365
qed
huffman@20554
   366
huffman@20554
   367
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   368
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   369
  by (rule Reals_cases) auto
huffman@20554
   370
huffman@20504
   371
huffman@20504
   372
subsection {* Real normed vector spaces *}
huffman@20504
   373
huffman@22636
   374
class norm = type +
huffman@22636
   375
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   376
huffman@22636
   377
instance real :: norm
huffman@22636
   378
  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>" ..
huffman@20554
   379
huffman@24520
   380
class sgn_div_norm = scaleR + norm + sgn +
huffman@24520
   381
  assumes sgn_div_norm: "sgn x = x /# norm x"
nipkow@24506
   382
nipkow@24506
   383
axclass real_normed_vector < real_vector, sgn_div_norm
huffman@20533
   384
  norm_ge_zero [simp]: "0 \<le> norm x"
huffman@20533
   385
  norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
huffman@20533
   386
  norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
huffman@21809
   387
  norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@20504
   388
huffman@20584
   389
axclass real_normed_algebra < real_algebra, real_normed_vector
huffman@20533
   390
  norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   391
huffman@22852
   392
axclass real_normed_algebra_1 < real_algebra_1, real_normed_algebra
huffman@22852
   393
  norm_one [simp]: "norm 1 = 1"
huffman@22852
   394
huffman@22852
   395
axclass real_normed_div_algebra < real_div_algebra, real_normed_vector
huffman@20533
   396
  norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   397
huffman@20584
   398
axclass real_normed_field < real_field, real_normed_div_algebra
huffman@20584
   399
huffman@22852
   400
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   401
proof
huffman@20554
   402
  fix x y :: 'a
huffman@20554
   403
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   404
    by (simp add: norm_mult)
huffman@22852
   405
next
huffman@22852
   406
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   407
    by (rule norm_mult)
huffman@22852
   408
  thus "norm (1::'a) = 1" by simp
huffman@20554
   409
qed
huffman@20554
   410
huffman@20584
   411
instance real :: real_normed_field
huffman@22852
   412
apply (intro_classes, unfold real_norm_def real_scaleR_def)
nipkow@24506
   413
apply (simp add: real_sgn_def)
huffman@20554
   414
apply (rule abs_ge_zero)
huffman@20554
   415
apply (rule abs_eq_0)
huffman@20554
   416
apply (rule abs_triangle_ineq)
huffman@22852
   417
apply (rule abs_mult)
huffman@20554
   418
apply (rule abs_mult)
huffman@20554
   419
done
huffman@20504
   420
huffman@22852
   421
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   422
by simp
huffman@20504
   423
huffman@22852
   424
lemma zero_less_norm_iff [simp]:
huffman@22852
   425
  fixes x :: "'a::real_normed_vector"
huffman@22852
   426
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   427
by (simp add: order_less_le)
huffman@20504
   428
huffman@22852
   429
lemma norm_not_less_zero [simp]:
huffman@22852
   430
  fixes x :: "'a::real_normed_vector"
huffman@22852
   431
  shows "\<not> norm x < 0"
huffman@20828
   432
by (simp add: linorder_not_less)
huffman@20828
   433
huffman@22852
   434
lemma norm_le_zero_iff [simp]:
huffman@22852
   435
  fixes x :: "'a::real_normed_vector"
huffman@22852
   436
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   437
by (simp add: order_le_less)
huffman@20828
   438
huffman@20504
   439
lemma norm_minus_cancel [simp]:
huffman@20584
   440
  fixes x :: "'a::real_normed_vector"
huffman@20584
   441
  shows "norm (- x) = norm x"
huffman@20504
   442
proof -
huffman@21809
   443
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   444
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   445
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   446
    by (rule norm_scaleR)
huffman@20504
   447
  finally show ?thesis by simp
huffman@20504
   448
qed
huffman@20504
   449
huffman@20504
   450
lemma norm_minus_commute:
huffman@20584
   451
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   452
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   453
proof -
huffman@22898
   454
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   455
    by (rule norm_minus_cancel)
huffman@22898
   456
  thus ?thesis by simp
huffman@20504
   457
qed
huffman@20504
   458
huffman@20504
   459
lemma norm_triangle_ineq2:
huffman@20584
   460
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   461
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   462
proof -
huffman@20533
   463
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   464
    by (rule norm_triangle_ineq)
huffman@22898
   465
  thus ?thesis by simp
huffman@20504
   466
qed
huffman@20504
   467
huffman@20584
   468
lemma norm_triangle_ineq3:
huffman@20584
   469
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   470
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   471
apply (subst abs_le_iff)
huffman@20584
   472
apply auto
huffman@20584
   473
apply (rule norm_triangle_ineq2)
huffman@20584
   474
apply (subst norm_minus_commute)
huffman@20584
   475
apply (rule norm_triangle_ineq2)
huffman@20584
   476
done
huffman@20584
   477
huffman@20504
   478
lemma norm_triangle_ineq4:
huffman@20584
   479
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   480
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   481
proof -
huffman@22898
   482
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   483
    by (rule norm_triangle_ineq)
huffman@22898
   484
  thus ?thesis
huffman@22898
   485
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   486
qed
huffman@22898
   487
huffman@22898
   488
lemma norm_diff_ineq:
huffman@22898
   489
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   490
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   491
proof -
huffman@22898
   492
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   493
    by (rule norm_triangle_ineq2)
huffman@22898
   494
  thus ?thesis by simp
huffman@20504
   495
qed
huffman@20504
   496
huffman@20551
   497
lemma norm_diff_triangle_ineq:
huffman@20551
   498
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   499
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   500
proof -
huffman@20551
   501
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   502
    by (simp add: diff_minus add_ac)
huffman@20551
   503
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   504
    by (rule norm_triangle_ineq)
huffman@20551
   505
  finally show ?thesis .
huffman@20551
   506
qed
huffman@20551
   507
huffman@22857
   508
lemma abs_norm_cancel [simp]:
huffman@22857
   509
  fixes a :: "'a::real_normed_vector"
huffman@22857
   510
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   511
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   512
huffman@22880
   513
lemma norm_add_less:
huffman@22880
   514
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   515
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   516
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   517
huffman@22880
   518
lemma norm_mult_less:
huffman@22880
   519
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   520
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   521
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   522
apply (simp add: mult_strict_mono')
huffman@22880
   523
done
huffman@22880
   524
huffman@22857
   525
lemma norm_of_real [simp]:
huffman@22857
   526
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@22852
   527
unfolding of_real_def by (simp add: norm_scaleR)
huffman@20560
   528
huffman@22876
   529
lemma norm_number_of [simp]:
huffman@22876
   530
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
huffman@22876
   531
    = \<bar>number_of w\<bar>"
huffman@22876
   532
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
huffman@22876
   533
huffman@22876
   534
lemma norm_of_int [simp]:
huffman@22876
   535
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   536
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   537
huffman@22876
   538
lemma norm_of_nat [simp]:
huffman@22876
   539
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   540
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   541
apply (subst norm_of_real, simp)
huffman@22876
   542
done
huffman@22876
   543
huffman@20504
   544
lemma nonzero_norm_inverse:
huffman@20504
   545
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   546
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   547
apply (rule inverse_unique [symmetric])
huffman@20504
   548
apply (simp add: norm_mult [symmetric])
huffman@20504
   549
done
huffman@20504
   550
huffman@20504
   551
lemma norm_inverse:
huffman@20504
   552
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
huffman@20533
   553
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   554
apply (case_tac "a = 0", simp)
huffman@20504
   555
apply (erule nonzero_norm_inverse)
huffman@20504
   556
done
huffman@20504
   557
huffman@20584
   558
lemma nonzero_norm_divide:
huffman@20584
   559
  fixes a b :: "'a::real_normed_field"
huffman@20584
   560
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   561
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   562
huffman@20584
   563
lemma norm_divide:
huffman@20584
   564
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
huffman@20584
   565
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   566
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   567
huffman@22852
   568
lemma norm_power_ineq:
huffman@22852
   569
  fixes x :: "'a::{real_normed_algebra_1,recpower}"
huffman@22852
   570
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   571
proof (induct n)
huffman@22852
   572
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   573
next
huffman@22852
   574
  case (Suc n)
huffman@22852
   575
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   576
    by (rule norm_mult_ineq)
huffman@22852
   577
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   578
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   579
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@22852
   580
    by (simp add: power_Suc)
huffman@22852
   581
qed
huffman@22852
   582
huffman@20684
   583
lemma norm_power:
huffman@20684
   584
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
huffman@20684
   585
  shows "norm (x ^ n) = norm x ^ n"
wenzelm@20772
   586
by (induct n) (simp_all add: power_Suc norm_mult)
huffman@20684
   587
huffman@22442
   588
huffman@22972
   589
subsection {* Sign function *}
huffman@22972
   590
nipkow@24506
   591
lemma norm_sgn:
nipkow@24506
   592
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
nipkow@24506
   593
by (simp add: sgn_div_norm norm_scaleR)
huffman@22972
   594
nipkow@24506
   595
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   596
by (simp add: sgn_div_norm)
huffman@22972
   597
nipkow@24506
   598
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   599
by (simp add: sgn_div_norm)
huffman@22972
   600
nipkow@24506
   601
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   602
by (simp add: sgn_div_norm)
huffman@22972
   603
nipkow@24506
   604
lemma sgn_scaleR:
nipkow@24506
   605
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
nipkow@24506
   606
by (simp add: sgn_div_norm norm_scaleR mult_ac)
huffman@22973
   607
huffman@22972
   608
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   609
by (simp add: sgn_div_norm)
huffman@22972
   610
huffman@22972
   611
lemma sgn_of_real:
huffman@22972
   612
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   613
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   614
huffman@22973
   615
lemma sgn_mult:
huffman@22973
   616
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   617
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   618
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   619
huffman@22972
   620
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   621
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   622
huffman@22972
   623
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   624
unfolding real_sgn_eq by simp
huffman@22972
   625
huffman@22972
   626
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   627
unfolding real_sgn_eq by simp
huffman@22972
   628
huffman@22972
   629
huffman@22442
   630
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   631
huffman@22442
   632
locale bounded_linear = additive +
huffman@22442
   633
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@22442
   634
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   635
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   636
huffman@22442
   637
lemma (in bounded_linear) pos_bounded:
huffman@22442
   638
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   639
proof -
huffman@22442
   640
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   641
    using bounded by fast
huffman@22442
   642
  show ?thesis
huffman@22442
   643
  proof (intro exI impI conjI allI)
huffman@22442
   644
    show "0 < max 1 K"
huffman@22442
   645
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   646
  next
huffman@22442
   647
    fix x
huffman@22442
   648
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   649
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   650
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   651
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   652
  qed
huffman@22442
   653
qed
huffman@22442
   654
huffman@22442
   655
lemma (in bounded_linear) nonneg_bounded:
huffman@22442
   656
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   657
proof -
huffman@22442
   658
  from pos_bounded
huffman@22442
   659
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   660
qed
huffman@22442
   661
huffman@22442
   662
locale bounded_bilinear =
huffman@22442
   663
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   664
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   665
    (infixl "**" 70)
huffman@22442
   666
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   667
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   668
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   669
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   670
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@22442
   671
huffman@22442
   672
lemma (in bounded_bilinear) pos_bounded:
huffman@22442
   673
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   674
apply (cut_tac bounded, erule exE)
huffman@22442
   675
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   676
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   677
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   678
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   679
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   680
done
huffman@22442
   681
huffman@22442
   682
lemma (in bounded_bilinear) nonneg_bounded:
huffman@22442
   683
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   684
proof -
huffman@22442
   685
  from pos_bounded
huffman@22442
   686
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   687
qed
huffman@22442
   688
huffman@22442
   689
lemma (in bounded_bilinear) additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
   690
by (rule additive.intro, rule add_right)
huffman@22442
   691
huffman@22442
   692
lemma (in bounded_bilinear) additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
   693
by (rule additive.intro, rule add_left)
huffman@22442
   694
huffman@22442
   695
lemma (in bounded_bilinear) zero_left: "prod 0 b = 0"
huffman@22442
   696
by (rule additive.zero [OF additive_left])
huffman@22442
   697
huffman@22442
   698
lemma (in bounded_bilinear) zero_right: "prod a 0 = 0"
huffman@22442
   699
by (rule additive.zero [OF additive_right])
huffman@22442
   700
huffman@22442
   701
lemma (in bounded_bilinear) minus_left: "prod (- a) b = - prod a b"
huffman@22442
   702
by (rule additive.minus [OF additive_left])
huffman@22442
   703
huffman@22442
   704
lemma (in bounded_bilinear) minus_right: "prod a (- b) = - prod a b"
huffman@22442
   705
by (rule additive.minus [OF additive_right])
huffman@22442
   706
huffman@22442
   707
lemma (in bounded_bilinear) diff_left:
huffman@22442
   708
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
   709
by (rule additive.diff [OF additive_left])
huffman@22442
   710
huffman@22442
   711
lemma (in bounded_bilinear) diff_right:
huffman@22442
   712
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
   713
by (rule additive.diff [OF additive_right])
huffman@22442
   714
huffman@22442
   715
lemma (in bounded_bilinear) bounded_linear_left:
huffman@22442
   716
  "bounded_linear (\<lambda>a. a ** b)"
huffman@22442
   717
apply (unfold_locales)
huffman@22442
   718
apply (rule add_left)
huffman@22442
   719
apply (rule scaleR_left)
huffman@22442
   720
apply (cut_tac bounded, safe)
huffman@22442
   721
apply (rule_tac x="norm b * K" in exI)
huffman@22442
   722
apply (simp add: mult_ac)
huffman@22442
   723
done
huffman@22442
   724
huffman@22442
   725
lemma (in bounded_bilinear) bounded_linear_right:
huffman@22442
   726
  "bounded_linear (\<lambda>b. a ** b)"
huffman@22442
   727
apply (unfold_locales)
huffman@22442
   728
apply (rule add_right)
huffman@22442
   729
apply (rule scaleR_right)
huffman@22442
   730
apply (cut_tac bounded, safe)
huffman@22442
   731
apply (rule_tac x="norm a * K" in exI)
huffman@22442
   732
apply (simp add: mult_ac)
huffman@22442
   733
done
huffman@22442
   734
huffman@22442
   735
lemma (in bounded_bilinear) prod_diff_prod:
huffman@22442
   736
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
   737
by (simp add: diff_left diff_right)
huffman@22442
   738
huffman@23127
   739
interpretation mult:
huffman@22442
   740
  bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
huffman@22442
   741
apply (rule bounded_bilinear.intro)
huffman@22442
   742
apply (rule left_distrib)
huffman@22442
   743
apply (rule right_distrib)
huffman@22442
   744
apply (rule mult_scaleR_left)
huffman@22442
   745
apply (rule mult_scaleR_right)
huffman@22442
   746
apply (rule_tac x="1" in exI)
huffman@22442
   747
apply (simp add: norm_mult_ineq)
huffman@22442
   748
done
huffman@22442
   749
huffman@23127
   750
interpretation mult_left:
huffman@22442
   751
  bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
huffman@23127
   752
by (rule mult.bounded_linear_left)
huffman@22442
   753
huffman@23127
   754
interpretation mult_right:
huffman@23127
   755
  bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
huffman@23127
   756
by (rule mult.bounded_linear_right)
huffman@23127
   757
huffman@23127
   758
interpretation divide:
huffman@23120
   759
  bounded_linear ["(\<lambda>x::'a::real_normed_field. x / y)"]
huffman@23127
   760
unfolding divide_inverse by (rule mult.bounded_linear_left)
huffman@23120
   761
huffman@23127
   762
interpretation scaleR: bounded_bilinear ["scaleR"]
huffman@22442
   763
apply (rule bounded_bilinear.intro)
huffman@22442
   764
apply (rule scaleR_left_distrib)
huffman@22442
   765
apply (rule scaleR_right_distrib)
huffman@22973
   766
apply simp
huffman@22442
   767
apply (rule scaleR_left_commute)
huffman@22442
   768
apply (rule_tac x="1" in exI)
huffman@22442
   769
apply (simp add: norm_scaleR)
huffman@22442
   770
done
huffman@22442
   771
huffman@23127
   772
interpretation scaleR_left: bounded_linear ["\<lambda>r. scaleR r x"]
huffman@23127
   773
by (rule scaleR.bounded_linear_left)
huffman@23127
   774
huffman@23127
   775
interpretation scaleR_right: bounded_linear ["\<lambda>x. scaleR r x"]
huffman@23127
   776
by (rule scaleR.bounded_linear_right)
huffman@23127
   777
huffman@23127
   778
interpretation of_real: bounded_linear ["\<lambda>r. of_real r"]
huffman@23127
   779
unfolding of_real_def by (rule scaleR.bounded_linear_left)
huffman@22625
   780
huffman@20504
   781
end