src/HOL/Real/RealVector.thy
author haftmann
Tue Oct 16 23:12:45 2007 +0200 (2007-10-16)
changeset 25062 af5ef0d4d655
parent 24901 d3cbf79769b9
child 25571 c9e39eafc7a0
permissions -rw-r--r--
global class syntax
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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" (infixr "*\<^sub>R" 75)
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begin
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abbreviation
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  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
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where
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  "x /\<^sub>R r == scaleR (inverse r) x"
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end
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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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class real_vector = scaleR + ab_group_add +
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  assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
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  and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
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  and scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
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  and scaleR_one [simp]: "scaleR 1 x = x"
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class real_algebra = real_vector + ring +
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  assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
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  and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
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class real_algebra_1 = real_algebra + ring_1
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class real_div_algebra = real_algebra_1 + division_ring
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class 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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  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)
huffman@20584
   327
apply (rule of_real_inverse [symmetric])
huffman@20584
   328
done
huffman@20584
   329
huffman@20584
   330
lemma nonzero_Reals_divide:
huffman@20584
   331
  fixes a b :: "'a::real_field"
huffman@20584
   332
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   333
apply (auto simp add: Reals_def)
huffman@20584
   334
apply (rule range_eqI)
huffman@20584
   335
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   336
done
huffman@20584
   337
huffman@20584
   338
lemma Reals_divide [simp]:
huffman@20584
   339
  fixes a b :: "'a::{real_field,division_by_zero}"
huffman@20584
   340
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   341
apply (auto simp add: Reals_def)
huffman@20584
   342
apply (rule range_eqI)
huffman@20584
   343
apply (rule of_real_divide [symmetric])
huffman@20584
   344
done
huffman@20584
   345
huffman@20722
   346
lemma Reals_power [simp]:
huffman@20722
   347
  fixes a :: "'a::{real_algebra_1,recpower}"
huffman@20722
   348
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   349
apply (auto simp add: Reals_def)
huffman@20722
   350
apply (rule range_eqI)
huffman@20722
   351
apply (rule of_real_power [symmetric])
huffman@20722
   352
done
huffman@20722
   353
huffman@20554
   354
lemma Reals_cases [cases set: Reals]:
huffman@20554
   355
  assumes "q \<in> \<real>"
huffman@20554
   356
  obtains (of_real) r where "q = of_real r"
huffman@20554
   357
  unfolding Reals_def
huffman@20554
   358
proof -
huffman@20554
   359
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   360
  then obtain r where "q = of_real r" ..
huffman@20554
   361
  then show thesis ..
huffman@20554
   362
qed
huffman@20554
   363
huffman@20554
   364
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   365
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   366
  by (rule Reals_cases) auto
huffman@20554
   367
huffman@20504
   368
huffman@20504
   369
subsection {* Real normed vector spaces *}
huffman@20504
   370
huffman@22636
   371
class norm = type +
huffman@22636
   372
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   373
huffman@22636
   374
instance real :: norm
huffman@22636
   375
  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>" ..
huffman@20554
   376
huffman@24520
   377
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   378
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   379
haftmann@24588
   380
class real_normed_vector = real_vector + sgn_div_norm +
haftmann@24588
   381
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
haftmann@25062
   382
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   383
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
haftmann@24588
   384
  and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@20504
   385
haftmann@24588
   386
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   387
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   388
haftmann@24588
   389
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   390
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   391
haftmann@24588
   392
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   393
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   394
haftmann@24588
   395
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   396
huffman@22852
   397
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   398
proof
huffman@20554
   399
  fix x y :: 'a
huffman@20554
   400
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   401
    by (simp add: norm_mult)
huffman@22852
   402
next
huffman@22852
   403
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   404
    by (rule norm_mult)
huffman@22852
   405
  thus "norm (1::'a) = 1" by simp
huffman@20554
   406
qed
huffman@20554
   407
huffman@20584
   408
instance real :: real_normed_field
huffman@22852
   409
apply (intro_classes, unfold real_norm_def real_scaleR_def)
nipkow@24506
   410
apply (simp add: real_sgn_def)
huffman@20554
   411
apply (rule abs_ge_zero)
huffman@20554
   412
apply (rule abs_eq_0)
huffman@20554
   413
apply (rule abs_triangle_ineq)
huffman@22852
   414
apply (rule abs_mult)
huffman@20554
   415
apply (rule abs_mult)
huffman@20554
   416
done
huffman@20504
   417
huffman@22852
   418
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   419
by simp
huffman@20504
   420
huffman@22852
   421
lemma zero_less_norm_iff [simp]:
huffman@22852
   422
  fixes x :: "'a::real_normed_vector"
huffman@22852
   423
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   424
by (simp add: order_less_le)
huffman@20504
   425
huffman@22852
   426
lemma norm_not_less_zero [simp]:
huffman@22852
   427
  fixes x :: "'a::real_normed_vector"
huffman@22852
   428
  shows "\<not> norm x < 0"
huffman@20828
   429
by (simp add: linorder_not_less)
huffman@20828
   430
huffman@22852
   431
lemma norm_le_zero_iff [simp]:
huffman@22852
   432
  fixes x :: "'a::real_normed_vector"
huffman@22852
   433
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   434
by (simp add: order_le_less)
huffman@20828
   435
huffman@20504
   436
lemma norm_minus_cancel [simp]:
huffman@20584
   437
  fixes x :: "'a::real_normed_vector"
huffman@20584
   438
  shows "norm (- x) = norm x"
huffman@20504
   439
proof -
huffman@21809
   440
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   441
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   442
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   443
    by (rule norm_scaleR)
huffman@20504
   444
  finally show ?thesis by simp
huffman@20504
   445
qed
huffman@20504
   446
huffman@20504
   447
lemma norm_minus_commute:
huffman@20584
   448
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   449
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   450
proof -
huffman@22898
   451
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   452
    by (rule norm_minus_cancel)
huffman@22898
   453
  thus ?thesis by simp
huffman@20504
   454
qed
huffman@20504
   455
huffman@20504
   456
lemma norm_triangle_ineq2:
huffman@20584
   457
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   458
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   459
proof -
huffman@20533
   460
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   461
    by (rule norm_triangle_ineq)
huffman@22898
   462
  thus ?thesis by simp
huffman@20504
   463
qed
huffman@20504
   464
huffman@20584
   465
lemma norm_triangle_ineq3:
huffman@20584
   466
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   467
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   468
apply (subst abs_le_iff)
huffman@20584
   469
apply auto
huffman@20584
   470
apply (rule norm_triangle_ineq2)
huffman@20584
   471
apply (subst norm_minus_commute)
huffman@20584
   472
apply (rule norm_triangle_ineq2)
huffman@20584
   473
done
huffman@20584
   474
huffman@20504
   475
lemma norm_triangle_ineq4:
huffman@20584
   476
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   477
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   478
proof -
huffman@22898
   479
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   480
    by (rule norm_triangle_ineq)
huffman@22898
   481
  thus ?thesis
huffman@22898
   482
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   483
qed
huffman@22898
   484
huffman@22898
   485
lemma norm_diff_ineq:
huffman@22898
   486
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   487
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   488
proof -
huffman@22898
   489
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   490
    by (rule norm_triangle_ineq2)
huffman@22898
   491
  thus ?thesis by simp
huffman@20504
   492
qed
huffman@20504
   493
huffman@20551
   494
lemma norm_diff_triangle_ineq:
huffman@20551
   495
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   496
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   497
proof -
huffman@20551
   498
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   499
    by (simp add: diff_minus add_ac)
huffman@20551
   500
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   501
    by (rule norm_triangle_ineq)
huffman@20551
   502
  finally show ?thesis .
huffman@20551
   503
qed
huffman@20551
   504
huffman@22857
   505
lemma abs_norm_cancel [simp]:
huffman@22857
   506
  fixes a :: "'a::real_normed_vector"
huffman@22857
   507
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   508
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   509
huffman@22880
   510
lemma norm_add_less:
huffman@22880
   511
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   512
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   513
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   514
huffman@22880
   515
lemma norm_mult_less:
huffman@22880
   516
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   517
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   518
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   519
apply (simp add: mult_strict_mono')
huffman@22880
   520
done
huffman@22880
   521
huffman@22857
   522
lemma norm_of_real [simp]:
huffman@22857
   523
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@22852
   524
unfolding of_real_def by (simp add: norm_scaleR)
huffman@20560
   525
huffman@22876
   526
lemma norm_number_of [simp]:
huffman@22876
   527
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
huffman@22876
   528
    = \<bar>number_of w\<bar>"
huffman@22876
   529
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
huffman@22876
   530
huffman@22876
   531
lemma norm_of_int [simp]:
huffman@22876
   532
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   533
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   534
huffman@22876
   535
lemma norm_of_nat [simp]:
huffman@22876
   536
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   537
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   538
apply (subst norm_of_real, simp)
huffman@22876
   539
done
huffman@22876
   540
huffman@20504
   541
lemma nonzero_norm_inverse:
huffman@20504
   542
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   543
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   544
apply (rule inverse_unique [symmetric])
huffman@20504
   545
apply (simp add: norm_mult [symmetric])
huffman@20504
   546
done
huffman@20504
   547
huffman@20504
   548
lemma norm_inverse:
huffman@20504
   549
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
huffman@20533
   550
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   551
apply (case_tac "a = 0", simp)
huffman@20504
   552
apply (erule nonzero_norm_inverse)
huffman@20504
   553
done
huffman@20504
   554
huffman@20584
   555
lemma nonzero_norm_divide:
huffman@20584
   556
  fixes a b :: "'a::real_normed_field"
huffman@20584
   557
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   558
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   559
huffman@20584
   560
lemma norm_divide:
huffman@20584
   561
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
huffman@20584
   562
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   563
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   564
huffman@22852
   565
lemma norm_power_ineq:
huffman@22852
   566
  fixes x :: "'a::{real_normed_algebra_1,recpower}"
huffman@22852
   567
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   568
proof (induct n)
huffman@22852
   569
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   570
next
huffman@22852
   571
  case (Suc n)
huffman@22852
   572
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   573
    by (rule norm_mult_ineq)
huffman@22852
   574
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   575
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   576
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@22852
   577
    by (simp add: power_Suc)
huffman@22852
   578
qed
huffman@22852
   579
huffman@20684
   580
lemma norm_power:
huffman@20684
   581
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
huffman@20684
   582
  shows "norm (x ^ n) = norm x ^ n"
wenzelm@20772
   583
by (induct n) (simp_all add: power_Suc norm_mult)
huffman@20684
   584
huffman@22442
   585
huffman@22972
   586
subsection {* Sign function *}
huffman@22972
   587
nipkow@24506
   588
lemma norm_sgn:
nipkow@24506
   589
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
nipkow@24506
   590
by (simp add: sgn_div_norm norm_scaleR)
huffman@22972
   591
nipkow@24506
   592
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   593
by (simp add: sgn_div_norm)
huffman@22972
   594
nipkow@24506
   595
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   596
by (simp add: sgn_div_norm)
huffman@22972
   597
nipkow@24506
   598
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   599
by (simp add: sgn_div_norm)
huffman@22972
   600
nipkow@24506
   601
lemma sgn_scaleR:
nipkow@24506
   602
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
nipkow@24506
   603
by (simp add: sgn_div_norm norm_scaleR mult_ac)
huffman@22973
   604
huffman@22972
   605
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   606
by (simp add: sgn_div_norm)
huffman@22972
   607
huffman@22972
   608
lemma sgn_of_real:
huffman@22972
   609
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   610
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   611
huffman@22973
   612
lemma sgn_mult:
huffman@22973
   613
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   614
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   615
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   616
huffman@22972
   617
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   618
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   619
huffman@22972
   620
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   621
unfolding real_sgn_eq by simp
huffman@22972
   622
huffman@22972
   623
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   624
unfolding real_sgn_eq by simp
huffman@22972
   625
huffman@22972
   626
huffman@22442
   627
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   628
huffman@22442
   629
locale bounded_linear = additive +
huffman@22442
   630
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@22442
   631
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   632
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   633
huffman@22442
   634
lemma (in bounded_linear) pos_bounded:
huffman@22442
   635
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   636
proof -
huffman@22442
   637
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   638
    using bounded by fast
huffman@22442
   639
  show ?thesis
huffman@22442
   640
  proof (intro exI impI conjI allI)
huffman@22442
   641
    show "0 < max 1 K"
huffman@22442
   642
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   643
  next
huffman@22442
   644
    fix x
huffman@22442
   645
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   646
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   647
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   648
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   649
  qed
huffman@22442
   650
qed
huffman@22442
   651
huffman@22442
   652
lemma (in bounded_linear) nonneg_bounded:
huffman@22442
   653
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   654
proof -
huffman@22442
   655
  from pos_bounded
huffman@22442
   656
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   657
qed
huffman@22442
   658
huffman@22442
   659
locale bounded_bilinear =
huffman@22442
   660
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   661
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   662
    (infixl "**" 70)
huffman@22442
   663
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   664
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   665
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   666
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   667
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@22442
   668
huffman@22442
   669
lemma (in bounded_bilinear) pos_bounded:
huffman@22442
   670
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   671
apply (cut_tac bounded, erule exE)
huffman@22442
   672
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   673
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   674
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   675
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   676
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   677
done
huffman@22442
   678
huffman@22442
   679
lemma (in bounded_bilinear) nonneg_bounded:
huffman@22442
   680
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   681
proof -
huffman@22442
   682
  from pos_bounded
huffman@22442
   683
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   684
qed
huffman@22442
   685
huffman@22442
   686
lemma (in bounded_bilinear) additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
   687
by (rule additive.intro, rule add_right)
huffman@22442
   688
huffman@22442
   689
lemma (in bounded_bilinear) additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
   690
by (rule additive.intro, rule add_left)
huffman@22442
   691
huffman@22442
   692
lemma (in bounded_bilinear) zero_left: "prod 0 b = 0"
huffman@22442
   693
by (rule additive.zero [OF additive_left])
huffman@22442
   694
huffman@22442
   695
lemma (in bounded_bilinear) zero_right: "prod a 0 = 0"
huffman@22442
   696
by (rule additive.zero [OF additive_right])
huffman@22442
   697
huffman@22442
   698
lemma (in bounded_bilinear) minus_left: "prod (- a) b = - prod a b"
huffman@22442
   699
by (rule additive.minus [OF additive_left])
huffman@22442
   700
huffman@22442
   701
lemma (in bounded_bilinear) minus_right: "prod a (- b) = - prod a b"
huffman@22442
   702
by (rule additive.minus [OF additive_right])
huffman@22442
   703
huffman@22442
   704
lemma (in bounded_bilinear) diff_left:
huffman@22442
   705
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
   706
by (rule additive.diff [OF additive_left])
huffman@22442
   707
huffman@22442
   708
lemma (in bounded_bilinear) diff_right:
huffman@22442
   709
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
   710
by (rule additive.diff [OF additive_right])
huffman@22442
   711
huffman@22442
   712
lemma (in bounded_bilinear) bounded_linear_left:
huffman@22442
   713
  "bounded_linear (\<lambda>a. a ** b)"
huffman@22442
   714
apply (unfold_locales)
huffman@22442
   715
apply (rule add_left)
huffman@22442
   716
apply (rule scaleR_left)
huffman@22442
   717
apply (cut_tac bounded, safe)
huffman@22442
   718
apply (rule_tac x="norm b * K" in exI)
huffman@22442
   719
apply (simp add: mult_ac)
huffman@22442
   720
done
huffman@22442
   721
huffman@22442
   722
lemma (in bounded_bilinear) bounded_linear_right:
huffman@22442
   723
  "bounded_linear (\<lambda>b. a ** b)"
huffman@22442
   724
apply (unfold_locales)
huffman@22442
   725
apply (rule add_right)
huffman@22442
   726
apply (rule scaleR_right)
huffman@22442
   727
apply (cut_tac bounded, safe)
huffman@22442
   728
apply (rule_tac x="norm a * K" in exI)
huffman@22442
   729
apply (simp add: mult_ac)
huffman@22442
   730
done
huffman@22442
   731
huffman@22442
   732
lemma (in bounded_bilinear) prod_diff_prod:
huffman@22442
   733
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
   734
by (simp add: diff_left diff_right)
huffman@22442
   735
huffman@23127
   736
interpretation mult:
huffman@22442
   737
  bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
huffman@22442
   738
apply (rule bounded_bilinear.intro)
huffman@22442
   739
apply (rule left_distrib)
huffman@22442
   740
apply (rule right_distrib)
huffman@22442
   741
apply (rule mult_scaleR_left)
huffman@22442
   742
apply (rule mult_scaleR_right)
huffman@22442
   743
apply (rule_tac x="1" in exI)
huffman@22442
   744
apply (simp add: norm_mult_ineq)
huffman@22442
   745
done
huffman@22442
   746
huffman@23127
   747
interpretation mult_left:
huffman@22442
   748
  bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
huffman@23127
   749
by (rule mult.bounded_linear_left)
huffman@22442
   750
huffman@23127
   751
interpretation mult_right:
huffman@23127
   752
  bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
huffman@23127
   753
by (rule mult.bounded_linear_right)
huffman@23127
   754
huffman@23127
   755
interpretation divide:
huffman@23120
   756
  bounded_linear ["(\<lambda>x::'a::real_normed_field. x / y)"]
huffman@23127
   757
unfolding divide_inverse by (rule mult.bounded_linear_left)
huffman@23120
   758
huffman@23127
   759
interpretation scaleR: bounded_bilinear ["scaleR"]
huffman@22442
   760
apply (rule bounded_bilinear.intro)
huffman@22442
   761
apply (rule scaleR_left_distrib)
huffman@22442
   762
apply (rule scaleR_right_distrib)
huffman@22973
   763
apply simp
huffman@22442
   764
apply (rule scaleR_left_commute)
huffman@22442
   765
apply (rule_tac x="1" in exI)
huffman@22442
   766
apply (simp add: norm_scaleR)
huffman@22442
   767
done
huffman@22442
   768
huffman@23127
   769
interpretation scaleR_left: bounded_linear ["\<lambda>r. scaleR r x"]
huffman@23127
   770
by (rule scaleR.bounded_linear_left)
huffman@23127
   771
huffman@23127
   772
interpretation scaleR_right: bounded_linear ["\<lambda>x. scaleR r x"]
huffman@23127
   773
by (rule scaleR.bounded_linear_right)
huffman@23127
   774
huffman@23127
   775
interpretation of_real: bounded_linear ["\<lambda>r. of_real r"]
huffman@23127
   776
unfolding of_real_def by (rule scaleR.bounded_linear_left)
huffman@22625
   777
huffman@20504
   778
end