author  huffman 
Thu, 14 Sep 2006 03:25:17 +0200  
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parent 20504  6342e872e71d 
child 20551  ba543692bfa1 
permissions  rwrr 
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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 RealDef 
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begin 
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subsection {* Locale for additive functions *} 
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locale additive = 
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fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add" 
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assumes add: "f (x + y) = f x + f y" 
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lemma (in additive) zero: "f 0 = 0" 
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proof  
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have "f 0 = f (0 + 0)" by simp 
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also have "\<dots> = f 0 + f 0" by (rule add) 
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finally show "f 0 = 0" by simp 
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qed 
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lemma (in additive) minus: "f ( x) =  f x" 
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proof  
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have "f ( x) + f x = f ( x + x)" by (rule add [symmetric]) 
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also have "\<dots> =  f x + f x" by (simp add: zero) 
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finally show "f ( x) =  f x" by (rule add_right_imp_eq) 
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qed 
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lemma (in additive) diff: "f (x  y) = f x  f y" 
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by (simp add: diff_def add minus) 
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subsection {* Real vector spaces *} 
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axclass scaleR < type 
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consts 
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scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a::scaleR" (infixr "*#" 75) 
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syntax (xsymbols) 
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scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a::scaleR" (infixr "*\<^sub>R" 75) 
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axclass real_vector < scaleR, ab_group_add 
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scaleR_right_distrib: "a *# (x + y) = a *# x + a *# y" 
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scaleR_left_distrib: "(a + b) *# x = a *# x + b *# x" 
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scaleR_assoc: "(a * b) *# x = a *# b *# x" 
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scaleR_one [simp]: "1 *# x = x" 
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axclass real_algebra < real_vector, ring 
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mult_scaleR_left: "a *# x * y = a *# (x * y)" 
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mult_scaleR_right: "x * a *# y = a *# (x * y)" 
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lemmas scaleR_scaleR = scaleR_assoc [symmetric] 
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lemma scaleR_left_commute: 
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fixes x :: "'a::real_vector" 
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shows "a *# b *# x = b *# a *# x" 
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by (simp add: scaleR_scaleR mult_commute) 
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lemma additive_scaleR_right: "additive (\<lambda>x. a *# x :: 'a::real_vector)" 
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by (rule additive.intro, rule scaleR_right_distrib) 
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lemma additive_scaleR_left: "additive (\<lambda>a. a *# x :: 'a::real_vector)" 
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by (rule additive.intro, rule scaleR_left_distrib) 
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lemmas scaleR_zero_left [simp] = 
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additive.zero [OF additive_scaleR_left, standard] 
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lemmas scaleR_zero_right [simp] = 
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additive.zero [OF additive_scaleR_right, standard] 
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lemmas scaleR_minus_left [simp] = 
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additive.minus [OF additive_scaleR_left, standard] 
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lemmas scaleR_minus_right [simp] = 
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additive.minus [OF additive_scaleR_right, standard] 
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lemmas scaleR_left_diff_distrib = 
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additive.diff [OF additive_scaleR_left, standard] 
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lemmas scaleR_right_diff_distrib = 
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additive.diff [OF additive_scaleR_right, standard] 
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subsection {* Real normed vector spaces *} 
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axclass norm < type 
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consts norm :: "'a::norm \<Rightarrow> real" 
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axclass real_normed_vector < real_vector, norm 
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norm_ge_zero [simp]: "0 \<le> norm x" 
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norm_eq_zero [simp]: "(norm x = 0) = (x = 0)" 

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norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y" 

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norm_scaleR: "norm (a *# x) = \<bar>a\<bar> * norm x" 

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axclass real_normed_algebra < real_normed_vector, real_algebra 
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norm_mult_ineq: "norm (x * y) \<le> norm x * norm y" 
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axclass real_normed_div_algebra 
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< real_normed_vector, real_algebra, division_ring 
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norm_mult: "norm (x * y) = norm x * norm y" 
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norm_one [simp]: "norm 1 = 1" 

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instance real_normed_div_algebra < real_normed_algebra 
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by (intro_classes, simp add: norm_mult) 
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lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" 
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by simp 
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lemma zero_less_norm_iff [simp]: 
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fixes x :: "'a::real_normed_vector" shows "(0 < norm x) = (x \<noteq> 0)" 
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by (simp add: order_less_le) 
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lemma norm_minus_cancel [simp]: 
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fixes x :: "'a::real_normed_vector" shows "norm ( x) = norm x" 
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proof  
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have "norm ( x) = norm ( 1 *# x)" 
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by (simp only: scaleR_minus_left scaleR_one) 
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also have "\<dots> = \<bar> 1\<bar> * norm x" 
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by (rule norm_scaleR) 
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finally show ?thesis by simp 
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qed 
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lemma norm_minus_commute: 
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fixes a b :: "'a::real_normed_vector" shows "norm (a  b) = norm (b  a)" 
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proof  
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have "norm (a  b) = norm ( (a  b))" 
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by (simp only: norm_minus_cancel) 

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also have "\<dots> = norm (b  a)" by simp 

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finally show ?thesis . 
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qed 
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lemma norm_triangle_ineq2: 
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fixes a :: "'a::real_normed_vector" 
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shows "norm a  norm b \<le> norm (a  b)" 

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proof  
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have "norm (a  b + b) \<le> norm (a  b) + norm b" 
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by (rule norm_triangle_ineq) 
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also have "(a  b + b) = a" 
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by simp 
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finally show ?thesis 
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by (simp add: compare_rls) 
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qed 
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lemma norm_triangle_ineq4: 
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fixes a :: "'a::real_normed_vector" 
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shows "norm (a  b) \<le> norm a + norm b" 

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proof  
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have "norm (a  b) = norm (a +  b)" 
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by (simp only: diff_minus) 
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also have "\<dots> \<le> norm a + norm ( b)" 
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by (rule norm_triangle_ineq) 
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finally show ?thesis 
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by simp 
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qed 
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lemma nonzero_norm_inverse: 
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fixes a :: "'a::real_normed_div_algebra" 
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shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)" 
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apply (rule inverse_unique [symmetric]) 
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apply (simp add: norm_mult [symmetric]) 
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done 
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lemma norm_inverse: 
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fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" 
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shows "norm (inverse a) = inverse (norm a)" 
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apply (case_tac "a = 0", simp) 
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apply (erule nonzero_norm_inverse) 
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done 
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subsection {* Instances for type @{typ real} *} 
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instance real :: scaleR .. 
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defs (overloaded) 
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real_scaleR_def: "a *# x \<equiv> a * x" 
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instance real :: real_algebra 
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apply (intro_classes, unfold real_scaleR_def) 
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apply (rule right_distrib) 
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apply (rule left_distrib) 
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apply (rule mult_assoc) 
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apply (rule mult_1_left) 
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apply (rule mult_assoc) 
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apply (rule mult_left_commute) 
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done 
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instance real :: norm .. 
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defs (overloaded) 
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real_norm_def: "norm r \<equiv> \<bar>r\<bar>" 
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instance real :: real_normed_div_algebra 
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apply (intro_classes, unfold real_norm_def real_scaleR_def) 
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apply (rule abs_ge_zero) 
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apply (rule abs_eq_0) 
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apply (rule abs_triangle_ineq) 
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apply (rule abs_mult) 
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apply (rule abs_mult) 
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apply (rule abs_one) 
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done 
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end 