author  huffman 
Sun, 17 Sep 2006 16:42:38 +0200  
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parent 20554  c433e78d4203 
child 20584  60b1d52a455d 
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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instance real :: scaleR .. 
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defs (overloaded) 
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real_scaleR_def: "a *# x \<equiv> a * x" 
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axclass real_vector < scaleR, ab_group_add 
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scaleR_right_distrib: "a *# (x + y) = a *# x + a *# y" 
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scaleR_left_distrib: "(a + b) *# x = a *# x + b *# x" 
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scaleR_assoc: "(a * b) *# x = a *# b *# x" 
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scaleR_one [simp]: "1 *# x = x" 
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axclass real_algebra < real_vector, ring 
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mult_scaleR_left: "a *# x * y = a *# (x * y)" 
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mult_scaleR_right: "x * a *# y = a *# (x * y)" 
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axclass real_algebra_1 < real_algebra, ring_1 
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instance real :: real_algebra_1 
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apply (intro_classes, unfold real_scaleR_def) 
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apply (rule right_distrib) 
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apply (rule left_distrib) 
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apply (rule mult_assoc) 
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apply (rule mult_1_left) 
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apply (rule mult_assoc) 
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apply (rule mult_left_commute) 
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done 
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lemmas scaleR_scaleR = scaleR_assoc [symmetric] 
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lemma scaleR_left_commute: 
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fixes x :: "'a::real_vector" 
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shows "a *# b *# x = b *# a *# x" 
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by (simp add: scaleR_scaleR mult_commute) 
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lemma additive_scaleR_right: "additive (\<lambda>x. a *# x :: 'a::real_vector)" 
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by (rule additive.intro, rule scaleR_right_distrib) 
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lemma additive_scaleR_left: "additive (\<lambda>a. a *# x :: 'a::real_vector)" 
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by (rule additive.intro, rule scaleR_left_distrib) 
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lemmas scaleR_zero_left [simp] = 
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additive.zero [OF additive_scaleR_left, standard] 
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lemmas scaleR_zero_right [simp] = 
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additive.zero [OF additive_scaleR_right, standard] 
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lemmas scaleR_minus_left [simp] = 
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additive.minus [OF additive_scaleR_left, standard] 
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lemmas scaleR_minus_right [simp] = 
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additive.minus [OF additive_scaleR_right, standard] 
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lemmas scaleR_left_diff_distrib = 
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additive.diff [OF additive_scaleR_left, standard] 
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lemmas scaleR_right_diff_distrib = 
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additive.diff [OF additive_scaleR_right, standard] 
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lemma scaleR_eq_0_iff: 
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fixes x :: "'a::real_vector" 
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shows "(a *# x = 0) = (a = 0 \<or> x = 0)" 
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proof cases 
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assume "a = 0" thus ?thesis by simp 
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next 
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assume anz [simp]: "a \<noteq> 0" 
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{ assume "a *# x = 0" 
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hence "inverse a *# a *# x = 0" by simp 
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hence "x = 0" by (simp (no_asm_use) add: scaleR_scaleR)} 
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thus ?thesis by force 
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qed 
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lemma scaleR_left_imp_eq: 
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fixes x y :: "'a::real_vector" 
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shows "\<lbrakk>a \<noteq> 0; a *# x = a *# y\<rbrakk> \<Longrightarrow> x = y" 
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proof  
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assume nonzero: "a \<noteq> 0" 
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assume "a *# x = a *# y" 
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hence "a *# (x  y) = 0" 
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by (simp add: scaleR_right_diff_distrib) 
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hence "x  y = 0" 
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by (simp add: scaleR_eq_0_iff nonzero) 
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thus "x = y" by simp 
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qed 
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lemma scaleR_right_imp_eq: 
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fixes x y :: "'a::real_vector" 
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shows "\<lbrakk>x \<noteq> 0; a *# x = b *# x\<rbrakk> \<Longrightarrow> a = b" 
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proof  
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assume nonzero: "x \<noteq> 0" 
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assume "a *# x = b *# x" 
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hence "(a  b) *# x = 0" 
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by (simp add: scaleR_left_diff_distrib) 
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hence "a  b = 0" 
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by (simp add: scaleR_eq_0_iff nonzero) 
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thus "a = b" by simp 
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qed 
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lemma scaleR_cancel_left: 
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fixes x y :: "'a::real_vector" 
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shows "(a *# x = a *# y) = (x = y \<or> a = 0)" 
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146 
by (auto intro: scaleR_left_imp_eq) 
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147 

c433e78d4203
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148 
lemma scaleR_cancel_right: 
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149 
fixes x y :: "'a::real_vector" 
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150 
shows "(a *# x = b *# x) = (a = b \<or> x = 0)" 
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151 
by (auto intro: scaleR_right_imp_eq) 
c433e78d4203
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152 

c433e78d4203
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153 

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154 
subsection {* Embedding of the Reals into any @{text real_algebra_1}: 
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155 
@{term of_real} *} 
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156 

c433e78d4203
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157 
definition 
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158 
of_real :: "real \<Rightarrow> 'a::real_algebra_1" 
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159 
"of_real r = r *# 1" 
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160 

c433e78d4203
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161 
lemma of_real_0 [simp]: "of_real 0 = 0" 
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162 
by (simp add: of_real_def) 
c433e78d4203
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163 

c433e78d4203
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164 
lemma of_real_1 [simp]: "of_real 1 = 1" 
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165 
by (simp add: of_real_def) 
c433e78d4203
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166 

c433e78d4203
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167 
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y" 
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168 
by (simp add: of_real_def scaleR_left_distrib) 
c433e78d4203
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169 

c433e78d4203
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170 
lemma of_real_minus [simp]: "of_real ( x) =  of_real x" 
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171 
by (simp add: of_real_def) 
c433e78d4203
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172 

c433e78d4203
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173 
lemma of_real_diff [simp]: "of_real (x  y) = of_real x  of_real y" 
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174 
by (simp add: of_real_def scaleR_left_diff_distrib) 
c433e78d4203
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175 

c433e78d4203
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176 
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y" 
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177 
by (simp add: of_real_def mult_scaleR_left scaleR_scaleR) 
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178 

c433e78d4203
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179 
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)" 
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180 
by (simp add: of_real_def scaleR_cancel_right) 
c433e78d4203
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181 

c433e78d4203
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182 
text{*Special cases where either operand is zero*} 
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183 
lemmas of_real_0_eq_iff = of_real_eq_iff [of 0, simplified] 
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184 
lemmas of_real_eq_0_iff = of_real_eq_iff [of _ 0, simplified] 
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185 
declare of_real_0_eq_iff [simp] 
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186 
declare of_real_eq_0_iff [simp] 
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187 

c433e78d4203
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188 
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)" 
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189 
proof 
c433e78d4203
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190 
fix r 
c433e78d4203
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191 
show "of_real r = id r" 
c433e78d4203
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192 
by (simp add: of_real_def real_scaleR_def) 
c433e78d4203
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193 
qed 
c433e78d4203
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194 

c433e78d4203
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195 
text{*Collapse nested embeddings*} 
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196 
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n" 
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197 
by (induct n, auto) 
c433e78d4203
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198 

c433e78d4203
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199 
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z" 
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200 
by (cases z rule: int_diff_cases, simp) 
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201 

c433e78d4203
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202 
lemma of_real_number_of_eq: 
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203 
"of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})" 
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204 
by (simp add: number_of_eq) 
c433e78d4203
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205 

c433e78d4203
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206 

c433e78d4203
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207 
subsection {* The Set of Real Numbers *} 
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208 

c433e78d4203
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209 
constdefs 
c433e78d4203
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210 
Reals :: "'a::real_algebra_1 set" 
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211 
"Reals \<equiv> range of_real" 
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212 

c433e78d4203
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213 
const_syntax (xsymbols) 
c433e78d4203
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214 
Reals ("\<real>") 
c433e78d4203
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215 

c433e78d4203
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216 
lemma of_real_in_Reals [simp]: "of_real r \<in> Reals" 
c433e78d4203
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217 
by (simp add: Reals_def) 
c433e78d4203
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218 

c433e78d4203
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219 
lemma Reals_0 [simp]: "0 \<in> Reals" 
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220 
apply (unfold Reals_def) 
c433e78d4203
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221 
apply (rule range_eqI) 
c433e78d4203
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222 
apply (rule of_real_0 [symmetric]) 
c433e78d4203
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223 
done 
c433e78d4203
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224 

c433e78d4203
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225 
lemma Reals_1 [simp]: "1 \<in> Reals" 
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226 
apply (unfold Reals_def) 
c433e78d4203
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227 
apply (rule range_eqI) 
c433e78d4203
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228 
apply (rule of_real_1 [symmetric]) 
c433e78d4203
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229 
done 
c433e78d4203
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230 

c433e78d4203
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231 
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a+b \<in> Reals" 
c433e78d4203
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232 
apply (auto simp add: Reals_def) 
c433e78d4203
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233 
apply (rule range_eqI) 
c433e78d4203
define new constant of_real for class real_algebra_1;
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diff
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234 
apply (rule of_real_add [symmetric]) 
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
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diff
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235 
done 
c433e78d4203
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diff
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236 

c433e78d4203
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237 
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a*b \<in> Reals" 
c433e78d4203
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huffman
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238 
apply (auto simp add: Reals_def) 
c433e78d4203
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239 
apply (rule range_eqI) 
c433e78d4203
define new constant of_real for class real_algebra_1;
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240 
apply (rule of_real_mult [symmetric]) 
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
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diff
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241 
done 
c433e78d4203
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huffman
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242 

c433e78d4203
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243 
lemma Reals_cases [cases set: Reals]: 
c433e78d4203
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244 
assumes "q \<in> \<real>" 
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245 
obtains (of_real) r where "q = of_real r" 
c433e78d4203
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246 
unfolding Reals_def 
c433e78d4203
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247 
proof  
c433e78d4203
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248 
from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def . 
c433e78d4203
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249 
then obtain r where "q = of_real r" .. 
c433e78d4203
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250 
then show thesis .. 
c433e78d4203
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251 
qed 
c433e78d4203
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252 

c433e78d4203
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253 
lemma Reals_induct [case_names of_real, induct set: Reals]: 
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254 
"q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q" 
c433e78d4203
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255 
by (rule Reals_cases) auto 
c433e78d4203
define new constant of_real for class real_algebra_1;
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256 

20504
6342e872e71d
formalization of vector spaces and algebras over the real numbers
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257 

6342e872e71d
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258 
subsection {* Real normed vector spaces *} 
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formalization of vector spaces and algebras over the real numbers
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259 

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260 
axclass norm < type 
20533  261 
consts norm :: "'a::norm \<Rightarrow> real" 
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262 

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263 
instance real :: norm .. 
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264 

c433e78d4203
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265 
defs (overloaded) 
c433e78d4203
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266 
real_norm_def: "norm r \<equiv> \<bar>r\<bar>" 
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267 

c433e78d4203
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268 
axclass normed < plus, zero, norm 
20533  269 
norm_ge_zero [simp]: "0 \<le> norm x" 
270 
norm_eq_zero [simp]: "(norm x = 0) = (x = 0)" 

271 
norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y" 

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272 

c433e78d4203
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273 
axclass real_normed_vector < real_vector, normed 
20533  274 
norm_scaleR: "norm (a *# x) = \<bar>a\<bar> * norm x" 
20504
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275 

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276 
axclass real_normed_algebra < real_normed_vector, real_algebra 
20533  277 
norm_mult_ineq: "norm (x * y) \<le> norm x * norm y" 
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278 

20554
c433e78d4203
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279 
axclass real_normed_div_algebra < normed, real_algebra_1, division_ring 
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280 
norm_of_real: "norm (of_real r) = abs r" 
20533  281 
norm_mult: "norm (x * y) = norm x * norm y" 
20504
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282 

6342e872e71d
formalization of vector spaces and algebras over the real numbers
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283 
instance real_normed_div_algebra < real_normed_algebra 
20554
c433e78d4203
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284 
proof 
c433e78d4203
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huffman
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285 
fix a :: real and x :: 'a 
c433e78d4203
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286 
have "norm (a *# x) = norm (of_real a * x)" 
c433e78d4203
define new constant of_real for class real_algebra_1;
huffman
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287 
by (simp add: of_real_def mult_scaleR_left) 
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also have "\<dots> = abs a * norm x" 
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289 
by (simp add: norm_mult norm_of_real) 
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finally show "norm (a *# x) = abs a * norm x" . 
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291 
next 
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292 
fix x y :: 'a 
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293 
show "norm (x * y) \<le> norm x * norm y" 
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294 
by (simp add: norm_mult) 
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295 
qed 
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296 

c433e78d4203
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instance real :: real_normed_div_algebra 
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apply (intro_classes, unfold real_norm_def) 
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299 
apply (rule abs_ge_zero) 
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300 
apply (rule abs_eq_0) 
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301 
apply (rule abs_triangle_ineq) 
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302 
apply simp 
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303 
apply (rule abs_mult) 
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304 
done 
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305 

20533  306 
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0" 
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by simp 
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308 

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lemma zero_less_norm_iff [simp]: 
20533  310 
fixes x :: "'a::real_normed_vector" shows "(0 < norm x) = (x \<noteq> 0)" 
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311 
by (simp add: order_less_le) 
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312 

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lemma norm_minus_cancel [simp]: 
20533  314 
fixes x :: "'a::real_normed_vector" shows "norm ( x) = norm x" 
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proof  
20533  316 
have "norm ( x) = norm ( 1 *# x)" 
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by (simp only: scaleR_minus_left scaleR_one) 
20533  318 
also have "\<dots> = \<bar> 1\<bar> * norm x" 
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319 
by (rule norm_scaleR) 
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320 
finally show ?thesis by simp 
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321 
qed 
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322 

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lemma norm_minus_commute: 
20533  324 
fixes a b :: "'a::real_normed_vector" shows "norm (a  b) = norm (b  a)" 
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325 
proof  
20533  326 
have "norm (a  b) = norm ( (a  b))" 
327 
by (simp only: norm_minus_cancel) 

328 
also have "\<dots> = norm (b  a)" by simp 

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329 
finally show ?thesis . 
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330 
qed 
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331 

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332 
lemma norm_triangle_ineq2: 
20533  333 
fixes a :: "'a::real_normed_vector" 
334 
shows "norm a  norm b \<le> norm (a  b)" 

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335 
proof  
20533  336 
have "norm (a  b + b) \<le> norm (a  b) + norm b" 
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337 
by (rule norm_triangle_ineq) 
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338 
also have "(a  b + b) = a" 
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339 
by simp 
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340 
finally show ?thesis 
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341 
by (simp add: compare_rls) 
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342 
qed 
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343 

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344 
lemma norm_triangle_ineq4: 
20533  345 
fixes a :: "'a::real_normed_vector" 
346 
shows "norm (a  b) \<le> norm a + norm b" 

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347 
proof  
20533  348 
have "norm (a  b) = norm (a +  b)" 
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349 
by (simp only: diff_minus) 
20533  350 
also have "\<dots> \<le> norm a + norm ( b)" 
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351 
by (rule norm_triangle_ineq) 
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352 
finally show ?thesis 
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353 
by simp 
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354 
qed 
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355 

20551  356 
lemma norm_diff_triangle_ineq: 
357 
fixes a b c d :: "'a::real_normed_vector" 

358 
shows "norm ((a + b)  (c + d)) \<le> norm (a  c) + norm (b  d)" 

359 
proof  

360 
have "norm ((a + b)  (c + d)) = norm ((a  c) + (b  d))" 

361 
by (simp add: diff_minus add_ac) 

362 
also have "\<dots> \<le> norm (a  c) + norm (b  d)" 

363 
by (rule norm_triangle_ineq) 

364 
finally show ?thesis . 

365 
qed 

366 

20560  367 
lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1" 
368 
proof  

369 
have "norm (of_real 1 :: 'a) = abs 1" 

370 
by (rule norm_of_real) 

371 
thus ?thesis by simp 

372 
qed 

373 

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374 
lemma nonzero_norm_inverse: 
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375 
fixes a :: "'a::real_normed_div_algebra" 
20533  376 
shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)" 
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377 
apply (rule inverse_unique [symmetric]) 
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378 
apply (simp add: norm_mult [symmetric]) 
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379 
done 
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380 

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381 
lemma norm_inverse: 
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382 
fixes a :: "'a::{real_normed_div_algebra,division_by_zero}" 
20533  383 
shows "norm (inverse a) = inverse (norm a)" 
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384 
apply (case_tac "a = 0", simp) 
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385 
apply (erule nonzero_norm_inverse) 
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386 
done 
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387 

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388 
end 