author  huffman 
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permissions  rwrr 
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(* Title : HOL/RealDef.thy 
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Author : Jacques D. Fleuriot, 1998 
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Polymorphic treatment of binary arithmetic using axclasses
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 
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Additional contributions by Jeremy Avigad 
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Construction of Cauchy Reals by Brian Huffman, 2010 
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*) 
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header {* Development of the Reals using Cauchy Sequences *} 
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15131  10 
theory RealDef 
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imports Rat 
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begin 
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text {* 
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This theory contains a formalization of the real numbers as 
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equivalence classes of Cauchy sequences of rationals. See 
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\url{HOL/ex/Dedekind_Real.thy} for an alternative construction 
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using Dedekind cuts. 
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*} 
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subsection {* Preliminary lemmas *} 
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lemma add_diff_add: 
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fixes a b c d :: "'a::ab_group_add" 
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shows "(a + c)  (b + d) = (a  b) + (c  d)" 
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by simp 
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lemma minus_diff_minus: 
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fixes a b :: "'a::ab_group_add" 
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shows " a   b =  (a  b)" 
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by simp 
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lemma mult_diff_mult: 
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fixes x y a b :: "'a::ring" 
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shows "(x * y  a * b) = x * (y  b) + (x  a) * b" 
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by (simp add: algebra_simps) 
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lemma inverse_diff_inverse: 
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fixes a b :: "'a::division_ring" 
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assumes "a \<noteq> 0" and "b \<noteq> 0" 
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shows "inverse a  inverse b =  (inverse a * (a  b) * inverse b)" 
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using assms by (simp add: algebra_simps) 
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lemma obtain_pos_sum: 
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fixes r :: rat assumes r: "0 < r" 
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obtains s t where "0 < s" and "0 < t" and "r = s + t" 
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proof 
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from r show "0 < r/2" by simp 
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from r show "0 < r/2" by simp 
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show "r = r/2 + r/2" by simp 
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qed 
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subsection {* Sequences that converge to zero *} 
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definition 
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vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool" 
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where 
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"vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)" 
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lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X" 
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unfolding vanishes_def by simp 
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lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r" 
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unfolding vanishes_def by simp 
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lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0" 
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unfolding vanishes_def 
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apply (cases "c = 0", auto) 
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apply (rule exI [where x="\<bar>c\<bar>"], auto) 
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done 
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lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n.  X n)" 
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unfolding vanishes_def by simp 
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lemma vanishes_add: 
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assumes X: "vanishes X" and Y: "vanishes Y" 
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shows "vanishes (\<lambda>n. X n + Y n)" 
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proof (rule vanishesI) 
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fix r :: rat assume "0 < r" 
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then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" 
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by (rule obtain_pos_sum) 
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obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s" 
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using vanishesD [OF X s] .. 
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obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t" 
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using vanishesD [OF Y t] .. 
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have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r" 
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proof (clarsimp) 
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fix n assume n: "i \<le> n" "j \<le> n" 
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have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq) 
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also have "\<dots> < s + t" by (simp add: add_strict_mono i j n) 
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finally show "\<bar>X n + Y n\<bar> < r" unfolding r . 
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qed 
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thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" .. 
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qed 
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lemma vanishes_diff: 
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assumes X: "vanishes X" and Y: "vanishes Y" 
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shows "vanishes (\<lambda>n. X n  Y n)" 
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unfolding diff_minus by (intro vanishes_add vanishes_minus X Y) 
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lemma vanishes_mult_bounded: 
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assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a" 
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assumes Y: "vanishes (\<lambda>n. Y n)" 
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shows "vanishes (\<lambda>n. X n * Y n)" 
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proof (rule vanishesI) 
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fix r :: rat assume r: "0 < r" 
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obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a" 
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using X by fast 
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obtain b where b: "0 < b" "r = a * b" 
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proof 
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show "0 < r / a" using r a by (simp add: divide_pos_pos) 
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show "r = a * (r / a)" using a by simp 
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qed 
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obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b" 
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using vanishesD [OF Y b(1)] .. 
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have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" 
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by (simp add: b(2) abs_mult mult_strict_mono' a k) 
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thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" .. 
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qed 
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subsection {* Cauchy sequences *} 
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cauchy :: "(nat \<Rightarrow> rat) set" 
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where 
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"cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m  X n\<bar> < r)" 
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lemma cauchyI: 
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"(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m  X n\<bar> < r) \<Longrightarrow> cauchy X" 
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unfolding cauchy_def by simp 
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lemma cauchyD: 
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"\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m  X n\<bar> < r" 
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unfolding cauchy_def by simp 
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lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)" 
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unfolding cauchy_def by simp 
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lemma cauchy_add [simp]: 
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assumes X: "cauchy X" and Y: "cauchy Y" 
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shows "cauchy (\<lambda>n. X n + Y n)" 
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proof (rule cauchyI) 
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fix r :: rat assume "0 < r" 
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then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" 
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by (rule obtain_pos_sum) 
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obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m  X n\<bar> < s" 
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using cauchyD [OF X s] .. 
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obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m  Y n\<bar> < t" 
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using cauchyD [OF Y t] .. 
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150 
have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m)  (X n + Y n)\<bar> < r" 
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parents:
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changeset

151 
proof (clarsimp) 
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parents:
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152 
fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n" 
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parents:
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changeset

153 
have "\<bar>(X m + Y m)  (X n + Y n)\<bar> \<le> \<bar>X m  X n\<bar> + \<bar>Y m  Y n\<bar>" 
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new construction of real numbers using Cauchy sequences
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parents:
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changeset

154 
unfolding add_diff_add by (rule abs_triangle_ineq) 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

155 
also have "\<dots> < s + t" 
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parents:
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changeset

156 
by (rule add_strict_mono, simp_all add: i j *) 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

157 
finally show "\<bar>(X m + Y m)  (X n + Y n)\<bar> < r" unfolding r . 
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parents:
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changeset

158 
qed 
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parents:
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changeset

159 
thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m)  (X n + Y n)\<bar> < r" .. 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

160 
qed 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

161 

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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

162 
lemma cauchy_minus [simp]: 
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new construction of real numbers using Cauchy sequences
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163 
assumes X: "cauchy X" 
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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

164 
shows "cauchy (\<lambda>n.  X n)" 
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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

165 
using assms unfolding cauchy_def 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

166 
unfolding minus_diff_minus abs_minus_cancel . 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

167 

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new construction of real numbers using Cauchy sequences
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parents:
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changeset

168 
lemma cauchy_diff [simp]: 
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169 
assumes X: "cauchy X" and Y: "cauchy Y" 
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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

170 
shows "cauchy (\<lambda>n. X n  Y n)" 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

171 
using assms unfolding diff_minus by simp 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

172 

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changeset

173 
lemma cauchy_imp_bounded: 
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174 
assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b" 
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parents:
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changeset

175 
proof  
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

176 
obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m  X n\<bar> < 1" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

177 
using cauchyD [OF assms zero_less_one] .. 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

178 
show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b" 
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parents:
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changeset

179 
proof (intro exI conjI allI) 
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new construction of real numbers using Cauchy sequences
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parents:
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changeset

180 
have "0 \<le> \<bar>X 0\<bar>" by simp 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

181 
also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp 
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huffman
parents:
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changeset

182 
finally have "0 \<le> Max (abs ` X ` {..k})" . 
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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

183 
thus "0 < Max (abs ` X ` {..k}) + 1" by simp 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

184 
next 
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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

185 
fix n :: nat 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

186 
show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

187 
proof (rule linorder_le_cases) 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

188 
assume "n \<le> k" 
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new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

189 
hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

190 
thus "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

191 
next 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

192 
assume "k \<le> n" 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

193 
have "\<bar>X n\<bar> = \<bar>X k + (X n  X k)\<bar>" by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

194 
also have "\<bar>X k + (X n  X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n  X k\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

195 
by (rule abs_triangle_ineq) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

196 
also have "\<dots> < Max (abs ` X ` {..k}) + 1" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

197 
by (rule add_le_less_mono, simp, simp add: k `k \<le> n`) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

198 
finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" . 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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changeset

199 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

200 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

201 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

202 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

203 
lemma cauchy_mult [simp]: 
e05e1283c550
new construction of real numbers using Cauchy sequences
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parents:
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diff
changeset

204 
assumes X: "cauchy X" and Y: "cauchy Y" 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

205 
shows "cauchy (\<lambda>n. X n * Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

206 
proof (rule cauchyI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

207 
fix r :: rat assume "0 < r" 
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new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

208 
then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

209 
by (rule obtain_pos_sum) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

210 
obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

211 
using cauchy_imp_bounded [OF X] by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

212 
obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

213 
using cauchy_imp_bounded [OF Y] by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

214 
obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

215 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

216 
show "0 < v/b" using v b(1) by (rule divide_pos_pos) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

217 
show "0 < u/a" using u a(1) by (rule divide_pos_pos) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

218 
show "r = a * (u/a) + (v/b) * b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

219 
using a(1) b(1) `r = u + v` by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

220 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

221 
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m  X n\<bar> < s" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

222 
using cauchyD [OF X s] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

223 
obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m  Y n\<bar> < t" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

224 
using cauchyD [OF Y t] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

225 
have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m  X n * Y n\<bar> < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

226 
proof (clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

227 
fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

228 
have "\<bar>X m * Y m  X n * Y n\<bar> = \<bar>X m * (Y m  Y n) + (X m  X n) * Y n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

229 
unfolding mult_diff_mult .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

230 
also have "\<dots> \<le> \<bar>X m * (Y m  Y n)\<bar> + \<bar>(X m  X n) * Y n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

231 
by (rule abs_triangle_ineq) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

232 
also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m  Y n\<bar> + \<bar>X m  X n\<bar> * \<bar>Y n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

233 
unfolding abs_mult .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

234 
also have "\<dots> < a * t + s * b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

235 
by (simp_all add: add_strict_mono mult_strict_mono' a b i j *) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

236 
finally show "\<bar>X m * Y m  X n * Y n\<bar> < r" unfolding r . 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

237 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

238 
thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m  X n * Y n\<bar> < r" .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

239 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

240 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

241 
lemma cauchy_not_vanishes_cases: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

242 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

243 
assumes nz: "\<not> vanishes X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

244 
shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b <  X n) \<or> (\<forall>n\<ge>k. b < X n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

245 
proof  
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

246 
obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

247 
using nz unfolding vanishes_def by (auto simp add: not_less) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

248 
obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

249 
using `0 < r` by (rule obtain_pos_sum) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

250 
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m  X n\<bar> < s" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

251 
using cauchyD [OF X s] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

252 
obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

253 
using r by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

254 
have k: "\<forall>n\<ge>k. \<bar>X n  X k\<bar> < s" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

255 
using i `i \<le> k` by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

256 
have "X k \<le>  r \<or> r \<le> X k" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

257 
using `r \<le> \<bar>X k\<bar>` by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

258 
hence "(\<forall>n\<ge>k. t <  X n) \<or> (\<forall>n\<ge>k. t < X n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

259 
unfolding `r = s + t` using k by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

260 
hence "\<exists>k. (\<forall>n\<ge>k. t <  X n) \<or> (\<forall>n\<ge>k. t < X n)" .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

261 
thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t <  X n) \<or> (\<forall>n\<ge>k. t < X n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

262 
using t by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

263 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

264 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

265 
lemma cauchy_not_vanishes: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

266 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

267 
assumes nz: "\<not> vanishes X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

268 
shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

269 
using cauchy_not_vanishes_cases [OF assms] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

270 
by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

271 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

272 
lemma cauchy_inverse [simp]: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

273 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

274 
assumes nz: "\<not> vanishes X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

275 
shows "cauchy (\<lambda>n. inverse (X n))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

276 
proof (rule cauchyI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

277 
fix r :: rat assume "0 < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

278 
obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

279 
using cauchy_not_vanishes [OF X nz] by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

280 
from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

281 
obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

282 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
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283 
show "0 < b * r * b" 
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284 
by (simp add: `0 < r` b mult_pos_pos) 
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285 
show "r = inverse b * (b * r * b) * inverse b" 
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286 
using b by simp 
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287 
qed 
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288 
obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m  X n\<bar> < s" 
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289 
using cauchyD [OF X s] .. 
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290 
have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m)  inverse (X n)\<bar> < r" 
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291 
proof (clarsimp) 
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292 
fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n" 
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293 
have "\<bar>inverse (X m)  inverse (X n)\<bar> = 
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294 
inverse \<bar>X m\<bar> * \<bar>X m  X n\<bar> * inverse \<bar>X n\<bar>" 
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295 
by (simp add: inverse_diff_inverse nz * abs_mult) 
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296 
also have "\<dots> < inverse b * s * inverse b" 
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297 
by (simp add: mult_strict_mono less_imp_inverse_less 
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298 
mult_pos_pos i j b * s) 
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299 
finally show "\<bar>inverse (X m)  inverse (X n)\<bar> < r" unfolding r . 
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300 
qed 
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301 
thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m)  inverse (X n)\<bar> < r" .. 
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302 
qed 
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303 

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304 
subsection {* Equivalence relation on Cauchy sequences *} 
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305 

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306 
definition 
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307 
realrel :: "((nat \<Rightarrow> rat) \<times> (nat \<Rightarrow> rat)) set" 
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308 
where 
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309 
"realrel = {(X, Y). cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n  Y n)}" 
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310 

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311 
lemma refl_realrel: "refl_on {X. cauchy X} realrel" 
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312 
unfolding realrel_def by (rule refl_onI, clarsimp, simp) 
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313 

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314 
lemma sym_realrel: "sym realrel" 
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315 
unfolding realrel_def 
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316 
by (rule symI, clarify, drule vanishes_minus, simp) 
14484  317 

36795
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318 
lemma trans_realrel: "trans realrel" 
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319 
unfolding realrel_def 
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320 
apply (rule transI, clarify) 
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321 
apply (drule (1) vanishes_add) 
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322 
apply (simp add: algebra_simps) 
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323 
done 
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324 

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325 
lemma equiv_realrel: "equiv {X. cauchy X} realrel" 
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326 
using refl_realrel sym_realrel trans_realrel 
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327 
by (rule equiv.intro) 
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328 

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329 
subsection {* The field of real numbers *} 
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330 

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331 
typedef (open) real = "{X. cauchy X} // realrel" 
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332 
by (fast intro: quotientI cauchy_const) 
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333 

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334 
definition 
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335 
Real :: "(nat \<Rightarrow> rat) \<Rightarrow> real" 
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336 
where 
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337 
"Real X = Abs_real (realrel `` {X})" 
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338 

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339 
definition 
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340 
real_case :: "((nat \<Rightarrow> rat) \<Rightarrow> 'a) \<Rightarrow> real \<Rightarrow> 'a" 
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341 
where 
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342 
"real_case f x = (THE y. \<forall>X\<in>Rep_real x. y = f X)" 
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343 

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344 
lemma Real_induct [induct type: real]: 
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345 
"(\<And>X. cauchy X \<Longrightarrow> P (Real X)) \<Longrightarrow> P x" 
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346 
unfolding Real_def 
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347 
apply (induct x) 
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348 
apply (erule quotientE) 
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349 
apply (simp) 
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350 
done 
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351 

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352 
lemma real_case_1: 
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353 
assumes f: "congruent realrel f" 
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354 
assumes X: "cauchy X" 
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355 
shows "real_case f (Real X) = f X" 
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356 
unfolding real_case_def Real_def 
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357 
apply (subst Abs_real_inverse) 
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358 
apply (simp add: quotientI X) 
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359 
apply (rule the_equality) 
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360 
apply clarsimp 
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361 
apply (erule congruent.congruent [OF f]) 
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362 
apply (erule bspec) 
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363 
apply simp 
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364 
apply (rule refl_onD [OF refl_realrel]) 
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365 
apply (simp add: X) 
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366 
done 
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367 

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368 
lemma real_case_2: 
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369 
assumes f: "congruent2 realrel realrel f" 
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370 
assumes X: "cauchy X" and Y: "cauchy Y" 
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371 
shows "real_case (\<lambda>X. real_case (\<lambda>Y. f X Y) (Real Y)) (Real X) = f X Y" 
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372 
apply (subst real_case_1 [OF _ X]) 
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373 
apply (rule congruent.intro) 
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374 
apply (subst real_case_1 [OF _ Y]) 
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375 
apply (rule congruent2_implies_congruent [OF equiv_realrel f]) 
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376 
apply (simp add: realrel_def) 
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377 
apply (subst real_case_1 [OF _ Y]) 
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378 
apply (rule congruent2_implies_congruent [OF equiv_realrel f]) 
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379 
apply (simp add: realrel_def) 
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380 
apply (erule congruent2.congruent2 [OF f]) 
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381 
apply (rule refl_onD [OF refl_realrel]) 
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parents:
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382 
apply (simp add: Y) 
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383 
apply (rule real_case_1 [OF _ Y]) 
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384 
apply (rule congruent2_implies_congruent [OF equiv_realrel f]) 
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385 
apply (simp add: X) 
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386 
done 
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387 

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388 
lemma eq_Real: 
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389 
"cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n  Y n)" 
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390 
unfolding Real_def 
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391 
apply (subst Abs_real_inject) 
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diff
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392 
apply (simp add: quotientI) 
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huffman
parents:
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diff
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393 
apply (simp add: quotientI) 
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huffman
parents:
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diff
changeset

394 
apply (simp add: eq_equiv_class_iff [OF equiv_realrel]) 
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395 
apply (simp add: realrel_def) 
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diff
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396 
done 
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diff
changeset

397 

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398 
lemma add_respects2_realrel: 
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399 
"(\<lambda>X Y. Real (\<lambda>n. X n + Y n)) respects2 realrel" 
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400 
proof (rule congruent2_commuteI [OF equiv_realrel, unfolded mem_Collect_eq]) 
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401 
fix X Y show "Real (\<lambda>n. X n + Y n) = Real (\<lambda>n. Y n + X n)" 
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402 
by (simp add: add_commute) 
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403 
next 
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404 
fix X assume X: "cauchy X" 
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changeset

405 
fix Y Z assume "(Y, Z) \<in> realrel" 
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406 
hence Y: "cauchy Y" and Z: "cauchy Z" and YZ: "vanishes (\<lambda>n. Y n  Z n)" 
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changeset

407 
unfolding realrel_def by simp_all 
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408 
show "Real (\<lambda>n. X n + Y n) = Real (\<lambda>n. X n + Z n)" 
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409 
proof (rule eq_Real [THEN iffD2]) 
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410 
show "cauchy (\<lambda>n. X n + Y n)" using X Y by (rule cauchy_add) 
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411 
show "cauchy (\<lambda>n. X n + Z n)" using X Z by (rule cauchy_add) 
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412 
show "vanishes (\<lambda>n. (X n + Y n)  (X n + Z n))" 
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changeset

413 
unfolding add_diff_add using YZ by simp 
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parents:
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diff
changeset

414 
qed 
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parents:
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415 
qed 
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parents:
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diff
changeset

416 

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417 
lemma minus_respects_realrel: 
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418 
"(\<lambda>X. Real (\<lambda>n.  X n)) respects realrel" 
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419 
proof (rule congruent.intro) 
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420 
fix X Y assume "(X, Y) \<in> realrel" 
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421 
hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n  Y n)" 
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422 
unfolding realrel_def by simp_all 
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423 
show "Real (\<lambda>n.  X n) = Real (\<lambda>n.  Y n)" 
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424 
proof (rule eq_Real [THEN iffD2]) 
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425 
show "cauchy (\<lambda>n.  X n)" using X by (rule cauchy_minus) 
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426 
show "cauchy (\<lambda>n.  Y n)" using Y by (rule cauchy_minus) 
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427 
show "vanishes (\<lambda>n. ( X n)  ( Y n))" 
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changeset

428 
unfolding minus_diff_minus using XY by (rule vanishes_minus) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

429 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

430 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

431 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

432 
lemma mult_respects2_realrel: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

433 
"(\<lambda>X Y. Real (\<lambda>n. X n * Y n)) respects2 realrel" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

434 
proof (rule congruent2_commuteI [OF equiv_realrel, unfolded mem_Collect_eq]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

435 
fix X Y 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

436 
show "Real (\<lambda>n. X n * Y n) = Real (\<lambda>n. Y n * X n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

437 
by (simp add: mult_commute) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

438 
next 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

439 
fix X assume X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

440 
fix Y Z assume "(Y, Z) \<in> realrel" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

441 
hence Y: "cauchy Y" and Z: "cauchy Z" and YZ: "vanishes (\<lambda>n. Y n  Z n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

442 
unfolding realrel_def by simp_all 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

443 
show "Real (\<lambda>n. X n * Y n) = Real (\<lambda>n. X n * Z n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

444 
proof (rule eq_Real [THEN iffD2]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

445 
show "cauchy (\<lambda>n. X n * Y n)" using X Y by (rule cauchy_mult) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

446 
show "cauchy (\<lambda>n. X n * Z n)" using X Z by (rule cauchy_mult) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

447 
have "vanishes (\<lambda>n. X n * (Y n  Z n))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

448 
by (intro vanishes_mult_bounded cauchy_imp_bounded X YZ) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

449 
thus "vanishes (\<lambda>n. X n * Y n  X n * Z n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

450 
by (simp add: right_diff_distrib) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

451 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

452 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

453 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

454 
lemma vanishes_diff_inverse: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

455 
assumes X: "cauchy X" "\<not> vanishes X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

456 
assumes Y: "cauchy Y" "\<not> vanishes Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

457 
assumes XY: "vanishes (\<lambda>n. X n  Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

458 
shows "vanishes (\<lambda>n. inverse (X n)  inverse (Y n))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

459 
proof (rule vanishesI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

460 
fix r :: rat assume r: "0 < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

461 
obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

462 
using cauchy_not_vanishes [OF X] by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

463 
obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

464 
using cauchy_not_vanishes [OF Y] by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

465 
obtain s where s: "0 < s" and "inverse a * s * inverse b = r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

466 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

467 
show "0 < a * r * b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

468 
using a r b by (simp add: mult_pos_pos) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

469 
show "inverse a * (a * r * b) * inverse b = r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

470 
using a r b by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

471 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

472 
obtain k where k: "\<forall>n\<ge>k. \<bar>X n  Y n\<bar> < s" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

473 
using vanishesD [OF XY s] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

474 
have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n)  inverse (Y n)\<bar> < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

475 
proof (clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

476 
fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

477 
have "X n \<noteq> 0" and "Y n \<noteq> 0" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

478 
using i j a b n by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

479 
hence "\<bar>inverse (X n)  inverse (Y n)\<bar> = 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

480 
inverse \<bar>X n\<bar> * \<bar>X n  Y n\<bar> * inverse \<bar>Y n\<bar>" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

481 
by (simp add: inverse_diff_inverse abs_mult) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

482 
also have "\<dots> < inverse a * s * inverse b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

483 
apply (intro mult_strict_mono' less_imp_inverse_less) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

484 
apply (simp_all add: a b i j k n mult_nonneg_nonneg) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

485 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

486 
also note `inverse a * s * inverse b = r` 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

487 
finally show "\<bar>inverse (X n)  inverse (Y n)\<bar> < r" . 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

488 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

489 
thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n)  inverse (Y n)\<bar> < r" .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

490 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

491 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

492 
lemma inverse_respects_realrel: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

493 
"(\<lambda>X. if vanishes X then c else Real (\<lambda>n. inverse (X n))) respects realrel" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

494 
(is "?inv respects realrel") 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

495 
proof (rule congruent.intro) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

496 
fix X Y assume "(X, Y) \<in> realrel" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

497 
hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n  Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

498 
unfolding realrel_def by simp_all 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

499 
have "vanishes X \<longleftrightarrow> vanishes Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

500 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

501 
assume "vanishes X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

502 
from vanishes_diff [OF this XY] show "vanishes Y" by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

503 
next 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

504 
assume "vanishes Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

505 
from vanishes_add [OF this XY] show "vanishes X" by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

506 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

507 
thus "?inv X = ?inv Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

508 
by (simp add: vanishes_diff_inverse eq_Real X Y XY) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

509 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

510 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

511 
instantiation real :: field_inverse_zero 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

512 
begin 
5588  513 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

514 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

515 
"0 = Real (\<lambda>n. 0)" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

516 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

517 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

518 
"1 = Real (\<lambda>n. 1)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

519 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

520 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

521 
"x + y = real_case (\<lambda>X. real_case (\<lambda>Y. Real (\<lambda>n. X n + Y n)) y) x" 
5588  522 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

523 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

524 
" x = real_case (\<lambda>X. Real (\<lambda>n.  X n)) x" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

525 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

526 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

527 
"x  y = (x::real) +  y" 
10606  528 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

529 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

530 
"x * y = real_case (\<lambda>X. real_case (\<lambda>Y. Real (\<lambda>n. X n * Y n)) y) x" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

531 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

532 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

533 
"inverse = 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

534 
real_case (\<lambda>X. if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))" 
14484  535 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

536 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

537 
"x / y = (x::real) * inverse y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

538 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

539 
lemma add_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

540 
assumes X: "cauchy X" and Y: "cauchy Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

541 
shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

542 
unfolding plus_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

543 
by (rule real_case_2 [OF add_respects2_realrel X Y]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

544 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

545 
lemma minus_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

546 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

547 
shows " Real X = Real (\<lambda>n.  X n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

548 
unfolding uminus_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

549 
by (rule real_case_1 [OF minus_respects_realrel X]) 
5588  550 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

551 
lemma diff_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

552 
assumes X: "cauchy X" and Y: "cauchy Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

553 
shows "Real X  Real Y = Real (\<lambda>n. X n  Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

554 
unfolding minus_real_def diff_minus 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

555 
by (simp add: minus_Real add_Real X Y) 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

556 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

557 
lemma mult_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

558 
assumes X: "cauchy X" and Y: "cauchy Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

559 
shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

560 
unfolding times_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

561 
by (rule real_case_2 [OF mult_respects2_realrel X Y]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

562 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

563 
lemma inverse_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

564 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

565 
shows "inverse (Real X) = 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

566 
(if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

567 
unfolding inverse_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

568 
by (rule real_case_1 [OF inverse_respects_realrel X]) 
14269  569 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

570 
instance proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

571 
fix a b c :: real 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

572 
show "a + b = b + a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

573 
by (induct a, induct b) (simp add: add_Real add_ac) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

574 
show "(a + b) + c = a + (b + c)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

575 
by (induct a, induct b, induct c) (simp add: add_Real add_ac) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

576 
show "0 + a = a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

577 
unfolding zero_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

578 
by (induct a) (simp add: add_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

579 
show " a + a = 0" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

580 
unfolding zero_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

581 
by (induct a) (simp add: minus_Real add_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

582 
show "a  b = a +  b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

583 
by (rule minus_real_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

584 
show "(a * b) * c = a * (b * c)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

585 
by (induct a, induct b, induct c) (simp add: mult_Real mult_ac) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

586 
show "a * b = b * a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

587 
by (induct a, induct b) (simp add: mult_Real mult_ac) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

588 
show "1 * a = a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

589 
unfolding one_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

590 
by (induct a) (simp add: mult_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

591 
show "(a + b) * c = a * c + b * c" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

592 
by (induct a, induct b, induct c) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

593 
(simp add: mult_Real add_Real algebra_simps) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

594 
show "(0\<Colon>real) \<noteq> (1\<Colon>real)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

595 
unfolding zero_real_def one_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

596 
by (simp add: eq_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

597 
show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

598 
unfolding zero_real_def one_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

599 
apply (induct a) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

600 
apply (simp add: eq_Real inverse_Real mult_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

601 
apply (rule vanishesI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

602 
apply (frule (1) cauchy_not_vanishes, clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

603 
apply (rule_tac x=k in exI, clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

604 
apply (drule_tac x=n in spec, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

605 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

606 
show "a / b = a * inverse b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

607 
by (rule divide_real_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

608 
show "inverse (0::real) = 0" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

609 
by (simp add: zero_real_def inverse_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

610 
qed 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

611 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

612 
end 
14334  613 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

614 
subsection {* Positive reals *} 
14269  615 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

616 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

617 
positive :: "real \<Rightarrow> bool" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

618 
where 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

619 
"positive = real_case (\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)" 
14269  620 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

621 
lemma bool_congruentI: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

622 
assumes sym: "sym r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

623 
assumes P: "\<And>x y. (x, y) \<in> r \<Longrightarrow> P x \<Longrightarrow> P y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

624 
shows "P respects r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

625 
apply (rule congruent.intro) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

626 
apply (rule iffI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

627 
apply (erule (1) P) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

628 
apply (erule (1) P [OF symD [OF sym]]) 
14269  629 
done 
630 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

631 
lemma positive_respects_realrel: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

632 
"(\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n) respects realrel" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

633 
proof (rule bool_congruentI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

634 
show "sym realrel" by (rule sym_realrel) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

635 
next 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

636 
fix X Y assume "(X, Y) \<in> realrel" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

637 
hence XY: "vanishes (\<lambda>n. X n  Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

638 
unfolding realrel_def by simp_all 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

639 
assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

640 
then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

641 
by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

642 
obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

643 
using `0 < r` by (rule obtain_pos_sum) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

644 
obtain j where j: "\<forall>n\<ge>j. \<bar>X n  Y n\<bar> < s" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

645 
using vanishesD [OF XY s] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

646 
have "\<forall>n\<ge>max i j. t < Y n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

647 
proof (clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

648 
fix n assume n: "i \<le> n" "j \<le> n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

649 
have "\<bar>X n  Y n\<bar> < s" and "r < X n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

650 
using i j n by simp_all 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

651 
thus "t < Y n" unfolding r by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

652 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

653 
thus "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast 
14484  654 
qed 
14269  655 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

656 
lemma positive_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

657 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

658 
shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

659 
unfolding positive_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

660 
by (rule real_case_1 [OF positive_respects_realrel X]) 
23287
063039db59dd
define (1::preal); clean up instance declarations
huffman
parents:
23031
diff
changeset

661 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

662 
lemma positive_zero: "\<not> positive 0" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

663 
unfolding zero_real_def by (auto simp add: positive_Real) 
14269  664 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

665 
lemma positive_add: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

666 
"positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

667 
apply (induct x, induct y, rename_tac Y X) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

668 
apply (simp add: add_Real positive_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

669 
apply (clarify, rename_tac a b i j) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

670 
apply (rule_tac x="a + b" in exI, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

671 
apply (rule_tac x="max i j" in exI, clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

672 
apply (simp add: add_strict_mono) 
14269  673 
done 
674 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

675 
lemma positive_mult: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

676 
"positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

677 
apply (induct x, induct y, rename_tac Y X) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

678 
apply (simp add: mult_Real positive_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

679 
apply (clarify, rename_tac a b i j) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

680 
apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

681 
apply (rule_tac x="max i j" in exI, clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

682 
apply (rule mult_strict_mono, auto) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

683 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

684 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

685 
lemma positive_minus: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

686 
"\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive ( x)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

687 
apply (induct x, rename_tac X) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

688 
apply (simp add: zero_real_def eq_Real minus_Real positive_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

689 
apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast) 
14269  690 
done 
14334  691 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

692 
instantiation real :: linordered_field_inverse_zero 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

693 
begin 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

694 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

695 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

696 
"x < y \<longleftrightarrow> positive (y  x)" 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14335
diff
changeset

697 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

698 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

699 
"x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y" 
14334  700 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

701 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

702 
"abs (a::real) = (if a < 0 then  a else a)" 
14269  703 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

704 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

705 
"sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else  1)" 
14269  706 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

707 
instance proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

708 
fix a b c :: real 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

709 
show "\<bar>a\<bar> = (if a < 0 then  a else a)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

710 
by (rule abs_real_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

711 
show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

712 
unfolding less_eq_real_def less_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

713 
by (auto, drule (1) positive_add, simp_all add: positive_zero) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

714 
show "a \<le> a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

715 
unfolding less_eq_real_def by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

716 
show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

717 
unfolding less_eq_real_def less_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

718 
by (auto, drule (1) positive_add, simp add: algebra_simps) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

719 
show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

720 
unfolding less_eq_real_def less_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

721 
by (auto, drule (1) positive_add, simp add: positive_zero) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

722 
show "a \<le> b \<Longrightarrow> c + a \<le> c + b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

723 
unfolding less_eq_real_def less_real_def by auto 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

724 
show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else  1)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

725 
by (rule sgn_real_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

726 
show "a \<le> b \<or> b \<le> a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

727 
unfolding less_eq_real_def less_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

728 
by (auto dest!: positive_minus) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

729 
show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

730 
unfolding less_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

731 
by (drule (1) positive_mult, simp add: algebra_simps) 
23288  732 
qed 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

733 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

734 
end 
14334  735 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

736 
instantiation real :: distrib_lattice 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

737 
begin 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

738 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

739 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

740 
"(inf :: real \<Rightarrow> real \<Rightarrow> real) = min" 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

741 

c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

742 
definition 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

743 
"(sup :: real \<Rightarrow> real \<Rightarrow> real) = max" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

744 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

745 
instance proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

746 
qed (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

747 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

748 
end 
25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

749 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

750 
instantiation real :: number_ring 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

751 
begin 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

752 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

753 
definition 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

754 
"(number_of x :: real) = of_int x" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

755 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

756 
instance proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

757 
qed (rule number_of_real_def) 
22456  758 

25571
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

759 
end 
c9e39eafc7a0
instantiation target rather than legacy instance
haftmann
parents:
25546
diff
changeset

760 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

761 
lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

762 
apply (induct x) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

763 
apply (simp add: zero_real_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

764 
apply (simp add: one_real_def add_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

765 
done 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

766 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

767 
lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

768 
apply (cases x rule: int_diff_cases) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

769 
apply (simp add: of_nat_Real diff_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

770 
done 
14334  771 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

772 
lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

773 
apply (induct x) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

774 
apply (simp add: Fract_of_int_quotient of_rat_divide) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

775 
apply (simp add: of_int_Real divide_inverse) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

776 
apply (simp add: inverse_Real mult_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

777 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

778 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

779 
instance real :: archimedean_field 
14334  780 
proof 
36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

781 
fix x :: real 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

782 
show "\<exists>z. x \<le> of_int z" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

783 
apply (induct x) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

784 
apply (frule cauchy_imp_bounded, clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

785 
apply (rule_tac x="ceiling b + 1" in exI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

786 
apply (rule less_imp_le) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

787 
apply (simp add: of_int_Real less_real_def diff_Real positive_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

788 
apply (rule_tac x=1 in exI, simp add: algebra_simps) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

789 
apply (rule_tac x=0 in exI, clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

790 
apply (rule le_less_trans [OF abs_ge_self]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

791 
apply (rule less_le_trans [OF _ le_of_int_ceiling]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

792 
apply simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

793 
done 
14334  794 
qed 
795 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

796 
subsection {* Completeness *} 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

797 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

798 
lemma not_positive_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

799 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

800 
shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

801 
unfolding positive_Real [OF X] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

802 
apply (auto, unfold not_less) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

803 
apply (erule obtain_pos_sum) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

804 
apply (drule_tac x=s in spec, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

805 
apply (drule_tac r=t in cauchyD [OF X], clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

806 
apply (drule_tac x=k in spec, clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

807 
apply (rule_tac x=n in exI, clarify, rename_tac m) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

808 
apply (drule_tac x=m in spec, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

809 
apply (drule_tac x=n in spec, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

810 
apply (drule spec, drule (1) mp, clarify, rename_tac i) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

811 
apply (rule_tac x="max i k" in exI, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

812 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

813 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

814 
lemma le_Real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

815 
assumes X: "cauchy X" and Y: "cauchy Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

816 
shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

817 
unfolding not_less [symmetric, where 'a=real] less_real_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

818 
apply (simp add: diff_Real not_positive_Real X Y) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

819 
apply (simp add: diff_le_eq add_ac) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

820 
done 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

821 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

822 
lemma le_RealI: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

823 
assumes Y: "cauchy Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

824 
shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

825 
proof (induct x) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

826 
fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

827 
hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

828 
by (simp add: of_rat_Real le_Real) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

829 
{ 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

830 
fix r :: rat assume "0 < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

831 
then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

832 
by (rule obtain_pos_sum) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

833 
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m  Y n\<bar> < s" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

834 
using cauchyD [OF Y s] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

835 
obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

836 
using le [OF t] .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

837 
have "\<forall>n\<ge>max i j. X n \<le> Y n + r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

838 
proof (clarsimp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

839 
fix n assume n: "i \<le> n" "j \<le> n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

840 
have "X n \<le> Y i + t" using n j by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

841 
moreover have "\<bar>Y i  Y n\<bar> < s" using n i by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

842 
ultimately show "X n \<le> Y n + r" unfolding r by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

843 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

844 
hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

845 
} 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

846 
thus "Real X \<le> Real Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

847 
by (simp add: of_rat_Real le_Real X Y) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

848 
qed 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

849 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

850 
lemma Real_leI: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

851 
assumes X: "cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

852 
assumes le: "\<forall>n. of_rat (X n) \<le> y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

853 
shows "Real X \<le> y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

854 
proof  
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

855 
have " y \<le>  Real X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

856 
by (simp add: minus_Real X le_RealI of_rat_minus le) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

857 
thus ?thesis by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

858 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

859 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

860 
lemma less_RealD: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

861 
assumes Y: "cauchy Y" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

862 
shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

863 
by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

864 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

865 
lemma of_nat_less_two_power: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

866 
"of_nat n < (2::'a::{linordered_idom,number_ring}) ^ n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

867 
apply (induct n) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

868 
apply simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

869 
apply (subgoal_tac "(1::'a) \<le> 2 ^ n") 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

870 
apply (drule (1) add_le_less_mono, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

871 
apply simp 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

872 
done 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

873 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

874 
lemma complete_real: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

875 
fixes S :: "real set" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

876 
assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

877 
shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

878 
proof  
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

879 
obtain x where x: "x \<in> S" using assms(1) .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

880 
obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) .. 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

881 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

882 
def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

883 
obtain a where a: "\<not> P a" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

884 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

885 
have "of_int (floor (x  1)) \<le> x  1" by (rule of_int_floor_le) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

886 
also have "x  1 < x" by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

887 
finally have "of_int (floor (x  1)) < x" . 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

888 
hence "\<not> x \<le> of_int (floor (x  1))" by (simp only: not_le) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

889 
then show "\<not> P (of_int (floor (x  1)))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

890 
unfolding P_def of_rat_of_int_eq using x by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

891 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

892 
obtain b where b: "P b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

893 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

894 
show "P (of_int (ceiling z))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

895 
unfolding P_def of_rat_of_int_eq 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

896 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

897 
fix y assume "y \<in> S" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

898 
hence "y \<le> z" using z by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

899 
also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

900 
finally show "y \<le> of_int (ceiling z)" . 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

901 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

902 
qed 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

903 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

904 
def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

905 
def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

906 
def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

907 
def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

908 
def C \<equiv> "\<lambda>n. avg (A n) (B n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

909 
have A_0 [simp]: "A 0 = a" unfolding A_def by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

910 
have B_0 [simp]: "B 0 = b" unfolding B_def by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

911 
have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

912 
unfolding A_def B_def C_def bisect_def split_def by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

913 
have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

914 
unfolding A_def B_def C_def bisect_def split_def by simp 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

915 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

916 
have width: "\<And>n. B n  A n = (b  a) / 2^n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

917 
apply (simp add: eq_divide_eq) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

918 
apply (induct_tac n, simp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

919 
apply (simp add: C_def avg_def algebra_simps) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

920 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

921 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

922 
have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

923 
apply (simp add: divide_less_eq) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

924 
apply (subst mult_commute) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

925 
apply (frule_tac y=y in ex_less_of_nat_mult) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

926 
apply clarify 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

927 
apply (rule_tac x=n in exI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

928 
apply (erule less_trans) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

929 
apply (rule mult_strict_right_mono) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

930 
apply (rule le_less_trans [OF _ of_nat_less_two_power]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

931 
apply simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

932 
apply assumption 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

933 
done 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

934 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

935 
have PA: "\<And>n. \<not> P (A n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

936 
by (induct_tac n, simp_all add: a) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

937 
have PB: "\<And>n. P (B n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

938 
by (induct_tac n, simp_all add: b) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

939 
have ab: "a < b" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

940 
using a b unfolding P_def 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

941 
apply (clarsimp simp add: not_le) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

942 
apply (drule (1) bspec) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

943 
apply (drule (1) less_le_trans) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

944 
apply (simp add: of_rat_less) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

945 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

946 
have AB: "\<And>n. A n < B n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

947 
by (induct_tac n, simp add: ab, simp add: C_def avg_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

948 
have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

949 
apply (auto simp add: le_less [where 'a=nat]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

950 
apply (erule less_Suc_induct) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

951 
apply (clarsimp simp add: C_def avg_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

952 
apply (simp add: add_divide_distrib [symmetric]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

953 
apply (rule AB [THEN less_imp_le]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

954 
apply simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

955 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

956 
have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

957 
apply (auto simp add: le_less [where 'a=nat]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

958 
apply (erule less_Suc_induct) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

959 
apply (clarsimp simp add: C_def avg_def) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

960 
apply (simp add: add_divide_distrib [symmetric]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

961 
apply (rule AB [THEN less_imp_le]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

962 
apply simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

963 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

964 
have cauchy_lemma: 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

965 
"\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

966 
apply (rule cauchyI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

967 
apply (drule twos [where y="b  a"]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

968 
apply (erule exE) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

969 
apply (rule_tac x=n in exI, clarify, rename_tac i j) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

970 
apply (rule_tac y="B n  A n" in le_less_trans) defer 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

971 
apply (simp add: width) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

972 
apply (drule_tac x=n in spec) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

973 
apply (frule_tac x=i in spec, drule (1) mp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

974 
apply (frule_tac x=j in spec, drule (1) mp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

975 
apply (frule A_mono, drule B_mono) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

976 
apply (frule A_mono, drule B_mono) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

977 
apply arith 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

978 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

979 
have "cauchy A" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

980 
apply (rule cauchy_lemma [rule_format]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

981 
apply (simp add: A_mono) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

982 
apply (erule order_trans [OF less_imp_le [OF AB] B_mono]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

983 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

984 
have "cauchy B" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

985 
apply (rule cauchy_lemma [rule_format]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

986 
apply (simp add: B_mono) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

987 
apply (erule order_trans [OF A_mono less_imp_le [OF AB]]) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

988 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

989 
have 1: "\<forall>x\<in>S. x \<le> Real B" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

990 
proof 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

991 
fix x assume "x \<in> S" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

992 
then show "x \<le> Real B" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

993 
using PB [unfolded P_def] `cauchy B` 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

994 
by (simp add: le_RealI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

995 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

996 
have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

997 
apply clarify 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

998 
apply (erule contrapos_pp) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

999 
apply (simp add: not_le) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1000 
apply (drule less_RealD [OF `cauchy A`], clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1001 
apply (subgoal_tac "\<not> P (A n)") 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1002 
apply (simp add: P_def not_le, clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1003 
apply (erule rev_bexI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1004 
apply (erule (1) less_trans) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1005 
apply (simp add: PA) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1006 
done 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1007 
have "vanishes (\<lambda>n. (b  a) / 2 ^ n)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1008 
proof (rule vanishesI) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1009 
fix r :: rat assume "0 < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1010 
then obtain k where k: "\<bar>b  a\<bar> / 2 ^ k < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1011 
using twos by fast 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1012 
have "\<forall>n\<ge>k. \<bar>(b  a) / 2 ^ n\<bar> < r" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1013 
proof (clarify) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1014 
fix n assume n: "k \<le> n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1015 
have "\<bar>(b  a) / 2 ^ n\<bar> = \<bar>b  a\<bar> / 2 ^ n" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1016 
by simp 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1017 
also have "\<dots> \<le> \<bar>b  a\<bar> / 2 ^ k" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1018 
using n by (simp add: divide_left_mono mult_pos_pos) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1019 
also note k 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

1020 
finally show "\<bar>(b  a) / 2 ^ n\<bar> < r" . 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1021 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1022 
thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b  a) / 2 ^ n\<bar> < r" .. 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1023 
qed 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1024 
hence 3: "Real B = Real A" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1025 
by (simp add: eq_Real `cauchy A` `cauchy B` width) 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1026 
show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)" 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1027 
using 1 2 3 by (rule_tac x="Real B" in exI, simp) 
14484  1028 
qed 
14365
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replacing HOL/Real/PRat, PNat by the rational number development
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parents:
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diff
changeset

1029 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

1030 
subsection {* Hiding implementation details *} 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

1031 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1032 
hide_const (open) vanishes cauchy positive Real real_case 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

1033 

36795
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1034 
declare Real_induct [induct del] 
e05e1283c550
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huffman
parents:
36776
diff
changeset

1035 
declare Abs_real_induct [induct del] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1036 
declare Abs_real_cases [cases del] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1037 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1038 
subsection {* Legacy theorem names *} 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1039 

e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
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diff
changeset

1040 
text {* TODO: Could we have a way to mark theorem names as deprecated, 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1041 
and have Isabelle issue a warning and display the preferred name? *} 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

1042 

36795
e05e1283c550
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huffman
parents:
36776
diff
changeset

1043 
lemmas real_diff_def = minus_real_def [of "r" "s", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1044 
lemmas real_divide_def = divide_real_def [of "R" "S", standard] 
e05e1283c550
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huffman
parents:
36776
diff
changeset

1045 
lemmas real_less_def = less_le [of "x::real" "y", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1046 
lemmas real_abs_def = abs_real_def [of "r", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1047 
lemmas real_sgn_def = sgn_real_def [of "x", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1048 
lemmas real_mult_assoc = mult_assoc [of "z1::real" "z2" "z3", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1049 
lemmas real_mult_1 = mult_1_left [of "z::real", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1050 
lemmas real_add_mult_distrib = left_distrib [of "z1::real" "z2" "w", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1051 
lemmas real_zero_not_eq_one = zero_neq_one [where 'a=real] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1052 
lemmas real_mult_inverse_left = left_inverse [of "x::real", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1053 
lemmas INVERSE_ZERO = inverse_zero [where 'a=real] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1054 
lemmas real_le_refl = order_refl [of "w::real", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1055 
lemmas real_le_antisym = order_antisym [of "z::real" "w", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1056 
lemmas real_le_trans = order_trans [of "i::real" "j" "k", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1057 
lemmas real_le_eq_diff = le_iff_diff_le_0 [of "x::real" "y", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1058 
lemmas real_add_left_mono = add_left_mono [of "x::real" "y" "z", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1059 
lemmas real_mult_order = mult_pos_pos [of "x::real" "y", standard] 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1060 
lemmas real_mult_less_mono2 = 
e05e1283c550
new construction of real numbers using Cauchy sequences
huffman
parents:
36776
diff
changeset

1061 
mult_strict_left_mono [of "x::real" "y" "z", COMP swap_prems_rl, standard] 
14334  1062 

1063 
subsection{*More Lemmas*} 

1064 

36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1065 
text {* BH: These lemmas should not be necessary; they should be 
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1066 
covered by existing simp rules and simplification procedures. *} 
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1067 

14334  1068 
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)" 
36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1069 
by simp (* redundant with mult_cancel_left *) 
14334  1070 

1071 
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)" 

36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1072 
by simp (* redundant with mult_cancel_right *) 
14334  1073 

1074 
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)" 

36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1075 
by simp (* solved by linordered_ring_less_cancel_factor simproc *) 
14334  1076 

1077 
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)" 

36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1078 
by simp (* solved by linordered_ring_le_cancel_factor simproc *) 
14334  1079 

1080 
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)" 

36776
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1081 
by (rule mult_le_cancel_left_pos) 
c137ae7673d3
remove a couple of redundant lemmas; simplify some proofs
huffman
parents:
36414
diff
changeset

1082 
(* BH: Why doesn't "simp" prove this one, like it does the last one? *) 
14334  1083 

1084 

24198  1085 
subsection {* Embedding numbers into the Reals *} 
1086 

1087 
abbreviation 

1088 
real_of_nat :: "nat \<Rightarrow> real" 

1089 
where 

1090 
"real_of_nat \<equiv> of_nat" 

1091 

1092 
abbreviation 

1093 
real_of_int :: "int \<Rightarrow> real" 

1094 
where 

1095 
"real_of_int \<equiv> of_int" 

1096 

1097 
abbreviation 

1098 
real_of_rat :: "rat \<Rightarrow> real" 

1099 
where 

1100 
"real_of_rat \<equiv> of_rat" 

1101 

1102 
consts 

1103 
(*overloaded constant for injecting other types into "real"*) 

1104 
real :: "'a => real" 

14365
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parents:
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diff
changeset

1105 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1106 
defs (overloaded) 
31998
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31952
diff
changeset

1107 
real_of_nat_def [code_unfold]: "real == real_of_nat" 
2c7a24f74db9
code attributes use common underscore convention
haftmann
parents:
31952
diff
changeset

1108 
real_of_int_def [code_unfold]: "real == real_of_int" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1109 

16819  1110 
lemma real_eq_of_nat: "real = of_nat" 
24198  1111 
unfolding real_of_nat_def .. 
16819  1112 

1113 
lemma real_eq_of_int: "real = of_int" 

24198  1114 
unfolding real_of_int_def .. 
16819  1115 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
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diff
changeset

1116 
lemma real_of_int_zero [simp]: "real (0::int) = 0" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1117 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1118 

3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
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diff
changeset

1119 
lemma real_of_one [simp]: "real (1::int) = (1::real)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1120 
by (simp add: real_of_int_def) 
14334  1121 

16819  1122 
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1123 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1124 

16819  1125 
lemma real_of_int_minus [simp]: "real(x) = real (x::int)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1126 
by (simp add: real_of_int_def) 
16819  1127 

1128 
lemma real_of_int_diff [simp]: "real(x  y) = real (x::int)  real y" 

1129 
by (simp add: real_of_int_def) 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1130 

16819  1131 
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1132 
by (simp add: real_of_int_def) 
14334  1133 

35344
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1134 
lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n" 
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1135 
by (simp add: real_of_int_def of_int_power) 
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1136 

e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1137 
lemmas power_real_of_int = real_of_int_power [symmetric] 
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1138 

16819  1139 
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))" 
1140 
apply (subst real_eq_of_int)+ 

1141 
apply (rule of_int_setsum) 

1142 
done 

1143 

1144 
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 

1145 
(PROD x:A. real(f x))" 

1146 
apply (subst real_eq_of_int)+ 

1147 
apply (rule of_int_setprod) 

1148 
done 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1149 

27668  1150 
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1151 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1152 

27668  1153 
lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1154 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1155 

27668  1156 
lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1157 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1158 

27668  1159 
lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1160 
by (simp add: real_of_int_def) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1161 

27668  1162 
lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)" 
16819  1163 
by (simp add: real_of_int_def) 
1164 

27668  1165 
lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)" 
16819  1166 
by (simp add: real_of_int_def) 
1167 

27668  1168 
lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" 
16819  1169 
by (simp add: real_of_int_def) 
1170 

27668  1171 
lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)" 
16819  1172 
by (simp add: real_of_int_def) 
1173 

16888  1174 
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))" 
1175 
by (auto simp add: abs_if) 

1176 

16819  1177 
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)" 
1178 
apply (subgoal_tac "real n + 1 = real (n + 1)") 

1179 
apply (simp del: real_of_int_add) 

1180 
apply auto 

1181 
done 

1182 

1183 
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)" 

1184 
apply (subgoal_tac "real m + 1 = real (m + 1)") 

1185 
apply (simp del: real_of_int_add) 

1186 
apply simp 

1187 
done 

1188 

1189 
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 

1190 
real (x div d) + (real (x mod d)) / (real d)" 

1191 
proof  

1192 
assume "d ~= 0" 

1193 
have "x = (x div d) * d + x mod d" 

1194 
by auto 

1195 
then have "real x = real (x div d) * real d + real(x mod d)" 

1196 
by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym]) 

1197 
then have "real x / real d = ... / real d" 

1198 
by simp 

1199 
then show ?thesis 

29667  1200 
by (auto simp add: add_divide_distrib algebra_simps prems) 
16819  1201 
qed 
1202 

1203 
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==> 

1204 
real(n div d) = real n / real d" 

1205 
apply (frule real_of_int_div_aux [of d n]) 

1206 
apply simp 

30042  1207 
apply (simp add: dvd_eq_mod_eq_0) 
16819  1208 
done 
1209 

1210 
lemma real_of_int_div2: 

1211 
"0 <= real (n::int) / real (x)  real (n div x)" 

1212 
apply (case_tac "x = 0") 

1213 
apply simp 

1214 
apply (case_tac "0 < x") 

29667  1215 
apply (simp add: algebra_simps) 
16819  1216 
apply (subst real_of_int_div_aux) 
1217 
apply simp 

1218 
apply simp 

1219 
apply (subst zero_le_divide_iff) 

1220 
apply auto 

29667  1221 
apply (simp add: algebra_simps) 
16819  1222 
apply (subst real_of_int_div_aux) 
1223 
apply simp 

1224 
apply simp 

1225 
apply (subst zero_le_divide_iff) 

1226 
apply auto 

1227 
done 

1228 

1229 
lemma real_of_int_div3: 

1230 
"real (n::int) / real (x)  real (n div x) <= 1" 

1231 
apply(case_tac "x = 0") 

1232 
apply simp 

29667  1233 
apply (simp add: algebra_simps) 
16819  1234 
apply (subst real_of_int_div_aux) 
1235 
apply assumption 

1236 
apply simp 

1237 
apply (subst divide_le_eq) 

1238 
apply clarsimp 

1239 
apply (rule conjI) 

1240 
apply (rule impI) 

1241 
apply (rule order_less_imp_le) 

1242 
apply simp 

1243 
apply (rule impI) 

1244 
apply (rule order_less_imp_le) 

1245 
apply simp 

1246 
done 

1247 

1248 
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 

27964  1249 
by (insert real_of_int_div2 [of n x], simp) 
1250 

35635  1251 
lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints" 
1252 
unfolding real_of_int_def by (rule Ints_of_int) 

1253 

27964  1254 

14365
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replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1255 
subsection{*Embedding the Naturals into the Reals*} 
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1256 

14334  1257 
lemma real_of_nat_zero [simp]: "real (0::nat) = 0" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1258 
by (simp add: real_of_nat_def) 
14334  1259 

30082
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
30042
diff
changeset

1260 
lemma real_of_nat_1 [simp]: "real (1::nat) = 1" 
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
30042
diff
changeset

1261 
by (simp add: real_of_nat_def) 
43c5b7bfc791
make more proofs work whether or not One_nat_def is a simp rule
huffman
parents:
30042
diff
changeset

1262 

14334  1263 
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)" 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1264 
by (simp add: real_of_nat_def) 
14334  1265 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1266 
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1267 
by (simp add: real_of_nat_def) 
14334  1268 

1269 
(*Not for addsimps: often the LHS is used to represent a positive natural*) 

1270 
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)" 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1271 
by (simp add: real_of_nat_def) 
14334  1272 

1273 
lemma real_of_nat_less_iff [iff]: 

1274 
"(real (n::nat) < real m) = (n < m)" 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1275 
by (simp add: real_of_nat_def) 
14334  1276 

1277 
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)" 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1278 
by (simp add: real_of_nat_def) 
14334  1279 

1280 
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)" 

14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1281 
by (simp add: real_of_nat_def zero_le_imp_of_nat) 
14334  1282 

14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1283 
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)" 
14378
69c4d5997669
generic of_nat and of_int functions, and generalization of iszero
paulson
parents:
14369
diff
changeset

1284 
by (simp add: real_of_nat_def del: of_nat_Suc) 
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14348
diff
changeset

1285 

14334  1286 
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n" 
23431
25ca91279a9b
change simp rules for of_nat to work like int did previously (reorient of_nat_Suc, remove of_nat_mult [simp]); preserve original variable names in legacy int theorems
huffman
parents:
23289
diff
changeset

1287 
by (simp add: real_of_nat_def of_nat_mult) 
14334  1288 

35344
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1289 
lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n" 
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1290 
by (simp add: real_of_nat_def of_nat_power) 
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1291 

e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1292 
lemmas power_real_of_nat = real_of_nat_power [symmetric] 
e0b46cd72414
moved some lemmas from RealPow to RealDef; changed orientation of real_of_int_power
huffman
parents:
35216
diff
changeset

1293 

16819  1294 
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
1295 
(SUM x:A. real(f x))" 

1296 
apply (subst real_eq_of_nat)+ 

1297 
apply (rule of_nat_setsum) 

1298 
done 

1299 

00d8f9300d13
Additions to the Rea 