author | boehmes |
Thu, 29 Sep 2016 20:54:44 +0200 | |
changeset 63960 | 3daf02070be5 |
parent 63680 | 6e1e8b5abbfa |
child 63961 | 2fd9656c4c82 |
permissions | -rw-r--r-- |
51523 | 1 |
(* Title: HOL/Real.thy |
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Author: Jacques D. Fleuriot, University of Edinburgh, 1998 |
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Author: Larry Paulson, University of Cambridge |
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Author: Jeremy Avigad, Carnegie Mellon University |
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Author: Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen |
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Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4 |
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Construction of Cauchy Reals by Brian Huffman, 2010 |
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*) |
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section \<open>Development of the Reals using Cauchy Sequences\<close> |
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theory Real |
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new proof method "argo" for a combination of quantifier-free propositional logic with equality and linear real arithmetic
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imports Rat Argo |
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begin |
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text \<open> |
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This theory contains a formalization of the real numbers as equivalence |
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classes of Cauchy sequences of rationals. See |
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\<^file>\<open>~~/src/HOL/ex/Dedekind_Real.thy\<close> for an alternative construction using |
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Dedekind cuts. |
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\<close> |
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subsection \<open>Preliminary lemmas\<close> |
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lemma inj_add_left [simp]: "inj (op + x)" |
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for x :: "'a::cancel_semigroup_add" |
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by (meson add_left_imp_eq injI) |
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lemma inj_mult_left [simp]: "inj (op * x) \<longleftrightarrow> x \<noteq> 0" |
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for x :: "'a::idom" |
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by (metis injI mult_cancel_left the_inv_f_f zero_neq_one) |
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||
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lemma add_diff_add: "(a + c) - (b + d) = (a - b) + (c - d)" |
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for a b c d :: "'a::ab_group_add" |
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by simp |
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||
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lemma minus_diff_minus: "- a - - b = - (a - b)" |
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for a b :: "'a::ab_group_add" |
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by simp |
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||
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lemma mult_diff_mult: "(x * y - a * b) = x * (y - b) + (x - a) * b" |
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for x y a b :: "'a::ring" |
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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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||
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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 \<open>Sequences that converge to zero\<close> |
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definition vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool" |
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where "vanishes X \<longleftrightarrow> (\<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: "vanishes X \<Longrightarrow> 0 < r \<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") |
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apply auto |
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apply (rule exI [where x = "\<bar>c\<bar>"]) |
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apply 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" |
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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 |
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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 |
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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>" |
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by (rule abs_triangle_ineq) |
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also have "\<dots> < s + t" |
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by (simp add: add_strict_mono i j n) |
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finally show "\<bar>X n + Y n\<bar> < r" |
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by (simp only: r) |
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qed |
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then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" .. |
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qed |
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111 |
lemma vanishes_diff: |
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assumes "vanishes X" "vanishes Y" |
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shows "vanishes (\<lambda>n. X n - Y n)" |
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unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus assms) |
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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 |
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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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Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
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parents:
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using X by blast |
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obtain b where b: "0 < b" "r = a * b" |
126 |
proof |
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show "0 < r / a" using r a by simp |
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show "r = a * (r / a)" using a by simp |
129 |
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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then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" .. |
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qed |
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||
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subsection \<open>Cauchy sequences\<close> |
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definition cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool" |
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where "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: "(\<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 |
145 |
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lemma cauchyD: "cauchy X \<Longrightarrow> 0 < r \<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 |
148 |
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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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63353 | 156 |
fix r :: rat |
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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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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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63353 | 165 |
proof clarsimp |
166 |
fix m n |
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assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n" |
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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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unfolding add_diff_add by (rule abs_triangle_ineq) |
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170 |
also have "\<dots> < s + t" |
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by (rule add_strict_mono) (simp_all add: i j *) |
172 |
finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" by (simp only: r) |
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qed |
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then show "\<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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qed |
176 |
||
177 |
lemma cauchy_minus [simp]: |
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178 |
assumes X: "cauchy X" |
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179 |
shows "cauchy (\<lambda>n. - X n)" |
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63353 | 180 |
using assms unfolding cauchy_def |
181 |
unfolding minus_diff_minus abs_minus_cancel . |
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51523 | 182 |
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lemma cauchy_diff [simp]: |
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assumes "cauchy X" "cauchy Y" |
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shows "cauchy (\<lambda>n. X n - Y n)" |
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more simplification rules on unary and binary minus
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using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff) |
51523 | 187 |
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188 |
lemma cauchy_imp_bounded: |
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assumes "cauchy X" |
190 |
shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b" |
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proof - |
192 |
obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1" |
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using cauchyD [OF assms zero_less_one] .. |
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show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b" |
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proof (intro exI conjI allI) |
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have "0 \<le> \<bar>X 0\<bar>" by simp |
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also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp |
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198 |
finally have "0 \<le> Max (abs ` X ` {..k})" . |
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then show "0 < Max (abs ` X ` {..k}) + 1" by simp |
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next |
201 |
fix n :: nat |
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202 |
show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" |
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203 |
proof (rule linorder_le_cases) |
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204 |
assume "n \<le> k" |
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then have "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp |
206 |
then show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp |
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next |
208 |
assume "k \<le> n" |
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209 |
have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp |
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210 |
also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>" |
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211 |
by (rule abs_triangle_ineq) |
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also have "\<dots> < Max (abs ` X ` {..k}) + 1" |
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by (rule add_le_less_mono) (simp_all add: k \<open>k \<le> n\<close>) |
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finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" . |
215 |
qed |
|
216 |
qed |
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217 |
qed |
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218 |
||
219 |
lemma cauchy_mult [simp]: |
|
220 |
assumes X: "cauchy X" and Y: "cauchy Y" |
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221 |
shows "cauchy (\<lambda>n. X n * Y n)" |
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222 |
proof (rule cauchyI) |
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223 |
fix r :: rat assume "0 < r" |
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224 |
then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v" |
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225 |
by (rule obtain_pos_sum) |
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226 |
obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a" |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
227 |
using cauchy_imp_bounded [OF X] by blast |
51523 | 228 |
obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
229 |
using cauchy_imp_bounded [OF Y] by blast |
51523 | 230 |
obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b" |
231 |
proof |
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56541 | 232 |
show "0 < v/b" using v b(1) by simp |
233 |
show "0 < u/a" using u a(1) by simp |
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51523 | 234 |
show "r = a * (u/a) + (v/b) * b" |
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using a(1) b(1) \<open>r = u + v\<close> by simp |
51523 | 236 |
qed |
237 |
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s" |
|
238 |
using cauchyD [OF X s] .. |
|
239 |
obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t" |
|
240 |
using cauchyD [OF Y t] .. |
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241 |
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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63353 | 242 |
proof clarsimp |
243 |
fix m n |
|
244 |
assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n" |
|
51523 | 245 |
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>" |
246 |
unfolding mult_diff_mult .. |
|
247 |
also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>" |
|
248 |
by (rule abs_triangle_ineq) |
|
249 |
also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>" |
|
250 |
unfolding abs_mult .. |
|
251 |
also have "\<dots> < a * t + s * b" |
|
252 |
by (simp_all add: add_strict_mono mult_strict_mono' a b i j *) |
|
63494 | 253 |
finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" |
254 |
by (simp only: r) |
|
51523 | 255 |
qed |
63353 | 256 |
then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" .. |
51523 | 257 |
qed |
258 |
||
259 |
lemma cauchy_not_vanishes_cases: |
|
260 |
assumes X: "cauchy X" |
|
261 |
assumes nz: "\<not> vanishes X" |
|
262 |
shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)" |
|
263 |
proof - |
|
264 |
obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>" |
|
265 |
using nz unfolding vanishes_def by (auto simp add: not_less) |
|
266 |
obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t" |
|
60758 | 267 |
using \<open>0 < r\<close> by (rule obtain_pos_sum) |
51523 | 268 |
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s" |
269 |
using cauchyD [OF X s] .. |
|
270 |
obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>" |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
271 |
using r by blast |
51523 | 272 |
have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s" |
60758 | 273 |
using i \<open>i \<le> k\<close> by auto |
51523 | 274 |
have "X k \<le> - r \<or> r \<le> X k" |
60758 | 275 |
using \<open>r \<le> \<bar>X k\<bar>\<close> by auto |
63353 | 276 |
then have "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" |
60758 | 277 |
unfolding \<open>r = s + t\<close> using k by auto |
63353 | 278 |
then have "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" .. |
279 |
then show "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" |
|
51523 | 280 |
using t by auto |
281 |
qed |
|
282 |
||
283 |
lemma cauchy_not_vanishes: |
|
284 |
assumes X: "cauchy X" |
|
63494 | 285 |
and nz: "\<not> vanishes X" |
51523 | 286 |
shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>" |
63353 | 287 |
using cauchy_not_vanishes_cases [OF assms] |
288 |
apply clarify |
|
289 |
apply (rule exI) |
|
290 |
apply (erule conjI) |
|
291 |
apply (rule_tac x = k in exI) |
|
292 |
apply auto |
|
293 |
done |
|
51523 | 294 |
|
295 |
lemma cauchy_inverse [simp]: |
|
296 |
assumes X: "cauchy X" |
|
63494 | 297 |
and nz: "\<not> vanishes X" |
51523 | 298 |
shows "cauchy (\<lambda>n. inverse (X n))" |
299 |
proof (rule cauchyI) |
|
63353 | 300 |
fix r :: rat |
301 |
assume "0 < r" |
|
51523 | 302 |
obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
303 |
using cauchy_not_vanishes [OF X nz] by blast |
51523 | 304 |
from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto |
305 |
obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b" |
|
306 |
proof |
|
60758 | 307 |
show "0 < b * r * b" by (simp add: \<open>0 < r\<close> b) |
51523 | 308 |
show "r = inverse b * (b * r * b) * inverse b" |
309 |
using b by simp |
|
310 |
qed |
|
311 |
obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s" |
|
312 |
using cauchyD [OF X s] .. |
|
313 |
have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r" |
|
63353 | 314 |
proof clarsimp |
315 |
fix m n |
|
316 |
assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n" |
|
317 |
have "\<bar>inverse (X m) - inverse (X n)\<bar> = inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>" |
|
51523 | 318 |
by (simp add: inverse_diff_inverse nz * abs_mult) |
319 |
also have "\<dots> < inverse b * s * inverse b" |
|
63353 | 320 |
by (simp add: mult_strict_mono less_imp_inverse_less i j b * s) |
321 |
finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" by (simp only: r) |
|
51523 | 322 |
qed |
63353 | 323 |
then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" .. |
51523 | 324 |
qed |
325 |
||
326 |
lemma vanishes_diff_inverse: |
|
327 |
assumes X: "cauchy X" "\<not> vanishes X" |
|
63353 | 328 |
and Y: "cauchy Y" "\<not> vanishes Y" |
329 |
and XY: "vanishes (\<lambda>n. X n - Y n)" |
|
51523 | 330 |
shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))" |
331 |
proof (rule vanishesI) |
|
63353 | 332 |
fix r :: rat |
333 |
assume r: "0 < r" |
|
51523 | 334 |
obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
335 |
using cauchy_not_vanishes [OF X] by blast |
51523 | 336 |
obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
337 |
using cauchy_not_vanishes [OF Y] by blast |
51523 | 338 |
obtain s where s: "0 < s" and "inverse a * s * inverse b = r" |
339 |
proof |
|
63494 | 340 |
show "0 < a * r * b" |
341 |
using a r b by simp |
|
342 |
show "inverse a * (a * r * b) * inverse b = r" |
|
343 |
using a r b by simp |
|
51523 | 344 |
qed |
345 |
obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s" |
|
346 |
using vanishesD [OF XY s] .. |
|
347 |
have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" |
|
63353 | 348 |
proof clarsimp |
349 |
fix n |
|
350 |
assume n: "i \<le> n" "j \<le> n" "k \<le> n" |
|
351 |
with i j a b have "X n \<noteq> 0" and "Y n \<noteq> 0" |
|
352 |
by auto |
|
353 |
then have "\<bar>inverse (X n) - inverse (Y n)\<bar> = inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>" |
|
51523 | 354 |
by (simp add: inverse_diff_inverse abs_mult) |
355 |
also have "\<dots> < inverse a * s * inverse b" |
|
63353 | 356 |
by (intro mult_strict_mono' less_imp_inverse_less) (simp_all add: a b i j k n) |
60758 | 357 |
also note \<open>inverse a * s * inverse b = r\<close> |
51523 | 358 |
finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" . |
359 |
qed |
|
63353 | 360 |
then show "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" .. |
51523 | 361 |
qed |
362 |
||
63353 | 363 |
|
60758 | 364 |
subsection \<open>Equivalence relation on Cauchy sequences\<close> |
51523 | 365 |
|
366 |
definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool" |
|
367 |
where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))" |
|
368 |
||
63353 | 369 |
lemma realrelI [intro?]: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> vanishes (\<lambda>n. X n - Y n) \<Longrightarrow> realrel X Y" |
370 |
by (simp add: realrel_def) |
|
51523 | 371 |
|
372 |
lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X" |
|
63353 | 373 |
by (simp add: realrel_def) |
51523 | 374 |
|
375 |
lemma symp_realrel: "symp realrel" |
|
376 |
unfolding realrel_def |
|
63353 | 377 |
apply (rule sympI) |
378 |
apply clarify |
|
379 |
apply (drule vanishes_minus) |
|
380 |
apply simp |
|
381 |
done |
|
51523 | 382 |
|
383 |
lemma transp_realrel: "transp realrel" |
|
384 |
unfolding realrel_def |
|
63353 | 385 |
apply (rule transpI) |
386 |
apply clarify |
|
51523 | 387 |
apply (drule (1) vanishes_add) |
388 |
apply (simp add: algebra_simps) |
|
389 |
done |
|
390 |
||
391 |
lemma part_equivp_realrel: "part_equivp realrel" |
|
63353 | 392 |
by (blast intro: part_equivpI symp_realrel transp_realrel realrel_refl cauchy_const) |
393 |
||
51523 | 394 |
|
60758 | 395 |
subsection \<open>The field of real numbers\<close> |
51523 | 396 |
|
397 |
quotient_type real = "nat \<Rightarrow> rat" / partial: realrel |
|
398 |
morphisms rep_real Real |
|
399 |
by (rule part_equivp_realrel) |
|
400 |
||
401 |
lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)" |
|
402 |
unfolding real.pcr_cr_eq cr_real_def realrel_def by auto |
|
403 |
||
404 |
lemma Real_induct [induct type: real]: (* TODO: generate automatically *) |
|
63353 | 405 |
assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" |
406 |
shows "P x" |
|
51523 | 407 |
proof (induct x) |
408 |
case (1 X) |
|
63353 | 409 |
then have "cauchy X" by (simp add: realrel_def) |
410 |
then show "P (Real X)" by (rule assms) |
|
51523 | 411 |
qed |
412 |
||
63353 | 413 |
lemma eq_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)" |
51523 | 414 |
using real.rel_eq_transfer |
55945 | 415 |
unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp |
51523 | 416 |
|
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51775
diff
changeset
|
417 |
lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy" |
63353 | 418 |
by (simp add: real.domain_eq realrel_def) |
51523 | 419 |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59587
diff
changeset
|
420 |
instantiation real :: field |
51523 | 421 |
begin |
422 |
||
423 |
lift_definition zero_real :: "real" is "\<lambda>n. 0" |
|
424 |
by (simp add: realrel_refl) |
|
425 |
||
426 |
lift_definition one_real :: "real" is "\<lambda>n. 1" |
|
427 |
by (simp add: realrel_refl) |
|
428 |
||
429 |
lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n" |
|
430 |
unfolding realrel_def add_diff_add |
|
431 |
by (simp only: cauchy_add vanishes_add simp_thms) |
|
432 |
||
433 |
lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n" |
|
434 |
unfolding realrel_def minus_diff_minus |
|
435 |
by (simp only: cauchy_minus vanishes_minus simp_thms) |
|
436 |
||
437 |
lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n" |
|
438 |
unfolding realrel_def mult_diff_mult |
|
63353 | 439 |
apply (subst (4) mult.commute) |
440 |
apply (simp only: cauchy_mult vanishes_add vanishes_mult_bounded cauchy_imp_bounded simp_thms) |
|
441 |
done |
|
51523 | 442 |
|
443 |
lift_definition inverse_real :: "real \<Rightarrow> real" |
|
444 |
is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))" |
|
445 |
proof - |
|
63353 | 446 |
fix X Y |
447 |
assume "realrel X Y" |
|
448 |
then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)" |
|
63494 | 449 |
by (simp_all add: realrel_def) |
51523 | 450 |
have "vanishes X \<longleftrightarrow> vanishes Y" |
451 |
proof |
|
452 |
assume "vanishes X" |
|
63494 | 453 |
from vanishes_diff [OF this XY] show "vanishes Y" |
454 |
by simp |
|
51523 | 455 |
next |
456 |
assume "vanishes Y" |
|
63494 | 457 |
from vanishes_add [OF this XY] show "vanishes X" |
458 |
by simp |
|
51523 | 459 |
qed |
63494 | 460 |
then show "?thesis X Y" |
461 |
by (simp add: vanishes_diff_inverse X Y XY realrel_def) |
|
51523 | 462 |
qed |
463 |
||
63353 | 464 |
definition "x - y = x + - y" for x y :: real |
51523 | 465 |
|
63353 | 466 |
definition "x div y = x * inverse y" for x y :: real |
467 |
||
468 |
lemma add_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X + Real Y = Real (\<lambda>n. X n + Y n)" |
|
469 |
using plus_real.transfer by (simp add: cr_real_eq rel_fun_def) |
|
51523 | 470 |
|
63353 | 471 |
lemma minus_Real: "cauchy X \<Longrightarrow> - Real X = Real (\<lambda>n. - X n)" |
472 |
using uminus_real.transfer by (simp add: cr_real_eq rel_fun_def) |
|
51523 | 473 |
|
63353 | 474 |
lemma diff_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X - Real Y = Real (\<lambda>n. X n - Y n)" |
475 |
by (simp add: minus_Real add_Real minus_real_def) |
|
51523 | 476 |
|
63353 | 477 |
lemma mult_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X * Real Y = Real (\<lambda>n. X n * Y n)" |
478 |
using times_real.transfer by (simp add: cr_real_eq rel_fun_def) |
|
51523 | 479 |
|
480 |
lemma inverse_Real: |
|
63353 | 481 |
"cauchy X \<Longrightarrow> inverse (Real X) = (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))" |
482 |
using inverse_real.transfer zero_real.transfer |
|
62390 | 483 |
unfolding cr_real_eq rel_fun_def by (simp split: if_split_asm, metis) |
51523 | 484 |
|
63353 | 485 |
instance |
486 |
proof |
|
51523 | 487 |
fix a b c :: real |
488 |
show "a + b = b + a" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
489 |
by transfer (simp add: ac_simps realrel_def) |
51523 | 490 |
show "(a + b) + c = a + (b + c)" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
491 |
by transfer (simp add: ac_simps realrel_def) |
51523 | 492 |
show "0 + a = a" |
493 |
by transfer (simp add: realrel_def) |
|
494 |
show "- a + a = 0" |
|
495 |
by transfer (simp add: realrel_def) |
|
496 |
show "a - b = a + - b" |
|
497 |
by (rule minus_real_def) |
|
498 |
show "(a * b) * c = a * (b * c)" |
|
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
499 |
by transfer (simp add: ac_simps realrel_def) |
51523 | 500 |
show "a * b = b * a" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
501 |
by transfer (simp add: ac_simps realrel_def) |
51523 | 502 |
show "1 * a = a" |
57514
bdc2c6b40bf2
prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents:
57512
diff
changeset
|
503 |
by transfer (simp add: ac_simps realrel_def) |
51523 | 504 |
show "(a + b) * c = a * c + b * c" |
505 |
by transfer (simp add: distrib_right realrel_def) |
|
61076 | 506 |
show "(0::real) \<noteq> (1::real)" |
51523 | 507 |
by transfer (simp add: realrel_def) |
508 |
show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1" |
|
509 |
apply transfer |
|
510 |
apply (simp add: realrel_def) |
|
511 |
apply (rule vanishesI) |
|
63494 | 512 |
apply (frule (1) cauchy_not_vanishes) |
513 |
apply clarify |
|
514 |
apply (rule_tac x=k in exI) |
|
515 |
apply clarify |
|
516 |
apply (drule_tac x=n in spec) |
|
517 |
apply simp |
|
51523 | 518 |
done |
60429
d3d1e185cd63
uniform _ div _ as infix syntax for ring division
haftmann
parents:
60352
diff
changeset
|
519 |
show "a div b = a * inverse b" |
51523 | 520 |
by (rule divide_real_def) |
521 |
show "inverse (0::real) = 0" |
|
522 |
by transfer (simp add: realrel_def) |
|
523 |
qed |
|
524 |
||
525 |
end |
|
526 |
||
63353 | 527 |
|
60758 | 528 |
subsection \<open>Positive reals\<close> |
51523 | 529 |
|
530 |
lift_definition positive :: "real \<Rightarrow> bool" |
|
531 |
is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n" |
|
532 |
proof - |
|
63353 | 533 |
have 1: "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" |
534 |
if *: "realrel X Y" and **: "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n" for X Y |
|
535 |
proof - |
|
536 |
from * have XY: "vanishes (\<lambda>n. X n - Y n)" |
|
537 |
by (simp_all add: realrel_def) |
|
538 |
from ** obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n" |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
539 |
by blast |
51523 | 540 |
obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" |
60758 | 541 |
using \<open>0 < r\<close> by (rule obtain_pos_sum) |
51523 | 542 |
obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s" |
543 |
using vanishesD [OF XY s] .. |
|
544 |
have "\<forall>n\<ge>max i j. t < Y n" |
|
63353 | 545 |
proof clarsimp |
546 |
fix n |
|
547 |
assume n: "i \<le> n" "j \<le> n" |
|
51523 | 548 |
have "\<bar>X n - Y n\<bar> < s" and "r < X n" |
549 |
using i j n by simp_all |
|
63353 | 550 |
then show "t < Y n" by (simp add: r) |
51523 | 551 |
qed |
63353 | 552 |
then show ?thesis using t by blast |
553 |
qed |
|
51523 | 554 |
fix X Y assume "realrel X Y" |
63353 | 555 |
then have "realrel X Y" and "realrel Y X" |
556 |
using symp_realrel by (auto simp: symp_def) |
|
557 |
then show "?thesis X Y" |
|
51523 | 558 |
by (safe elim!: 1) |
559 |
qed |
|
560 |
||
63353 | 561 |
lemma positive_Real: "cauchy X \<Longrightarrow> positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)" |
562 |
using positive.transfer by (simp add: cr_real_eq rel_fun_def) |
|
51523 | 563 |
|
564 |
lemma positive_zero: "\<not> positive 0" |
|
565 |
by transfer auto |
|
566 |
||
63353 | 567 |
lemma positive_add: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)" |
568 |
apply transfer |
|
569 |
apply clarify |
|
570 |
apply (rename_tac a b i j) |
|
571 |
apply (rule_tac x = "a + b" in exI) |
|
572 |
apply simp |
|
573 |
apply (rule_tac x = "max i j" in exI) |
|
574 |
apply clarsimp |
|
575 |
apply (simp add: add_strict_mono) |
|
576 |
done |
|
51523 | 577 |
|
63353 | 578 |
lemma positive_mult: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)" |
579 |
apply transfer |
|
580 |
apply clarify |
|
581 |
apply (rename_tac a b i j) |
|
582 |
apply (rule_tac x = "a * b" in exI) |
|
583 |
apply simp |
|
584 |
apply (rule_tac x = "max i j" in exI) |
|
585 |
apply clarsimp |
|
586 |
apply (rule mult_strict_mono) |
|
63494 | 587 |
apply auto |
63353 | 588 |
done |
51523 | 589 |
|
63353 | 590 |
lemma positive_minus: "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)" |
591 |
apply transfer |
|
592 |
apply (simp add: realrel_def) |
|
63494 | 593 |
apply (drule (1) cauchy_not_vanishes_cases) |
594 |
apply safe |
|
595 |
apply blast+ |
|
63353 | 596 |
done |
51523 | 597 |
|
59867
58043346ca64
given up separate type classes demanding `inverse 0 = 0`
haftmann
parents:
59587
diff
changeset
|
598 |
instantiation real :: linordered_field |
51523 | 599 |
begin |
600 |
||
63353 | 601 |
definition "x < y \<longleftrightarrow> positive (y - x)" |
51523 | 602 |
|
63353 | 603 |
definition "x \<le> y \<longleftrightarrow> x < y \<or> x = y" for x y :: real |
51523 | 604 |
|
63353 | 605 |
definition "\<bar>a\<bar> = (if a < 0 then - a else a)" for a :: real |
51523 | 606 |
|
63353 | 607 |
definition "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" for a :: real |
51523 | 608 |
|
63353 | 609 |
instance |
610 |
proof |
|
51523 | 611 |
fix a b c :: real |
612 |
show "\<bar>a\<bar> = (if a < 0 then - a else a)" |
|
613 |
by (rule abs_real_def) |
|
614 |
show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a" |
|
615 |
unfolding less_eq_real_def less_real_def |
|
63353 | 616 |
apply auto |
63494 | 617 |
apply (drule (1) positive_add) |
618 |
apply (simp_all add: positive_zero) |
|
63353 | 619 |
done |
51523 | 620 |
show "a \<le> a" |
621 |
unfolding less_eq_real_def by simp |
|
622 |
show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" |
|
623 |
unfolding less_eq_real_def less_real_def |
|
63353 | 624 |
apply auto |
625 |
apply (drule (1) positive_add) |
|
626 |
apply (simp add: algebra_simps) |
|
627 |
done |
|
51523 | 628 |
show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b" |
629 |
unfolding less_eq_real_def less_real_def |
|
63353 | 630 |
apply auto |
631 |
apply (drule (1) positive_add) |
|
632 |
apply (simp add: positive_zero) |
|
633 |
done |
|
51523 | 634 |
show "a \<le> b \<Longrightarrow> c + a \<le> c + b" |
63353 | 635 |
by (auto simp: less_eq_real_def less_real_def) |
51523 | 636 |
(* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *) |
637 |
(* Should produce c + b - (c + a) \<equiv> b - a *) |
|
638 |
show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" |
|
639 |
by (rule sgn_real_def) |
|
640 |
show "a \<le> b \<or> b \<le> a" |
|
63353 | 641 |
by (auto dest!: positive_minus simp: less_eq_real_def less_real_def) |
51523 | 642 |
show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b" |
643 |
unfolding less_real_def |
|
63353 | 644 |
apply (drule (1) positive_mult) |
645 |
apply (simp add: algebra_simps) |
|
646 |
done |
|
51523 | 647 |
qed |
648 |
||
649 |
end |
|
650 |
||
651 |
instantiation real :: distrib_lattice |
|
652 |
begin |
|
653 |
||
63353 | 654 |
definition "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min" |
51523 | 655 |
|
63353 | 656 |
definition "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max" |
51523 | 657 |
|
63494 | 658 |
instance |
659 |
by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2) |
|
51523 | 660 |
|
661 |
end |
|
662 |
||
663 |
lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)" |
|
63353 | 664 |
by (induct x) (simp_all add: zero_real_def one_real_def add_Real) |
51523 | 665 |
|
666 |
lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)" |
|
63353 | 667 |
by (cases x rule: int_diff_cases) (simp add: of_nat_Real diff_Real) |
51523 | 668 |
|
669 |
lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)" |
|
63353 | 670 |
apply (induct x) |
671 |
apply (simp add: Fract_of_int_quotient of_rat_divide) |
|
672 |
apply (simp add: of_int_Real divide_inverse) |
|
673 |
apply (simp add: inverse_Real mult_Real) |
|
674 |
done |
|
51523 | 675 |
|
676 |
instance real :: archimedean_field |
|
677 |
proof |
|
63494 | 678 |
show "\<exists>z. x \<le> of_int z" for x :: real |
51523 | 679 |
apply (induct x) |
680 |
apply (frule cauchy_imp_bounded, clarify) |
|
61942 | 681 |
apply (rule_tac x="\<lceil>b\<rceil> + 1" in exI) |
51523 | 682 |
apply (rule less_imp_le) |
683 |
apply (simp add: of_int_Real less_real_def diff_Real positive_Real) |
|
63494 | 684 |
apply (rule_tac x=1 in exI) |
685 |
apply (simp add: algebra_simps) |
|
686 |
apply (rule_tac x=0 in exI) |
|
687 |
apply clarsimp |
|
51523 | 688 |
apply (rule le_less_trans [OF abs_ge_self]) |
689 |
apply (rule less_le_trans [OF _ le_of_int_ceiling]) |
|
690 |
apply simp |
|
691 |
done |
|
692 |
qed |
|
693 |
||
694 |
instantiation real :: floor_ceiling |
|
695 |
begin |
|
696 |
||
63353 | 697 |
definition [code del]: "\<lfloor>x::real\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))" |
51523 | 698 |
|
61942 | 699 |
instance |
700 |
proof |
|
63353 | 701 |
show "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)" for x :: real |
51523 | 702 |
unfolding floor_real_def using floor_exists1 by (rule theI') |
703 |
qed |
|
704 |
||
705 |
end |
|
706 |
||
63353 | 707 |
|
60758 | 708 |
subsection \<open>Completeness\<close> |
51523 | 709 |
|
63494 | 710 |
lemma not_positive_Real: "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)" if "cauchy X" |
711 |
apply (simp only: positive_Real [OF that]) |
|
63353 | 712 |
apply auto |
63494 | 713 |
apply (unfold not_less) |
714 |
apply (erule obtain_pos_sum) |
|
715 |
apply (drule_tac x=s in spec) |
|
716 |
apply simp |
|
717 |
apply (drule_tac r=t in cauchyD [OF that]) |
|
718 |
apply clarify |
|
719 |
apply (drule_tac x=k in spec) |
|
720 |
apply clarsimp |
|
721 |
apply (rule_tac x=n in exI) |
|
722 |
apply clarify |
|
723 |
apply (rename_tac m) |
|
724 |
apply (drule_tac x=m in spec) |
|
725 |
apply simp |
|
726 |
apply (drule_tac x=n in spec) |
|
727 |
apply simp |
|
63353 | 728 |
apply (drule spec) |
729 |
apply (drule (1) mp) |
|
730 |
apply clarify |
|
731 |
apply (rename_tac i) |
|
732 |
apply (rule_tac x = "max i k" in exI) |
|
733 |
apply simp |
|
734 |
done |
|
51523 | 735 |
|
736 |
lemma le_Real: |
|
63353 | 737 |
assumes "cauchy X" "cauchy Y" |
51523 | 738 |
shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)" |
63353 | 739 |
unfolding not_less [symmetric, where 'a=real] less_real_def |
740 |
apply (simp add: diff_Real not_positive_Real assms) |
|
741 |
apply (simp add: diff_le_eq ac_simps) |
|
742 |
done |
|
51523 | 743 |
|
744 |
lemma le_RealI: |
|
745 |
assumes Y: "cauchy Y" |
|
746 |
shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y" |
|
747 |
proof (induct x) |
|
63353 | 748 |
fix X |
749 |
assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)" |
|
750 |
then have le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r" |
|
51523 | 751 |
by (simp add: of_rat_Real le_Real) |
63353 | 752 |
then have "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" if "0 < r" for r :: rat |
753 |
proof - |
|
754 |
from that obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t" |
|
51523 | 755 |
by (rule obtain_pos_sum) |
756 |
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s" |
|
757 |
using cauchyD [OF Y s] .. |
|
758 |
obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t" |
|
759 |
using le [OF t] .. |
|
760 |
have "\<forall>n\<ge>max i j. X n \<le> Y n + r" |
|
63353 | 761 |
proof clarsimp |
762 |
fix n |
|
763 |
assume n: "i \<le> n" "j \<le> n" |
|
63494 | 764 |
have "X n \<le> Y i + t" |
765 |
using n j by simp |
|
766 |
moreover have "\<bar>Y i - Y n\<bar> < s" |
|
767 |
using n i by simp |
|
768 |
ultimately show "X n \<le> Y n + r" |
|
769 |
unfolding r by simp |
|
51523 | 770 |
qed |
63353 | 771 |
then show ?thesis .. |
772 |
qed |
|
773 |
then show "Real X \<le> Real Y" |
|
51523 | 774 |
by (simp add: of_rat_Real le_Real X Y) |
775 |
qed |
|
776 |
||
777 |
lemma Real_leI: |
|
778 |
assumes X: "cauchy X" |
|
779 |
assumes le: "\<forall>n. of_rat (X n) \<le> y" |
|
780 |
shows "Real X \<le> y" |
|
781 |
proof - |
|
782 |
have "- y \<le> - Real X" |
|
783 |
by (simp add: minus_Real X le_RealI of_rat_minus le) |
|
63353 | 784 |
then show ?thesis by simp |
51523 | 785 |
qed |
786 |
||
787 |
lemma less_RealD: |
|
63353 | 788 |
assumes "cauchy Y" |
51523 | 789 |
shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)" |
63353 | 790 |
apply (erule contrapos_pp) |
791 |
apply (simp add: not_less) |
|
792 |
apply (erule Real_leI [OF assms]) |
|
793 |
done |
|
51523 | 794 |
|
63353 | 795 |
lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n" |
796 |
apply (induct n) |
|
63494 | 797 |
apply simp |
63353 | 798 |
apply (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc) |
799 |
done |
|
51523 | 800 |
|
801 |
lemma complete_real: |
|
802 |
fixes S :: "real set" |
|
803 |
assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z" |
|
804 |
shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)" |
|
805 |
proof - |
|
806 |
obtain x where x: "x \<in> S" using assms(1) .. |
|
807 |
obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) .. |
|
808 |
||
63040 | 809 |
define P where "P x \<longleftrightarrow> (\<forall>y\<in>S. y \<le> of_rat x)" for x |
51523 | 810 |
obtain a where a: "\<not> P a" |
811 |
proof |
|
61942 | 812 |
have "of_int \<lfloor>x - 1\<rfloor> \<le> x - 1" by (rule of_int_floor_le) |
51523 | 813 |
also have "x - 1 < x" by simp |
61942 | 814 |
finally have "of_int \<lfloor>x - 1\<rfloor> < x" . |
63353 | 815 |
then have "\<not> x \<le> of_int \<lfloor>x - 1\<rfloor>" by (simp only: not_le) |
61942 | 816 |
then show "\<not> P (of_int \<lfloor>x - 1\<rfloor>)" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
817 |
unfolding P_def of_rat_of_int_eq using x by blast |
51523 | 818 |
qed |
819 |
obtain b where b: "P b" |
|
820 |
proof |
|
61942 | 821 |
show "P (of_int \<lceil>z\<rceil>)" |
51523 | 822 |
unfolding P_def of_rat_of_int_eq |
823 |
proof |
|
824 |
fix y assume "y \<in> S" |
|
63353 | 825 |
then have "y \<le> z" using z by simp |
61942 | 826 |
also have "z \<le> of_int \<lceil>z\<rceil>" by (rule le_of_int_ceiling) |
827 |
finally show "y \<le> of_int \<lceil>z\<rceil>" . |
|
51523 | 828 |
qed |
829 |
qed |
|
830 |
||
63040 | 831 |
define avg where "avg x y = x/2 + y/2" for x y :: rat |
832 |
define bisect where "bisect = (\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))" |
|
833 |
define A where "A n = fst ((bisect ^^ n) (a, b))" for n |
|
834 |
define B where "B n = snd ((bisect ^^ n) (a, b))" for n |
|
835 |
define C where "C n = avg (A n) (B n)" for n |
|
51523 | 836 |
have A_0 [simp]: "A 0 = a" unfolding A_def by simp |
837 |
have B_0 [simp]: "B 0 = b" unfolding B_def by simp |
|
838 |
have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)" |
|
839 |
unfolding A_def B_def C_def bisect_def split_def by simp |
|
840 |
have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)" |
|
841 |
unfolding A_def B_def C_def bisect_def split_def by simp |
|
842 |
||
63353 | 843 |
have width: "B n - A n = (b - a) / 2^n" for n |
844 |
apply (induct n) |
|
63494 | 845 |
apply (simp_all add: eq_divide_eq) |
63353 | 846 |
apply (simp_all add: C_def avg_def algebra_simps) |
51523 | 847 |
done |
848 |
||
63353 | 849 |
have twos: "0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r" for y r :: rat |
51523 | 850 |
apply (simp add: divide_less_eq) |
57512
cc97b347b301
reduced name variants for assoc and commute on plus and mult
haftmann
parents:
57447
diff
changeset
|
851 |
apply (subst mult.commute) |
51523 | 852 |
apply (frule_tac y=y in ex_less_of_nat_mult) |
853 |
apply clarify |
|
854 |
apply (rule_tac x=n in exI) |
|
855 |
apply (erule less_trans) |
|
856 |
apply (rule mult_strict_right_mono) |
|
63494 | 857 |
apply (rule le_less_trans [OF _ of_nat_less_two_power]) |
858 |
apply simp |
|
51523 | 859 |
apply assumption |
860 |
done |
|
861 |
||
63494 | 862 |
have PA: "\<not> P (A n)" for n |
863 |
by (induct n) (simp_all add: a) |
|
864 |
have PB: "P (B n)" for n |
|
865 |
by (induct n) (simp_all add: b) |
|
51523 | 866 |
have ab: "a < b" |
867 |
using a b unfolding P_def |
|
868 |
apply (clarsimp simp add: not_le) |
|
869 |
apply (drule (1) bspec) |
|
870 |
apply (drule (1) less_le_trans) |
|
871 |
apply (simp add: of_rat_less) |
|
872 |
done |
|
63494 | 873 |
have AB: "A n < B n" for n |
874 |
by (induct n) (simp_all add: ab C_def avg_def) |
|
51523 | 875 |
have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j" |
876 |
apply (auto simp add: le_less [where 'a=nat]) |
|
877 |
apply (erule less_Suc_induct) |
|
63494 | 878 |
apply (clarsimp simp add: C_def avg_def) |
879 |
apply (simp add: add_divide_distrib [symmetric]) |
|
880 |
apply (rule AB [THEN less_imp_le]) |
|
51523 | 881 |
apply simp |
882 |
done |
|
883 |
have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i" |
|
884 |
apply (auto simp add: le_less [where 'a=nat]) |
|
885 |
apply (erule less_Suc_induct) |
|
63494 | 886 |
apply (clarsimp simp add: C_def avg_def) |
887 |
apply (simp add: add_divide_distrib [symmetric]) |
|
888 |
apply (rule AB [THEN less_imp_le]) |
|
51523 | 889 |
apply simp |
890 |
done |
|
63353 | 891 |
have cauchy_lemma: "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X" |
51523 | 892 |
apply (rule cauchyI) |
893 |
apply (drule twos [where y="b - a"]) |
|
894 |
apply (erule exE) |
|
895 |
apply (rule_tac x=n in exI, clarify, rename_tac i j) |
|
896 |
apply (rule_tac y="B n - A n" in le_less_trans) defer |
|
63494 | 897 |
apply (simp add: width) |
51523 | 898 |
apply (drule_tac x=n in spec) |
899 |
apply (frule_tac x=i in spec, drule (1) mp) |
|
900 |
apply (frule_tac x=j in spec, drule (1) mp) |
|
901 |
apply (frule A_mono, drule B_mono) |
|
902 |
apply (frule A_mono, drule B_mono) |
|
903 |
apply arith |
|
904 |
done |
|
905 |
have "cauchy A" |
|
906 |
apply (rule cauchy_lemma [rule_format]) |
|
907 |
apply (simp add: A_mono) |
|
908 |
apply (erule order_trans [OF less_imp_le [OF AB] B_mono]) |
|
909 |
done |
|
910 |
have "cauchy B" |
|
911 |
apply (rule cauchy_lemma [rule_format]) |
|
912 |
apply (simp add: B_mono) |
|
913 |
apply (erule order_trans [OF A_mono less_imp_le [OF AB]]) |
|
914 |
done |
|
915 |
have 1: "\<forall>x\<in>S. x \<le> Real B" |
|
916 |
proof |
|
63353 | 917 |
fix x |
918 |
assume "x \<in> S" |
|
51523 | 919 |
then show "x \<le> Real B" |
60758 | 920 |
using PB [unfolded P_def] \<open>cauchy B\<close> |
51523 | 921 |
by (simp add: le_RealI) |
922 |
qed |
|
923 |
have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z" |
|
924 |
apply clarify |
|
925 |
apply (erule contrapos_pp) |
|
926 |
apply (simp add: not_le) |
|
63494 | 927 |
apply (drule less_RealD [OF \<open>cauchy A\<close>]) |
928 |
apply clarify |
|
51523 | 929 |
apply (subgoal_tac "\<not> P (A n)") |
63494 | 930 |
apply (simp add: P_def not_le) |
931 |
apply clarify |
|
932 |
apply (erule rev_bexI) |
|
933 |
apply (erule (1) less_trans) |
|
51523 | 934 |
apply (simp add: PA) |
935 |
done |
|
936 |
have "vanishes (\<lambda>n. (b - a) / 2 ^ n)" |
|
937 |
proof (rule vanishesI) |
|
63353 | 938 |
fix r :: rat |
939 |
assume "0 < r" |
|
51523 | 940 |
then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r" |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
941 |
using twos by blast |
51523 | 942 |
have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" |
63353 | 943 |
proof clarify |
944 |
fix n |
|
945 |
assume n: "k \<le> n" |
|
51523 | 946 |
have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n" |
947 |
by simp |
|
948 |
also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k" |
|
56544 | 949 |
using n by (simp add: divide_left_mono) |
51523 | 950 |
also note k |
951 |
finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" . |
|
952 |
qed |
|
63353 | 953 |
then show "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" .. |
51523 | 954 |
qed |
63353 | 955 |
then have 3: "Real B = Real A" |
60758 | 956 |
by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width) |
51523 | 957 |
show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)" |
63353 | 958 |
apply (rule exI [where x = "Real B"]) |
959 |
using 1 2 3 |
|
960 |
apply simp |
|
961 |
done |
|
51523 | 962 |
qed |
963 |
||
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51773
diff
changeset
|
964 |
instantiation real :: linear_continuum |
51523 | 965 |
begin |
966 |
||
63353 | 967 |
subsection \<open>Supremum of a set of reals\<close> |
51523 | 968 |
|
54281 | 969 |
definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)" |
63353 | 970 |
definition "Inf X = - Sup (uminus ` X)" for X :: "real set" |
51523 | 971 |
|
972 |
instance |
|
973 |
proof |
|
63494 | 974 |
show Sup_upper: "x \<le> Sup X" |
975 |
if "x \<in> X" "bdd_above X" |
|
976 |
for x :: real and X :: "real set" |
|
63353 | 977 |
proof - |
978 |
from that obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z" |
|
54258
adfc759263ab
use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents:
54230
diff
changeset
|
979 |
using complete_real[of X] unfolding bdd_above_def by blast |
63494 | 980 |
then show ?thesis |
981 |
unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that) |
|
63353 | 982 |
qed |
63494 | 983 |
show Sup_least: "Sup X \<le> z" |
984 |
if "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z" |
|
63353 | 985 |
for z :: real and X :: "real set" |
986 |
proof - |
|
987 |
from that obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z" |
|
988 |
using complete_real [of X] by blast |
|
51523 | 989 |
then have "Sup X = s" |
61284
2314c2f62eb1
real_of_nat_Suc is now a simprule
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
990 |
unfolding Sup_real_def by (best intro: Least_equality) |
63353 | 991 |
also from s z have "\<dots> \<le> z" |
51523 | 992 |
by blast |
63353 | 993 |
finally show ?thesis . |
994 |
qed |
|
63494 | 995 |
show "Inf X \<le> x" if "x \<in> X" "bdd_below X" |
996 |
for x :: real and X :: "real set" |
|
63353 | 997 |
using Sup_upper [of "-x" "uminus ` X"] by (auto simp: Inf_real_def that) |
63494 | 998 |
show "z \<le> Inf X" if "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" |
999 |
for z :: real and X :: "real set" |
|
63353 | 1000 |
using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that) |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51773
diff
changeset
|
1001 |
show "\<exists>a b::real. a \<noteq> b" |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51773
diff
changeset
|
1002 |
using zero_neq_one by blast |
51523 | 1003 |
qed |
63353 | 1004 |
|
51523 | 1005 |
end |
1006 |
||
63353 | 1007 |
|
60758 | 1008 |
subsection \<open>Hiding implementation details\<close> |
51523 | 1009 |
|
1010 |
hide_const (open) vanishes cauchy positive Real |
|
1011 |
||
1012 |
declare Real_induct [induct del] |
|
1013 |
declare Abs_real_induct [induct del] |
|
1014 |
declare Abs_real_cases [cases del] |
|
1015 |
||
53652
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents:
53374
diff
changeset
|
1016 |
lifting_update real.lifting |
18fbca265e2e
use lifting_forget for deregistering numeric types as a quotient type
kuncar
parents:
53374
diff
changeset
|
1017 |
lifting_forget real.lifting |
61284
2314c2f62eb1
real_of_nat_Suc is now a simprule
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1018 |
|
63353 | 1019 |
|
1020 |
subsection \<open>More Lemmas\<close> |
|
51523 | 1021 |
|
60758 | 1022 |
text \<open>BH: These lemmas should not be necessary; they should be |
63353 | 1023 |
covered by existing simp rules and simplification procedures.\<close> |
51523 | 1024 |
|
63494 | 1025 |
lemma real_mult_less_iff1 [simp]: "0 < z \<Longrightarrow> x * z < y * z \<longleftrightarrow> x < y" |
1026 |
for x y z :: real |
|
63353 | 1027 |
by simp (* solved by linordered_ring_less_cancel_factor simproc *) |
51523 | 1028 |
|
63494 | 1029 |
lemma real_mult_le_cancel_iff1 [simp]: "0 < z \<Longrightarrow> x * z \<le> y * z \<longleftrightarrow> x \<le> y" |
1030 |
for x y z :: real |
|
63353 | 1031 |
by simp (* solved by linordered_ring_le_cancel_factor simproc *) |
51523 | 1032 |
|
63494 | 1033 |
lemma real_mult_le_cancel_iff2 [simp]: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y" |
1034 |
for x y z :: real |
|
63353 | 1035 |
by simp (* solved by linordered_ring_le_cancel_factor simproc *) |
51523 | 1036 |
|
1037 |
||
60758 | 1038 |
subsection \<open>Embedding numbers into the Reals\<close> |
51523 | 1039 |
|
63353 | 1040 |
abbreviation real_of_nat :: "nat \<Rightarrow> real" |
1041 |
where "real_of_nat \<equiv> of_nat" |
|
51523 | 1042 |
|
63353 | 1043 |
abbreviation real :: "nat \<Rightarrow> real" |
1044 |
where "real \<equiv> of_nat" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1045 |
|
63353 | 1046 |
abbreviation real_of_int :: "int \<Rightarrow> real" |
1047 |
where "real_of_int \<equiv> of_int" |
|
51523 | 1048 |
|
63353 | 1049 |
abbreviation real_of_rat :: "rat \<Rightarrow> real" |
1050 |
where "real_of_rat \<equiv> of_rat" |
|
51523 | 1051 |
|
1052 |
declare [[coercion_enabled]] |
|
59000 | 1053 |
|
1054 |
declare [[coercion "of_nat :: nat \<Rightarrow> int"]] |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1055 |
declare [[coercion "of_nat :: nat \<Rightarrow> real"]] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1056 |
declare [[coercion "of_int :: int \<Rightarrow> real"]] |
59000 | 1057 |
|
1058 |
(* We do not add rat to the coerced types, this has often unpleasant side effects when writing |
|
1059 |
inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *) |
|
51523 | 1060 |
|
1061 |
declare [[coercion_map map]] |
|
59000 | 1062 |
declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]] |
1063 |
declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]] |
|
51523 | 1064 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1065 |
declare of_int_eq_0_iff [algebra, presburger] |
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1066 |
declare of_int_eq_1_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1067 |
declare of_int_eq_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1068 |
declare of_int_less_0_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1069 |
declare of_int_less_1_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1070 |
declare of_int_less_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1071 |
declare of_int_le_0_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1072 |
declare of_int_le_1_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1073 |
declare of_int_le_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1074 |
declare of_int_0_less_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1075 |
declare of_int_0_le_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1076 |
declare of_int_1_less_iff [algebra, presburger] |
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1077 |
declare of_int_1_le_iff [algebra, presburger] |
51523 | 1078 |
|
63353 | 1079 |
lemma int_less_real_le: "n < m \<longleftrightarrow> real_of_int n + 1 \<le> real_of_int m" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1080 |
proof - |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1081 |
have "(0::real) \<le> 1" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1082 |
by (metis less_eq_real_def zero_less_one) |
63353 | 1083 |
then show ?thesis |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
1084 |
by (metis floor_of_int less_floor_iff) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1085 |
qed |
51523 | 1086 |
|
63353 | 1087 |
lemma int_le_real_less: "n \<le> m \<longleftrightarrow> real_of_int n < real_of_int m + 1" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1088 |
by (meson int_less_real_le not_le) |
51523 | 1089 |
|
63353 | 1090 |
lemma real_of_int_div_aux: |
1091 |
"(real_of_int x) / (real_of_int d) = |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1092 |
real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)" |
51523 | 1093 |
proof - |
1094 |
have "x = (x div d) * d + x mod d" |
|
1095 |
by auto |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1096 |
then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1097 |
by (metis of_int_add of_int_mult) |
63353 | 1098 |
then have "real_of_int x / real_of_int d = \<dots> / real_of_int d" |
51523 | 1099 |
by simp |
1100 |
then show ?thesis |
|
1101 |
by (auto simp add: add_divide_distrib algebra_simps) |
|
1102 |
qed |
|
1103 |
||
58834 | 1104 |
lemma real_of_int_div: |
63353 | 1105 |
"d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int |
58834 | 1106 |
by (simp add: real_of_int_div_aux) |
51523 | 1107 |
|
63353 | 1108 |
lemma real_of_int_div2: "0 \<le> real_of_int n / real_of_int x - real_of_int (n div x)" |
1109 |
apply (cases "x = 0") |
|
63494 | 1110 |
apply simp |
63353 | 1111 |
apply (cases "0 < x") |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1112 |
apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1113 |
apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1114 |
done |
51523 | 1115 |
|
63353 | 1116 |
lemma real_of_int_div3: "real_of_int n / real_of_int x - real_of_int (n div x) \<le> 1" |
51523 | 1117 |
apply (simp add: algebra_simps) |
1118 |
apply (subst real_of_int_div_aux) |
|
1119 |
apply (auto simp add: divide_le_eq intro: order_less_imp_le) |
|
63353 | 1120 |
done |
51523 | 1121 |
|
63353 | 1122 |
lemma real_of_int_div4: "real_of_int (n div x) \<le> real_of_int n / real_of_int x" |
1123 |
using real_of_int_div2 [of n x] by simp |
|
51523 | 1124 |
|
1125 |
||
63353 | 1126 |
subsection \<open>Embedding the Naturals into the Reals\<close> |
51523 | 1127 |
|
63353 | 1128 |
lemma real_of_card: "real (card A) = setsum (\<lambda>x. 1) A" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1129 |
by simp |
51523 | 1130 |
|
63353 | 1131 |
lemma nat_less_real_le: "n < m \<longleftrightarrow> real n + 1 \<le> real m" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1132 |
by (metis discrete of_nat_1 of_nat_add of_nat_le_iff) |
51523 | 1133 |
|
63494 | 1134 |
lemma nat_le_real_less: "n \<le> m \<longleftrightarrow> real n < real m + 1" |
1135 |
for m n :: nat |
|
61284
2314c2f62eb1
real_of_nat_Suc is now a simprule
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1136 |
by (meson nat_less_real_le not_le) |
51523 | 1137 |
|
63353 | 1138 |
lemma real_of_nat_div_aux: "real x / real d = real (x div d) + real (x mod d) / real d" |
51523 | 1139 |
proof - |
1140 |
have "x = (x div d) * d + x mod d" |
|
1141 |
by auto |
|
1142 |
then have "real x = real (x div d) * real d + real(x mod d)" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1143 |
by (metis of_nat_add of_nat_mult) |
51523 | 1144 |
then have "real x / real d = \<dots> / real d" |
1145 |
by simp |
|
1146 |
then show ?thesis |
|
1147 |
by (auto simp add: add_divide_distrib algebra_simps) |
|
1148 |
qed |
|
1149 |
||
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1150 |
lemma real_of_nat_div: "d dvd n \<Longrightarrow> real(n div d) = real n / real d" |
63353 | 1151 |
by (subst real_of_nat_div_aux) (auto simp add: dvd_eq_mod_eq_0 [symmetric]) |
51523 | 1152 |
|
63353 | 1153 |
lemma real_of_nat_div2: "0 \<le> real n / real x - real (n div x)" for n x :: nat |
1154 |
apply (simp add: algebra_simps) |
|
1155 |
apply (subst real_of_nat_div_aux) |
|
1156 |
apply simp |
|
1157 |
done |
|
51523 | 1158 |
|
63353 | 1159 |
lemma real_of_nat_div3: "real n / real x - real (n div x) \<le> 1" for n x :: nat |
1160 |
apply (cases "x = 0") |
|
63494 | 1161 |
apply simp |
63353 | 1162 |
apply (simp add: algebra_simps) |
1163 |
apply (subst real_of_nat_div_aux) |
|
1164 |
apply simp |
|
1165 |
done |
|
51523 | 1166 |
|
63353 | 1167 |
lemma real_of_nat_div4: "real (n div x) \<le> real n / real x" for n x :: nat |
1168 |
using real_of_nat_div2 [of n x] by simp |
|
1169 |
||
51523 | 1170 |
|
60758 | 1171 |
subsection \<open>The Archimedean Property of the Reals\<close> |
51523 | 1172 |
|
62623
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
1173 |
lemma real_arch_inverse: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)" |
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
1174 |
using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat] |
dbc62f86a1a9
rationalisation of theorem names esp about "real Archimedian" etc.
paulson <lp15@cam.ac.uk>
parents:
62398
diff
changeset
|
1175 |
by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc) |
51523 | 1176 |
|
63494 | 1177 |
lemma reals_Archimedean3: "0 < x \<Longrightarrow> \<forall>y. \<exists>n. y < real n * x" |
1178 |
by (auto intro: ex_less_of_nat_mult) |
|
51523 | 1179 |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62348
diff
changeset
|
1180 |
lemma real_archimedian_rdiv_eq_0: |
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62348
diff
changeset
|
1181 |
assumes x0: "x \<ge> 0" |
63353 | 1182 |
and c: "c \<ge> 0" |
1183 |
and xc: "\<And>m::nat. m > 0 \<Longrightarrow> real m * x \<le> c" |
|
1184 |
shows "x = 0" |
|
1185 |
by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc) |
|
62397
5ae24f33d343
Substantial new material for multivariate analysis. Also removal of some duplicates.
paulson <lp15@cam.ac.uk>
parents:
62348
diff
changeset
|
1186 |
|
51523 | 1187 |
|
63353 | 1188 |
subsection \<open>Rationals\<close> |
51523 | 1189 |
|
63353 | 1190 |
lemma Rats_eq_int_div_int: "\<rat> = {real_of_int i / real_of_int j | i j. j \<noteq> 0}" (is "_ = ?S") |
51523 | 1191 |
proof |
1192 |
show "\<rat> \<subseteq> ?S" |
|
1193 |
proof |
|
63353 | 1194 |
fix x :: real |
1195 |
assume "x \<in> \<rat>" |
|
1196 |
then obtain r where "x = of_rat r" |
|
1197 |
unfolding Rats_def .. |
|
1198 |
have "of_rat r \<in> ?S" |
|
1199 |
by (cases r) (auto simp add: of_rat_rat) |
|
1200 |
then show "x \<in> ?S" |
|
1201 |
using \<open>x = of_rat r\<close> by simp |
|
51523 | 1202 |
qed |
1203 |
next |
|
1204 |
show "?S \<subseteq> \<rat>" |
|
63353 | 1205 |
proof (auto simp: Rats_def) |
1206 |
fix i j :: int |
|
1207 |
assume "j \<noteq> 0" |
|
1208 |
then have "real_of_int i / real_of_int j = of_rat (Fract i j)" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1209 |
by (simp add: of_rat_rat) |
63353 | 1210 |
then show "real_of_int i / real_of_int j \<in> range of_rat" |
1211 |
by blast |
|
51523 | 1212 |
qed |
1213 |
qed |
|
1214 |
||
63353 | 1215 |
lemma Rats_eq_int_div_nat: "\<rat> = { real_of_int i / real n | i n. n \<noteq> 0}" |
1216 |
proof (auto simp: Rats_eq_int_div_int) |
|
1217 |
fix i j :: int |
|
1218 |
assume "j \<noteq> 0" |
|
1219 |
show "\<exists>(i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i' / real n \<and> 0 < n" |
|
1220 |
proof (cases "j > 0") |
|
1221 |
case True |
|
1222 |
then have "real_of_int i / real_of_int j = real_of_int i / real (nat j) \<and> 0 < nat j" |
|
1223 |
by simp |
|
1224 |
then show ?thesis by blast |
|
51523 | 1225 |
next |
63353 | 1226 |
case False |
1227 |
with \<open>j \<noteq> 0\<close> |
|
1228 |
have "real_of_int i / real_of_int j = real_of_int (- i) / real (nat (- j)) \<and> 0 < nat (- j)" |
|
1229 |
by simp |
|
1230 |
then show ?thesis by blast |
|
51523 | 1231 |
qed |
1232 |
next |
|
63353 | 1233 |
fix i :: int and n :: nat |
1234 |
assume "0 < n" |
|
1235 |
then have "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0" |
|
1236 |
by simp |
|
1237 |
then show "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int j \<and> j \<noteq> 0" |
|
1238 |
by blast |
|
51523 | 1239 |
qed |
1240 |
||
1241 |
lemma Rats_abs_nat_div_natE: |
|
1242 |
assumes "x \<in> \<rat>" |
|
63353 | 1243 |
obtains m n :: nat where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1" |
51523 | 1244 |
proof - |
63353 | 1245 |
from \<open>x \<in> \<rat>\<close> obtain i :: int and n :: nat where "n \<noteq> 0" and "x = real_of_int i / real n" |
1246 |
by (auto simp add: Rats_eq_int_div_nat) |
|
1247 |
then have "\<bar>x\<bar> = real (nat \<bar>i\<bar>) / real n" by simp |
|
51523 | 1248 |
then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast |
1249 |
let ?gcd = "gcd m n" |
|
63353 | 1250 |
from \<open>n \<noteq> 0\<close> have gcd: "?gcd \<noteq> 0" by simp |
51523 | 1251 |
let ?k = "m div ?gcd" |
1252 |
let ?l = "n div ?gcd" |
|
1253 |
let ?gcd' = "gcd ?k ?l" |
|
63353 | 1254 |
have "?gcd dvd m" .. |
1255 |
then have gcd_k: "?gcd * ?k = m" |
|
51523 | 1256 |
by (rule dvd_mult_div_cancel) |
63353 | 1257 |
have "?gcd dvd n" .. |
1258 |
then have gcd_l: "?gcd * ?l = n" |
|
51523 | 1259 |
by (rule dvd_mult_div_cancel) |
63353 | 1260 |
from \<open>n \<noteq> 0\<close> and gcd_l have "?gcd * ?l \<noteq> 0" by simp |
61284
2314c2f62eb1
real_of_nat_Suc is now a simprule
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1261 |
then have "?l \<noteq> 0" by (blast dest!: mult_not_zero) |
51523 | 1262 |
moreover |
1263 |
have "\<bar>x\<bar> = real ?k / real ?l" |
|
1264 |
proof - |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1265 |
from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1266 |
by (simp add: real_of_nat_div) |
51523 | 1267 |
also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp |
1268 |
also from x_rat have "\<dots> = \<bar>x\<bar>" .. |
|
1269 |
finally show ?thesis .. |
|
1270 |
qed |
|
1271 |
moreover |
|
1272 |
have "?gcd' = 1" |
|
1273 |
proof - |
|
1274 |
have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)" |
|
1275 |
by (rule gcd_mult_distrib_nat) |
|
1276 |
with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp |
|
1277 |
with gcd show ?thesis by auto |
|
1278 |
qed |
|
1279 |
ultimately show ?thesis .. |
|
1280 |
qed |
|
1281 |
||
63353 | 1282 |
|
1283 |
subsection \<open>Density of the Rational Reals in the Reals\<close> |
|
51523 | 1284 |
|
63353 | 1285 |
text \<open> |
1286 |
This density proof is due to Stefan Richter and was ported by TN. The |
|
63494 | 1287 |
original source is \<^emph>\<open>Real Analysis\<close> by H.L. Royden. |
63353 | 1288 |
It employs the Archimedean property of the reals.\<close> |
51523 | 1289 |
|
1290 |
lemma Rats_dense_in_real: |
|
1291 |
fixes x :: real |
|
63353 | 1292 |
assumes "x < y" |
1293 |
shows "\<exists>r\<in>\<rat>. x < r \<and> r < y" |
|
51523 | 1294 |
proof - |
63353 | 1295 |
from \<open>x < y\<close> have "0 < y - x" by simp |
1296 |
with reals_Archimedean obtain q :: nat where q: "inverse (real q) < y - x" and "0 < q" |
|
1297 |
by blast |
|
63040 | 1298 |
define p where "p = \<lceil>y * real q\<rceil> - 1" |
1299 |
define r where "r = of_int p / real q" |
|
63494 | 1300 |
from q have "x < y - inverse (real q)" |
1301 |
by simp |
|
1302 |
also from \<open>0 < q\<close> have "y - inverse (real q) \<le> r" |
|
1303 |
by (simp add: r_def p_def le_divide_eq left_diff_distrib) |
|
51523 | 1304 |
finally have "x < r" . |
63494 | 1305 |
moreover from \<open>0 < q\<close> have "r < y" |
1306 |
by (simp add: r_def p_def divide_less_eq diff_less_eq less_ceiling_iff [symmetric]) |
|
1307 |
moreover have "r \<in> \<rat>" |
|
1308 |
by (simp add: r_def) |
|
61649
268d88ec9087
Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
paulson <lp15@cam.ac.uk>
parents:
61609
diff
changeset
|
1309 |
ultimately show ?thesis by blast |
51523 | 1310 |
qed |
1311 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1312 |
lemma of_rat_dense: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1313 |
fixes x y :: real |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1314 |
assumes "x < y" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1315 |
shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y" |
63353 | 1316 |
using Rats_dense_in_real [OF \<open>x < y\<close>] |
1317 |
by (auto elim: Rats_cases) |
|
51523 | 1318 |
|
1319 |
||
63353 | 1320 |
subsection \<open>Numerals and Arithmetic\<close> |
51523 | 1321 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1322 |
lemma [code_abbrev]: (*FIXME*) |
51523 | 1323 |
"real_of_int (numeral k) = numeral k" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54281
diff
changeset
|
1324 |
"real_of_int (- numeral k) = - numeral k" |
51523 | 1325 |
by simp_all |
1326 |
||
60758 | 1327 |
declaration \<open> |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1328 |
K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1329 |
(* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1330 |
#> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2] |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1331 |
(* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *) |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1332 |
#> Lin_Arith.add_simps [@{thm of_nat_0}, @{thm of_nat_Suc}, @{thm of_nat_add}, |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1333 |
@{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1}, |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1334 |
@{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff}, |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1335 |
@{thm of_int_mult}, @{thm of_int_of_nat_eq}, |
62348 | 1336 |
@{thm of_nat_numeral}, @{thm of_nat_numeral}, @{thm of_int_neg_numeral}] |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1337 |
#> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"}) |
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1338 |
#> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"})) |
60758 | 1339 |
\<close> |
51523 | 1340 |
|
63353 | 1341 |
|
1342 |
subsection \<open>Simprules combining \<open>x + y\<close> and \<open>0\<close>\<close> (* FIXME ARE THEY NEEDED? *) |
|
51523 | 1343 |
|
63494 | 1344 |
lemma real_add_minus_iff [simp]: "x + - a = 0 \<longleftrightarrow> x = a" |
1345 |
for x a :: real |
|
63353 | 1346 |
by arith |
51523 | 1347 |
|
63494 | 1348 |
lemma real_add_less_0_iff: "x + y < 0 \<longleftrightarrow> y < - x" |
1349 |
for x y :: real |
|
63353 | 1350 |
by auto |
51523 | 1351 |
|
63494 | 1352 |
lemma real_0_less_add_iff: "0 < x + y \<longleftrightarrow> - x < y" |
1353 |
for x y :: real |
|
63353 | 1354 |
by auto |
51523 | 1355 |
|
63494 | 1356 |
lemma real_add_le_0_iff: "x + y \<le> 0 \<longleftrightarrow> y \<le> - x" |
1357 |
for x y :: real |
|
63353 | 1358 |
by auto |
51523 | 1359 |
|
63494 | 1360 |
lemma real_0_le_add_iff: "0 \<le> x + y \<longleftrightarrow> - x \<le> y" |
1361 |
for x y :: real |
|
63353 | 1362 |
by auto |
1363 |
||
51523 | 1364 |
|
60758 | 1365 |
subsection \<open>Lemmas about powers\<close> |
51523 | 1366 |
|
1367 |
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1368 |
by simp |
51523 | 1369 |
|
63353 | 1370 |
(* FIXME: declare this [simp] for all types, or not at all *) |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1371 |
declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp] |
51523 | 1372 |
|
63494 | 1373 |
lemma real_minus_mult_self_le [simp]: "- (u * u) \<le> x * x" |
1374 |
for u x :: real |
|
63353 | 1375 |
by (rule order_trans [where y = 0]) auto |
51523 | 1376 |
|
63494 | 1377 |
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> x\<^sup>2" |
1378 |
for u x :: real |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1379 |
by (auto simp add: power2_eq_square) |
51523 | 1380 |
|
63353 | 1381 |
lemma numeral_power_eq_real_of_int_cancel_iff [simp]: |
1382 |
"numeral x ^ n = real_of_int y \<longleftrightarrow> numeral x ^ n = y" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1383 |
by (metis of_int_eq_iff of_int_numeral of_int_power) |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1384 |
|
63353 | 1385 |
lemma real_of_int_eq_numeral_power_cancel_iff [simp]: |
1386 |
"real_of_int y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n" |
|
1387 |
using numeral_power_eq_real_of_int_cancel_iff [of x n y] by metis |
|
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1388 |
|
63353 | 1389 |
lemma numeral_power_eq_real_of_nat_cancel_iff [simp]: |
1390 |
"numeral x ^ n = real y \<longleftrightarrow> numeral x ^ n = y" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1391 |
using of_nat_eq_iff by fastforce |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1392 |
|
63353 | 1393 |
lemma real_of_nat_eq_numeral_power_cancel_iff [simp]: |
1394 |
"real y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n" |
|
1395 |
using numeral_power_eq_real_of_nat_cancel_iff [of x n y] by metis |
|
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1396 |
|
63353 | 1397 |
lemma numeral_power_le_real_of_nat_cancel_iff [simp]: |
1398 |
"(numeral x :: real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a" |
|
1399 |
by (metis of_nat_le_iff of_nat_numeral of_nat_power) |
|
51523 | 1400 |
|
63353 | 1401 |
lemma real_of_nat_le_numeral_power_cancel_iff [simp]: |
51523 | 1402 |
"real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n" |
63353 | 1403 |
by (metis of_nat_le_iff of_nat_numeral of_nat_power) |
51523 | 1404 |
|
63353 | 1405 |
lemma numeral_power_le_real_of_int_cancel_iff [simp]: |
1406 |
"(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1407 |
by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power) |
51523 | 1408 |
|
63353 | 1409 |
lemma real_of_int_le_numeral_power_cancel_iff [simp]: |
1410 |
"real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1411 |
by (metis floor_of_int le_floor_iff of_int_numeral of_int_power) |
51523 | 1412 |
|
63353 | 1413 |
lemma numeral_power_less_real_of_nat_cancel_iff [simp]: |
1414 |
"(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a" |
|
1415 |
by (metis of_nat_less_iff of_nat_numeral of_nat_power) |
|
1416 |
||
1417 |
lemma real_of_nat_less_numeral_power_cancel_iff [simp]: |
|
1418 |
"real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1419 |
by (metis of_nat_less_iff of_nat_numeral of_nat_power) |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1420 |
|
63353 | 1421 |
lemma numeral_power_less_real_of_int_cancel_iff [simp]: |
1422 |
"(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1423 |
by (meson not_less real_of_int_le_numeral_power_cancel_iff) |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1424 |
|
63353 | 1425 |
lemma real_of_int_less_numeral_power_cancel_iff [simp]: |
1426 |
"real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1427 |
by (meson not_less numeral_power_le_real_of_int_cancel_iff) |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1428 |
|
63353 | 1429 |
lemma neg_numeral_power_le_real_of_int_cancel_iff [simp]: |
1430 |
"(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1431 |
by (metis of_int_le_iff of_int_neg_numeral of_int_power) |
51523 | 1432 |
|
63353 | 1433 |
lemma real_of_int_le_neg_numeral_power_cancel_iff [simp]: |
1434 |
"real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1435 |
by (metis of_int_le_iff of_int_neg_numeral of_int_power) |
51523 | 1436 |
|
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56571
diff
changeset
|
1437 |
|
63353 | 1438 |
subsection \<open>Density of the Reals\<close> |
1439 |
||
63494 | 1440 |
lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2" |
1441 |
for d1 d2 :: real |
|
63353 | 1442 |
by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def) |
51523 | 1443 |
|
63353 | 1444 |
text \<open>Similar results are proved in @{theory Fields}\<close> |
63494 | 1445 |
lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2" |
1446 |
for x y :: real |
|
63353 | 1447 |
by auto |
1448 |
||
63494 | 1449 |
lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y" |
1450 |
for x y :: real |
|
63353 | 1451 |
by auto |
1452 |
||
63494 | 1453 |
lemma real_sum_of_halves: "x / 2 + x / 2 = x" |
1454 |
for x :: real |
|
63353 | 1455 |
by simp |
51523 | 1456 |
|
1457 |
||
63353 | 1458 |
subsection \<open>Floor and Ceiling Functions from the Reals to the Integers\<close> |
51523 | 1459 |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1460 |
(* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *) |
51523 | 1461 |
|
63494 | 1462 |
lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \<longleftrightarrow> n < numeral w" |
1463 |
for n :: nat |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1464 |
by (metis of_nat_less_iff of_nat_numeral) |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56571
diff
changeset
|
1465 |
|
63494 | 1466 |
lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n \<longleftrightarrow> numeral w < n" |
1467 |
for n :: nat |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1468 |
by (metis of_nat_less_iff of_nat_numeral) |
56889
48a745e1bde7
avoid the Complex constructor, use the more natural Re/Im view; moved csqrt to Complex.
hoelzl
parents:
56571
diff
changeset
|
1469 |
|
63494 | 1470 |
lemma numeral_le_real_of_nat_iff [simp]: "numeral n \<le> real m \<longleftrightarrow> numeral n \<le> m" |
1471 |
for m :: nat |
|
63353 | 1472 |
by (metis not_le real_of_nat_less_numeral_iff) |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59000
diff
changeset
|
1473 |
|
63353 | 1474 |
declare of_int_floor_le [simp] (* FIXME duplicate!? *) |
51523 | 1475 |
|
63353 | 1476 |
lemma of_int_floor_cancel [simp]: "of_int \<lfloor>x\<rfloor> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1477 |
by (metis floor_of_int) |
51523 | 1478 |
|
63353 | 1479 |
lemma floor_eq: "real_of_int n < x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1480 |
by linarith |
51523 | 1481 |
|
63353 | 1482 |
lemma floor_eq2: "real_of_int n \<le> x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1483 |
by linarith |
51523 | 1484 |
|
63353 | 1485 |
lemma floor_eq3: "real n < x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1486 |
by linarith |
51523 | 1487 |
|
63353 | 1488 |
lemma floor_eq4: "real n \<le> x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1489 |
by linarith |
51523 | 1490 |
|
61942 | 1491 |
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int \<lfloor>r\<rfloor>" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1492 |
by linarith |
51523 | 1493 |
|
61942 | 1494 |
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int \<lfloor>r\<rfloor>" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1495 |
by linarith |
51523 | 1496 |
|
61942 | 1497 |
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int \<lfloor>r\<rfloor> + 1" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1498 |
by linarith |
51523 | 1499 |
|
61942 | 1500 |
lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int \<lfloor>r\<rfloor> + 1" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1501 |
by linarith |
51523 | 1502 |
|
61942 | 1503 |
lemma floor_eq_iff: "\<lfloor>x\<rfloor> = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)" |
1504 |
by (simp add: floor_unique_iff) |
|
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1505 |
|
63353 | 1506 |
lemma floor_divide_real_eq_div: |
1507 |
assumes "0 \<le> b" |
|
1508 |
shows "\<lfloor>a / real_of_int b\<rfloor> = \<lfloor>a\<rfloor> div b" |
|
1509 |
proof (cases "b = 0") |
|
1510 |
case True |
|
1511 |
then show ?thesis by simp |
|
1512 |
next |
|
1513 |
case False |
|
1514 |
with assms have b: "b > 0" by simp |
|
1515 |
have "j = i div b" |
|
1516 |
if "real_of_int i \<le> a" "a < 1 + real_of_int i" |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1517 |
"real_of_int j * real_of_int b \<le> a" "a < real_of_int b + real_of_int j * real_of_int b" |
63353 | 1518 |
for i j :: int |
1519 |
proof - |
|
1520 |
from that have "i < b + j * b" |
|
1521 |
by (metis le_less_trans of_int_add of_int_less_iff of_int_mult) |
|
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1522 |
moreover have "j * b < 1 + i" |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1523 |
proof - |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1524 |
have "real_of_int (j * b) < real_of_int i + 1" |
61799 | 1525 |
using \<open>a < 1 + real_of_int i\<close> \<open>real_of_int j * real_of_int b \<le> a\<close> by force |
63597 | 1526 |
then show "j * b < 1 + i" by linarith |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1527 |
qed |
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1528 |
ultimately have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b" |
58788
d17b3844b726
generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents:
58134
diff
changeset
|
1529 |
by (auto simp: field_simps) |
d17b3844b726
generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents:
58134
diff
changeset
|
1530 |
then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b" |
63353 | 1531 |
using pos_mod_bound [OF b, of i] pos_mod_sign [OF b, of i] |
1532 |
by linarith+ |
|
63597 | 1533 |
then show ?thesis using b unfolding mult_less_cancel_right by auto |
63353 | 1534 |
qed |
63597 | 1535 |
with b show ?thesis by (auto split: floor_split simp: field_simps) |
63353 | 1536 |
qed |
58788
d17b3844b726
generalize natfloor_div_nat, add floor variant: floor_divide_real_eq_div
hoelzl
parents:
58134
diff
changeset
|
1537 |
|
63601 | 1538 |
lemma floor_one_divide_eq_div_numeral [simp]: |
1539 |
"\<lfloor>1 / numeral b::real\<rfloor> = 1 div numeral b" |
|
1540 |
by (metis floor_divide_of_int_eq of_int_1 of_int_numeral) |
|
1541 |
||
1542 |
lemma floor_minus_one_divide_eq_div_numeral [simp]: |
|
1543 |
"\<lfloor>- (1 / numeral b)::real\<rfloor> = - 1 div numeral b" |
|
1544 |
by (metis (mono_tags, hide_lams) div_minus_right minus_divide_right |
|
1545 |
floor_divide_of_int_eq of_int_neg_numeral of_int_1) |
|
1546 |
||
63597 | 1547 |
lemma floor_divide_eq_div_numeral [simp]: |
1548 |
"\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b" |
|
1549 |
by (metis floor_divide_of_int_eq of_int_numeral) |
|
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1550 |
|
63353 | 1551 |
lemma floor_minus_divide_eq_div_numeral [simp]: |
1552 |
"\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b" |
|
63597 | 1553 |
by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral) |
51523 | 1554 |
|
63353 | 1555 |
lemma of_int_ceiling_cancel [simp]: "of_int \<lceil>x\<rceil> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1556 |
using ceiling_of_int by metis |
51523 | 1557 |
|
63353 | 1558 |
lemma ceiling_eq: "of_int n < x \<Longrightarrow> x \<le> of_int n + 1 \<Longrightarrow> \<lceil>x\<rceil> = n + 1" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
1559 |
by (simp add: ceiling_unique) |
51523 | 1560 |
|
61942 | 1561 |
lemma of_int_ceiling_diff_one_le [simp]: "of_int \<lceil>r\<rceil> - 1 \<le> r" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1562 |
by linarith |
51523 | 1563 |
|
61942 | 1564 |
lemma of_int_ceiling_le_add_one [simp]: "of_int \<lceil>r\<rceil> \<le> r + 1" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1565 |
by linarith |
51523 | 1566 |
|
63353 | 1567 |
lemma ceiling_le: "x \<le> of_int a \<Longrightarrow> \<lceil>x\<rceil> \<le> a" |
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
1568 |
by (simp add: ceiling_le_iff) |
51523 | 1569 |
|
61694
6571c78c9667
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
paulson <lp15@cam.ac.uk>
parents:
61649
diff
changeset
|
1570 |
lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)" |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1571 |
by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus) |
58097
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1572 |
|
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1573 |
lemma ceiling_divide_eq_div_numeral [simp]: |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1574 |
"\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)" |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1575 |
using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1576 |
|
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1577 |
lemma ceiling_minus_divide_eq_div_numeral [simp]: |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1578 |
"\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)" |
cfd3cff9387b
add simp rules for divisions of numerals in floor and ceiling.
hoelzl
parents:
58061
diff
changeset
|
1579 |
using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp |
51523 | 1580 |
|
63353 | 1581 |
text \<open> |
1582 |
The following lemmas are remnants of the erstwhile functions natfloor |
|
1583 |
and natceiling. |
|
1584 |
\<close> |
|
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1585 |
|
63494 | 1586 |
lemma nat_floor_neg: "x \<le> 0 \<Longrightarrow> nat \<lfloor>x\<rfloor> = 0" |
1587 |
for x :: real |
|
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1588 |
by linarith |
51523 | 1589 |
|
63353 | 1590 |
lemma le_nat_floor: "real x \<le> a \<Longrightarrow> x \<le> nat \<lfloor>a\<rfloor>" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1591 |
by linarith |
51523 | 1592 |
|
61942 | 1593 |
lemma le_mult_nat_floor: "nat \<lfloor>a\<rfloor> * nat \<lfloor>b\<rfloor> \<le> nat \<lfloor>a * b\<rfloor>" |
63353 | 1594 |
by (cases "0 \<le> a \<and> 0 \<le> b") |
59587
8ea7b22525cb
Removed the obsolete functions "natfloor" and "natceiling"
nipkow
parents:
59000
diff
changeset
|
1595 |
(auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor) |
51523 | 1596 |
|
63353 | 1597 |
lemma nat_ceiling_le_eq [simp]: "nat \<lceil>x\<rceil> \<le> a \<longleftrightarrow> x \<le> real a" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1598 |
by linarith |
51523 | 1599 |
|
63353 | 1600 |
lemma real_nat_ceiling_ge: "x \<le> real (nat \<lceil>x\<rceil>)" |
58040
9a867afaab5a
better linarith support for floor, ceiling, natfloor, and natceiling
hoelzl
parents:
57514
diff
changeset
|
1601 |
by linarith |
51523 | 1602 |
|
63494 | 1603 |
lemma Rats_no_top_le: "\<exists>q \<in> \<rat>. x \<le> q" |
1604 |
for x :: real |
|
61942 | 1605 |
by (auto intro!: bexI[of _ "of_nat (nat \<lceil>x\<rceil>)"]) linarith |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
56889
diff
changeset
|
1606 |
|
63353 | 1607 |
lemma Rats_no_bot_less: "\<exists>q \<in> \<rat>. q < x" for x :: real |
61942 | 1608 |
apply (auto intro!: bexI[of _ "of_int (\<lfloor>x\<rfloor> - 1)"]) |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1609 |
apply (rule less_le_trans[OF _ of_int_floor_le]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1610 |
apply simp |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1611 |
done |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57275
diff
changeset
|
1612 |
|
63353 | 1613 |
|
60758 | 1614 |
subsection \<open>Exponentiation with floor\<close> |
51523 | 1615 |
|
1616 |
lemma floor_power: |
|
61942 | 1617 |
assumes "x = of_int \<lfloor>x\<rfloor>" |
1618 |
shows "\<lfloor>x ^ n\<rfloor> = \<lfloor>x\<rfloor> ^ n" |
|
51523 | 1619 |
proof - |
61942 | 1620 |
have "x ^ n = of_int (\<lfloor>x\<rfloor> ^ n)" |
51523 | 1621 |
using assms by (induct n arbitrary: x) simp_all |
62626
de25474ce728
Contractible sets. Also removal of obsolete theorems and refactoring
paulson <lp15@cam.ac.uk>
parents:
62623
diff
changeset
|
1622 |
then show ?thesis by (metis floor_of_int) |
51523 | 1623 |
qed |
61609
77b453bd616f
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
paulson <lp15@cam.ac.uk>
parents:
61284
diff
changeset
|
1624 |
|
63353 | 1625 |
lemma floor_numeral_power [simp]: "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n" |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1626 |
by (metis floor_of_int of_int_numeral of_int_power) |
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1627 |
|
63353 | 1628 |
lemma ceiling_numeral_power [simp]: "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n" |
58983
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1629 |
by (metis ceiling_of_int of_int_numeral of_int_power) |
9c390032e4eb
cancel real of power of numeral also for equality and strict inequality;
immler
parents:
58889
diff
changeset
|
1630 |
|
63353 | 1631 |
|
60758 | 1632 |
subsection \<open>Implementation of rational real numbers\<close> |
51523 | 1633 |
|
60758 | 1634 |
text \<open>Formal constructor\<close> |
51523 | 1635 |
|
63353 | 1636 |
definition Ratreal :: "rat \<Rightarrow> real" |
1637 |
where [code_abbrev, simp]: "Ratreal = of_rat" |
|
51523 | 1638 |
|
1639 |
code_datatype Ratreal |
|
1640 |
||
1641 |
||
60758 | 1642 |
text \<open>Numerals\<close> |
51523 | 1643 |
|
63353 | 1644 |
lemma [code_abbrev]: "(of_rat (of_int a) :: real) = of_int a" |
51523 | 1645 |
by simp |
1646 |
||
63353 | 1647 |
lemma [code_abbrev]: "(of_rat 0 :: real) = 0" |
51523 | 1648 |
by simp |
1649 |
||
63353 | 1650 |
lemma [code_abbrev]: "(of_rat 1 :: real) = 1" |
51523 | 1651 |
by simp |
1652 |
||
63353 | 1653 |
lemma [code_abbrev]: "(of_rat (- 1) :: real) = - 1" |
58134
b563ec62d04e
more convenient printing of real numbers after evaluation
haftmann
parents:
58097
diff
changeset
|
1654 |
by simp |
b563ec62d04e
more convenient printing of real numbers after evaluation
haftmann
parents:
58097
diff
changeset
|
1655 |
|
63353 | 1656 |
lemma [code_abbrev]: "(of_rat (numeral k) :: real) = numeral k" |
51523 | 1657 |
by simp |
1658 |
||
63353 | 1659 |
lemma [code_abbrev]: "(of_rat (- numeral k) :: real) = - numeral k" |
51523 | 1660 |
by simp |
1661 |
||
1662 |
lemma [code_post]: |
|
1663 |
"(of_rat (1 / numeral k) :: real) = 1 / numeral k" |
|
58134
b563ec62d04e
more convenient printing of real numbers after evaluation
haftmann
parents:
58097
diff
changeset
|
1664 |
"(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l" |
b563ec62d04e
more convenient printing of real numbers after evaluation
haftmann
parents:
58097
diff
changeset
|
1665 |
"(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)" |
b563ec62d04e
more convenient printing of real numbers after evaluation
haftmann
parents:
58097
diff
changeset
|
1666 |
"(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)" |
54489
03ff4d1e6784
eliminiated neg_numeral in favour of - (numeral _)
haftmann
parents:
54281
diff
changeset
|
1667 |
by (simp_all add: of_rat_divide of_rat_minus) |
51523 | 1668 |
|
1669 |
||
60758 | 1670 |
text \<open>Operations\<close> |
51523 | 1671 |
|
63353 | 1672 |
lemma zero_real_code [code]: "0 = Ratreal 0" |
63494 | 1673 |
by simp |
51523 | 1674 |
|
63353 | 1675 |
lemma one_real_code [code]: "1 = Ratreal 1" |
63494 | 1676 |
by simp |
51523 | 1677 |
|
1678 |
instantiation real :: equal |
|
1679 |
begin |
|
1680 |
||
63353 | 1681 |
definition "HOL.equal x y \<longleftrightarrow> x - y = 0" for x :: real |
51523 | 1682 |
|
63353 | 1683 |
instance by standard (simp add: equal_real_def) |
51523 | 1684 |
|
63353 | 1685 |
lemma real_equal_code [code]: "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y" |
51523 | 1686 |
by (simp add: equal_real_def equal) |
1687 |
||
63494 | 1688 |
lemma [code nbe]: "HOL.equal x x \<longleftrightarrow> True" |
1689 |
for x :: real |
|
51523 | 1690 |
by (rule equal_refl) |
1691 |
||
1692 |
end |
|
1693 |
||
1694 |
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y" |
|
1695 |
by (simp add: of_rat_less_eq) |
|
1696 |
||
1697 |
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y" |
|
1698 |
by (simp add: of_rat_less) |
|
1699 |
||
1700 |
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)" |
|
1701 |
by (simp add: of_rat_add) |
|
1702 |
||
1703 |
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)" |
|
1704 |
by (simp add: of_rat_mult) |
|
1705 |
||
1706 |
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)" |
|
1707 |
by (simp add: of_rat_minus) |
|
1708 |
||
1709 |
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)" |
|
1710 |
by (simp add: of_rat_diff) |
|
1711 |
||
1712 |
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)" |
|
1713 |
by (simp add: of_rat_inverse) |
|
61284
2314c2f62eb1
real_of_nat_Suc is now a simprule
paulson <lp15@cam.ac.uk>
parents:
61204
diff
changeset
|
1714 |
|
51523 | 1715 |
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)" |
1716 |
by (simp add: of_rat_divide) |
|
1717 |
||
61942 | 1718 |
lemma real_floor_code [code]: "\<lfloor>Ratreal x\<rfloor> = \<lfloor>x\<rfloor>" |
63353 | 1719 |
by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff |
1720 |
of_int_floor_le of_rat_of_int_eq real_less_eq_code) |
|
51523 | 1721 |
|
1722 |
||
60758 | 1723 |
text \<open>Quickcheck\<close> |
51523 | 1724 |
|
1725 |
definition (in term_syntax) |
|
63353 | 1726 |
valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" |
1727 |
where [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k" |
|
51523 | 1728 |
|
1729 |
notation fcomp (infixl "\<circ>>" 60) |
|
1730 |
notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
1731 |
||
1732 |
instantiation real :: random |
|
1733 |
begin |
|
1734 |
||
1735 |
definition |
|
1736 |
"Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))" |
|
1737 |
||
1738 |
instance .. |
|
1739 |
||
1740 |
end |
|
1741 |
||
1742 |
no_notation fcomp (infixl "\<circ>>" 60) |
|
1743 |
no_notation scomp (infixl "\<circ>\<rightarrow>" 60) |
|
1744 |
||
1745 |
instantiation real :: exhaustive |
|
1746 |
begin |
|
1747 |
||
1748 |
definition |
|
63353 | 1749 |
"exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (\<lambda>r. f (Ratreal r)) d" |
51523 | 1750 |
|
1751 |
instance .. |
|
1752 |
||
1753 |
end |
|
1754 |
||
1755 |
instantiation real :: full_exhaustive |
|
1756 |
begin |
|
1757 |
||
1758 |
definition |
|
63353 | 1759 |
"full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (\<lambda>r. f (valterm_ratreal r)) d" |
51523 | 1760 |
|
1761 |
instance .. |
|
1762 |
||
1763 |
end |
|
1764 |
||
1765 |
instantiation real :: narrowing |
|
1766 |
begin |
|
1767 |
||
1768 |
definition |
|
63353 | 1769 |
"narrowing_real = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing" |
51523 | 1770 |
|
1771 |
instance .. |
|
1772 |
||
1773 |
end |
|
1774 |
||
1775 |
||
60758 | 1776 |
subsection \<open>Setup for Nitpick\<close> |
51523 | 1777 |
|
60758 | 1778 |
declaration \<open> |
51523 | 1779 |
Nitpick_HOL.register_frac_type @{type_name real} |
62079 | 1780 |
[(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}), |
1781 |
(@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}), |
|
1782 |
(@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}), |
|
1783 |
(@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}), |
|
1784 |
(@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}), |
|
1785 |
(@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}), |
|
1786 |
(@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}), |
|
1787 |
(@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})] |
|
60758 | 1788 |
\<close> |
51523 | 1789 |
|
1790 |
lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real |
|
63353 | 1791 |
ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real |
1792 |
times_real_inst.times_real uminus_real_inst.uminus_real |
|
1793 |
zero_real_inst.zero_real |
|
51523 | 1794 |
|
56078
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1795 |
|
60758 | 1796 |
subsection \<open>Setup for SMT\<close> |
56078
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1797 |
|
58061 | 1798 |
ML_file "Tools/SMT/smt_real.ML" |
1799 |
ML_file "Tools/SMT/z3_real.ML" |
|
56078
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1800 |
|
58061 | 1801 |
lemma [z3_rule]: |
63353 | 1802 |
"0 + x = x" |
56078
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1803 |
"x + 0 = x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1804 |
"0 * x = 0" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1805 |
"1 * x = x" |
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1806 |
"x + y = y + x" |
63353 | 1807 |
for x y :: real |
56078
624faeda77b5
moved 'SMT2' (SMT-LIB-2-based SMT module) into Isabelle
blanchet
parents:
55945
diff
changeset
|
1808 |
by auto |
51523 | 1809 |
|
63960
3daf02070be5
new proof method "argo" for a combination of quantifier-free propositional logic with equality and linear real arithmetic
boehmes
parents:
63680
diff
changeset
|
1810 |
|
3daf02070be5
new proof method "argo" for a combination of quantifier-free propositional logic with equality and linear real arithmetic
boehmes
parents:
63680
diff
changeset
|
1811 |
subsection \<open>Setup for Argo\<close> |
3daf02070be5
new proof method "argo" for a combination of quantifier-free propositional logic with equality and linear real arithmetic
boehmes
parents:
63680
diff
changeset
|
1812 |
|
3daf02070be5
new proof method "argo" for a combination of quantifier-free propositional logic with equality and linear real arithmetic
boehmes
parents:
63680
diff
changeset
|
1813 |
ML_file "Tools/Argo/argo_real.ML" |
3daf02070be5
new proof method "argo" for a combination of quantifier-free propositional logic with equality and linear real arithmetic
boehmes
parents:
63680
diff
changeset
|
1814 |
|
51523 | 1815 |
end |