src/HOL/Real.thy
author wenzelm
Thu Jun 23 23:08:37 2016 +0200 (2016-06-23)
changeset 63353 176d1f408696
parent 63331 247eac9758dd
child 63494 ac0a3b9c6dae
permissions -rw-r--r--
misc tuning and modernization;
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(*  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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imports Rat
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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
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  equivalence classes of Cauchy sequences of rationals.  See
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  @{file "~~/src/HOL/ex/Dedekind_Real.thy"} for an alternative
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  construction using 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)" 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" 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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lemma add_diff_add: "(a + c) - (b + d) = (a - b) + (c - d)" for a b c d :: "'a::ab_group_add"
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  by simp
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lemma minus_diff_minus: "- a - - b = - (a - b)" for a b :: "'a::ab_group_add"
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  by simp
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lemma mult_diff_mult: "(x * y - a * b) = x * (y - b) + (x - a) * b" 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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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>" by (rule abs_triangle_ineq)
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    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
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    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
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  qed
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  then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
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qed
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lemma vanishes_diff:
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  assumes "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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    using X by blast
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  obtain b where b: "0 < b" "r = a * b"
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  proof
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    show "0 < r / a" using r a by simp
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    show "r = a * (r / a)" using a by simp
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  qed
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  obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
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    using vanishesD [OF Y b(1)] ..
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  have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
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    by (simp add: b(2) abs_mult mult_strict_mono' a k)
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  then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
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qed
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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
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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
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lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
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  unfolding cauchy_def by simp
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lemma cauchy_add [simp]:
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  assumes X: "cauchy X" and Y: "cauchy Y"
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  shows "cauchy (\<lambda>n. X n + Y n)"
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proof (rule cauchyI)
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  fix r :: rat
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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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  proof clarsimp
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    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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    also have "\<dots> < s + t"
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      by (rule add_strict_mono) (simp_all add: i j *)
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    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
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lemma cauchy_minus [simp]:
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  assumes X: "cauchy X"
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  shows "cauchy (\<lambda>n. - X n)"
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  using assms unfolding cauchy_def
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  unfolding minus_diff_minus abs_minus_cancel .
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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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  using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff)
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lemma cauchy_imp_bounded:
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  assumes "cauchy X"
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  shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
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proof -
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  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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    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
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    fix n :: nat
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    show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1"
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    proof (rule linorder_le_cases)
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      assume "n \<le> k"
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      then have "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
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      then show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp
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    next
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      assume "k \<le> n"
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      have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
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      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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        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" .
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    qed
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  qed
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qed
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lemma cauchy_mult [simp]:
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  assumes X: "cauchy X" and Y: "cauchy Y"
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  shows "cauchy (\<lambda>n. X n * Y n)"
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proof (rule cauchyI)
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  fix r :: rat assume "0 < r"
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  then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
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    by (rule obtain_pos_sum)
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  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
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    using cauchy_imp_bounded [OF X] by blast
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  obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
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    using cauchy_imp_bounded [OF Y] by blast
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  obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
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  proof
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    show "0 < v/b" using v b(1) by simp
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    show "0 < u/a" using u a(1) by simp
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    show "r = a * (u/a) + (v/b) * b"
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      using a(1) b(1) \<open>r = u + v\<close> by simp
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  qed
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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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  proof clarsimp
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    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> = \<bar>X m * (Y m - Y n) + (X m - X n) * Y n\<bar>"
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      unfolding mult_diff_mult ..
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    also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
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      by (rule abs_triangle_ineq)
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    also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
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      unfolding abs_mult ..
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    also have "\<dots> < a * t + s * b"
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      by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
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    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
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lemma cauchy_not_vanishes_cases:
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  assumes X: "cauchy X"
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  assumes nz: "\<not> vanishes X"
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  shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
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proof -
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  obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
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    using nz unfolding vanishes_def by (auto simp add: not_less)
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  obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
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    using \<open>0 < r\<close> 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 k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
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    using r by blast
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  have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
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    using i \<open>i \<le> k\<close> by auto
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  have "X k \<le> - r \<or> r \<le> X k"
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    using \<open>r \<le> \<bar>X k\<bar>\<close> by auto
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  then have "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
wenzelm@60758
   268
    unfolding \<open>r = s + t\<close> using k by auto
wenzelm@63353
   269
  then have "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
wenzelm@63353
   270
  then show "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
hoelzl@51523
   271
    using t by auto
hoelzl@51523
   272
qed
hoelzl@51523
   273
hoelzl@51523
   274
lemma cauchy_not_vanishes:
hoelzl@51523
   275
  assumes X: "cauchy X"
hoelzl@51523
   276
  assumes nz: "\<not> vanishes X"
hoelzl@51523
   277
  shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
wenzelm@63353
   278
  using cauchy_not_vanishes_cases [OF assms]
wenzelm@63353
   279
  apply clarify
wenzelm@63353
   280
  apply (rule exI)
wenzelm@63353
   281
  apply (erule conjI)
wenzelm@63353
   282
  apply (rule_tac x = k in exI)
wenzelm@63353
   283
  apply auto
wenzelm@63353
   284
  done
hoelzl@51523
   285
hoelzl@51523
   286
lemma cauchy_inverse [simp]:
hoelzl@51523
   287
  assumes X: "cauchy X"
hoelzl@51523
   288
  assumes nz: "\<not> vanishes X"
hoelzl@51523
   289
  shows "cauchy (\<lambda>n. inverse (X n))"
hoelzl@51523
   290
proof (rule cauchyI)
wenzelm@63353
   291
  fix r :: rat
wenzelm@63353
   292
  assume "0 < r"
hoelzl@51523
   293
  obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
lp15@61649
   294
    using cauchy_not_vanishes [OF X nz] by blast
hoelzl@51523
   295
  from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
hoelzl@51523
   296
  obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
hoelzl@51523
   297
  proof
wenzelm@60758
   298
    show "0 < b * r * b" by (simp add: \<open>0 < r\<close> b)
hoelzl@51523
   299
    show "r = inverse b * (b * r * b) * inverse b"
hoelzl@51523
   300
      using b by simp
hoelzl@51523
   301
  qed
hoelzl@51523
   302
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
hoelzl@51523
   303
    using cauchyD [OF X s] ..
hoelzl@51523
   304
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
wenzelm@63353
   305
  proof clarsimp
wenzelm@63353
   306
    fix m n
wenzelm@63353
   307
    assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
wenzelm@63353
   308
    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>"
hoelzl@51523
   309
      by (simp add: inverse_diff_inverse nz * abs_mult)
hoelzl@51523
   310
    also have "\<dots> < inverse b * s * inverse b"
wenzelm@63353
   311
      by (simp add: mult_strict_mono less_imp_inverse_less i j b * s)
wenzelm@63353
   312
    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" by (simp only: r)
hoelzl@51523
   313
  qed
wenzelm@63353
   314
  then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
hoelzl@51523
   315
qed
hoelzl@51523
   316
hoelzl@51523
   317
lemma vanishes_diff_inverse:
hoelzl@51523
   318
  assumes X: "cauchy X" "\<not> vanishes X"
wenzelm@63353
   319
    and Y: "cauchy Y" "\<not> vanishes Y"
wenzelm@63353
   320
    and XY: "vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   321
  shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
hoelzl@51523
   322
proof (rule vanishesI)
wenzelm@63353
   323
  fix r :: rat
wenzelm@63353
   324
  assume r: "0 < r"
hoelzl@51523
   325
  obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
lp15@61649
   326
    using cauchy_not_vanishes [OF X] by blast
hoelzl@51523
   327
  obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
lp15@61649
   328
    using cauchy_not_vanishes [OF Y] by blast
hoelzl@51523
   329
  obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
hoelzl@51523
   330
  proof
wenzelm@63353
   331
    show "0 < a * r * b" using a r b by simp
wenzelm@63353
   332
    show "inverse a * (a * r * b) * inverse b = r" using a r b by simp
hoelzl@51523
   333
  qed
hoelzl@51523
   334
  obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
hoelzl@51523
   335
    using vanishesD [OF XY s] ..
hoelzl@51523
   336
  have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
wenzelm@63353
   337
  proof clarsimp
wenzelm@63353
   338
    fix n
wenzelm@63353
   339
    assume n: "i \<le> n" "j \<le> n" "k \<le> n"
wenzelm@63353
   340
    with i j a b have "X n \<noteq> 0" and "Y n \<noteq> 0"
wenzelm@63353
   341
      by auto
wenzelm@63353
   342
    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>"
hoelzl@51523
   343
      by (simp add: inverse_diff_inverse abs_mult)
hoelzl@51523
   344
    also have "\<dots> < inverse a * s * inverse b"
wenzelm@63353
   345
      by (intro mult_strict_mono' less_imp_inverse_less) (simp_all add: a b i j k n)
wenzelm@60758
   346
    also note \<open>inverse a * s * inverse b = r\<close>
hoelzl@51523
   347
    finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
hoelzl@51523
   348
  qed
wenzelm@63353
   349
  then show "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
hoelzl@51523
   350
qed
hoelzl@51523
   351
wenzelm@63353
   352
wenzelm@60758
   353
subsection \<open>Equivalence relation on Cauchy sequences\<close>
hoelzl@51523
   354
hoelzl@51523
   355
definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
hoelzl@51523
   356
  where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
hoelzl@51523
   357
wenzelm@63353
   358
lemma realrelI [intro?]: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> vanishes (\<lambda>n. X n - Y n) \<Longrightarrow> realrel X Y"
wenzelm@63353
   359
  by (simp add: realrel_def)
hoelzl@51523
   360
hoelzl@51523
   361
lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
wenzelm@63353
   362
  by (simp add: realrel_def)
hoelzl@51523
   363
hoelzl@51523
   364
lemma symp_realrel: "symp realrel"
hoelzl@51523
   365
  unfolding realrel_def
wenzelm@63353
   366
  apply (rule sympI)
wenzelm@63353
   367
  apply clarify
wenzelm@63353
   368
  apply (drule vanishes_minus)
wenzelm@63353
   369
  apply simp
wenzelm@63353
   370
  done
hoelzl@51523
   371
hoelzl@51523
   372
lemma transp_realrel: "transp realrel"
hoelzl@51523
   373
  unfolding realrel_def
wenzelm@63353
   374
  apply (rule transpI)
wenzelm@63353
   375
  apply clarify
hoelzl@51523
   376
  apply (drule (1) vanishes_add)
hoelzl@51523
   377
  apply (simp add: algebra_simps)
hoelzl@51523
   378
  done
hoelzl@51523
   379
hoelzl@51523
   380
lemma part_equivp_realrel: "part_equivp realrel"
wenzelm@63353
   381
  by (blast intro: part_equivpI symp_realrel transp_realrel realrel_refl cauchy_const)
wenzelm@63353
   382
hoelzl@51523
   383
wenzelm@60758
   384
subsection \<open>The field of real numbers\<close>
hoelzl@51523
   385
hoelzl@51523
   386
quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
hoelzl@51523
   387
  morphisms rep_real Real
hoelzl@51523
   388
  by (rule part_equivp_realrel)
hoelzl@51523
   389
hoelzl@51523
   390
lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
hoelzl@51523
   391
  unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
hoelzl@51523
   392
hoelzl@51523
   393
lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
wenzelm@63353
   394
  assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)"
wenzelm@63353
   395
  shows "P x"
hoelzl@51523
   396
proof (induct x)
hoelzl@51523
   397
  case (1 X)
wenzelm@63353
   398
  then have "cauchy X" by (simp add: realrel_def)
wenzelm@63353
   399
  then show "P (Real X)" by (rule assms)
hoelzl@51523
   400
qed
hoelzl@51523
   401
wenzelm@63353
   402
lemma eq_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   403
  using real.rel_eq_transfer
blanchet@55945
   404
  unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp
hoelzl@51523
   405
kuncar@51956
   406
lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
wenzelm@63353
   407
  by (simp add: real.domain_eq realrel_def)
hoelzl@51523
   408
haftmann@59867
   409
instantiation real :: field
hoelzl@51523
   410
begin
hoelzl@51523
   411
hoelzl@51523
   412
lift_definition zero_real :: "real" is "\<lambda>n. 0"
hoelzl@51523
   413
  by (simp add: realrel_refl)
hoelzl@51523
   414
hoelzl@51523
   415
lift_definition one_real :: "real" is "\<lambda>n. 1"
hoelzl@51523
   416
  by (simp add: realrel_refl)
hoelzl@51523
   417
hoelzl@51523
   418
lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
hoelzl@51523
   419
  unfolding realrel_def add_diff_add
hoelzl@51523
   420
  by (simp only: cauchy_add vanishes_add simp_thms)
hoelzl@51523
   421
hoelzl@51523
   422
lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
hoelzl@51523
   423
  unfolding realrel_def minus_diff_minus
hoelzl@51523
   424
  by (simp only: cauchy_minus vanishes_minus simp_thms)
hoelzl@51523
   425
hoelzl@51523
   426
lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
hoelzl@51523
   427
  unfolding realrel_def mult_diff_mult
wenzelm@63353
   428
  apply (subst (4) mult.commute)
wenzelm@63353
   429
  apply (simp only: cauchy_mult vanishes_add vanishes_mult_bounded cauchy_imp_bounded simp_thms)
wenzelm@63353
   430
  done
hoelzl@51523
   431
hoelzl@51523
   432
lift_definition inverse_real :: "real \<Rightarrow> real"
hoelzl@51523
   433
  is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
hoelzl@51523
   434
proof -
wenzelm@63353
   435
  fix X Y
wenzelm@63353
   436
  assume "realrel X Y"
wenzelm@63353
   437
  then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   438
    unfolding realrel_def by simp_all
hoelzl@51523
   439
  have "vanishes X \<longleftrightarrow> vanishes Y"
hoelzl@51523
   440
  proof
hoelzl@51523
   441
    assume "vanishes X"
hoelzl@51523
   442
    from vanishes_diff [OF this XY] show "vanishes Y" by simp
hoelzl@51523
   443
  next
hoelzl@51523
   444
    assume "vanishes Y"
hoelzl@51523
   445
    from vanishes_add [OF this XY] show "vanishes X" by simp
hoelzl@51523
   446
  qed
wenzelm@63353
   447
  then show "?thesis X Y" by (simp add: vanishes_diff_inverse X Y XY realrel_def)
hoelzl@51523
   448
qed
hoelzl@51523
   449
wenzelm@63353
   450
definition "x - y = x + - y" for x y :: real
hoelzl@51523
   451
wenzelm@63353
   452
definition "x div y = x * inverse y" for x y :: real
wenzelm@63353
   453
wenzelm@63353
   454
lemma add_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X + Real Y = Real (\<lambda>n. X n + Y n)"
wenzelm@63353
   455
  using plus_real.transfer by (simp add: cr_real_eq rel_fun_def)
hoelzl@51523
   456
wenzelm@63353
   457
lemma minus_Real: "cauchy X \<Longrightarrow> - Real X = Real (\<lambda>n. - X n)"
wenzelm@63353
   458
  using uminus_real.transfer by (simp add: cr_real_eq rel_fun_def)
hoelzl@51523
   459
wenzelm@63353
   460
lemma diff_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X - Real Y = Real (\<lambda>n. X n - Y n)"
wenzelm@63353
   461
  by (simp add: minus_Real add_Real minus_real_def)
hoelzl@51523
   462
wenzelm@63353
   463
lemma mult_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X * Real Y = Real (\<lambda>n. X n * Y n)"
wenzelm@63353
   464
  using times_real.transfer by (simp add: cr_real_eq rel_fun_def)
hoelzl@51523
   465
hoelzl@51523
   466
lemma inverse_Real:
wenzelm@63353
   467
  "cauchy X \<Longrightarrow> inverse (Real X) = (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
wenzelm@63353
   468
  using inverse_real.transfer zero_real.transfer
nipkow@62390
   469
  unfolding cr_real_eq rel_fun_def by (simp split: if_split_asm, metis)
hoelzl@51523
   470
wenzelm@63353
   471
instance
wenzelm@63353
   472
proof
hoelzl@51523
   473
  fix a b c :: real
hoelzl@51523
   474
  show "a + b = b + a"
haftmann@57514
   475
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   476
  show "(a + b) + c = a + (b + c)"
haftmann@57514
   477
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   478
  show "0 + a = a"
hoelzl@51523
   479
    by transfer (simp add: realrel_def)
hoelzl@51523
   480
  show "- a + a = 0"
hoelzl@51523
   481
    by transfer (simp add: realrel_def)
hoelzl@51523
   482
  show "a - b = a + - b"
hoelzl@51523
   483
    by (rule minus_real_def)
hoelzl@51523
   484
  show "(a * b) * c = a * (b * c)"
haftmann@57514
   485
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   486
  show "a * b = b * a"
haftmann@57514
   487
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   488
  show "1 * a = a"
haftmann@57514
   489
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   490
  show "(a + b) * c = a * c + b * c"
hoelzl@51523
   491
    by transfer (simp add: distrib_right realrel_def)
wenzelm@61076
   492
  show "(0::real) \<noteq> (1::real)"
hoelzl@51523
   493
    by transfer (simp add: realrel_def)
hoelzl@51523
   494
  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
hoelzl@51523
   495
    apply transfer
hoelzl@51523
   496
    apply (simp add: realrel_def)
hoelzl@51523
   497
    apply (rule vanishesI)
hoelzl@51523
   498
    apply (frule (1) cauchy_not_vanishes, clarify)
hoelzl@51523
   499
    apply (rule_tac x=k in exI, clarify)
hoelzl@51523
   500
    apply (drule_tac x=n in spec, simp)
hoelzl@51523
   501
    done
haftmann@60429
   502
  show "a div b = a * inverse b"
hoelzl@51523
   503
    by (rule divide_real_def)
hoelzl@51523
   504
  show "inverse (0::real) = 0"
hoelzl@51523
   505
    by transfer (simp add: realrel_def)
hoelzl@51523
   506
qed
hoelzl@51523
   507
hoelzl@51523
   508
end
hoelzl@51523
   509
wenzelm@63353
   510
wenzelm@60758
   511
subsection \<open>Positive reals\<close>
hoelzl@51523
   512
hoelzl@51523
   513
lift_definition positive :: "real \<Rightarrow> bool"
hoelzl@51523
   514
  is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
hoelzl@51523
   515
proof -
wenzelm@63353
   516
  have 1: "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n"
wenzelm@63353
   517
    if *: "realrel X Y" and **: "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n" for X Y
wenzelm@63353
   518
  proof -
wenzelm@63353
   519
    from * have XY: "vanishes (\<lambda>n. X n - Y n)"
wenzelm@63353
   520
      by (simp_all add: realrel_def)
wenzelm@63353
   521
    from ** obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
lp15@61649
   522
      by blast
hoelzl@51523
   523
    obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
wenzelm@60758
   524
      using \<open>0 < r\<close> by (rule obtain_pos_sum)
hoelzl@51523
   525
    obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
hoelzl@51523
   526
      using vanishesD [OF XY s] ..
hoelzl@51523
   527
    have "\<forall>n\<ge>max i j. t < Y n"
wenzelm@63353
   528
    proof clarsimp
wenzelm@63353
   529
      fix n
wenzelm@63353
   530
      assume n: "i \<le> n" "j \<le> n"
hoelzl@51523
   531
      have "\<bar>X n - Y n\<bar> < s" and "r < X n"
hoelzl@51523
   532
        using i j n by simp_all
wenzelm@63353
   533
      then show "t < Y n" by (simp add: r)
hoelzl@51523
   534
    qed
wenzelm@63353
   535
    then show ?thesis using t by blast
wenzelm@63353
   536
  qed
hoelzl@51523
   537
  fix X Y assume "realrel X Y"
wenzelm@63353
   538
  then have "realrel X Y" and "realrel Y X"
wenzelm@63353
   539
    using symp_realrel by (auto simp: symp_def)
wenzelm@63353
   540
  then show "?thesis X Y"
hoelzl@51523
   541
    by (safe elim!: 1)
hoelzl@51523
   542
qed
hoelzl@51523
   543
wenzelm@63353
   544
lemma positive_Real: "cauchy X \<Longrightarrow> positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
wenzelm@63353
   545
  using positive.transfer by (simp add: cr_real_eq rel_fun_def)
hoelzl@51523
   546
hoelzl@51523
   547
lemma positive_zero: "\<not> positive 0"
hoelzl@51523
   548
  by transfer auto
hoelzl@51523
   549
wenzelm@63353
   550
lemma positive_add: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
wenzelm@63353
   551
  apply transfer
wenzelm@63353
   552
  apply clarify
wenzelm@63353
   553
  apply (rename_tac a b i j)
wenzelm@63353
   554
  apply (rule_tac x = "a + b" in exI)
wenzelm@63353
   555
  apply simp
wenzelm@63353
   556
  apply (rule_tac x = "max i j" in exI)
wenzelm@63353
   557
  apply clarsimp
wenzelm@63353
   558
  apply (simp add: add_strict_mono)
wenzelm@63353
   559
  done
hoelzl@51523
   560
wenzelm@63353
   561
lemma positive_mult: "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
wenzelm@63353
   562
  apply transfer
wenzelm@63353
   563
  apply clarify
wenzelm@63353
   564
  apply (rename_tac a b i j)
wenzelm@63353
   565
  apply (rule_tac x = "a * b" in exI)
wenzelm@63353
   566
  apply simp
wenzelm@63353
   567
  apply (rule_tac x = "max i j" in exI)
wenzelm@63353
   568
  apply clarsimp
wenzelm@63353
   569
  apply (rule mult_strict_mono)
wenzelm@63353
   570
  apply auto
wenzelm@63353
   571
  done
hoelzl@51523
   572
wenzelm@63353
   573
lemma positive_minus: "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
wenzelm@63353
   574
  apply transfer
wenzelm@63353
   575
  apply (simp add: realrel_def)
wenzelm@63353
   576
  apply (drule (1) cauchy_not_vanishes_cases, safe)
wenzelm@63353
   577
  apply blast+
wenzelm@63353
   578
  done
hoelzl@51523
   579
haftmann@59867
   580
instantiation real :: linordered_field
hoelzl@51523
   581
begin
hoelzl@51523
   582
wenzelm@63353
   583
definition "x < y \<longleftrightarrow> positive (y - x)"
hoelzl@51523
   584
wenzelm@63353
   585
definition "x \<le> y \<longleftrightarrow> x < y \<or> x = y" for x y :: real
hoelzl@51523
   586
wenzelm@63353
   587
definition "\<bar>a\<bar> = (if a < 0 then - a else a)" for a :: real
hoelzl@51523
   588
wenzelm@63353
   589
definition "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)" for a :: real
hoelzl@51523
   590
wenzelm@63353
   591
instance
wenzelm@63353
   592
proof
hoelzl@51523
   593
  fix a b c :: real
hoelzl@51523
   594
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
hoelzl@51523
   595
    by (rule abs_real_def)
hoelzl@51523
   596
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
hoelzl@51523
   597
    unfolding less_eq_real_def less_real_def
wenzelm@63353
   598
    apply auto
wenzelm@63353
   599
    apply (drule (1) positive_add)
wenzelm@63353
   600
    apply (simp_all add: positive_zero)
wenzelm@63353
   601
    done
hoelzl@51523
   602
  show "a \<le> a"
hoelzl@51523
   603
    unfolding less_eq_real_def by simp
hoelzl@51523
   604
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
hoelzl@51523
   605
    unfolding less_eq_real_def less_real_def
wenzelm@63353
   606
    apply auto
wenzelm@63353
   607
    apply (drule (1) positive_add)
wenzelm@63353
   608
    apply (simp add: algebra_simps)
wenzelm@63353
   609
    done
hoelzl@51523
   610
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
hoelzl@51523
   611
    unfolding less_eq_real_def less_real_def
wenzelm@63353
   612
    apply auto
wenzelm@63353
   613
    apply (drule (1) positive_add)
wenzelm@63353
   614
    apply (simp add: positive_zero)
wenzelm@63353
   615
    done
hoelzl@51523
   616
  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
wenzelm@63353
   617
    by (auto simp: less_eq_real_def less_real_def)
hoelzl@51523
   618
    (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
hoelzl@51523
   619
    (* Should produce c + b - (c + a) \<equiv> b - a *)
hoelzl@51523
   620
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
hoelzl@51523
   621
    by (rule sgn_real_def)
hoelzl@51523
   622
  show "a \<le> b \<or> b \<le> a"
wenzelm@63353
   623
    by (auto dest!: positive_minus simp: less_eq_real_def less_real_def)
hoelzl@51523
   624
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
hoelzl@51523
   625
    unfolding less_real_def
wenzelm@63353
   626
    apply (drule (1) positive_mult)
wenzelm@63353
   627
    apply (simp add: algebra_simps)
wenzelm@63353
   628
    done
hoelzl@51523
   629
qed
hoelzl@51523
   630
hoelzl@51523
   631
end
hoelzl@51523
   632
hoelzl@51523
   633
instantiation real :: distrib_lattice
hoelzl@51523
   634
begin
hoelzl@51523
   635
wenzelm@63353
   636
definition "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
hoelzl@51523
   637
wenzelm@63353
   638
definition "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
hoelzl@51523
   639
wenzelm@63353
   640
instance by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)
hoelzl@51523
   641
hoelzl@51523
   642
end
hoelzl@51523
   643
hoelzl@51523
   644
lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
wenzelm@63353
   645
  by (induct x) (simp_all add: zero_real_def one_real_def add_Real)
hoelzl@51523
   646
hoelzl@51523
   647
lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
wenzelm@63353
   648
  by (cases x rule: int_diff_cases) (simp add: of_nat_Real diff_Real)
hoelzl@51523
   649
hoelzl@51523
   650
lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
wenzelm@63353
   651
  apply (induct x)
wenzelm@63353
   652
  apply (simp add: Fract_of_int_quotient of_rat_divide)
wenzelm@63353
   653
  apply (simp add: of_int_Real divide_inverse)
wenzelm@63353
   654
  apply (simp add: inverse_Real mult_Real)
wenzelm@63353
   655
  done
hoelzl@51523
   656
hoelzl@51523
   657
instance real :: archimedean_field
hoelzl@51523
   658
proof
hoelzl@51523
   659
  fix x :: real
hoelzl@51523
   660
  show "\<exists>z. x \<le> of_int z"
hoelzl@51523
   661
    apply (induct x)
hoelzl@51523
   662
    apply (frule cauchy_imp_bounded, clarify)
wenzelm@61942
   663
    apply (rule_tac x="\<lceil>b\<rceil> + 1" in exI)
hoelzl@51523
   664
    apply (rule less_imp_le)
hoelzl@51523
   665
    apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
hoelzl@51523
   666
    apply (rule_tac x=1 in exI, simp add: algebra_simps)
hoelzl@51523
   667
    apply (rule_tac x=0 in exI, clarsimp)
hoelzl@51523
   668
    apply (rule le_less_trans [OF abs_ge_self])
hoelzl@51523
   669
    apply (rule less_le_trans [OF _ le_of_int_ceiling])
hoelzl@51523
   670
    apply simp
hoelzl@51523
   671
    done
hoelzl@51523
   672
qed
hoelzl@51523
   673
hoelzl@51523
   674
instantiation real :: floor_ceiling
hoelzl@51523
   675
begin
hoelzl@51523
   676
wenzelm@63353
   677
definition [code del]: "\<lfloor>x::real\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
hoelzl@51523
   678
wenzelm@61942
   679
instance
wenzelm@61942
   680
proof
wenzelm@63353
   681
  show "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 1)" for x :: real
hoelzl@51523
   682
    unfolding floor_real_def using floor_exists1 by (rule theI')
hoelzl@51523
   683
qed
hoelzl@51523
   684
hoelzl@51523
   685
end
hoelzl@51523
   686
wenzelm@63353
   687
wenzelm@60758
   688
subsection \<open>Completeness\<close>
hoelzl@51523
   689
hoelzl@51523
   690
lemma not_positive_Real:
hoelzl@51523
   691
  assumes X: "cauchy X"
hoelzl@51523
   692
  shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
wenzelm@63353
   693
  unfolding positive_Real [OF X]
wenzelm@63353
   694
  apply auto
wenzelm@63353
   695
  apply (unfold not_less)
wenzelm@63353
   696
  apply (erule obtain_pos_sum)
wenzelm@63353
   697
  apply (drule_tac x=s in spec)
wenzelm@63353
   698
  apply simp
wenzelm@63353
   699
  apply (drule_tac r=t in cauchyD [OF X])
wenzelm@63353
   700
  apply clarify
wenzelm@63353
   701
  apply (drule_tac x=k in spec)
wenzelm@63353
   702
  apply clarsimp
wenzelm@63353
   703
  apply (rule_tac x=n in exI)
wenzelm@63353
   704
  apply clarify
wenzelm@63353
   705
  apply (rename_tac m)
wenzelm@63353
   706
  apply (drule_tac x=m in spec)
wenzelm@63353
   707
  apply simp
wenzelm@63353
   708
  apply (drule_tac x=n in spec)
wenzelm@63353
   709
  apply simp
wenzelm@63353
   710
  apply (drule spec)
wenzelm@63353
   711
  apply (drule (1) mp)
wenzelm@63353
   712
  apply clarify
wenzelm@63353
   713
  apply (rename_tac i)
wenzelm@63353
   714
  apply (rule_tac x = "max i k" in exI)
wenzelm@63353
   715
  apply simp
wenzelm@63353
   716
  done
hoelzl@51523
   717
hoelzl@51523
   718
lemma le_Real:
wenzelm@63353
   719
  assumes "cauchy X" "cauchy Y"
hoelzl@51523
   720
  shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
wenzelm@63353
   721
  unfolding not_less [symmetric, where 'a=real] less_real_def
wenzelm@63353
   722
  apply (simp add: diff_Real not_positive_Real assms)
wenzelm@63353
   723
  apply (simp add: diff_le_eq ac_simps)
wenzelm@63353
   724
  done
hoelzl@51523
   725
hoelzl@51523
   726
lemma le_RealI:
hoelzl@51523
   727
  assumes Y: "cauchy Y"
hoelzl@51523
   728
  shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
hoelzl@51523
   729
proof (induct x)
wenzelm@63353
   730
  fix X
wenzelm@63353
   731
  assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
wenzelm@63353
   732
  then have le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
hoelzl@51523
   733
    by (simp add: of_rat_Real le_Real)
wenzelm@63353
   734
  then have "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" if "0 < r" for r :: rat
wenzelm@63353
   735
  proof -
wenzelm@63353
   736
    from that obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
hoelzl@51523
   737
      by (rule obtain_pos_sum)
hoelzl@51523
   738
    obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
hoelzl@51523
   739
      using cauchyD [OF Y s] ..
hoelzl@51523
   740
    obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
hoelzl@51523
   741
      using le [OF t] ..
hoelzl@51523
   742
    have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
wenzelm@63353
   743
    proof clarsimp
wenzelm@63353
   744
      fix n
wenzelm@63353
   745
      assume n: "i \<le> n" "j \<le> n"
hoelzl@51523
   746
      have "X n \<le> Y i + t" using n j by simp
hoelzl@51523
   747
      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
hoelzl@51523
   748
      ultimately show "X n \<le> Y n + r" unfolding r by simp
hoelzl@51523
   749
    qed
wenzelm@63353
   750
    then show ?thesis ..
wenzelm@63353
   751
  qed
wenzelm@63353
   752
  then show "Real X \<le> Real Y"
hoelzl@51523
   753
    by (simp add: of_rat_Real le_Real X Y)
hoelzl@51523
   754
qed
hoelzl@51523
   755
hoelzl@51523
   756
lemma Real_leI:
hoelzl@51523
   757
  assumes X: "cauchy X"
hoelzl@51523
   758
  assumes le: "\<forall>n. of_rat (X n) \<le> y"
hoelzl@51523
   759
  shows "Real X \<le> y"
hoelzl@51523
   760
proof -
hoelzl@51523
   761
  have "- y \<le> - Real X"
hoelzl@51523
   762
    by (simp add: minus_Real X le_RealI of_rat_minus le)
wenzelm@63353
   763
  then show ?thesis by simp
hoelzl@51523
   764
qed
hoelzl@51523
   765
hoelzl@51523
   766
lemma less_RealD:
wenzelm@63353
   767
  assumes "cauchy Y"
hoelzl@51523
   768
  shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
wenzelm@63353
   769
  apply (erule contrapos_pp)
wenzelm@63353
   770
  apply (simp add: not_less)
wenzelm@63353
   771
  apply (erule Real_leI [OF assms])
wenzelm@63353
   772
  done
hoelzl@51523
   773
wenzelm@63353
   774
lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n"
wenzelm@63353
   775
  apply (induct n)
wenzelm@63353
   776
  apply simp
wenzelm@63353
   777
  apply (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
wenzelm@63353
   778
  done
hoelzl@51523
   779
hoelzl@51523
   780
lemma complete_real:
hoelzl@51523
   781
  fixes S :: "real set"
hoelzl@51523
   782
  assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
hoelzl@51523
   783
  shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
hoelzl@51523
   784
proof -
hoelzl@51523
   785
  obtain x where x: "x \<in> S" using assms(1) ..
hoelzl@51523
   786
  obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
hoelzl@51523
   787
wenzelm@63040
   788
  define P where "P x \<longleftrightarrow> (\<forall>y\<in>S. y \<le> of_rat x)" for x
hoelzl@51523
   789
  obtain a where a: "\<not> P a"
hoelzl@51523
   790
  proof
wenzelm@61942
   791
    have "of_int \<lfloor>x - 1\<rfloor> \<le> x - 1" by (rule of_int_floor_le)
hoelzl@51523
   792
    also have "x - 1 < x" by simp
wenzelm@61942
   793
    finally have "of_int \<lfloor>x - 1\<rfloor> < x" .
wenzelm@63353
   794
    then have "\<not> x \<le> of_int \<lfloor>x - 1\<rfloor>" by (simp only: not_le)
wenzelm@61942
   795
    then show "\<not> P (of_int \<lfloor>x - 1\<rfloor>)"
lp15@61649
   796
      unfolding P_def of_rat_of_int_eq using x by blast
hoelzl@51523
   797
  qed
hoelzl@51523
   798
  obtain b where b: "P b"
hoelzl@51523
   799
  proof
wenzelm@61942
   800
    show "P (of_int \<lceil>z\<rceil>)"
hoelzl@51523
   801
    unfolding P_def of_rat_of_int_eq
hoelzl@51523
   802
    proof
hoelzl@51523
   803
      fix y assume "y \<in> S"
wenzelm@63353
   804
      then have "y \<le> z" using z by simp
wenzelm@61942
   805
      also have "z \<le> of_int \<lceil>z\<rceil>" by (rule le_of_int_ceiling)
wenzelm@61942
   806
      finally show "y \<le> of_int \<lceil>z\<rceil>" .
hoelzl@51523
   807
    qed
hoelzl@51523
   808
  qed
hoelzl@51523
   809
wenzelm@63040
   810
  define avg where "avg x y = x/2 + y/2" for x y :: rat
wenzelm@63040
   811
  define bisect where "bisect = (\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))"
wenzelm@63040
   812
  define A where "A n = fst ((bisect ^^ n) (a, b))" for n
wenzelm@63040
   813
  define B where "B n = snd ((bisect ^^ n) (a, b))" for n
wenzelm@63040
   814
  define C where "C n = avg (A n) (B n)" for n
hoelzl@51523
   815
  have A_0 [simp]: "A 0 = a" unfolding A_def by simp
hoelzl@51523
   816
  have B_0 [simp]: "B 0 = b" unfolding B_def by simp
hoelzl@51523
   817
  have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
hoelzl@51523
   818
    unfolding A_def B_def C_def bisect_def split_def by simp
hoelzl@51523
   819
  have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
hoelzl@51523
   820
    unfolding A_def B_def C_def bisect_def split_def by simp
hoelzl@51523
   821
wenzelm@63353
   822
  have width: "B n - A n = (b - a) / 2^n" for n
wenzelm@63353
   823
    apply (induct n)
wenzelm@63353
   824
    apply (simp_all add: eq_divide_eq)
wenzelm@63353
   825
    apply (simp_all add: C_def avg_def algebra_simps)
hoelzl@51523
   826
    done
hoelzl@51523
   827
wenzelm@63353
   828
  have twos: "0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r" for y r :: rat
hoelzl@51523
   829
    apply (simp add: divide_less_eq)
haftmann@57512
   830
    apply (subst mult.commute)
hoelzl@51523
   831
    apply (frule_tac y=y in ex_less_of_nat_mult)
hoelzl@51523
   832
    apply clarify
hoelzl@51523
   833
    apply (rule_tac x=n in exI)
hoelzl@51523
   834
    apply (erule less_trans)
hoelzl@51523
   835
    apply (rule mult_strict_right_mono)
hoelzl@51523
   836
    apply (rule le_less_trans [OF _ of_nat_less_two_power])
hoelzl@51523
   837
    apply simp
hoelzl@51523
   838
    apply assumption
hoelzl@51523
   839
    done
hoelzl@51523
   840
wenzelm@63353
   841
  have PA: "\<not> P (A n)" for n by (induct n) (simp_all add: a)
wenzelm@63353
   842
  have PB: "P (B n)" for n by (induct n) (simp_all add: b)
hoelzl@51523
   843
  have ab: "a < b"
hoelzl@51523
   844
    using a b unfolding P_def
hoelzl@51523
   845
    apply (clarsimp simp add: not_le)
hoelzl@51523
   846
    apply (drule (1) bspec)
hoelzl@51523
   847
    apply (drule (1) less_le_trans)
hoelzl@51523
   848
    apply (simp add: of_rat_less)
hoelzl@51523
   849
    done
wenzelm@63353
   850
  have AB: "A n < B n" for n by (induct n) (simp_all add: ab C_def avg_def)
hoelzl@51523
   851
  have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
hoelzl@51523
   852
    apply (auto simp add: le_less [where 'a=nat])
hoelzl@51523
   853
    apply (erule less_Suc_induct)
hoelzl@51523
   854
    apply (clarsimp simp add: C_def avg_def)
hoelzl@51523
   855
    apply (simp add: add_divide_distrib [symmetric])
hoelzl@51523
   856
    apply (rule AB [THEN less_imp_le])
hoelzl@51523
   857
    apply simp
hoelzl@51523
   858
    done
hoelzl@51523
   859
  have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
hoelzl@51523
   860
    apply (auto simp add: le_less [where 'a=nat])
hoelzl@51523
   861
    apply (erule less_Suc_induct)
hoelzl@51523
   862
    apply (clarsimp simp add: C_def avg_def)
hoelzl@51523
   863
    apply (simp add: add_divide_distrib [symmetric])
hoelzl@51523
   864
    apply (rule AB [THEN less_imp_le])
hoelzl@51523
   865
    apply simp
hoelzl@51523
   866
    done
wenzelm@63353
   867
  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"
hoelzl@51523
   868
    apply (rule cauchyI)
hoelzl@51523
   869
    apply (drule twos [where y="b - a"])
hoelzl@51523
   870
    apply (erule exE)
hoelzl@51523
   871
    apply (rule_tac x=n in exI, clarify, rename_tac i j)
hoelzl@51523
   872
    apply (rule_tac y="B n - A n" in le_less_trans) defer
hoelzl@51523
   873
    apply (simp add: width)
hoelzl@51523
   874
    apply (drule_tac x=n in spec)
hoelzl@51523
   875
    apply (frule_tac x=i in spec, drule (1) mp)
hoelzl@51523
   876
    apply (frule_tac x=j in spec, drule (1) mp)
hoelzl@51523
   877
    apply (frule A_mono, drule B_mono)
hoelzl@51523
   878
    apply (frule A_mono, drule B_mono)
hoelzl@51523
   879
    apply arith
hoelzl@51523
   880
    done
hoelzl@51523
   881
  have "cauchy A"
hoelzl@51523
   882
    apply (rule cauchy_lemma [rule_format])
hoelzl@51523
   883
    apply (simp add: A_mono)
hoelzl@51523
   884
    apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
hoelzl@51523
   885
    done
hoelzl@51523
   886
  have "cauchy B"
hoelzl@51523
   887
    apply (rule cauchy_lemma [rule_format])
hoelzl@51523
   888
    apply (simp add: B_mono)
hoelzl@51523
   889
    apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
hoelzl@51523
   890
    done
hoelzl@51523
   891
  have 1: "\<forall>x\<in>S. x \<le> Real B"
hoelzl@51523
   892
  proof
wenzelm@63353
   893
    fix x
wenzelm@63353
   894
    assume "x \<in> S"
hoelzl@51523
   895
    then show "x \<le> Real B"
wenzelm@60758
   896
      using PB [unfolded P_def] \<open>cauchy B\<close>
hoelzl@51523
   897
      by (simp add: le_RealI)
hoelzl@51523
   898
  qed
hoelzl@51523
   899
  have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
hoelzl@51523
   900
    apply clarify
hoelzl@51523
   901
    apply (erule contrapos_pp)
hoelzl@51523
   902
    apply (simp add: not_le)
wenzelm@60758
   903
    apply (drule less_RealD [OF \<open>cauchy A\<close>], clarify)
hoelzl@51523
   904
    apply (subgoal_tac "\<not> P (A n)")
hoelzl@51523
   905
    apply (simp add: P_def not_le, clarify)
hoelzl@51523
   906
    apply (erule rev_bexI)
hoelzl@51523
   907
    apply (erule (1) less_trans)
hoelzl@51523
   908
    apply (simp add: PA)
hoelzl@51523
   909
    done
hoelzl@51523
   910
  have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
hoelzl@51523
   911
  proof (rule vanishesI)
wenzelm@63353
   912
    fix r :: rat
wenzelm@63353
   913
    assume "0 < r"
hoelzl@51523
   914
    then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
lp15@61649
   915
      using twos by blast
hoelzl@51523
   916
    have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
wenzelm@63353
   917
    proof clarify
wenzelm@63353
   918
      fix n
wenzelm@63353
   919
      assume n: "k \<le> n"
hoelzl@51523
   920
      have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
hoelzl@51523
   921
        by simp
hoelzl@51523
   922
      also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
nipkow@56544
   923
        using n by (simp add: divide_left_mono)
hoelzl@51523
   924
      also note k
hoelzl@51523
   925
      finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
hoelzl@51523
   926
    qed
wenzelm@63353
   927
    then show "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
hoelzl@51523
   928
  qed
wenzelm@63353
   929
  then have 3: "Real B = Real A"
wenzelm@60758
   930
    by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width)
hoelzl@51523
   931
  show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
wenzelm@63353
   932
    apply (rule exI [where x = "Real B"])
wenzelm@63353
   933
    using 1 2 3
wenzelm@63353
   934
    apply simp
wenzelm@63353
   935
    done
hoelzl@51523
   936
qed
hoelzl@51523
   937
hoelzl@51775
   938
instantiation real :: linear_continuum
hoelzl@51523
   939
begin
hoelzl@51523
   940
wenzelm@63353
   941
subsection \<open>Supremum of a set of reals\<close>
hoelzl@51523
   942
hoelzl@54281
   943
definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
wenzelm@63353
   944
definition "Inf X = - Sup (uminus ` X)" for X :: "real set"
hoelzl@51523
   945
hoelzl@51523
   946
instance
hoelzl@51523
   947
proof
wenzelm@63353
   948
  show Sup_upper: "x \<le> Sup X" if "x \<in> X" "bdd_above X" for x :: real and X :: "real set"
wenzelm@63353
   949
  proof -
wenzelm@63353
   950
    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"
hoelzl@54258
   951
      using complete_real[of X] unfolding bdd_above_def by blast
wenzelm@63353
   952
    then show ?thesis unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that)
wenzelm@63353
   953
  qed
wenzelm@63353
   954
  show Sup_least: "Sup X \<le> z" if "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
wenzelm@63353
   955
    for z :: real and X :: "real set"
wenzelm@63353
   956
  proof -
wenzelm@63353
   957
    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"
wenzelm@63353
   958
      using complete_real [of X] by blast
hoelzl@51523
   959
    then have "Sup X = s"
lp15@61284
   960
      unfolding Sup_real_def by (best intro: Least_equality)
wenzelm@63353
   961
    also from s z have "\<dots> \<le> z"
hoelzl@51523
   962
      by blast
wenzelm@63353
   963
    finally show ?thesis .
wenzelm@63353
   964
  qed
wenzelm@63353
   965
  show "Inf X \<le> x" if "x \<in> X" "bdd_below X" for x :: real and X :: "real set"
wenzelm@63353
   966
    using Sup_upper [of "-x" "uminus ` X"] by (auto simp: Inf_real_def that)
wenzelm@63353
   967
  show "z \<le> Inf X" if "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" for z :: real and X :: "real set"
wenzelm@63353
   968
    using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that)
hoelzl@51775
   969
  show "\<exists>a b::real. a \<noteq> b"
hoelzl@51775
   970
    using zero_neq_one by blast
hoelzl@51523
   971
qed
wenzelm@63353
   972
hoelzl@51523
   973
end
hoelzl@51523
   974
wenzelm@63353
   975
wenzelm@60758
   976
subsection \<open>Hiding implementation details\<close>
hoelzl@51523
   977
hoelzl@51523
   978
hide_const (open) vanishes cauchy positive Real
hoelzl@51523
   979
hoelzl@51523
   980
declare Real_induct [induct del]
hoelzl@51523
   981
declare Abs_real_induct [induct del]
hoelzl@51523
   982
declare Abs_real_cases [cases del]
hoelzl@51523
   983
kuncar@53652
   984
lifting_update real.lifting
kuncar@53652
   985
lifting_forget real.lifting
lp15@61284
   986
wenzelm@63353
   987
wenzelm@63353
   988
subsection \<open>More Lemmas\<close>
hoelzl@51523
   989
wenzelm@60758
   990
text \<open>BH: These lemmas should not be necessary; they should be
wenzelm@63353
   991
  covered by existing simp rules and simplification procedures.\<close>
hoelzl@51523
   992
wenzelm@63353
   993
lemma real_mult_less_iff1 [simp]: "0 < z \<Longrightarrow> x * z < y * z \<longleftrightarrow> x < y" for x y z :: real
wenzelm@63353
   994
  by simp (* solved by linordered_ring_less_cancel_factor simproc *)
hoelzl@51523
   995
wenzelm@63353
   996
lemma real_mult_le_cancel_iff1 [simp]: "0 < z \<Longrightarrow> x * z \<le> y * z \<longleftrightarrow> x \<le> y" for x y z :: real
wenzelm@63353
   997
  by simp (* solved by linordered_ring_le_cancel_factor simproc *)
hoelzl@51523
   998
wenzelm@63353
   999
lemma real_mult_le_cancel_iff2 [simp]: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y" for x y z :: real
wenzelm@63353
  1000
  by simp (* solved by linordered_ring_le_cancel_factor simproc *)
hoelzl@51523
  1001
hoelzl@51523
  1002
wenzelm@60758
  1003
subsection \<open>Embedding numbers into the Reals\<close>
hoelzl@51523
  1004
wenzelm@63353
  1005
abbreviation real_of_nat :: "nat \<Rightarrow> real"
wenzelm@63353
  1006
  where "real_of_nat \<equiv> of_nat"
hoelzl@51523
  1007
wenzelm@63353
  1008
abbreviation real :: "nat \<Rightarrow> real"
wenzelm@63353
  1009
  where "real \<equiv> of_nat"
lp15@61609
  1010
wenzelm@63353
  1011
abbreviation real_of_int :: "int \<Rightarrow> real"
wenzelm@63353
  1012
  where "real_of_int \<equiv> of_int"
hoelzl@51523
  1013
wenzelm@63353
  1014
abbreviation real_of_rat :: "rat \<Rightarrow> real"
wenzelm@63353
  1015
  where "real_of_rat \<equiv> of_rat"
hoelzl@51523
  1016
hoelzl@51523
  1017
declare [[coercion_enabled]]
hoelzl@59000
  1018
hoelzl@59000
  1019
declare [[coercion "of_nat :: nat \<Rightarrow> int"]]
lp15@61609
  1020
declare [[coercion "of_nat :: nat \<Rightarrow> real"]]
lp15@61609
  1021
declare [[coercion "of_int :: int \<Rightarrow> real"]]
hoelzl@59000
  1022
hoelzl@59000
  1023
(* We do not add rat to the coerced types, this has often unpleasant side effects when writing
hoelzl@59000
  1024
inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)
hoelzl@51523
  1025
hoelzl@51523
  1026
declare [[coercion_map map]]
hoelzl@59000
  1027
declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]
hoelzl@59000
  1028
declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]
hoelzl@51523
  1029
lp15@61609
  1030
declare of_int_eq_0_iff [algebra, presburger]
lp15@61649
  1031
declare of_int_eq_1_iff [algebra, presburger]
lp15@61649
  1032
declare of_int_eq_iff [algebra, presburger]
lp15@61649
  1033
declare of_int_less_0_iff [algebra, presburger]
lp15@61649
  1034
declare of_int_less_1_iff [algebra, presburger]
lp15@61649
  1035
declare of_int_less_iff [algebra, presburger]
lp15@61649
  1036
declare of_int_le_0_iff [algebra, presburger]
lp15@61649
  1037
declare of_int_le_1_iff [algebra, presburger]
lp15@61649
  1038
declare of_int_le_iff [algebra, presburger]
lp15@61649
  1039
declare of_int_0_less_iff [algebra, presburger]
lp15@61649
  1040
declare of_int_0_le_iff [algebra, presburger]
lp15@61649
  1041
declare of_int_1_less_iff [algebra, presburger]
lp15@61649
  1042
declare of_int_1_le_iff [algebra, presburger]
hoelzl@51523
  1043
wenzelm@63353
  1044
lemma int_less_real_le: "n < m \<longleftrightarrow> real_of_int n + 1 \<le> real_of_int m"
lp15@61609
  1045
proof -
lp15@61609
  1046
  have "(0::real) \<le> 1"
lp15@61609
  1047
    by (metis less_eq_real_def zero_less_one)
wenzelm@63353
  1048
  then show ?thesis
lp15@61694
  1049
    by (metis floor_of_int less_floor_iff)
lp15@61609
  1050
qed
hoelzl@51523
  1051
wenzelm@63353
  1052
lemma int_le_real_less: "n \<le> m \<longleftrightarrow> real_of_int n < real_of_int m + 1"
lp15@61609
  1053
  by (meson int_less_real_le not_le)
hoelzl@51523
  1054
wenzelm@63353
  1055
lemma real_of_int_div_aux:
wenzelm@63353
  1056
  "(real_of_int x) / (real_of_int d) =
lp15@61609
  1057
    real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)"
hoelzl@51523
  1058
proof -
hoelzl@51523
  1059
  have "x = (x div d) * d + x mod d"
hoelzl@51523
  1060
    by auto
lp15@61609
  1061
  then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)"
lp15@61609
  1062
    by (metis of_int_add of_int_mult)
wenzelm@63353
  1063
  then have "real_of_int x / real_of_int d = \<dots> / real_of_int d"
hoelzl@51523
  1064
    by simp
hoelzl@51523
  1065
  then show ?thesis
hoelzl@51523
  1066
    by (auto simp add: add_divide_distrib algebra_simps)
hoelzl@51523
  1067
qed
hoelzl@51523
  1068
haftmann@58834
  1069
lemma real_of_int_div:
wenzelm@63353
  1070
  "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int
haftmann@58834
  1071
  by (simp add: real_of_int_div_aux)
hoelzl@51523
  1072
wenzelm@63353
  1073
lemma real_of_int_div2: "0 \<le> real_of_int n / real_of_int x - real_of_int (n div x)"
wenzelm@63353
  1074
  apply (cases "x = 0")
wenzelm@63353
  1075
  apply simp
wenzelm@63353
  1076
  apply (cases "0 < x")
lp15@61609
  1077
   apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
lp15@61609
  1078
  apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
lp15@61609
  1079
  done
hoelzl@51523
  1080
wenzelm@63353
  1081
lemma real_of_int_div3: "real_of_int n / real_of_int x - real_of_int (n div x) \<le> 1"
hoelzl@51523
  1082
  apply (simp add: algebra_simps)
hoelzl@51523
  1083
  apply (subst real_of_int_div_aux)
hoelzl@51523
  1084
  apply (auto simp add: divide_le_eq intro: order_less_imp_le)
wenzelm@63353
  1085
  done
hoelzl@51523
  1086
wenzelm@63353
  1087
lemma real_of_int_div4: "real_of_int (n div x) \<le> real_of_int n / real_of_int x"
wenzelm@63353
  1088
  using real_of_int_div2 [of n x] by simp
hoelzl@51523
  1089
hoelzl@51523
  1090
wenzelm@63353
  1091
subsection \<open>Embedding the Naturals into the Reals\<close>
hoelzl@51523
  1092
wenzelm@63353
  1093
lemma real_of_card: "real (card A) = setsum (\<lambda>x. 1) A"
lp15@61609
  1094
  by simp
hoelzl@51523
  1095
wenzelm@63353
  1096
lemma nat_less_real_le: "n < m \<longleftrightarrow> real n + 1 \<le> real m"
lp15@61609
  1097
  by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)
hoelzl@51523
  1098
wenzelm@63353
  1099
lemma nat_le_real_less: "n \<le> m \<longleftrightarrow> real n < real m + 1" for m n :: nat
lp15@61284
  1100
  by (meson nat_less_real_le not_le)
hoelzl@51523
  1101
wenzelm@63353
  1102
lemma real_of_nat_div_aux: "real x / real d = real (x div d) + real (x mod d) / real d"
hoelzl@51523
  1103
proof -
hoelzl@51523
  1104
  have "x = (x div d) * d + x mod d"
hoelzl@51523
  1105
    by auto
hoelzl@51523
  1106
  then have "real x = real (x div d) * real d + real(x mod d)"
lp15@61609
  1107
    by (metis of_nat_add of_nat_mult)
hoelzl@51523
  1108
  then have "real x / real d = \<dots> / real d"
hoelzl@51523
  1109
    by simp
hoelzl@51523
  1110
  then show ?thesis
hoelzl@51523
  1111
    by (auto simp add: add_divide_distrib algebra_simps)
hoelzl@51523
  1112
qed
hoelzl@51523
  1113
lp15@61609
  1114
lemma real_of_nat_div: "d dvd n \<Longrightarrow> real(n div d) = real n / real d"
wenzelm@63353
  1115
  by (subst real_of_nat_div_aux) (auto simp add: dvd_eq_mod_eq_0 [symmetric])
hoelzl@51523
  1116
wenzelm@63353
  1117
lemma real_of_nat_div2: "0 \<le> real n / real x - real (n div x)" for n x :: nat
wenzelm@63353
  1118
  apply (simp add: algebra_simps)
wenzelm@63353
  1119
  apply (subst real_of_nat_div_aux)
wenzelm@63353
  1120
  apply simp
wenzelm@63353
  1121
  done
hoelzl@51523
  1122
wenzelm@63353
  1123
lemma real_of_nat_div3: "real n / real x - real (n div x) \<le> 1" for n x :: nat
wenzelm@63353
  1124
  apply (cases "x = 0")
wenzelm@63353
  1125
  apply simp
wenzelm@63353
  1126
  apply (simp add: algebra_simps)
wenzelm@63353
  1127
  apply (subst real_of_nat_div_aux)
wenzelm@63353
  1128
  apply simp
wenzelm@63353
  1129
  done
hoelzl@51523
  1130
wenzelm@63353
  1131
lemma real_of_nat_div4: "real (n div x) \<le> real n / real x" for n x :: nat
wenzelm@63353
  1132
  using real_of_nat_div2 [of n x] by simp
wenzelm@63353
  1133
hoelzl@51523
  1134
wenzelm@60758
  1135
subsection \<open>The Archimedean Property of the Reals\<close>
hoelzl@51523
  1136
lp15@62623
  1137
lemma real_arch_inverse: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
lp15@62623
  1138
  using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
lp15@62623
  1139
  by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)
hoelzl@51523
  1140
hoelzl@51523
  1141
lemma reals_Archimedean3:
hoelzl@51523
  1142
  assumes x_greater_zero: "0 < x"
lp15@61609
  1143
  shows "\<forall>y. \<exists>n. y < real n * x"
lp15@61609
  1144
  using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)
hoelzl@51523
  1145
lp15@62397
  1146
lemma real_archimedian_rdiv_eq_0:
lp15@62397
  1147
  assumes x0: "x \<ge> 0"
wenzelm@63353
  1148
    and c: "c \<ge> 0"
wenzelm@63353
  1149
    and xc: "\<And>m::nat. m > 0 \<Longrightarrow> real m * x \<le> c"
wenzelm@63353
  1150
  shows "x = 0"
wenzelm@63353
  1151
  by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc)
lp15@62397
  1152
hoelzl@51523
  1153
wenzelm@63353
  1154
subsection \<open>Rationals\<close>
hoelzl@51523
  1155
wenzelm@63353
  1156
lemma Rats_eq_int_div_int: "\<rat> = {real_of_int i / real_of_int j | i j. j \<noteq> 0}"  (is "_ = ?S")
hoelzl@51523
  1157
proof
hoelzl@51523
  1158
  show "\<rat> \<subseteq> ?S"
hoelzl@51523
  1159
  proof
wenzelm@63353
  1160
    fix x :: real
wenzelm@63353
  1161
    assume "x \<in> \<rat>"
wenzelm@63353
  1162
    then obtain r where "x = of_rat r"
wenzelm@63353
  1163
      unfolding Rats_def ..
wenzelm@63353
  1164
    have "of_rat r \<in> ?S"
wenzelm@63353
  1165
      by (cases r) (auto simp add: of_rat_rat)
wenzelm@63353
  1166
    then show "x \<in> ?S"
wenzelm@63353
  1167
      using \<open>x = of_rat r\<close> by simp
hoelzl@51523
  1168
  qed
hoelzl@51523
  1169
next
hoelzl@51523
  1170
  show "?S \<subseteq> \<rat>"
wenzelm@63353
  1171
  proof (auto simp: Rats_def)
wenzelm@63353
  1172
    fix i j :: int
wenzelm@63353
  1173
    assume "j \<noteq> 0"
wenzelm@63353
  1174
    then have "real_of_int i / real_of_int j = of_rat (Fract i j)"
lp15@61609
  1175
      by (simp add: of_rat_rat)
wenzelm@63353
  1176
    then show "real_of_int i / real_of_int j \<in> range of_rat"
wenzelm@63353
  1177
      by blast
hoelzl@51523
  1178
  qed
hoelzl@51523
  1179
qed
hoelzl@51523
  1180
wenzelm@63353
  1181
lemma Rats_eq_int_div_nat: "\<rat> = { real_of_int i / real n | i n. n \<noteq> 0}"
wenzelm@63353
  1182
proof (auto simp: Rats_eq_int_div_int)
wenzelm@63353
  1183
  fix i j :: int
wenzelm@63353
  1184
  assume "j \<noteq> 0"
wenzelm@63353
  1185
  show "\<exists>(i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i' / real n \<and> 0 < n"
wenzelm@63353
  1186
  proof (cases "j > 0")
wenzelm@63353
  1187
    case True
wenzelm@63353
  1188
    then have "real_of_int i / real_of_int j = real_of_int i / real (nat j) \<and> 0 < nat j"
wenzelm@63353
  1189
      by simp
wenzelm@63353
  1190
    then show ?thesis by blast
hoelzl@51523
  1191
  next
wenzelm@63353
  1192
    case False
wenzelm@63353
  1193
    with \<open>j \<noteq> 0\<close>
wenzelm@63353
  1194
    have "real_of_int i / real_of_int j = real_of_int (- i) / real (nat (- j)) \<and> 0 < nat (- j)"
wenzelm@63353
  1195
      by simp
wenzelm@63353
  1196
    then show ?thesis by blast
hoelzl@51523
  1197
  qed
hoelzl@51523
  1198
next
wenzelm@63353
  1199
  fix i :: int and n :: nat
wenzelm@63353
  1200
  assume "0 < n"
wenzelm@63353
  1201
  then have "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0"
wenzelm@63353
  1202
    by simp
wenzelm@63353
  1203
  then show "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int j \<and> j \<noteq> 0"
wenzelm@63353
  1204
    by blast
hoelzl@51523
  1205
qed
hoelzl@51523
  1206
hoelzl@51523
  1207
lemma Rats_abs_nat_div_natE:
hoelzl@51523
  1208
  assumes "x \<in> \<rat>"
wenzelm@63353
  1209
  obtains m n :: nat where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
hoelzl@51523
  1210
proof -
wenzelm@63353
  1211
  from \<open>x \<in> \<rat>\<close> obtain i :: int and n :: nat where "n \<noteq> 0" and "x = real_of_int i / real n"
wenzelm@63353
  1212
    by (auto simp add: Rats_eq_int_div_nat)
wenzelm@63353
  1213
  then have "\<bar>x\<bar> = real (nat \<bar>i\<bar>) / real n" by simp
hoelzl@51523
  1214
  then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
hoelzl@51523
  1215
  let ?gcd = "gcd m n"
wenzelm@63353
  1216
  from \<open>n \<noteq> 0\<close> have gcd: "?gcd \<noteq> 0" by simp
hoelzl@51523
  1217
  let ?k = "m div ?gcd"
hoelzl@51523
  1218
  let ?l = "n div ?gcd"
hoelzl@51523
  1219
  let ?gcd' = "gcd ?k ?l"
wenzelm@63353
  1220
  have "?gcd dvd m" ..
wenzelm@63353
  1221
  then have gcd_k: "?gcd * ?k = m"
hoelzl@51523
  1222
    by (rule dvd_mult_div_cancel)
wenzelm@63353
  1223
  have "?gcd dvd n" ..
wenzelm@63353
  1224
  then have gcd_l: "?gcd * ?l = n"
hoelzl@51523
  1225
    by (rule dvd_mult_div_cancel)
wenzelm@63353
  1226
  from \<open>n \<noteq> 0\<close> and gcd_l have "?gcd * ?l \<noteq> 0" by simp
lp15@61284
  1227
  then have "?l \<noteq> 0" by (blast dest!: mult_not_zero)
hoelzl@51523
  1228
  moreover
hoelzl@51523
  1229
  have "\<bar>x\<bar> = real ?k / real ?l"
hoelzl@51523
  1230
  proof -
lp15@61609
  1231
    from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
lp15@61609
  1232
      by (simp add: real_of_nat_div)
hoelzl@51523
  1233
    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
hoelzl@51523
  1234
    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
hoelzl@51523
  1235
    finally show ?thesis ..
hoelzl@51523
  1236
  qed
hoelzl@51523
  1237
  moreover
hoelzl@51523
  1238
  have "?gcd' = 1"
hoelzl@51523
  1239
  proof -
hoelzl@51523
  1240
    have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
hoelzl@51523
  1241
      by (rule gcd_mult_distrib_nat)
hoelzl@51523
  1242
    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
hoelzl@51523
  1243
    with gcd show ?thesis by auto
hoelzl@51523
  1244
  qed
hoelzl@51523
  1245
  ultimately show ?thesis ..
hoelzl@51523
  1246
qed
hoelzl@51523
  1247
wenzelm@63353
  1248
wenzelm@63353
  1249
subsection \<open>Density of the Rational Reals in the Reals\<close>
hoelzl@51523
  1250
wenzelm@63353
  1251
text \<open>
wenzelm@63353
  1252
  This density proof is due to Stefan Richter and was ported by TN.  The
wenzelm@63353
  1253
  original source is \emph{Real Analysis} by H.L. Royden.
wenzelm@63353
  1254
  It employs the Archimedean property of the reals.\<close>
hoelzl@51523
  1255
hoelzl@51523
  1256
lemma Rats_dense_in_real:
hoelzl@51523
  1257
  fixes x :: real
wenzelm@63353
  1258
  assumes "x < y"
wenzelm@63353
  1259
  shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
hoelzl@51523
  1260
proof -
wenzelm@63353
  1261
  from \<open>x < y\<close> have "0 < y - x" by simp
wenzelm@63353
  1262
  with reals_Archimedean obtain q :: nat where q: "inverse (real q) < y - x" and "0 < q"
wenzelm@63353
  1263
    by blast
wenzelm@63040
  1264
  define p where "p = \<lceil>y * real q\<rceil> - 1"
wenzelm@63040
  1265
  define r where "r = of_int p / real q"
hoelzl@51523
  1266
  from q have "x < y - inverse (real q)" by simp
hoelzl@51523
  1267
  also have "y - inverse (real q) \<le> r"
hoelzl@51523
  1268
    unfolding r_def p_def
wenzelm@60758
  1269
    by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling \<open>0 < q\<close>)
hoelzl@51523
  1270
  finally have "x < r" .
hoelzl@51523
  1271
  moreover have "r < y"
hoelzl@51523
  1272
    unfolding r_def p_def
wenzelm@63353
  1273
    by (simp add: divide_less_eq diff_less_eq \<open>0 < q\<close> less_ceiling_iff [symmetric])
hoelzl@51523
  1274
  moreover from r_def have "r \<in> \<rat>" by simp
lp15@61649
  1275
  ultimately show ?thesis by blast
hoelzl@51523
  1276
qed
hoelzl@51523
  1277
hoelzl@57447
  1278
lemma of_rat_dense:
hoelzl@57447
  1279
  fixes x y :: real
hoelzl@57447
  1280
  assumes "x < y"
hoelzl@57447
  1281
  shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"
wenzelm@63353
  1282
  using Rats_dense_in_real [OF \<open>x < y\<close>]
wenzelm@63353
  1283
  by (auto elim: Rats_cases)
hoelzl@51523
  1284
hoelzl@51523
  1285
wenzelm@63353
  1286
subsection \<open>Numerals and Arithmetic\<close>
hoelzl@51523
  1287
lp15@61609
  1288
lemma [code_abbrev]:   (*FIXME*)
hoelzl@51523
  1289
  "real_of_int (numeral k) = numeral k"
haftmann@54489
  1290
  "real_of_int (- numeral k) = - numeral k"
hoelzl@51523
  1291
  by simp_all
hoelzl@51523
  1292
wenzelm@60758
  1293
declaration \<open>
lp15@61609
  1294
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
lp15@61609
  1295
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
lp15@61609
  1296
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
lp15@61609
  1297
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
lp15@61609
  1298
  #> Lin_Arith.add_simps [@{thm of_nat_0}, @{thm of_nat_Suc}, @{thm of_nat_add},
lp15@61609
  1299
      @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1},
lp15@61609
  1300
      @{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff},
lp15@61609
  1301
      @{thm of_int_mult}, @{thm of_int_of_nat_eq},
haftmann@62348
  1302
      @{thm of_nat_numeral}, @{thm of_nat_numeral}, @{thm of_int_neg_numeral}]
hoelzl@58040
  1303
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
hoelzl@58040
  1304
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
wenzelm@60758
  1305
\<close>
hoelzl@51523
  1306
wenzelm@63353
  1307
wenzelm@63353
  1308
subsection \<open>Simprules combining \<open>x + y\<close> and \<open>0\<close>\<close> (* FIXME ARE THEY NEEDED? *)
hoelzl@51523
  1309
wenzelm@63353
  1310
lemma real_add_minus_iff [simp]: "x + - a = 0 \<longleftrightarrow> x = a" for x a :: real
wenzelm@63353
  1311
  by arith
hoelzl@51523
  1312
wenzelm@63353
  1313
lemma real_add_less_0_iff: "x + y < 0 \<longleftrightarrow> y < - x" for x y :: real
wenzelm@63353
  1314
  by auto
hoelzl@51523
  1315
wenzelm@63353
  1316
lemma real_0_less_add_iff: "0 < x + y \<longleftrightarrow> - x < y" for x y :: real
wenzelm@63353
  1317
  by auto
hoelzl@51523
  1318
wenzelm@63353
  1319
lemma real_add_le_0_iff: "x + y \<le> 0 \<longleftrightarrow> y \<le> - x" for x y :: real
wenzelm@63353
  1320
  by auto
hoelzl@51523
  1321
wenzelm@63353
  1322
lemma real_0_le_add_iff: "0 \<le> x + y \<longleftrightarrow> - x \<le> y" for x y :: real
wenzelm@63353
  1323
  by auto
wenzelm@63353
  1324
hoelzl@51523
  1325
wenzelm@60758
  1326
subsection \<open>Lemmas about powers\<close>
hoelzl@51523
  1327
hoelzl@51523
  1328
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
lp15@61609
  1329
  by simp
hoelzl@51523
  1330
wenzelm@63353
  1331
(* FIXME: declare this [simp] for all types, or not at all *)
lp15@61609
  1332
declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
hoelzl@51523
  1333
wenzelm@63353
  1334
lemma real_minus_mult_self_le [simp]: "- (u * u) \<le> x * x" for u x :: real
wenzelm@63353
  1335
  by (rule order_trans [where y = 0]) auto
hoelzl@51523
  1336
wenzelm@63353
  1337
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> x\<^sup>2" for u x :: real
lp15@61609
  1338
  by (auto simp add: power2_eq_square)
hoelzl@51523
  1339
wenzelm@63353
  1340
lemma numeral_power_eq_real_of_int_cancel_iff [simp]:
wenzelm@63353
  1341
  "numeral x ^ n = real_of_int y \<longleftrightarrow> numeral x ^ n = y"
lp15@61609
  1342
  by (metis of_int_eq_iff of_int_numeral of_int_power)
immler@58983
  1343
wenzelm@63353
  1344
lemma real_of_int_eq_numeral_power_cancel_iff [simp]:
wenzelm@63353
  1345
  "real_of_int y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
wenzelm@63353
  1346
  using numeral_power_eq_real_of_int_cancel_iff [of x n y] by metis
immler@58983
  1347
wenzelm@63353
  1348
lemma numeral_power_eq_real_of_nat_cancel_iff [simp]:
wenzelm@63353
  1349
  "numeral x ^ n = real y \<longleftrightarrow> numeral x ^ n = y"
lp15@61609
  1350
  using of_nat_eq_iff by fastforce
immler@58983
  1351
wenzelm@63353
  1352
lemma real_of_nat_eq_numeral_power_cancel_iff [simp]:
wenzelm@63353
  1353
  "real y = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
wenzelm@63353
  1354
  using numeral_power_eq_real_of_nat_cancel_iff [of x n y] by metis
immler@58983
  1355
wenzelm@63353
  1356
lemma numeral_power_le_real_of_nat_cancel_iff [simp]:
wenzelm@63353
  1357
  "(numeral x :: real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
wenzelm@63353
  1358
  by (metis of_nat_le_iff of_nat_numeral of_nat_power)
hoelzl@51523
  1359
wenzelm@63353
  1360
lemma real_of_nat_le_numeral_power_cancel_iff [simp]:
hoelzl@51523
  1361
  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
wenzelm@63353
  1362
  by (metis of_nat_le_iff of_nat_numeral of_nat_power)
hoelzl@51523
  1363
wenzelm@63353
  1364
lemma numeral_power_le_real_of_int_cancel_iff [simp]:
wenzelm@63353
  1365
  "(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
lp15@61609
  1366
  by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power)
hoelzl@51523
  1367
wenzelm@63353
  1368
lemma real_of_int_le_numeral_power_cancel_iff [simp]:
wenzelm@63353
  1369
  "real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
lp15@61609
  1370
  by (metis floor_of_int le_floor_iff of_int_numeral of_int_power)
hoelzl@51523
  1371
wenzelm@63353
  1372
lemma numeral_power_less_real_of_nat_cancel_iff [simp]:
wenzelm@63353
  1373
  "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
wenzelm@63353
  1374
  by (metis of_nat_less_iff of_nat_numeral of_nat_power)
wenzelm@63353
  1375
wenzelm@63353
  1376
lemma real_of_nat_less_numeral_power_cancel_iff [simp]:
wenzelm@63353
  1377
  "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"
lp15@61609
  1378
  by (metis of_nat_less_iff of_nat_numeral of_nat_power)
immler@58983
  1379
wenzelm@63353
  1380
lemma numeral_power_less_real_of_int_cancel_iff [simp]:
wenzelm@63353
  1381
  "(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a"
lp15@61609
  1382
  by (meson not_less real_of_int_le_numeral_power_cancel_iff)
immler@58983
  1383
wenzelm@63353
  1384
lemma real_of_int_less_numeral_power_cancel_iff [simp]:
wenzelm@63353
  1385
  "real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
lp15@61609
  1386
  by (meson not_less numeral_power_le_real_of_int_cancel_iff)
immler@58983
  1387
wenzelm@63353
  1388
lemma neg_numeral_power_le_real_of_int_cancel_iff [simp]:
wenzelm@63353
  1389
  "(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
lp15@61609
  1390
  by (metis of_int_le_iff of_int_neg_numeral of_int_power)
hoelzl@51523
  1391
wenzelm@63353
  1392
lemma real_of_int_le_neg_numeral_power_cancel_iff [simp]:
wenzelm@63353
  1393
  "real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
lp15@61609
  1394
  by (metis of_int_le_iff of_int_neg_numeral of_int_power)
hoelzl@51523
  1395
hoelzl@56889
  1396
wenzelm@63353
  1397
subsection \<open>Density of the Reals\<close>
wenzelm@63353
  1398
wenzelm@63353
  1399
lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2" for d1 d2 :: real
wenzelm@63353
  1400
  by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def)
hoelzl@51523
  1401
wenzelm@63353
  1402
text \<open>Similar results are proved in @{theory Fields}\<close>
wenzelm@63353
  1403
lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2" for x y :: real
wenzelm@63353
  1404
  by auto
wenzelm@63353
  1405
wenzelm@63353
  1406
lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y" for x y :: real
wenzelm@63353
  1407
  by auto
wenzelm@63353
  1408
wenzelm@63353
  1409
lemma real_sum_of_halves: "x / 2 + x / 2 = x" for x :: real
wenzelm@63353
  1410
  by simp
hoelzl@51523
  1411
hoelzl@51523
  1412
wenzelm@63353
  1413
subsection \<open>Floor and Ceiling Functions from the Reals to the Integers\<close>
hoelzl@51523
  1414
lp15@61609
  1415
(* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)
hoelzl@51523
  1416
wenzelm@63353
  1417
lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \<longleftrightarrow> n < numeral w" for n :: nat
lp15@61609
  1418
  by (metis of_nat_less_iff of_nat_numeral)
hoelzl@56889
  1419
wenzelm@63353
  1420
lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n \<longleftrightarrow> numeral w < n" for n :: nat
lp15@61609
  1421
  by (metis of_nat_less_iff of_nat_numeral)
hoelzl@56889
  1422
wenzelm@63353
  1423
lemma numeral_le_real_of_nat_iff [simp]: "numeral n \<le> real m \<longleftrightarrow> numeral n \<le> m" for m :: nat
wenzelm@63353
  1424
  by (metis not_le real_of_nat_less_numeral_iff)
nipkow@59587
  1425
wenzelm@63353
  1426
declare of_int_floor_le [simp]  (* FIXME duplicate!? *)
hoelzl@51523
  1427
wenzelm@63353
  1428
lemma of_int_floor_cancel [simp]: "of_int \<lfloor>x\<rfloor> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)"
lp15@61609
  1429
  by (metis floor_of_int)
hoelzl@51523
  1430
wenzelm@63353
  1431
lemma floor_eq: "real_of_int n < x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n"
hoelzl@58040
  1432
  by linarith
hoelzl@51523
  1433
wenzelm@63353
  1434
lemma floor_eq2: "real_of_int n \<le> x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n"
hoelzl@58040
  1435
  by linarith
hoelzl@51523
  1436
wenzelm@63353
  1437
lemma floor_eq3: "real n < x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n"
hoelzl@58040
  1438
  by linarith
hoelzl@51523
  1439
wenzelm@63353
  1440
lemma floor_eq4: "real n \<le> x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n"
hoelzl@58040
  1441
  by linarith
hoelzl@51523
  1442
wenzelm@61942
  1443
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int \<lfloor>r\<rfloor>"
hoelzl@58040
  1444
  by linarith
hoelzl@51523
  1445
wenzelm@61942
  1446
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int \<lfloor>r\<rfloor>"
hoelzl@58040
  1447
  by linarith
hoelzl@51523
  1448
wenzelm@61942
  1449
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int \<lfloor>r\<rfloor> + 1"
hoelzl@58040
  1450
  by linarith
hoelzl@51523
  1451
wenzelm@61942
  1452
lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int \<lfloor>r\<rfloor> + 1"
hoelzl@58040
  1453
  by linarith
hoelzl@51523
  1454
wenzelm@61942
  1455
lemma floor_eq_iff: "\<lfloor>x\<rfloor> = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)"
wenzelm@61942
  1456
  by (simp add: floor_unique_iff)
immler@58983
  1457
wenzelm@61942
  1458
lemma floor_add2[simp]: "\<lfloor>of_int a + x\<rfloor> = a + \<lfloor>x\<rfloor>"
lp15@61609
  1459
  by (simp add: add.commute)
hoelzl@51523
  1460
wenzelm@63353
  1461
lemma floor_divide_real_eq_div:
wenzelm@63353
  1462
  assumes "0 \<le> b"
wenzelm@63353
  1463
  shows "\<lfloor>a / real_of_int b\<rfloor> = \<lfloor>a\<rfloor> div b"
wenzelm@63353
  1464
proof (cases "b = 0")
wenzelm@63353
  1465
  case True
wenzelm@63353
  1466
  then show ?thesis by simp
wenzelm@63353
  1467
next
wenzelm@63353
  1468
  case False
wenzelm@63353
  1469
  with assms have b: "b > 0" by simp
wenzelm@63353
  1470
  have "j = i div b"
wenzelm@63353
  1471
    if "real_of_int i \<le> a" "a < 1 + real_of_int i"
lp15@61609
  1472
      "real_of_int j * real_of_int b \<le> a" "a < real_of_int b + real_of_int j * real_of_int b"
wenzelm@63353
  1473
    for i j :: int
wenzelm@63353
  1474
  proof -
wenzelm@63353
  1475
    from that have "i < b + j * b"
wenzelm@63353
  1476
      by (metis le_less_trans of_int_add of_int_less_iff of_int_mult)
lp15@61609
  1477
    moreover have "j * b < 1 + i"
lp15@61609
  1478
    proof -
lp15@61609
  1479
      have "real_of_int (j * b) < real_of_int i + 1"
wenzelm@61799
  1480
        using \<open>a < 1 + real_of_int i\<close> \<open>real_of_int j * real_of_int b \<le> a\<close> by force
wenzelm@63353
  1481
      then show "j * b < 1 + i"
lp15@61609
  1482
        by linarith
lp15@61609
  1483
    qed
lp15@61609
  1484
    ultimately have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"
hoelzl@58788
  1485
      by (auto simp: field_simps)
hoelzl@58788
  1486
    then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
wenzelm@63353
  1487
      using pos_mod_bound [OF b, of i] pos_mod_sign [OF b, of i]
wenzelm@63353
  1488
      by linarith+
wenzelm@63353
  1489
    then show ?thesis
wenzelm@63353
  1490
      using b unfolding mult_less_cancel_right by auto
wenzelm@63353
  1491
  qed
wenzelm@63353
  1492
  with b show ?thesis
hoelzl@58788
  1493
    by (auto split: floor_split simp: field_simps)
wenzelm@63353
  1494
qed
hoelzl@58788
  1495
wenzelm@63353
  1496
lemma floor_divide_eq_div_numeral [simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
lp15@61609
  1497
  by (metis floor_divide_of_int_eq of_int_numeral)
hoelzl@58097
  1498
wenzelm@63353
  1499
lemma floor_minus_divide_eq_div_numeral [simp]:
wenzelm@63353
  1500
  "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
lp15@61609
  1501
  by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)
hoelzl@51523
  1502
wenzelm@63353
  1503
lemma of_int_ceiling_cancel [simp]: "of_int \<lceil>x\<rceil> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)"
lp15@61609
  1504
  using ceiling_of_int by metis
hoelzl@51523
  1505
wenzelm@63353
  1506
lemma ceiling_eq: "of_int n < x \<Longrightarrow> x \<le> of_int n + 1 \<Longrightarrow> \<lceil>x\<rceil> = n + 1"
lp15@61694
  1507
  by (simp add: ceiling_unique)
hoelzl@51523
  1508
wenzelm@61942
  1509
lemma of_int_ceiling_diff_one_le [simp]: "of_int \<lceil>r\<rceil> - 1 \<le> r"
hoelzl@58040
  1510
  by linarith
hoelzl@51523
  1511
wenzelm@61942
  1512
lemma of_int_ceiling_le_add_one [simp]: "of_int \<lceil>r\<rceil> \<le> r + 1"
hoelzl@58040
  1513
  by linarith
hoelzl@51523
  1514
wenzelm@63353
  1515
lemma ceiling_le: "x \<le> of_int a \<Longrightarrow> \<lceil>x\<rceil> \<le> a"
lp15@61694
  1516
  by (simp add: ceiling_le_iff)
hoelzl@51523
  1517
lp15@61694
  1518
lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)"
lp15@61609
  1519
  by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)
hoelzl@58097
  1520
hoelzl@58097
  1521
lemma ceiling_divide_eq_div_numeral [simp]:
hoelzl@58097
  1522
  "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"
hoelzl@58097
  1523
  using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp
hoelzl@58097
  1524
hoelzl@58097
  1525
lemma ceiling_minus_divide_eq_div_numeral [simp]:
hoelzl@58097
  1526
  "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"
hoelzl@58097
  1527
  using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp
hoelzl@51523
  1528
wenzelm@63353
  1529
text \<open>
wenzelm@63353
  1530
  The following lemmas are remnants of the erstwhile functions natfloor
wenzelm@63353
  1531
  and natceiling.
wenzelm@63353
  1532
\<close>
hoelzl@58040
  1533
wenzelm@63353
  1534
lemma nat_floor_neg: "x \<le> 0 \<Longrightarrow> nat \<lfloor>x\<rfloor> = 0" for x :: real
hoelzl@58040
  1535
  by linarith
hoelzl@51523
  1536
wenzelm@63353
  1537
lemma le_nat_floor: "real x \<le> a \<Longrightarrow> x \<le> nat \<lfloor>a\<rfloor>"
hoelzl@58040
  1538
  by linarith
hoelzl@51523
  1539
wenzelm@61942
  1540
lemma le_mult_nat_floor: "nat \<lfloor>a\<rfloor> * nat \<lfloor>b\<rfloor> \<le> nat \<lfloor>a * b\<rfloor>"
wenzelm@63353
  1541
  by (cases "0 \<le> a \<and> 0 \<le> b")
nipkow@59587
  1542
     (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
hoelzl@51523
  1543
wenzelm@63353
  1544
lemma nat_ceiling_le_eq [simp]: "nat \<lceil>x\<rceil> \<le> a \<longleftrightarrow> x \<le> real a"
hoelzl@58040
  1545
  by linarith
hoelzl@51523
  1546
wenzelm@63353
  1547
lemma real_nat_ceiling_ge: "x \<le> real (nat \<lceil>x\<rceil>)"
hoelzl@58040
  1548
  by linarith
hoelzl@51523
  1549
wenzelm@63353
  1550
lemma Rats_no_top_le: "\<exists>q \<in> \<rat>. x \<le> q" for x :: real
wenzelm@61942
  1551
  by (auto intro!: bexI[of _ "of_nat (nat \<lceil>x\<rceil>)"]) linarith
hoelzl@57275
  1552
wenzelm@63353
  1553
lemma Rats_no_bot_less: "\<exists>q \<in> \<rat>. q < x" for x :: real
wenzelm@61942
  1554
  apply (auto intro!: bexI[of _ "of_int (\<lfloor>x\<rfloor> - 1)"])
hoelzl@57447
  1555
  apply (rule less_le_trans[OF _ of_int_floor_le])
hoelzl@57447
  1556
  apply simp
hoelzl@57447
  1557
  done
hoelzl@57447
  1558
wenzelm@63353
  1559
wenzelm@60758
  1560
subsection \<open>Exponentiation with floor\<close>
hoelzl@51523
  1561
hoelzl@51523
  1562
lemma floor_power:
wenzelm@61942
  1563
  assumes "x = of_int \<lfloor>x\<rfloor>"
wenzelm@61942
  1564
  shows "\<lfloor>x ^ n\<rfloor> = \<lfloor>x\<rfloor> ^ n"
hoelzl@51523
  1565
proof -
wenzelm@61942
  1566
  have "x ^ n = of_int (\<lfloor>x\<rfloor> ^ n)"
hoelzl@51523
  1567
    using assms by (induct n arbitrary: x) simp_all
lp15@62626
  1568
  then show ?thesis by (metis floor_of_int)
hoelzl@51523
  1569
qed
lp15@61609
  1570
wenzelm@63353
  1571
lemma floor_numeral_power [simp]: "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
immler@58983
  1572
  by (metis floor_of_int of_int_numeral of_int_power)
immler@58983
  1573
wenzelm@63353
  1574
lemma ceiling_numeral_power [simp]: "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n"
immler@58983
  1575
  by (metis ceiling_of_int of_int_numeral of_int_power)
immler@58983
  1576
wenzelm@63353
  1577
wenzelm@60758
  1578
subsection \<open>Implementation of rational real numbers\<close>
hoelzl@51523
  1579
wenzelm@60758
  1580
text \<open>Formal constructor\<close>
hoelzl@51523
  1581
wenzelm@63353
  1582
definition Ratreal :: "rat \<Rightarrow> real"
wenzelm@63353
  1583
  where [code_abbrev, simp]: "Ratreal = of_rat"
hoelzl@51523
  1584
hoelzl@51523
  1585
code_datatype Ratreal
hoelzl@51523
  1586
hoelzl@51523
  1587
wenzelm@60758
  1588
text \<open>Numerals\<close>
hoelzl@51523
  1589
wenzelm@63353
  1590
lemma [code_abbrev]: "(of_rat (of_int a) :: real) = of_int a"
hoelzl@51523
  1591
  by simp
hoelzl@51523
  1592
wenzelm@63353
  1593
lemma [code_abbrev]: "(of_rat 0 :: real) = 0"
hoelzl@51523
  1594
  by simp
hoelzl@51523
  1595
wenzelm@63353
  1596
lemma [code_abbrev]: "(of_rat 1 :: real) = 1"
hoelzl@51523
  1597
  by simp
hoelzl@51523
  1598
wenzelm@63353
  1599
lemma [code_abbrev]: "(of_rat (- 1) :: real) = - 1"
haftmann@58134
  1600
  by simp
haftmann@58134
  1601
wenzelm@63353
  1602
lemma [code_abbrev]: "(of_rat (numeral k) :: real) = numeral k"
hoelzl@51523
  1603
  by simp
hoelzl@51523
  1604
wenzelm@63353
  1605
lemma [code_abbrev]: "(of_rat (- numeral k) :: real) = - numeral k"
hoelzl@51523
  1606
  by simp
hoelzl@51523
  1607
hoelzl@51523
  1608
lemma [code_post]:
hoelzl@51523
  1609
  "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
haftmann@58134
  1610
  "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
haftmann@58134
  1611
  "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"
haftmann@58134
  1612
  "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"
haftmann@54489
  1613
  by (simp_all add: of_rat_divide of_rat_minus)
hoelzl@51523
  1614
hoelzl@51523
  1615
wenzelm@60758
  1616
text \<open>Operations\<close>
hoelzl@51523
  1617
wenzelm@63353
  1618
lemma zero_real_code [code]: "0 = Ratreal 0"
hoelzl@51523
  1619
by simp
hoelzl@51523
  1620
wenzelm@63353
  1621
lemma one_real_code [code]: "1 = Ratreal 1"
hoelzl@51523
  1622
by simp
hoelzl@51523
  1623
hoelzl@51523
  1624
instantiation real :: equal
hoelzl@51523
  1625
begin
hoelzl@51523
  1626
wenzelm@63353
  1627
definition "HOL.equal x y \<longleftrightarrow> x - y = 0" for x :: real
hoelzl@51523
  1628
wenzelm@63353
  1629
instance by standard (simp add: equal_real_def)
hoelzl@51523
  1630
wenzelm@63353
  1631
lemma real_equal_code [code]: "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
hoelzl@51523
  1632
  by (simp add: equal_real_def equal)
hoelzl@51523
  1633
wenzelm@63353
  1634
lemma [code nbe]: "HOL.equal x x \<longleftrightarrow> True" for x :: real
hoelzl@51523
  1635
  by (rule equal_refl)
hoelzl@51523
  1636
hoelzl@51523
  1637
end
hoelzl@51523
  1638
hoelzl@51523
  1639
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
hoelzl@51523
  1640
  by (simp add: of_rat_less_eq)
hoelzl@51523
  1641
hoelzl@51523
  1642
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
hoelzl@51523
  1643
  by (simp add: of_rat_less)
hoelzl@51523
  1644
hoelzl@51523
  1645
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
hoelzl@51523
  1646
  by (simp add: of_rat_add)
hoelzl@51523
  1647
hoelzl@51523
  1648
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
hoelzl@51523
  1649
  by (simp add: of_rat_mult)
hoelzl@51523
  1650
hoelzl@51523
  1651
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
hoelzl@51523
  1652
  by (simp add: of_rat_minus)
hoelzl@51523
  1653
hoelzl@51523
  1654
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
hoelzl@51523
  1655
  by (simp add: of_rat_diff)
hoelzl@51523
  1656
hoelzl@51523
  1657
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
hoelzl@51523
  1658
  by (simp add: of_rat_inverse)
lp15@61284
  1659
hoelzl@51523
  1660
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
hoelzl@51523
  1661
  by (simp add: of_rat_divide)
hoelzl@51523
  1662
wenzelm@61942
  1663
lemma real_floor_code [code]: "\<lfloor>Ratreal x\<rfloor> = \<lfloor>x\<rfloor>"
wenzelm@63353
  1664
  by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff
wenzelm@63353
  1665
      of_int_floor_le of_rat_of_int_eq real_less_eq_code)
hoelzl@51523
  1666
hoelzl@51523
  1667
wenzelm@60758
  1668
text \<open>Quickcheck\<close>
hoelzl@51523
  1669
hoelzl@51523
  1670
definition (in term_syntax)
wenzelm@63353
  1671
  valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)"
wenzelm@63353
  1672
  where [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
hoelzl@51523
  1673
hoelzl@51523
  1674
notation fcomp (infixl "\<circ>>" 60)
hoelzl@51523
  1675
notation scomp (infixl "\<circ>\<rightarrow>" 60)
hoelzl@51523
  1676
hoelzl@51523
  1677
instantiation real :: random
hoelzl@51523
  1678
begin
hoelzl@51523
  1679
hoelzl@51523
  1680
definition
hoelzl@51523
  1681
  "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
hoelzl@51523
  1682
hoelzl@51523
  1683
instance ..
hoelzl@51523
  1684
hoelzl@51523
  1685
end
hoelzl@51523
  1686
hoelzl@51523
  1687
no_notation fcomp (infixl "\<circ>>" 60)
hoelzl@51523
  1688
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
hoelzl@51523
  1689
hoelzl@51523
  1690
instantiation real :: exhaustive
hoelzl@51523
  1691
begin
hoelzl@51523
  1692
hoelzl@51523
  1693
definition
wenzelm@63353
  1694
  "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (\<lambda>r. f (Ratreal r)) d"
hoelzl@51523
  1695
hoelzl@51523
  1696
instance ..
hoelzl@51523
  1697
hoelzl@51523
  1698
end
hoelzl@51523
  1699
hoelzl@51523
  1700
instantiation real :: full_exhaustive
hoelzl@51523
  1701
begin
hoelzl@51523
  1702
hoelzl@51523
  1703
definition
wenzelm@63353
  1704
  "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (\<lambda>r. f (valterm_ratreal r)) d"
hoelzl@51523
  1705
hoelzl@51523
  1706
instance ..
hoelzl@51523
  1707
hoelzl@51523
  1708
end
hoelzl@51523
  1709
hoelzl@51523
  1710
instantiation real :: narrowing
hoelzl@51523
  1711
begin
hoelzl@51523
  1712
hoelzl@51523
  1713
definition
wenzelm@63353
  1714
  "narrowing_real = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
hoelzl@51523
  1715
hoelzl@51523
  1716
instance ..
hoelzl@51523
  1717
hoelzl@51523
  1718
end
hoelzl@51523
  1719
hoelzl@51523
  1720
wenzelm@60758
  1721
subsection \<open>Setup for Nitpick\<close>
hoelzl@51523
  1722
wenzelm@60758
  1723
declaration \<open>
hoelzl@51523
  1724
  Nitpick_HOL.register_frac_type @{type_name real}
blanchet@62079
  1725
    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
blanchet@62079
  1726
     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
blanchet@62079
  1727
     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
blanchet@62079
  1728
     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
blanchet@62079
  1729
     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
blanchet@62079
  1730
     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
blanchet@62079
  1731
     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
blanchet@62079
  1732
     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
wenzelm@60758
  1733
\<close>
hoelzl@51523
  1734
hoelzl@51523
  1735
lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
wenzelm@63353
  1736
  ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
wenzelm@63353
  1737
  times_real_inst.times_real uminus_real_inst.uminus_real
wenzelm@63353
  1738
  zero_real_inst.zero_real
hoelzl@51523
  1739
blanchet@56078
  1740
wenzelm@60758
  1741
subsection \<open>Setup for SMT\<close>
blanchet@56078
  1742
blanchet@58061
  1743
ML_file "Tools/SMT/smt_real.ML"
blanchet@58061
  1744
ML_file "Tools/SMT/z3_real.ML"
blanchet@56078
  1745
blanchet@58061
  1746
lemma [z3_rule]:
wenzelm@63353
  1747
  "0 + x = x"
blanchet@56078
  1748
  "x + 0 = x"
blanchet@56078
  1749
  "0 * x = 0"
blanchet@56078
  1750
  "1 * x = x"
blanchet@56078
  1751
  "x + y = y + x"
wenzelm@63353
  1752
  for x y :: real
blanchet@56078
  1753
  by auto
hoelzl@51523
  1754
hoelzl@51523
  1755
end