src/HOL/Real.thy
author paulson <lp15@cam.ac.uk>
Tue Nov 10 14:18:41 2015 +0000 (2015-11-10)
changeset 61609 77b453bd616f
parent 61284 2314c2f62eb1
child 61649 268d88ec9087
permissions -rw-r--r--
Coercion "real" now has type nat => real only and is no longer overloaded. Type class "real_of" is gone. Many duplicate theorems removed.
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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 Conditionally_Complete_Lattices
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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]:
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  fixes x :: "'a::cancel_semigroup_add" shows "inj (op+ x)"
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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::'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:
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  fixes a b c d :: "'a::ab_group_add"
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  shows "(a + c) - (b + d) = (a - b) + (c - d)"
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  by simp
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lemma minus_diff_minus:
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  fixes a b :: "'a::ab_group_add"
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  shows "- a - - b = - (a - b)"
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  by simp
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lemma mult_diff_mult:
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  fixes x y a b :: "'a::ring"
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  shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
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  by (simp add: algebra_simps)
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lemma inverse_diff_inverse:
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  fixes a b :: "'a::division_ring"
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  assumes "a \<noteq> 0" and "b \<noteq> 0"
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  shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
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  using assms by (simp add: algebra_simps)
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lemma obtain_pos_sum:
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  fixes r :: rat assumes r: "0 < r"
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  obtains s t where "0 < s" and "0 < t" and "r = s + t"
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proof
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    from r show "0 < r/2" by simp
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    from r show "0 < r/2" by simp
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    show "r = r/2 + r/2" by simp
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qed
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subsection \<open>Sequences that converge to zero\<close>
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definition
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  vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
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where
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  "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"
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lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
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  unfolding vanishes_def by simp
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lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
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  unfolding vanishes_def by simp
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lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
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  unfolding vanishes_def
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  apply (cases "c = 0", auto)
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  apply (rule exI [where x="\<bar>c\<bar>"], auto)
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  done
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lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
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  unfolding vanishes_def by simp
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lemma vanishes_add:
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  assumes X: "vanishes X" and Y: "vanishes Y"
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  shows "vanishes (\<lambda>n. X n + Y n)"
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proof (rule vanishesI)
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  fix r :: rat assume "0 < r"
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  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
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    by (rule obtain_pos_sum)
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  obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
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    using vanishesD [OF X s] ..
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  obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
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    using vanishesD [OF Y t] ..
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  have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
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  proof (clarsimp)
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    fix n assume n: "i \<le> n" "j \<le> n"
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    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
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    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
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    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
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  qed
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  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
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qed
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lemma vanishes_diff:
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  assumes X: "vanishes X" and Y: "vanishes Y"
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  shows "vanishes (\<lambda>n. X n - Y n)"
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  unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)
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lemma vanishes_mult_bounded:
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  assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
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  assumes Y: "vanishes (\<lambda>n. Y n)"
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  shows "vanishes (\<lambda>n. X n * Y n)"
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proof (rule vanishesI)
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  fix r :: rat assume r: "0 < r"
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  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
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    using X by fast
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  obtain b where b: "0 < b" "r = a * b"
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  proof
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    show "0 < r / a" using r a by simp
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    show "r = a * (r / a)" using a by simp
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  qed
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  obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
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    using vanishesD [OF Y b(1)] ..
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  have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
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    by (simp add: b(2) abs_mult mult_strict_mono' a k)
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  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
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qed
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subsection \<open>Cauchy sequences\<close>
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definition
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  cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
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where
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  "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
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lemma cauchyI:
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  "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r) \<Longrightarrow> cauchy X"
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  unfolding cauchy_def by simp
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lemma cauchyD:
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  "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
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  unfolding cauchy_def by simp
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lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
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  unfolding cauchy_def by simp
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lemma cauchy_add [simp]:
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  assumes X: "cauchy X" and Y: "cauchy Y"
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  shows "cauchy (\<lambda>n. X n + Y n)"
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proof (rule cauchyI)
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  fix r :: rat assume "0 < r"
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  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
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    by (rule obtain_pos_sum)
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  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
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    using cauchyD [OF X s] ..
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  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
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    using cauchyD [OF Y t] ..
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  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 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" unfolding r .
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  qed
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  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
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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 X: "cauchy X" and Y: "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" 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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    thus "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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      hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
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      thus "\<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, simp 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 fast
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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 fast
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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 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" unfolding r .
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  qed
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  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
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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 fast
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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
hoelzl@51523
   265
  have "X k \<le> - r \<or> r \<le> X k"
wenzelm@60758
   266
    using \<open>r \<le> \<bar>X k\<bar>\<close> by auto
hoelzl@51523
   267
  hence "(\<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
hoelzl@51523
   269
  hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
hoelzl@51523
   270
  thus "\<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>"
hoelzl@51523
   278
using cauchy_not_vanishes_cases [OF assms]
hoelzl@51523
   279
by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)
hoelzl@51523
   280
hoelzl@51523
   281
lemma cauchy_inverse [simp]:
hoelzl@51523
   282
  assumes X: "cauchy X"
hoelzl@51523
   283
  assumes nz: "\<not> vanishes X"
hoelzl@51523
   284
  shows "cauchy (\<lambda>n. inverse (X n))"
hoelzl@51523
   285
proof (rule cauchyI)
hoelzl@51523
   286
  fix r :: rat assume "0 < r"
hoelzl@51523
   287
  obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
hoelzl@51523
   288
    using cauchy_not_vanishes [OF X nz] by fast
hoelzl@51523
   289
  from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
hoelzl@51523
   290
  obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
hoelzl@51523
   291
  proof
wenzelm@60758
   292
    show "0 < b * r * b" by (simp add: \<open>0 < r\<close> b)
hoelzl@51523
   293
    show "r = inverse b * (b * r * b) * inverse b"
hoelzl@51523
   294
      using b by simp
hoelzl@51523
   295
  qed
hoelzl@51523
   296
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
hoelzl@51523
   297
    using cauchyD [OF X s] ..
hoelzl@51523
   298
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
hoelzl@51523
   299
  proof (clarsimp)
hoelzl@51523
   300
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
hoelzl@51523
   301
    have "\<bar>inverse (X m) - inverse (X n)\<bar> =
hoelzl@51523
   302
          inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
hoelzl@51523
   303
      by (simp add: inverse_diff_inverse nz * abs_mult)
hoelzl@51523
   304
    also have "\<dots> < inverse b * s * inverse b"
hoelzl@51523
   305
      by (simp add: mult_strict_mono less_imp_inverse_less
nipkow@56544
   306
                    i j b * s)
hoelzl@51523
   307
    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
hoelzl@51523
   308
  qed
hoelzl@51523
   309
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
hoelzl@51523
   310
qed
hoelzl@51523
   311
hoelzl@51523
   312
lemma vanishes_diff_inverse:
hoelzl@51523
   313
  assumes X: "cauchy X" "\<not> vanishes X"
hoelzl@51523
   314
  assumes Y: "cauchy Y" "\<not> vanishes Y"
hoelzl@51523
   315
  assumes XY: "vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   316
  shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
hoelzl@51523
   317
proof (rule vanishesI)
hoelzl@51523
   318
  fix r :: rat assume r: "0 < r"
hoelzl@51523
   319
  obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
hoelzl@51523
   320
    using cauchy_not_vanishes [OF X] by fast
hoelzl@51523
   321
  obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
hoelzl@51523
   322
    using cauchy_not_vanishes [OF Y] by fast
hoelzl@51523
   323
  obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
hoelzl@51523
   324
  proof
hoelzl@51523
   325
    show "0 < a * r * b"
nipkow@56544
   326
      using a r b by simp
hoelzl@51523
   327
    show "inverse a * (a * r * b) * inverse b = r"
hoelzl@51523
   328
      using a r b by simp
hoelzl@51523
   329
  qed
hoelzl@51523
   330
  obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
hoelzl@51523
   331
    using vanishesD [OF XY s] ..
hoelzl@51523
   332
  have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
hoelzl@51523
   333
  proof (clarsimp)
hoelzl@51523
   334
    fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
hoelzl@51523
   335
    have "X n \<noteq> 0" and "Y n \<noteq> 0"
hoelzl@51523
   336
      using i j a b n by auto
hoelzl@51523
   337
    hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
hoelzl@51523
   338
        inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
hoelzl@51523
   339
      by (simp add: inverse_diff_inverse abs_mult)
hoelzl@51523
   340
    also have "\<dots> < inverse a * s * inverse b"
hoelzl@51523
   341
      apply (intro mult_strict_mono' less_imp_inverse_less)
nipkow@56536
   342
      apply (simp_all add: a b i j k n)
hoelzl@51523
   343
      done
wenzelm@60758
   344
    also note \<open>inverse a * s * inverse b = r\<close>
hoelzl@51523
   345
    finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
hoelzl@51523
   346
  qed
hoelzl@51523
   347
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
hoelzl@51523
   348
qed
hoelzl@51523
   349
wenzelm@60758
   350
subsection \<open>Equivalence relation on Cauchy sequences\<close>
hoelzl@51523
   351
hoelzl@51523
   352
definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
hoelzl@51523
   353
  where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"
hoelzl@51523
   354
hoelzl@51523
   355
lemma realrelI [intro?]:
hoelzl@51523
   356
  assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   357
  shows "realrel X Y"
hoelzl@51523
   358
  using assms unfolding realrel_def by simp
hoelzl@51523
   359
hoelzl@51523
   360
lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
hoelzl@51523
   361
  unfolding realrel_def by simp
hoelzl@51523
   362
hoelzl@51523
   363
lemma symp_realrel: "symp realrel"
hoelzl@51523
   364
  unfolding realrel_def
hoelzl@51523
   365
  by (rule sympI, clarify, drule vanishes_minus, simp)
hoelzl@51523
   366
hoelzl@51523
   367
lemma transp_realrel: "transp realrel"
hoelzl@51523
   368
  unfolding realrel_def
hoelzl@51523
   369
  apply (rule transpI, clarify)
hoelzl@51523
   370
  apply (drule (1) vanishes_add)
hoelzl@51523
   371
  apply (simp add: algebra_simps)
hoelzl@51523
   372
  done
hoelzl@51523
   373
hoelzl@51523
   374
lemma part_equivp_realrel: "part_equivp realrel"
hoelzl@51523
   375
  by (fast intro: part_equivpI symp_realrel transp_realrel
hoelzl@51523
   376
    realrel_refl cauchy_const)
hoelzl@51523
   377
wenzelm@60758
   378
subsection \<open>The field of real numbers\<close>
hoelzl@51523
   379
hoelzl@51523
   380
quotient_type real = "nat \<Rightarrow> rat" / partial: realrel
hoelzl@51523
   381
  morphisms rep_real Real
hoelzl@51523
   382
  by (rule part_equivp_realrel)
hoelzl@51523
   383
hoelzl@51523
   384
lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
hoelzl@51523
   385
  unfolding real.pcr_cr_eq cr_real_def realrel_def by auto
hoelzl@51523
   386
hoelzl@51523
   387
lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
hoelzl@51523
   388
  assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"
hoelzl@51523
   389
proof (induct x)
hoelzl@51523
   390
  case (1 X)
hoelzl@51523
   391
  hence "cauchy X" by (simp add: realrel_def)
hoelzl@51523
   392
  thus "P (Real X)" by (rule assms)
hoelzl@51523
   393
qed
hoelzl@51523
   394
hoelzl@51523
   395
lemma eq_Real:
hoelzl@51523
   396
  "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   397
  using real.rel_eq_transfer
blanchet@55945
   398
  unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp
hoelzl@51523
   399
kuncar@51956
   400
lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
kuncar@51956
   401
by (simp add: real.domain_eq realrel_def)
hoelzl@51523
   402
haftmann@59867
   403
instantiation real :: field
hoelzl@51523
   404
begin
hoelzl@51523
   405
hoelzl@51523
   406
lift_definition zero_real :: "real" is "\<lambda>n. 0"
hoelzl@51523
   407
  by (simp add: realrel_refl)
hoelzl@51523
   408
hoelzl@51523
   409
lift_definition one_real :: "real" is "\<lambda>n. 1"
hoelzl@51523
   410
  by (simp add: realrel_refl)
hoelzl@51523
   411
hoelzl@51523
   412
lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n + Y n"
hoelzl@51523
   413
  unfolding realrel_def add_diff_add
hoelzl@51523
   414
  by (simp only: cauchy_add vanishes_add simp_thms)
hoelzl@51523
   415
hoelzl@51523
   416
lift_definition uminus_real :: "real \<Rightarrow> real" is "\<lambda>X n. - X n"
hoelzl@51523
   417
  unfolding realrel_def minus_diff_minus
hoelzl@51523
   418
  by (simp only: cauchy_minus vanishes_minus simp_thms)
hoelzl@51523
   419
hoelzl@51523
   420
lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
hoelzl@51523
   421
  unfolding realrel_def mult_diff_mult
haftmann@57512
   422
  by (subst (4) mult.commute, simp only: cauchy_mult vanishes_add
hoelzl@51523
   423
    vanishes_mult_bounded cauchy_imp_bounded simp_thms)
hoelzl@51523
   424
hoelzl@51523
   425
lift_definition inverse_real :: "real \<Rightarrow> real"
hoelzl@51523
   426
  is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
hoelzl@51523
   427
proof -
hoelzl@51523
   428
  fix X Y assume "realrel X Y"
hoelzl@51523
   429
  hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   430
    unfolding realrel_def by simp_all
hoelzl@51523
   431
  have "vanishes X \<longleftrightarrow> vanishes Y"
hoelzl@51523
   432
  proof
hoelzl@51523
   433
    assume "vanishes X"
hoelzl@51523
   434
    from vanishes_diff [OF this XY] show "vanishes Y" by simp
hoelzl@51523
   435
  next
hoelzl@51523
   436
    assume "vanishes Y"
hoelzl@51523
   437
    from vanishes_add [OF this XY] show "vanishes X" by simp
hoelzl@51523
   438
  qed
hoelzl@51523
   439
  thus "?thesis X Y"
hoelzl@51523
   440
    unfolding realrel_def
hoelzl@51523
   441
    by (simp add: vanishes_diff_inverse X Y XY)
hoelzl@51523
   442
qed
hoelzl@51523
   443
hoelzl@51523
   444
definition
hoelzl@51523
   445
  "x - y = (x::real) + - y"
hoelzl@51523
   446
hoelzl@51523
   447
definition
haftmann@60429
   448
  "x div y = (x::real) * inverse y"
hoelzl@51523
   449
hoelzl@51523
   450
lemma add_Real:
hoelzl@51523
   451
  assumes X: "cauchy X" and Y: "cauchy Y"
hoelzl@51523
   452
  shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
hoelzl@51523
   453
  using assms plus_real.transfer
blanchet@55945
   454
  unfolding cr_real_eq rel_fun_def by simp
hoelzl@51523
   455
hoelzl@51523
   456
lemma minus_Real:
hoelzl@51523
   457
  assumes X: "cauchy X"
hoelzl@51523
   458
  shows "- Real X = Real (\<lambda>n. - X n)"
hoelzl@51523
   459
  using assms uminus_real.transfer
blanchet@55945
   460
  unfolding cr_real_eq rel_fun_def by simp
hoelzl@51523
   461
hoelzl@51523
   462
lemma diff_Real:
hoelzl@51523
   463
  assumes X: "cauchy X" and Y: "cauchy Y"
hoelzl@51523
   464
  shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
haftmann@54230
   465
  unfolding minus_real_def
hoelzl@51523
   466
  by (simp add: minus_Real add_Real X Y)
hoelzl@51523
   467
hoelzl@51523
   468
lemma mult_Real:
hoelzl@51523
   469
  assumes X: "cauchy X" and Y: "cauchy Y"
hoelzl@51523
   470
  shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
hoelzl@51523
   471
  using assms times_real.transfer
blanchet@55945
   472
  unfolding cr_real_eq rel_fun_def by simp
hoelzl@51523
   473
hoelzl@51523
   474
lemma inverse_Real:
hoelzl@51523
   475
  assumes X: "cauchy X"
hoelzl@51523
   476
  shows "inverse (Real X) =
hoelzl@51523
   477
    (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
hoelzl@51523
   478
  using assms inverse_real.transfer zero_real.transfer
blanchet@55945
   479
  unfolding cr_real_eq rel_fun_def by (simp split: split_if_asm, metis)
hoelzl@51523
   480
hoelzl@51523
   481
instance proof
hoelzl@51523
   482
  fix a b c :: real
hoelzl@51523
   483
  show "a + b = b + a"
haftmann@57514
   484
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   485
  show "(a + b) + c = a + (b + c)"
haftmann@57514
   486
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   487
  show "0 + a = a"
hoelzl@51523
   488
    by transfer (simp add: realrel_def)
hoelzl@51523
   489
  show "- a + a = 0"
hoelzl@51523
   490
    by transfer (simp add: realrel_def)
hoelzl@51523
   491
  show "a - b = a + - b"
hoelzl@51523
   492
    by (rule minus_real_def)
hoelzl@51523
   493
  show "(a * b) * c = a * (b * c)"
haftmann@57514
   494
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   495
  show "a * b = b * a"
haftmann@57514
   496
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   497
  show "1 * a = a"
haftmann@57514
   498
    by transfer (simp add: ac_simps realrel_def)
hoelzl@51523
   499
  show "(a + b) * c = a * c + b * c"
hoelzl@51523
   500
    by transfer (simp add: distrib_right realrel_def)
wenzelm@61076
   501
  show "(0::real) \<noteq> (1::real)"
hoelzl@51523
   502
    by transfer (simp add: realrel_def)
hoelzl@51523
   503
  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
hoelzl@51523
   504
    apply transfer
hoelzl@51523
   505
    apply (simp add: realrel_def)
hoelzl@51523
   506
    apply (rule vanishesI)
hoelzl@51523
   507
    apply (frule (1) cauchy_not_vanishes, clarify)
hoelzl@51523
   508
    apply (rule_tac x=k in exI, clarify)
hoelzl@51523
   509
    apply (drule_tac x=n in spec, simp)
hoelzl@51523
   510
    done
haftmann@60429
   511
  show "a div b = a * inverse b"
hoelzl@51523
   512
    by (rule divide_real_def)
hoelzl@51523
   513
  show "inverse (0::real) = 0"
hoelzl@51523
   514
    by transfer (simp add: realrel_def)
hoelzl@51523
   515
qed
hoelzl@51523
   516
hoelzl@51523
   517
end
hoelzl@51523
   518
wenzelm@60758
   519
subsection \<open>Positive reals\<close>
hoelzl@51523
   520
hoelzl@51523
   521
lift_definition positive :: "real \<Rightarrow> bool"
hoelzl@51523
   522
  is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
hoelzl@51523
   523
proof -
hoelzl@51523
   524
  { fix X Y
hoelzl@51523
   525
    assume "realrel X Y"
hoelzl@51523
   526
    hence XY: "vanishes (\<lambda>n. X n - Y n)"
hoelzl@51523
   527
      unfolding realrel_def by simp_all
hoelzl@51523
   528
    assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
hoelzl@51523
   529
    then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
hoelzl@51523
   530
      by fast
hoelzl@51523
   531
    obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
wenzelm@60758
   532
      using \<open>0 < r\<close> by (rule obtain_pos_sum)
hoelzl@51523
   533
    obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
hoelzl@51523
   534
      using vanishesD [OF XY s] ..
hoelzl@51523
   535
    have "\<forall>n\<ge>max i j. t < Y n"
hoelzl@51523
   536
    proof (clarsimp)
hoelzl@51523
   537
      fix n assume n: "i \<le> n" "j \<le> n"
hoelzl@51523
   538
      have "\<bar>X n - Y n\<bar> < s" and "r < X n"
hoelzl@51523
   539
        using i j n by simp_all
hoelzl@51523
   540
      thus "t < Y n" unfolding r by simp
hoelzl@51523
   541
    qed
hoelzl@51523
   542
    hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
hoelzl@51523
   543
  } note 1 = this
hoelzl@51523
   544
  fix X Y assume "realrel X Y"
hoelzl@51523
   545
  hence "realrel X Y" and "realrel Y X"
hoelzl@51523
   546
    using symp_realrel unfolding symp_def by auto
hoelzl@51523
   547
  thus "?thesis X Y"
hoelzl@51523
   548
    by (safe elim!: 1)
hoelzl@51523
   549
qed
hoelzl@51523
   550
hoelzl@51523
   551
lemma positive_Real:
hoelzl@51523
   552
  assumes X: "cauchy X"
hoelzl@51523
   553
  shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
hoelzl@51523
   554
  using assms positive.transfer
blanchet@55945
   555
  unfolding cr_real_eq rel_fun_def by simp
hoelzl@51523
   556
hoelzl@51523
   557
lemma positive_zero: "\<not> positive 0"
hoelzl@51523
   558
  by transfer auto
hoelzl@51523
   559
hoelzl@51523
   560
lemma positive_add:
hoelzl@51523
   561
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
hoelzl@51523
   562
apply transfer
hoelzl@51523
   563
apply (clarify, rename_tac a b i j)
hoelzl@51523
   564
apply (rule_tac x="a + b" in exI, simp)
hoelzl@51523
   565
apply (rule_tac x="max i j" in exI, clarsimp)
hoelzl@51523
   566
apply (simp add: add_strict_mono)
hoelzl@51523
   567
done
hoelzl@51523
   568
hoelzl@51523
   569
lemma positive_mult:
hoelzl@51523
   570
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
hoelzl@51523
   571
apply transfer
hoelzl@51523
   572
apply (clarify, rename_tac a b i j)
nipkow@56544
   573
apply (rule_tac x="a * b" in exI, simp)
hoelzl@51523
   574
apply (rule_tac x="max i j" in exI, clarsimp)
hoelzl@51523
   575
apply (rule mult_strict_mono, auto)
hoelzl@51523
   576
done
hoelzl@51523
   577
hoelzl@51523
   578
lemma positive_minus:
hoelzl@51523
   579
  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
hoelzl@51523
   580
apply transfer
hoelzl@51523
   581
apply (simp add: realrel_def)
hoelzl@51523
   582
apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
hoelzl@51523
   583
done
hoelzl@51523
   584
haftmann@59867
   585
instantiation real :: linordered_field
hoelzl@51523
   586
begin
hoelzl@51523
   587
hoelzl@51523
   588
definition
hoelzl@51523
   589
  "x < y \<longleftrightarrow> positive (y - x)"
hoelzl@51523
   590
hoelzl@51523
   591
definition
hoelzl@51523
   592
  "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"
hoelzl@51523
   593
hoelzl@51523
   594
definition
hoelzl@51523
   595
  "abs (a::real) = (if a < 0 then - a else a)"
hoelzl@51523
   596
hoelzl@51523
   597
definition
hoelzl@51523
   598
  "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
hoelzl@51523
   599
hoelzl@51523
   600
instance proof
hoelzl@51523
   601
  fix a b c :: real
hoelzl@51523
   602
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
hoelzl@51523
   603
    by (rule abs_real_def)
hoelzl@51523
   604
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
hoelzl@51523
   605
    unfolding less_eq_real_def less_real_def
hoelzl@51523
   606
    by (auto, drule (1) positive_add, simp_all add: positive_zero)
hoelzl@51523
   607
  show "a \<le> a"
hoelzl@51523
   608
    unfolding less_eq_real_def by simp
hoelzl@51523
   609
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
hoelzl@51523
   610
    unfolding less_eq_real_def less_real_def
hoelzl@51523
   611
    by (auto, drule (1) positive_add, simp add: algebra_simps)
hoelzl@51523
   612
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
hoelzl@51523
   613
    unfolding less_eq_real_def less_real_def
hoelzl@51523
   614
    by (auto, drule (1) positive_add, simp add: positive_zero)
hoelzl@51523
   615
  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
haftmann@54230
   616
    unfolding less_eq_real_def less_real_def by auto
hoelzl@51523
   617
    (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)
hoelzl@51523
   618
    (* Should produce c + b - (c + a) \<equiv> b - a *)
hoelzl@51523
   619
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
hoelzl@51523
   620
    by (rule sgn_real_def)
hoelzl@51523
   621
  show "a \<le> b \<or> b \<le> a"
hoelzl@51523
   622
    unfolding less_eq_real_def less_real_def
hoelzl@51523
   623
    by (auto dest!: positive_minus)
hoelzl@51523
   624
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
hoelzl@51523
   625
    unfolding less_real_def
hoelzl@51523
   626
    by (drule (1) positive_mult, simp add: algebra_simps)
hoelzl@51523
   627
qed
hoelzl@51523
   628
hoelzl@51523
   629
end
hoelzl@51523
   630
hoelzl@51523
   631
instantiation real :: distrib_lattice
hoelzl@51523
   632
begin
hoelzl@51523
   633
hoelzl@51523
   634
definition
hoelzl@51523
   635
  "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"
hoelzl@51523
   636
hoelzl@51523
   637
definition
hoelzl@51523
   638
  "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"
hoelzl@51523
   639
hoelzl@51523
   640
instance proof
haftmann@54863
   641
qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)
hoelzl@51523
   642
hoelzl@51523
   643
end
hoelzl@51523
   644
hoelzl@51523
   645
lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
hoelzl@51523
   646
apply (induct x)
hoelzl@51523
   647
apply (simp add: zero_real_def)
hoelzl@51523
   648
apply (simp add: one_real_def add_Real)
hoelzl@51523
   649
done
hoelzl@51523
   650
hoelzl@51523
   651
lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
hoelzl@51523
   652
apply (cases x rule: int_diff_cases)
hoelzl@51523
   653
apply (simp add: of_nat_Real diff_Real)
hoelzl@51523
   654
done
hoelzl@51523
   655
hoelzl@51523
   656
lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
hoelzl@51523
   657
apply (induct x)
hoelzl@51523
   658
apply (simp add: Fract_of_int_quotient of_rat_divide)
hoelzl@51523
   659
apply (simp add: of_int_Real divide_inverse)
hoelzl@51523
   660
apply (simp add: inverse_Real mult_Real)
hoelzl@51523
   661
done
hoelzl@51523
   662
hoelzl@51523
   663
instance real :: archimedean_field
hoelzl@51523
   664
proof
hoelzl@51523
   665
  fix x :: real
hoelzl@51523
   666
  show "\<exists>z. x \<le> of_int z"
hoelzl@51523
   667
    apply (induct x)
hoelzl@51523
   668
    apply (frule cauchy_imp_bounded, clarify)
hoelzl@51523
   669
    apply (rule_tac x="ceiling b + 1" in exI)
hoelzl@51523
   670
    apply (rule less_imp_le)
hoelzl@51523
   671
    apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
hoelzl@51523
   672
    apply (rule_tac x=1 in exI, simp add: algebra_simps)
hoelzl@51523
   673
    apply (rule_tac x=0 in exI, clarsimp)
hoelzl@51523
   674
    apply (rule le_less_trans [OF abs_ge_self])
hoelzl@51523
   675
    apply (rule less_le_trans [OF _ le_of_int_ceiling])
hoelzl@51523
   676
    apply simp
hoelzl@51523
   677
    done
hoelzl@51523
   678
qed
hoelzl@51523
   679
hoelzl@51523
   680
instantiation real :: floor_ceiling
hoelzl@51523
   681
begin
hoelzl@51523
   682
hoelzl@51523
   683
definition [code del]:
hoelzl@51523
   684
  "floor (x::real) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
hoelzl@51523
   685
hoelzl@51523
   686
instance proof
hoelzl@51523
   687
  fix x :: real
hoelzl@51523
   688
  show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
hoelzl@51523
   689
    unfolding floor_real_def using floor_exists1 by (rule theI')
hoelzl@51523
   690
qed
hoelzl@51523
   691
hoelzl@51523
   692
end
hoelzl@51523
   693
wenzelm@60758
   694
subsection \<open>Completeness\<close>
hoelzl@51523
   695
hoelzl@51523
   696
lemma not_positive_Real:
hoelzl@51523
   697
  assumes X: "cauchy X"
hoelzl@51523
   698
  shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
hoelzl@51523
   699
unfolding positive_Real [OF X]
hoelzl@51523
   700
apply (auto, unfold not_less)
hoelzl@51523
   701
apply (erule obtain_pos_sum)
hoelzl@51523
   702
apply (drule_tac x=s in spec, simp)
hoelzl@51523
   703
apply (drule_tac r=t in cauchyD [OF X], clarify)
hoelzl@51523
   704
apply (drule_tac x=k in spec, clarsimp)
hoelzl@51523
   705
apply (rule_tac x=n in exI, clarify, rename_tac m)
hoelzl@51523
   706
apply (drule_tac x=m in spec, simp)
hoelzl@51523
   707
apply (drule_tac x=n in spec, simp)
hoelzl@51523
   708
apply (drule spec, drule (1) mp, clarify, rename_tac i)
hoelzl@51523
   709
apply (rule_tac x="max i k" in exI, simp)
hoelzl@51523
   710
done
hoelzl@51523
   711
hoelzl@51523
   712
lemma le_Real:
hoelzl@51523
   713
  assumes X: "cauchy X" and Y: "cauchy Y"
hoelzl@51523
   714
  shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
hoelzl@51523
   715
unfolding not_less [symmetric, where 'a=real] less_real_def
hoelzl@51523
   716
apply (simp add: diff_Real not_positive_Real X Y)
haftmann@57514
   717
apply (simp add: diff_le_eq ac_simps)
hoelzl@51523
   718
done
hoelzl@51523
   719
hoelzl@51523
   720
lemma le_RealI:
hoelzl@51523
   721
  assumes Y: "cauchy Y"
hoelzl@51523
   722
  shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
hoelzl@51523
   723
proof (induct x)
hoelzl@51523
   724
  fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
hoelzl@51523
   725
  hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
hoelzl@51523
   726
    by (simp add: of_rat_Real le_Real)
hoelzl@51523
   727
  {
hoelzl@51523
   728
    fix r :: rat assume "0 < r"
hoelzl@51523
   729
    then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
hoelzl@51523
   730
      by (rule obtain_pos_sum)
hoelzl@51523
   731
    obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
hoelzl@51523
   732
      using cauchyD [OF Y s] ..
hoelzl@51523
   733
    obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
hoelzl@51523
   734
      using le [OF t] ..
hoelzl@51523
   735
    have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
hoelzl@51523
   736
    proof (clarsimp)
hoelzl@51523
   737
      fix n assume n: "i \<le> n" "j \<le> n"
hoelzl@51523
   738
      have "X n \<le> Y i + t" using n j by simp
hoelzl@51523
   739
      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
hoelzl@51523
   740
      ultimately show "X n \<le> Y n + r" unfolding r by simp
hoelzl@51523
   741
    qed
hoelzl@51523
   742
    hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
hoelzl@51523
   743
  }
hoelzl@51523
   744
  thus "Real X \<le> Real Y"
hoelzl@51523
   745
    by (simp add: of_rat_Real le_Real X Y)
hoelzl@51523
   746
qed
hoelzl@51523
   747
hoelzl@51523
   748
lemma Real_leI:
hoelzl@51523
   749
  assumes X: "cauchy X"
hoelzl@51523
   750
  assumes le: "\<forall>n. of_rat (X n) \<le> y"
hoelzl@51523
   751
  shows "Real X \<le> y"
hoelzl@51523
   752
proof -
hoelzl@51523
   753
  have "- y \<le> - Real X"
hoelzl@51523
   754
    by (simp add: minus_Real X le_RealI of_rat_minus le)
hoelzl@51523
   755
  thus ?thesis by simp
hoelzl@51523
   756
qed
hoelzl@51523
   757
hoelzl@51523
   758
lemma less_RealD:
hoelzl@51523
   759
  assumes Y: "cauchy Y"
hoelzl@51523
   760
  shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
hoelzl@51523
   761
by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])
hoelzl@51523
   762
lp15@61609
   763
lemma of_nat_less_two_power [simp]:
hoelzl@51523
   764
  "of_nat n < (2::'a::linordered_idom) ^ n"
lp15@61609
   765
apply (induct n, simp)
lp15@60162
   766
by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
hoelzl@51523
   767
hoelzl@51523
   768
lemma complete_real:
hoelzl@51523
   769
  fixes S :: "real set"
hoelzl@51523
   770
  assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
hoelzl@51523
   771
  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
   772
proof -
hoelzl@51523
   773
  obtain x where x: "x \<in> S" using assms(1) ..
hoelzl@51523
   774
  obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..
hoelzl@51523
   775
hoelzl@51523
   776
  def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
hoelzl@51523
   777
  obtain a where a: "\<not> P a"
hoelzl@51523
   778
  proof
hoelzl@51523
   779
    have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
hoelzl@51523
   780
    also have "x - 1 < x" by simp
hoelzl@51523
   781
    finally have "of_int (floor (x - 1)) < x" .
hoelzl@51523
   782
    hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
hoelzl@51523
   783
    then show "\<not> P (of_int (floor (x - 1)))"
hoelzl@51523
   784
      unfolding P_def of_rat_of_int_eq using x by fast
hoelzl@51523
   785
  qed
hoelzl@51523
   786
  obtain b where b: "P b"
hoelzl@51523
   787
  proof
hoelzl@51523
   788
    show "P (of_int (ceiling z))"
hoelzl@51523
   789
    unfolding P_def of_rat_of_int_eq
hoelzl@51523
   790
    proof
hoelzl@51523
   791
      fix y assume "y \<in> S"
hoelzl@51523
   792
      hence "y \<le> z" using z by simp
hoelzl@51523
   793
      also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
hoelzl@51523
   794
      finally show "y \<le> of_int (ceiling z)" .
hoelzl@51523
   795
    qed
hoelzl@51523
   796
  qed
hoelzl@51523
   797
hoelzl@51523
   798
  def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
hoelzl@51523
   799
  def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
hoelzl@51523
   800
  def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
hoelzl@51523
   801
  def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
hoelzl@51523
   802
  def C \<equiv> "\<lambda>n. avg (A n) (B n)"
hoelzl@51523
   803
  have A_0 [simp]: "A 0 = a" unfolding A_def by simp
hoelzl@51523
   804
  have B_0 [simp]: "B 0 = b" unfolding B_def by simp
hoelzl@51523
   805
  have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
hoelzl@51523
   806
    unfolding A_def B_def C_def bisect_def split_def by simp
hoelzl@51523
   807
  have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
hoelzl@51523
   808
    unfolding A_def B_def C_def bisect_def split_def by simp
hoelzl@51523
   809
hoelzl@51523
   810
  have width: "\<And>n. B n - A n = (b - a) / 2^n"
hoelzl@51523
   811
    apply (simp add: eq_divide_eq)
hoelzl@51523
   812
    apply (induct_tac n, simp)
lp15@60162
   813
    apply (simp add: C_def avg_def power_Suc algebra_simps)
hoelzl@51523
   814
    done
hoelzl@51523
   815
hoelzl@51523
   816
  have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
hoelzl@51523
   817
    apply (simp add: divide_less_eq)
haftmann@57512
   818
    apply (subst mult.commute)
hoelzl@51523
   819
    apply (frule_tac y=y in ex_less_of_nat_mult)
hoelzl@51523
   820
    apply clarify
hoelzl@51523
   821
    apply (rule_tac x=n in exI)
hoelzl@51523
   822
    apply (erule less_trans)
hoelzl@51523
   823
    apply (rule mult_strict_right_mono)
hoelzl@51523
   824
    apply (rule le_less_trans [OF _ of_nat_less_two_power])
hoelzl@51523
   825
    apply simp
hoelzl@51523
   826
    apply assumption
hoelzl@51523
   827
    done
hoelzl@51523
   828
hoelzl@51523
   829
  have PA: "\<And>n. \<not> P (A n)"
hoelzl@51523
   830
    by (induct_tac n, simp_all add: a)
hoelzl@51523
   831
  have PB: "\<And>n. P (B n)"
hoelzl@51523
   832
    by (induct_tac n, simp_all add: b)
hoelzl@51523
   833
  have ab: "a < b"
hoelzl@51523
   834
    using a b unfolding P_def
hoelzl@51523
   835
    apply (clarsimp simp add: not_le)
hoelzl@51523
   836
    apply (drule (1) bspec)
hoelzl@51523
   837
    apply (drule (1) less_le_trans)
hoelzl@51523
   838
    apply (simp add: of_rat_less)
hoelzl@51523
   839
    done
hoelzl@51523
   840
  have AB: "\<And>n. A n < B n"
hoelzl@51523
   841
    by (induct_tac n, simp add: ab, simp add: C_def avg_def)
hoelzl@51523
   842
  have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
hoelzl@51523
   843
    apply (auto simp add: le_less [where 'a=nat])
hoelzl@51523
   844
    apply (erule less_Suc_induct)
hoelzl@51523
   845
    apply (clarsimp simp add: C_def avg_def)
hoelzl@51523
   846
    apply (simp add: add_divide_distrib [symmetric])
hoelzl@51523
   847
    apply (rule AB [THEN less_imp_le])
hoelzl@51523
   848
    apply simp
hoelzl@51523
   849
    done
hoelzl@51523
   850
  have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
hoelzl@51523
   851
    apply (auto simp add: le_less [where 'a=nat])
hoelzl@51523
   852
    apply (erule less_Suc_induct)
hoelzl@51523
   853
    apply (clarsimp simp add: C_def avg_def)
hoelzl@51523
   854
    apply (simp add: add_divide_distrib [symmetric])
hoelzl@51523
   855
    apply (rule AB [THEN less_imp_le])
hoelzl@51523
   856
    apply simp
hoelzl@51523
   857
    done
hoelzl@51523
   858
  have cauchy_lemma:
hoelzl@51523
   859
    "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
hoelzl@51523
   860
    apply (rule cauchyI)
hoelzl@51523
   861
    apply (drule twos [where y="b - a"])
hoelzl@51523
   862
    apply (erule exE)
hoelzl@51523
   863
    apply (rule_tac x=n in exI, clarify, rename_tac i j)
hoelzl@51523
   864
    apply (rule_tac y="B n - A n" in le_less_trans) defer
hoelzl@51523
   865
    apply (simp add: width)
hoelzl@51523
   866
    apply (drule_tac x=n in spec)
hoelzl@51523
   867
    apply (frule_tac x=i in spec, drule (1) mp)
hoelzl@51523
   868
    apply (frule_tac x=j in spec, drule (1) mp)
hoelzl@51523
   869
    apply (frule A_mono, drule B_mono)
hoelzl@51523
   870
    apply (frule A_mono, drule B_mono)
hoelzl@51523
   871
    apply arith
hoelzl@51523
   872
    done
hoelzl@51523
   873
  have "cauchy A"
hoelzl@51523
   874
    apply (rule cauchy_lemma [rule_format])
hoelzl@51523
   875
    apply (simp add: A_mono)
hoelzl@51523
   876
    apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
hoelzl@51523
   877
    done
hoelzl@51523
   878
  have "cauchy B"
hoelzl@51523
   879
    apply (rule cauchy_lemma [rule_format])
hoelzl@51523
   880
    apply (simp add: B_mono)
hoelzl@51523
   881
    apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
hoelzl@51523
   882
    done
hoelzl@51523
   883
  have 1: "\<forall>x\<in>S. x \<le> Real B"
hoelzl@51523
   884
  proof
hoelzl@51523
   885
    fix x assume "x \<in> S"
hoelzl@51523
   886
    then show "x \<le> Real B"
wenzelm@60758
   887
      using PB [unfolded P_def] \<open>cauchy B\<close>
hoelzl@51523
   888
      by (simp add: le_RealI)
hoelzl@51523
   889
  qed
hoelzl@51523
   890
  have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
hoelzl@51523
   891
    apply clarify
hoelzl@51523
   892
    apply (erule contrapos_pp)
hoelzl@51523
   893
    apply (simp add: not_le)
wenzelm@60758
   894
    apply (drule less_RealD [OF \<open>cauchy A\<close>], clarify)
hoelzl@51523
   895
    apply (subgoal_tac "\<not> P (A n)")
hoelzl@51523
   896
    apply (simp add: P_def not_le, clarify)
hoelzl@51523
   897
    apply (erule rev_bexI)
hoelzl@51523
   898
    apply (erule (1) less_trans)
hoelzl@51523
   899
    apply (simp add: PA)
hoelzl@51523
   900
    done
hoelzl@51523
   901
  have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
hoelzl@51523
   902
  proof (rule vanishesI)
hoelzl@51523
   903
    fix r :: rat assume "0 < r"
hoelzl@51523
   904
    then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
hoelzl@51523
   905
      using twos by fast
hoelzl@51523
   906
    have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
hoelzl@51523
   907
    proof (clarify)
hoelzl@51523
   908
      fix n assume n: "k \<le> n"
hoelzl@51523
   909
      have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
hoelzl@51523
   910
        by simp
hoelzl@51523
   911
      also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
nipkow@56544
   912
        using n by (simp add: divide_left_mono)
hoelzl@51523
   913
      also note k
hoelzl@51523
   914
      finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
hoelzl@51523
   915
    qed
hoelzl@51523
   916
    thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
hoelzl@51523
   917
  qed
hoelzl@51523
   918
  hence 3: "Real B = Real A"
wenzelm@60758
   919
    by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width)
hoelzl@51523
   920
  show "\<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
   921
    using 1 2 3 by (rule_tac x="Real B" in exI, simp)
hoelzl@51523
   922
qed
hoelzl@51523
   923
hoelzl@51775
   924
instantiation real :: linear_continuum
hoelzl@51523
   925
begin
hoelzl@51523
   926
wenzelm@60758
   927
subsection\<open>Supremum of a set of reals\<close>
hoelzl@51523
   928
hoelzl@54281
   929
definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
hoelzl@54281
   930
definition "Inf (X::real set) = - Sup (uminus ` X)"
hoelzl@51523
   931
hoelzl@51523
   932
instance
hoelzl@51523
   933
proof
hoelzl@54258
   934
  { fix x :: real and X :: "real set"
hoelzl@54258
   935
    assume x: "x \<in> X" "bdd_above X"
hoelzl@51523
   936
    then 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
   937
      using complete_real[of X] unfolding bdd_above_def by blast
hoelzl@51523
   938
    then show "x \<le> Sup X"
hoelzl@51523
   939
      unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
hoelzl@51523
   940
  note Sup_upper = this
hoelzl@51523
   941
hoelzl@51523
   942
  { fix z :: real and X :: "real set"
hoelzl@51523
   943
    assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
hoelzl@51523
   944
    then 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@51523
   945
      using complete_real[of X] by blast
hoelzl@51523
   946
    then have "Sup X = s"
lp15@61284
   947
      unfolding Sup_real_def by (best intro: Least_equality)
wenzelm@53374
   948
    also from s z have "... \<le> z"
hoelzl@51523
   949
      by blast
hoelzl@51523
   950
    finally show "Sup X \<le> z" . }
hoelzl@51523
   951
  note Sup_least = this
hoelzl@51523
   952
hoelzl@54281
   953
  { fix x :: real and X :: "real set" assume x: "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
hoelzl@54281
   954
      using Sup_upper[of "-x" "uminus ` X"] by (auto simp: Inf_real_def) }
hoelzl@54281
   955
  { fix z :: real and X :: "real set" assume "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" then show "z \<le> Inf X"
hoelzl@54281
   956
      using Sup_least[of "uminus ` X" "- z"] by (force simp: Inf_real_def) }
hoelzl@51775
   957
  show "\<exists>a b::real. a \<noteq> b"
hoelzl@51775
   958
    using zero_neq_one by blast
hoelzl@51523
   959
qed
hoelzl@51523
   960
end
hoelzl@51523
   961
hoelzl@51523
   962
wenzelm@60758
   963
subsection \<open>Hiding implementation details\<close>
hoelzl@51523
   964
hoelzl@51523
   965
hide_const (open) vanishes cauchy positive Real
hoelzl@51523
   966
hoelzl@51523
   967
declare Real_induct [induct del]
hoelzl@51523
   968
declare Abs_real_induct [induct del]
hoelzl@51523
   969
declare Abs_real_cases [cases del]
hoelzl@51523
   970
kuncar@53652
   971
lifting_update real.lifting
kuncar@53652
   972
lifting_forget real.lifting
lp15@61284
   973
wenzelm@60758
   974
subsection\<open>More Lemmas\<close>
hoelzl@51523
   975
wenzelm@60758
   976
text \<open>BH: These lemmas should not be necessary; they should be
wenzelm@60758
   977
covered by existing simp rules and simplification procedures.\<close>
hoelzl@51523
   978
hoelzl@51523
   979
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
hoelzl@51523
   980
by simp (* solved by linordered_ring_less_cancel_factor simproc *)
hoelzl@51523
   981
hoelzl@51523
   982
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
hoelzl@51523
   983
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
hoelzl@51523
   984
hoelzl@51523
   985
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
hoelzl@51523
   986
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
hoelzl@51523
   987
hoelzl@51523
   988
wenzelm@60758
   989
subsection \<open>Embedding numbers into the Reals\<close>
hoelzl@51523
   990
hoelzl@51523
   991
abbreviation
hoelzl@51523
   992
  real_of_nat :: "nat \<Rightarrow> real"
hoelzl@51523
   993
where
hoelzl@51523
   994
  "real_of_nat \<equiv> of_nat"
hoelzl@51523
   995
hoelzl@51523
   996
abbreviation
lp15@61609
   997
  real :: "nat \<Rightarrow> real"
lp15@61609
   998
where
lp15@61609
   999
  "real \<equiv> of_nat"
lp15@61609
  1000
lp15@61609
  1001
abbreviation
hoelzl@51523
  1002
  real_of_int :: "int \<Rightarrow> real"
hoelzl@51523
  1003
where
hoelzl@51523
  1004
  "real_of_int \<equiv> of_int"
hoelzl@51523
  1005
hoelzl@51523
  1006
abbreviation
hoelzl@51523
  1007
  real_of_rat :: "rat \<Rightarrow> real"
hoelzl@51523
  1008
where
hoelzl@51523
  1009
  "real_of_rat \<equiv> of_rat"
hoelzl@51523
  1010
hoelzl@51523
  1011
declare [[coercion_enabled]]
hoelzl@59000
  1012
hoelzl@59000
  1013
declare [[coercion "of_nat :: nat \<Rightarrow> int"]]
lp15@61609
  1014
declare [[coercion "of_nat :: nat \<Rightarrow> real"]]
lp15@61609
  1015
declare [[coercion "of_int :: int \<Rightarrow> real"]]
hoelzl@59000
  1016
hoelzl@59000
  1017
(* We do not add rat to the coerced types, this has often unpleasant side effects when writing
hoelzl@59000
  1018
inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)
hoelzl@51523
  1019
hoelzl@51523
  1020
declare [[coercion_map map]]
hoelzl@59000
  1021
declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]
hoelzl@59000
  1022
declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]
hoelzl@51523
  1023
lp15@61609
  1024
declare of_int_eq_0_iff [algebra, presburger]
lp15@61609
  1025
declare of_int_eq_iff [iff, algebra, presburger] (*FIXME*)
lp15@61609
  1026
declare of_int_eq_iff [iff, algebra, presburger] (*FIXME*)
lp15@61609
  1027
declare of_int_less_iff [iff, algebra, presburger] (*FIXME*)
lp15@61609
  1028
declare of_int_le_iff [iff, algebra, presburger] (*FIXME*)
hoelzl@51523
  1029
lp15@61609
  1030
declare of_int_0_less_iff [presburger]
lp15@61609
  1031
declare of_int_0_le_iff [presburger]
lp15@61609
  1032
declare of_int_less_0_iff [presburger]
lp15@61609
  1033
declare of_int_le_0_iff [presburger]
hoelzl@51523
  1034
lp15@61609
  1035
lemma real_of_int_abs [simp]: "real_of_int (abs x) = abs(real_of_int x)"
lp15@61609
  1036
  by (auto simp add: abs_if)
hoelzl@51523
  1037
lp15@61609
  1038
lemma int_less_real_le: "(n < m) = (real_of_int n + 1 <= real_of_int m)"
lp15@61609
  1039
proof -
lp15@61609
  1040
  have "(0::real) \<le> 1"
lp15@61609
  1041
    by (metis less_eq_real_def zero_less_one)
lp15@61609
  1042
  thus ?thesis
lp15@61609
  1043
    by (metis floor_unique less_add_one less_imp_le not_less of_int_le_iff order_trans)
lp15@61609
  1044
qed
hoelzl@51523
  1045
lp15@61609
  1046
lemma int_le_real_less: "(n \<le> m) = (real_of_int n < real_of_int m + 1)"
lp15@61609
  1047
  by (meson int_less_real_le not_le)
hoelzl@51523
  1048
hoelzl@51523
  1049
lp15@61609
  1050
lemma real_of_int_div_aux: "(real_of_int x) / (real_of_int d) =
lp15@61609
  1051
    real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)"
hoelzl@51523
  1052
proof -
hoelzl@51523
  1053
  have "x = (x div d) * d + x mod d"
hoelzl@51523
  1054
    by auto
lp15@61609
  1055
  then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)"
lp15@61609
  1056
    by (metis of_int_add of_int_mult)
lp15@61609
  1057
  then have "real_of_int x / real_of_int d = ... / real_of_int d"
hoelzl@51523
  1058
    by simp
hoelzl@51523
  1059
  then show ?thesis
hoelzl@51523
  1060
    by (auto simp add: add_divide_distrib algebra_simps)
hoelzl@51523
  1061
qed
hoelzl@51523
  1062
haftmann@58834
  1063
lemma real_of_int_div:
haftmann@58834
  1064
  fixes d n :: int
lp15@61609
  1065
  shows "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d"
haftmann@58834
  1066
  by (simp add: real_of_int_div_aux)
hoelzl@51523
  1067
hoelzl@51523
  1068
lemma real_of_int_div2:
lp15@61609
  1069
  "0 <= real_of_int n / real_of_int x - real_of_int (n div x)"
lp15@61609
  1070
  apply (case_tac "x = 0", simp)
hoelzl@51523
  1071
  apply (case_tac "0 < x")
lp15@61609
  1072
   apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
lp15@61609
  1073
  apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
lp15@61609
  1074
  done
hoelzl@51523
  1075
hoelzl@51523
  1076
lemma real_of_int_div3:
lp15@61609
  1077
  "real_of_int (n::int) / real_of_int (x) - real_of_int (n div x) <= 1"
hoelzl@51523
  1078
  apply (simp add: algebra_simps)
hoelzl@51523
  1079
  apply (subst real_of_int_div_aux)
hoelzl@51523
  1080
  apply (auto simp add: divide_le_eq intro: order_less_imp_le)
hoelzl@51523
  1081
done
hoelzl@51523
  1082
lp15@61609
  1083
lemma real_of_int_div4: "real_of_int (n div x) <= real_of_int (n::int) / real_of_int x"
hoelzl@51523
  1084
by (insert real_of_int_div2 [of n x], simp)
hoelzl@51523
  1085
hoelzl@51523
  1086
wenzelm@60758
  1087
subsection\<open>Embedding the Naturals into the Reals\<close>
hoelzl@51523
  1088
lp15@61609
  1089
declare of_nat_less_iff [iff] (*FIXME*)
lp15@61609
  1090
declare of_nat_le_iff [iff] (*FIXME*)
lp15@61609
  1091
declare of_nat_0_le_iff [iff] (*FIXME*)
lp15@61609
  1092
declare of_nat_less_iff [iff] (*FIXME*)
lp15@61609
  1093
declare of_nat_less_iff [iff] (*FIXME*)
hoelzl@51523
  1094
lp15@61609
  1095
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"  (*NEEDED?*)
lp15@61609
  1096
   using of_nat_0_less_iff by blast
hoelzl@51523
  1097
lp15@61609
  1098
declare of_nat_mult [simp] (*FIXME*)
lp15@61609
  1099
declare of_nat_power [simp] (*FIXME*)
hoelzl@51523
  1100
lp15@61609
  1101
lemmas power_real_of_nat = of_nat_power [symmetric]
hoelzl@51523
  1102
lp15@61609
  1103
declare of_nat_setsum [simp] (*FIXME*)
lp15@61609
  1104
declare of_nat_setprod [simp] (*FIXME*)
hoelzl@51523
  1105
lp15@61609
  1106
lemma real_of_card: "real (card A) = setsum (\<lambda>x.1) A"
lp15@61609
  1107
  by simp
hoelzl@51523
  1108
lp15@61609
  1109
declare of_nat_eq_iff [iff] (*FIXME*)
lp15@61609
  1110
declare of_nat_eq_0_iff [iff] (*FIXME*)
hoelzl@51523
  1111
lp15@61609
  1112
lemma nat_less_real_le: "(n < m) = (real n + 1 \<le> real m)"
lp15@61609
  1113
  by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)
hoelzl@51523
  1114
hoelzl@51523
  1115
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
lp15@61284
  1116
  by (meson nat_less_real_le not_le)
hoelzl@51523
  1117
lp15@61609
  1118
lemma real_of_nat_div_aux: "(real x) / (real d) =
hoelzl@51523
  1119
    real (x div d) + (real (x mod d)) / (real d)"
hoelzl@51523
  1120
proof -
hoelzl@51523
  1121
  have "x = (x div d) * d + x mod d"
hoelzl@51523
  1122
    by auto
hoelzl@51523
  1123
  then have "real x = real (x div d) * real d + real(x mod d)"
lp15@61609
  1124
    by (metis of_nat_add of_nat_mult)
hoelzl@51523
  1125
  then have "real x / real d = \<dots> / real d"
hoelzl@51523
  1126
    by simp
hoelzl@51523
  1127
  then show ?thesis
hoelzl@51523
  1128
    by (auto simp add: add_divide_distrib algebra_simps)
hoelzl@51523
  1129
qed
hoelzl@51523
  1130
lp15@61609
  1131
lemma real_of_nat_div: "d dvd n \<Longrightarrow> real(n div d) = real n / real d"
hoelzl@51523
  1132
  by (subst real_of_nat_div_aux)
hoelzl@51523
  1133
    (auto simp add: dvd_eq_mod_eq_0 [symmetric])
hoelzl@51523
  1134
hoelzl@51523
  1135
lemma real_of_nat_div2:
hoelzl@51523
  1136
  "0 <= real (n::nat) / real (x) - real (n div x)"
hoelzl@51523
  1137
apply (simp add: algebra_simps)
hoelzl@51523
  1138
apply (subst real_of_nat_div_aux)
hoelzl@51523
  1139
apply simp
hoelzl@51523
  1140
done
hoelzl@51523
  1141
hoelzl@51523
  1142
lemma real_of_nat_div3:
hoelzl@51523
  1143
  "real (n::nat) / real (x) - real (n div x) <= 1"
hoelzl@51523
  1144
apply(case_tac "x = 0")
hoelzl@51523
  1145
apply (simp)
hoelzl@51523
  1146
apply (simp add: algebra_simps)
hoelzl@51523
  1147
apply (subst real_of_nat_div_aux)
hoelzl@51523
  1148
apply simp
hoelzl@51523
  1149
done
hoelzl@51523
  1150
lp15@61284
  1151
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
hoelzl@51523
  1152
by (insert real_of_nat_div2 [of n x], simp)
hoelzl@51523
  1153
lp15@61609
  1154
lemma of_nat_nat [simp]: "0 <= x ==> real(nat x) = real x"
lp15@61609
  1155
  by force
hoelzl@51523
  1156
wenzelm@60758
  1157
subsection \<open>The Archimedean Property of the Reals\<close>
hoelzl@51523
  1158
lp15@61609
  1159
lemmas reals_Archimedean = ex_inverse_of_nat_Suc_less  (*FIXME*)
lp15@61609
  1160
lemmas reals_Archimedean2 = ex_less_of_nat
hoelzl@51523
  1161
hoelzl@51523
  1162
lemma reals_Archimedean3:
hoelzl@51523
  1163
  assumes x_greater_zero: "0 < x"
lp15@61609
  1164
  shows "\<forall>y. \<exists>n. y < real n * x"
lp15@61609
  1165
  using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)
hoelzl@51523
  1166
hoelzl@51523
  1167
wenzelm@60758
  1168
subsection\<open>Rationals\<close>
hoelzl@51523
  1169
hoelzl@51523
  1170
lemma Rats_eq_int_div_int:
lp15@61609
  1171
  "\<rat> = { real_of_int i / real_of_int j |i j. j \<noteq> 0}" (is "_ = ?S")
hoelzl@51523
  1172
proof
hoelzl@51523
  1173
  show "\<rat> \<subseteq> ?S"
hoelzl@51523
  1174
  proof
hoelzl@51523
  1175
    fix x::real assume "x : \<rat>"
hoelzl@51523
  1176
    then obtain r where "x = of_rat r" unfolding Rats_def ..
hoelzl@51523
  1177
    have "of_rat r : ?S"
lp15@61609
  1178
      by (cases r) (auto simp add:of_rat_rat)
wenzelm@60758
  1179
    thus "x : ?S" using \<open>x = of_rat r\<close> by simp
hoelzl@51523
  1180
  qed
hoelzl@51523
  1181
next
hoelzl@51523
  1182
  show "?S \<subseteq> \<rat>"
hoelzl@51523
  1183
  proof(auto simp:Rats_def)
hoelzl@51523
  1184
    fix i j :: int assume "j \<noteq> 0"
lp15@61609
  1185
    hence "real_of_int i / real_of_int j = of_rat(Fract i j)"
lp15@61609
  1186
      by (simp add: of_rat_rat)
lp15@61609
  1187
    thus "real_of_int i / real_of_int j \<in> range of_rat" by blast
hoelzl@51523
  1188
  qed
hoelzl@51523
  1189
qed
hoelzl@51523
  1190
hoelzl@51523
  1191
lemma Rats_eq_int_div_nat:
lp15@61609
  1192
  "\<rat> = { real_of_int i / real n |i n. n \<noteq> 0}"
hoelzl@51523
  1193
proof(auto simp:Rats_eq_int_div_int)
hoelzl@51523
  1194
  fix i j::int assume "j \<noteq> 0"
lp15@61609
  1195
  show "EX (i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i'/real n \<and> 0<n"
hoelzl@51523
  1196
  proof cases
hoelzl@51523
  1197
    assume "j>0"
lp15@61609
  1198
    hence "real_of_int i / real_of_int j = real_of_int i/real(nat j) \<and> 0<nat j"
lp15@61609
  1199
      by (simp add: of_nat_nat)
hoelzl@51523
  1200
    thus ?thesis by blast
hoelzl@51523
  1201
  next
hoelzl@51523
  1202
    assume "~ j>0"
lp15@61609
  1203
    hence "real_of_int i / real_of_int j = real_of_int(-i) / real(nat(-j)) \<and> 0<nat(-j)" using \<open>j\<noteq>0\<close>
lp15@61609
  1204
      by (simp add: of_nat_nat)
hoelzl@51523
  1205
    thus ?thesis by blast
hoelzl@51523
  1206
  qed
hoelzl@51523
  1207
next
hoelzl@51523
  1208
  fix i::int and n::nat assume "0 < n"
lp15@61609
  1209
  hence "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0" by simp
lp15@61609
  1210
  thus "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int j \<and> j \<noteq> 0" by blast
hoelzl@51523
  1211
qed
hoelzl@51523
  1212
hoelzl@51523
  1213
lemma Rats_abs_nat_div_natE:
hoelzl@51523
  1214
  assumes "x \<in> \<rat>"
hoelzl@51523
  1215
  obtains m n :: nat
hoelzl@51523
  1216
  where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
hoelzl@51523
  1217
proof -
lp15@61609
  1218
  from \<open>x \<in> \<rat>\<close> obtain i::int and n::nat where "n \<noteq> 0" and "x = real_of_int i / real n"
hoelzl@51523
  1219
    by(auto simp add: Rats_eq_int_div_nat)
lp15@61609
  1220
  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by (simp add: of_nat_nat)
hoelzl@51523
  1221
  then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
hoelzl@51523
  1222
  let ?gcd = "gcd m n"
wenzelm@60758
  1223
  from \<open>n\<noteq>0\<close> have gcd: "?gcd \<noteq> 0" by simp
hoelzl@51523
  1224
  let ?k = "m div ?gcd"
hoelzl@51523
  1225
  let ?l = "n div ?gcd"
hoelzl@51523
  1226
  let ?gcd' = "gcd ?k ?l"
hoelzl@51523
  1227
  have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
hoelzl@51523
  1228
    by (rule dvd_mult_div_cancel)
hoelzl@51523
  1229
  have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
hoelzl@51523
  1230
    by (rule dvd_mult_div_cancel)
wenzelm@60758
  1231
  from \<open>n \<noteq> 0\<close> and gcd_l
haftmann@58834
  1232
  have "?gcd * ?l \<noteq> 0" by simp
lp15@61284
  1233
  then have "?l \<noteq> 0" by (blast dest!: mult_not_zero)
hoelzl@51523
  1234
  moreover
hoelzl@51523
  1235
  have "\<bar>x\<bar> = real ?k / real ?l"
hoelzl@51523
  1236
  proof -
lp15@61609
  1237
    from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
lp15@61609
  1238
      by (simp add: real_of_nat_div)
hoelzl@51523
  1239
    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
hoelzl@51523
  1240
    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
hoelzl@51523
  1241
    finally show ?thesis ..
hoelzl@51523
  1242
  qed
hoelzl@51523
  1243
  moreover
hoelzl@51523
  1244
  have "?gcd' = 1"
hoelzl@51523
  1245
  proof -
hoelzl@51523
  1246
    have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
hoelzl@51523
  1247
      by (rule gcd_mult_distrib_nat)
hoelzl@51523
  1248
    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
hoelzl@51523
  1249
    with gcd show ?thesis by auto
hoelzl@51523
  1250
  qed
hoelzl@51523
  1251
  ultimately show ?thesis ..
hoelzl@51523
  1252
qed
hoelzl@51523
  1253
wenzelm@60758
  1254
subsection\<open>Density of the Rational Reals in the Reals\<close>
hoelzl@51523
  1255
wenzelm@60758
  1256
text\<open>This density proof is due to Stefan Richter and was ported by TN.  The
hoelzl@51523
  1257
original source is \emph{Real Analysis} by H.L. Royden.
wenzelm@60758
  1258
It employs the Archimedean property of the reals.\<close>
hoelzl@51523
  1259
hoelzl@51523
  1260
lemma Rats_dense_in_real:
hoelzl@51523
  1261
  fixes x :: real
hoelzl@51523
  1262
  assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"
hoelzl@51523
  1263
proof -
wenzelm@60758
  1264
  from \<open>x<y\<close> have "0 < y-x" by simp
lp15@61284
  1265
  with reals_Archimedean obtain q::nat
lp15@61609
  1266
    where q: "inverse (real q) < y-x" and "0 < q" by blast
hoelzl@51523
  1267
  def p \<equiv> "ceiling (y * real q) - 1"
hoelzl@51523
  1268
  def r \<equiv> "of_int p / real q"
hoelzl@51523
  1269
  from q have "x < y - inverse (real q)" by simp
hoelzl@51523
  1270
  also have "y - inverse (real q) \<le> r"
hoelzl@51523
  1271
    unfolding r_def p_def
wenzelm@60758
  1272
    by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling \<open>0 < q\<close>)
hoelzl@51523
  1273
  finally have "x < r" .
hoelzl@51523
  1274
  moreover have "r < y"
hoelzl@51523
  1275
    unfolding r_def p_def
wenzelm@60758
  1276
    by (simp add: divide_less_eq diff_less_eq \<open>0 < q\<close>
hoelzl@51523
  1277
      less_ceiling_iff [symmetric])
hoelzl@51523
  1278
  moreover from r_def have "r \<in> \<rat>" by simp
hoelzl@51523
  1279
  ultimately show ?thesis by fast
hoelzl@51523
  1280
qed
hoelzl@51523
  1281
hoelzl@57447
  1282
lemma of_rat_dense:
hoelzl@57447
  1283
  fixes x y :: real
hoelzl@57447
  1284
  assumes "x < y"
hoelzl@57447
  1285
  shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"
wenzelm@60758
  1286
using Rats_dense_in_real [OF \<open>x < y\<close>]
hoelzl@57447
  1287
by (auto elim: Rats_cases)
hoelzl@51523
  1288
hoelzl@51523
  1289
wenzelm@60758
  1290
subsection\<open>Numerals and Arithmetic\<close>
hoelzl@51523
  1291
lp15@61609
  1292
lemma [code_abbrev]:   (*FIXME*)
hoelzl@51523
  1293
  "real_of_int (numeral k) = numeral k"
haftmann@54489
  1294
  "real_of_int (- numeral k) = - numeral k"
hoelzl@51523
  1295
  by simp_all
hoelzl@51523
  1296
hoelzl@51523
  1297
lp15@61609
  1298
  (*FIXME*)
wenzelm@60758
  1299
declaration \<open>
lp15@61609
  1300
  K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
lp15@61609
  1301
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
lp15@61609
  1302
  #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
lp15@61609
  1303
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
lp15@61609
  1304
  #> Lin_Arith.add_simps [@{thm of_nat_0}, @{thm of_nat_Suc}, @{thm of_nat_add},
lp15@61609
  1305
      @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1},
lp15@61609
  1306
      @{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff},
lp15@61609
  1307
      @{thm of_int_mult}, @{thm of_int_of_nat_eq},
lp15@61609
  1308
      @{thm of_nat_numeral}, @{thm int_numeral}, @{thm of_int_neg_numeral}]
hoelzl@58040
  1309
  #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
hoelzl@58040
  1310
  #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
wenzelm@60758
  1311
\<close>
hoelzl@51523
  1312
wenzelm@60758
  1313
subsection\<open>Simprules combining x+y and 0: ARE THEY NEEDED?\<close>
hoelzl@51523
  1314
lp15@61284
  1315
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
hoelzl@51523
  1316
by arith
hoelzl@51523
  1317
hoelzl@51523
  1318
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
hoelzl@51523
  1319
by auto
hoelzl@51523
  1320
hoelzl@51523
  1321
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
hoelzl@51523
  1322
by auto
hoelzl@51523
  1323
hoelzl@51523
  1324
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
hoelzl@51523
  1325
by auto
hoelzl@51523
  1326
hoelzl@51523
  1327
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
hoelzl@51523
  1328
by auto
hoelzl@51523
  1329
wenzelm@60758
  1330
subsection \<open>Lemmas about powers\<close>
hoelzl@51523
  1331
wenzelm@60758
  1332
text \<open>FIXME: declare this in Rings.thy or not at all\<close>
hoelzl@51523
  1333
declare abs_mult_self [simp]
hoelzl@51523
  1334
hoelzl@51523
  1335
(* used by Import/HOL/real.imp *)
hoelzl@51523
  1336
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
lp15@61609
  1337
  by simp
hoelzl@51523
  1338
lp15@61609
  1339
text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>
hoelzl@51523
  1340
lemma realpow_Suc_le_self:
hoelzl@51523
  1341
  fixes r :: "'a::linordered_semidom"
hoelzl@51523
  1342
  shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
hoelzl@51523
  1343
by (insert power_decreasing [of 1 "Suc n" r], simp)
hoelzl@51523
  1344
lp15@61609
  1345
text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>
hoelzl@51523
  1346
lemma realpow_minus_mult:
hoelzl@51523
  1347
  fixes x :: "'a::monoid_mult"
hoelzl@51523
  1348
  shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
lp15@60162
  1349
by (simp add: power_Suc power_commutes split add: nat_diff_split)
hoelzl@51523
  1350
wenzelm@60758
  1351
text \<open>FIXME: declare this [simp] for all types, or not at all\<close>
lp15@61609
  1352
declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
hoelzl@51523
  1353
hoelzl@51523
  1354
lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
hoelzl@51523
  1355
by (rule_tac y = 0 in order_trans, auto)
hoelzl@51523
  1356
wenzelm@53076
  1357
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
lp15@61609
  1358
  by (auto simp add: power2_eq_square)
hoelzl@51523
  1359
immler@58983
  1360
lemma numeral_power_eq_real_of_int_cancel_iff[simp]:
lp15@61609
  1361
     "numeral x ^ n = real_of_int (y::int) \<longleftrightarrow> numeral x ^ n = y"
lp15@61609
  1362
  by (metis of_int_eq_iff of_int_numeral of_int_power)
immler@58983
  1363
immler@58983
  1364
lemma real_of_int_eq_numeral_power_cancel_iff[simp]:
lp15@61609
  1365
     "real_of_int (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
immler@58983
  1366
  using numeral_power_eq_real_of_int_cancel_iff[of x n y]
immler@58983
  1367
  by metis
immler@58983
  1368
immler@58983
  1369
lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:
lp15@61609
  1370
     "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"
lp15@61609
  1371
  using of_nat_eq_iff by fastforce
immler@58983
  1372
immler@58983
  1373
lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:
immler@58983
  1374
  "real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
immler@58983
  1375
  using numeral_power_eq_real_of_nat_cancel_iff[of x n y]
immler@58983
  1376
  by metis
immler@58983
  1377
hoelzl@51523
  1378
lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
hoelzl@51523
  1379
  "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
lp15@61609
  1380
by (metis of_nat_le_iff of_nat_numeral of_nat_power)
hoelzl@51523
  1381
hoelzl@51523
  1382
lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
hoelzl@51523
  1383
  "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
lp15@61609
  1384
by (metis of_nat_le_iff of_nat_numeral of_nat_power)
hoelzl@51523
  1385
hoelzl@51523
  1386
lemma numeral_power_le_real_of_int_cancel_iff[simp]:
lp15@61609
  1387
    "(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
lp15@61609
  1388
  by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power)
hoelzl@51523
  1389
hoelzl@51523
  1390
lemma real_of_int_le_numeral_power_cancel_iff[simp]:
lp15@61609
  1391
    "real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
lp15@61609
  1392
  by (metis floor_of_int le_floor_iff of_int_numeral of_int_power)
hoelzl@51523
  1393
immler@58983
  1394
lemma numeral_power_less_real_of_nat_cancel_iff[simp]:
lp15@61609
  1395
    "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
lp15@61609
  1396
  by (metis of_nat_less_iff of_nat_numeral of_nat_power)
immler@58983
  1397
immler@58983
  1398
lemma real_of_nat_less_numeral_power_cancel_iff[simp]:
immler@58983
  1399
  "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"
lp15@61609
  1400
by (metis of_nat_less_iff of_nat_numeral of_nat_power)
immler@58983
  1401
immler@58983
  1402
lemma numeral_power_less_real_of_int_cancel_iff[simp]:
lp15@61609
  1403
    "(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a"
lp15@61609
  1404
  by (meson not_less real_of_int_le_numeral_power_cancel_iff)
immler@58983
  1405
immler@58983
  1406
lemma real_of_int_less_numeral_power_cancel_iff[simp]:
lp15@61609
  1407
     "real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
lp15@61609
  1408
  by (meson not_less numeral_power_le_real_of_int_cancel_iff)
immler@58983
  1409
hoelzl@51523
  1410
lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
lp15@61609
  1411
    "(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
lp15@61609
  1412
  by (metis of_int_le_iff of_int_neg_numeral of_int_power)
hoelzl@51523
  1413
hoelzl@51523
  1414
lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
lp15@61609
  1415
     "real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
lp15@61609
  1416
  by (metis of_int_le_iff of_int_neg_numeral of_int_power)
hoelzl@51523
  1417
hoelzl@56889
  1418
wenzelm@60758
  1419
subsection\<open>Density of the Reals\<close>
hoelzl@51523
  1420
hoelzl@51523
  1421
lemma real_lbound_gt_zero:
hoelzl@51523
  1422
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
hoelzl@51523
  1423
apply (rule_tac x = " (min d1 d2) /2" in exI)
hoelzl@51523
  1424
apply (simp add: min_def)
hoelzl@51523
  1425
done
hoelzl@51523
  1426
hoelzl@51523
  1427
wenzelm@60758
  1428
text\<open>Similar results are proved in @{text Fields}\<close>
hoelzl@51523
  1429
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
hoelzl@51523
  1430
  by auto
hoelzl@51523
  1431
hoelzl@51523
  1432
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
hoelzl@51523
  1433
  by auto
hoelzl@51523
  1434
hoelzl@51523
  1435
lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"
hoelzl@51523
  1436
  by simp
hoelzl@51523
  1437
wenzelm@60758
  1438
subsection\<open>Absolute Value Function for the Reals\<close>
hoelzl@51523
  1439
hoelzl@51523
  1440
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
hoelzl@51523
  1441
by (simp add: abs_if)
hoelzl@51523
  1442
hoelzl@51523
  1443
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
hoelzl@51523
  1444
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
hoelzl@51523
  1445
by (force simp add: abs_le_iff)
hoelzl@51523
  1446
hoelzl@51523
  1447
lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
hoelzl@51523
  1448
by (simp add: abs_if)
hoelzl@51523
  1449
hoelzl@51523
  1450
lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
hoelzl@51523
  1451
by simp
lp15@61284
  1452
hoelzl@51523
  1453
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
hoelzl@51523
  1454
by simp
hoelzl@51523
  1455
hoelzl@51523
  1456
wenzelm@60758
  1457
subsection\<open>Floor and Ceiling Functions from the Reals to the Integers\<close>
hoelzl@51523
  1458
lp15@61609
  1459
(* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)
hoelzl@51523
  1460
hoelzl@56889
  1461
lemma real_of_nat_less_numeral_iff [simp]:
lp15@61609
  1462
     "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"
lp15@61609
  1463
  by (metis of_nat_less_iff of_nat_numeral)
hoelzl@56889
  1464
hoelzl@56889
  1465
lemma numeral_less_real_of_nat_iff [simp]:
lp15@61609
  1466
     "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
lp15@61609
  1467
  by (metis of_nat_less_iff of_nat_numeral)
hoelzl@56889
  1468
nipkow@59587
  1469
lemma numeral_le_real_of_nat_iff[simp]:
nipkow@59587
  1470
  "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"
nipkow@59587
  1471
by (metis not_le real_of_nat_less_numeral_iff)
nipkow@59587
  1472
lp15@61609
  1473
declare of_int_floor_le [simp] (* FIXME*)
hoelzl@51523
  1474
lp15@61609
  1475
lemma of_int_floor_cancel [simp]:
lp15@61609
  1476
    "(of_int (floor x) = x) = (\<exists>n::int. x = of_int n)"
lp15@61609
  1477
  by (metis floor_of_int)
hoelzl@51523
  1478
lp15@61609
  1479
lemma floor_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> floor x = n"
hoelzl@58040
  1480
  by linarith
hoelzl@51523
  1481
lp15@61609
  1482
lemma floor_eq2: "[| real_of_int n \<le> x; x < real_of_int n + 1 |] ==> floor x = n"
hoelzl@58040
  1483
  by linarith
hoelzl@51523
  1484
hoelzl@51523
  1485
lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
hoelzl@58040
  1486
  by linarith
hoelzl@51523
  1487
hoelzl@51523
  1488
lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
hoelzl@58040
  1489
  by linarith
hoelzl@51523
  1490
lp15@61609
  1491
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int(floor r)"
hoelzl@58040
  1492
  by linarith
hoelzl@51523
  1493
lp15@61609
  1494
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int(floor r)"
hoelzl@58040
  1495
  by linarith
hoelzl@51523
  1496
lp15@61609
  1497
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int(floor r) + 1"
hoelzl@58040
  1498
  by linarith
hoelzl@51523
  1499
lp15@61609
  1500
lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int(floor r) + 1"
hoelzl@58040
  1501
  by linarith
hoelzl@51523
  1502
lp15@61609
  1503
lemma floor_eq_iff: "floor x = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)"
lp15@61609
  1504
by (simp add: floor_unique_iff)
immler@58983
  1505
lp15@61609
  1506
lemma floor_add2[simp]: "floor (of_int a + x) = a + floor x"
lp15@61609
  1507
  by (simp add: add.commute)
hoelzl@51523
  1508
lp15@61609
  1509
lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real_of_int b) = floor a div b"
hoelzl@58788
  1510
proof cases
hoelzl@58788
  1511
  assume "0 < b"
lp15@61609
  1512
  { fix i j :: int assume "real_of_int i \<le> a" "a < 1 + real_of_int i"
lp15@61609
  1513
      "real_of_int j * real_of_int b \<le> a" "a < real_of_int b + real_of_int j * real_of_int b"
lp15@61609
  1514
    then have "i < b + j * b"
lp15@61609
  1515
      by (metis linorder_not_less of_int_add of_int_le_iff of_int_mult order_trans_rules(21))
lp15@61609
  1516
    moreover have "j * b < 1 + i"
lp15@61609
  1517
    proof -
lp15@61609
  1518
      have "real_of_int (j * b) < real_of_int i + 1"
lp15@61609
  1519
        using `a < 1 + real_of_int i` `real_of_int j * real_of_int b \<le> a` by force
lp15@61609
  1520
      thus "j * b < 1 + i"
lp15@61609
  1521
        by linarith
lp15@61609
  1522
    qed
lp15@61609
  1523
    ultimately have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"
hoelzl@58788
  1524
      by (auto simp: field_simps)
hoelzl@58788
  1525
    then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
wenzelm@60758
  1526
      using pos_mod_bound[OF \<open>0<b\<close>, of i] pos_mod_sign[OF \<open>0<b\<close>, of i] by linarith+
hoelzl@58788
  1527
    then have "j = i div b"
lp15@61609
  1528
      using \<open>0 < b\<close> unfolding mult_less_cancel_right by auto
lp15@61609
  1529
  }
wenzelm@60758
  1530
  with \<open>0 < b\<close> show ?thesis
hoelzl@58788
  1531
    by (auto split: floor_split simp: field_simps)
hoelzl@58788
  1532
qed auto
hoelzl@58788
  1533
hoelzl@58097
  1534
lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
lp15@61609
  1535
  by (metis floor_divide_of_int_eq of_int_numeral)
hoelzl@58097
  1536
hoelzl@58097
  1537
lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
lp15@61609
  1538
  by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)
hoelzl@51523
  1539
hoelzl@51523
  1540
lemma real_of_int_ceiling_cancel [simp]:
lp15@61609
  1541
     "(real_of_int (ceiling x) = x) = (\<exists>n::int. x = real_of_int n)"
lp15@61609
  1542
  using ceiling_of_int by metis
hoelzl@51523
  1543
lp15@61609
  1544
lemma ceiling_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> ceiling x = n + 1"
hoelzl@58040
  1545
  by linarith
hoelzl@51523
  1546
lp15@61609
  1547
lemma ceiling_eq2: "[| real_of_int n < x; x \<le> real_of_int n + 1 |] ==> ceiling x = n + 1"
hoelzl@58040
  1548
  by linarith
hoelzl@51523
  1549
lp15@61609
  1550
lemma real_of_int_ceiling_diff_one_le [simp]: "real_of_int (ceiling r) - 1 \<le> r"
hoelzl@58040
  1551
  by linarith
hoelzl@51523
  1552
lp15@61609
  1553
lemma real_of_int_ceiling_le_add_one [simp]: "real_of_int (ceiling r) \<le> r + 1"
hoelzl@58040
  1554
  by linarith
hoelzl@51523
  1555
lp15@61609
  1556
lemma ceiling_le: "x <= real_of_int a ==> ceiling x <= a"
hoelzl@58040
  1557
  by linarith
hoelzl@51523
  1558
lp15@61609
  1559
lemma ceiling_le_real: "ceiling x <= a ==> x <= real_of_int a"
hoelzl@58040
  1560
  by linarith
hoelzl@51523
  1561
lp15@61609
  1562
lemma ceiling_divide_eq_div: "\<lceil>real_of_int a / real_of_int b\<rceil> = - (- a div b)"
lp15@61609
  1563
  by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)
hoelzl@58097
  1564
hoelzl@58097
  1565
lemma ceiling_divide_eq_div_numeral [simp]:
hoelzl@58097
  1566
  "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"
hoelzl@58097
  1567
  using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp
hoelzl@58097
  1568
hoelzl@58097
  1569
lemma ceiling_minus_divide_eq_div_numeral [simp]:
hoelzl@58097
  1570
  "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"
hoelzl@58097
  1571
  using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp
hoelzl@51523
  1572
wenzelm@60758
  1573
text\<open>The following lemmas are remnants of the erstwhile functions natfloor
wenzelm@60758
  1574
and natceiling.\<close>
hoelzl@58040
  1575
nipkow@59587
  1576
lemma nat_floor_neg: "(x::real) <= 0 ==> nat(floor x) = 0"
hoelzl@58040
  1577
  by linarith
hoelzl@51523
  1578
nipkow@59587
  1579
lemma le_nat_floor: "real x <= a ==> x <= nat(floor a)"
hoelzl@58040
  1580
  by linarith
hoelzl@51523
  1581
nipkow@59587
  1582
lemma le_mult_nat_floor:
nipkow@59587
  1583
  shows "nat(floor a) * nat(floor b) \<le> nat(floor (a * b))"
nipkow@59587
  1584
  by (cases "0 <= a & 0 <= b")
nipkow@59587
  1585
     (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
hoelzl@51523
  1586
lp15@61609
  1587
lemma nat_ceiling_le_eq [simp]: "(nat(ceiling x) <= a) = (x <= real a)"
hoelzl@58040
  1588
  by linarith
hoelzl@51523
  1589
nipkow@59587
  1590
lemma real_nat_ceiling_ge: "x <= real(nat(ceiling x))"
hoelzl@58040
  1591
  by linarith
hoelzl@51523
  1592
hoelzl@57447
  1593
lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
nipkow@59587
  1594
  by (auto intro!: bexI[of _ "of_nat (nat(ceiling x))"]) linarith
hoelzl@57275
  1595
hoelzl@57447
  1596
lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"
hoelzl@57447
  1597
  apply (auto intro!: bexI[of _ "of_int (floor x - 1)"])
hoelzl@57447
  1598
  apply (rule less_le_trans[OF _ of_int_floor_le])
hoelzl@57447
  1599
  apply simp
hoelzl@57447
  1600
  done
hoelzl@57447
  1601
wenzelm@60758
  1602
subsection \<open>Exponentiation with floor\<close>
hoelzl@51523
  1603
hoelzl@51523
  1604
lemma floor_power:
lp15@61609
  1605
  assumes "x = real_of_int (floor x)"
hoelzl@51523
  1606
  shows "floor (x ^ n) = floor x ^ n"
hoelzl@51523
  1607
proof -
lp15@61609
  1608
  have "x ^ n = real_of_int (floor x ^ n)"
hoelzl@51523
  1609
    using assms by (induct n arbitrary: x) simp_all
lp15@61609
  1610
  then show ?thesis  by linarith
hoelzl@51523
  1611
qed
nipkow@59587
  1612
(*
hoelzl@51523
  1613
lemma natfloor_power:
hoelzl@51523
  1614
  assumes "x = real (natfloor x)"
hoelzl@51523
  1615
  shows "natfloor (x ^ n) = natfloor x ^ n"
hoelzl@51523
  1616
proof -
hoelzl@51523
  1617
  from assms have "0 \<le> floor x" by auto
lp15@61609
  1618
  note assms[unfolded natfloor_def of_nat_nat[OF `0 \<le> floor x`]]
hoelzl@51523
  1619
  from floor_power[OF this]
hoelzl@51523
  1620
  show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
hoelzl@51523
  1621
    by simp
hoelzl@51523
  1622
qed
nipkow@59587
  1623
*)
lp15@61609
  1624
immler@58983
  1625
lemma floor_numeral_power[simp]:
immler@58983
  1626
  "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
immler@58983
  1627
  by (metis floor_of_int of_int_numeral of_int_power)
immler@58983
  1628
immler@58983
  1629
lemma ceiling_numeral_power[simp]:
immler@58983
  1630
  "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n"
immler@58983
  1631
  by (metis ceiling_of_int of_int_numeral of_int_power)
immler@58983
  1632
hoelzl@51523
  1633
wenzelm@60758
  1634
subsection \<open>Implementation of rational real numbers\<close>
hoelzl@51523
  1635
wenzelm@60758
  1636
text \<open>Formal constructor\<close>
hoelzl@51523
  1637
hoelzl@51523
  1638
definition Ratreal :: "rat \<Rightarrow> real" where
hoelzl@51523
  1639
  [code_abbrev, simp]: "Ratreal = of_rat"
hoelzl@51523
  1640
hoelzl@51523
  1641
code_datatype Ratreal
hoelzl@51523
  1642
hoelzl@51523
  1643
wenzelm@60758
  1644
text \<open>Numerals\<close>
hoelzl@51523
  1645
hoelzl@51523
  1646
lemma [code_abbrev]:
hoelzl@51523
  1647
  "(of_rat (of_int a) :: real) = of_int a"
hoelzl@51523
  1648
  by simp
hoelzl@51523
  1649
hoelzl@51523
  1650
lemma [code_abbrev]:
hoelzl@51523
  1651
  "(of_rat 0 :: real) = 0"
hoelzl@51523
  1652
  by simp
hoelzl@51523
  1653
hoelzl@51523
  1654
lemma [code_abbrev]:
hoelzl@51523
  1655
  "(of_rat 1 :: real) = 1"
hoelzl@51523
  1656
  by simp
hoelzl@51523
  1657
hoelzl@51523
  1658
lemma [code_abbrev]:
haftmann@58134
  1659
  "(of_rat (- 1) :: real) = - 1"
haftmann@58134
  1660
  by simp
haftmann@58134
  1661
haftmann@58134
  1662
lemma [code_abbrev]:
hoelzl@51523
  1663
  "(of_rat (numeral k) :: real) = numeral k"
hoelzl@51523
  1664
  by simp
hoelzl@51523
  1665
hoelzl@51523
  1666
lemma [code_abbrev]:
haftmann@54489
  1667
  "(of_rat (- numeral k) :: real) = - numeral k"
hoelzl@51523
  1668
  by simp
hoelzl@51523
  1669
hoelzl@51523
  1670
lemma [code_post]:
hoelzl@51523
  1671
  "(of_rat (1 / numeral k) :: real) = 1 / numeral k"
haftmann@58134
  1672
  "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
haftmann@58134
  1673
  "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"
haftmann@58134
  1674
  "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"
haftmann@54489
  1675
  by (simp_all add: of_rat_divide of_rat_minus)
hoelzl@51523
  1676
hoelzl@51523
  1677
wenzelm@60758
  1678
text \<open>Operations\<close>
hoelzl@51523
  1679
hoelzl@51523
  1680
lemma zero_real_code [code]:
hoelzl@51523
  1681
  "0 = Ratreal 0"
hoelzl@51523
  1682
by simp
hoelzl@51523
  1683
hoelzl@51523
  1684
lemma one_real_code [code]:
hoelzl@51523
  1685
  "1 = Ratreal 1"
hoelzl@51523
  1686
by simp
hoelzl@51523
  1687
hoelzl@51523
  1688
instantiation real :: equal
hoelzl@51523
  1689
begin
hoelzl@51523
  1690
wenzelm@61076
  1691
definition "HOL.equal (x::real) y \<longleftrightarrow> x - y = 0"
hoelzl@51523
  1692
hoelzl@51523
  1693
instance proof
hoelzl@51523
  1694
qed (simp add: equal_real_def)
hoelzl@51523
  1695
hoelzl@51523
  1696
lemma real_equal_code [code]:
hoelzl@51523
  1697
  "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
hoelzl@51523
  1698
  by (simp add: equal_real_def equal)
hoelzl@51523
  1699
hoelzl@51523
  1700
lemma [code nbe]:
hoelzl@51523
  1701
  "HOL.equal (x::real) x \<longleftrightarrow> True"
hoelzl@51523
  1702
  by (rule equal_refl)
hoelzl@51523
  1703
hoelzl@51523
  1704
end
hoelzl@51523
  1705
hoelzl@51523
  1706
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
hoelzl@51523
  1707
  by (simp add: of_rat_less_eq)
hoelzl@51523
  1708
hoelzl@51523
  1709
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
hoelzl@51523
  1710
  by (simp add: of_rat_less)
hoelzl@51523
  1711
hoelzl@51523
  1712
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
hoelzl@51523
  1713
  by (simp add: of_rat_add)
hoelzl@51523
  1714
hoelzl@51523
  1715
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
hoelzl@51523
  1716
  by (simp add: of_rat_mult)
hoelzl@51523
  1717
hoelzl@51523
  1718
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
hoelzl@51523
  1719
  by (simp add: of_rat_minus)
hoelzl@51523
  1720
hoelzl@51523
  1721
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
hoelzl@51523
  1722
  by (simp add: of_rat_diff)
hoelzl@51523
  1723
hoelzl@51523
  1724
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
hoelzl@51523
  1725
  by (simp add: of_rat_inverse)
lp15@61284
  1726
hoelzl@51523
  1727
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
hoelzl@51523
  1728
  by (simp add: of_rat_divide)
hoelzl@51523
  1729
hoelzl@51523
  1730
lemma real_floor_code [code]: "floor (Ratreal x) = floor x"
hoelzl@51523
  1731
  by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff of_int_floor_le of_rat_of_int_eq real_less_eq_code)
hoelzl@51523
  1732
hoelzl@51523
  1733
wenzelm@60758
  1734
text \<open>Quickcheck\<close>
hoelzl@51523
  1735
hoelzl@51523
  1736
definition (in term_syntax)
hoelzl@51523
  1737
  valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
hoelzl@51523
  1738
  [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"
hoelzl@51523
  1739
hoelzl@51523
  1740
notation fcomp (infixl "\<circ>>" 60)
hoelzl@51523
  1741
notation scomp (infixl "\<circ>\<rightarrow>" 60)
hoelzl@51523
  1742
hoelzl@51523
  1743
instantiation real :: random
hoelzl@51523
  1744
begin
hoelzl@51523
  1745
hoelzl@51523
  1746
definition
hoelzl@51523
  1747
  "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"
hoelzl@51523
  1748
hoelzl@51523
  1749
instance ..
hoelzl@51523
  1750
hoelzl@51523
  1751
end
hoelzl@51523
  1752
hoelzl@51523
  1753
no_notation fcomp (infixl "\<circ>>" 60)
hoelzl@51523
  1754
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)
hoelzl@51523
  1755
hoelzl@51523
  1756
instantiation real :: exhaustive
hoelzl@51523
  1757
begin
hoelzl@51523
  1758
hoelzl@51523
  1759
definition
hoelzl@51523
  1760
  "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
hoelzl@51523
  1761
hoelzl@51523
  1762
instance ..
hoelzl@51523
  1763
hoelzl@51523
  1764
end
hoelzl@51523
  1765
hoelzl@51523
  1766
instantiation real :: full_exhaustive
hoelzl@51523
  1767
begin
hoelzl@51523
  1768
hoelzl@51523
  1769
definition
hoelzl@51523
  1770
  "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
hoelzl@51523
  1771
hoelzl@51523
  1772
instance ..
hoelzl@51523
  1773
hoelzl@51523
  1774
end
hoelzl@51523
  1775
hoelzl@51523
  1776
instantiation real :: narrowing
hoelzl@51523
  1777
begin
hoelzl@51523
  1778
hoelzl@51523
  1779
definition
hoelzl@51523
  1780
  "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"
hoelzl@51523
  1781
hoelzl@51523
  1782
instance ..
hoelzl@51523
  1783
hoelzl@51523
  1784
end
hoelzl@51523
  1785
hoelzl@51523
  1786
wenzelm@60758
  1787
subsection \<open>Setup for Nitpick\<close>
hoelzl@51523
  1788
wenzelm@60758
  1789
declaration \<open>
hoelzl@51523
  1790
  Nitpick_HOL.register_frac_type @{type_name real}
hoelzl@51523
  1791
   [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
hoelzl@51523
  1792
    (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
hoelzl@51523
  1793
    (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
hoelzl@51523
  1794
    (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
hoelzl@51523
  1795
    (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
hoelzl@51523
  1796
    (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
hoelzl@51523
  1797
    (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),
hoelzl@51523
  1798
    (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
wenzelm@60758
  1799
\<close>
hoelzl@51523
  1800
hoelzl@51523
  1801
lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real
hoelzl@51523
  1802
    ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
hoelzl@51523
  1803
    times_real_inst.times_real uminus_real_inst.uminus_real
hoelzl@51523
  1804
    zero_real_inst.zero_real
hoelzl@51523
  1805
blanchet@56078
  1806
wenzelm@60758
  1807
subsection \<open>Setup for SMT\<close>
blanchet@56078
  1808
blanchet@58061
  1809
ML_file "Tools/SMT/smt_real.ML"
blanchet@58061
  1810
ML_file "Tools/SMT/z3_real.ML"
blanchet@56078
  1811
blanchet@58061
  1812
lemma [z3_rule]:
blanchet@56078
  1813
  "0 + (x::real) = x"
blanchet@56078
  1814
  "x + 0 = x"
blanchet@56078
  1815
  "0 * x = 0"
blanchet@56078
  1816
  "1 * x = x"
blanchet@56078
  1817
  "x + y = y + x"
blanchet@56078
  1818
  by auto
hoelzl@51523
  1819
hoelzl@51523
  1820
end