src/HOL/RealDef.thy

replace "setup" with "declaration"

(*  Title       : HOL/RealDef.thy
    Author      : Jacques D. Fleuriot, 1998
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
    Additional contributions by Jeremy Avigad
    Construction of Cauchy Reals by Brian Huffman, 2010
*)

header {* Development of the Reals using Cauchy Sequences *}

theory RealDef
imports Rat
begin

text {*
  This theory contains a formalization of the real numbers as
  equivalence classes of Cauchy sequences of rationals.  See
  \url{HOL/ex/Dedekind_Real.thy} for an alternative construction
  using Dedekind cuts.
*}

subsection {* Preliminary lemmas *}

lemma add_diff_add:
  fixes a b c d :: "'a::ab_group_add"
  shows "(a + c) - (b + d) = (a - b) + (c - d)"
  by simp

lemma minus_diff_minus:
  fixes a b :: "'a::ab_group_add"
  shows "- a - - b = - (a - b)"
  by simp

lemma mult_diff_mult:
  fixes x y a b :: "'a::ring"
  shows "(x * y - a * b) = x * (y - b) + (x - a) * b"
  by (simp add: algebra_simps)

lemma inverse_diff_inverse:
  fixes a b :: "'a::division_ring"
  assumes "a \<noteq> 0" and "b \<noteq> 0"
  shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
  using assms by (simp add: algebra_simps)

lemma obtain_pos_sum:
  fixes r :: rat assumes r: "0 < r"
  obtains s t where "0 < s" and "0 < t" and "r = s + t"
proof
    from r show "0 < r/2" by simp
    from r show "0 < r/2" by simp
    show "r = r/2 + r/2" by simp
qed

subsection {* Sequences that converge to zero *}

definition
  vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
where
  "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"

lemma vanishesI: "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r) \<Longrightarrow> vanishes X"
  unfolding vanishes_def by simp

lemma vanishesD: "\<lbrakk>vanishes X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
  unfolding vanishes_def by simp

lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
  unfolding vanishes_def
  apply (cases "c = 0", auto)
  apply (rule exI [where x="\<bar>c\<bar>"], auto)
  done

lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"
  unfolding vanishes_def by simp

lemma vanishes_add:
  assumes X: "vanishes X" and Y: "vanishes Y"
  shows "vanishes (\<lambda>n. X n + Y n)"
proof (rule vanishesI)
  fix r :: rat assume "0 < r"
  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
    by (rule obtain_pos_sum)
  obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"
    using vanishesD [OF X s] ..
  obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"
    using vanishesD [OF Y t] ..
  have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"
  proof (clarsimp)
    fix n assume n: "i \<le> n" "j \<le> n"
    have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)
    also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)
    finally show "\<bar>X n + Y n\<bar> < r" unfolding r .
  qed
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
qed

lemma vanishes_diff:
  assumes X: "vanishes X" and Y: "vanishes Y"
  shows "vanishes (\<lambda>n. X n - Y n)"
unfolding diff_minus by (intro vanishes_add vanishes_minus X Y)

lemma vanishes_mult_bounded:
  assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
  assumes Y: "vanishes (\<lambda>n. Y n)"
  shows "vanishes (\<lambda>n. X n * Y n)"
proof (rule vanishesI)
  fix r :: rat assume r: "0 < r"
  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
    using X by fast
  obtain b where b: "0 < b" "r = a * b"
  proof
    show "0 < r / a" using r a by (simp add: divide_pos_pos)
    show "r = a * (r / a)" using a by simp
  qed
  obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"
    using vanishesD [OF Y b(1)] ..
  have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"
    by (simp add: b(2) abs_mult mult_strict_mono' a k)
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
qed

subsection {* Cauchy sequences *}

definition
  cauchy :: "(nat \<Rightarrow> rat) set"
where
  "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"

lemma cauchyI:
  "(\<And>r. 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r) \<Longrightarrow> cauchy X"
  unfolding cauchy_def by simp

lemma cauchyD:
  "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
  unfolding cauchy_def by simp

lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
  unfolding cauchy_def by simp

lemma cauchy_add [simp]:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "cauchy (\<lambda>n. X n + Y n)"
proof (rule cauchyI)
  fix r :: rat assume "0 < r"
  then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
    by (rule obtain_pos_sum)
  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
    using cauchyD [OF X s] ..
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
    using cauchyD [OF Y t] ..
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"
  proof (clarsimp)
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
    have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"
      unfolding add_diff_add by (rule abs_triangle_ineq)
    also have "\<dots> < s + t"
      by (rule add_strict_mono, simp_all add: i j *)
    finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
  qed
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
qed

lemma cauchy_minus [simp]:
  assumes X: "cauchy X"
  shows "cauchy (\<lambda>n. - X n)"
using assms unfolding cauchy_def
unfolding minus_diff_minus abs_minus_cancel .

lemma cauchy_diff [simp]:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "cauchy (\<lambda>n. X n - Y n)"
using assms unfolding diff_minus by simp

lemma cauchy_imp_bounded:
  assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
proof -
  obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"
    using cauchyD [OF assms zero_less_one] ..
  show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
  proof (intro exI conjI allI)
    have "0 \<le> \<bar>X 0\<bar>" by simp
    also have "\<bar>X 0\<bar> \<le> Max (abs ` X ` {..k})" by simp
    finally have "0 \<le> Max (abs ` X ` {..k})" .
    thus "0 < Max (abs ` X ` {..k}) + 1" by simp
  next
    fix n :: nat
    show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1"
    proof (rule linorder_le_cases)
      assume "n \<le> k"
      hence "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
      thus "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" by simp
    next
      assume "k \<le> n"
      have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
      also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
        by (rule abs_triangle_ineq)
      also have "\<dots> < Max (abs ` X ` {..k}) + 1"
        by (rule add_le_less_mono, simp, simp add: k `k \<le> n`)
      finally show "\<bar>X n\<bar> < Max (abs ` X ` {..k}) + 1" .
    qed
  qed
qed

lemma cauchy_mult [simp]:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "cauchy (\<lambda>n. X n * Y n)"
proof (rule cauchyI)
  fix r :: rat assume "0 < r"
  then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
    by (rule obtain_pos_sum)
  obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
    using cauchy_imp_bounded [OF X] by fast
  obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
    using cauchy_imp_bounded [OF Y] by fast
  obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
  proof
    show "0 < v/b" using v b(1) by (rule divide_pos_pos)
    show "0 < u/a" using u a(1) by (rule divide_pos_pos)
    show "r = a * (u/a) + (v/b) * b"
      using a(1) b(1) `r = u + v` by simp
  qed
  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
    using cauchyD [OF X s] ..
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
    using cauchyD [OF Y t] ..
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
  proof (clarsimp)
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
    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>"
      unfolding mult_diff_mult ..
    also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
      by (rule abs_triangle_ineq)
    also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
      unfolding abs_mult ..
    also have "\<dots> < a * t + s * b"
      by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
    finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
  qed
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
qed

lemma cauchy_not_vanishes_cases:
  assumes X: "cauchy X"
  assumes nz: "\<not> vanishes X"
  shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"
proof -
  obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"
    using nz unfolding vanishes_def by (auto simp add: not_less)
  obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"
    using `0 < r` by (rule obtain_pos_sum)
  obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
    using cauchyD [OF X s] ..
  obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"
    using r by fast
  have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
    using i `i \<le> k` by auto
  have "X k \<le> - r \<or> r \<le> X k"
    using `r \<le> \<bar>X k\<bar>` by auto
  hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
    unfolding `r = s + t` using k by auto
  hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
  thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
    using t by auto
qed

lemma cauchy_not_vanishes:
  assumes X: "cauchy X"
  assumes nz: "\<not> vanishes X"
  shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"
using cauchy_not_vanishes_cases [OF assms]
by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)

lemma cauchy_inverse [simp]:
  assumes X: "cauchy X"
  assumes nz: "\<not> vanishes X"
  shows "cauchy (\<lambda>n. inverse (X n))"
proof (rule cauchyI)
  fix r :: rat assume "0 < r"
  obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"
    using cauchy_not_vanishes [OF X nz] by fast
  from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto
  obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"
  proof
    show "0 < b * r * b"
      by (simp add: `0 < r` b mult_pos_pos)
    show "r = inverse b * (b * r * b) * inverse b"
      using b by simp
  qed
  obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"
    using cauchyD [OF X s] ..
  have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"
  proof (clarsimp)
    fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
    have "\<bar>inverse (X m) - inverse (X n)\<bar> =
          inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"
      by (simp add: inverse_diff_inverse nz * abs_mult)
    also have "\<dots> < inverse b * s * inverse b"
      by (simp add: mult_strict_mono less_imp_inverse_less
                    mult_pos_pos i j b * s)
    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .
  qed
  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
qed

subsection {* Equivalence relation on Cauchy sequences *}

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

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

lemma sym_realrel: "sym realrel"
  unfolding realrel_def
  by (rule symI, clarify, drule vanishes_minus, simp)

lemma trans_realrel: "trans realrel"
  unfolding realrel_def
  apply (rule transI, clarify)
  apply (drule (1) vanishes_add)
  apply (simp add: algebra_simps)
  done

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

subsection {* The field of real numbers *}

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

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

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

lemma Real_induct [induct type: real]:
  "(\<And>X. cauchy X \<Longrightarrow> P (Real X)) \<Longrightarrow> P x"
  unfolding Real_def
  apply (induct x)
  apply (erule quotientE)
  apply (simp)
  done

lemma real_case_1:
  assumes f: "congruent realrel f"
  assumes X: "cauchy X"
  shows "real_case f (Real X) = f X"
  unfolding real_case_def Real_def
  apply (subst Abs_real_inverse)
  apply (simp add: quotientI X)
  apply (rule the_equality)
  apply clarsimp
  apply (erule congruent.congruent [OF f])
  apply (erule bspec)
  apply simp
  apply (rule refl_onD [OF refl_realrel])
  apply (simp add: X)
  done

lemma real_case_2:
  assumes f: "congruent2 realrel realrel f"
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "real_case (\<lambda>X. real_case (\<lambda>Y. f X Y) (Real Y)) (Real X) = f X Y"
 apply (subst real_case_1 [OF _ X])
  apply (rule congruent.intro)
  apply (subst real_case_1 [OF _ Y])
   apply (rule congruent2_implies_congruent [OF equiv_realrel f])
   apply (simp add: realrel_def)
  apply (subst real_case_1 [OF _ Y])
   apply (rule congruent2_implies_congruent [OF equiv_realrel f])
   apply (simp add: realrel_def)
  apply (erule congruent2.congruent2 [OF f])
  apply (rule refl_onD [OF refl_realrel])
  apply (simp add: Y)
  apply (rule real_case_1 [OF _ Y])
  apply (rule congruent2_implies_congruent [OF equiv_realrel f])
  apply (simp add: X)
  done

lemma eq_Real:
  "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
  unfolding Real_def
  apply (subst Abs_real_inject)
  apply (simp add: quotientI)
  apply (simp add: quotientI)
  apply (simp add: eq_equiv_class_iff [OF equiv_realrel])
  apply (simp add: realrel_def)
  done

lemma add_respects2_realrel:
  "(\<lambda>X Y. Real (\<lambda>n. X n + Y n)) respects2 realrel"
proof (rule congruent2_commuteI [OF equiv_realrel, unfolded mem_Collect_eq])
  fix X Y show "Real (\<lambda>n. X n + Y n) = Real (\<lambda>n. Y n + X n)"
    by (simp add: add_commute)
next
  fix X assume X: "cauchy X"
  fix Y Z assume "(Y, Z) \<in> realrel"
  hence Y: "cauchy Y" and Z: "cauchy Z" and YZ: "vanishes (\<lambda>n. Y n - Z n)"
    unfolding realrel_def by simp_all
  show "Real (\<lambda>n. X n + Y n) = Real (\<lambda>n. X n + Z n)"
  proof (rule eq_Real [THEN iffD2])
    show "cauchy (\<lambda>n. X n + Y n)" using X Y by (rule cauchy_add)
    show "cauchy (\<lambda>n. X n + Z n)" using X Z by (rule cauchy_add)
    show "vanishes (\<lambda>n. (X n + Y n) - (X n + Z n))"
      unfolding add_diff_add using YZ by simp
  qed
qed

lemma minus_respects_realrel:
  "(\<lambda>X. Real (\<lambda>n. - X n)) respects realrel"
proof (rule congruent.intro)
  fix X Y assume "(X, Y) \<in> realrel"
  hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
    unfolding realrel_def by simp_all
  show "Real (\<lambda>n. - X n) = Real (\<lambda>n. - Y n)"
  proof (rule eq_Real [THEN iffD2])
    show "cauchy (\<lambda>n. - X n)" using X by (rule cauchy_minus)
    show "cauchy (\<lambda>n. - Y n)" using Y by (rule cauchy_minus)
    show "vanishes (\<lambda>n. (- X n) - (- Y n))"
      unfolding minus_diff_minus using XY by (rule vanishes_minus)
  qed
qed

lemma mult_respects2_realrel:
  "(\<lambda>X Y. Real (\<lambda>n. X n * Y n)) respects2 realrel"
proof (rule congruent2_commuteI [OF equiv_realrel, unfolded mem_Collect_eq])
  fix X Y
  show "Real (\<lambda>n. X n * Y n) = Real (\<lambda>n. Y n * X n)"
    by (simp add: mult_commute)
next
  fix X assume X: "cauchy X"
  fix Y Z assume "(Y, Z) \<in> realrel"
  hence Y: "cauchy Y" and Z: "cauchy Z" and YZ: "vanishes (\<lambda>n. Y n - Z n)"
    unfolding realrel_def by simp_all
  show "Real (\<lambda>n. X n * Y n) = Real (\<lambda>n. X n * Z n)"
  proof (rule eq_Real [THEN iffD2])
    show "cauchy (\<lambda>n. X n * Y n)" using X Y by (rule cauchy_mult)
    show "cauchy (\<lambda>n. X n * Z n)" using X Z by (rule cauchy_mult)
    have "vanishes (\<lambda>n. X n * (Y n - Z n))"
      by (intro vanishes_mult_bounded cauchy_imp_bounded X YZ)
    thus "vanishes (\<lambda>n. X n * Y n - X n * Z n)"
      by (simp add: right_diff_distrib)
  qed
qed

lemma vanishes_diff_inverse:
  assumes X: "cauchy X" "\<not> vanishes X"
  assumes Y: "cauchy Y" "\<not> vanishes Y"
  assumes XY: "vanishes (\<lambda>n. X n - Y n)"
  shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
proof (rule vanishesI)
  fix r :: rat assume r: "0 < r"
  obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"
    using cauchy_not_vanishes [OF X] by fast
  obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"
    using cauchy_not_vanishes [OF Y] by fast
  obtain s where s: "0 < s" and "inverse a * s * inverse b = r"
  proof
    show "0 < a * r * b"
      using a r b by (simp add: mult_pos_pos)
    show "inverse a * (a * r * b) * inverse b = r"
      using a r b by simp
  qed
  obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"
    using vanishesD [OF XY s] ..
  have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"
  proof (clarsimp)
    fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"
    have "X n \<noteq> 0" and "Y n \<noteq> 0"
      using i j a b n by auto
    hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =
        inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
      by (simp add: inverse_diff_inverse abs_mult)
    also have "\<dots> < inverse a * s * inverse b"
      apply (intro mult_strict_mono' less_imp_inverse_less)
      apply (simp_all add: a b i j k n mult_nonneg_nonneg)
      done
    also note `inverse a * s * inverse b = r`
    finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
  qed
  thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
qed

lemma inverse_respects_realrel:
  "(\<lambda>X. if vanishes X then c else Real (\<lambda>n. inverse (X n))) respects realrel"
    (is "?inv respects realrel")
proof (rule congruent.intro)
  fix X Y assume "(X, Y) \<in> realrel"
  hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
    unfolding realrel_def by simp_all
  have "vanishes X \<longleftrightarrow> vanishes Y"
  proof
    assume "vanishes X"
    from vanishes_diff [OF this XY] show "vanishes Y" by simp
  next
    assume "vanishes Y"
    from vanishes_add [OF this XY] show "vanishes X" by simp
  qed
  thus "?inv X = ?inv Y"
    by (simp add: vanishes_diff_inverse eq_Real X Y XY)
qed

instantiation real :: field_inverse_zero
begin

definition
  "0 = Real (\<lambda>n. 0)"

definition
  "1 = Real (\<lambda>n. 1)"

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

definition
  "- x = real_case (\<lambda>X. Real (\<lambda>n. - X n)) x"

definition
  "x - y = (x::real) + - y"

definition
  "x * y = real_case (\<lambda>X. real_case (\<lambda>Y. Real (\<lambda>n. X n * Y n)) y) x"

definition
  "inverse =
    real_case (\<lambda>X. if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"

definition
  "x / y = (x::real) * inverse y"

lemma add_Real:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
  unfolding plus_real_def
  by (rule real_case_2 [OF add_respects2_realrel X Y])

lemma minus_Real:
  assumes X: "cauchy X"
  shows "- Real X = Real (\<lambda>n. - X n)"
  unfolding uminus_real_def
  by (rule real_case_1 [OF minus_respects_realrel X])

lemma diff_Real:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "Real X - Real Y = Real (\<lambda>n. X n - Y n)"
  unfolding minus_real_def diff_minus
  by (simp add: minus_Real add_Real X Y)

lemma mult_Real:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
  unfolding times_real_def
  by (rule real_case_2 [OF mult_respects2_realrel X Y])

lemma inverse_Real:
  assumes X: "cauchy X"
  shows "inverse (Real X) =
    (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
  unfolding inverse_real_def
  by (rule real_case_1 [OF inverse_respects_realrel X])

instance proof
  fix a b c :: real
  show "a + b = b + a"
    by (induct a, induct b) (simp add: add_Real add_ac)
  show "(a + b) + c = a + (b + c)"
    by (induct a, induct b, induct c) (simp add: add_Real add_ac)
  show "0 + a = a"
    unfolding zero_real_def
    by (induct a) (simp add: add_Real)
  show "- a + a = 0"
    unfolding zero_real_def
    by (induct a) (simp add: minus_Real add_Real)
  show "a - b = a + - b"
    by (rule minus_real_def)
  show "(a * b) * c = a * (b * c)"
    by (induct a, induct b, induct c) (simp add: mult_Real mult_ac)
  show "a * b = b * a"
    by (induct a, induct b) (simp add: mult_Real mult_ac)
  show "1 * a = a"
    unfolding one_real_def
    by (induct a) (simp add: mult_Real)
  show "(a + b) * c = a * c + b * c"
    by (induct a, induct b, induct c)
       (simp add: mult_Real add_Real algebra_simps)
  show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
    unfolding zero_real_def one_real_def
    by (simp add: eq_Real)
  show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
    unfolding zero_real_def one_real_def
    apply (induct a)
    apply (simp add: eq_Real inverse_Real mult_Real)
    apply (rule vanishesI)
    apply (frule (1) cauchy_not_vanishes, clarify)
    apply (rule_tac x=k in exI, clarify)
    apply (drule_tac x=n in spec, simp)
    done
  show "a / b = a * inverse b"
    by (rule divide_real_def)
  show "inverse (0::real) = 0"
    by (simp add: zero_real_def inverse_Real)
qed

end

subsection {* Positive reals *}

definition
  positive :: "real \<Rightarrow> bool"
where
  "positive = real_case (\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"

lemma bool_congruentI:
  assumes sym: "sym r"
  assumes P: "\<And>x y. (x, y) \<in> r \<Longrightarrow> P x \<Longrightarrow> P y"
  shows "P respects r"
apply (rule congruent.intro)
apply (rule iffI)
apply (erule (1) P)
apply (erule (1) P [OF symD [OF sym]])
done

lemma positive_respects_realrel:
  "(\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n) respects realrel"
proof (rule bool_congruentI)
  show "sym realrel" by (rule sym_realrel)
next
  fix X Y assume "(X, Y) \<in> realrel"
  hence XY: "vanishes (\<lambda>n. X n - Y n)"
    unfolding realrel_def by simp_all
  assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
  then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
    by fast
  obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
    using `0 < r` by (rule obtain_pos_sum)
  obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
    using vanishesD [OF XY s] ..
  have "\<forall>n\<ge>max i j. t < Y n"
  proof (clarsimp)
    fix n assume n: "i \<le> n" "j \<le> n"
    have "\<bar>X n - Y n\<bar> < s" and "r < X n"
      using i j n by simp_all
    thus "t < Y n" unfolding r by simp
  qed
  thus "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
qed

lemma positive_Real:
  assumes X: "cauchy X"
  shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
unfolding positive_def
by (rule real_case_1 [OF positive_respects_realrel X])

lemma positive_zero: "\<not> positive 0"
unfolding zero_real_def by (auto simp add: positive_Real)

lemma positive_add:
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"
apply (induct x, induct y, rename_tac Y X)
apply (simp add: add_Real positive_Real)
apply (clarify, rename_tac a b i j)
apply (rule_tac x="a + b" in exI, simp)
apply (rule_tac x="max i j" in exI, clarsimp)
apply (simp add: add_strict_mono)
done

lemma positive_mult:
  "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"
apply (induct x, induct y, rename_tac Y X)
apply (simp add: mult_Real positive_Real)
apply (clarify, rename_tac a b i j)
apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos)
apply (rule_tac x="max i j" in exI, clarsimp)
apply (rule mult_strict_mono, auto)
done

lemma positive_minus:
  "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
apply (induct x, rename_tac X)
apply (simp add: zero_real_def eq_Real minus_Real positive_Real)
apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
done

instantiation real :: linordered_field_inverse_zero
begin

definition
  "x < y \<longleftrightarrow> positive (y - x)"

definition
  "x \<le> (y::real) \<longleftrightarrow> x < y \<or> x = y"

definition
  "abs (a::real) = (if a < 0 then - a else a)"

definition
  "sgn (a::real) = (if a = 0 then 0 else if 0 < a then 1 else - 1)"

instance proof
  fix a b c :: real
  show "\<bar>a\<bar> = (if a < 0 then - a else a)"
    by (rule abs_real_def)
  show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"
    unfolding less_eq_real_def less_real_def
    by (auto, drule (1) positive_add, simp_all add: positive_zero)
  show "a \<le> a"
    unfolding less_eq_real_def by simp
  show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
    unfolding less_eq_real_def less_real_def
    by (auto, drule (1) positive_add, simp add: algebra_simps)
  show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
    unfolding less_eq_real_def less_real_def
    by (auto, drule (1) positive_add, simp add: positive_zero)
  show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
    unfolding less_eq_real_def less_real_def by auto
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
    by (rule sgn_real_def)
  show "a \<le> b \<or> b \<le> a"
    unfolding less_eq_real_def less_real_def
    by (auto dest!: positive_minus)
  show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
    unfolding less_real_def
    by (drule (1) positive_mult, simp add: algebra_simps)
qed

end

instantiation real :: distrib_lattice
begin

definition
  "(inf :: real \<Rightarrow> real \<Rightarrow> real) = min"

definition
  "(sup :: real \<Rightarrow> real \<Rightarrow> real) = max"

instance proof
qed (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)

end

instantiation real :: number_ring
begin

definition
  "(number_of x :: real) = of_int x"

instance proof
qed (rule number_of_real_def)

end

lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
apply (induct x)
apply (simp add: zero_real_def)
apply (simp add: one_real_def add_Real)
done

lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
apply (cases x rule: int_diff_cases)
apply (simp add: of_nat_Real diff_Real)
done

lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
apply (induct x)
apply (simp add: Fract_of_int_quotient of_rat_divide)
apply (simp add: of_int_Real divide_inverse)
apply (simp add: inverse_Real mult_Real)
done

instance real :: archimedean_field
proof
  fix x :: real
  show "\<exists>z. x \<le> of_int z"
    apply (induct x)
    apply (frule cauchy_imp_bounded, clarify)
    apply (rule_tac x="ceiling b + 1" in exI)
    apply (rule less_imp_le)
    apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
    apply (rule_tac x=1 in exI, simp add: algebra_simps)
    apply (rule_tac x=0 in exI, clarsimp)
    apply (rule le_less_trans [OF abs_ge_self])
    apply (rule less_le_trans [OF _ le_of_int_ceiling])
    apply simp
    done
qed

subsection {* Completeness *}

lemma not_positive_Real:
  assumes X: "cauchy X"
  shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"
unfolding positive_Real [OF X]
apply (auto, unfold not_less)
apply (erule obtain_pos_sum)
apply (drule_tac x=s in spec, simp)
apply (drule_tac r=t in cauchyD [OF X], clarify)
apply (drule_tac x=k in spec, clarsimp)
apply (rule_tac x=n in exI, clarify, rename_tac m)
apply (drule_tac x=m in spec, simp)
apply (drule_tac x=n in spec, simp)
apply (drule spec, drule (1) mp, clarify, rename_tac i)
apply (rule_tac x="max i k" in exI, simp)
done

lemma le_Real:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
unfolding not_less [symmetric, where 'a=real] less_real_def
apply (simp add: diff_Real not_positive_Real X Y)
apply (simp add: diff_le_eq add_ac)
done

lemma le_RealI:
  assumes Y: "cauchy Y"
  shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
proof (induct x)
  fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
  hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
    by (simp add: of_rat_Real le_Real)
  {
    fix r :: rat assume "0 < r"
    then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
      by (rule obtain_pos_sum)
    obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
      using cauchyD [OF Y s] ..
    obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
      using le [OF t] ..
    have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
    proof (clarsimp)
      fix n assume n: "i \<le> n" "j \<le> n"
      have "X n \<le> Y i + t" using n j by simp
      moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
      ultimately show "X n \<le> Y n + r" unfolding r by simp
    qed
    hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
  }
  thus "Real X \<le> Real Y"
    by (simp add: of_rat_Real le_Real X Y)
qed

lemma Real_leI:
  assumes X: "cauchy X"
  assumes le: "\<forall>n. of_rat (X n) \<le> y"
  shows "Real X \<le> y"
proof -
  have "- y \<le> - Real X"
    by (simp add: minus_Real X le_RealI of_rat_minus le)
  thus ?thesis by simp
qed

lemma less_RealD:
  assumes Y: "cauchy Y"
  shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])

lemma of_nat_less_two_power:
  "of_nat n < (2::'a::{linordered_idom,number_ring}) ^ n"
apply (induct n)
apply simp
apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
apply (drule (1) add_le_less_mono, simp)
apply simp
done

lemma complete_real:
  fixes S :: "real set"
  assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"
  shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
proof -
  obtain x where x: "x \<in> S" using assms(1) ..
  obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..

  def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"
  obtain a where a: "\<not> P a"
  proof
    have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)
    also have "x - 1 < x" by simp
    finally have "of_int (floor (x - 1)) < x" .
    hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)
    then show "\<not> P (of_int (floor (x - 1)))"
      unfolding P_def of_rat_of_int_eq using x by fast
  qed
  obtain b where b: "P b"
  proof
    show "P (of_int (ceiling z))"
    unfolding P_def of_rat_of_int_eq
    proof
      fix y assume "y \<in> S"
      hence "y \<le> z" using z by simp
      also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)
      finally show "y \<le> of_int (ceiling z)" .
    qed
  qed

  def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"
  def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"
  def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"
  def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"
  def C \<equiv> "\<lambda>n. avg (A n) (B n)"
  have A_0 [simp]: "A 0 = a" unfolding A_def by simp
  have B_0 [simp]: "B 0 = b" unfolding B_def by simp
  have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"
    unfolding A_def B_def C_def bisect_def split_def by simp
  have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"
    unfolding A_def B_def C_def bisect_def split_def by simp

  have width: "\<And>n. B n - A n = (b - a) / 2^n"
    apply (simp add: eq_divide_eq)
    apply (induct_tac n, simp)
    apply (simp add: C_def avg_def algebra_simps)
    done

  have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
    apply (simp add: divide_less_eq)
    apply (subst mult_commute)
    apply (frule_tac y=y in ex_less_of_nat_mult)
    apply clarify
    apply (rule_tac x=n in exI)
    apply (erule less_trans)
    apply (rule mult_strict_right_mono)
    apply (rule le_less_trans [OF _ of_nat_less_two_power])
    apply simp
    apply assumption
    done

  have PA: "\<And>n. \<not> P (A n)"
    by (induct_tac n, simp_all add: a)
  have PB: "\<And>n. P (B n)"
    by (induct_tac n, simp_all add: b)
  have ab: "a < b"
    using a b unfolding P_def
    apply (clarsimp simp add: not_le)
    apply (drule (1) bspec)
    apply (drule (1) less_le_trans)
    apply (simp add: of_rat_less)
    done
  have AB: "\<And>n. A n < B n"
    by (induct_tac n, simp add: ab, simp add: C_def avg_def)
  have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"
    apply (auto simp add: le_less [where 'a=nat])
    apply (erule less_Suc_induct)
    apply (clarsimp simp add: C_def avg_def)
    apply (simp add: add_divide_distrib [symmetric])
    apply (rule AB [THEN less_imp_le])
    apply simp
    done
  have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"
    apply (auto simp add: le_less [where 'a=nat])
    apply (erule less_Suc_induct)
    apply (clarsimp simp add: C_def avg_def)
    apply (simp add: add_divide_distrib [symmetric])
    apply (rule AB [THEN less_imp_le])
    apply simp
    done
  have cauchy_lemma:
    "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"
    apply (rule cauchyI)
    apply (drule twos [where y="b - a"])
    apply (erule exE)
    apply (rule_tac x=n in exI, clarify, rename_tac i j)
    apply (rule_tac y="B n - A n" in le_less_trans) defer
    apply (simp add: width)
    apply (drule_tac x=n in spec)
    apply (frule_tac x=i in spec, drule (1) mp)
    apply (frule_tac x=j in spec, drule (1) mp)
    apply (frule A_mono, drule B_mono)
    apply (frule A_mono, drule B_mono)
    apply arith
    done
  have "cauchy A"
    apply (rule cauchy_lemma [rule_format])
    apply (simp add: A_mono)
    apply (erule order_trans [OF less_imp_le [OF AB] B_mono])
    done
  have "cauchy B"
    apply (rule cauchy_lemma [rule_format])
    apply (simp add: B_mono)
    apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
    done
  have 1: "\<forall>x\<in>S. x \<le> Real B"
  proof
    fix x assume "x \<in> S"
    then show "x \<le> Real B"
      using PB [unfolded P_def] `cauchy B`
      by (simp add: le_RealI)
  qed
  have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
    apply clarify
    apply (erule contrapos_pp)
    apply (simp add: not_le)
    apply (drule less_RealD [OF `cauchy A`], clarify)
    apply (subgoal_tac "\<not> P (A n)")
    apply (simp add: P_def not_le, clarify)
    apply (erule rev_bexI)
    apply (erule (1) less_trans)
    apply (simp add: PA)
    done
  have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
  proof (rule vanishesI)
    fix r :: rat assume "0 < r"
    then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
      using twos by fast
    have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
    proof (clarify)
      fix n assume n: "k \<le> n"
      have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
        by simp
      also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
        using n by (simp add: divide_left_mono mult_pos_pos)
      also note k
      finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
    qed
    thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
  qed
  hence 3: "Real B = Real A"
    by (simp add: eq_Real `cauchy A` `cauchy B` width)
  show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
    using 1 2 3 by (rule_tac x="Real B" in exI, simp)
qed

subsection {* Hiding implementation details *}

hide_const (open) vanishes cauchy positive Real real_case

declare Real_induct [induct del]
declare Abs_real_induct [induct del]
declare Abs_real_cases [cases del]

subsection {* Legacy theorem names *}

text {* TODO: Could we have a way to mark theorem names as deprecated,
and have Isabelle issue a warning and display the preferred name? *}

lemmas real_diff_def = minus_real_def [of "r" "s", standard]
lemmas real_divide_def = divide_real_def [of "R" "S", standard]
lemmas real_less_def = less_le [of "x::real" "y", standard]
lemmas real_abs_def = abs_real_def [of "r", standard]
lemmas real_sgn_def = sgn_real_def [of "x", standard]
lemmas real_mult_commute = mult_commute [of "z::real" "w", standard]
lemmas real_mult_assoc = mult_assoc [of "z1::real" "z2" "z3", standard]
lemmas real_mult_1 = mult_1_left [of "z::real", standard]
lemmas real_add_mult_distrib = left_distrib [of "z1::real" "z2" "w", standard]
lemmas real_zero_not_eq_one = zero_neq_one [where 'a=real]
lemmas real_mult_inverse_left = left_inverse [of "x::real", standard]
lemmas INVERSE_ZERO = inverse_zero [where 'a=real]
lemmas real_le_refl = order_refl [of "w::real", standard]
lemmas real_le_antisym = order_antisym [of "z::real" "w", standard]
lemmas real_le_trans = order_trans [of "i::real" "j" "k", standard]
lemmas real_le_linear = linear [of "z::real" "w", standard]
lemmas real_le_eq_diff = le_iff_diff_le_0 [of "x::real" "y", standard]
lemmas real_add_left_mono = add_left_mono [of "x::real" "y" "z", standard]
lemmas real_mult_order = mult_pos_pos [of "x::real" "y", standard]
lemmas real_mult_less_mono2 =
  mult_strict_left_mono [of "x::real" "y" "z", COMP swap_prems_rl, standard]

subsection{*More Lemmas*}

text {* BH: These lemmas should not be necessary; they should be
covered by existing simp rules and simplification procedures. *}

lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
by simp (* redundant with mult_cancel_left *)

lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
by simp (* redundant with mult_cancel_right *)

lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
by simp (* solved by linordered_ring_less_cancel_factor simproc *)

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

lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
by (rule mult_le_cancel_left_pos)
(* BH: Why doesn't "simp" prove this one, like it does the last one? *)


subsection {* Embedding numbers into the Reals *}

abbreviation
  real_of_nat :: "nat \<Rightarrow> real"
where
  "real_of_nat \<equiv> of_nat"

abbreviation
  real_of_int :: "int \<Rightarrow> real"
where
  "real_of_int \<equiv> of_int"

abbreviation
  real_of_rat :: "rat \<Rightarrow> real"
where
  "real_of_rat \<equiv> of_rat"

consts
  (*overloaded constant for injecting other types into "real"*)
  real :: "'a => real"

defs (overloaded)
  real_of_nat_def [code_unfold]: "real == real_of_nat"
  real_of_int_def [code_unfold]: "real == real_of_int"

lemma real_eq_of_nat: "real = of_nat"
  unfolding real_of_nat_def ..

lemma real_eq_of_int: "real = of_int"
  unfolding real_of_int_def ..

lemma real_of_int_zero [simp]: "real (0::int) = 0"  
by (simp add: real_of_int_def) 

lemma real_of_one [simp]: "real (1::int) = (1::real)"
by (simp add: real_of_int_def) 

lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
by (simp add: real_of_int_def) 

lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
by (simp add: real_of_int_def) 

lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
by (simp add: real_of_int_def) 

lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
by (simp add: real_of_int_def) 

lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"
by (simp add: real_of_int_def of_int_power)

lemmas power_real_of_int = real_of_int_power [symmetric]

lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
  apply (subst real_eq_of_int)+
  apply (rule of_int_setsum)
done

lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
    (PROD x:A. real(f x))"
  apply (subst real_eq_of_int)+
  apply (rule of_int_setprod)
done

lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
by (simp add: real_of_int_def) 

lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
by (simp add: real_of_int_def) 

lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
by (simp add: real_of_int_def) 

lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
by (simp add: real_of_int_def) 

lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
by (simp add: real_of_int_def) 

lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
by (simp add: real_of_int_def) 

lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" 
by (simp add: real_of_int_def)

lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
by (simp add: real_of_int_def)

lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
by (auto simp add: abs_if)

lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
  apply (subgoal_tac "real n + 1 = real (n + 1)")
  apply (simp del: real_of_int_add)
  apply auto
done

lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
  apply (subgoal_tac "real m + 1 = real (m + 1)")
  apply (simp del: real_of_int_add)
  apply simp
done

lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
    real (x div d) + (real (x mod d)) / (real d)"
proof -
  assume "d ~= 0"
  have "x = (x div d) * d + x mod d"
    by auto
  then have "real x = real (x div d) * real d + real(x mod d)"
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
  then have "real x / real d = ... / real d"
    by simp
  then show ?thesis
    by (auto simp add: add_divide_distrib algebra_simps prems)
qed

lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
    real(n div d) = real n / real d"
  apply (frule real_of_int_div_aux [of d n])
  apply simp
  apply (simp add: dvd_eq_mod_eq_0)
done

lemma real_of_int_div2:
  "0 <= real (n::int) / real (x) - real (n div x)"
  apply (case_tac "x = 0")
  apply simp
  apply (case_tac "0 < x")
  apply (simp add: algebra_simps)
  apply (subst real_of_int_div_aux)
  apply simp
  apply simp
  apply (subst zero_le_divide_iff)
  apply auto
  apply (simp add: algebra_simps)
  apply (subst real_of_int_div_aux)
  apply simp
  apply simp
  apply (subst zero_le_divide_iff)
  apply auto
done

lemma real_of_int_div3:
  "real (n::int) / real (x) - real (n div x) <= 1"
  apply(case_tac "x = 0")
  apply simp
  apply (simp add: algebra_simps)
  apply (subst real_of_int_div_aux)
  apply assumption
  apply simp
  apply (subst divide_le_eq)
  apply clarsimp
  apply (rule conjI)
  apply (rule impI)
  apply (rule order_less_imp_le)
  apply simp
  apply (rule impI)
  apply (rule order_less_imp_le)
  apply simp
done

lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
by (insert real_of_int_div2 [of n x], simp)

lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"
unfolding real_of_int_def by (rule Ints_of_int)


subsection{*Embedding the Naturals into the Reals*}

lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
by (simp add: real_of_nat_def)

lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
by (simp add: real_of_nat_def)

lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
by (simp add: real_of_nat_def)

lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
by (simp add: real_of_nat_def)

(*Not for addsimps: often the LHS is used to represent a positive natural*)
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
by (simp add: real_of_nat_def)

lemma real_of_nat_less_iff [iff]: 
     "(real (n::nat) < real m) = (n < m)"
by (simp add: real_of_nat_def)

lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
by (simp add: real_of_nat_def)

lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
by (simp add: real_of_nat_def zero_le_imp_of_nat)

lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
by (simp add: real_of_nat_def del: of_nat_Suc)

lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
by (simp add: real_of_nat_def of_nat_mult)

lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"
by (simp add: real_of_nat_def of_nat_power)

lemmas power_real_of_nat = real_of_nat_power [symmetric]

lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
    (SUM x:A. real(f x))"
  apply (subst real_eq_of_nat)+
  apply (rule of_nat_setsum)
done

lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
    (PROD x:A. real(f x))"
  apply (subst real_eq_of_nat)+
  apply (rule of_nat_setprod)
done

lemma real_of_card: "real (card A) = setsum (%x.1) A"
  apply (subst card_eq_setsum)
  apply (subst real_of_nat_setsum)
  apply simp
done

lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
by (simp add: real_of_nat_def)

lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
by (simp add: real_of_nat_def)

lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
by (simp add: add: real_of_nat_def of_nat_diff)

lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
by (auto simp: real_of_nat_def)

lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
by (simp add: add: real_of_nat_def)

lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
by (simp add: add: real_of_nat_def)

lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
  apply (subgoal_tac "real n + 1 = real (Suc n)")
  apply simp
  apply (auto simp add: real_of_nat_Suc)
done

lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
  apply (subgoal_tac "real m + 1 = real (Suc m)")
  apply (simp add: less_Suc_eq_le)
  apply (simp add: real_of_nat_Suc)
done

lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
    real (x div d) + (real (x mod d)) / (real d)"
proof -
  assume "0 < d"
  have "x = (x div d) * d + x mod d"
    by auto
  then have "real x = real (x div d) * real d + real(x mod d)"
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
  then have "real x / real d = \<dots> / real d"
    by simp
  then show ?thesis
    by (auto simp add: add_divide_distrib algebra_simps prems)
qed

lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
    real(n div d) = real n / real d"
  apply (frule real_of_nat_div_aux [of d n])
  apply simp
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
  apply assumption
done

lemma real_of_nat_div2:
  "0 <= real (n::nat) / real (x) - real (n div x)"
apply(case_tac "x = 0")
 apply (simp)
apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
 apply simp
apply simp
apply (subst zero_le_divide_iff)
apply simp
done

lemma real_of_nat_div3:
  "real (n::nat) / real (x) - real (n div x) <= 1"
apply(case_tac "x = 0")
apply (simp)
apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
 apply simp
apply simp
done

lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
by (insert real_of_nat_div2 [of n x], simp)

lemma real_of_int_real_of_nat: "real (int n) = real n"
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)

lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
by (simp add: real_of_int_def real_of_nat_def)

lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
  apply force
  apply (simp only: real_of_int_real_of_nat)
done

lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"
unfolding real_of_nat_def by (rule of_nat_in_Nats)

lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"
unfolding real_of_nat_def by (rule Ints_of_nat)


subsection{* Rationals *}

lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
by (simp add: real_eq_of_nat)


lemma Rats_eq_int_div_int:
  "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
proof
  show "\<rat> \<subseteq> ?S"
  proof
    fix x::real assume "x : \<rat>"
    then obtain r where "x = of_rat r" unfolding Rats_def ..
    have "of_rat r : ?S"
      by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
    thus "x : ?S" using `x = of_rat r` by simp
  qed
next
  show "?S \<subseteq> \<rat>"
  proof(auto simp:Rats_def)
    fix i j :: int assume "j \<noteq> 0"
    hence "real i / real j = of_rat(Fract i j)"
      by (simp add:of_rat_rat real_eq_of_int)
    thus "real i / real j \<in> range of_rat" by blast
  qed
qed

lemma Rats_eq_int_div_nat:
  "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
proof(auto simp:Rats_eq_int_div_int)
  fix i j::int assume "j \<noteq> 0"
  show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
  proof cases
    assume "j>0"
    hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
    thus ?thesis by blast
  next
    assume "~ j>0"
    hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
    thus ?thesis by blast
  qed
next
  fix i::int and n::nat assume "0 < n"
  hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
  thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
qed

lemma Rats_abs_nat_div_natE:
  assumes "x \<in> \<rat>"
  obtains m n :: nat
  where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
proof -
  from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
    by(auto simp add: Rats_eq_int_div_nat)
  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
  then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
  let ?gcd = "gcd m n"
  from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by simp
  let ?k = "m div ?gcd"
  let ?l = "n div ?gcd"
  let ?gcd' = "gcd ?k ?l"
  have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
    by (rule dvd_mult_div_cancel)
  have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
    by (rule dvd_mult_div_cancel)
  from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
  moreover
  have "\<bar>x\<bar> = real ?k / real ?l"
  proof -
    from gcd have "real ?k / real ?l =
        real (?gcd * ?k) / real (?gcd * ?l)" by simp
    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
    finally show ?thesis ..
  qed
  moreover
  have "?gcd' = 1"
  proof -
    have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
      by (rule gcd_mult_distrib_nat)
    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
    with gcd show ?thesis by auto
  qed
  ultimately show ?thesis ..
qed


subsection{*Numerals and Arithmetic*}

lemma [code_unfold_post]:
  "number_of k = real_of_int (number_of k)"
  unfolding number_of_is_id number_of_real_def ..


text{*Collapse applications of @{term real} to @{term number_of}*}
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
by (simp add: real_of_int_def)

lemma real_of_nat_number_of [simp]:
     "real (number_of v :: nat) =  
        (if neg (number_of v :: int) then 0  
         else (number_of v :: real))"
by (simp add: real_of_int_real_of_nat [symmetric])

declaration {*
  K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
    (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
  #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
    (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
  #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},
      @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
      @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
      @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
      @{thm real_of_nat_number_of}, @{thm real_number_of}]
  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
  #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))
*}


subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}

text{*Needed in this non-standard form by Hyperreal/Transcendental*}
lemma real_0_le_divide_iff:
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
by (auto simp add: zero_le_divide_iff)

lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
by arith

text {* FIXME: redundant with @{text add_eq_0_iff} below *}
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
by auto

lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
by auto

lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
by auto

lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
by auto

lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
by auto

subsection {* Lemmas about powers *}

text {* FIXME: declare this in Rings.thy or not at all *}
declare abs_mult_self [simp]

(* used by Import/HOL/real.imp *)
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
by simp

lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"
apply (induct "n")
apply (auto simp add: real_of_nat_Suc)
apply (subst mult_2)
apply (erule add_less_le_mono)
apply (rule two_realpow_ge_one)
done

text {* TODO: no longer real-specific; rename and move elsewhere *}
lemma realpow_Suc_le_self:
  fixes r :: "'a::linordered_semidom"
  shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"
by (insert power_decreasing [of 1 "Suc n" r], simp)

text {* TODO: no longer real-specific; rename and move elsewhere *}
lemma realpow_minus_mult:
  fixes x :: "'a::monoid_mult"
  shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
by (simp add: power_commutes split add: nat_diff_split)

text {* TODO: no longer real-specific; rename and move elsewhere *}
lemma realpow_two_diff:
  fixes x :: "'a::comm_ring_1"
  shows "x^Suc (Suc 0) - y^Suc (Suc 0) = (x - y) * (x + y)"
by (simp add: algebra_simps)

text {* TODO: move elsewhere *}
lemma add_eq_0_iff:
  fixes x y :: "'a::group_add"
  shows "x + y = 0 \<longleftrightarrow> y = - x"
by (auto dest: minus_unique)

text {* TODO: no longer real-specific; rename and move elsewhere *}
lemma realpow_two_disj:
  fixes x :: "'a::idom"
  shows "(x^Suc (Suc 0) = y^Suc (Suc 0)) = (x = y | x = -y)"
using realpow_two_diff [of x y]
by (auto simp add: add_eq_0_iff)

text {* FIXME: declare this [simp] for all types, or not at all *}
lemma real_two_squares_add_zero_iff [simp]:
  "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"
by (rule sum_squares_eq_zero_iff)

text {* TODO: no longer real-specific; rename and move elsewhere *}
lemma real_squared_diff_one_factored:
  fixes x :: "'a::ring_1"
  shows "x * x - 1 = (x + 1) * (x - 1)"
by (simp add: algebra_simps)

text {* FIXME: declare this [simp] for all types, or not at all *}
lemma realpow_two_sum_zero_iff [simp]:
     "(x ^ 2 + y ^ 2 = (0::real)) = (x = 0 & y = 0)"
by (rule sum_power2_eq_zero_iff)

lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"
by (rule_tac y = 0 in order_trans, auto)

lemma realpow_square_minus_le [simp]: "-(u ^ 2) \<le> (x::real) ^ 2"
by (auto simp add: power2_eq_square)


subsection{*Density of the Reals*}

lemma real_lbound_gt_zero:
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
apply (rule_tac x = " (min d1 d2) /2" in exI)
apply (simp add: min_def)
done


text{*Similar results are proved in @{text Fields}*}
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
  by auto

lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
  by auto


subsection{*Absolute Value Function for the Reals*}

lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
by (simp add: abs_if)

(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
by (force simp add: abs_le_iff)

lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
by (simp add: abs_if)

lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])

lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
by simp
 
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
by simp


subsection {* Implementation of rational real numbers *}

definition Ratreal :: "rat \<Rightarrow> real" where
  [simp]: "Ratreal = of_rat"

code_datatype Ratreal

lemma Ratreal_number_collapse [code_post]:
  "Ratreal 0 = 0"
  "Ratreal 1 = 1"
  "Ratreal (number_of k) = number_of k"
by simp_all

lemma zero_real_code [code, code_unfold]:
  "0 = Ratreal 0"
by simp

lemma one_real_code [code, code_unfold]:
  "1 = Ratreal 1"
by simp

lemma number_of_real_code [code_unfold]:
  "number_of k = Ratreal (number_of k)"
by simp

lemma Ratreal_number_of_quotient [code_post]:
  "Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s"
by simp

lemma Ratreal_number_of_quotient2 [code_post]:
  "Ratreal (number_of r / number_of s) = number_of r / number_of s"
unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide ..

instantiation real :: eq
begin

definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0"

instance by default (simp add: eq_real_def)

lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y"
  by (simp add: eq_real_def eq)

lemma real_eq_refl [code nbe]:
  "eq_class.eq (x::real) x \<longleftrightarrow> True"
  by (rule HOL.eq_refl)

end

lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
  by (simp add: of_rat_less_eq)

lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
  by (simp add: of_rat_less)

lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
  by (simp add: of_rat_add)

lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
  by (simp add: of_rat_mult)

lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
  by (simp add: of_rat_minus)

lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
  by (simp add: of_rat_diff)

lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
  by (simp add: of_rat_inverse)
 
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
  by (simp add: of_rat_divide)

definition (in term_syntax)
  valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where
  [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"

notation fcomp (infixl "\<circ>>" 60)
notation scomp (infixl "\<circ>\<rightarrow>" 60)

instantiation real :: random
begin

definition
  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"

instance ..

end

no_notation fcomp (infixl "\<circ>>" 60)
no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

text {* Setup for SML code generator *}

types_code
  real ("(int */ int)")
attach (term_of) {*
fun term_of_real (p, q) =
  let
    val rT = HOLogic.realT
  in
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
    else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
  end;
*}
attach (test) {*
fun gen_real i =
  let
    val p = random_range 0 i;
    val q = random_range 1 (i + 1);
    val g = Integer.gcd p q;
    val p' = p div g;
    val q' = q div g;
    val r = (if one_of [true, false] then p' else ~ p',
      if p' = 0 then 1 else q')
  in
    (r, fn () => term_of_real r)
  end;
*}

consts_code
  Ratreal ("(_)")

consts_code
  "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
attach {*
fun real_of_int i = (i, 1);
*}

declaration {*
  Nitpick_HOL.register_frac_type @{type_name real}
   [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),
    (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),
    (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),
    (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),
    (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),
    (@{const_name number_real_inst.number_of_real}, @{const_name Nitpick.number_of_frac}),
    (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),
    (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_eq_frac}),
    (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]
*}

lemmas [nitpick_def] = inverse_real_inst.inverse_real
    number_real_inst.number_of_real one_real_inst.one_real
    ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real
    times_real_inst.times_real uminus_real_inst.uminus_real
    zero_real_inst.zero_real

end