SSL Real.thy Sprache: Isabelle

(*  Title:      HOL/Real.thy
    Author:     Jacques D. Fleuriot, University of Edinburgh, 1998
    Author:     Larry Paulson, University of Cambridge
    Author:     Jeremy Avigad, Carnegie Mellon University
    Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
    Construction of Cauchy Reals by Brian Huffman, 2010
*)

section \<open>Development of the Reals using Cauchy Sequences\<close>

theory Real
imports Rat
begin

text \<open>
  This theory contains a formalization of the real numbers as equivalence
  classes of Cauchy sequences of rationals. See the AFP entry
  @{text Dedekind_Real} for an alternative construction using
  Dedekind cuts.
\<close>

subsection \<open>Preliminary lemmas\<close>

text\<open>Useful in convergence arguments\<close>
lemma inverse_of_nat_le:
  fixes n::nat shows "\n \ m; n\0\ \ 1 / of_nat m \ (1::'a::linordered_field) / of_nat n"
  by (simp add: frac_le)

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

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

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

lemma inverse_diff_inverse:
  fixes a b :: "'a::division_ring"
  assumes "a \ 0" and "b \ 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 \<open>Sequences that converge to zero\<close>

definition vanishes :: "(nat \ rat) \ bool"
  where "vanishes X \ (\r>0. \k. \n\k. \X n\ < r)"

lemma vanishesI: "(\r. 0 < r \ \k. \n\k. \X n\ < r) \ vanishes X"
  unfolding vanishes_def by simp

lemma vanishesD: "vanishes X \ 0 < r \ \k. \n\k. \X n\ < r"
  unfolding vanishes_def by simp

lemma vanishes_const [simp]: "vanishes (\n. c) \ c = 0"
proof (cases "c = 0")
  case True
  then show ?thesis
    by (simp add: vanishesI)
next
  case False
  then show ?thesis
    unfolding vanishes_def
    using zero_less_abs_iff by blast
qed

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

lemma vanishes_add:
  assumes X: "vanishes X"
    and Y: "vanishes Y"
  shows "vanishes (\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: "\n\i. \X n\ < s"
    using vanishesD [OF X s] ..
  obtain j where j: "\n\j. \Y n\ < t"
    using vanishesD [OF Y t] ..
  have "\n\max i j. \X n + Y n\ < r"
  proof clarsimp
    fix n
    assume n: "i \ n" "j \ n"
    have "\X n + Y n\ \ \X n\ + \Y n\"
      by (rule abs_triangle_ineq)
    also have "\ < s + t"
      by (simp add: add_strict_mono i j n)
    finally show "\X n + Y n\ < r"
      by (simp only: r)
  qed
  then show "\k. \n\k. \X n + Y n\ < r" ..
qed

lemma vanishes_diff:
  assumes "vanishes X" "vanishes Y"
  shows "vanishes (\n. X n - Y n)"
  unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus assms)

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

subsection \<open>Cauchy sequences\<close>

definition cauchy :: "(nat \ rat) \ bool"
  where "cauchy X \ (\r>0. \k. \m\k. \n\k. \X m - X n\ < r)"

lemma cauchyI: "(\r. 0 < r \ \k. \m\k. \n\k. \X m - X n\ < r) \ cauchy X"
  unfolding cauchy_def by simp

lemma cauchyD: "cauchy X \ 0 < r \ \k. \m\k. \n\k. \X m - X n\ < r"
  unfolding cauchy_def by simp

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

lemma cauchy_add [simp]:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "cauchy (\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: "\m\i. \n\i. \X m - X n\ < s"
    using cauchyD [OF X s] ..
  obtain j where j: "\m\j. \n\j. \Y m - Y n\ < t"
    using cauchyD [OF Y t] ..
  have "\m\max i j. \n\max i j. \(X m + Y m) - (X n + Y n)\ < r"
  proof clarsimp
    fix m n
    assume *: "i \ m" "j \ m" "i \ n" "j \ n"
    have "\(X m + Y m) - (X n + Y n)\ \ \X m - X n\ + \Y m - Y n\"
      unfolding add_diff_add by (rule abs_triangle_ineq)
    also have "\ < s + t"
      by (rule add_strict_mono) (simp_all add: i j *)
    finally show "\(X m + Y m) - (X n + Y n)\ < r" by (simp only: r)
  qed
  then show "\k. \m\k. \n\k. \(X m + Y m) - (X n + Y n)\ < r" ..
qed

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

lemma cauchy_diff [simp]:
  assumes "cauchy X" "cauchy Y"
  shows "cauchy (\n. X n - Y n)"
  using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff)

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

lemma cauchy_mult [simp]:
  assumes X: "cauchy X" and Y: "cauchy Y"
  shows "cauchy (\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" "\n. \X n\ < a"
    using cauchy_imp_bounded [OF X] by blast
  obtain b where b: "0 < b" "\n. \Y n\ < b"
    using cauchy_imp_bounded [OF Y] by blast
  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 simp
    show "0 < u/a" using u a(1) by simp
    show "r = a * (u/a) + (v/b) * b"
      using a(1) b(1) \<open>r = u + v\<close> by simp
  qed
  obtain i where i: "\m\i. \n\i. \X m - X n\ < s"
    using cauchyD [OF X s] ..
  obtain j where j: "\m\j. \n\j. \Y m - Y n\ < t"
    using cauchyD [OF Y t] ..
  have "\m\max i j. \n\max i j. \X m * Y m - X n * Y n\ < r"
  proof clarsimp
    fix m n
    assume *: "i \ m" "j \ m" "i \ n" "j \ n"
    have "\X m * Y m - X n * Y n\ = \X m * (Y m - Y n) + (X m - X n) * Y n\"
      unfolding mult_diff_mult ..
    also have "\ \ \X m * (Y m - Y n)\ + \(X m - X n) * Y n\"
      by (rule abs_triangle_ineq)
    also have "\ = \X m\ * \Y m - Y n\ + \X m - X n\ * \Y n\"
      unfolding abs_mult ..
    also have "\ < a * t + s * b"
      by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)
    finally show "\X m * Y m - X n * Y n\ < r"
      by (simp only: r)
  qed
  then show "\k. \m\k. \n\k. \X m * Y m - X n * Y n\ < r" ..
qed

lemma cauchy_not_vanishes_cases:
  assumes X: "cauchy X"
  assumes nz: "\ vanishes X"
  shows "\b>0. \k. (\n\k. b < - X n) \ (\n\k. b < X n)"
proof -
  obtain r where "0 < r" and r: "\k. \n\k. r \ \X n\"
    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 \<open>0 < r\<close> by (rule obtain_pos_sum)
  obtain i where i: "\m\i. \n\i. \X m - X n\ < s"
    using cauchyD [OF X s] ..
  obtain k where "i \ k" and "r \ \X k\"
    using r by blast
  have k: "\n\k. \X n - X k\ < s"
    using i \<open>i \<le> k\<close> by auto
  have "X k \ - r \ r \ X k"
    using \<open>r \<le> \<bar>X k\<bar>\<close> by auto
  then have "(\n\k. t < - X n) \ (\n\k. t < X n)"
    unfolding \<open>r = s + t\<close> using k by auto
  then have "\k. (\n\k. t < - X n) \ (\n\k. t < X n)" ..
  then show "\t>0. \k. (\n\k. t < - X n) \ (\n\k. t < X n)"
    using t by auto
qed

lemma cauchy_not_vanishes:
  assumes X: "cauchy X"
    and nz: "\ vanishes X"
  shows "\b>0. \k. \n\k. b < \X n\"
  using cauchy_not_vanishes_cases [OF assms]
  by (elim ex_forward conj_forward asm_rl) auto

lemma cauchy_inverse [simp]:
  assumes X: "cauchy X"
    and nz: "\ vanishes X"
  shows "cauchy (\n. inverse (X n))"
proof (rule cauchyI)
  fix r :: rat
  assume "0 < r"
  obtain b i where b: "0 < b" and i: "\n\i. b < \X n\"
    using cauchy_not_vanishes [OF X nz] by blast
  from b i have nz: "\n\i. X n \ 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: \<open>0 < r\<close> b)
    show "r = inverse b * (b * r * b) * inverse b"
      using b by simp
  qed
  obtain j where j: "\m\j. \n\j. \X m - X n\ < s"
    using cauchyD [OF X s] ..
  have "\m\max i j. \n\max i j. \inverse (X m) - inverse (X n)\ < r"
  proof clarsimp
    fix m n
    assume *: "i \ m" "j \ m" "i \ n" "j \ n"
    have "\inverse (X m) - inverse (X n)\ = inverse \X m\ * \X m - X n\ * inverse \X n\"
      by (simp add: inverse_diff_inverse nz * abs_mult)
    also have "\ < inverse b * s * inverse b"
      by (simp add: mult_strict_mono less_imp_inverse_less i j b * s)
    finally show "\inverse (X m) - inverse (X n)\ < r" by (simp only: r)
  qed
  then show "\k. \m\k. \n\k. \inverse (X m) - inverse (X n)\ < r" ..
qed

lemma vanishes_diff_inverse:
  assumes X: "cauchy X" "\ vanishes X"
    and Y: "cauchy Y" "\ vanishes Y"
    and XY: "vanishes (\n. X n - Y n)"
  shows "vanishes (\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: "\n\i. a < \X n\"
    using cauchy_not_vanishes [OF X] by blast
  obtain b j where b: "0 < b" and j: "\n\j. b < \Y n\"
    using cauchy_not_vanishes [OF Y] by blast
  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
    show "inverse a * (a * r * b) * inverse b = r"
      using a r b by simp
  qed
  obtain k where k: "\n\k. \X n - Y n\ < s"
    using vanishesD [OF XY s] ..
  have "\n\max (max i j) k. \inverse (X n) - inverse (Y n)\ < r"
  proof clarsimp
    fix n
    assume n: "i \ n" "j \ n" "k \ n"
    with i j a b have "X n \ 0" and "Y n \ 0"
      by auto
    then have "\inverse (X n) - inverse (Y n)\ = inverse \X n\ * \X n - Y n\ * inverse \Y n\"
      by (simp add: inverse_diff_inverse abs_mult)
    also have "\ < inverse a * s * inverse b"
      by (intro mult_strict_mono' less_imp_inverse_less) (simp_all add: a b i j k n)
    also note \<open>inverse a * s * inverse b = r\<close>
    finally show "\inverse (X n) - inverse (Y n)\ < r" .
  qed
  then show "\k. \n\k. \inverse (X n) - inverse (Y n)\ < r" ..
qed

subsection \<open>Equivalence relation on Cauchy sequences\<close>

definition realrel :: "(nat \ rat) \ (nat \ rat) \ bool"
  where "realrel = (\X Y. cauchy X \ cauchy Y \ vanishes (\n. X n - Y n))"

lemma realrelI [intro?]: "cauchy X \ cauchy Y \ vanishes (\n. X n - Y n) \ realrel X Y"
  by (simp add: realrel_def)

lemma realrel_refl: "cauchy X \ realrel X X"
  by (simp add: realrel_def)

lemma symp_realrel: "symp realrel"
  by (simp add: abs_minus_commute realrel_def symp_def vanishes_def)

lemma transp_realrel: "transp realrel"
  unfolding realrel_def
  by (rule transpI) (force simp add: dest: vanishes_add)

lemma part_equivp_realrel: "part_equivp realrel"
  by (blast intro: part_equivpI symp_realrel transp_realrel realrel_refl cauchy_const)

subsection \<open>The field of real numbers\<close>

quotient_type real = "nat \ rat" / partial: realrel
  morphisms rep_real Real
  by (rule part_equivp_realrel)

lemma cr_real_eq: "pcr_real = (\x y. cauchy x \ Real x = y)"
  unfolding real.pcr_cr_eq cr_real_def realrel_def by auto

lemma Real_induct [induct type: real]: (* TODO: generate automatically *)
  assumes "\X. cauchy X \ P (Real X)"
  shows "P x"
proof (induct x)
  case (1 X)
  then have "cauchy X" by (simp add: realrel_def)
  then show "P (Real X)" by (rule assms)
qed

lemma eq_Real: "cauchy X \ cauchy Y \ Real X = Real Y \ vanishes (\n. X n - Y n)"
  using real.rel_eq_transfer
  unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp

lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"
  by (simp add: real.domain_eq realrel_def)

instantiation real :: field
begin

lift_definition zero_real :: "real" is "\n. 0"
  by (simp add: realrel_refl)

lift_definition one_real :: "real" is "\n. 1"
  by (simp add: realrel_refl)

lift_definition plus_real :: "real \ real \ real" is "\X Y n. X n + Y n"
  unfolding realrel_def add_diff_add
  by (simp only: cauchy_add vanishes_add simp_thms)

lift_definition uminus_real :: "real \ real" is "\X n. - X n"
  unfolding realrel_def minus_diff_minus
  by (simp only: cauchy_minus vanishes_minus simp_thms)

lift_definition times_real :: "real \ real \ real" is "\X Y n. X n * Y n"
proof -
  fix f1 f2 f3 f4
  have "\cauchy f1; cauchy f4; vanishes (\n. f1 n - f2 n); vanishes (\n. f3 n - f4 n)\
       \<Longrightarrow> vanishes (\<lambda>n. f1 n * (f3 n - f4 n) + f4 n * (f1 n - f2 n))"
    by (simp add: vanishes_add vanishes_mult_bounded cauchy_imp_bounded)
  then show "\realrel f1 f2; realrel f3 f4\ \ realrel (\n. f1 n * f3 n) (\n. f2 n * f4 n)"
    by (simp add: mult.commute realrel_def mult_diff_mult)
qed

lift_definition inverse_real :: "real \ real"
  is "\X. if vanishes X then (\n. 0) else (\n. inverse (X n))"
proof -
  fix X Y
  assume "realrel X Y"
  then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\n. X n - Y n)"
    by (simp_all add: realrel_def)
  have "vanishes X \ 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
  then show "?thesis X Y"
    by (simp add: vanishes_diff_inverse X Y XY realrel_def)
qed

definition "x - y = x + - y" for x y :: real

definition "x div y = x * inverse y" for x y :: real

lemma add_Real: "cauchy X \ cauchy Y \ Real X + Real Y = Real (\n. X n + Y n)"
  using plus_real.transfer by (simp add: cr_real_eq rel_fun_def)

lemma minus_Real: "cauchy X \ - Real X = Real (\n. - X n)"
  using uminus_real.transfer by (simp add: cr_real_eq rel_fun_def)

lemma diff_Real: "cauchy X \ cauchy Y \ Real X - Real Y = Real (\n. X n - Y n)"
  by (simp add: minus_Real add_Real minus_real_def)

lemma mult_Real: "cauchy X \ cauchy Y \ Real X * Real Y = Real (\n. X n * Y n)"
  using times_real.transfer by (simp add: cr_real_eq rel_fun_def)

lemma inverse_Real:
  "cauchy X \ inverse (Real X) = (if vanishes X then 0 else Real (\n. inverse (X n)))"
  using inverse_real.transfer zero_real.transfer
  unfolding cr_real_eq rel_fun_def by (simp split: if_split_asm, metis)

instance
proof
  fix a b c :: real
  show "a + b = b + a"
    by transfer (simp add: ac_simps realrel_def)
  show "(a + b) + c = a + (b + c)"
    by transfer (simp add: ac_simps realrel_def)
  show "0 + a = a"
    by transfer (simp add: realrel_def)
  show "- a + a = 0"
    by transfer (simp add: realrel_def)
  show "a - b = a + - b"
    by (rule minus_real_def)
  show "(a * b) * c = a * (b * c)"
    by transfer (simp add: ac_simps realrel_def)
  show "a * b = b * a"
    by transfer (simp add: ac_simps realrel_def)
  show "1 * a = a"
    by transfer (simp add: ac_simps realrel_def)
  show "(a + b) * c = a * c + b * c"
    by transfer (simp add: distrib_right realrel_def)
  show "(0::real) \ (1::real)"
    by transfer (simp add: realrel_def)
  have "vanishes (\n. inverse (X n) * X n - 1)" if X: "cauchy X" "\ vanishes X" for X
  proof (rule vanishesI)
    fix r::rat
    assume "0 < r"
    obtain b k where "b>0" "\n\k. b < \X n\"
      using X cauchy_not_vanishes by blast
    then show "\k. \n\k. \inverse (X n) * X n - 1\ < r"
      using \<open>0 < r\<close> by force
  qed
  then show "a \ 0 \ inverse a * a = 1"
    by transfer (simp add: realrel_def)
  show "a div b = a * inverse b"
    by (rule divide_real_def)
  show "inverse (0::real) = 0"
    by transfer (simp add: realrel_def)
qed

end

subsection \<open>Positive reals\<close>

lift_definition positive :: "real \ bool"
  is "\X. \r>0. \k. \n\k. r < X n"
proof -
  have 1: "\r>0. \k. \n\k. r < Y n"
    if *: "realrel X Y" and **: "\r>0. \k. \n\k. r < X n" for X Y
  proof -
    from * have XY: "vanishes (\n. X n - Y n)"
      by (simp_all add: realrel_def)
    from ** obtain r i where "0 < r" and i: "\n\i. r < X n"
      by blast
    obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
      using \<open>0 < r\<close> by (rule obtain_pos_sum)
    obtain j where j: "\n\j. \X n - Y n\ < s"
      using vanishesD [OF XY s] ..
    have "\n\max i j. t < Y n"
    proof clarsimp
      fix n
      assume n: "i \ n" "j \ n"
      have "\X n - Y n\ < s" and "r < X n"
        using i j n by simp_all
      then show "t < Y n" by (simp add: r)
    qed
    then show ?thesis using t by blast
  qed
  fix X Y assume "realrel X Y"
  then have "realrel X Y" and "realrel Y X"
    using symp_realrel by (auto simp: symp_def)
  then show "?thesis X Y"
    by (safe elim!: 1)
qed

lemma positive_Real: "cauchy X \ positive (Real X) \ (\r>0. \k. \n\k. r < X n)"
  using positive.transfer by (simp add: cr_real_eq rel_fun_def)

lemma positive_zero: "\ positive 0"
  by transfer auto

lemma positive_add:
  assumes "positive x" "positive y" shows "positive (x + y)"
proof -
  have *: "\\n\i. a < x n; \n\j. b < y n; 0 < a; 0 < b; n \ max i j\
       \<Longrightarrow> a+b < x n + y n" for x y and a b::rat and i j n::nat
    by (simp add: add_strict_mono)
  show ?thesis
    using assms
    by transfer (blast intro: * pos_add_strict)
qed

lemma positive_mult:
  assumes "positive x" "positive y" shows "positive (x * y)"
proof -
  have *: "\\n\i. a < x n; \n\j. b < y n; 0 < a; 0 < b; n \ max i j\
       \<Longrightarrow> a*b < x n * y n" for x y and a b::rat and i j n::nat
    by (simp add: mult_strict_mono')
  show ?thesis
    using assms
    by transfer (blast intro: *  mult_pos_pos)
qed

lemma positive_minus: "\ positive x \ x \ 0 \ positive (- x)"
  apply transfer
  apply (simp add: realrel_def)
  apply (blast dest: cauchy_not_vanishes_cases)
  done

instantiation real :: linordered_field
begin

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

definition "x \ y \ x < y \ x = y" for x y :: real

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

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

instance
proof
  fix a b c :: real
  show "\a\ = (if a < 0 then - a else a)"
    by (rule abs_real_def)
  show "a < b \ a \ b \ \ b \ a"
       "a \ b \ b \ c \ a \ c" "a \ a"
       "a \ b \ b \ a \ a = b"
       "a \ b \ c + a \ c + b"
    unfolding less_eq_real_def less_real_def
    by (force simp add: positive_zero dest: positive_add)+
  show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"
    by (rule sgn_real_def)
  show "a \ b \ b \ a"
    by (auto dest!: positive_minus simp: less_eq_real_def less_real_def)
  show "a < b \ 0 < c \ c * a < c * b"
    unfolding less_real_def
    by (force simp add: algebra_simps dest: positive_mult)
qed

end

instantiation real :: distrib_lattice
begin

definition "(inf :: real \ real \ real) = min"

definition "(sup :: real \ real \ real) = max"

instance
  by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)

end

lemma of_nat_Real: "of_nat x = Real (\n. of_nat x)"
  by (induct x) (simp_all add: zero_real_def one_real_def add_Real)

lemma of_int_Real: "of_int x = Real (\n. of_int x)"
  by (cases x rule: int_diff_cases) (simp add: of_nat_Real diff_Real)

lemma of_rat_Real: "of_rat x = Real (\n. x)"
proof (induct x)
  case (Fract a b)
  then show ?case
  apply (simp add: Fract_of_int_quotient of_rat_divide)
  apply (simp add: of_int_Real divide_inverse inverse_Real mult_Real)
  done
qed

instance real :: archimedean_field
proof
  show "\z. x \ of_int z" for x :: real
  proof (induct x)
    case (1 X)
    then obtain b where "0 < b" and b: "\n. \X n\ < b"
      by (blast dest: cauchy_imp_bounded)
    then have "Real X < of_int (\b\ + 1)"
      using 1
      apply (simp add: of_int_Real less_real_def diff_Real positive_Real)
      apply (rule_tac x=1 in exI)
      apply (simp add: algebra_simps)
      by (metis abs_ge_self le_less_trans le_of_int_ceiling less_le)
    then show ?case
      using less_eq_real_def by blast
  qed
qed

instantiation real :: floor_ceiling
begin

definition [code del]: "\x::real\ = (THE z. of_int z \ x \ x < of_int (z + 1))"

instance
proof
  show "of_int \x\ \ x \ x < of_int (\x\ + 1)" for x :: real
    unfolding floor_real_def using floor_exists1 by (rule theI')
qed

end

subsection \<open>Completeness\<close>

lemma not_positive_Real:
  assumes "cauchy X" shows "\ positive (Real X) \ (\r>0. \k. \n\k. X n \ r)" (is "?lhs = ?rhs")
  unfolding positive_Real [OF assms]
proof (intro iffI allI notI impI)
  show "\k. \n\k. X n \ r" if r: "\ (\r>0. \k. \n\k. r < X n)" and "0 < r" for r
  proof -
    obtain s t where "s > 0" "t > 0" "r = s+t"
      using \<open>r > 0\<close> obtain_pos_sum by blast
    obtain k where k: "\m n. \m\k; n\k\ \ \X m - X n\ < t"
      using cauchyD [OF assms \<open>t > 0\<close>] by blast
    obtain n where "n \ k" "X n \ s"
      by (meson r \<open>0 < s\<close> not_less)
    then have "X l \ r" if "l \ n" for l
      using k [OF \<open>n \<ge> k\<close>, of l] that \<open>r = s+t\<close> by linarith
    then show ?thesis
      by blast
    qed
qed (meson le_cases not_le)

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

lemma le_RealI:
  assumes Y: "cauchy Y"
  shows "\n. x \ of_rat (Y n) \ x \ Real Y"
proof (induct x)
  fix X
  assume X: "cauchy X" and "\n. Real X \ of_rat (Y n)"
  then have le: "\m r. 0 < r \ \k. \n\k. X n \ Y m + r"
    by (simp add: of_rat_Real le_Real)
  then have "\k. \n\k. X n \ Y n + r" if "0 < r" for r :: rat
  proof -
    from that 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: "\m\i. \n\i. \Y m - Y n\ < s"
      using cauchyD [OF Y s] ..
    obtain j where j: "\n\j. X n \ Y i + t"
      using le [OF t] ..
    have "\n\max i j. X n \ Y n + r"
    proof clarsimp
      fix n
      assume n: "i \ n" "j \ n"
      have "X n \ Y i + t"
        using n j by simp
      moreover have "\Y i - Y n\ < s"
        using n i by simp
      ultimately show "X n \ Y n + r"
        unfolding r by simp
    qed
    then show ?thesis ..
  qed
  then show "Real X \ Real Y"
    by (simp add: of_rat_Real le_Real X Y)
qed

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

lemma less_RealD:
  assumes "cauchy Y"
  shows "x < Real Y \ \n. x < of_rat (Y n)"
  by (meson Real_leI assms leD leI)

lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n"
  by auto

lemma complete_real:
  fixes S :: "real set"
  assumes "\x. x \ S" and "\z. \x\S. x \ z"
  shows "\y. (\x\S. x \ y) \ (\z. (\x\S. x \ z) \ y \ z)"
proof -
  obtain x where x: "x \ S" using assms(1) ..
  obtain z where z: "\x\S. x \ z" using assms(2) ..

  define P where "P x \ (\y\S. y \ of_rat x)" for x
  obtain a where a: "\ P a"
  proof
    have "of_int \x - 1\ \ x - 1" by (rule of_int_floor_le)
    also have "x - 1 < x" by simp
    finally have "of_int \x - 1\ < x" .
    then have "\ x \ of_int \x - 1\" by (simp only: not_le)
    then show "\ P (of_int \x - 1\)"
      unfolding P_def of_rat_of_int_eq using x by blast
  qed
  obtain b where b: "P b"
  proof
    show "P (of_int \z\)"
    unfolding P_def of_rat_of_int_eq
    proof
      fix y assume "y \ S"
      then have "y \ z" using z by simp
      also have "z \ of_int \z\" by (rule le_of_int_ceiling)
      finally show "y \ of_int \z\" .
    qed
  qed

  define avg where "avg x y = x/2 + y/2" for x y :: rat
  define bisect where "bisect = (\(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y))"
  define A where "A n = fst ((bisect ^^ n) (a, b))" for n
  define B where "B n = snd ((bisect ^^ n) (a, b))" for n
  define C where "C n = avg (A n) (B n)" for 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]: "\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]: "\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: "B n - A n = (b - a) / 2^n" for n
  proof (induct n)
    case (Suc n)
    then show ?case
      by (simp add: C_def eq_divide_eq avg_def algebra_simps)
  qed simp
  have twos: "\n. y / 2 ^ n < r" if "0 < r" for y r :: rat
  proof -
    obtain n where "y / r < rat_of_nat n"
      using \<open>0 < r\<close> reals_Archimedean2 by blast
    then have "\n. y < r * 2 ^ n"
      by (metis divide_less_eq less_trans mult.commute of_nat_less_two_power that)
    then show ?thesis
      by (simp add: field_split_simps)
  qed
  have PA: "\ P (A n)" for n
    by (induct n) (simp_all add: a)
  have PB: "P (B n)" for n
    by (induct n) (simp_all add: b)
  have ab: "a < b"
    using a b unfolding P_def
    by (meson leI less_le_trans of_rat_less)
  have AB: "A n < B n" for n
    by (induct n) (simp_all add: ab C_def avg_def)
  have  "A i \ A j \ B j \ B i" if "i < j" for i j
    using that
  proof (induction rule: less_Suc_induct)
    case (1 i)
    then show ?case
      apply (clarsimp simp add: C_def avg_def add_divide_distrib [symmetric])
      apply (rule AB [THEN less_imp_le])
      done
  qed simp
  then have A_mono: "A i \ A j" and B_mono: "B j \ B i" if "i \ j" for i j
    by (metis eq_refl le_neq_implies_less that)+
  have cauchy_lemma: "cauchy X" if *: "\n i. i\n \ A n \ X i \ X i \ B n" for X
  proof (rule cauchyI)
    fix r::rat
    assume "0 < r"
    then obtain k where k: "(b - a) / 2 ^ k < r"
      using twos by blast
    have "\X m - X n\ < r" if "m\k" "n\k" for m n
    proof -
      have "\X m - X n\ \ B k - A k"
        by (simp add: * abs_rat_def diff_mono that)
      also have "... < r"
        by (simp add: k width)
      finally show ?thesis .
    qed
    then show "\k. \m\k. \n\k. \X m - X n\ < r"
      by blast
  qed
  have "cauchy A"
    by (rule cauchy_lemma) (meson AB A_mono B_mono dual_order.strict_implies_order less_le_trans)
  have "cauchy B"
    by (rule cauchy_lemma) (meson AB A_mono B_mono dual_order.strict_implies_order le_less_trans)
  have "\x\S. x \ Real B"
  proof
    fix x
    assume "x \ S"
    then show "x \ Real B"
      using PB [unfolded P_def] \<open>cauchy B\<close>
      by (simp add: le_RealI)
  qed
  moreover have "\z. (\x\S. x \ z) \ Real A \ z"
    by (meson PA Real_leI P_def \<open>cauchy A\<close> le_cases order.trans)
  moreover have "vanishes (\n. (b - a) / 2 ^ n)"
  proof (rule vanishesI)
    fix r :: rat
    assume "0 < r"
    then obtain k where k: "\b - a\ / 2 ^ k < r"
      using twos by blast
    have "\n\k. \(b - a) / 2 ^ n\ < r"
    proof clarify
      fix n
      assume n: "k \ n"
      have "\(b - a) / 2 ^ n\ = \b - a\ / 2 ^ n"
        by simp
      also have "\ \ \b - a\ / 2 ^ k"
        using n by (simp add: divide_left_mono)
      also note k
      finally show "\(b - a) / 2 ^ n\ < r" .
    qed
    then show "\k. \n\k. \(b - a) / 2 ^ n\ < r" ..
  qed
  then have "Real B = Real A"
    by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width)
  ultimately show "\y. (\x\S. x \ y) \ (\z. (\x\S. x \ z) \ y \ z)"
    by force
qed

instantiation real :: linear_continuum
begin

subsection \<open>Supremum of a set of reals\<close>

definition "Sup X = (LEAST z::real. \x\X. x \ z)"
definition "Inf X = - Sup (uminus ` X)" for X :: "real set"

instance
proof
  show Sup_upper: "x \ Sup X"
    if "x \ X" "bdd_above X"
    for x :: real and X :: "real set"
  proof -
    from that obtain s where s: "\y\X. y \ s" "\z. \y\X. y \ z \ s \ z"
      using complete_real[of X] unfolding bdd_above_def by blast
    then show ?thesis
      unfolding Sup_real_def by (rule LeastI2_order) (auto simp: that)
  qed
  show Sup_least: "Sup X \ z"
    if "X \ {}" and z: "\x. x \ X \ x \ z"
    for z :: real and X :: "real set"
  proof -
    from that obtain s where s: "\y\X. y \ s" "\z. \y\X. y \ z \ s \ z"
      using complete_real [of X] by blast
    then have "Sup X = s"
      unfolding Sup_real_def by (best intro: Least_equality)
    also from s z have "\ \ z"
      by blast
    finally show ?thesis .
  qed
  show "Inf X \ x" if "x \ X" "bdd_below X"
    for x :: real and X :: "real set"
    using Sup_upper [of "-x" "uminus ` X"] by (auto simp: Inf_real_def that)
  show "z \ Inf X" if "X \ {}" "\x. x \ X \ z \ x"
    for z :: real and X :: "real set"
    using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that)
  show "\a b::real. a \ b"
    using zero_neq_one by blast
qed

end

subsection \<open>Hiding implementation details\<close>

hide_const (open) vanishes cauchy positive Real

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

lifting_update real.lifting
lifting_forget real.lifting

subsection \<open>Embedding numbers into the Reals\<close>

abbreviation real_of_nat :: "nat \ real"
  where "real_of_nat \ of_nat"

abbreviation real :: "nat \ real"
  where "real \ of_nat"

abbreviation real_of_int :: "int \ real"
  where "real_of_int \ of_int"

abbreviation real_of_rat :: "rat \ real"
  where "real_of_rat \ of_rat"

declare [[coercion_enabled]]

declare [[coercion "of_nat :: nat \ int"]]
declare [[coercion "of_nat :: nat \ real"]]
declare [[coercion "of_int :: int \ real"]]

(* We do not add rat to the coerced types, this has often unpleasant side effects when writing
inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)

declare [[coercion_map map]]
declare [[coercion_map "\f g h x. g (h (f x))"]]
declare [[coercion_map "\f g (x,y). (f x, g y)"]]

declare of_int_eq_0_iff [algebra, presburger]
declare of_int_eq_1_iff [algebra, presburger]
declare of_int_eq_iff [algebra, presburger]
declare of_int_less_0_iff [algebra, presburger]
declare of_int_less_1_iff [algebra, presburger]
declare of_int_less_iff [algebra, presburger]
declare of_int_le_0_iff [algebra, presburger]
declare of_int_le_1_iff [algebra, presburger]
declare of_int_le_iff [algebra, presburger]
declare of_int_0_less_iff [algebra, presburger]
declare of_int_0_le_iff [algebra, presburger]
declare of_int_1_less_iff [algebra, presburger]
declare of_int_1_le_iff [algebra, presburger]

lemma int_less_real_le: "n < m \ real_of_int n + 1 \ real_of_int m"
proof -
  have "(0::real) \ 1"
    by (metis less_eq_real_def zero_less_one)
  then show ?thesis
    by (metis floor_of_int less_floor_iff)
qed

lemma int_le_real_less: "n \ m \ real_of_int n < real_of_int m + 1"
  by (meson int_less_real_le not_le)

lemma (in field_char_0) of_int_div_aux:
  "(of_int x) / (of_int d) =
    of_int (x div d) + (of_int (x mod d)) / (of_int d)"
proof -
  have "x = (x div d) * d + x mod d"
    by auto
  then have "of_int x = of_int (x div d) * of_int d + of_int(x mod d)"
    by (metis local.of_int_add local.of_int_mult)
  then show ?thesis
    by (simp add: divide_simps)
qed

lemma real_of_int_div:
  "d dvd n \ real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int
  by auto

lemma real_of_int_div2: "0 \ real_of_int n / real_of_int x - real_of_int (n div x)"
proof (cases "x = 0")
  case False
  then show ?thesis
    by (metis diff_ge_0_iff_ge floor_divide_of_int_eq of_int_floor_le)
qed simp

lemma real_of_int_div3: "real_of_int n / real_of_int x - real_of_int (n div x) \ 1"
  apply (simp add: algebra_simps)
  by (metis add.commute floor_correct floor_divide_of_int_eq less_eq_real_def of_int_1 of_int_add)

lemma real_of_int_div4: "real_of_int (n div x) \ real_of_int n / real_of_int x"
  using real_of_int_div2 [of n x] by simp

subsection \<open>Embedding the Naturals into the Reals\<close>

lemma (in field_char_0) of_nat_of_nat_div_aux:
  "of_nat x / of_nat d = of_nat (x div d) + of_nat (x mod d) / of_nat d"
  by (metis add_divide_distrib diff_add_cancel of_nat_div)

lemma(in field_char_0) of_nat_of_nat_div: "d dvd n \ of_nat(n div d) = of_nat n / of_nat d"
  by auto

lemma (in linordered_field) of_nat_div_le_of_nat: "of_nat (n div x) \ of_nat n / of_nat x"
  by (metis le_add_same_cancel1 of_nat_0_le_iff of_nat_of_nat_div_aux zero_le_divide_iff)

lemma real_of_card: "real (card A) = sum (\x. 1) A"
  by simp

lemma nat_less_real_le: "n < m \ real n + 1 \ real m"
  by (metis less_iff_succ_less_eq of_nat_1 of_nat_add of_nat_le_iff)

lemma nat_le_real_less: "n \ m \ real n < real m + 1"
  by (meson nat_less_real_le not_le)

lemma real_of_nat_div: "d dvd n \ real(n div d) = real n / real d"
  by auto

lemma real_binomial_eq_mult_binomial_Suc:
  assumes "k \ n"
  shows "real(n choose k) = (n + 1 - k) / (n + 1) * (Suc n choose k)"
  using assms
  by (simp add: of_nat_binomial_eq_mult_binomial_Suc [of k n] add.commute)

subsection \<open>The Archimedean Property of the Reals\<close>

text \<open>Not actually the reals any more!\<close>
lemma real_arch_inverse:
  fixes e::"'a::archimedean_field"
  shows "0 < e \ (\n::nat. n \ 0 \ 0 < inverse (real n) \ inverse (of_nat n) < e)"
  using reals_Archimedean[of e] less_trans[of 0 "1 / of_nat n" e for n::nat]
  by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)

lemma reals_Archimedean3:
  fixes x::"'a::archimedean_field"
  shows "0 < x \ \y. \n. y < of_nat n * x"
  by (auto intro: ex_less_of_nat_mult)

lemma real_archimedian_rdiv_eq_0:
  fixes x::"'a::archimedean_field"
  assumes "x \ 0" and "\m::nat. m > 0 \ of_nat m * x \ c"
  shows "x = 0"
  by (metis (no_types, opaque_lifting) reals_Archimedean3 order.order_iff_strict le0 le_less_trans not_le assms)

lemma inverse_Suc: "inverse (of_nat (Suc n)) > (0::'a::archimedean_field)"
  using of_nat_0_less_iff positive_imp_inverse_positive zero_less_Suc by blast

lemma Archimedean_eventually_inverse:
  fixes \<epsilon>::"'a::archimedean_field" shows "(\<forall>\<^sub>F n in sequentially. inverse (of_nat (Suc n)) < \<epsilon>) \<longleftrightarrow> 0 < \<epsilon>"
  (is "?lhs=?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding eventually_at_top_dense
    by (metis (no_types, lifting) gt_ex inverse_Suc nat.distinct(1) real_arch_inverse)
next
  assume ?rhs
  then obtain N where "inverse (of_nat (Suc N)) < \"
    using reals_Archimedean by blast
  then have "inverse (of_nat (Suc n)) < \" if "n \ N" for n
    using that Suc_le_mono inverse_Suc inverse_less_imp_less
    by (meson inverse_positive_iff_positive linorder_not_less of_nat_less_iff order_le_less_trans)
  then show ?lhs
    unfolding eventually_sequentially by blast
qed

(*HOL Light's FORALL_POS_MONO_1_EQ*)
text \<open>On the relationship between two different ways of converting to 0\<close>
lemma Inter_eq_Inter_inverse_Suc:
  assumes "\r' r. r' < r \ A r' \ A r"
  shows "\ (A ` {0<..}) = (\n. A(inverse(Suc n)))"
proof
  have "x \ A \"
    if x: "\n. x \ A (inverse (Suc n))" and "\>0" for x and \ :: real
  proof -
    obtain n where "inverse (Suc n) < \"
      using \<open>\<epsilon>>0\<close> reals_Archimedean by blast
    with assms x show ?thesis
      by blast
  qed
  then show "(\n. A(inverse(Suc n))) \ (\\\{0<..}. A \)"
    by auto
qed (use inverse_Suc in fastforce)

subsection \<open>Rationals\<close>

lemma Rats_abs_iff[simp]:
  "\(x::real)\ \ \ \ x \ \"
by(simp add: abs_real_def split: if_splits)

lemma Rats_eq_int_div_int: "\ = {real_of_int i / real_of_int j | i j. j \ 0}" (is "_ = ?S")
proof
  show "\ \ ?S"
  proof
    fix x :: real
    assume "x \ \"
    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)
    then show "x \ ?S"
      using \<open>x = of_rat r\<close> by simp
  qed
next
  show "?S \ \"
  proof (auto simp: Rats_def)
    fix i j :: int
    assume "j \ 0"
    then have "real_of_int i / real_of_int j = of_rat (Fract i j)"
      by (simp add: of_rat_rat)
    then show "real_of_int i / real_of_int j \ range of_rat"
      by blast
  qed
qed

lemma Rats_eq_int_div_nat: "\ = { real_of_int i / real n | i n. n \ 0}"
proof (auto simp: Rats_eq_int_div_int)
  fix i j :: int
  assume "j \ 0"
  show "\(i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i' / real n \ 0 < n"
  proof (cases "j > 0")
    case True
    then have "real_of_int i / real_of_int j = real_of_int i / real (nat j) \ 0 < nat j"
      by simp
    then show ?thesis by blast
  next
    case False
    with \<open>j \<noteq> 0\<close>
    have "real_of_int i / real_of_int j = real_of_int (- i) / real (nat (- j)) \ 0 < nat (- j)"
      by simp
    then show ?thesis by blast
  qed
next
  fix i :: int and n :: nat
  assume "0 < n"
  then have "real_of_int i / real n = real_of_int i / real_of_int(int n) \ int n \ 0"
    by simp
  then show "\i' j. real_of_int i / real n = real_of_int i' / real_of_int j \ j \ 0"
    by blast
qed

lemma Rats_abs_nat_div_natE:
  assumes "x \ \"
  obtains m n :: nat where "n \ 0" and "\x\ = real m / real n" and "coprime m n"
proof -
  from \<open>x \<in> \<rat>\<close> obtain i :: int and n :: nat where "n \<noteq> 0" and "x = real_of_int i / real n"
    by (auto simp add: Rats_eq_int_div_nat)
  then have "\x\ = real (nat \i\) / real n" by simp
  then obtain m :: nat where x_rat: "\x\ = real m / real n" by blast
  let ?gcd = "gcd m n"
  from \<open>n \<noteq> 0\<close> 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 \<open>n \<noteq> 0\<close> and gcd_l have "?gcd * ?l \<noteq> 0" by simp
  then have "?l \ 0" by (blast dest!: mult_not_zero)
  moreover
  have "\x\ = real ?k / real ?l"
  proof -
    from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"
      by (simp add: real_of_nat_div)
    also from gcd_k and gcd_l have "\ = real m / real n" by simp
    also from x_rat have "\ = \x\" ..
    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
  then have "coprime ?k ?l"
    by (simp only: coprime_iff_gcd_eq_1)
  ultimately show ?thesis ..
qed

subsection \<open>Density of the Rational Reals in the Reals\<close>

text \<open>
  This density proof is due to Stefan Richter and was ported by TN.  The
  original source is \<^emph>\<open>Real Analysis\<close> by H.L. Royden.
  It employs the Archimedean property of the reals.\<close>

lemma Rats_dense_in_real:
  fixes x :: real
  assumes "x < y"
  shows "\r\\. x < r \ r < y"
proof -
  from \<open>x < y\<close> have "0 < y - x" by simp
  with reals_Archimedean obtain q :: nat where q: "inverse (real q) < y - x" and "0 < q"
    by blast
  define p where "p = \y * real q\ - 1"
  define r where "r = of_int p / real q"
  from q have "x < y - inverse (real q)"
    by simp
  also from \<open>0 < q\<close> have "y - inverse (real q) \<le> r"
    by (simp add: r_def p_def le_divide_eq left_diff_distrib)
  finally have "x < r" .
  moreover from \<open>0 < q\<close> have "r < y"
    by (simp add: r_def p_def divide_less_eq diff_less_eq less_ceiling_iff [symmetric])
  moreover have "r \ \"
    by (simp add: r_def)
  ultimately show ?thesis by blast
qed

lemma of_rat_dense:
  fixes x y :: real
  assumes "x < y"
  shows "\q :: rat. x < of_rat q \ of_rat q < y"
  using Rats_dense_in_real [OF \<open>x < y\<close>]
  by (auto elim: Rats_cases)

subsection \<open>Numerals and Arithmetic\<close>

declaration \<open>
  K (Lin_Arith.add_inj_const (\<^const_name>\<open>of_nat\<close>, \<^typ>\<open>nat \<Rightarrow> real\<close>)
  #> Lin_Arith.add_inj_const (\<^const_name>\<open>of_int\<close>, \<^typ>\<open>int \<Rightarrow> real\<close>))
\<close>

subsection \<open>Simprules combining \<open>x + y\<close> and \<open>0\<close>\<close> (* FIXME ARE THEY NEEDED? *)

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

lemma real_add_less_0_iff: "x + y < 0 \ y < - x"
  for x y :: real
  by auto

lemma real_0_less_add_iff: "0 < x + y \ - x < y"
  for x y :: real
  by auto

lemma real_add_le_0_iff: "x + y \ 0 \ y \ - x"
  for x y :: real
  by auto

lemma real_0_le_add_iff: "0 \ x + y \ - x \ y"
  for x y :: real
  by auto

lemma mult_ge1_I: "\x\1; y\1\ \ x*y \ (1::real)"
  using mult_mono by fastforce

subsection \<open>Lemmas about powers\<close>

lemma two_realpow_ge_one: "(1::real) \ 2 ^ n"
  by simp

declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]

lemma real_minus_mult_self_le [simp]: "- (u * u) \ x * x"
  for u x :: real
  by (rule order_trans [where y = 0]) auto

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

subsection \<open>Density of the Reals\<close>

lemma field_lbound_gt_zero: "0 < d1 \ 0 < d2 \ \e. 0 < e \ e < d1 \ e < d2"
  for d1 d2 :: "'a::linordered_field"
  by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def)

lemma field_less_half_sum: "x < y \ x < (x + y) / 2"
  for x y :: "'a::linordered_field"
  by auto

lemma field_sum_of_halves: "x / 2 + x / 2 = x"
  for x :: "'a::linordered_field"
  by simp

subsection \<open>Archimedean properties and useful consequences\<close>

text\<open>Bernoulli's inequality\<close>
proposition Bernoulli_inequality:
  fixes x :: "'a :: linordered_field"
  assumes "-1 \ x"
    shows "1 + of_nat n * x \ (1 + x) ^ n"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "1 + of_nat (Suc n) * x \ 1 + of_nat(Suc n) * x + of_nat n * x^2"
    by simp
  also have "... = (1 + x) * (1 + of_nat n * x)"
    by (auto simp: power2_eq_square algebra_simps)
  also have "\ \ (1 + x) ^ Suc n"
    using Suc.hyps assms mult_left_mono by fastforce
  finally show ?case .
qed

corollary Bernoulli_inequality_even:
  fixes x :: "'a :: linordered_field"
  assumes "even n"
    shows "1 + of_nat n * x \ (1 + x) ^ n"
proof (cases "-1 \ x \ n=0")
  case True
  then show ?thesis
    by (auto simp: Bernoulli_inequality)
next
  case False
  then have "of_nat n \ (1::'a)"
    by simp
  with False have "of_nat n * x \ -1"
    by (metis linear minus_zero mult.commute mult.left_neutral mult_left_mono_neg neg_le_iff_le order_trans zero_le_one)
  then have "1 + of_nat n * x \ 0"
    by auto
  also have "... \ (1 + x) ^ n"
    using assms zero_le_even_power by blast
  finally show ?thesis .
qed

corollary real_arch_pow:
  fixes x :: real
  assumes x: "1 < x"
  shows "\n. y < x^n"
proof -
  from x have x0: "x - 1 > 0"
    by arith
  from reals_Archimedean3[OF x0, rule_format, of y]
  obtain n :: nat where n: "y < real n * (x - 1)" by metis
  from x0 have x00: "x- 1 \ -1" by arith
  from Bernoulli_inequality[OF x00, of n] n
  have "y < x^n" by auto
  then show ?thesis by metis
qed

corollary real_arch_pow_inv:
  fixes x y :: real
  assumes y: "y > 0"
    and x1: "x < 1"
  shows "\n. x^n < y"
proof (cases "x > 0")
  case True
  with x1 have ix: "1 < 1/x" by (simp add: field_simps)
  from real_arch_pow[OF ix, of "1/y"]
  obtain n where n: "1/y < (1/x)^n" by blast
  then show ?thesis using y \<open>x > 0\<close>
    by (auto simp add: field_simps)
next
  case False
  with y x1 show ?thesis
    by (metis less_le_trans not_less power_one_right)
qed

lemma forall_pos_mono:
  "(\d e::real. d < e \ P d \ P e) \
    (\<And>n::nat. n \<noteq> 0 \<Longrightarrow> P (inverse (real n))) \<Longrightarrow> (\<And>e. 0 < e \<Longrightarrow> P e)"
  by (metis real_arch_inverse)

lemma forall_pos_mono_1:
  "(\d e::real. d < e \ P d \ P e) \
    (\<And>n. P (inverse (real (Suc n)))) \<Longrightarrow> 0 < e \<Longrightarrow> P e"
  using reals_Archimedean by blast

lemma Archimedean_eventually_pow:
  fixes x::real
  assumes "1 < x"
  shows "\\<^sub>F n in sequentially. b < x ^ n"
proof -
  obtain N where "\n. n\N \ b < x ^ n"
    by (metis assms le_less order_less_trans power_strict_increasing_iff real_arch_pow)
  then show ?thesis
    using eventually_sequentially by blast
qed

lemma Archimedean_eventually_pow_inverse:
  fixes x::real
  assumes "\x\ < 1" "\ > 0"
  shows "\\<^sub>F n in sequentially. \x^n\ < \"
proof (cases "x = 0")
  case True
  then show ?thesis
    by (simp add: assms eventually_at_top_dense zero_power)
next
  case False
  then have "\\<^sub>F n in sequentially. inverse \ < inverse \x\ ^ n"
    by (simp add: Archimedean_eventually_pow assms(1) one_less_inverse)
  then show ?thesis
    by eventually_elim (metis \<open>\<epsilon> > 0\<close> inverse_less_imp_less power_abs power_inverse)
qed

subsection \<open>Floor and Ceiling Functions from the Reals to the Integers\<close>

(* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)

lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \ n < numeral w"
  for n :: nat
  by (metis of_nat_less_iff of_nat_numeral)

lemma numeral_less_real_of_nat_iff [simp]: "numeral w < real n \ numeral w < n"
  for n :: nat
  by (metis of_nat_less_iff of_nat_numeral)

lemma numeral_le_real_of_nat_iff [simp]: "numeral n \ real m \ numeral n \ m"
  for m :: nat
  by (metis not_le real_of_nat_less_numeral_iff)

lemma of_int_floor_cancel [simp]: "of_int \x\ = x \ (\n::int. x = of_int n)"
  by (metis floor_of_int)

lemma of_int_floor [simp]: "a \ \ \ of_int (floor a) = a"
  by (metis Ints_cases of_int_floor_cancel)

lemma floor_frac [simp]: "\frac r\ = 0"
  by (simp add: frac_def)

lemma frac_1 [simp]: "frac 1 = 0"
  by (simp add: frac_def)

lemma frac_in_Rats_iff [simp]:
  fixes r::"'a::{floor_ceiling,field_char_0}"
  shows "frac r \ \ \ r \ \"
  by (metis Rats_add Rats_diff Rats_of_int diff_add_cancel frac_def)

lemma floor_eq: "real_of_int n < x \ x < real_of_int n + 1 \ \x\ = n"
  by linarith

lemma floor_eq2: "real_of_int n \ x \ x < real_of_int n + 1 \ \x\ = n"
  by (fact floor_unique)

lemma floor_eq3: "real n < x \ x < real (Suc n) \ nat \x\ = n"
  by linarith

lemma floor_eq4: "real n \ x \ x < real (Suc n) \ nat \x\ = n"
  by linarith

lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \ real_of_int \r\"
  by linarith

lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int \r\"
  by linarith

lemma real_of_int_floor_add_one_ge [simp]: "r \ real_of_int \r\ + 1"
  by linarith

lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int \r\ + 1"
  by linarith

lemma floor_divide_real_eq_div:
  assumes "0 \ b"
  shows "\a / real_of_int b\ = \a\ div b"
proof (cases "b = 0")
  case True
  then show ?thesis by simp
next
  case False
  with assms have b: "b > 0" by simp
  have "j = i div b"
    if "real_of_int i \ a" "a < 1 + real_of_int i"
      "real_of_int j * real_of_int b \ a" "a < real_of_int b + real_of_int j * real_of_int b"
    for i j :: int
  proof -
    from that have "i < b + j * b"
      by (metis le_less_trans of_int_add of_int_less_iff of_int_mult)
    moreover have "j * b < 1 + i"
    proof -
      have "real_of_int (j * b) < real_of_int i + 1"
        using \<open>a < 1 + real_of_int i\<close> \<open>real_of_int j * real_of_int b \<le> a\<close> by force
      then show "j * b < 1 + i" by linarith
    qed
    ultimately have "(j - i div b) * b \ i mod b" "i mod b < ((j - i div b) + 1) * b"
      by (auto simp: field_simps)
    then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"
      using pos_mod_bound [OF b, of i] pos_mod_sign [OF b, of i]
      by linarith+
    then show ?thesis using b unfolding mult_less_cancel_right by auto
  qed
  with b show ?thesis by (auto split: floor_split simp: field_simps)
qed

lemma floor_one_divide_eq_div_numeral [simp]:
  "\1 / numeral b::real\ = 1 div numeral b"
by (metis floor_divide_of_int_eq of_int_1 of_int_numeral)

lemma floor_minus_one_divide_eq_div_numeral [simp]:
  "\- (1 / numeral b)::real\ = - 1 div numeral b"
by (metis (mono_tags, opaque_lifting) div_minus_right minus_divide_right
    floor_divide_of_int_eq of_int_neg_numeral of_int_1)

lemma floor_divide_eq_div_numeral [simp]:
  "\numeral a / numeral b::real\ = numeral a div numeral b"
by (metis floor_divide_of_int_eq of_int_numeral)

lemma floor_minus_divide_eq_div_numeral [simp]:
  "\- (numeral a / numeral b)::real\ = - numeral a div numeral b"
by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)

lemma of_int_ceiling_cancel [simp]: "of_int \x\ = x \ (\n::int. x = of_int n)"
  using ceiling_of_int by metis

lemma of_int_ceiling [simp]: "a \ \ \ of_int (ceiling a) = a"
  by (metis Ints_cases of_int_ceiling_cancel)

lemma ceiling_eq: "of_int n < x \ x \ of_int n + 1 \ \x\ = n + 1"
  by (simp add: ceiling_unique)

lemma of_int_ceiling_diff_one_le [simp]: "of_int \r\ - 1 \ r"
  by linarith

lemma of_int_ceiling_le_add_one [simp]: "of_int \r\ \ r + 1"
  by linarith

lemma ceiling_le: "x \ of_int a \ \x\ \ a"
  by (simp add: ceiling_le_iff)

lemma ceiling_divide_eq_div: "\of_int a / of_int b\ = - (- a div b)"
  by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)

lemma ceiling_divide_eq_div_numeral [simp]:
  "\numeral a / numeral b :: real\ = - (- numeral a div numeral b)"
  using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp

lemma ceiling_minus_divide_eq_div_numeral [simp]:
  "\- (numeral a / numeral b :: real)\ = - (numeral a div numeral b)"
  using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp

text \<open>
  The following lemmas are remnants of the erstwhile functions natfloor
  and natceiling.
\<close>

lemma nat_floor_neg: "x \ 0 \ nat \x\ = 0"
  for x :: real
  by linarith

lemma le_nat_floor: "real x \ a \ x \ nat \a\"
  by linarith

lemma le_mult_nat_floor: "nat \a\ * nat \b\ \ nat \a * b\"
  by (cases "0 \ a \ 0 \ b")
     (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)

lemma nat_ceiling_le_eq [simp]: "nat \x\ \ a \ x \ real a"
  by linarith

lemma real_nat_ceiling_ge: "x \ real (nat \x\)"
  by linarith

lemma Rats_no_top_le: "\q \ \. x \ q"
  for x :: real
  by (auto intro!: bexI[of _ "of_nat (nat \x\)"]) linarith

lemma Rats_no_bot_less: "\q \ \. q < x" for x :: real
  by (auto intro!: bexI[of _ "of_int (\x\ - 1)"]) linarith

lemma floor_ceiling_diff_le: "0 \ r \ nat\real k - r\ \ k - nat\r\"
  by linarith

lemma floor_ceiling_diff_le': "nat\r - real k\ \ nat\r\ - k"
  by linarith

lemma ceiling_floor_diff_ge: "nat\r - real k\ \ nat\r\ - k"
  by linarith

lemma ceiling_floor_diff_ge': "r \ k \ nat\r - real k\ \ k - nat\r\"
  by linarith

subsection \<open>Exponentiation with floor\<close>

lemma floor_power:
  assumes "x = of_int \x\"
  shows "\x ^ n\ = \x\ ^ n"
proof -
  have "x ^ n = of_int (\x\ ^ n)"
    using assms by (induct n arbitrary: x) simp_all
  then show ?thesis by (metis floor_of_int)
qed

lemma floor_numeral_power [simp]: "\numeral x ^ n\ = numeral x ^ n"
  by (metis floor_of_int of_int_numeral of_int_power)

lemma ceiling_numeral_power [simp]: "\numeral x ^ n\ = numeral x ^ n"
  by (metis ceiling_of_int of_int_numeral of_int_power)

subsection \<open>Implementation of rational real numbers\<close>

text \<open>Formal constructor\<close>

definition Ratreal :: "rat \ real"
  where [code_abbrev, simp]: "Ratreal = real_of_rat"

code_datatype Ratreal

text \<open>Quasi-Numerals\<close>

lemma [code_abbrev]:
  "real_of_rat (numeral k) = numeral k"
  "real_of_rat (- numeral k) = - numeral k"
  "real_of_rat (rat_of_int a) = real_of_int a"
  by simp_all

lemma [code_post]:
  "real_of_rat 0 = 0"
  "real_of_rat 1 = 1"
  "real_of_rat (- 1) = - 1"
  "real_of_rat (1 / numeral k) = 1 / numeral k"
  "real_of_rat (numeral k / numeral l) = numeral k / numeral l"
  "real_of_rat (- (1 / numeral k)) = - (1 / numeral k)"
  "real_of_rat (- (numeral k / numeral l)) = - (numeral k / numeral l)"
  by (simp_all add: of_rat_divide of_rat_minus)

text \<open>Operations\<close>

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

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

instantiation real :: equal
begin

definition "HOL.equal x y \ x - y = 0" for x :: real

instance by standard (simp add: equal_real_def)

lemma real_equal_code [code]: "HOL.equal (Ratreal x) (Ratreal y) \ HOL.equal x y"
  by (simp add: equal_real_def equal)

lemma [code nbe]: "HOL.equal x x \ True"
  for x :: real
  by (rule equal_refl)

end

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

lemma real_less_code [code]: "Ratreal x < Ratreal y \ 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)

lemma real_floor_code [code]: "\Ratreal x\ = \x\"
  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)

text \<open>Quickcheck\<close>

context
  includes term_syntax
begin

definition
  valterm_ratreal :: "rat \ (unit \ Code_Evaluation.term) \ real \ (unit \ Code_Evaluation.term)"
  where [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\} k"

end

instantiation real :: random
begin

context
  includes state_combinator_syntax
begin

definition
  "Quickcheck_Random.random i = Quickcheck_Random.random i \\ (\r. Pair (valterm_ratreal r))"

instance ..

end

end

instantiation real :: exhaustive
begin

definition
  "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (\r. f (Ratreal r)) d"

instance ..

end

instantiation real :: full_exhaustive
begin

definition
  "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (\r. f (valterm_ratreal r)) d"

instance ..

end

instantiation real :: narrowing
begin

definition
  "narrowing_real = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"

instance ..

end

subsection \<open>Setup for Nitpick\<close>

declaration \<open>
  Nitpick_HOL.register_frac_type \<^type_name>\<open>real\<close>
    [(\<^const_name>\<open>zero_real_inst.zero_real\<close>, \<^const_name>\<open>Nitpick.zero_frac\<close>),
     (\<^const_name>\<open>one_real_inst.one_real\<close>, \<^const_name>\<open>Nitpick.one_frac\<close>),
     (\<^const_name>\<open>plus_real_inst.plus_real\<close>, \<^const_name>\<open>Nitpick.plus_frac\<close>),
     (\<^const_name>\<open>times_real_inst.times_real\<close>, \<^const_name>\<open>Nitpick.times_frac\<close>),
     (\<^const_name>\<open>uminus_real_inst.uminus_real\<close>, \<^const_name>\<open>Nitpick.uminus_frac\<close>),
--> --------------------

--> maximum size reached

--> --------------------

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