(*  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

 *)

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

 theory Real

 imports Rat Conditionally_Complete_Lattices

 begin

 text {*

   This theory contains a formalization of the real numbers as

   equivalence classes of Cauchy sequences of rationals.  See

   @{file "~~/src/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_conv_add_uminus 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) \<Rightarrow> bool"

 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_conv_add_uminus by (simp del: add_uminus_conv_diff)

 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

 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

 subsection {* Equivalence relation on Cauchy sequences *}

 definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"

   where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"

 lemma realrelI [intro?]:

   assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"

   shows "realrel X Y"

   using assms unfolding realrel_def by simp

 lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"

   unfolding realrel_def by simp

 lemma symp_realrel: "symp realrel"

   unfolding realrel_def

   by (rule sympI, clarify, drule vanishes_minus, simp)

 lemma transp_realrel: "transp realrel"

   unfolding realrel_def

   apply (rule transpI, clarify)

   apply (drule (1) vanishes_add)

   apply (simp add: algebra_simps)

   done

 lemma part_equivp_realrel: "part_equivp realrel"

   by (fast intro: part_equivpI symp_realrel transp_realrel

     realrel_refl cauchy_const)

 subsection {* The field of real numbers *}

 quotient_type real = "nat \<Rightarrow> rat" / partial: realrel

   morphisms rep_real Real

   by (rule part_equivp_realrel)

 lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> 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 "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"

 proof (induct x)

   case (1 X)

   hence "cauchy X" by (simp add: realrel_def)

   thus "P (Real X)" by (rule assms)

 qed

 lemma eq_Real:

   "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"

   using real.rel_eq_transfer

   unfolding real.pcr_cr_eq cr_real_def fun_rel_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_inverse_zero

 begin

 lift_definition zero_real :: "real" is "\<lambda>n. 0"

   by (simp add: realrel_refl)

 lift_definition one_real :: "real" is "\<lambda>n. 1"

   by (simp add: realrel_refl)

 lift_definition plus_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>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 \<Rightarrow> real" is "\<lambda>X n. - X n"

   unfolding realrel_def minus_diff_minus

   by (simp only: cauchy_minus vanishes_minus simp_thms)

 lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"

   unfolding realrel_def mult_diff_mult

   by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add

     vanishes_mult_bounded cauchy_imp_bounded simp_thms)

 lift_definition inverse_real :: "real \<Rightarrow> real"

   is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"

 proof -

   fix X Y assume "realrel X Y"

   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 "?thesis X Y"

     unfolding realrel_def

     by (simp add: vanishes_diff_inverse X Y XY)

 qed

 definition

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

 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)"

   using assms plus_real.transfer

   unfolding cr_real_eq fun_rel_def by simp

 lemma minus_Real:

   assumes X: "cauchy X"

   shows "- Real X = Real (\<lambda>n. - X n)"

   using assms uminus_real.transfer

   unfolding cr_real_eq fun_rel_def by simp

 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

   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)"

   using assms times_real.transfer

   unfolding cr_real_eq fun_rel_def by simp

 lemma inverse_Real:

   assumes X: "cauchy X"

   shows "inverse (Real X) =

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

   using assms inverse_real.transfer zero_real.transfer

   unfolding cr_real_eq fun_rel_def by (simp split: split_if_asm, metis)

 instance proof

   fix a b c :: real

   show "a + b = b + a"

     by transfer (simp add: add_ac realrel_def)

   show "(a + b) + c = a + (b + c)"

     by transfer (simp add: add_ac 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: mult_ac realrel_def)

   show "a * b = b * a"

     by transfer (simp add: mult_ac realrel_def)

   show "1 * a = a"

     by transfer (simp add: mult_ac realrel_def)

   show "(a + b) * c = a * c + b * c"

     by transfer (simp add: distrib_right realrel_def)

   show "(0\<Colon>real) \<noteq> (1\<Colon>real)"

     by transfer (simp add: realrel_def)

   show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"

     apply transfer

     apply (simp add: realrel_def)

     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 transfer (simp add: realrel_def)

 qed

 end

 subsection {* Positive reals *}

 lift_definition positive :: "real \<Rightarrow> bool"

   is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"

 proof -

   { fix X Y

     assume "realrel X Y"

     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

     hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast

   } note 1 = this

   fix X Y assume "realrel X Y"

   hence "realrel X Y" and "realrel Y X"

     using symp_realrel unfolding symp_def by auto

   thus "?thesis X Y"

     by (safe elim!: 1)

 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)"

   using assms positive.transfer

   unfolding cr_real_eq fun_rel_def by simp

 lemma positive_zero: "\<not> positive 0"

   by transfer auto

 lemma positive_add:

   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x + y)"

 apply transfer

 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 transfer

 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 transfer

 apply (simp add: realrel_def)

 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

     (* FIXME: Procedure int_combine_numerals: c + b - (c + a) \<equiv> b + - a *)

     (* Should produce c + b - (c + a) \<equiv> b - a *)

   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

 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

 instantiation real :: floor_ceiling

 begin

 definition [code del]:

   "floor (x::real) = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"

 instance proof

   fix x :: real

   show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"

     unfolding floor_real_def using floor_exists1 by (rule theI')

 qed

 end

 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) ^ 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

 instantiation real :: linear_continuum

 begin

 subsection{*Supremum of a set of reals*}

 definition

   Sup_real_def: "Sup X \<equiv> LEAST z::real. \<forall>x\<in>X. x\<le>z"

 definition

   Inf_real_def: "Inf (X::real set) \<equiv> - Sup (uminus ` X)"

 instance

 proof

   { fix x :: real and X :: "real set"

     assume x: "x \<in> X" "bdd_above X"

     then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"

       using complete_real[of X] unfolding bdd_above_def by blast

     then show "x \<le> Sup X"

       unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }

   note Sup_upper = this

   { fix z :: real and X :: "real set"

     assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"

     then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"

       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 "... \<le> z"

       by blast

     finally show "Sup X \<le> z" . }

   note Sup_least = this

   { fix x z :: real and X :: "real set"

     assume x: "x \<in> X" and [simp]: "bdd_below X"

     have "-x \<le> Sup (uminus ` X)"

       by (rule Sup_upper) (auto simp add: image_iff x)

     then show "Inf X \<le> x" 

       by (auto simp add: Inf_real_def) }

   { fix z :: real and X :: "real set"

     assume x: "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"

     have "Sup (uminus ` X) \<le> -z"

       using x z by (force intro: Sup_least)

     then show "z \<le> Inf X" 

         by (auto simp add: Inf_real_def) }

   show "\<exists>a b::real. a \<noteq> b"

     using zero_neq_one by blast

 qed

 end

 subsection {* Hiding implementation details *}

 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{*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 simp (* solved by linordered_ring_le_cancel_factor simproc *)

 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"

 declare [[coercion_enabled]]

 declare [[coercion "real::nat\<Rightarrow>real"]]

 declare [[coercion "real::int\<Rightarrow>real"]]

 declare [[coercion "int"]]

 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)"]]

 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 one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"

   unfolding real_of_one[symmetric] real_of_int_less_iff ..

 lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"

   unfolding real_of_one[symmetric] real_of_int_le_iff ..

 lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"

   unfolding real_of_one[symmetric] real_of_int_less_iff ..

 lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"

   unfolding real_of_one[symmetric] real_of_int_le_iff ..

 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: "(real (x::int)) / (real d) = 

     real (x div d) + (real (x mod d)) / (real d)"

 proof -

   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)

 qed

 lemma real_of_int_div: "(d :: int) dvd n ==>

     real(n div d) = real n / real d"

   apply (subst real_of_int_div_aux)

   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 (subst zero_le_divide_iff)

   apply auto

   apply (simp add: algebra_simps)

   apply (subst real_of_int_div_aux)

   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 (simp add: algebra_simps)

   apply (subst real_of_int_div_aux)

   apply (auto simp add: divide_le_eq intro: order_less_imp_le)

 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)

 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: "(real (x::nat)) / (real d) = 

     real (x div d) + (real (x mod d)) / (real d)"

 proof -

   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)

 qed

 lemma real_of_nat_div: "(d :: nat) dvd n ==>

     real(n div d) = real n / real d"

   by (subst real_of_nat_div_aux)

     (auto simp add: dvd_eq_mod_eq_0 [symmetric])

 lemma real_of_nat_div2:

   "0 <= real (n::nat) / real (x) - real (n div x)"

 apply (simp add: algebra_simps)

 apply (subst real_of_nat_div_aux)

 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

 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_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_of_nat_eq)

 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 {* The Archimedean Property of the Reals *}

 theorem reals_Archimedean:

   assumes x_pos: "0 < x"

   shows "\<exists>n. inverse (real (Suc n)) < x"

   unfolding real_of_nat_def using x_pos

   by (rule ex_inverse_of_nat_Suc_less)

 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"

   unfolding real_of_nat_def by (rule ex_less_of_nat)

 lemma reals_Archimedean3:

   assumes x_greater_zero: "0 < x"

   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"

   unfolding real_of_nat_def using `0 < x`

   by (auto intro: ex_less_of_nat_mult)

 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{*Density of the Rational Reals in the Reals*}

 text{* This density proof is due to Stefan Richter and was ported by TN.  The

 original source is \emph{Real Analysis} by H.L. Royden.

 It employs the Archimedean property of the reals. *}

 lemma Rats_dense_in_real:

   fixes x :: real

   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"

 proof -

   from `x<y` have "0 < y-x" by simp

   with reals_Archimedean obtain q::nat 

     where q: "inverse (real q) < y-x" and "0 < q" by auto

   def p \<equiv> "ceiling (y * real q) - 1"

   def r \<equiv> "of_int p / real q"

   from q have "x < y - inverse (real q)" by simp

   also have "y - inverse (real q) \<le> r"

     unfolding r_def p_def

     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling `0 < q`)

   finally have "x < r" .

   moreover have "r < y"

     unfolding r_def p_def

     by (simp add: divide_less_eq diff_less_eq `0 < q`

       less_ceiling_iff [symmetric])

   moreover from r_def have "r \<in> \<rat>" by simp

   ultimately show ?thesis by fast

 qed

 subsection{*Numerals and Arithmetic*}

 lemma [code_abbrev]:

   "real_of_int (numeral k) = numeral k"

   "real_of_int (neg_numeral k) = neg_numeral k"

   by simp_all

 text{*Collapse applications of @{term real} to @{term number_of}*}

 lemma real_numeral [simp]:

   "real (numeral v :: int) = numeral v"

   "real (neg_numeral v :: int) = neg_numeral v"

 by (simp_all add: real_of_int_def)

 lemma real_of_nat_numeral [simp]:

   "real (numeral v :: nat) = numeral v"

 by (simp add: real_of_nat_def)

 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_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)}]

   #> 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?*}

 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 {* 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 {* FIXME: declare this [simp] for all types, or not at all *}

 lemma realpow_two_sum_zero_iff [simp]:

      "(x\<^sup>2 + y\<^sup>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\<^sup>2 \<le> (x::real)\<^sup>2"

 by (auto simp add: power2_eq_square)

 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:

   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"

   unfolding real_of_nat_le_iff[symmetric] by simp

 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:

   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"

   unfolding real_of_nat_le_iff[symmetric] by simp

 lemma numeral_power_le_real_of_int_cancel_iff[simp]:

   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"

   unfolding real_of_int_le_iff[symmetric] by simp

 lemma real_of_int_le_numeral_power_cancel_iff[simp]:

   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"

   unfolding real_of_int_le_iff[symmetric] by simp

 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:

   "(neg_numeral x::real) ^ n \<le> real a \<longleftrightarrow> (neg_numeral x::int) ^ n \<le> a"

   unfolding real_of_int_le_iff[symmetric] by simp

 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:

   "real a \<le> (neg_numeral x::real) ^ n \<longleftrightarrow> a \<le> (neg_numeral x::int) ^ n"

   unfolding real_of_int_le_iff[symmetric] by simp

 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

 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

   by simp

 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: "(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: "~ 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{*Floor and Ceiling Functions from the Reals to the Integers*}

 (* FIXME: theorems for negative numerals *)

 lemma numeral_less_real_of_int_iff [simp]:

      "((numeral n) < real (m::int)) = (numeral n < m)"

 apply auto

 apply (rule real_of_int_less_iff [THEN iffD1])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

 done

 lemma numeral_less_real_of_int_iff2 [simp]:

      "(real (m::int) < (numeral n)) = (m < numeral n)"

 apply auto

 apply (rule real_of_int_less_iff [THEN iffD1])

 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

 done

 lemma numeral_le_real_of_int_iff [simp]:

      "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"

 by (simp add: linorder_not_less [symmetric])

 lemma numeral_le_real_of_int_iff2 [simp]:

      "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"

 by (simp add: linorder_not_less [symmetric])

 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"

 unfolding real_of_nat_def by simp

 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"

 unfolding real_of_nat_def by (simp add: floor_minus)

 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"

 unfolding real_of_int_def by simp

 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"

 unfolding real_of_int_def by (simp add: floor_minus)

 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"

 unfolding real_of_int_def by (rule floor_exists)

 lemma lemma_floor:

   assumes a1: "real m \<le> r" and a2: "r < real n + 1"

   shows "m \<le> (n::int)"

 proof -

   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)

   also have "... = real (n + 1)" by simp

   finally have "m < n + 1" by (simp only: real_of_int_less_iff)

   thus ?thesis by arith

 qed

 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"

 unfolding real_of_int_def by (rule of_int_floor_le)

 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"

 by (auto intro: lemma_floor)

 lemma real_of_int_floor_cancel [simp]:

     "(real (floor x) = x) = (\<exists>n::int. x = real n)"

   using floor_real_of_int by metis

 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"

   unfolding real_of_int_def using floor_unique [of n x] by simp

 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"

   unfolding real_of_int_def by (rule floor_unique)

 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"

 apply (rule inj_int [THEN injD])

 apply (simp add: real_of_nat_Suc)

 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])

 done

 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"

 apply (drule order_le_imp_less_or_eq)

 apply (auto intro: floor_eq3)

 done

 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"

   unfolding real_of_int_def using floor_correct [of r] by simp

 lemma le_floor: "real a <= x ==> a <= floor x"

   unfolding real_of_int_def by (simp add: le_floor_iff)

 lemma real_le_floor: "a <= floor x ==> real a <= x"

   unfolding real_of_int_def by (simp add: le_floor_iff)

 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"

   unfolding real_of_int_def by (rule le_floor_iff)

 lemma floor_less_eq: "(floor x < a) = (x < real a)"

   unfolding real_of_int_def by (rule floor_less_iff)

 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"

   unfolding real_of_int_def by (rule less_floor_iff)

 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"

   unfolding real_of_int_def by (rule floor_le_iff)

 lemma floor_add [simp]: "floor (x + real a) = floor x + a"

   unfolding real_of_int_def by (rule floor_add_of_int)

 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"

   unfolding real_of_int_def by (rule floor_diff_of_int)

 lemma le_mult_floor:

   assumes "0 \<le> (a :: real)" and "0 \<le> b"

   shows "floor a * floor b \<le> floor (a * b)"

 proof -

   have "real (floor a) \<le> a"

     and "real (floor b) \<le> b" by auto

   hence "real (floor a * floor b) \<le> a * b"

     using assms by (auto intro!: mult_mono)

   also have "a * b < real (floor (a * b) + 1)" by auto

   finally show ?thesis unfolding real_of_int_less_iff by simp

 qed

 lemma floor_divide_eq_div:

   "floor (real a / real b) = a div b"

 proof cases

   assume "b \<noteq> 0 \<or> b dvd a"

   with real_of_int_div3[of a b] show ?thesis

     by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)

        (metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject

               real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)

 qed (auto simp: real_of_int_div)

 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"

   unfolding real_of_nat_def by simp

 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"

   unfolding real_of_int_def by (rule le_of_int_ceiling)

 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"

   unfolding real_of_int_def by simp

 lemma real_of_int_ceiling_cancel [simp]:

      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"

   using ceiling_real_of_int by metis

 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"

   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"

   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"

   unfolding real_of_int_def using ceiling_unique [of n x] by simp

 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"

   unfolding real_of_int_def using ceiling_correct [of r] by simp

 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"

   unfolding real_of_int_def using ceiling_correct [of r] by simp

 lemma ceiling_le: "x <= real a ==> ceiling x <= a"

   unfolding real_of_int_def by (simp add: ceiling_le_iff)

 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"

   unfolding real_of_int_def by (simp add: ceiling_le_iff)

 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"

   unfolding real_of_int_def by (rule ceiling_le_iff)

 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"

   unfolding real_of_int_def by (rule less_ceiling_iff)

 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"

   unfolding real_of_int_def by (rule ceiling_less_iff)

 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"

   unfolding real_of_int_def by (rule le_ceiling_iff)

 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"

   unfolding real_of_int_def by (rule ceiling_add_of_int)

 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"

   unfolding real_of_int_def by (rule ceiling_diff_of_int)

 subsubsection {* Versions for the natural numbers *}

 definition

   natfloor :: "real => nat" where

   "natfloor x = nat(floor x)"

 definition

   natceiling :: "real => nat" where

   "natceiling x = nat(ceiling x)"

 lemma natfloor_zero [simp]: "natfloor 0 = 0"

   by (unfold natfloor_def, simp)

 lemma natfloor_one [simp]: "natfloor 1 = 1"

   by (unfold natfloor_def, simp)

 lemma zero_le_natfloor [simp]: "0 <= natfloor x"

   by (unfold natfloor_def, simp)

 lemma natfloor_numeral_eq [simp]: "natfloor (numeral n) = numeral n"

   by (unfold natfloor_def, simp)

 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"

   by (unfold natfloor_def, simp)

 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"

   by (unfold natfloor_def, simp)

 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"

   unfolding natfloor_def by simp

 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"

   unfolding natfloor_def by (intro nat_mono floor_mono)

 lemma le_natfloor: "real x <= a ==> x <= natfloor a"

   apply (unfold natfloor_def)

   apply (subst nat_int [THEN sym])

   apply (rule nat_mono)

   apply (rule le_floor)

   apply simp

 done

 lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"

   unfolding natfloor_def real_of_nat_def

   by (simp add: nat_less_iff floor_less_iff)

 lemma less_natfloor:

   assumes "0 \<le> x" and "x < real (n :: nat)"

   shows "natfloor x < n"

   using assms by (simp add: natfloor_less_iff)

 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"

   apply (rule iffI)

   apply (rule order_trans)

   prefer 2

   apply (erule real_natfloor_le)

   apply (subst real_of_nat_le_iff)

   apply assumption

   apply (erule le_natfloor)

 done

 lemma le_natfloor_eq_numeral [simp]:

     "~ neg((numeral n)::int) ==> 0 <= x ==>

       (numeral n <= natfloor x) = (numeral n <= x)"

   apply (subst le_natfloor_eq, assumption)

   apply simp

 done

 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"

   apply (case_tac "0 <= x")

   apply (subst le_natfloor_eq, assumption, simp)

   apply (rule iffI)

   apply (subgoal_tac "natfloor x <= natfloor 0")

   apply simp

   apply (rule natfloor_mono)

   apply simp

   apply simp

 done

 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"

   unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])

 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"

   apply (case_tac "0 <= x")

   apply (unfold natfloor_def)

   apply simp

   apply simp_all

 done

 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"

 using real_natfloor_add_one_gt by (simp add: algebra_simps)

 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"

   apply (subgoal_tac "z < real(natfloor z) + 1")

   apply arith

   apply (rule real_natfloor_add_one_gt)

 done

 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"

   unfolding natfloor_def

   unfolding real_of_int_of_nat_eq [symmetric] floor_add

   by (simp add: nat_add_distrib)

 lemma natfloor_add_numeral [simp]:

     "~neg ((numeral n)::int) ==> 0 <= x ==>

       natfloor (x + numeral n) = natfloor x + numeral n"

   by (simp add: natfloor_add [symmetric])

 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"

   by (simp add: natfloor_add [symmetric] del: One_nat_def)

 lemma natfloor_subtract [simp]:

     "natfloor(x - real a) = natfloor x - a"

   unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract

   by simp

 lemma natfloor_div_nat:

   assumes "1 <= x" and "y > 0"

   shows "natfloor (x / real y) = natfloor x div y"

 proof (rule natfloor_eq)

   have "(natfloor x) div y * y \<le> natfloor x"

     by (rule add_leD1 [where k="natfloor x mod y"], simp)

   thus "real (natfloor x div y) \<le> x / real y"

     using assms by (simp add: le_divide_eq le_natfloor_eq)

   have "natfloor x < (natfloor x) div y * y + y"

     apply (subst mod_div_equality [symmetric])

     apply (rule add_strict_left_mono)

     apply (rule mod_less_divisor)

     apply fact

     done

   thus "x / real y < real (natfloor x div y) + 1"

     using assms

     by (simp add: divide_less_eq natfloor_less_iff distrib_right)

 qed

 lemma le_mult_natfloor:

   shows "natfloor a * natfloor b \<le> natfloor (a * b)"

   by (cases "0 <= a & 0 <= b")

     (auto simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le natfloor_neg)

 lemma natceiling_zero [simp]: "natceiling 0 = 0"

   by (unfold natceiling_def, simp)

 lemma natceiling_one [simp]: "natceiling 1 = 1"

   by (unfold natceiling_def, simp)

 lemma zero_le_natceiling [simp]: "0 <= natceiling x"

   by (unfold natceiling_def, simp)

 lemma natceiling_numeral_eq [simp]: "natceiling (numeral n) = numeral n"

   by (unfold natceiling_def, simp)

 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"

   by (unfold natceiling_def, simp)

 lemma real_natceiling_ge: "x <= real(natceiling x)"

   unfolding natceiling_def by (cases "x < 0", simp_all)

 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"

   unfolding natceiling_def by simp

 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"

   unfolding natceiling_def by (intro nat_mono ceiling_mono)

 lemma natceiling_le: "x <= real a ==> natceiling x <= a"

   unfolding natceiling_def real_of_nat_def

   by (simp add: nat_le_iff ceiling_le_iff)

 lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"

   unfolding natceiling_def real_of_nat_def

   by (simp add: nat_le_iff ceiling_le_iff)

 lemma natceiling_le_eq_numeral [simp]:

     "~ neg((numeral n)::int) ==>

       (natceiling x <= numeral n) = (x <= numeral n)"

   by (simp add: natceiling_le_eq)

 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"

   unfolding natceiling_def

   by (simp add: nat_le_iff ceiling_le_iff)

 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"

   unfolding natceiling_def

   by (simp add: ceiling_eq2 [where n="int n"])

 lemma natceiling_add [simp]: "0 <= x ==>

     natceiling (x + real a) = natceiling x + a"

   unfolding natceiling_def

   unfolding real_of_int_of_nat_eq [symmetric] ceiling_add

   by (simp add: nat_add_distrib)

 lemma natceiling_add_numeral [simp]:

     "~ neg ((numeral n)::int) ==> 0 <= x ==>

       natceiling (x + numeral n) = natceiling x + numeral n"

   by (simp add: natceiling_add [symmetric])

 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

   by (simp add: natceiling_add [symmetric] del: One_nat_def)

 lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"

   unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract

   by simp

 subsection {* Exponentiation with floor *}

 lemma floor_power:

   assumes "x = real (floor x)"

   shows "floor (x ^ n) = floor x ^ n"

 proof -

   have *: "x ^ n = real (floor x ^ n)"

     using assms by (induct n arbitrary: x) simp_all

   show ?thesis unfolding real_of_int_inject[symmetric]

     unfolding * floor_real_of_int ..

 qed

 lemma natfloor_power:

   assumes "x = real (natfloor x)"

   shows "natfloor (x ^ n) = natfloor x ^ n"

 proof -

   from assms have "0 \<le> floor x" by auto

   note assms[unfolded natfloor_def real_nat_eq_real[OF `0 \<le> floor x`]]

   from floor_power[OF this]

   show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]

     by simp

 qed

 subsection {* Implementation of rational real numbers *}

 text {* Formal constructor *}

 definition Ratreal :: "rat \<Rightarrow> real" where

   [code_abbrev, simp]: "Ratreal = of_rat"

 code_datatype Ratreal

 text {* Numerals *}

 lemma [code_abbrev]:

   "(of_rat (of_int a) :: real) = of_int a"

   by simp

 lemma [code_abbrev]:

   "(of_rat 0 :: real) = 0"

   by simp

 lemma [code_abbrev]:

   "(of_rat 1 :: real) = 1"

   by simp

 lemma [code_abbrev]:

   "(of_rat (numeral k) :: real) = numeral k"

   by simp

 lemma [code_abbrev]:

   "(of_rat (neg_numeral k) :: real) = neg_numeral k"

   by simp

 lemma [code_post]:

   "(of_rat (0 / r)  :: real) = 0"

   "(of_rat (r / 0)  :: real) = 0"

   "(of_rat (1 / 1)  :: real) = 1"

   "(of_rat (numeral k / 1) :: real) = numeral k"

   "(of_rat (neg_numeral k / 1) :: real) = neg_numeral k"

   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"

   "(of_rat (1 / neg_numeral k) :: real) = 1 / neg_numeral k"

   "(of_rat (numeral k / numeral l)  :: real) = numeral k / numeral l"

   "(of_rat (numeral k / neg_numeral l)  :: real) = numeral k / neg_numeral l"

   "(of_rat (neg_numeral k / numeral l)  :: real) = neg_numeral k / numeral l"

   "(of_rat (neg_numeral k / neg_numeral l)  :: real) = neg_numeral k / neg_numeral l"

   by (simp_all add: of_rat_divide)

 text {* Operations *}

 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\<Colon>real) y \<longleftrightarrow> x - y = 0"

 instance proof

 qed (simp add: equal_real_def)

 lemma real_equal_code [code]:

   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"

   by (simp add: equal_real_def equal)

 lemma [code nbe]:

   "HOL.equal (x::real) x \<longleftrightarrow> True"

   by (rule equal_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)

 lemma real_floor_code [code]: "floor (Ratreal x) = floor 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 {* Quickcheck *}

 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.random i = Quickcheck_Random.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)

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

 instance ..

 end

 subsection {* Setup for Nitpick *}

 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 inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),

     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),

     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]

 *}

 lemmas [nitpick_unfold] = inverse_real_inst.inverse_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

 ML_file "Tools/SMT/smt_real.ML"

 setup SMT_Real.setup

 end