(* 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
\<^file>\<open>~~/src/HOL/ex/Dedekind_Real.thy\<close> 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 "\<lbrakk>n \<le> m; n\<noteq>0\<rbrakk> \<Longrightarrow> 1 / of_nat m \<le> (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 \<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 \<open>Sequences that converge to zero\<close>
definition vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
where "vanishes X \<longleftrightarrow> (\<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: "vanishes X \<Longrightarrow> 0 < r \<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"
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 \<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"
by (simp only: r)
qed
then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..
qed
lemma vanishes_diff:
assumes "vanishes X" "vanishes Y"
shows "vanishes (\<lambda>n. X n - Y n)"
unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus assms)
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 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: "\<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)
then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
qed
subsection \<open>Cauchy sequences\<close>
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: "cauchy X \<Longrightarrow> 0 < r \<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" by (simp only: r)
qed
then show "\<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 "cauchy X" "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})" .
then show "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"
then have "\<bar>X n\<bar> \<le> Max (abs ` X ` {..k})" by simp
then show "\<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_all add: k \<open>k \<le> n\<close>)
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 blast
obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < 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: "\<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"
by (simp only: r)
qed
then show "\<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 \<open>0 < r\<close> 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 blast
have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"
using i \<open>i \<le> k\<close> by auto
have "X k \<le> - r \<or> r \<le> X k"
using \<open>r \<le> \<bar>X k\<bar>\<close> by auto
then have "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
unfolding \<open>r = s + t\<close> using k by auto
then have "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
then show "\<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"
and 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 (elim ex_forward conj_forward asm_rl) auto
lemma cauchy_inverse [simp]:
assumes X: "cauchy X"
and 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 blast
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: \<open>0 < r\<close> b)
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 i j b * s)
finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" by (simp only: r)
qed
then show "\<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"
and Y: "cauchy Y" "\<not> vanishes Y"
and 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 blast
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 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: "\<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"
with i j a b have "X n \<noteq> 0" and "Y n \<noteq> 0"
by auto
then have "\<bar>inverse (X n) - inverse (Y n)\<bar> = inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
by (simp add: inverse_diff_inverse abs_mult)
also have "\<dots> < 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 "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
qed
then show "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
qed
subsection \<open>Equivalence relation on Cauchy sequences\<close>
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?]: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> vanishes (\<lambda>n. X n - Y n) \<Longrightarrow> realrel X Y"
by (simp add: realrel_def)
lemma realrel_refl: "cauchy X \<Longrightarrow> 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 \<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)
then have "cauchy X" by (simp add: realrel_def)
then show "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 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 "\<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"
proof -
fix f1 f2 f3 f4
have "\<lbrakk>cauchy f1; cauchy f4; vanishes (\<lambda>n. f1 n - f2 n); vanishes (\<lambda>n. f3 n - f4 n)\<rbrakk>
\<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 "\<lbrakk>realrel f1 f2; realrel f3 f4\<rbrakk> \<Longrightarrow> realrel (\<lambda>n. f1 n * f3 n) (\<lambda>n. f2 n * f4 n)"
by (simp add: mult.commute realrel_def mult_diff_mult)
qed
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"
then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
by (simp_all add: realrel_def)
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
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 \<Longrightarrow> cauchy Y \<Longrightarrow> Real X + Real Y = Real (\<lambda>n. X n + Y n)"
using plus_real.transfer by (simp add: cr_real_eq rel_fun_def)
lemma minus_Real: "cauchy X \<Longrightarrow> - Real X = Real (\<lambda>n. - X n)"
using uminus_real.transfer by (simp add: cr_real_eq rel_fun_def)
lemma diff_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X - Real Y = Real (\<lambda>n. X n - Y n)"
by (simp add: minus_Real add_Real minus_real_def)
lemma mult_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X * Real Y = Real (\<lambda>n. X n * Y n)"
using times_real.transfer by (simp add: cr_real_eq rel_fun_def)
lemma inverse_Real:
"cauchy X \<Longrightarrow> inverse (Real X) = (if vanishes X then 0 else Real (\<lambda>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) \<noteq> (1::real)"
by transfer (simp add: realrel_def)
have "vanishes (\<lambda>n. inverse (X n) * X n - 1)" if X: "cauchy X" "\<not> vanishes X" for X
proof (rule vanishesI)
fix r::rat
assume "0 < r"
obtain b k where "b>0" "\<forall>n\<ge>k. b < \<bar>X n\<bar>"
using X cauchy_not_vanishes by blast
then show "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) * X n - 1\<bar> < r"
using \<open>0 < r\<close> by force
qed
then show "a \<noteq> 0 \<Longrightarrow> 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 \<Rightarrow> bool"
is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
proof -
have 1: "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n"
if *: "realrel X Y" and **: "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n" for X Y
proof -
from * have XY: "vanishes (\<lambda>n. X n - Y n)"
by (simp_all add: realrel_def)
from ** obtain r i where "0 < r" and i: "\<forall>n\<ge>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: "\<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
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 \<Longrightarrow> positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
using positive.transfer by (simp add: cr_real_eq rel_fun_def)
lemma positive_zero: "\<not> positive 0"
by transfer auto
lemma positive_add:
assumes "positive x" "positive y" shows "positive (x + y)"
proof -
have *: "\<lbrakk>\<forall>n\<ge>i. a < x n; \<forall>n\<ge>j. b < y n; 0 < a; 0 < b; n \<ge> max i j\<rbrakk>
\<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 *: "\<lbrakk>\<forall>n\<ge>i. a < x n; \<forall>n\<ge>j. b < y n; 0 < a; 0 < b; n \<ge> max i j\<rbrakk>
\<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: "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> 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 \<longleftrightarrow> positive (y - x)"
definition "x \<le> y \<longleftrightarrow> x < y \<or> x = y" for x y :: real
definition "\<bar>a\<bar> = (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 "\<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"
"a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c" "a \<le> a"
"a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"
"a \<le> b \<Longrightarrow> c + a \<le> 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 \<le> b \<or> b \<le> a"
by (auto dest!: positive_minus simp: less_eq_real_def less_real_def)
show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> 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 \<Rightarrow> real \<Rightarrow> real) = min"
definition "(sup :: real \<Rightarrow> real \<Rightarrow> 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 (\<lambda>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 (\<lambda>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 (\<lambda>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 "\<exists>z. x \<le> of_int z" for x :: real
proof (induct x)
case (1 X)
then obtain b where "0 < b" and b: "\<And>n. \<bar>X n\<bar> < b"
by (blast dest: cauchy_imp_bounded)
then have "Real X < of_int (\<lceil>b\<rceil> + 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]: "\<lfloor>x::real\<rfloor> = (THE z. of_int z \<le> x \<and> x < of_int (z + 1))"
instance
proof
show "of_int \<lfloor>x\<rfloor> \<le> x \<and> x < of_int (\<lfloor>x\<rfloor> + 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 "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)" (is "?lhs = ?rhs")
unfolding positive_Real [OF assms]
proof (intro iffI allI notI impI)
show "\<exists>k. \<forall>n\<ge>k. X n \<le> r" if r: "\<not> (\<exists>r>0. \<exists>k. \<forall>n\<ge>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: "\<And>m n. \<lbrakk>m\<ge>k; n\<ge>k\<rbrakk> \<Longrightarrow> \<bar>X m - X n\<bar> < t"
using cauchyD [OF assms \<open>t > 0\<close>] by blast
obtain n where "n \<ge> k" "X n \<le> s"
by (meson r \<open>0 < s\<close> not_less)
then have "X l \<le> r" if "l \<ge> 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 \<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 assms)
apply (simp add: diff_le_eq ac_simps)
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)"
then have 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)
then have "\<exists>k. \<forall>n\<ge>k. X n \<le> 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: "\<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
then show ?thesis ..
qed
then show "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)
then show ?thesis by simp
qed
lemma less_RealD:
assumes "cauchy Y"
shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"
apply (erule contrapos_pp)
apply (simp add: not_less)
apply (erule Real_leI [OF assms])
done
lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n"
apply (induct n)
apply simp
apply (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
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) ..
define P where "P x \<longleftrightarrow> (\<forall>y\<in>S. y \<le> of_rat x)" for x
obtain a where a: "\<not> P a"
proof
have "of_int \<lfloor>x - 1\<rfloor> \<le> x - 1" by (rule of_int_floor_le)
also have "x - 1 < x" by simp
finally have "of_int \<lfloor>x - 1\<rfloor> < x" .
then have "\<not> x \<le> of_int \<lfloor>x - 1\<rfloor>" by (simp only: not_le)
then show "\<not> P (of_int \<lfloor>x - 1\<rfloor>)"
unfolding P_def of_rat_of_int_eq using x by blast
qed
obtain b where b: "P b"
proof
show "P (of_int \<lceil>z\<rceil>)"
unfolding P_def of_rat_of_int_eq
proof
fix y assume "y \<in> S"
then have "y \<le> z" using z by simp
also have "z \<le> of_int \<lceil>z\<rceil>" by (rule le_of_int_ceiling)
finally show "y \<le> of_int \<lceil>z\<rceil>" .
qed
qed
define avg where "avg x y = x/2 + y/2" for x y :: rat
define bisect where "bisect = (\<lambda>(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]: "\<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: "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: "\<exists>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 "\<exists>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: divide_simps)
qed
have PA: "\<not> 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 \<le> A j \<and> B j \<le> 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 \<le> A j" and B_mono: "B j \<le> B i" if "i \<le> j" for i j
by (metis eq_refl le_neq_implies_less that)+
have cauchy_lemma: "cauchy X" if *: "\<And>n i. i\<ge>n \<Longrightarrow> A n \<le> X i \<and> X i \<le> 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 "\<bar>X m - X n\<bar> < r" if "m\<ge>k" "n\<ge>k" for m n
proof -
have "\<bar>X m - X n\<bar> \<le> 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 "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 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 "\<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] \<open>cauchy B\<close>
by (simp add: le_RealI)
qed
moreover have "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
by (meson PA Real_leI P_def \<open>cauchy A\<close> le_cases order.trans)
moreover 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 blast
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)
also note k
finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
qed
then show "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
qed
then have "Real B = Real A"
by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width)
ultimately show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
by force
qed
instantiation real :: linear_continuum
begin
subsection \<open>Supremum of a set of reals\<close>
definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
definition "Inf X = - Sup (uminus ` X)" for X :: "real set"
instance
proof
show Sup_upper: "x \<le> Sup X"
if "x \<in> X" "bdd_above X"
for x :: real and X :: "real set"
proof -
from that obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
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 \<le> z"
if "X \<noteq> {}" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
for z :: real and X :: "real set"
proof -
from that obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
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 "\<dots> \<le> z"
by blast
finally show ?thesis .
qed
show "Inf X \<le> x" if "x \<in> 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 \<le> Inf X" if "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"
for z :: real and X :: "real set"
using Sup_least [of "uminus ` X" "- z"] by (force simp: Inf_real_def that)
show "\<exists>a b::real. a \<noteq> 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>More Lemmas\<close>
text \<open>BH: These lemmas should not be necessary; they should be
covered by existing simp rules and simplification procedures.\<close>
lemma real_mult_less_iff1 [simp]: "0 < z \<Longrightarrow> x * z < y * z \<longleftrightarrow> x < y"
for x y z :: real
by simp (* solved by linordered_ring_less_cancel_factor simproc *)
lemma real_mult_le_cancel_iff1 [simp]: "0 < z \<Longrightarrow> x * z \<le> y * z \<longleftrightarrow> x \<le> y"
for x y z :: real
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
lemma real_mult_le_cancel_iff2 [simp]: "0 < z \<Longrightarrow> z * x \<le> z * y \<longleftrightarrow> x \<le> y"
for x y z :: real
by simp (* solved by linordered_ring_le_cancel_factor simproc *)
subsection \<open>Embedding numbers into the Reals\<close>
abbreviation real_of_nat :: "nat \<Rightarrow> real"
where "real_of_nat \<equiv> of_nat"
abbreviation real :: "nat \<Rightarrow> real"
where "real \<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"
declare [[coercion_enabled]]
declare [[coercion "of_nat :: nat \<Rightarrow> int"]]
declare [[coercion "of_nat :: nat \<Rightarrow> real"]]
declare [[coercion "of_int :: int \<Rightarrow> 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 "\<lambda>f g h x. g (h (f x))"]]
declare [[coercion_map "\<lambda>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 \<longleftrightarrow> real_of_int n + 1 \<le> real_of_int m"
proof -
have "(0::real) \<le> 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 \<le> m \<longleftrightarrow> real_of_int n < real_of_int m + 1"
by (meson int_less_real_le not_le)
lemma real_of_int_div_aux:
"(real_of_int x) / (real_of_int d) =
real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)"
proof -
have "x = (x div d) * d + x mod d"
by auto
then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)"
by (metis of_int_add of_int_mult)
then have "real_of_int x / real_of_int d = \<dots> / real_of_int d"
by simp
then show ?thesis
by (auto simp add: add_divide_distrib algebra_simps)
qed
lemma real_of_int_div:
"d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int
by (simp add: real_of_int_div_aux)
lemma real_of_int_div2: "0 \<le> 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) \<le> 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) \<le> 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 real_of_card: "real (card A) = sum (\<lambda>x. 1) A"
by simp
lemma nat_less_real_le: "n < m \<longleftrightarrow> real n + 1 \<le> real m"
by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)
lemma nat_le_real_less: "n \<le> m \<longleftrightarrow> real n < real m + 1"
for m n :: nat
by (meson nat_less_real_le not_le)
lemma real_of_nat_div_aux: "real x / 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 (metis of_nat_add of_nat_mult)
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 dvd n \<Longrightarrow> 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 \<le> real n / real x - real (n div x)" for n x :: nat
apply (simp add: algebra_simps)
by (metis floor_divide_of_nat_eq of_int_floor_le of_int_of_nat_eq)
lemma real_of_nat_div3: "real n / real x - real (n div x) \<le> 1" for n x :: nat
proof (cases "x = 0")
case False
then show ?thesis
by (metis of_int_of_nat_eq real_of_int_div3 zdiv_int)
qed auto
lemma real_of_nat_div4: "real (n div x) \<le> real n / real x" for n x :: nat
using real_of_nat_div2 [of n x] by simp
subsection \<open>The Archimedean Property of the Reals\<close>
lemma real_arch_inverse: "0 < e \<longleftrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
using reals_Archimedean[of e] less_trans[of 0 "1 / real n" e for n::nat]
by (auto simp add: field_simps cong: conj_cong simp del: of_nat_Suc)
lemma reals_Archimedean3: "0 < x \<Longrightarrow> \<forall>y. \<exists>n. y < real n * x"
by (auto intro: ex_less_of_nat_mult)
lemma real_archimedian_rdiv_eq_0:
assumes x0: "x \<ge> 0"
and c: "c \<ge> 0"
and xc: "\<And>m::nat. m > 0 \<Longrightarrow> real m * x \<le> c"
shows "x = 0"
by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc)
subsection \<open>Rationals\<close>
lemma Rats_abs_iff[simp]:
"\<bar>(x::real)\<bar> \<in> \<rat> \<longleftrightarrow> x \<in> \<rat>"
by(simp add: abs_real_def split: if_splits)
lemma Rats_eq_int_div_int: "\<rat> = {real_of_int i / real_of_int j | i j. j \<noteq> 0}" (is "_ = ?S")
proof
show "\<rat> \<subseteq> ?S"
proof
fix x :: real
assume "x \<in> \<rat>"
then obtain r where "x = of_rat r"
unfolding Rats_def ..
have "of_rat r \<in> ?S"
by (cases r) (auto simp add: of_rat_rat)
then show "x \<in> ?S"
using \<open>x = of_rat r\<close> by simp
qed
next
show "?S \<subseteq> \<rat>"
proof (auto simp: Rats_def)
fix i j :: int
assume "j \<noteq> 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 \<in> range of_rat"
by blast
qed
qed
lemma Rats_eq_int_div_nat: "\<rat> = { real_of_int i / real n | i n. n \<noteq> 0}"
proof (auto simp: Rats_eq_int_div_int)
fix i j :: int
assume "j \<noteq> 0"
show "\<exists>(i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i' / real n \<and> 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) \<and> 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)) \<and> 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) \<and> int n \<noteq> 0"
by simp
then show "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int 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 "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 "\<bar>x\<bar> = real (nat \<bar>i\<bar>) / 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 \<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 \<noteq> 0" by (blast dest!: mult_not_zero)
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 add: real_of_nat_div)
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
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 "\<exists>r\<in>\<rat>. x < r \<and> 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 = \<lceil>y * real q\<rceil> - 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 \<in> \<rat>"
by (simp add: r_def)
ultimately show ?thesis by blast
qed
lemma of_rat_dense:
fixes x y :: real
assumes "x < y"
shows "\<exists>q :: rat. x < of_rat q \<and> 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_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
(* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)
#> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]
(* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)
#> Lin_Arith.add_simps [@{thm of_nat_0}, @{thm of_nat_Suc}, @{thm of_nat_add},
@{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1},
@{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff},
@{thm of_int_mult}, @{thm of_int_of_nat_eq},
@{thm of_nat_numeral}, @{thm of_int_numeral}, @{thm of_int_neg_numeral}]
#> 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 \<longleftrightarrow> x = a"
for x a :: real
by arith
lemma real_add_less_0_iff: "x + y < 0 \<longleftrightarrow> y < - x"
for x y :: real
by auto
lemma real_0_less_add_iff: "0 < x + y \<longleftrightarrow> - x < y"
for x y :: real
by auto
lemma real_add_le_0_iff: "x + y \<le> 0 \<longleftrightarrow> y \<le> - x"
for x y :: real
by auto
lemma real_0_le_add_iff: "0 \<le> x + y \<longleftrightarrow> - x \<le> y"
for x y :: real
by auto
subsection \<open>Lemmas about powers\<close>
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
by simp
(* FIXME: declare this [simp] for all types, or not at all *)
declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]
lemma real_minus_mult_self_le [simp]: "- (u * u) \<le> x * x"
for u x :: real
by (rule order_trans [where y = 0]) auto
lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> 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 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> 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 \<Longrightarrow> 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>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 \<longleftrightarrow> 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 \<longleftrightarrow> 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 \<le> real m \<longleftrightarrow> numeral n \<le> m"
for m :: nat
by (metis not_le real_of_nat_less_numeral_iff)
lemma of_int_floor_cancel [simp]: "of_int \<lfloor>x\<rfloor> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)"
by (metis floor_of_int)
lemma floor_eq: "real_of_int n < x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n"
by linarith
lemma floor_eq2: "real_of_int n \<le> x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n"
by (fact floor_unique)
lemma floor_eq3: "real n < x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n"
by linarith
lemma floor_eq4: "real n \<le> x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n"
by linarith
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int \<lfloor>r\<rfloor>"
by linarith
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int \<lfloor>r\<rfloor>"
by linarith
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int \<lfloor>r\<rfloor> + 1"
by linarith
lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int \<lfloor>r\<rfloor> + 1"
by linarith
lemma floor_divide_real_eq_div:
assumes "0 \<le> b"
shows "\<lfloor>a / real_of_int b\<rfloor> = \<lfloor>a\<rfloor> 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 \<le> a" "a < 1 + real_of_int i"
"real_of_int j * real_of_int b \<le> 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 \<le> 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]:
"\<lfloor>1 / numeral b::real\<rfloor> = 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]:
"\<lfloor>- (1 / numeral b)::real\<rfloor> = - 1 div numeral b"
by (metis (mono_tags, hide_lams) 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]:
"\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"
by (metis floor_divide_of_int_eq of_int_numeral)
lemma floor_minus_divide_eq_div_numeral [simp]:
"\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - 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 \<lceil>x\<rceil> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)"
using ceiling_of_int by metis
lemma ceiling_eq: "of_int n < x \<Longrightarrow> x \<le> of_int n + 1 \<Longrightarrow> \<lceil>x\<rceil> = n + 1"
by (simp add: ceiling_unique)
lemma of_int_ceiling_diff_one_le [simp]: "of_int \<lceil>r\<rceil> - 1 \<le> r"
by linarith
lemma of_int_ceiling_le_add_one [simp]: "of_int \<lceil>r\<rceil> \<le> r + 1"
by linarith
lemma ceiling_le: "x \<le> of_int a \<Longrightarrow> \<lceil>x\<rceil> \<le> a"
by (simp add: ceiling_le_iff)
lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- 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]:
"\<lceil>numeral a / numeral b :: real\<rceil> = - (- 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]:
"\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (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 \<le> 0 \<Longrightarrow> nat \<lfloor>x\<rfloor> = 0"
for x :: real
by linarith
lemma le_nat_floor: "real x \<le> a \<Longrightarrow> x \<le> nat \<lfloor>a\<rfloor>"
by linarith
lemma le_mult_nat_floor: "nat \<lfloor>a\<rfloor> * nat \<lfloor>b\<rfloor> \<le> nat \<lfloor>a * b\<rfloor>"
by (cases "0 \<le> a \<and> 0 \<le> b")
(auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
lemma nat_ceiling_le_eq [simp]: "nat \<lceil>x\<rceil> \<le> a \<longleftrightarrow> x \<le> real a"
by linarith
lemma real_nat_ceiling_ge: "x \<le> real (nat \<lceil>x\<rceil>)"
by linarith
lemma Rats_no_top_le: "\<exists>q \<in> \<rat>. x \<le> q"
for x :: real
by (auto intro!: bexI[of _ "of_nat (nat \<lceil>x\<rceil>)"]) linarith
lemma Rats_no_bot_less: "\<exists>q \<in> \<rat>. q < x" for x :: real
by (auto intro!: bexI[of _ "of_int (\<lfloor>x\<rfloor> - 1)"]) linarith
subsection \<open>Exponentiation with floor\<close>
lemma floor_power:
assumes "x = of_int \<lfloor>x\<rfloor>"
shows "\<lfloor>x ^ n\<rfloor> = \<lfloor>x\<rfloor> ^ n"
proof -
have "x ^ n = of_int (\<lfloor>x\<rfloor> ^ n)"
using assms by (induct n arbitrary: x) simp_all
then show ?thesis by (metis floor_of_int)
qed
lemma floor_numeral_power [simp]: "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
by (metis floor_of_int of_int_numeral of_int_power)
lemma ceiling_numeral_power [simp]: "\<lceil>numeral x ^ n\<rceil> = 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 \<Rightarrow> 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 \<longleftrightarrow> 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) \<longleftrightarrow> HOL.equal x y"
by (simp add: equal_real_def equal)
lemma [code nbe]: "HOL.equal x x \<longleftrightarrow> True"
for x :: real
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]: "\<lfloor>Ratreal x\<rfloor> = \<lfloor>x\<rfloor>"
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>
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 (\<lambda>r. f (Ratreal r)) d"
instance ..
end
instantiation real :: full_exhaustive
begin
definition
"full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (\<lambda>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>),
(\<^const_name>\<open>inverse_real_inst.inverse_real\<close>, \<^const_name>\<open>Nitpick.inverse_frac\<close>),
(\<^const_name>\<open>ord_real_inst.less_real\<close>, \<^const_name>\<open>Nitpick.less_frac\<close>),
(\<^const_name>\<open>ord_real_inst.less_eq_real\<close>, \<^const_name>\<open>Nitpick.less_eq_frac\<close>)]
\<close>
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
subsection \<open>Setup for SMT\<close>
ML_file \<open>Tools/SMT/smt_real.ML\<close>
ML_file \<open>Tools/SMT/z3_real.ML\<close>
lemma [z3_rule]:
"0 + x = x"
"x + 0 = x"
"0 * x = 0"
"1 * x = x"
"-x = -1 * x"
"x + y = y + x"
for x y :: real
by auto
subsection \<open>Setup for Argo\<close>
ML_file \<open>Tools/Argo/argo_real.ML\<close>
end