Tweaks for "real": Removal of [iff] status for some lemmas, adding [simp] for others. Plus fixes.
(* 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 Conditionally_Complete_Lattices
begin
text \<open>
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.
\<close>
subsection \<open>Preliminary lemmas\<close>
lemma inj_add_left [simp]:
fixes x :: "'a::cancel_semigroup_add" shows "inj (op+ x)"
by (meson add_left_imp_eq injI)
lemma inj_mult_left [simp]: "inj (op* x) \<longleftrightarrow> x \<noteq> (0::'a::idom)"
by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)
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 \<open>Sequences that converge to zero\<close>
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 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)
thus "\<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:
"\<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 \<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" 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 \<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
hence "(\<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
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 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" 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 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"
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)
done
also note \<open>inverse a * s * inverse b = r\<close>
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 \<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?]:
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 (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)
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 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"
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 div 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 rel_fun_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 rel_fun_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 rel_fun_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 rel_fun_def by (simp split: split_if_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)
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 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 -
{ 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 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
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 blast
} 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 rel_fun_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)
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)
apply blast+
done
instantiation real :: linordered_field
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 max_min_distrib2)
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 \<open>Completeness\<close>
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 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)"
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 [simp]:
"of_nat n < (2::'a::linordered_idom) ^ n"
apply (induct n, simp)
by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
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 blast
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] \<open>cauchy B\<close>
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 \<open>cauchy A\<close>], 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 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
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 \<open>cauchy A\<close> \<open>cauchy B\<close> 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\<open>Supremum of a set of reals\<close>
definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"
definition "Inf (X::real set) = - 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 :: real and X :: "real set" assume x: "x \<in> X" "bdd_below X" then show "Inf X \<le> x"
using Sup_upper[of "-x" "uminus ` X"] by (auto simp: Inf_real_def) }
{ fix z :: real and X :: "real set" assume "X \<noteq> {}" "\<And>x. x \<in> X \<Longrightarrow> z \<le> x" then show "z \<le> Inf X"
using Sup_least[of "uminus ` X" "- z"] by (force simp: Inf_real_def) }
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::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 \<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 of_int_abs [simp]: "of_int (abs x) = (abs(of_int x) :: 'a::linordered_idom)"
by (auto simp add: abs_if)
lemma int_less_real_le: "(n < m) = (real_of_int n + 1 <= real_of_int m)"
proof -
have "(0::real) \<le> 1"
by (metis less_eq_real_def zero_less_one)
thus ?thesis
by (metis floor_unique less_add_one less_imp_le not_less of_int_le_iff order_trans)
qed
lemma int_le_real_less: "(n \<le> m) = (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 = ... / real_of_int d"
by simp
then show ?thesis
by (auto simp add: add_divide_distrib algebra_simps)
qed
lemma real_of_int_div:
fixes d n :: int
shows "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d"
by (simp add: real_of_int_div_aux)
lemma real_of_int_div2:
"0 <= real_of_int n / real_of_int x - real_of_int (n div x)"
apply (case_tac "x = 0", simp)
apply (case_tac "0 < x")
apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)
done
lemma real_of_int_div3:
"real_of_int (n::int) / real_of_int (x) - real_of_int (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_of_int (n div x) <= real_of_int (n::int) / real_of_int x"
by (insert real_of_int_div2 [of n x], simp)
subsection\<open>Embedding the Naturals into the Reals\<close>
lemma real_of_card: "real (card A) = setsum (\<lambda>x.1) A"
by simp
lemma nat_less_real_le: "(n < m) = (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::nat) <= m) = (real n < real m + 1)"
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 <= real (n::nat) / real (x) - real (n div x)"
apply (simp add: algebra_simps)
apply (subst real_of_nat_div_aux)
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)
subsection \<open>The Archimedean Property of the Reals\<close>
lemmas reals_Archimedean = ex_inverse_of_nat_Suc_less (*FIXME*)
lemmas reals_Archimedean2 = ex_less_of_nat
lemma reals_Archimedean3:
assumes x_greater_zero: "0 < x"
shows "\<forall>y. \<exists>n. y < real n * x"
using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)
subsection\<open>Rationals\<close>
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 : \<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)
thus "x : ?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"
hence "real_of_int i / real_of_int j = of_rat(Fract i j)"
by (simp add: of_rat_rat)
thus "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 "EX (i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i'/real n \<and> 0<n"
proof cases
assume "j>0"
hence "real_of_int i / real_of_int j = real_of_int i/real(nat j) \<and> 0<nat j"
by (simp add: of_nat_nat)
thus ?thesis by blast
next
assume "~ j>0"
hence "real_of_int i / real_of_int j = real_of_int(-i) / real(nat(-j)) \<and> 0<nat(-j)" using \<open>j\<noteq>0\<close>
by (simp add: of_nat_nat)
thus ?thesis by blast
qed
next
fix i::int and n::nat assume "0 < n"
hence "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0" by simp
thus "\<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 "gcd m n = 1"
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)
hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by (simp add: of_nat_nat)
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
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{Real Analysis} 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
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 \<open>0 < q\<close>)
finally have "x < r" .
moreover have "r < y"
unfolding r_def p_def
by (simp add: divide_less_eq diff_less_eq \<open>0 < q\<close>
less_ceiling_iff [symmetric])
moreover from r_def have "r \<in> \<rat>" by simp
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>
lemma [code_abbrev]: (*FIXME*)
"real_of_int (numeral k) = numeral k"
"real_of_int (- numeral k) = - numeral k"
by simp_all
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 int_numeral}, @{thm of_int_neg_numeral}]
#> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})
#> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))
\<close>
subsection\<open>Simprules combining x+y and 0: ARE THEY NEEDED?\<close>
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
by arith
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 \<open>Lemmas about powers\<close>
(* used by Import/HOL/real.imp *)
lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"
by simp
text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>
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 \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>
lemma realpow_minus_mult:
fixes x :: "'a::monoid_mult"
shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"
by (simp add: power_Suc power_commutes split add: nat_diff_split)
text \<open>FIXME: declare this [simp] for all types, or not at all\<close>
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::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_eq_real_of_int_cancel_iff[simp]:
"numeral x ^ n = real_of_int (y::int) \<longleftrightarrow> numeral x ^ n = y"
by (metis of_int_eq_iff of_int_numeral of_int_power)
lemma real_of_int_eq_numeral_power_cancel_iff[simp]:
"real_of_int (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
using numeral_power_eq_real_of_int_cancel_iff[of x n y]
by metis
lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:
"numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"
using of_nat_eq_iff by fastforce
lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:
"real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"
using numeral_power_eq_real_of_nat_cancel_iff[of x n y]
by metis
lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
"(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
by (metis of_nat_le_iff of_nat_numeral of_nat_power)
lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
"real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
by (metis of_nat_le_iff of_nat_numeral of_nat_power)
lemma numeral_power_le_real_of_int_cancel_iff[simp]:
"(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power)
lemma real_of_int_le_numeral_power_cancel_iff[simp]:
"real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
by (metis floor_of_int le_floor_iff of_int_numeral of_int_power)
lemma numeral_power_less_real_of_nat_cancel_iff[simp]:
"(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"
by (metis of_nat_less_iff of_nat_numeral of_nat_power)
lemma real_of_nat_less_numeral_power_cancel_iff[simp]:
"real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"
by (metis of_nat_less_iff of_nat_numeral of_nat_power)
lemma numeral_power_less_real_of_int_cancel_iff[simp]:
"(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a"
by (meson not_less real_of_int_le_numeral_power_cancel_iff)
lemma real_of_int_less_numeral_power_cancel_iff[simp]:
"real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"
by (meson not_less numeral_power_le_real_of_int_cancel_iff)
lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
"(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"
by (metis of_int_le_iff of_int_neg_numeral of_int_power)
lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
"real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"
by (metis of_int_le_iff of_int_neg_numeral of_int_power)
subsection\<open>Density of the Reals\<close>
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\<open>Similar results are proved in @{text Fields}\<close>
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\<open>Absolute Value Function for the Reals\<close>
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
by (simp add: abs_if)
lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"
by (simp add: abs_if)
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\<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::nat) < numeral w \<longleftrightarrow> n < numeral w"
by (metis of_nat_less_iff of_nat_numeral)
lemma numeral_less_real_of_nat_iff [simp]:
"numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"
by (metis of_nat_less_iff of_nat_numeral)
lemma numeral_le_real_of_nat_iff[simp]:
"(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"
by (metis not_le real_of_nat_less_numeral_iff)
declare of_int_floor_le [simp] (* FIXME*)
lemma of_int_floor_cancel [simp]:
"(of_int (floor x) = x) = (\<exists>n::int. x = of_int n)"
by (metis floor_of_int)
lemma floor_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> floor x = n"
by linarith
lemma floor_eq2: "[| real_of_int n \<le> x; x < real_of_int n + 1 |] ==> floor x = n"
by linarith
lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"
by linarith
lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"
by linarith
lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int(floor r)"
by linarith
lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int(floor r)"
by linarith
lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int(floor r) + 1"
by linarith
lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int(floor r) + 1"
by linarith
lemma floor_eq_iff: "floor x = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)"
by (simp add: floor_unique_iff)
lemma floor_add2[simp]: "floor (of_int a + x) = a + floor x"
by (simp add: add.commute)
lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real_of_int b) = floor a div b"
proof cases
assume "0 < b"
{ fix i j :: int assume "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"
then have "i < b + j * b"
by (metis linorder_not_less of_int_add of_int_le_iff of_int_mult order_trans_rules(21))
moreover have "j * b < 1 + i"
proof -
have "real_of_int (j * b) < real_of_int i + 1"
using `a < 1 + real_of_int i` `real_of_int j * real_of_int b \<le> a` by force
thus "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 \<open>0<b\<close>, of i] pos_mod_sign[OF \<open>0<b\<close>, of i] by linarith+
then have "j = i div b"
using \<open>0 < b\<close> unfolding mult_less_cancel_right by auto
}
with \<open>0 < b\<close> show ?thesis
by (auto split: floor_split simp: field_simps)
qed auto
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 real_of_int_ceiling_cancel [simp]:
"(real_of_int (ceiling x) = x) = (\<exists>n::int. x = real_of_int n)"
using ceiling_of_int by metis
lemma ceiling_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> ceiling x = n + 1"
by linarith
lemma ceiling_eq2: "[| real_of_int n < x; x \<le> real_of_int n + 1 |] ==> ceiling x = n + 1"
by linarith
lemma real_of_int_ceiling_diff_one_le [simp]: "real_of_int (ceiling r) - 1 \<le> r"
by linarith
lemma real_of_int_ceiling_le_add_one [simp]: "real_of_int (ceiling r) \<le> r + 1"
by linarith
lemma ceiling_le: "x <= real_of_int a ==> ceiling x <= a"
by linarith
lemma ceiling_le_real: "ceiling x <= a ==> x <= real_of_int a"
by linarith
lemma ceiling_divide_eq_div: "\<lceil>real_of_int a / real_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::real) <= 0 ==> nat(floor x) = 0"
by linarith
lemma le_nat_floor: "real x <= a ==> x <= nat(floor a)"
by linarith
lemma le_mult_nat_floor:
shows "nat(floor a) * nat(floor b) \<le> nat(floor (a * b))"
by (cases "0 <= a & 0 <= b")
(auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)
lemma nat_ceiling_le_eq [simp]: "(nat(ceiling x) <= a) = (x <= real a)"
by linarith
lemma real_nat_ceiling_ge: "x <= real(nat(ceiling x))"
by linarith
lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
by (auto intro!: bexI[of _ "of_nat (nat(ceiling x))"]) linarith
lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"
apply (auto intro!: bexI[of _ "of_int (floor x - 1)"])
apply (rule less_le_trans[OF _ of_int_floor_le])
apply simp
done
subsection \<open>Exponentiation with floor\<close>
lemma floor_power:
assumes "x = real_of_int (floor x)"
shows "floor (x ^ n) = floor x ^ n"
proof -
have "x ^ n = real_of_int (floor x ^ n)"
using assms by (induct n arbitrary: x) simp_all
then show ?thesis by linarith
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 of_nat_nat[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
*)
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 = of_rat"
code_datatype Ratreal
text \<open>Numerals\<close>
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 (- 1) :: real) = - 1"
by simp
lemma [code_abbrev]:
"(of_rat (numeral k) :: real) = numeral k"
by simp
lemma [code_abbrev]:
"(of_rat (- numeral k) :: real) = - numeral k"
by simp
lemma [code_post]:
"(of_rat (1 / numeral k) :: real) = 1 / numeral k"
"(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"
"(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"
"(of_rat (- (numeral k / numeral l)) :: real) = - (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::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 \<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 (%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 \<open>Setup for Nitpick\<close>
declaration \<open>
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})]
\<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 "Tools/SMT/smt_real.ML"
ML_file "Tools/SMT/z3_real.ML"
lemma [z3_rule]:
"0 + (x::real) = x"
"x + 0 = x"
"0 * x = 0"
"1 * x = x"
"x + y = y + x"
by auto
end