src/HOL/RealDef.thy
 author webertj Fri, 19 Oct 2012 15:12:52 +0200 changeset 49962 a8cc904a6820 parent 47902 34a9e81e5bfd child 51126 df86080de4cb permissions -rw-r--r--
Renamed {left,right}_distrib to distrib_{right,left}.
```
(*  Title       : HOL/RealDef.thy
Author      : Jacques D. Fleuriot, 1998
Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
Construction of Cauchy Reals by Brian Huffman, 2010
*)

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

theory RealDef
imports Rat
begin

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

subsection {* Preliminary lemmas *}

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

lemma minus_diff_minus:
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"

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

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

subsection {* Sequences that converge to zero *}

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

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

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

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

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

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

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

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

subsection {* Cauchy sequences *}

definition
cauchy :: "(nat \<Rightarrow> rat) \<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

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>"
also have "\<dots> < s + t"
finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .
qed
thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
qed

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

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

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

lemma cauchy_mult [simp]:
assumes X: "cauchy X" and Y: "cauchy Y"
shows "cauchy (\<lambda>n. X n * Y n)"
proof (rule cauchyI)
fix r :: rat assume "0 < r"
then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"
by (rule obtain_pos_sum)
obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"
using cauchy_imp_bounded [OF X] by fast
obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"
using cauchy_imp_bounded [OF Y] by fast
obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"
proof
show "0 < v/b" using v b(1) by (rule divide_pos_pos)
show "0 < u/a" using u a(1) by (rule divide_pos_pos)
show "r = a * (u/a) + (v/b) * b"
using a(1) b(1) `r = u + v` by simp
qed
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"
using cauchyD [OF X s] ..
obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"
using cauchyD [OF Y t] ..
have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"
proof (clarsimp)
fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"
have "\<bar>X m * Y m - X n * Y n\<bar> = \<bar>X m * (Y m - Y n) + (X m - X n) * Y n\<bar>"
unfolding mult_diff_mult ..
also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"
by (rule abs_triangle_ineq)
also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"
unfolding abs_mult ..
also have "\<dots> < a * t + s * b"
finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .
qed
thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
qed

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

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

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

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

subsection {* Equivalence relation on Cauchy sequences *}

definition realrel :: "(nat \<Rightarrow> rat) \<Rightarrow> (nat \<Rightarrow> rat) \<Rightarrow> bool"
where "realrel = (\<lambda>X Y. cauchy X \<and> cauchy Y \<and> vanishes (\<lambda>n. X n - Y n))"

lemma realrelI [intro?]:
assumes "cauchy X" and "cauchy Y" and "vanishes (\<lambda>n. X n - Y n)"
shows "realrel X Y"
using assms unfolding realrel_def by simp

lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
unfolding realrel_def by simp

lemma symp_realrel: "symp realrel"
unfolding realrel_def
by (rule sympI, clarify, drule vanishes_minus, simp)

lemma transp_realrel: "transp realrel"
unfolding realrel_def
apply (rule transpI, clarify)
done

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

subsection {* The field of real numbers *}

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

lemma cr_real_eq: "cr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"
unfolding cr_real_def realrel_def by simp

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 cr_real_def fun_rel_def realrel_def by simp

declare real.forall_transfer [transfer_rule del]

lemma forall_real_transfer [transfer_rule]: (* TODO: generate automatically *)
"(fun_rel (fun_rel cr_real op =) op =)
(transfer_bforall cauchy) transfer_forall"
using Quotient_forall_transfer [OF Quotient_real]

instantiation real :: field_inverse_zero
begin

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

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

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

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

lift_definition times_real :: "real \<Rightarrow> real \<Rightarrow> real" is "\<lambda>X Y n. X n * Y n"
unfolding realrel_def mult_diff_mult
by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add
vanishes_mult_bounded cauchy_imp_bounded simp_thms)

lift_definition inverse_real :: "real \<Rightarrow> real"
is "\<lambda>X. if vanishes X then (\<lambda>n. 0) else (\<lambda>n. inverse (X n))"
proof -
fix X Y assume "realrel X Y"
hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
unfolding realrel_def by simp_all
have "vanishes X \<longleftrightarrow> vanishes Y"
proof
assume "vanishes X"
from vanishes_diff [OF this XY] show "vanishes Y" by simp
next
assume "vanishes Y"
from vanishes_add [OF this XY] show "vanishes X" by simp
qed
thus "?thesis X Y"
unfolding realrel_def
by (simp add: vanishes_diff_inverse X Y XY)
qed

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

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

assumes X: "cauchy X" and Y: "cauchy Y"
shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"
using assms plus_real.transfer
unfolding cr_real_eq fun_rel_def by simp

lemma minus_Real:
assumes X: "cauchy X"
shows "- Real X = Real (\<lambda>n. - X n)"
using assms uminus_real.transfer
unfolding cr_real_eq fun_rel_def by simp

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

lemma mult_Real:
assumes X: "cauchy X" and Y: "cauchy Y"
shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"
using assms times_real.transfer
unfolding cr_real_eq fun_rel_def by simp

lemma inverse_Real:
assumes X: "cauchy X"
shows "inverse (Real X) =
(if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
using assms inverse_real.transfer zero_real.transfer
unfolding cr_real_eq fun_rel_def by (simp split: split_if_asm, metis)

instance proof
fix a b c :: real
show "a + b = b + a"
show "(a + b) + c = a + (b + c)"
show "0 + a = a"
show "- a + a = 0"
show "a - b = a + - b"
by (rule minus_real_def)
show "(a * b) * c = a * (b * c)"
by transfer (simp add: mult_ac realrel_def)
show "a * b = b * a"
by transfer (simp add: mult_ac realrel_def)
show "1 * a = a"
by transfer (simp add: mult_ac realrel_def)
show "(a + b) * c = a * c + b * c"
by transfer (simp add: distrib_right realrel_def)
show "(0\<Colon>real) \<noteq> (1\<Colon>real)"
show "a \<noteq> 0 \<Longrightarrow> inverse a * a = 1"
apply transfer
apply (rule vanishesI)
apply (frule (1) cauchy_not_vanishes, clarify)
apply (rule_tac x=k in exI, clarify)
apply (drule_tac x=n in spec, simp)
done
show "a / b = a * inverse b"
by (rule divide_real_def)
show "inverse (0::real) = 0"
qed

end

subsection {* Positive reals *}

lift_definition positive :: "real \<Rightarrow> bool"
is "\<lambda>X. \<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
proof -
{ fix X Y
assume "realrel X Y"
hence XY: "vanishes (\<lambda>n. X n - Y n)"
unfolding realrel_def by simp_all
assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"
then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"
by fast
obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
using `0 < r` by (rule obtain_pos_sum)
obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"
using vanishesD [OF XY s] ..
have "\<forall>n\<ge>max i j. t < Y n"
proof (clarsimp)
fix n assume n: "i \<le> n" "j \<le> n"
have "\<bar>X n - Y n\<bar> < s" and "r < X n"
using i j n by simp_all
thus "t < Y n" unfolding r by simp
qed
hence "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < Y n" using t by fast
} note 1 = this
fix X Y assume "realrel X Y"
hence "realrel X Y" and "realrel Y X"
using symp_realrel unfolding symp_def by auto
thus "?thesis X Y"
by (safe elim!: 1)
qed

lemma positive_Real:
assumes X: "cauchy X"
shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
using assms positive.transfer
unfolding cr_real_eq fun_rel_def by simp

lemma positive_zero: "\<not> positive 0"
by transfer auto

"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)
done

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

lemma positive_minus:
"\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"
apply transfer
apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)
done

instantiation real :: linordered_field_inverse_zero
begin

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

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

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

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

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

end

instantiation real :: distrib_lattice
begin

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

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

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

end

lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
apply (induct x)
done

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

lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
apply (induct x)
done

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

instantiation real :: floor_ceiling
begin

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

instance proof
fix x :: real
show "of_int (floor x) \<le> x \<and> x < of_int (floor x + 1)"
unfolding floor_real_def using floor_exists1 by (rule theI')
qed

end

subsection {* Completeness *}

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

lemma le_Real:
assumes X: "cauchy X" and Y: "cauchy Y"
shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"
unfolding not_less [symmetric, where 'a=real] less_real_def
apply (simp add: diff_Real not_positive_Real X Y)
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"
{
fix r :: rat assume "0 < r"
then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
by (rule obtain_pos_sum)
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
using cauchyD [OF Y s] ..
obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
using le [OF t] ..
have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
proof (clarsimp)
fix n assume n: "i \<le> n" "j \<le> n"
have "X n \<le> Y i + t" using n j by simp
moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp
ultimately show "X n \<le> Y n + r" unfolding r by simp
qed
hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..
}
thus "Real X \<le> Real Y"
by (simp add: of_rat_Real le_Real X Y)
qed

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

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

lemma of_nat_less_two_power:
"of_nat n < (2::'a::linordered_idom) ^ n"
apply (induct n)
apply simp
apply (subgoal_tac "(1::'a) \<le> 2 ^ n")
apply simp
done

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

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

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

have width: "\<And>n. B n - A n = (b - a) / 2^n"
apply (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 (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 (drule (1) bspec)
apply (drule (1) less_le_trans)
done
have AB: "\<And>n. A n < B n"
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 (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 (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 (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 (erule order_trans [OF less_imp_le [OF AB] B_mono])
done
have "cauchy B"
apply (rule cauchy_lemma [rule_format])
apply (erule order_trans [OF A_mono less_imp_le [OF AB]])
done
have 1: "\<forall>x\<in>S. x \<le> Real B"
proof
fix x assume "x \<in> S"
then show "x \<le> Real B"
using PB [unfolded P_def] `cauchy B`
qed
have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"
apply clarify
apply (erule contrapos_pp)
apply (drule less_RealD [OF `cauchy A`], clarify)
apply (subgoal_tac "\<not> P (A n)")
apply (simp add: P_def not_le, clarify)
apply (erule rev_bexI)
apply (erule (1) less_trans)
done
have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
proof (rule vanishesI)
fix r :: rat assume "0 < r"
then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"
using twos by fast
have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"
proof (clarify)
fix n assume n: "k \<le> n"
have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"
by simp
also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"
using n by (simp add: divide_left_mono mult_pos_pos)
also note k
finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .
qed
thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
qed
hence 3: "Real B = Real A"
by (simp add: eq_Real `cauchy A` `cauchy B` width)
show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"
using 1 2 3 by (rule_tac x="Real B" in exI, simp)
qed

subsection {* Hiding implementation details *}

hide_const (open) vanishes cauchy positive Real

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

lemmas [transfer_rule del] =
real.All_transfer real.Ex_transfer real.rel_eq_transfer forall_real_transfer
zero_real.transfer one_real.transfer plus_real.transfer uminus_real.transfer
times_real.transfer inverse_real.transfer positive.transfer

subsection{*More Lemmas*}

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

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

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

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

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

lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
by simp (* solved by linordered_ring_le_cancel_factor simproc *)

subsection {* Embedding numbers into the Reals *}

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

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

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

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

real_of_nat_def [code_unfold]: "real == real_of_nat"
real_of_int_def [code_unfold]: "real == real_of_int"

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

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

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

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

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

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

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

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

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

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

lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"

lemmas power_real_of_int = real_of_int_power [symmetric]

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

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

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

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

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

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

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

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

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

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

lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"
unfolding real_of_one[symmetric] real_of_int_less_iff ..

lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"
unfolding real_of_one[symmetric] real_of_int_le_iff ..

lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"
unfolding real_of_one[symmetric] real_of_int_less_iff ..

lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"
unfolding real_of_one[symmetric] real_of_int_le_iff ..

lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"

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

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

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

lemma real_of_int_div: "(d :: int) dvd n ==>
real(n div d) = real n / real d"
apply (subst real_of_int_div_aux)
apply simp
done

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

lemma real_of_int_div3:
"real (n::int) / real (x) - real (n div x) <= 1"
apply (subst real_of_int_div_aux)
apply (auto simp add: divide_le_eq intro: order_less_imp_le)
done

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

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

subsection{*Embedding the Naturals into the Reals*}

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

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

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

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

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

lemma real_of_nat_less_iff [iff]:
"(real (n::nat) < real m) = (n < m)"

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

lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"

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

lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"

lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"

lemmas power_real_of_nat = real_of_nat_power [symmetric]

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

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

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

lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"

lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"

lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"

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

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

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

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

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

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

lemma real_of_nat_div: "(d :: nat) dvd n ==>
real(n div d) = real n / real d"
by (subst real_of_nat_div_aux)

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

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

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

lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"

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

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

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

subsection{* Rationals *}

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

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

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

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

subsection{*Numerals and Arithmetic*}

lemma [code_abbrev]:
"real_of_int (numeral k) = numeral k"
"real_of_int (neg_numeral k) = neg_numeral k"
by simp_all

text{*Collapse applications of @{term real} to @{term number_of}*}
lemma real_numeral [simp]:
"real (numeral v :: int) = numeral v"
"real (neg_numeral v :: int) = neg_numeral v"

lemma real_of_nat_numeral [simp]:
"real (numeral v :: nat) = numeral v"

declaration {*
K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]
(* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)
#> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]
(* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)
@{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},
@{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},
@{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},
@{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)}]
#> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})
#> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))
*}

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

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

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

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

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

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

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

subsection {* Lemmas about powers *}

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

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

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

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

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

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

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

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

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

lemma numeral_power_le_real_of_nat_cancel_iff[simp]:
"(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"
unfolding real_of_nat_le_iff[symmetric] by simp

lemma real_of_nat_le_numeral_power_cancel_iff[simp]:
"real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"
unfolding real_of_nat_le_iff[symmetric] by simp

lemma numeral_power_le_real_of_int_cancel_iff[simp]:
"(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"
unfolding real_of_int_le_iff[symmetric] by simp

lemma real_of_int_le_numeral_power_cancel_iff[simp]:
"real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"
unfolding real_of_int_le_iff[symmetric] by simp

lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:
"(neg_numeral x::real) ^ n \<le> real a \<longleftrightarrow> (neg_numeral x::int) ^ n \<le> a"
unfolding real_of_int_le_iff[symmetric] by simp

lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:
"real a \<le> (neg_numeral x::real) ^ n \<longleftrightarrow> a \<le> (neg_numeral x::int) ^ n"
unfolding real_of_int_le_iff[symmetric] by simp

subsection{*Density of the Reals*}

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

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

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

subsection{*Absolute Value Function for the Reals*}

lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"

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

lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"

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

lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"
by simp

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

subsection {* Implementation of rational real numbers *}

text {* Formal constructor *}

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

code_datatype Ratreal

text {* Numerals *}

lemma [code_abbrev]:
"(of_rat (of_int a) :: real) = of_int a"
by simp

lemma [code_abbrev]:
"(of_rat 0 :: real) = 0"
by simp

lemma [code_abbrev]:
"(of_rat 1 :: real) = 1"
by simp

lemma [code_abbrev]:
"(of_rat (numeral k) :: real) = numeral k"
by simp

lemma [code_abbrev]:
"(of_rat (neg_numeral k) :: real) = neg_numeral k"
by simp

lemma [code_post]:
"(of_rat (0 / r)  :: real) = 0"
"(of_rat (r / 0)  :: real) = 0"
"(of_rat (1 / 1)  :: real) = 1"
"(of_rat (numeral k / 1) :: real) = numeral k"
"(of_rat (neg_numeral k / 1) :: real) = neg_numeral k"
"(of_rat (1 / numeral k) :: real) = 1 / numeral k"
"(of_rat (1 / neg_numeral k) :: real) = 1 / neg_numeral k"
"(of_rat (numeral k / numeral l)  :: real) = numeral k / numeral l"
"(of_rat (numeral k / neg_numeral l)  :: real) = numeral k / neg_numeral l"
"(of_rat (neg_numeral k / numeral l)  :: real) = neg_numeral k / numeral l"
"(of_rat (neg_numeral k / neg_numeral l)  :: real) = neg_numeral k / neg_numeral l"

text {* Operations *}

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

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

instantiation real :: equal
begin

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

instance proof

lemma real_equal_code [code]:
"HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"

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"

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

lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"

lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"

lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"

lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"

lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"

lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"

lemma real_floor_code [code]: "floor (Ratreal x) = floor x"
by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff of_int_floor_le of_rat_of_int_eq real_less_eq_code)

text {* Quickcheck *}

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

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

instantiation real :: random
begin

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

instance ..

end

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

instantiation real :: exhaustive
begin

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

instance ..

end

instantiation real :: full_exhaustive
begin

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

instance ..

end

instantiation real :: narrowing
begin

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

instance ..

end

subsection {* Setup for Nitpick *}

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

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

end
```