src/HOL/Real.thy
 changeset 63353 176d1f408696 parent 63331 247eac9758dd child 63494 ac0a3b9c6dae
--- a/src/HOL/Real.thy	Thu Jun 23 11:01:14 2016 +0200
+++ b/src/HOL/Real.thy	Thu Jun 23 23:08:37 2016 +0200
@@ -20,28 +20,22 @@
construction using Dedekind cuts.
\<close>

+
subsection \<open>Preliminary lemmas\<close>

-  fixes x :: "'a::cancel_semigroup_add" shows "inj (op+ x)"
-by (meson add_left_imp_eq injI)
+lemma inj_add_left [simp]: "inj (op + x)" for x :: "'a::cancel_semigroup_add"
+  by (meson add_left_imp_eq injI)

-lemma inj_mult_left [simp]: "inj (op* x) \<longleftrightarrow> x \<noteq> (0::'a::idom)"
+lemma inj_mult_left [simp]: "inj (op * x) \<longleftrightarrow> x \<noteq> 0" for x :: "'a::idom"
by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)

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

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

lemma inverse_diff_inverse:
@@ -54,38 +48,41 @@
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
+  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)"
+definition vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
+  where "vanishes X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"

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

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

lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"
unfolding vanishes_def
-  apply (cases "c = 0", auto)
-  apply (rule exI [where x="\<bar>c\<bar>"], auto)
+  apply (cases "c = 0")
+  apply auto
+  apply (rule exI [where x = "\<bar>c\<bar>"])
+  apply 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"
+  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"
+  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"
@@ -93,26 +90,28 @@
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"
+  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" ..
+  then show "\<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"
+  assumes "vanishes X" "vanishes Y"
shows "vanishes (\<lambda>n. X n - Y n)"
-  unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)
+  unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus assms)

lemma vanishes_mult_bounded:
assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"
assumes Y: "vanishes (\<lambda>n. Y n)"
shows "vanishes (\<lambda>n. X n * Y n)"
proof (rule vanishesI)
-  fix r :: rat assume r: "0 < r"
+  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"
@@ -124,22 +123,19 @@
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" ..
+  then show "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..
qed

+
subsection \<open>Cauchy sequences\<close>

-definition
-  cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"
-where
-  "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"
+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"
+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"
+lemma cauchyD: "cauchy X \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"
unfolding cauchy_def by simp

lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"
@@ -149,7 +145,8 @@
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"
+  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"
@@ -157,30 +154,32 @@
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"
+  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"
-      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 .
+      by (rule add_strict_mono) (simp_all add: i j *)
+    finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" by (simp only: r)
qed
-  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
+  then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..
qed

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

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

lemma cauchy_imp_bounded:
-  assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"
+  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] ..
@@ -189,21 +188,21 @@
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
+    then show "0 < Max (abs  X  {..k}) + 1" by simp
next
fix n :: nat
show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1"
proof (rule linorder_le_cases)
assume "n \<le> k"
-      hence "\<bar>X n\<bar> \<le> Max (abs  X  {..k})" by simp
-      thus "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" by simp
+      then have "\<bar>X n\<bar> \<le> Max (abs  X  {..k})" by simp
+      then show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" by simp
next
assume "k \<le> n"
have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp
also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"
by (rule abs_triangle_ineq)
also have "\<dots> < Max (abs  X  {..k}) + 1"
-        by (rule add_le_less_mono, simp, simp add: k \<open>k \<le> n\<close>)
+        by (rule add_le_less_mono) (simp_all add: k \<open>k \<le> n\<close>)
finally show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" .
qed
qed
@@ -232,8 +231,9 @@
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"
+  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>"
@@ -242,9 +242,9 @@
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 .
+    finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" by (simp only: r)
qed
-  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
+  then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..
qed

lemma cauchy_not_vanishes_cases:
@@ -264,10 +264,10 @@
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)"
+  then have "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
unfolding \<open>r = s + t\<close> using k by auto
-  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)"
+  then have "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..
+  then show "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"
using t by auto
qed

@@ -275,15 +275,21 @@
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)
+  using cauchy_not_vanishes_cases [OF assms]
+  apply clarify
+  apply (rule exI)
+  apply (erule conjI)
+  apply (rule_tac x = k in exI)
+  apply auto
+  done

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"
+  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
@@ -296,84 +302,84 @@
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>"
+  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 .
+      by (simp add: mult_strict_mono less_imp_inverse_less i j b * s)
+    finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" by (simp only: r)
qed
-  thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
+  then show "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..
qed

lemma vanishes_diff_inverse:
assumes X: "cauchy X" "\<not> vanishes X"
-  assumes Y: "cauchy Y" "\<not> vanishes Y"
-  assumes XY: "vanishes (\<lambda>n. X n - Y n)"
+    and Y: "cauchy Y" "\<not> vanishes Y"
+    and XY: "vanishes (\<lambda>n. X n - Y n)"
shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"
proof (rule vanishesI)
-  fix r :: rat assume r: "0 < r"
+  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
+    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>"
+  proof clarsimp
+    fix n
+    assume n: "i \<le> n" "j \<le> n" "k \<le> n"
+    with i j a b have "X n \<noteq> 0" and "Y n \<noteq> 0"
+      by auto
+    then have "\<bar>inverse (X n) - inverse (Y n)\<bar> = inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"
by (simp add: inverse_diff_inverse abs_mult)
also have "\<dots> < inverse a * s * inverse b"
-      apply (intro mult_strict_mono' less_imp_inverse_less)
-      apply (simp_all add: a b i j k n)
-      done
+      by (intro mult_strict_mono' less_imp_inverse_less) (simp_all add: a b i j k n)
also note \<open>inverse a * s * inverse b = r\<close>
finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .
qed
-  thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
+  then show "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..
qed

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

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

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

lemma realrel_refl: "cauchy X \<Longrightarrow> realrel X X"
-  unfolding realrel_def by simp
+  by (simp add: realrel_def)

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

lemma transp_realrel: "transp realrel"
unfolding realrel_def
-  apply (rule transpI, clarify)
+  apply (rule transpI)
+  apply 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)
+  by (blast intro: part_equivpI symp_realrel transp_realrel realrel_refl cauchy_const)
+

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

@@ -385,20 +391,20 @@
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"
+  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)
+  then have "cauchy X" by (simp add: realrel_def)
+  then show "P (Real X)" by (rule assms)
qed

-lemma eq_Real:
-  "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X = Real Y \<longleftrightarrow> vanishes (\<lambda>n. X n - Y n)"
+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)
+  by (simp add: real.domain_eq realrel_def)

instantiation real :: field
begin
@@ -419,14 +425,16 @@

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)
+  apply (subst (4) mult.commute)
+  apply (simp only: cauchy_mult vanishes_add vanishes_mult_bounded cauchy_imp_bounded simp_thms)
+  done

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)"
+  fix X Y
+  assume "realrel X Y"
+  then have X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"
unfolding realrel_def by simp_all
have "vanishes X \<longleftrightarrow> vanishes Y"
proof
@@ -436,49 +444,32 @@
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)
+  then show "?thesis X Y" by (simp add: vanishes_diff_inverse X Y XY realrel_def)
qed

-definition
-  "x - y = (x::real) + - y"
-
-definition
-  "x div y = (x::real) * inverse y"
+definition "x - y = x + - y" for x y :: 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
+definition "x div y = x * inverse y" for x y :: real
+
+lemma add_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X + Real Y = Real (\<lambda>n. X n + Y n)"
+  using plus_real.transfer by (simp add: cr_real_eq rel_fun_def)

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

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

-lemma mult_Real:
-  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 mult_Real: "cauchy X \<Longrightarrow> cauchy Y \<Longrightarrow> Real X * Real Y = Real (\<lambda>n. X n * Y n)"
+  using times_real.transfer by (simp add: cr_real_eq rel_fun_def)

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

-instance proof
+instance
+proof
fix a b c :: real
show "a + b = b + a"
by transfer (simp add: ac_simps realrel_def)
@@ -516,115 +507,125 @@

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

-lemma positive_Real:
-  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_Real: "cauchy X \<Longrightarrow> positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"
+  using positive.transfer by (simp add: cr_real_eq rel_fun_def)

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

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

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

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

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

-instance proof
+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)
+    apply auto
+    apply (drule (1) positive_add)
+    apply (simp_all add: positive_zero)
+    done
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)
+    apply auto
+    apply (drule (1) positive_add)
+    apply (simp add: algebra_simps)
+    done
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)
+    apply auto
+    apply (drule (1) positive_add)
+    apply (simp add: positive_zero)
+    done
show "a \<le> b \<Longrightarrow> c + a \<le> c + b"
-    unfolding less_eq_real_def less_real_def by auto
+    by (auto simp: less_eq_real_def less_real_def)
(* 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)
+    by (auto dest!: positive_minus simp: less_eq_real_def less_real_def)
show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"
unfolding less_real_def
-    by (drule (1) positive_mult, simp add: algebra_simps)
+    apply (drule (1) positive_mult)
+    apply (simp add: algebra_simps)
+    done
qed

end
@@ -632,34 +633,26 @@
instantiation real :: distrib_lattice
begin

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

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

-instance proof
-qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)
+instance by standard (auto simp add: inf_real_def sup_real_def max_min_distrib2)

end

lemma of_nat_Real: "of_nat x = Real (\<lambda>n. of_nat x)"
-apply (induct x)
-apply (simp add: zero_real_def)
-done
+  by (induct x) (simp_all add: zero_real_def one_real_def add_Real)

lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"
-apply (cases x rule: int_diff_cases)
-apply (simp add: of_nat_Real diff_Real)
-done
+  by (cases x rule: int_diff_cases) (simp add: of_nat_Real diff_Real)

lemma of_rat_Real: "of_rat x = Real (\<lambda>n. x)"
-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
+  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
@@ -681,69 +674,82 @@
instantiation real :: floor_ceiling
begin

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

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

end

+
subsection \<open>Completeness\<close>

lemma not_positive_Real:
assumes 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
+  unfolding positive_Real [OF X]
+  apply auto
+  apply (unfold not_less)
+  apply (erule obtain_pos_sum)
+  apply (drule_tac x=s in spec)
+  apply simp
+  apply (drule_tac r=t in cauchyD [OF X])
+  apply clarify
+  apply (drule_tac x=k in spec)
+  apply clarsimp
+  apply (rule_tac x=n in exI)
+  apply clarify
+  apply (rename_tac m)
+  apply (drule_tac x=m in spec)
+  apply simp
+  apply (drule_tac x=n in spec)
+  apply simp
+  apply (drule spec)
+  apply (drule (1) mp)
+  apply clarify
+  apply (rename_tac i)
+  apply (rule_tac x = "max i k" in exI)
+  apply simp
+  done

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

lemma le_RealI:
assumes Y: "cauchy Y"
shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"
proof (induct x)
-  fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
-  hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
+  fix X
+  assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"
+  then have le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"
by (simp add: of_rat_Real le_Real)
-  {
-    fix r :: rat assume "0 < r"
-    then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
+  then have "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" if "0 < r" for r :: rat
+  proof -
+    from that obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"
by (rule obtain_pos_sum)
obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"
using cauchyD [OF Y s] ..
obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"
using le [OF t] ..
have "\<forall>n\<ge>max i j. X n \<le> Y n + r"
-    proof (clarsimp)
-      fix n assume n: "i \<le> n" "j \<le> n"
+    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"
+    then show ?thesis ..
+  qed
+  then show "Real X \<le> Real Y"
by (simp add: of_rat_Real le_Real X Y)
qed

@@ -754,18 +760,22 @@
proof -
have "- y \<le> - Real X"
by (simp add: minus_Real X le_RealI of_rat_minus le)
-  thus ?thesis by simp
+  then show ?thesis by simp
qed

lemma less_RealD:
-  assumes Y: "cauchy Y"
+  assumes "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])
+  apply (erule contrapos_pp)
+  apply (simp add: not_less)
+  apply (erule Real_leI [OF assms])
+  done

-lemma of_nat_less_two_power [simp]:
-  "of_nat n < (2::'a::linordered_idom) ^ n"
-apply (induct n, simp)
-by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
+lemma of_nat_less_two_power [simp]: "of_nat n < (2::'a::linordered_idom) ^ n"
+  apply (induct n)
+  apply simp
+  apply (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)
+  done

lemma complete_real:
fixes S :: "real set"
@@ -781,7 +791,7 @@
have "of_int \<lfloor>x - 1\<rfloor> \<le> x - 1" by (rule of_int_floor_le)
also have "x - 1 < x" by simp
finally have "of_int \<lfloor>x - 1\<rfloor> < x" .
-    hence "\<not> x \<le> of_int \<lfloor>x - 1\<rfloor>" by (simp only: not_le)
+    then have "\<not> x \<le> of_int \<lfloor>x - 1\<rfloor>" by (simp only: not_le)
then show "\<not> P (of_int \<lfloor>x - 1\<rfloor>)"
unfolding P_def of_rat_of_int_eq using x by blast
qed
@@ -791,7 +801,7 @@
unfolding P_def of_rat_of_int_eq
proof
fix y assume "y \<in> S"
-      hence "y \<le> z" using z by simp
+      then have "y \<le> z" using z by simp
also have "z \<le> of_int \<lceil>z\<rceil>" by (rule le_of_int_ceiling)
finally show "y \<le> of_int \<lceil>z\<rceil>" .
qed
@@ -809,13 +819,13 @@
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)
+  have width: "B n - A n = (b - a) / 2^n" for n
+    apply (induct n)
+    apply (simp_all add: eq_divide_eq)
+    apply (simp_all add: C_def avg_def algebra_simps)
done

-  have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"
+  have twos: "0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r" for y r :: rat
apply (simp add: divide_less_eq)
apply (subst mult.commute)
apply (frule_tac y=y in ex_less_of_nat_mult)
@@ -828,10 +838,8 @@
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 PA: "\<not> P (A n)" for n by (induct n) (simp_all add: a)
+  have PB: "P (B n)" for n by (induct n) (simp_all add: b)
have ab: "a < b"
using a b unfolding P_def
apply (clarsimp simp add: not_le)
@@ -839,8 +847,7 @@
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 AB: "A n < B n" for n by (induct n) (simp_all add: ab 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)
@@ -857,8 +864,7 @@
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"
+  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)
@@ -884,7 +890,8 @@
done
have 1: "\<forall>x\<in>S. x \<le> Real B"
proof
-    fix x assume "x \<in> S"
+    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)
@@ -902,12 +909,14 @@
done
have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"
proof (rule vanishesI)
-    fix r :: rat assume "0 < r"
+    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"
+    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"
@@ -915,52 +924,55 @@
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" ..
+    then show "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..
qed
-  hence 3: "Real B = Real A"
+  then have 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)
+    apply (rule exI [where x = "Real B"])
+    using 1 2 3
+    apply simp
+    done
qed

instantiation real :: linear_continuum
begin

-subsection\<open>Supremum of a set of reals\<close>
+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)"
+definition "Inf X = - Sup (uminus  X)" for X :: "real set"

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

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

hide_const (open) vanishes cauchy positive Real
@@ -972,42 +984,35 @@
lifting_update real.lifting
lifting_forget real.lifting

-subsection\<open>More Lemmas\<close>
+
+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>
+  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_less_iff1 [simp]: "0 < z \<Longrightarrow> x * z < y * z \<longleftrightarrow> x < y" for x y z :: real
+  by simp (* solved by linordered_ring_less_cancel_factor simproc *)

-lemma real_mult_le_cancel_iff1 [simp]: "(0::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_iff1 [simp]: "0 < z \<Longrightarrow> x * z \<le> y * z \<longleftrightarrow> x \<le> y" for x y z :: real
+  by simp (* solved by linordered_ring_le_cancel_factor simproc *)

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

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

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

-abbreviation
-  real :: "nat \<Rightarrow> real"
-where
-  "real \<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_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"
+abbreviation real_of_rat :: "rat \<Rightarrow> real"
+  where "real_of_rat \<equiv> of_rat"

declare [[coercion_enabled]]

@@ -1036,68 +1041,65 @@
declare of_int_1_less_iff [algebra, presburger]
declare of_int_1_le_iff [algebra, presburger]

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

-lemma int_le_real_less: "(n \<le> m) = (real_of_int n < real_of_int m + 1)"
+lemma int_le_real_less: "n \<le> m \<longleftrightarrow> real_of_int n < real_of_int m + 1"
by (meson int_less_real_le not_le)

-
-lemma real_of_int_div_aux: "(real_of_int x) / (real_of_int d) =
+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"
+  then have "real_of_int x / real_of_int d = \<dots> / real_of_int d"
by simp
then show ?thesis
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"
+  "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d" for d n :: int
by (simp add: real_of_int_div_aux)

-lemma real_of_int_div2:
-  "0 <= 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")
+lemma real_of_int_div2: "0 \<le> real_of_int n / real_of_int x - real_of_int (n div x)"
+  apply (cases "x = 0")
+  apply simp
+  apply (cases "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"
+lemma real_of_int_div3: "real_of_int n / real_of_int x - real_of_int (n div x) \<le> 1"
apply (simp add: algebra_simps)
apply (subst real_of_int_div_aux)
apply (auto simp add: divide_le_eq intro: order_less_imp_le)
-done
+  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)
+lemma real_of_int_div4: "real_of_int (n div x) \<le> real_of_int n / real_of_int x"
+  using real_of_int_div2 [of n x] by simp

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

-lemma real_of_card: "real (card A) = setsum (\<lambda>x.1) A"
+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)"
+lemma nat_less_real_le: "n < m \<longleftrightarrow> real n + 1 \<le> real m"
by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)

-lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
+lemma nat_le_real_less: "n \<le> m \<longleftrightarrow> real n < real m + 1" for m n :: nat
by (meson nat_less_real_le not_le)

-lemma real_of_nat_div_aux: "(real x) / (real d) =
-    real (x div d) + (real (x mod d)) / (real d)"
+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
@@ -1110,27 +1112,25 @@
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])
+  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_div2: "0 \<le> real n / real x - real (n div x)" for n x :: nat
+  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_div3: "real n / real x - real (n div x) \<le> 1" for n x :: nat
+  apply (cases "x = 0")
+  apply simp
+  apply (simp add: algebra_simps)
+  apply (subst real_of_nat_div_aux)
+  apply simp
+  done

-lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"
-by (insert real_of_nat_div2 [of n x], simp)
+lemma real_of_nat_div4: "real (n div x) \<le> real n / real x" for n x :: nat
+  using real_of_nat_div2 [of n x] by simp
+

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

@@ -1145,77 +1145,85 @@

lemma real_archimedian_rdiv_eq_0:
assumes x0: "x \<ge> 0"
-      and c: "c \<ge> 0"
-      and xc: "\<And>m::nat. m > 0 \<Longrightarrow> real m * x \<le> c"
-    shows "x = 0"
-by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc)
+    and c: "c \<ge> 0"
+    and xc: "\<And>m::nat. m > 0 \<Longrightarrow> real m * x \<le> c"
+  shows "x = 0"
+  by (metis reals_Archimedean3 dual_order.order_iff_strict le0 le_less_trans not_le x0 xc)

-subsection\<open>Rationals\<close>
+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")
+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
+    fix x :: real
+    assume "x \<in> \<rat>"
+    then obtain r where "x = of_rat r"
+      unfolding Rats_def ..
+    have "of_rat r \<in> ?S"
+      by (cases r) (auto simp add: of_rat_rat)
+    then show "x \<in> ?S"
+      using \<open>x = of_rat r\<close> by simp
qed
next
show "?S \<subseteq> \<rat>"
-  proof(auto simp:Rats_def)
-    fix i j :: int assume "j \<noteq> 0"
-    hence "real_of_int i / real_of_int j = of_rat(Fract i j)"
+  proof (auto simp: Rats_def)
+    fix i j :: int
+    assume "j \<noteq> 0"
+    then have "real_of_int i / real_of_int j = of_rat (Fract i j)"
by (simp add: of_rat_rat)
-    thus "real_of_int i / real_of_int j \<in> range of_rat" by blast
+    then show "real_of_int i / real_of_int j \<in> range of_rat"
+      by blast
qed
qed

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

lemma Rats_abs_nat_div_natE:
assumes "x \<in> \<rat>"
-  obtains m n :: nat
-  where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
+  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 \<bar>i\<bar>) / real n" by (simp add: of_nat_nat)
+  from \<open>x \<in> \<rat>\<close> obtain i :: int and n :: nat where "n \<noteq> 0" and "x = real_of_int i / real n"
+    by (auto simp add: Rats_eq_int_div_nat)
+  then have "\<bar>x\<bar> = real (nat \<bar>i\<bar>) / real n" by simp
then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
let ?gcd = "gcd m n"
-  from \<open>n\<noteq>0\<close> have gcd: "?gcd \<noteq> 0" by simp
+  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"
+  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"
+  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
+  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"
@@ -1237,19 +1245,22 @@
ultimately show ?thesis ..
qed

-subsection\<open>Density of the Rational Reals in the Reals\<close>
+
+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>
+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"
+  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
+  from \<open>x < y\<close> have "0 < y - x" by simp
+  with reals_Archimedean obtain q :: nat where q: "inverse (real q) < y - x" and "0 < q"
+    by blast
define p where "p = \<lceil>y * real q\<rceil> - 1"
define r where "r = of_int p / real q"
from q have "x < y - inverse (real q)" by simp
@@ -1259,8 +1270,7 @@
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])
+    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
@@ -1269,11 +1279,11 @@
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)
+  using Rats_dense_in_real [OF \<open>x < y\<close>]
+  by (auto elim: Rats_cases)

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

lemma [code_abbrev]:   (*FIXME*)
"real_of_int (numeral k) = numeral k"
@@ -1294,147 +1304,140 @@
#> 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>
+
+subsection \<open>Simprules combining \<open>x + y\<close> and \<open>0\<close>\<close> (* FIXME ARE THEY NEEDED? *)

-lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"
-by arith
+lemma real_add_minus_iff [simp]: "x + - a = 0 \<longleftrightarrow> x = a" for x a :: real
+  by arith

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

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

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

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

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

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

-text \<open>FIXME: declare this [simp] for all types, or not at all\<close>
+(* FIXME: declare this [simp] for all types, or not at all *)
declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]

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

-lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"
+lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> x\<^sup>2" for u x :: real
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"
+lemma numeral_power_eq_real_of_int_cancel_iff [simp]:
+  "numeral x ^ n = real_of_int y \<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 real_of_int_eq_numeral_power_cancel_iff [simp]:
+  "real_of_int y = 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"
+lemma numeral_power_eq_real_of_nat_cancel_iff [simp]:
+  "numeral x ^ n = real y \<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 real_of_nat_eq_numeral_power_cancel_iff [simp]:
+  "real y = 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 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]:
+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)
+  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"
+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"
+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"
+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 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"
+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"
+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"
+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"
+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>
+subsection \<open>Density of the Reals\<close>
+
+lemma real_lbound_gt_zero: "0 < d1 \<Longrightarrow> 0 < d2 \<Longrightarrow> \<exists>e. 0 < e \<and> e < d1 \<and> e < d2" for d1 d2 :: real
+  by (rule exI [where x = "min d1 d2 / 2"]) (simp add: min_def)

-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 @{theory Fields}\<close>
+lemma real_less_half_sum: "x < y \<Longrightarrow> x < (x + y) / 2" for x y :: real
+  by auto
+
+lemma real_gt_half_sum: "x < y \<Longrightarrow> (x + y) / 2 < y" for x y :: real
+  by auto
+
+lemma real_sum_of_halves: "x / 2 + x / 2 = x" for x :: real
+  by simp

-text\<open>Similar results are proved in \<open>Fields\<close>\<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>Floor and Ceiling Functions from the Reals to the Integers\<close>
+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"
+lemma real_of_nat_less_numeral_iff [simp]: "real n < numeral w \<longleftrightarrow> n < numeral w" for n :: nat
by (metis of_nat_less_iff of_nat_numeral)

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

-lemma numeral_le_real_of_nat_iff[simp]:
-  "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"
-by (metis not_le real_of_nat_less_numeral_iff)
+lemma numeral_le_real_of_nat_iff [simp]: "numeral n \<le> real m \<longleftrightarrow> numeral n \<le> m" for m :: nat
+  by (metis not_le real_of_nat_less_numeral_iff)

-declare of_int_floor_le [simp] (* FIXME*)
+declare of_int_floor_le [simp]  (* FIXME duplicate!? *)

-lemma of_int_floor_cancel [simp]:
-    "(of_int \<lfloor>x\<rfloor> = x) = (\<exists>n::int. x = of_int n)"
+lemma of_int_floor_cancel [simp]: "of_int \<lfloor>x\<rfloor> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)"
by (metis floor_of_int)

-lemma floor_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> \<lfloor>x\<rfloor> = n"
+lemma floor_eq: "real_of_int n < x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n"
by linarith

-lemma floor_eq2: "[| real_of_int n \<le> x; x < real_of_int n + 1 |] ==> \<lfloor>x\<rfloor> = n"
+lemma floor_eq2: "real_of_int n \<le> x \<Longrightarrow> x < real_of_int n + 1 \<Longrightarrow> \<lfloor>x\<rfloor> = n"
by linarith

-lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat \<lfloor>x\<rfloor> = n"
+lemma floor_eq3: "real n < x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n"
by linarith

-lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat \<lfloor>x\<rfloor> = n"
+lemma floor_eq4: "real n \<le> x \<Longrightarrow> x < real (Suc n) \<Longrightarrow> nat \<lfloor>x\<rfloor> = n"
by linarith

lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int \<lfloor>r\<rfloor>"
@@ -1455,41 +1458,52 @@
lemma floor_add2[simp]: "\<lfloor>of_int a + x\<rfloor> = a + \<lfloor>x\<rfloor>"

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

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

-lemma of_int_ceiling_cancel [simp]: "(of_int \<lceil>x\<rceil> = x) = (\<exists>n::int. x = of_int n)"
+lemma of_int_ceiling_cancel [simp]: "of_int \<lceil>x\<rceil> = x \<longleftrightarrow> (\<exists>n::int. x = of_int n)"
using ceiling_of_int by metis

-lemma ceiling_eq: "[| of_int n < x; x \<le> of_int n + 1 |] ==> \<lceil>x\<rceil> = n + 1"
+lemma ceiling_eq: "of_int n < x \<Longrightarrow> x \<le> of_int n + 1 \<Longrightarrow> \<lceil>x\<rceil> = n + 1"
by (simp add: ceiling_unique)

lemma of_int_ceiling_diff_one_le [simp]: "of_int \<lceil>r\<rceil> - 1 \<le> r"
@@ -1498,7 +1512,7 @@
lemma of_int_ceiling_le_add_one [simp]: "of_int \<lceil>r\<rceil> \<le> r + 1"
by linarith

-lemma ceiling_le: "x <= of_int a ==> \<lceil>x\<rceil> <= a"
+lemma ceiling_le: "x \<le> of_int a \<Longrightarrow> \<lceil>x\<rceil> \<le> a"
by (simp add: ceiling_le_iff)

lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)"
@@ -1512,34 +1526,37 @@
"\<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>
+text \<open>
+  The following lemmas are remnants of the erstwhile functions natfloor
+  and natceiling.
+\<close>

-lemma nat_floor_neg: "(x::real) <= 0 ==> nat \<lfloor>x\<rfloor> = 0"
+lemma nat_floor_neg: "x \<le> 0 \<Longrightarrow> nat \<lfloor>x\<rfloor> = 0" for x :: real
by linarith

-lemma le_nat_floor: "real x <= a ==> x <= nat \<lfloor>a\<rfloor>"
+lemma le_nat_floor: "real x \<le> a \<Longrightarrow> x \<le> nat \<lfloor>a\<rfloor>"
by linarith

lemma le_mult_nat_floor: "nat \<lfloor>a\<rfloor> * nat \<lfloor>b\<rfloor> \<le> nat \<lfloor>a * b\<rfloor>"
-  by (cases "0 <= a & 0 <= b")
+  by (cases "0 \<le> a \<and> 0 \<le> b")
(auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)

-lemma nat_ceiling_le_eq [simp]: "(nat \<lceil>x\<rceil> <= a) = (x <= real a)"
+lemma nat_ceiling_le_eq [simp]: "nat \<lceil>x\<rceil> \<le> a \<longleftrightarrow> x \<le> real a"
by linarith

-lemma real_nat_ceiling_ge: "x <= real (nat \<lceil>x\<rceil>)"
+lemma real_nat_ceiling_ge: "x \<le> real (nat \<lceil>x\<rceil>)"
by linarith

-lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"
+lemma Rats_no_top_le: "\<exists>q \<in> \<rat>. x \<le> q" for x :: real
by (auto intro!: bexI[of _ "of_nat (nat \<lceil>x\<rceil>)"]) linarith

-lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"
+lemma Rats_no_bot_less: "\<exists>q \<in> \<rat>. q < x" for x :: real
apply (auto intro!: bexI[of _ "of_int (\<lfloor>x\<rfloor> - 1)"])
apply (rule less_le_trans[OF _ of_int_floor_le])
apply simp
done

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

lemma floor_power:
@@ -1551,48 +1568,41 @@
then show ?thesis by (metis floor_of_int)
qed

-lemma floor_numeral_power[simp]:
-  "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
+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"
+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"
+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"
+lemma [code_abbrev]: "(of_rat (of_int a) :: real) = of_int a"
by simp

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

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

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

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

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

lemma [code_post]:
@@ -1605,28 +1615,23 @@

text \<open>Operations\<close>

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

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

instantiation real :: equal
begin

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

-instance proof
-qed (simp add: equal_real_def)
+instance by standard (simp add: equal_real_def)

-lemma real_equal_code [code]:
-  "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"
+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"
+lemma [code nbe]: "HOL.equal x x \<longleftrightarrow> True" for x :: real
by (rule equal_refl)

end
@@ -1656,14 +1661,15 @@
by (simp add: of_rat_divide)

lemma real_floor_code [code]: "\<lfloor>Ratreal x\<rfloor> = \<lfloor>x\<rfloor>"
-  by (metis Ratreal_def floor_le_iff floor_unique le_floor_iff of_int_floor_le of_rat_of_int_eq real_less_eq_code)
+  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"
+  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)
@@ -1685,7 +1691,7 @@
begin

definition
-  "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"
+  "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (\<lambda>r. f (Ratreal r)) d"

instance ..

@@ -1695,7 +1701,7 @@
begin

definition
-  "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"
+  "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (\<lambda>r. f (valterm_ratreal r)) d"

instance ..

@@ -1705,7 +1711,7 @@
begin

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

instance ..

@@ -1727,9 +1733,9 @@
\<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
+  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>
@@ -1738,11 +1744,12 @@
ML_file "Tools/SMT/z3_real.ML"

lemma [z3_rule]:
-  "0 + (x::real) = x"
+  "0 + x = x"
"x + 0 = x"
"0 * x = 0"
"1 * x = x"
"x + y = y + x"
+  for x y :: real
by auto

end