src/HOL/Real.thy
 author paulson Tue Nov 17 12:32:08 2015 +0000 (2015-11-17) changeset 61694 6571c78c9667 parent 61649 268d88ec9087 child 61799 4cf66f21b764 permissions -rw-r--r--
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths
     1 (*  Title:      HOL/Real.thy

     2     Author:     Jacques D. Fleuriot, University of Edinburgh, 1998

     3     Author:     Larry Paulson, University of Cambridge

     4     Author:     Jeremy Avigad, Carnegie Mellon University

     5     Author:     Florian Zuleger, Johannes Hoelzl, and Simon Funke, TU Muenchen

     6     Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4

     7     Construction of Cauchy Reals by Brian Huffman, 2010

     8 *)

     9

    10 section \<open>Development of the Reals using Cauchy Sequences\<close>

    11

    12 theory Real

    13 imports Rat Conditionally_Complete_Lattices

    14 begin

    15

    16 text \<open>

    17   This theory contains a formalization of the real numbers as

    18   equivalence classes of Cauchy sequences of rationals.  See

    19   @{file "~~/src/HOL/ex/Dedekind_Real.thy"} for an alternative

    20   construction using Dedekind cuts.

    21 \<close>

    22

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

    24

    25 lemma inj_add_left [simp]:

    26   fixes x :: "'a::cancel_semigroup_add" shows "inj (op+ x)"

    27 by (meson add_left_imp_eq injI)

    28

    29 lemma inj_mult_left [simp]: "inj (op* x) \<longleftrightarrow> x \<noteq> (0::'a::idom)"

    30   by (metis injI mult_cancel_left the_inv_f_f zero_neq_one)

    31

    32 lemma add_diff_add:

    33   fixes a b c d :: "'a::ab_group_add"

    34   shows "(a + c) - (b + d) = (a - b) + (c - d)"

    35   by simp

    36

    37 lemma minus_diff_minus:

    38   fixes a b :: "'a::ab_group_add"

    39   shows "- a - - b = - (a - b)"

    40   by simp

    41

    42 lemma mult_diff_mult:

    43   fixes x y a b :: "'a::ring"

    44   shows "(x * y - a * b) = x * (y - b) + (x - a) * b"

    45   by (simp add: algebra_simps)

    46

    47 lemma inverse_diff_inverse:

    48   fixes a b :: "'a::division_ring"

    49   assumes "a \<noteq> 0" and "b \<noteq> 0"

    50   shows "inverse a - inverse b = - (inverse a * (a - b) * inverse b)"

    51   using assms by (simp add: algebra_simps)

    52

    53 lemma obtain_pos_sum:

    54   fixes r :: rat assumes r: "0 < r"

    55   obtains s t where "0 < s" and "0 < t" and "r = s + t"

    56 proof

    57     from r show "0 < r/2" by simp

    58     from r show "0 < r/2" by simp

    59     show "r = r/2 + r/2" by simp

    60 qed

    61

    62 subsection \<open>Sequences that converge to zero\<close>

    63

    64 definition

    65   vanishes :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"

    66 where

    67   "vanishes X = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r)"

    68

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

    70   unfolding vanishes_def by simp

    71

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

    73   unfolding vanishes_def by simp

    74

    75 lemma vanishes_const [simp]: "vanishes (\<lambda>n. c) \<longleftrightarrow> c = 0"

    76   unfolding vanishes_def

    77   apply (cases "c = 0", auto)

    78   apply (rule exI [where x="\<bar>c\<bar>"], auto)

    79   done

    80

    81 lemma vanishes_minus: "vanishes X \<Longrightarrow> vanishes (\<lambda>n. - X n)"

    82   unfolding vanishes_def by simp

    83

    84 lemma vanishes_add:

    85   assumes X: "vanishes X" and Y: "vanishes Y"

    86   shows "vanishes (\<lambda>n. X n + Y n)"

    87 proof (rule vanishesI)

    88   fix r :: rat assume "0 < r"

    89   then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"

    90     by (rule obtain_pos_sum)

    91   obtain i where i: "\<forall>n\<ge>i. \<bar>X n\<bar> < s"

    92     using vanishesD [OF X s] ..

    93   obtain j where j: "\<forall>n\<ge>j. \<bar>Y n\<bar> < t"

    94     using vanishesD [OF Y t] ..

    95   have "\<forall>n\<ge>max i j. \<bar>X n + Y n\<bar> < r"

    96   proof (clarsimp)

    97     fix n assume n: "i \<le> n" "j \<le> n"

    98     have "\<bar>X n + Y n\<bar> \<le> \<bar>X n\<bar> + \<bar>Y n\<bar>" by (rule abs_triangle_ineq)

    99     also have "\<dots> < s + t" by (simp add: add_strict_mono i j n)

   100     finally show "\<bar>X n + Y n\<bar> < r" unfolding r .

   101   qed

   102   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n + Y n\<bar> < r" ..

   103 qed

   104

   105 lemma vanishes_diff:

   106   assumes X: "vanishes X" and Y: "vanishes Y"

   107   shows "vanishes (\<lambda>n. X n - Y n)"

   108   unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)

   109

   110 lemma vanishes_mult_bounded:

   111   assumes X: "\<exists>a>0. \<forall>n. \<bar>X n\<bar> < a"

   112   assumes Y: "vanishes (\<lambda>n. Y n)"

   113   shows "vanishes (\<lambda>n. X n * Y n)"

   114 proof (rule vanishesI)

   115   fix r :: rat assume r: "0 < r"

   116   obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"

   117     using X by blast

   118   obtain b where b: "0 < b" "r = a * b"

   119   proof

   120     show "0 < r / a" using r a by simp

   121     show "r = a * (r / a)" using a by simp

   122   qed

   123   obtain k where k: "\<forall>n\<ge>k. \<bar>Y n\<bar> < b"

   124     using vanishesD [OF Y b(1)] ..

   125   have "\<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r"

   126     by (simp add: b(2) abs_mult mult_strict_mono' a k)

   127   thus "\<exists>k. \<forall>n\<ge>k. \<bar>X n * Y n\<bar> < r" ..

   128 qed

   129

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

   131

   132 definition

   133   cauchy :: "(nat \<Rightarrow> rat) \<Rightarrow> bool"

   134 where

   135   "cauchy X \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r)"

   136

   137 lemma cauchyI:

   138   "(\<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"

   139   unfolding cauchy_def by simp

   140

   141 lemma cauchyD:

   142   "\<lbrakk>cauchy X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < r"

   143   unfolding cauchy_def by simp

   144

   145 lemma cauchy_const [simp]: "cauchy (\<lambda>n. x)"

   146   unfolding cauchy_def by simp

   147

   148 lemma cauchy_add [simp]:

   149   assumes X: "cauchy X" and Y: "cauchy Y"

   150   shows "cauchy (\<lambda>n. X n + Y n)"

   151 proof (rule cauchyI)

   152   fix r :: rat assume "0 < r"

   153   then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"

   154     by (rule obtain_pos_sum)

   155   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"

   156     using cauchyD [OF X s] ..

   157   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"

   158     using cauchyD [OF Y t] ..

   159   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r"

   160   proof (clarsimp)

   161     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"

   162     have "\<bar>(X m + Y m) - (X n + Y n)\<bar> \<le> \<bar>X m - X n\<bar> + \<bar>Y m - Y n\<bar>"

   163       unfolding add_diff_add by (rule abs_triangle_ineq)

   164     also have "\<dots> < s + t"

   165       by (rule add_strict_mono, simp_all add: i j *)

   166     finally show "\<bar>(X m + Y m) - (X n + Y n)\<bar> < r" unfolding r .

   167   qed

   168   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>(X m + Y m) - (X n + Y n)\<bar> < r" ..

   169 qed

   170

   171 lemma cauchy_minus [simp]:

   172   assumes X: "cauchy X"

   173   shows "cauchy (\<lambda>n. - X n)"

   174 using assms unfolding cauchy_def

   175 unfolding minus_diff_minus abs_minus_cancel .

   176

   177 lemma cauchy_diff [simp]:

   178   assumes X: "cauchy X" and Y: "cauchy Y"

   179   shows "cauchy (\<lambda>n. X n - Y n)"

   180   using assms unfolding diff_conv_add_uminus by (simp del: add_uminus_conv_diff)

   181

   182 lemma cauchy_imp_bounded:

   183   assumes "cauchy X" shows "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"

   184 proof -

   185   obtain k where k: "\<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m - X n\<bar> < 1"

   186     using cauchyD [OF assms zero_less_one] ..

   187   show "\<exists>b>0. \<forall>n. \<bar>X n\<bar> < b"

   188   proof (intro exI conjI allI)

   189     have "0 \<le> \<bar>X 0\<bar>" by simp

   190     also have "\<bar>X 0\<bar> \<le> Max (abs  X  {..k})" by simp

   191     finally have "0 \<le> Max (abs  X  {..k})" .

   192     thus "0 < Max (abs  X  {..k}) + 1" by simp

   193   next

   194     fix n :: nat

   195     show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1"

   196     proof (rule linorder_le_cases)

   197       assume "n \<le> k"

   198       hence "\<bar>X n\<bar> \<le> Max (abs  X  {..k})" by simp

   199       thus "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" by simp

   200     next

   201       assume "k \<le> n"

   202       have "\<bar>X n\<bar> = \<bar>X k + (X n - X k)\<bar>" by simp

   203       also have "\<bar>X k + (X n - X k)\<bar> \<le> \<bar>X k\<bar> + \<bar>X n - X k\<bar>"

   204         by (rule abs_triangle_ineq)

   205       also have "\<dots> < Max (abs  X  {..k}) + 1"

   206         by (rule add_le_less_mono, simp, simp add: k \<open>k \<le> n\<close>)

   207       finally show "\<bar>X n\<bar> < Max (abs  X  {..k}) + 1" .

   208     qed

   209   qed

   210 qed

   211

   212 lemma cauchy_mult [simp]:

   213   assumes X: "cauchy X" and Y: "cauchy Y"

   214   shows "cauchy (\<lambda>n. X n * Y n)"

   215 proof (rule cauchyI)

   216   fix r :: rat assume "0 < r"

   217   then obtain u v where u: "0 < u" and v: "0 < v" and "r = u + v"

   218     by (rule obtain_pos_sum)

   219   obtain a where a: "0 < a" "\<forall>n. \<bar>X n\<bar> < a"

   220     using cauchy_imp_bounded [OF X] by blast

   221   obtain b where b: "0 < b" "\<forall>n. \<bar>Y n\<bar> < b"

   222     using cauchy_imp_bounded [OF Y] by blast

   223   obtain s t where s: "0 < s" and t: "0 < t" and r: "r = a * t + s * b"

   224   proof

   225     show "0 < v/b" using v b(1) by simp

   226     show "0 < u/a" using u a(1) by simp

   227     show "r = a * (u/a) + (v/b) * b"

   228       using a(1) b(1) \<open>r = u + v\<close> by simp

   229   qed

   230   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"

   231     using cauchyD [OF X s] ..

   232   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>Y m - Y n\<bar> < t"

   233     using cauchyD [OF Y t] ..

   234   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>X m * Y m - X n * Y n\<bar> < r"

   235   proof (clarsimp)

   236     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"

   237     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>"

   238       unfolding mult_diff_mult ..

   239     also have "\<dots> \<le> \<bar>X m * (Y m - Y n)\<bar> + \<bar>(X m - X n) * Y n\<bar>"

   240       by (rule abs_triangle_ineq)

   241     also have "\<dots> = \<bar>X m\<bar> * \<bar>Y m - Y n\<bar> + \<bar>X m - X n\<bar> * \<bar>Y n\<bar>"

   242       unfolding abs_mult ..

   243     also have "\<dots> < a * t + s * b"

   244       by (simp_all add: add_strict_mono mult_strict_mono' a b i j *)

   245     finally show "\<bar>X m * Y m - X n * Y n\<bar> < r" unfolding r .

   246   qed

   247   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>X m * Y m - X n * Y n\<bar> < r" ..

   248 qed

   249

   250 lemma cauchy_not_vanishes_cases:

   251   assumes X: "cauchy X"

   252   assumes nz: "\<not> vanishes X"

   253   shows "\<exists>b>0. \<exists>k. (\<forall>n\<ge>k. b < - X n) \<or> (\<forall>n\<ge>k. b < X n)"

   254 proof -

   255   obtain r where "0 < r" and r: "\<forall>k. \<exists>n\<ge>k. r \<le> \<bar>X n\<bar>"

   256     using nz unfolding vanishes_def by (auto simp add: not_less)

   257   obtain s t where s: "0 < s" and t: "0 < t" and "r = s + t"

   258     using \<open>0 < r\<close> by (rule obtain_pos_sum)

   259   obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>X m - X n\<bar> < s"

   260     using cauchyD [OF X s] ..

   261   obtain k where "i \<le> k" and "r \<le> \<bar>X k\<bar>"

   262     using r by blast

   263   have k: "\<forall>n\<ge>k. \<bar>X n - X k\<bar> < s"

   264     using i \<open>i \<le> k\<close> by auto

   265   have "X k \<le> - r \<or> r \<le> X k"

   266     using \<open>r \<le> \<bar>X k\<bar>\<close> by auto

   267   hence "(\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"

   268     unfolding \<open>r = s + t\<close> using k by auto

   269   hence "\<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)" ..

   270   thus "\<exists>t>0. \<exists>k. (\<forall>n\<ge>k. t < - X n) \<or> (\<forall>n\<ge>k. t < X n)"

   271     using t by auto

   272 qed

   273

   274 lemma cauchy_not_vanishes:

   275   assumes X: "cauchy X"

   276   assumes nz: "\<not> vanishes X"

   277   shows "\<exists>b>0. \<exists>k. \<forall>n\<ge>k. b < \<bar>X n\<bar>"

   278 using cauchy_not_vanishes_cases [OF assms]

   279 by clarify (rule exI, erule conjI, rule_tac x=k in exI, auto)

   280

   281 lemma cauchy_inverse [simp]:

   282   assumes X: "cauchy X"

   283   assumes nz: "\<not> vanishes X"

   284   shows "cauchy (\<lambda>n. inverse (X n))"

   285 proof (rule cauchyI)

   286   fix r :: rat assume "0 < r"

   287   obtain b i where b: "0 < b" and i: "\<forall>n\<ge>i. b < \<bar>X n\<bar>"

   288     using cauchy_not_vanishes [OF X nz] by blast

   289   from b i have nz: "\<forall>n\<ge>i. X n \<noteq> 0" by auto

   290   obtain s where s: "0 < s" and r: "r = inverse b * s * inverse b"

   291   proof

   292     show "0 < b * r * b" by (simp add: \<open>0 < r\<close> b)

   293     show "r = inverse b * (b * r * b) * inverse b"

   294       using b by simp

   295   qed

   296   obtain j where j: "\<forall>m\<ge>j. \<forall>n\<ge>j. \<bar>X m - X n\<bar> < s"

   297     using cauchyD [OF X s] ..

   298   have "\<forall>m\<ge>max i j. \<forall>n\<ge>max i j. \<bar>inverse (X m) - inverse (X n)\<bar> < r"

   299   proof (clarsimp)

   300     fix m n assume *: "i \<le> m" "j \<le> m" "i \<le> n" "j \<le> n"

   301     have "\<bar>inverse (X m) - inverse (X n)\<bar> =

   302           inverse \<bar>X m\<bar> * \<bar>X m - X n\<bar> * inverse \<bar>X n\<bar>"

   303       by (simp add: inverse_diff_inverse nz * abs_mult)

   304     also have "\<dots> < inverse b * s * inverse b"

   305       by (simp add: mult_strict_mono less_imp_inverse_less

   306                     i j b * s)

   307     finally show "\<bar>inverse (X m) - inverse (X n)\<bar> < r" unfolding r .

   308   qed

   309   thus "\<exists>k. \<forall>m\<ge>k. \<forall>n\<ge>k. \<bar>inverse (X m) - inverse (X n)\<bar> < r" ..

   310 qed

   311

   312 lemma vanishes_diff_inverse:

   313   assumes X: "cauchy X" "\<not> vanishes X"

   314   assumes Y: "cauchy Y" "\<not> vanishes Y"

   315   assumes XY: "vanishes (\<lambda>n. X n - Y n)"

   316   shows "vanishes (\<lambda>n. inverse (X n) - inverse (Y n))"

   317 proof (rule vanishesI)

   318   fix r :: rat assume r: "0 < r"

   319   obtain a i where a: "0 < a" and i: "\<forall>n\<ge>i. a < \<bar>X n\<bar>"

   320     using cauchy_not_vanishes [OF X] by blast

   321   obtain b j where b: "0 < b" and j: "\<forall>n\<ge>j. b < \<bar>Y n\<bar>"

   322     using cauchy_not_vanishes [OF Y] by blast

   323   obtain s where s: "0 < s" and "inverse a * s * inverse b = r"

   324   proof

   325     show "0 < a * r * b"

   326       using a r b by simp

   327     show "inverse a * (a * r * b) * inverse b = r"

   328       using a r b by simp

   329   qed

   330   obtain k where k: "\<forall>n\<ge>k. \<bar>X n - Y n\<bar> < s"

   331     using vanishesD [OF XY s] ..

   332   have "\<forall>n\<ge>max (max i j) k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r"

   333   proof (clarsimp)

   334     fix n assume n: "i \<le> n" "j \<le> n" "k \<le> n"

   335     have "X n \<noteq> 0" and "Y n \<noteq> 0"

   336       using i j a b n by auto

   337     hence "\<bar>inverse (X n) - inverse (Y n)\<bar> =

   338         inverse \<bar>X n\<bar> * \<bar>X n - Y n\<bar> * inverse \<bar>Y n\<bar>"

   339       by (simp add: inverse_diff_inverse abs_mult)

   340     also have "\<dots> < inverse a * s * inverse b"

   341       apply (intro mult_strict_mono' less_imp_inverse_less)

   342       apply (simp_all add: a b i j k n)

   343       done

   344     also note \<open>inverse a * s * inverse b = r\<close>

   345     finally show "\<bar>inverse (X n) - inverse (Y n)\<bar> < r" .

   346   qed

   347   thus "\<exists>k. \<forall>n\<ge>k. \<bar>inverse (X n) - inverse (Y n)\<bar> < r" ..

   348 qed

   349

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

   351

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

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

   354

   355 lemma realrelI [intro?]:

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

   357   shows "realrel X Y"

   358   using assms unfolding realrel_def by simp

   359

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

   361   unfolding realrel_def by simp

   362

   363 lemma symp_realrel: "symp realrel"

   364   unfolding realrel_def

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

   366

   367 lemma transp_realrel: "transp realrel"

   368   unfolding realrel_def

   369   apply (rule transpI, clarify)

   370   apply (drule (1) vanishes_add)

   371   apply (simp add: algebra_simps)

   372   done

   373

   374 lemma part_equivp_realrel: "part_equivp realrel"

   375   by (blast intro: part_equivpI symp_realrel transp_realrel

   376     realrel_refl cauchy_const)

   377

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

   379

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

   381   morphisms rep_real Real

   382   by (rule part_equivp_realrel)

   383

   384 lemma cr_real_eq: "pcr_real = (\<lambda>x y. cauchy x \<and> Real x = y)"

   385   unfolding real.pcr_cr_eq cr_real_def realrel_def by auto

   386

   387 lemma Real_induct [induct type: real]: (* TODO: generate automatically *)

   388   assumes "\<And>X. cauchy X \<Longrightarrow> P (Real X)" shows "P x"

   389 proof (induct x)

   390   case (1 X)

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

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

   393 qed

   394

   395 lemma eq_Real:

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

   397   using real.rel_eq_transfer

   398   unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp

   399

   400 lemma Domainp_pcr_real [transfer_domain_rule]: "Domainp pcr_real = cauchy"

   401 by (simp add: real.domain_eq realrel_def)

   402

   403 instantiation real :: field

   404 begin

   405

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

   407   by (simp add: realrel_refl)

   408

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

   410   by (simp add: realrel_refl)

   411

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

   413   unfolding realrel_def add_diff_add

   414   by (simp only: cauchy_add vanishes_add simp_thms)

   415

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

   417   unfolding realrel_def minus_diff_minus

   418   by (simp only: cauchy_minus vanishes_minus simp_thms)

   419

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

   421   unfolding realrel_def mult_diff_mult

   422   by (subst (4) mult.commute, simp only: cauchy_mult vanishes_add

   423     vanishes_mult_bounded cauchy_imp_bounded simp_thms)

   424

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

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

   427 proof -

   428   fix X Y assume "realrel X Y"

   429   hence X: "cauchy X" and Y: "cauchy Y" and XY: "vanishes (\<lambda>n. X n - Y n)"

   430     unfolding realrel_def by simp_all

   431   have "vanishes X \<longleftrightarrow> vanishes Y"

   432   proof

   433     assume "vanishes X"

   434     from vanishes_diff [OF this XY] show "vanishes Y" by simp

   435   next

   436     assume "vanishes Y"

   437     from vanishes_add [OF this XY] show "vanishes X" by simp

   438   qed

   439   thus "?thesis X Y"

   440     unfolding realrel_def

   441     by (simp add: vanishes_diff_inverse X Y XY)

   442 qed

   443

   444 definition

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

   446

   447 definition

   448   "x div y = (x::real) * inverse y"

   449

   450 lemma add_Real:

   451   assumes X: "cauchy X" and Y: "cauchy Y"

   452   shows "Real X + Real Y = Real (\<lambda>n. X n + Y n)"

   453   using assms plus_real.transfer

   454   unfolding cr_real_eq rel_fun_def by simp

   455

   456 lemma minus_Real:

   457   assumes X: "cauchy X"

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

   459   using assms uminus_real.transfer

   460   unfolding cr_real_eq rel_fun_def by simp

   461

   462 lemma diff_Real:

   463   assumes X: "cauchy X" and Y: "cauchy Y"

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

   465   unfolding minus_real_def

   466   by (simp add: minus_Real add_Real X Y)

   467

   468 lemma mult_Real:

   469   assumes X: "cauchy X" and Y: "cauchy Y"

   470   shows "Real X * Real Y = Real (\<lambda>n. X n * Y n)"

   471   using assms times_real.transfer

   472   unfolding cr_real_eq rel_fun_def by simp

   473

   474 lemma inverse_Real:

   475   assumes X: "cauchy X"

   476   shows "inverse (Real X) =

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

   478   using assms inverse_real.transfer zero_real.transfer

   479   unfolding cr_real_eq rel_fun_def by (simp split: split_if_asm, metis)

   480

   481 instance proof

   482   fix a b c :: real

   483   show "a + b = b + a"

   484     by transfer (simp add: ac_simps realrel_def)

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

   486     by transfer (simp add: ac_simps realrel_def)

   487   show "0 + a = a"

   488     by transfer (simp add: realrel_def)

   489   show "- a + a = 0"

   490     by transfer (simp add: realrel_def)

   491   show "a - b = a + - b"

   492     by (rule minus_real_def)

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

   494     by transfer (simp add: ac_simps realrel_def)

   495   show "a * b = b * a"

   496     by transfer (simp add: ac_simps realrel_def)

   497   show "1 * a = a"

   498     by transfer (simp add: ac_simps realrel_def)

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

   500     by transfer (simp add: distrib_right realrel_def)

   501   show "(0::real) \<noteq> (1::real)"

   502     by transfer (simp add: realrel_def)

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

   504     apply transfer

   505     apply (simp add: realrel_def)

   506     apply (rule vanishesI)

   507     apply (frule (1) cauchy_not_vanishes, clarify)

   508     apply (rule_tac x=k in exI, clarify)

   509     apply (drule_tac x=n in spec, simp)

   510     done

   511   show "a div b = a * inverse b"

   512     by (rule divide_real_def)

   513   show "inverse (0::real) = 0"

   514     by transfer (simp add: realrel_def)

   515 qed

   516

   517 end

   518

   519 subsection \<open>Positive reals\<close>

   520

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

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

   523 proof -

   524   { fix X Y

   525     assume "realrel X Y"

   526     hence XY: "vanishes (\<lambda>n. X n - Y n)"

   527       unfolding realrel_def by simp_all

   528     assume "\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n"

   529     then obtain r i where "0 < r" and i: "\<forall>n\<ge>i. r < X n"

   530       by blast

   531     obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"

   532       using \<open>0 < r\<close> by (rule obtain_pos_sum)

   533     obtain j where j: "\<forall>n\<ge>j. \<bar>X n - Y n\<bar> < s"

   534       using vanishesD [OF XY s] ..

   535     have "\<forall>n\<ge>max i j. t < Y n"

   536     proof (clarsimp)

   537       fix n assume n: "i \<le> n" "j \<le> n"

   538       have "\<bar>X n - Y n\<bar> < s" and "r < X n"

   539         using i j n by simp_all

   540       thus "t < Y n" unfolding r by simp

   541     qed

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

   543   } note 1 = this

   544   fix X Y assume "realrel X Y"

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

   546     using symp_realrel unfolding symp_def by auto

   547   thus "?thesis X Y"

   548     by (safe elim!: 1)

   549 qed

   550

   551 lemma positive_Real:

   552   assumes X: "cauchy X"

   553   shows "positive (Real X) \<longleftrightarrow> (\<exists>r>0. \<exists>k. \<forall>n\<ge>k. r < X n)"

   554   using assms positive.transfer

   555   unfolding cr_real_eq rel_fun_def by simp

   556

   557 lemma positive_zero: "\<not> positive 0"

   558   by transfer auto

   559

   560 lemma positive_add:

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

   562 apply transfer

   563 apply (clarify, rename_tac a b i j)

   564 apply (rule_tac x="a + b" in exI, simp)

   565 apply (rule_tac x="max i j" in exI, clarsimp)

   566 apply (simp add: add_strict_mono)

   567 done

   568

   569 lemma positive_mult:

   570   "positive x \<Longrightarrow> positive y \<Longrightarrow> positive (x * y)"

   571 apply transfer

   572 apply (clarify, rename_tac a b i j)

   573 apply (rule_tac x="a * b" in exI, simp)

   574 apply (rule_tac x="max i j" in exI, clarsimp)

   575 apply (rule mult_strict_mono, auto)

   576 done

   577

   578 lemma positive_minus:

   579   "\<not> positive x \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> positive (- x)"

   580 apply transfer

   581 apply (simp add: realrel_def)

   582 apply (drule (1) cauchy_not_vanishes_cases, safe)

   583 apply blast+

   584 done

   585

   586 instantiation real :: linordered_field

   587 begin

   588

   589 definition

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

   591

   592 definition

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

   594

   595 definition

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

   597

   598 definition

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

   600

   601 instance proof

   602   fix a b c :: real

   603   show "\<bar>a\<bar> = (if a < 0 then - a else a)"

   604     by (rule abs_real_def)

   605   show "a < b \<longleftrightarrow> a \<le> b \<and> \<not> b \<le> a"

   606     unfolding less_eq_real_def less_real_def

   607     by (auto, drule (1) positive_add, simp_all add: positive_zero)

   608   show "a \<le> a"

   609     unfolding less_eq_real_def by simp

   610   show "a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"

   611     unfolding less_eq_real_def less_real_def

   612     by (auto, drule (1) positive_add, simp add: algebra_simps)

   613   show "a \<le> b \<Longrightarrow> b \<le> a \<Longrightarrow> a = b"

   614     unfolding less_eq_real_def less_real_def

   615     by (auto, drule (1) positive_add, simp add: positive_zero)

   616   show "a \<le> b \<Longrightarrow> c + a \<le> c + b"

   617     unfolding less_eq_real_def less_real_def by auto

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

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

   620   show "sgn a = (if a = 0 then 0 else if 0 < a then 1 else - 1)"

   621     by (rule sgn_real_def)

   622   show "a \<le> b \<or> b \<le> a"

   623     unfolding less_eq_real_def less_real_def

   624     by (auto dest!: positive_minus)

   625   show "a < b \<Longrightarrow> 0 < c \<Longrightarrow> c * a < c * b"

   626     unfolding less_real_def

   627     by (drule (1) positive_mult, simp add: algebra_simps)

   628 qed

   629

   630 end

   631

   632 instantiation real :: distrib_lattice

   633 begin

   634

   635 definition

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

   637

   638 definition

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

   640

   641 instance proof

   642 qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)

   643

   644 end

   645

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

   647 apply (induct x)

   648 apply (simp add: zero_real_def)

   649 apply (simp add: one_real_def add_Real)

   650 done

   651

   652 lemma of_int_Real: "of_int x = Real (\<lambda>n. of_int x)"

   653 apply (cases x rule: int_diff_cases)

   654 apply (simp add: of_nat_Real diff_Real)

   655 done

   656

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

   658 apply (induct x)

   659 apply (simp add: Fract_of_int_quotient of_rat_divide)

   660 apply (simp add: of_int_Real divide_inverse)

   661 apply (simp add: inverse_Real mult_Real)

   662 done

   663

   664 instance real :: archimedean_field

   665 proof

   666   fix x :: real

   667   show "\<exists>z. x \<le> of_int z"

   668     apply (induct x)

   669     apply (frule cauchy_imp_bounded, clarify)

   670     apply (rule_tac x="ceiling b + 1" in exI)

   671     apply (rule less_imp_le)

   672     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)

   673     apply (rule_tac x=1 in exI, simp add: algebra_simps)

   674     apply (rule_tac x=0 in exI, clarsimp)

   675     apply (rule le_less_trans [OF abs_ge_self])

   676     apply (rule less_le_trans [OF _ le_of_int_ceiling])

   677     apply simp

   678     done

   679 qed

   680

   681 instantiation real :: floor_ceiling

   682 begin

   683

   684 definition [code del]:

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

   686

   687 instance proof

   688   fix x :: real

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

   690     unfolding floor_real_def using floor_exists1 by (rule theI')

   691 qed

   692

   693 end

   694

   695 subsection \<open>Completeness\<close>

   696

   697 lemma not_positive_Real:

   698   assumes X: "cauchy X"

   699   shows "\<not> positive (Real X) \<longleftrightarrow> (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> r)"

   700 unfolding positive_Real [OF X]

   701 apply (auto, unfold not_less)

   702 apply (erule obtain_pos_sum)

   703 apply (drule_tac x=s in spec, simp)

   704 apply (drule_tac r=t in cauchyD [OF X], clarify)

   705 apply (drule_tac x=k in spec, clarsimp)

   706 apply (rule_tac x=n in exI, clarify, rename_tac m)

   707 apply (drule_tac x=m in spec, simp)

   708 apply (drule_tac x=n in spec, simp)

   709 apply (drule spec, drule (1) mp, clarify, rename_tac i)

   710 apply (rule_tac x="max i k" in exI, simp)

   711 done

   712

   713 lemma le_Real:

   714   assumes X: "cauchy X" and Y: "cauchy Y"

   715   shows "Real X \<le> Real Y = (\<forall>r>0. \<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r)"

   716 unfolding not_less [symmetric, where 'a=real] less_real_def

   717 apply (simp add: diff_Real not_positive_Real X Y)

   718 apply (simp add: diff_le_eq ac_simps)

   719 done

   720

   721 lemma le_RealI:

   722   assumes Y: "cauchy Y"

   723   shows "\<forall>n. x \<le> of_rat (Y n) \<Longrightarrow> x \<le> Real Y"

   724 proof (induct x)

   725   fix X assume X: "cauchy X" and "\<forall>n. Real X \<le> of_rat (Y n)"

   726   hence le: "\<And>m r. 0 < r \<Longrightarrow> \<exists>k. \<forall>n\<ge>k. X n \<le> Y m + r"

   727     by (simp add: of_rat_Real le_Real)

   728   {

   729     fix r :: rat assume "0 < r"

   730     then obtain s t where s: "0 < s" and t: "0 < t" and r: "r = s + t"

   731       by (rule obtain_pos_sum)

   732     obtain i where i: "\<forall>m\<ge>i. \<forall>n\<ge>i. \<bar>Y m - Y n\<bar> < s"

   733       using cauchyD [OF Y s] ..

   734     obtain j where j: "\<forall>n\<ge>j. X n \<le> Y i + t"

   735       using le [OF t] ..

   736     have "\<forall>n\<ge>max i j. X n \<le> Y n + r"

   737     proof (clarsimp)

   738       fix n assume n: "i \<le> n" "j \<le> n"

   739       have "X n \<le> Y i + t" using n j by simp

   740       moreover have "\<bar>Y i - Y n\<bar> < s" using n i by simp

   741       ultimately show "X n \<le> Y n + r" unfolding r by simp

   742     qed

   743     hence "\<exists>k. \<forall>n\<ge>k. X n \<le> Y n + r" ..

   744   }

   745   thus "Real X \<le> Real Y"

   746     by (simp add: of_rat_Real le_Real X Y)

   747 qed

   748

   749 lemma Real_leI:

   750   assumes X: "cauchy X"

   751   assumes le: "\<forall>n. of_rat (X n) \<le> y"

   752   shows "Real X \<le> y"

   753 proof -

   754   have "- y \<le> - Real X"

   755     by (simp add: minus_Real X le_RealI of_rat_minus le)

   756   thus ?thesis by simp

   757 qed

   758

   759 lemma less_RealD:

   760   assumes Y: "cauchy Y"

   761   shows "x < Real Y \<Longrightarrow> \<exists>n. x < of_rat (Y n)"

   762 by (erule contrapos_pp, simp add: not_less, erule Real_leI [OF Y])

   763

   764 lemma of_nat_less_two_power [simp]:

   765   "of_nat n < (2::'a::linordered_idom) ^ n"

   766 apply (induct n, simp)

   767 by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)

   768

   769 lemma complete_real:

   770   fixes S :: "real set"

   771   assumes "\<exists>x. x \<in> S" and "\<exists>z. \<forall>x\<in>S. x \<le> z"

   772   shows "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"

   773 proof -

   774   obtain x where x: "x \<in> S" using assms(1) ..

   775   obtain z where z: "\<forall>x\<in>S. x \<le> z" using assms(2) ..

   776

   777   def P \<equiv> "\<lambda>x. \<forall>y\<in>S. y \<le> of_rat x"

   778   obtain a where a: "\<not> P a"

   779   proof

   780     have "of_int (floor (x - 1)) \<le> x - 1" by (rule of_int_floor_le)

   781     also have "x - 1 < x" by simp

   782     finally have "of_int (floor (x - 1)) < x" .

   783     hence "\<not> x \<le> of_int (floor (x - 1))" by (simp only: not_le)

   784     then show "\<not> P (of_int (floor (x - 1)))"

   785       unfolding P_def of_rat_of_int_eq using x by blast

   786   qed

   787   obtain b where b: "P b"

   788   proof

   789     show "P (of_int (ceiling z))"

   790     unfolding P_def of_rat_of_int_eq

   791     proof

   792       fix y assume "y \<in> S"

   793       hence "y \<le> z" using z by simp

   794       also have "z \<le> of_int (ceiling z)" by (rule le_of_int_ceiling)

   795       finally show "y \<le> of_int (ceiling z)" .

   796     qed

   797   qed

   798

   799   def avg \<equiv> "\<lambda>x y :: rat. x/2 + y/2"

   800   def bisect \<equiv> "\<lambda>(x, y). if P (avg x y) then (x, avg x y) else (avg x y, y)"

   801   def A \<equiv> "\<lambda>n. fst ((bisect ^^ n) (a, b))"

   802   def B \<equiv> "\<lambda>n. snd ((bisect ^^ n) (a, b))"

   803   def C \<equiv> "\<lambda>n. avg (A n) (B n)"

   804   have A_0 [simp]: "A 0 = a" unfolding A_def by simp

   805   have B_0 [simp]: "B 0 = b" unfolding B_def by simp

   806   have A_Suc [simp]: "\<And>n. A (Suc n) = (if P (C n) then A n else C n)"

   807     unfolding A_def B_def C_def bisect_def split_def by simp

   808   have B_Suc [simp]: "\<And>n. B (Suc n) = (if P (C n) then C n else B n)"

   809     unfolding A_def B_def C_def bisect_def split_def by simp

   810

   811   have width: "\<And>n. B n - A n = (b - a) / 2^n"

   812     apply (simp add: eq_divide_eq)

   813     apply (induct_tac n, simp)

   814     apply (simp add: C_def avg_def algebra_simps)

   815     done

   816

   817   have twos: "\<And>y r :: rat. 0 < r \<Longrightarrow> \<exists>n. y / 2 ^ n < r"

   818     apply (simp add: divide_less_eq)

   819     apply (subst mult.commute)

   820     apply (frule_tac y=y in ex_less_of_nat_mult)

   821     apply clarify

   822     apply (rule_tac x=n in exI)

   823     apply (erule less_trans)

   824     apply (rule mult_strict_right_mono)

   825     apply (rule le_less_trans [OF _ of_nat_less_two_power])

   826     apply simp

   827     apply assumption

   828     done

   829

   830   have PA: "\<And>n. \<not> P (A n)"

   831     by (induct_tac n, simp_all add: a)

   832   have PB: "\<And>n. P (B n)"

   833     by (induct_tac n, simp_all add: b)

   834   have ab: "a < b"

   835     using a b unfolding P_def

   836     apply (clarsimp simp add: not_le)

   837     apply (drule (1) bspec)

   838     apply (drule (1) less_le_trans)

   839     apply (simp add: of_rat_less)

   840     done

   841   have AB: "\<And>n. A n < B n"

   842     by (induct_tac n, simp add: ab, simp add: C_def avg_def)

   843   have A_mono: "\<And>i j. i \<le> j \<Longrightarrow> A i \<le> A j"

   844     apply (auto simp add: le_less [where 'a=nat])

   845     apply (erule less_Suc_induct)

   846     apply (clarsimp simp add: C_def avg_def)

   847     apply (simp add: add_divide_distrib [symmetric])

   848     apply (rule AB [THEN less_imp_le])

   849     apply simp

   850     done

   851   have B_mono: "\<And>i j. i \<le> j \<Longrightarrow> B j \<le> B i"

   852     apply (auto simp add: le_less [where 'a=nat])

   853     apply (erule less_Suc_induct)

   854     apply (clarsimp simp add: C_def avg_def)

   855     apply (simp add: add_divide_distrib [symmetric])

   856     apply (rule AB [THEN less_imp_le])

   857     apply simp

   858     done

   859   have cauchy_lemma:

   860     "\<And>X. \<forall>n. \<forall>i\<ge>n. A n \<le> X i \<and> X i \<le> B n \<Longrightarrow> cauchy X"

   861     apply (rule cauchyI)

   862     apply (drule twos [where y="b - a"])

   863     apply (erule exE)

   864     apply (rule_tac x=n in exI, clarify, rename_tac i j)

   865     apply (rule_tac y="B n - A n" in le_less_trans) defer

   866     apply (simp add: width)

   867     apply (drule_tac x=n in spec)

   868     apply (frule_tac x=i in spec, drule (1) mp)

   869     apply (frule_tac x=j in spec, drule (1) mp)

   870     apply (frule A_mono, drule B_mono)

   871     apply (frule A_mono, drule B_mono)

   872     apply arith

   873     done

   874   have "cauchy A"

   875     apply (rule cauchy_lemma [rule_format])

   876     apply (simp add: A_mono)

   877     apply (erule order_trans [OF less_imp_le [OF AB] B_mono])

   878     done

   879   have "cauchy B"

   880     apply (rule cauchy_lemma [rule_format])

   881     apply (simp add: B_mono)

   882     apply (erule order_trans [OF A_mono less_imp_le [OF AB]])

   883     done

   884   have 1: "\<forall>x\<in>S. x \<le> Real B"

   885   proof

   886     fix x assume "x \<in> S"

   887     then show "x \<le> Real B"

   888       using PB [unfolded P_def] \<open>cauchy B\<close>

   889       by (simp add: le_RealI)

   890   qed

   891   have 2: "\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> Real A \<le> z"

   892     apply clarify

   893     apply (erule contrapos_pp)

   894     apply (simp add: not_le)

   895     apply (drule less_RealD [OF \<open>cauchy A\<close>], clarify)

   896     apply (subgoal_tac "\<not> P (A n)")

   897     apply (simp add: P_def not_le, clarify)

   898     apply (erule rev_bexI)

   899     apply (erule (1) less_trans)

   900     apply (simp add: PA)

   901     done

   902   have "vanishes (\<lambda>n. (b - a) / 2 ^ n)"

   903   proof (rule vanishesI)

   904     fix r :: rat assume "0 < r"

   905     then obtain k where k: "\<bar>b - a\<bar> / 2 ^ k < r"

   906       using twos by blast

   907     have "\<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r"

   908     proof (clarify)

   909       fix n assume n: "k \<le> n"

   910       have "\<bar>(b - a) / 2 ^ n\<bar> = \<bar>b - a\<bar> / 2 ^ n"

   911         by simp

   912       also have "\<dots> \<le> \<bar>b - a\<bar> / 2 ^ k"

   913         using n by (simp add: divide_left_mono)

   914       also note k

   915       finally show "\<bar>(b - a) / 2 ^ n\<bar> < r" .

   916     qed

   917     thus "\<exists>k. \<forall>n\<ge>k. \<bar>(b - a) / 2 ^ n\<bar> < r" ..

   918   qed

   919   hence 3: "Real B = Real A"

   920     by (simp add: eq_Real \<open>cauchy A\<close> \<open>cauchy B\<close> width)

   921   show "\<exists>y. (\<forall>x\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>x\<in>S. x \<le> z) \<longrightarrow> y \<le> z)"

   922     using 1 2 3 by (rule_tac x="Real B" in exI, simp)

   923 qed

   924

   925 instantiation real :: linear_continuum

   926 begin

   927

   928 subsection\<open>Supremum of a set of reals\<close>

   929

   930 definition "Sup X = (LEAST z::real. \<forall>x\<in>X. x \<le> z)"

   931 definition "Inf (X::real set) = - Sup (uminus  X)"

   932

   933 instance

   934 proof

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

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

   937     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"

   938       using complete_real[of X] unfolding bdd_above_def by blast

   939     then show "x \<le> Sup X"

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

   941   note Sup_upper = this

   942

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

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

   945     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"

   946       using complete_real[of X] by blast

   947     then have "Sup X = s"

   948       unfolding Sup_real_def by (best intro: Least_equality)

   949     also from s z have "... \<le> z"

   950       by blast

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

   952   note Sup_least = this

   953

   954   { fix x :: real and X :: "real set" assume x: "x \<in> X" "bdd_below X" then show "Inf X \<le> x"

   955       using Sup_upper[of "-x" "uminus  X"] by (auto simp: Inf_real_def) }

   956   { 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"

   957       using Sup_least[of "uminus  X" "- z"] by (force simp: Inf_real_def) }

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

   959     using zero_neq_one by blast

   960 qed

   961 end

   962

   963

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

   965

   966 hide_const (open) vanishes cauchy positive Real

   967

   968 declare Real_induct [induct del]

   969 declare Abs_real_induct [induct del]

   970 declare Abs_real_cases [cases del]

   971

   972 lifting_update real.lifting

   973 lifting_forget real.lifting

   974

   975 subsection\<open>More Lemmas\<close>

   976

   977 text \<open>BH: These lemmas should not be necessary; they should be

   978 covered by existing simp rules and simplification procedures.\<close>

   979

   980 lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"

   981 by simp (* solved by linordered_ring_less_cancel_factor simproc *)

   982

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

   984 by simp (* solved by linordered_ring_le_cancel_factor simproc *)

   985

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

   987 by simp (* solved by linordered_ring_le_cancel_factor simproc *)

   988

   989

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

   991

   992 abbreviation

   993   real_of_nat :: "nat \<Rightarrow> real"

   994 where

   995   "real_of_nat \<equiv> of_nat"

   996

   997 abbreviation

   998   real :: "nat \<Rightarrow> real"

   999 where

  1000   "real \<equiv> of_nat"

  1001

  1002 abbreviation

  1003   real_of_int :: "int \<Rightarrow> real"

  1004 where

  1005   "real_of_int \<equiv> of_int"

  1006

  1007 abbreviation

  1008   real_of_rat :: "rat \<Rightarrow> real"

  1009 where

  1010   "real_of_rat \<equiv> of_rat"

  1011

  1012 declare [[coercion_enabled]]

  1013

  1014 declare [[coercion "of_nat :: nat \<Rightarrow> int"]]

  1015 declare [[coercion "of_nat :: nat \<Rightarrow> real"]]

  1016 declare [[coercion "of_int :: int \<Rightarrow> real"]]

  1017

  1018 (* We do not add rat to the coerced types, this has often unpleasant side effects when writing

  1019 inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)

  1020

  1021 declare [[coercion_map map]]

  1022 declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]

  1023 declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]

  1024

  1025 declare of_int_eq_0_iff [algebra, presburger]

  1026 declare of_int_eq_1_iff [algebra, presburger]

  1027 declare of_int_eq_iff [algebra, presburger]

  1028 declare of_int_less_0_iff [algebra, presburger]

  1029 declare of_int_less_1_iff [algebra, presburger]

  1030 declare of_int_less_iff [algebra, presburger]

  1031 declare of_int_le_0_iff [algebra, presburger]

  1032 declare of_int_le_1_iff [algebra, presburger]

  1033 declare of_int_le_iff [algebra, presburger]

  1034 declare of_int_0_less_iff [algebra, presburger]

  1035 declare of_int_0_le_iff [algebra, presburger]

  1036 declare of_int_1_less_iff [algebra, presburger]

  1037 declare of_int_1_le_iff [algebra, presburger]

  1038

  1039 lemma of_int_abs [simp]: "of_int (abs x) = (abs(of_int x) :: 'a::linordered_idom)"

  1040   by (auto simp add: abs_if)

  1041

  1042 lemma int_less_real_le: "(n < m) = (real_of_int n + 1 <= real_of_int m)"

  1043 proof -

  1044   have "(0::real) \<le> 1"

  1045     by (metis less_eq_real_def zero_less_one)

  1046   thus ?thesis

  1047     by (metis floor_of_int less_floor_iff)

  1048 qed

  1049

  1050 lemma int_le_real_less: "(n \<le> m) = (real_of_int n < real_of_int m + 1)"

  1051   by (meson int_less_real_le not_le)

  1052

  1053

  1054 lemma real_of_int_div_aux: "(real_of_int x) / (real_of_int d) =

  1055     real_of_int (x div d) + (real_of_int (x mod d)) / (real_of_int d)"

  1056 proof -

  1057   have "x = (x div d) * d + x mod d"

  1058     by auto

  1059   then have "real_of_int x = real_of_int (x div d) * real_of_int d + real_of_int(x mod d)"

  1060     by (metis of_int_add of_int_mult)

  1061   then have "real_of_int x / real_of_int d = ... / real_of_int d"

  1062     by simp

  1063   then show ?thesis

  1064     by (auto simp add: add_divide_distrib algebra_simps)

  1065 qed

  1066

  1067 lemma real_of_int_div:

  1068   fixes d n :: int

  1069   shows "d dvd n \<Longrightarrow> real_of_int (n div d) = real_of_int n / real_of_int d"

  1070   by (simp add: real_of_int_div_aux)

  1071

  1072 lemma real_of_int_div2:

  1073   "0 <= real_of_int n / real_of_int x - real_of_int (n div x)"

  1074   apply (case_tac "x = 0", simp)

  1075   apply (case_tac "0 < x")

  1076    apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)

  1077   apply (metis add.left_neutral floor_correct floor_divide_of_int_eq le_diff_eq)

  1078   done

  1079

  1080 lemma real_of_int_div3:

  1081   "real_of_int (n::int) / real_of_int (x) - real_of_int (n div x) <= 1"

  1082   apply (simp add: algebra_simps)

  1083   apply (subst real_of_int_div_aux)

  1084   apply (auto simp add: divide_le_eq intro: order_less_imp_le)

  1085 done

  1086

  1087 lemma real_of_int_div4: "real_of_int (n div x) <= real_of_int (n::int) / real_of_int x"

  1088 by (insert real_of_int_div2 [of n x], simp)

  1089

  1090

  1091 subsection\<open>Embedding the Naturals into the Reals\<close>

  1092

  1093 lemma real_of_card: "real (card A) = setsum (\<lambda>x.1) A"

  1094   by simp

  1095

  1096 lemma nat_less_real_le: "(n < m) = (real n + 1 \<le> real m)"

  1097   by (metis discrete of_nat_1 of_nat_add of_nat_le_iff)

  1098

  1099 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"

  1100   by (meson nat_less_real_le not_le)

  1101

  1102 lemma real_of_nat_div_aux: "(real x) / (real d) =

  1103     real (x div d) + (real (x mod d)) / (real d)"

  1104 proof -

  1105   have "x = (x div d) * d + x mod d"

  1106     by auto

  1107   then have "real x = real (x div d) * real d + real(x mod d)"

  1108     by (metis of_nat_add of_nat_mult)

  1109   then have "real x / real d = \<dots> / real d"

  1110     by simp

  1111   then show ?thesis

  1112     by (auto simp add: add_divide_distrib algebra_simps)

  1113 qed

  1114

  1115 lemma real_of_nat_div: "d dvd n \<Longrightarrow> real(n div d) = real n / real d"

  1116   by (subst real_of_nat_div_aux)

  1117     (auto simp add: dvd_eq_mod_eq_0 [symmetric])

  1118

  1119 lemma real_of_nat_div2:

  1120   "0 <= real (n::nat) / real (x) - real (n div x)"

  1121 apply (simp add: algebra_simps)

  1122 apply (subst real_of_nat_div_aux)

  1123 apply simp

  1124 done

  1125

  1126 lemma real_of_nat_div3:

  1127   "real (n::nat) / real (x) - real (n div x) <= 1"

  1128 apply(case_tac "x = 0")

  1129 apply (simp)

  1130 apply (simp add: algebra_simps)

  1131 apply (subst real_of_nat_div_aux)

  1132 apply simp

  1133 done

  1134

  1135 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"

  1136 by (insert real_of_nat_div2 [of n x], simp)

  1137

  1138 subsection \<open>The Archimedean Property of the Reals\<close>

  1139

  1140 lemmas reals_Archimedean = ex_inverse_of_nat_Suc_less  (*FIXME*)

  1141 lemmas reals_Archimedean2 = ex_less_of_nat

  1142

  1143 lemma reals_Archimedean3:

  1144   assumes x_greater_zero: "0 < x"

  1145   shows "\<forall>y. \<exists>n. y < real n * x"

  1146   using \<open>0 < x\<close> by (auto intro: ex_less_of_nat_mult)

  1147

  1148

  1149 subsection\<open>Rationals\<close>

  1150

  1151 lemma Rats_eq_int_div_int:

  1152   "\<rat> = { real_of_int i / real_of_int j |i j. j \<noteq> 0}" (is "_ = ?S")

  1153 proof

  1154   show "\<rat> \<subseteq> ?S"

  1155   proof

  1156     fix x::real assume "x : \<rat>"

  1157     then obtain r where "x = of_rat r" unfolding Rats_def ..

  1158     have "of_rat r : ?S"

  1159       by (cases r) (auto simp add:of_rat_rat)

  1160     thus "x : ?S" using \<open>x = of_rat r\<close> by simp

  1161   qed

  1162 next

  1163   show "?S \<subseteq> \<rat>"

  1164   proof(auto simp:Rats_def)

  1165     fix i j :: int assume "j \<noteq> 0"

  1166     hence "real_of_int i / real_of_int j = of_rat(Fract i j)"

  1167       by (simp add: of_rat_rat)

  1168     thus "real_of_int i / real_of_int j \<in> range of_rat" by blast

  1169   qed

  1170 qed

  1171

  1172 lemma Rats_eq_int_div_nat:

  1173   "\<rat> = { real_of_int i / real n |i n. n \<noteq> 0}"

  1174 proof(auto simp:Rats_eq_int_div_int)

  1175   fix i j::int assume "j \<noteq> 0"

  1176   show "EX (i'::int) (n::nat). real_of_int i / real_of_int j = real_of_int i'/real n \<and> 0<n"

  1177   proof cases

  1178     assume "j>0"

  1179     hence "real_of_int i / real_of_int j = real_of_int i/real(nat j) \<and> 0<nat j"

  1180       by (simp add: of_nat_nat)

  1181     thus ?thesis by blast

  1182   next

  1183     assume "~ j>0"

  1184     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>

  1185       by (simp add: of_nat_nat)

  1186     thus ?thesis by blast

  1187   qed

  1188 next

  1189   fix i::int and n::nat assume "0 < n"

  1190   hence "real_of_int i / real n = real_of_int i / real_of_int(int n) \<and> int n \<noteq> 0" by simp

  1191   thus "\<exists>i' j. real_of_int i / real n = real_of_int i' / real_of_int j \<and> j \<noteq> 0" by blast

  1192 qed

  1193

  1194 lemma Rats_abs_nat_div_natE:

  1195   assumes "x \<in> \<rat>"

  1196   obtains m n :: nat

  1197   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"

  1198 proof -

  1199   from \<open>x \<in> \<rat>\<close> obtain i::int and n::nat where "n \<noteq> 0" and "x = real_of_int i / real n"

  1200     by(auto simp add: Rats_eq_int_div_nat)

  1201   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by (simp add: of_nat_nat)

  1202   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast

  1203   let ?gcd = "gcd m n"

  1204   from \<open>n\<noteq>0\<close> have gcd: "?gcd \<noteq> 0" by simp

  1205   let ?k = "m div ?gcd"

  1206   let ?l = "n div ?gcd"

  1207   let ?gcd' = "gcd ?k ?l"

  1208   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"

  1209     by (rule dvd_mult_div_cancel)

  1210   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"

  1211     by (rule dvd_mult_div_cancel)

  1212   from \<open>n \<noteq> 0\<close> and gcd_l

  1213   have "?gcd * ?l \<noteq> 0" by simp

  1214   then have "?l \<noteq> 0" by (blast dest!: mult_not_zero)

  1215   moreover

  1216   have "\<bar>x\<bar> = real ?k / real ?l"

  1217   proof -

  1218     from gcd have "real ?k / real ?l = real (?gcd * ?k) / real (?gcd * ?l)"

  1219       by (simp add: real_of_nat_div)

  1220     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp

  1221     also from x_rat have "\<dots> = \<bar>x\<bar>" ..

  1222     finally show ?thesis ..

  1223   qed

  1224   moreover

  1225   have "?gcd' = 1"

  1226   proof -

  1227     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"

  1228       by (rule gcd_mult_distrib_nat)

  1229     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp

  1230     with gcd show ?thesis by auto

  1231   qed

  1232   ultimately show ?thesis ..

  1233 qed

  1234

  1235 subsection\<open>Density of the Rational Reals in the Reals\<close>

  1236

  1237 text\<open>This density proof is due to Stefan Richter and was ported by TN.  The

  1238 original source is \emph{Real Analysis} by H.L. Royden.

  1239 It employs the Archimedean property of the reals.\<close>

  1240

  1241 lemma Rats_dense_in_real:

  1242   fixes x :: real

  1243   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"

  1244 proof -

  1245   from \<open>x<y\<close> have "0 < y-x" by simp

  1246   with reals_Archimedean obtain q::nat

  1247     where q: "inverse (real q) < y-x" and "0 < q" by blast

  1248   def p \<equiv> "ceiling (y * real q) - 1"

  1249   def r \<equiv> "of_int p / real q"

  1250   from q have "x < y - inverse (real q)" by simp

  1251   also have "y - inverse (real q) \<le> r"

  1252     unfolding r_def p_def

  1253     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling \<open>0 < q\<close>)

  1254   finally have "x < r" .

  1255   moreover have "r < y"

  1256     unfolding r_def p_def

  1257     by (simp add: divide_less_eq diff_less_eq \<open>0 < q\<close>

  1258       less_ceiling_iff [symmetric])

  1259   moreover from r_def have "r \<in> \<rat>" by simp

  1260   ultimately show ?thesis by blast

  1261 qed

  1262

  1263 lemma of_rat_dense:

  1264   fixes x y :: real

  1265   assumes "x < y"

  1266   shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"

  1267 using Rats_dense_in_real [OF \<open>x < y\<close>]

  1268 by (auto elim: Rats_cases)

  1269

  1270

  1271 subsection\<open>Numerals and Arithmetic\<close>

  1272

  1273 lemma [code_abbrev]:   (*FIXME*)

  1274   "real_of_int (numeral k) = numeral k"

  1275   "real_of_int (- numeral k) = - numeral k"

  1276   by simp_all

  1277

  1278 declaration \<open>

  1279   K (Lin_Arith.add_inj_thms [@{thm of_nat_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]

  1280     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: of_nat_less_iff RS iffD2 *)

  1281   #> Lin_Arith.add_inj_thms [@{thm of_int_le_iff} RS iffD2, @{thm of_nat_eq_iff} RS iffD2]

  1282     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: of_int_less_iff RS iffD2 *)

  1283   #> Lin_Arith.add_simps [@{thm of_nat_0}, @{thm of_nat_Suc}, @{thm of_nat_add},

  1284       @{thm of_nat_mult}, @{thm of_int_0}, @{thm of_int_1},

  1285       @{thm of_int_add}, @{thm of_int_minus}, @{thm of_int_diff},

  1286       @{thm of_int_mult}, @{thm of_int_of_nat_eq},

  1287       @{thm of_nat_numeral}, @{thm int_numeral}, @{thm of_int_neg_numeral}]

  1288   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})

  1289   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))

  1290 \<close>

  1291

  1292 subsection\<open>Simprules combining x+y and 0: ARE THEY NEEDED?\<close>

  1293

  1294 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"

  1295 by arith

  1296

  1297 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"

  1298 by auto

  1299

  1300 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"

  1301 by auto

  1302

  1303 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"

  1304 by auto

  1305

  1306 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"

  1307 by auto

  1308

  1309 subsection \<open>Lemmas about powers\<close>

  1310

  1311 (* used by Import/HOL/real.imp *)

  1312 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"

  1313   by simp

  1314

  1315 text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>

  1316 lemma realpow_Suc_le_self:

  1317   fixes r :: "'a::linordered_semidom"

  1318   shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"

  1319 by (insert power_decreasing [of 1 "Suc n" r], simp)

  1320

  1321 text \<open>FIXME: no longer real-specific; rename and move elsewhere\<close>

  1322 lemma realpow_minus_mult:

  1323   fixes x :: "'a::monoid_mult"

  1324   shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"

  1325 by (simp add: power_Suc power_commutes split add: nat_diff_split)

  1326

  1327 text \<open>FIXME: declare this [simp] for all types, or not at all\<close>

  1328 declare sum_squares_eq_zero_iff [simp] sum_power2_eq_zero_iff [simp]

  1329

  1330 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"

  1331 by (rule_tac y = 0 in order_trans, auto)

  1332

  1333 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"

  1334   by (auto simp add: power2_eq_square)

  1335

  1336 lemma numeral_power_eq_real_of_int_cancel_iff[simp]:

  1337      "numeral x ^ n = real_of_int (y::int) \<longleftrightarrow> numeral x ^ n = y"

  1338   by (metis of_int_eq_iff of_int_numeral of_int_power)

  1339

  1340 lemma real_of_int_eq_numeral_power_cancel_iff[simp]:

  1341      "real_of_int (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"

  1342   using numeral_power_eq_real_of_int_cancel_iff[of x n y]

  1343   by metis

  1344

  1345 lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:

  1346      "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"

  1347   using of_nat_eq_iff by fastforce

  1348

  1349 lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:

  1350   "real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"

  1351   using numeral_power_eq_real_of_nat_cancel_iff[of x n y]

  1352   by metis

  1353

  1354 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:

  1355   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"

  1356 by (metis of_nat_le_iff of_nat_numeral of_nat_power)

  1357

  1358 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:

  1359   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"

  1360 by (metis of_nat_le_iff of_nat_numeral of_nat_power)

  1361

  1362 lemma numeral_power_le_real_of_int_cancel_iff[simp]:

  1363     "(numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"

  1364   by (metis ceiling_le_iff ceiling_of_int of_int_numeral of_int_power)

  1365

  1366 lemma real_of_int_le_numeral_power_cancel_iff[simp]:

  1367     "real_of_int a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"

  1368   by (metis floor_of_int le_floor_iff of_int_numeral of_int_power)

  1369

  1370 lemma numeral_power_less_real_of_nat_cancel_iff[simp]:

  1371     "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"

  1372   by (metis of_nat_less_iff of_nat_numeral of_nat_power)

  1373

  1374 lemma real_of_nat_less_numeral_power_cancel_iff[simp]:

  1375   "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"

  1376 by (metis of_nat_less_iff of_nat_numeral of_nat_power)

  1377

  1378 lemma numeral_power_less_real_of_int_cancel_iff[simp]:

  1379     "(numeral x::real) ^ n < real_of_int a \<longleftrightarrow> (numeral x::int) ^ n < a"

  1380   by (meson not_less real_of_int_le_numeral_power_cancel_iff)

  1381

  1382 lemma real_of_int_less_numeral_power_cancel_iff[simp]:

  1383      "real_of_int a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"

  1384   by (meson not_less numeral_power_le_real_of_int_cancel_iff)

  1385

  1386 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:

  1387     "(- numeral x::real) ^ n \<le> real_of_int a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"

  1388   by (metis of_int_le_iff of_int_neg_numeral of_int_power)

  1389

  1390 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:

  1391      "real_of_int a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"

  1392   by (metis of_int_le_iff of_int_neg_numeral of_int_power)

  1393

  1394

  1395 subsection\<open>Density of the Reals\<close>

  1396

  1397 lemma real_lbound_gt_zero:

  1398      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"

  1399 apply (rule_tac x = " (min d1 d2) /2" in exI)

  1400 apply (simp add: min_def)

  1401 done

  1402

  1403

  1404 text\<open>Similar results are proved in @{text Fields}\<close>

  1405 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"

  1406   by auto

  1407

  1408 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"

  1409   by auto

  1410

  1411 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

  1412   by simp

  1413

  1414 subsection\<open>Absolute Value Function for the Reals\<close>

  1415

  1416 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"

  1417 by (simp add: abs_if)

  1418

  1419 lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"

  1420 by (simp add: abs_if)

  1421

  1422 lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"

  1423 by simp

  1424

  1425 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"

  1426 by simp

  1427

  1428

  1429 subsection\<open>Floor and Ceiling Functions from the Reals to the Integers\<close>

  1430

  1431 (* FIXME: theorems for negative numerals. Many duplicates, e.g. from Archimedean_Field.thy. *)

  1432

  1433 lemma real_of_nat_less_numeral_iff [simp]:

  1434      "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"

  1435   by (metis of_nat_less_iff of_nat_numeral)

  1436

  1437 lemma numeral_less_real_of_nat_iff [simp]:

  1438      "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"

  1439   by (metis of_nat_less_iff of_nat_numeral)

  1440

  1441 lemma numeral_le_real_of_nat_iff[simp]:

  1442   "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"

  1443 by (metis not_le real_of_nat_less_numeral_iff)

  1444

  1445 declare of_int_floor_le [simp] (* FIXME*)

  1446

  1447 lemma of_int_floor_cancel [simp]:

  1448     "(of_int (floor x) = x) = (\<exists>n::int. x = of_int n)"

  1449   by (metis floor_of_int)

  1450

  1451 lemma floor_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> floor x = n"

  1452   by linarith

  1453

  1454 lemma floor_eq2: "[| real_of_int n \<le> x; x < real_of_int n + 1 |] ==> floor x = n"

  1455   by linarith

  1456

  1457 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"

  1458   by linarith

  1459

  1460 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"

  1461   by linarith

  1462

  1463 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real_of_int(floor r)"

  1464   by linarith

  1465

  1466 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real_of_int(floor r)"

  1467   by linarith

  1468

  1469 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real_of_int(floor r) + 1"

  1470   by linarith

  1471

  1472 lemma real_of_int_floor_add_one_gt [simp]: "r < real_of_int(floor r) + 1"

  1473   by linarith

  1474

  1475 lemma floor_eq_iff: "floor x = b \<longleftrightarrow> of_int b \<le> x \<and> x < of_int (b + 1)"

  1476 by (simp add: floor_unique_iff)

  1477

  1478 lemma floor_add2[simp]: "floor (of_int a + x) = a + floor x"

  1479   by (simp add: add.commute)

  1480

  1481 lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real_of_int b) = floor a div b"

  1482 proof cases

  1483   assume "0 < b"

  1484   { fix i j :: int assume "real_of_int i \<le> a" "a < 1 + real_of_int i"

  1485       "real_of_int j * real_of_int b \<le> a" "a < real_of_int b + real_of_int j * real_of_int b"

  1486     then have "i < b + j * b"

  1487       by (metis linorder_not_less of_int_add of_int_le_iff of_int_mult order_trans_rules(21))

  1488     moreover have "j * b < 1 + i"

  1489     proof -

  1490       have "real_of_int (j * b) < real_of_int i + 1"

  1491         using a < 1 + real_of_int i real_of_int j * real_of_int b \<le> a by force

  1492       thus "j * b < 1 + i"

  1493         by linarith

  1494     qed

  1495     ultimately have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"

  1496       by (auto simp: field_simps)

  1497     then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"

  1498       using pos_mod_bound[OF \<open>0<b\<close>, of i] pos_mod_sign[OF \<open>0<b\<close>, of i] by linarith+

  1499     then have "j = i div b"

  1500       using \<open>0 < b\<close> unfolding mult_less_cancel_right by auto

  1501   }

  1502   with \<open>0 < b\<close> show ?thesis

  1503     by (auto split: floor_split simp: field_simps)

  1504 qed auto

  1505

  1506 lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"

  1507   by (metis floor_divide_of_int_eq of_int_numeral)

  1508

  1509 lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"

  1510   by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)

  1511

  1512 lemma of_int_ceiling_cancel [simp]:

  1513      "(of_int (ceiling x) = x) = (\<exists>n::int. x = of_int n)"

  1514   using ceiling_of_int by metis

  1515

  1516 lemma ceiling_eq: "[| of_int n < x; x \<le> of_int n + 1 |] ==> ceiling x = n + 1"

  1517   by (simp add: ceiling_unique)

  1518

  1519 lemma of_int_ceiling_diff_one_le [simp]: "of_int (ceiling r) - 1 \<le> r"

  1520   by linarith

  1521

  1522 lemma of_int_ceiling_le_add_one [simp]: "of_int (ceiling r) \<le> r + 1"

  1523   by linarith

  1524

  1525 lemma ceiling_le: "x <= of_int a ==> ceiling x <= a"

  1526   by (simp add: ceiling_le_iff)

  1527

  1528 lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)"

  1529   by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)

  1530

  1531 lemma ceiling_divide_eq_div_numeral [simp]:

  1532   "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"

  1533   using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp

  1534

  1535 lemma ceiling_minus_divide_eq_div_numeral [simp]:

  1536   "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"

  1537   using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp

  1538

  1539 text\<open>The following lemmas are remnants of the erstwhile functions natfloor

  1540 and natceiling.\<close>

  1541

  1542 lemma nat_floor_neg: "(x::real) <= 0 ==> nat(floor x) = 0"

  1543   by linarith

  1544

  1545 lemma le_nat_floor: "real x <= a ==> x <= nat(floor a)"

  1546   by linarith

  1547

  1548 lemma le_mult_nat_floor:

  1549   shows "nat(floor a) * nat(floor b) \<le> nat(floor (a * b))"

  1550   by (cases "0 <= a & 0 <= b")

  1551      (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)

  1552

  1553 lemma nat_ceiling_le_eq [simp]: "(nat(ceiling x) <= a) = (x <= real a)"

  1554   by linarith

  1555

  1556 lemma real_nat_ceiling_ge: "x <= real(nat(ceiling x))"

  1557   by linarith

  1558

  1559 lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"

  1560   by (auto intro!: bexI[of _ "of_nat (nat(ceiling x))"]) linarith

  1561

  1562 lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"

  1563   apply (auto intro!: bexI[of _ "of_int (floor x - 1)"])

  1564   apply (rule less_le_trans[OF _ of_int_floor_le])

  1565   apply simp

  1566   done

  1567

  1568 subsection \<open>Exponentiation with floor\<close>

  1569

  1570 lemma floor_power:

  1571   assumes "x = of_int (floor x)"

  1572   shows "floor (x ^ n) = floor x ^ n"

  1573 proof -

  1574   have "x ^ n = of_int (floor x ^ n)"

  1575     using assms by (induct n arbitrary: x) simp_all

  1576   then show ?thesis by (metis floor_of_int)

  1577 qed

  1578

  1579 lemma floor_numeral_power[simp]:

  1580   "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"

  1581   by (metis floor_of_int of_int_numeral of_int_power)

  1582

  1583 lemma ceiling_numeral_power[simp]:

  1584   "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n"

  1585   by (metis ceiling_of_int of_int_numeral of_int_power)

  1586

  1587

  1588 subsection \<open>Implementation of rational real numbers\<close>

  1589

  1590 text \<open>Formal constructor\<close>

  1591

  1592 definition Ratreal :: "rat \<Rightarrow> real" where

  1593   [code_abbrev, simp]: "Ratreal = of_rat"

  1594

  1595 code_datatype Ratreal

  1596

  1597

  1598 text \<open>Numerals\<close>

  1599

  1600 lemma [code_abbrev]:

  1601   "(of_rat (of_int a) :: real) = of_int a"

  1602   by simp

  1603

  1604 lemma [code_abbrev]:

  1605   "(of_rat 0 :: real) = 0"

  1606   by simp

  1607

  1608 lemma [code_abbrev]:

  1609   "(of_rat 1 :: real) = 1"

  1610   by simp

  1611

  1612 lemma [code_abbrev]:

  1613   "(of_rat (- 1) :: real) = - 1"

  1614   by simp

  1615

  1616 lemma [code_abbrev]:

  1617   "(of_rat (numeral k) :: real) = numeral k"

  1618   by simp

  1619

  1620 lemma [code_abbrev]:

  1621   "(of_rat (- numeral k) :: real) = - numeral k"

  1622   by simp

  1623

  1624 lemma [code_post]:

  1625   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"

  1626   "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"

  1627   "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"

  1628   "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"

  1629   by (simp_all add: of_rat_divide of_rat_minus)

  1630

  1631

  1632 text \<open>Operations\<close>

  1633

  1634 lemma zero_real_code [code]:

  1635   "0 = Ratreal 0"

  1636 by simp

  1637

  1638 lemma one_real_code [code]:

  1639   "1 = Ratreal 1"

  1640 by simp

  1641

  1642 instantiation real :: equal

  1643 begin

  1644

  1645 definition "HOL.equal (x::real) y \<longleftrightarrow> x - y = 0"

  1646

  1647 instance proof

  1648 qed (simp add: equal_real_def)

  1649

  1650 lemma real_equal_code [code]:

  1651   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"

  1652   by (simp add: equal_real_def equal)

  1653

  1654 lemma [code nbe]:

  1655   "HOL.equal (x::real) x \<longleftrightarrow> True"

  1656   by (rule equal_refl)

  1657

  1658 end

  1659

  1660 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"

  1661   by (simp add: of_rat_less_eq)

  1662

  1663 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"

  1664   by (simp add: of_rat_less)

  1665

  1666 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"

  1667   by (simp add: of_rat_add)

  1668

  1669 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"

  1670   by (simp add: of_rat_mult)

  1671

  1672 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"

  1673   by (simp add: of_rat_minus)

  1674

  1675 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"

  1676   by (simp add: of_rat_diff)

  1677

  1678 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"

  1679   by (simp add: of_rat_inverse)

  1680

  1681 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"

  1682   by (simp add: of_rat_divide)

  1683

  1684 lemma real_floor_code [code]: "floor (Ratreal x) = floor x"

  1685   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)

  1686

  1687

  1688 text \<open>Quickcheck\<close>

  1689

  1690 definition (in term_syntax)

  1691   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

  1692   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"

  1693

  1694 notation fcomp (infixl "\<circ>>" 60)

  1695 notation scomp (infixl "\<circ>\<rightarrow>" 60)

  1696

  1697 instantiation real :: random

  1698 begin

  1699

  1700 definition

  1701   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"

  1702

  1703 instance ..

  1704

  1705 end

  1706

  1707 no_notation fcomp (infixl "\<circ>>" 60)

  1708 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

  1709

  1710 instantiation real :: exhaustive

  1711 begin

  1712

  1713 definition

  1714   "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"

  1715

  1716 instance ..

  1717

  1718 end

  1719

  1720 instantiation real :: full_exhaustive

  1721 begin

  1722

  1723 definition

  1724   "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"

  1725

  1726 instance ..

  1727

  1728 end

  1729

  1730 instantiation real :: narrowing

  1731 begin

  1732

  1733 definition

  1734   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"

  1735

  1736 instance ..

  1737

  1738 end

  1739

  1740

  1741 subsection \<open>Setup for Nitpick\<close>

  1742

  1743 declaration \<open>

  1744   Nitpick_HOL.register_frac_type @{type_name real}

  1745    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),

  1746     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),

  1747     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),

  1748     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),

  1749     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),

  1750     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),

  1751     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),

  1752     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]

  1753 \<close>

  1754

  1755 lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real

  1756     ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real

  1757     times_real_inst.times_real uminus_real_inst.uminus_real

  1758     zero_real_inst.zero_real

  1759

  1760

  1761 subsection \<open>Setup for SMT\<close>

  1762

  1763 ML_file "Tools/SMT/smt_real.ML"

  1764 ML_file "Tools/SMT/z3_real.ML"

  1765

  1766 lemma [z3_rule]:

  1767   "0 + (x::real) = x"

  1768   "x + 0 = x"

  1769   "0 * x = 0"

  1770   "1 * x = x"

  1771   "x + y = y + x"

  1772   by auto

  1773

  1774 end
`