src/HOL/Real.thy
 author hoelzl Tue Nov 05 21:23:42 2013 +0100 (2013-11-05) changeset 54281 b01057e72233 parent 54263 c4159fe6fa46 child 54489 03ff4d1e6784 permissions -rw-r--r--
int and nat are conditionally_complete_lattices
     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 header {* Development of the Reals using Cauchy Sequences *}

    11

    12 theory Real

    13 imports Rat Conditionally_Complete_Lattices

    14 begin

    15

    16 text {*

    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 *}

    22

    23 subsection {* Preliminary lemmas *}

    24

    25 lemma add_diff_add:

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

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

    28   by simp

    29

    30 lemma minus_diff_minus:

    31   fixes a b :: "'a::ab_group_add"

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

    33   by simp

    34

    35 lemma mult_diff_mult:

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

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

    38   by (simp add: algebra_simps)

    39

    40 lemma inverse_diff_inverse:

    41   fixes a b :: "'a::division_ring"

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

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

    44   using assms by (simp add: algebra_simps)

    45

    46 lemma obtain_pos_sum:

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

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

    49 proof

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

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

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

    53 qed

    54

    55 subsection {* Sequences that converge to zero *}

    56

    57 definition

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

    59 where

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

    61

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

    63   unfolding vanishes_def by simp

    64

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

    66   unfolding vanishes_def by simp

    67

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

    69   unfolding vanishes_def

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

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

    72   done

    73

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

    75   unfolding vanishes_def by simp

    76

    77 lemma vanishes_add:

    78   assumes X: "vanishes X" and Y: "vanishes Y"

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

    80 proof (rule vanishesI)

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

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

    83     by (rule obtain_pos_sum)

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

    85     using vanishesD [OF X s] ..

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

    87     using vanishesD [OF Y t] ..

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

    89   proof (clarsimp)

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

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

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

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

    94   qed

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

    96 qed

    97

    98 lemma vanishes_diff:

    99   assumes X: "vanishes X" and Y: "vanishes Y"

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

   101   unfolding diff_conv_add_uminus by (intro vanishes_add vanishes_minus X Y)

   102

   103 lemma vanishes_mult_bounded:

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

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

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

   107 proof (rule vanishesI)

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

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

   110     using X by fast

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

   112   proof

   113     show "0 < r / a" using r a by (simp add: divide_pos_pos)

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

   115   qed

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

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

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

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

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

   121 qed

   122

   123 subsection {* Cauchy sequences *}

   124

   125 definition

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

   127 where

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

   129

   130 lemma cauchyI:

   131   "(\<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"

   132   unfolding cauchy_def by simp

   133

   134 lemma cauchyD:

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

   136   unfolding cauchy_def by simp

   137

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

   139   unfolding cauchy_def by simp

   140

   141 lemma cauchy_add [simp]:

   142   assumes X: "cauchy X" and Y: "cauchy Y"

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

   144 proof (rule cauchyI)

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

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

   147     by (rule obtain_pos_sum)

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

   149     using cauchyD [OF X s] ..

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

   151     using cauchyD [OF Y t] ..

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

   153   proof (clarsimp)

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

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

   156       unfolding add_diff_add by (rule abs_triangle_ineq)

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

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

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

   160   qed

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

   162 qed

   163

   164 lemma cauchy_minus [simp]:

   165   assumes X: "cauchy X"

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

   167 using assms unfolding cauchy_def

   168 unfolding minus_diff_minus abs_minus_cancel .

   169

   170 lemma cauchy_diff [simp]:

   171   assumes X: "cauchy X" and Y: "cauchy Y"

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

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

   174

   175 lemma cauchy_imp_bounded:

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

   177 proof -

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

   179     using cauchyD [OF assms zero_less_one] ..

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

   181   proof (intro exI conjI allI)

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

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

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

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

   186   next

   187     fix n :: nat

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

   189     proof (rule linorder_le_cases)

   190       assume "n \<le> k"

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

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

   193     next

   194       assume "k \<le> n"

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

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

   197         by (rule abs_triangle_ineq)

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

   199         by (rule add_le_less_mono, simp, simp add: k k \<le> n)

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

   201     qed

   202   qed

   203 qed

   204

   205 lemma cauchy_mult [simp]:

   206   assumes X: "cauchy X" and Y: "cauchy Y"

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

   208 proof (rule cauchyI)

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

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

   211     by (rule obtain_pos_sum)

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

   213     using cauchy_imp_bounded [OF X] by fast

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

   215     using cauchy_imp_bounded [OF Y] by fast

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

   217   proof

   218     show "0 < v/b" using v b(1) by (rule divide_pos_pos)

   219     show "0 < u/a" using u a(1) by (rule divide_pos_pos)

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

   221       using a(1) b(1) r = u + v by simp

   222   qed

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

   224     using cauchyD [OF X s] ..

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

   226     using cauchyD [OF Y t] ..

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

   228   proof (clarsimp)

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

   230     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>"

   231       unfolding mult_diff_mult ..

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

   233       by (rule abs_triangle_ineq)

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

   235       unfolding abs_mult ..

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

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

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

   239   qed

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

   241 qed

   242

   243 lemma cauchy_not_vanishes_cases:

   244   assumes X: "cauchy X"

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

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

   247 proof -

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

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

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

   251     using 0 < r by (rule obtain_pos_sum)

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

   253     using cauchyD [OF X s] ..

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

   255     using r by fast

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

   257     using i i \<le> k by auto

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

   259     using r \<le> \<bar>X k\<bar> by auto

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

   261     unfolding r = s + t using k by auto

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

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

   264     using t by auto

   265 qed

   266

   267 lemma cauchy_not_vanishes:

   268   assumes X: "cauchy X"

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

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

   271 using cauchy_not_vanishes_cases [OF assms]

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

   273

   274 lemma cauchy_inverse [simp]:

   275   assumes X: "cauchy X"

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

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

   278 proof (rule cauchyI)

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

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

   281     using cauchy_not_vanishes [OF X nz] by fast

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

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

   284   proof

   285     show "0 < b * r * b"

   286       by (simp add: 0 < r b mult_pos_pos)

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

   288       using b by simp

   289   qed

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

   291     using cauchyD [OF X s] ..

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

   293   proof (clarsimp)

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

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

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

   297       by (simp add: inverse_diff_inverse nz * abs_mult)

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

   299       by (simp add: mult_strict_mono less_imp_inverse_less

   300                     mult_pos_pos i j b * s)

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

   302   qed

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

   304 qed

   305

   306 lemma vanishes_diff_inverse:

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

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

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

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

   311 proof (rule vanishesI)

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

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

   314     using cauchy_not_vanishes [OF X] by fast

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

   316     using cauchy_not_vanishes [OF Y] by fast

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

   318   proof

   319     show "0 < a * r * b"

   320       using a r b by (simp add: mult_pos_pos)

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

   322       using a r b by simp

   323   qed

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

   325     using vanishesD [OF XY s] ..

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

   327   proof (clarsimp)

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

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

   330       using i j a b n by auto

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

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

   333       by (simp add: inverse_diff_inverse abs_mult)

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

   335       apply (intro mult_strict_mono' less_imp_inverse_less)

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

   337       done

   338     also note inverse a * s * inverse b = r

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

   340   qed

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

   342 qed

   343

   344 subsection {* Equivalence relation on Cauchy sequences *}

   345

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

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

   348

   349 lemma realrelI [intro?]:

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

   351   shows "realrel X Y"

   352   using assms unfolding realrel_def by simp

   353

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

   355   unfolding realrel_def by simp

   356

   357 lemma symp_realrel: "symp realrel"

   358   unfolding realrel_def

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

   360

   361 lemma transp_realrel: "transp realrel"

   362   unfolding realrel_def

   363   apply (rule transpI, clarify)

   364   apply (drule (1) vanishes_add)

   365   apply (simp add: algebra_simps)

   366   done

   367

   368 lemma part_equivp_realrel: "part_equivp realrel"

   369   by (fast intro: part_equivpI symp_realrel transp_realrel

   370     realrel_refl cauchy_const)

   371

   372 subsection {* The field of real numbers *}

   373

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

   375   morphisms rep_real Real

   376   by (rule part_equivp_realrel)

   377

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

   379   unfolding real.pcr_cr_eq cr_real_def realrel_def by auto

   380

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

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

   383 proof (induct x)

   384   case (1 X)

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

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

   387 qed

   388

   389 lemma eq_Real:

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

   391   using real.rel_eq_transfer

   392   unfolding real.pcr_cr_eq cr_real_def fun_rel_def realrel_def by simp

   393

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

   395 by (simp add: real.domain_eq realrel_def)

   396

   397 instantiation real :: field_inverse_zero

   398 begin

   399

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

   401   by (simp add: realrel_refl)

   402

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

   404   by (simp add: realrel_refl)

   405

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

   407   unfolding realrel_def add_diff_add

   408   by (simp only: cauchy_add vanishes_add simp_thms)

   409

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

   411   unfolding realrel_def minus_diff_minus

   412   by (simp only: cauchy_minus vanishes_minus simp_thms)

   413

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

   415   unfolding realrel_def mult_diff_mult

   416   by (subst (4) mult_commute, simp only: cauchy_mult vanishes_add

   417     vanishes_mult_bounded cauchy_imp_bounded simp_thms)

   418

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

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

   421 proof -

   422   fix X Y assume "realrel X Y"

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

   424     unfolding realrel_def by simp_all

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

   426   proof

   427     assume "vanishes X"

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

   429   next

   430     assume "vanishes Y"

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

   432   qed

   433   thus "?thesis X Y"

   434     unfolding realrel_def

   435     by (simp add: vanishes_diff_inverse X Y XY)

   436 qed

   437

   438 definition

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

   440

   441 definition

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

   443

   444 lemma add_Real:

   445   assumes X: "cauchy X" and Y: "cauchy Y"

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

   447   using assms plus_real.transfer

   448   unfolding cr_real_eq fun_rel_def by simp

   449

   450 lemma minus_Real:

   451   assumes X: "cauchy X"

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

   453   using assms uminus_real.transfer

   454   unfolding cr_real_eq fun_rel_def by simp

   455

   456 lemma diff_Real:

   457   assumes X: "cauchy X" and Y: "cauchy Y"

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

   459   unfolding minus_real_def

   460   by (simp add: minus_Real add_Real X Y)

   461

   462 lemma mult_Real:

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

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

   465   using assms times_real.transfer

   466   unfolding cr_real_eq fun_rel_def by simp

   467

   468 lemma inverse_Real:

   469   assumes X: "cauchy X"

   470   shows "inverse (Real X) =

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

   472   using assms inverse_real.transfer zero_real.transfer

   473   unfolding cr_real_eq fun_rel_def by (simp split: split_if_asm, metis)

   474

   475 instance proof

   476   fix a b c :: real

   477   show "a + b = b + a"

   478     by transfer (simp add: add_ac realrel_def)

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

   480     by transfer (simp add: add_ac realrel_def)

   481   show "0 + a = a"

   482     by transfer (simp add: realrel_def)

   483   show "- a + a = 0"

   484     by transfer (simp add: realrel_def)

   485   show "a - b = a + - b"

   486     by (rule minus_real_def)

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

   488     by transfer (simp add: mult_ac realrel_def)

   489   show "a * b = b * a"

   490     by transfer (simp add: mult_ac realrel_def)

   491   show "1 * a = a"

   492     by transfer (simp add: mult_ac realrel_def)

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

   494     by transfer (simp add: distrib_right realrel_def)

   495   show "(0\<Colon>real) \<noteq> (1\<Colon>real)"

   496     by transfer (simp add: realrel_def)

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

   498     apply transfer

   499     apply (simp add: realrel_def)

   500     apply (rule vanishesI)

   501     apply (frule (1) cauchy_not_vanishes, clarify)

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

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

   504     done

   505   show "a / b = a * inverse b"

   506     by (rule divide_real_def)

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

   508     by transfer (simp add: realrel_def)

   509 qed

   510

   511 end

   512

   513 subsection {* Positive reals *}

   514

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

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

   517 proof -

   518   { fix X Y

   519     assume "realrel X Y"

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

   521       unfolding realrel_def by simp_all

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

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

   524       by fast

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

   526       using 0 < r by (rule obtain_pos_sum)

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

   528       using vanishesD [OF XY s] ..

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

   530     proof (clarsimp)

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

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

   533         using i j n by simp_all

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

   535     qed

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

   537   } note 1 = this

   538   fix X Y assume "realrel X Y"

   539   hence "realrel X Y" and "realrel Y X"

   540     using symp_realrel unfolding symp_def by auto

   541   thus "?thesis X Y"

   542     by (safe elim!: 1)

   543 qed

   544

   545 lemma positive_Real:

   546   assumes X: "cauchy X"

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

   548   using assms positive.transfer

   549   unfolding cr_real_eq fun_rel_def by simp

   550

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

   552   by transfer auto

   553

   554 lemma positive_add:

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

   556 apply transfer

   557 apply (clarify, rename_tac a b i j)

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

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

   560 apply (simp add: add_strict_mono)

   561 done

   562

   563 lemma positive_mult:

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

   565 apply transfer

   566 apply (clarify, rename_tac a b i j)

   567 apply (rule_tac x="a * b" in exI, simp add: mult_pos_pos)

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

   569 apply (rule mult_strict_mono, auto)

   570 done

   571

   572 lemma positive_minus:

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

   574 apply transfer

   575 apply (simp add: realrel_def)

   576 apply (drule (1) cauchy_not_vanishes_cases, safe, fast, fast)

   577 done

   578

   579 instantiation real :: linordered_field_inverse_zero

   580 begin

   581

   582 definition

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

   584

   585 definition

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

   587

   588 definition

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

   590

   591 definition

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

   593

   594 instance proof

   595   fix a b c :: real

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

   597     by (rule abs_real_def)

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

   599     unfolding less_eq_real_def less_real_def

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

   601   show "a \<le> a"

   602     unfolding less_eq_real_def by simp

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

   604     unfolding less_eq_real_def less_real_def

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

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

   607     unfolding less_eq_real_def less_real_def

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

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

   610     unfolding less_eq_real_def less_real_def by auto

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

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

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

   614     by (rule sgn_real_def)

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

   616     unfolding less_eq_real_def less_real_def

   617     by (auto dest!: positive_minus)

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

   619     unfolding less_real_def

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

   621 qed

   622

   623 end

   624

   625 instantiation real :: distrib_lattice

   626 begin

   627

   628 definition

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

   630

   631 definition

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

   633

   634 instance proof

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

   636

   637 end

   638

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

   640 apply (induct x)

   641 apply (simp add: zero_real_def)

   642 apply (simp add: one_real_def add_Real)

   643 done

   644

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

   646 apply (cases x rule: int_diff_cases)

   647 apply (simp add: of_nat_Real diff_Real)

   648 done

   649

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

   651 apply (induct x)

   652 apply (simp add: Fract_of_int_quotient of_rat_divide)

   653 apply (simp add: of_int_Real divide_inverse)

   654 apply (simp add: inverse_Real mult_Real)

   655 done

   656

   657 instance real :: archimedean_field

   658 proof

   659   fix x :: real

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

   661     apply (induct x)

   662     apply (frule cauchy_imp_bounded, clarify)

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

   664     apply (rule less_imp_le)

   665     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)

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

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

   668     apply (rule le_less_trans [OF abs_ge_self])

   669     apply (rule less_le_trans [OF _ le_of_int_ceiling])

   670     apply simp

   671     done

   672 qed

   673

   674 instantiation real :: floor_ceiling

   675 begin

   676

   677 definition [code del]:

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

   679

   680 instance proof

   681   fix x :: real

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

   683     unfolding floor_real_def using floor_exists1 by (rule theI')

   684 qed

   685

   686 end

   687

   688 subsection {* Completeness *}

   689

   690 lemma not_positive_Real:

   691   assumes X: "cauchy X"

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

   693 unfolding positive_Real [OF X]

   694 apply (auto, unfold not_less)

   695 apply (erule obtain_pos_sum)

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

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

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

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

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

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

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

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

   704 done

   705

   706 lemma le_Real:

   707   assumes X: "cauchy X" and Y: "cauchy Y"

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

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

   710 apply (simp add: diff_Real not_positive_Real X Y)

   711 apply (simp add: diff_le_eq add_ac)

   712 done

   713

   714 lemma le_RealI:

   715   assumes Y: "cauchy Y"

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

   717 proof (induct x)

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

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

   720     by (simp add: of_rat_Real le_Real)

   721   {

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

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

   724       by (rule obtain_pos_sum)

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

   726       using cauchyD [OF Y s] ..

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

   728       using le [OF t] ..

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

   730     proof (clarsimp)

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

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

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

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

   735     qed

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

   737   }

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

   739     by (simp add: of_rat_Real le_Real X Y)

   740 qed

   741

   742 lemma Real_leI:

   743   assumes X: "cauchy X"

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

   745   shows "Real X \<le> y"

   746 proof -

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

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

   749   thus ?thesis by simp

   750 qed

   751

   752 lemma less_RealD:

   753   assumes Y: "cauchy Y"

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

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

   756

   757 lemma of_nat_less_two_power:

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

   759 apply (induct n)

   760 apply simp

   761 apply (subgoal_tac "(1::'a) \<le> 2 ^ n")

   762 apply (drule (1) add_le_less_mono, simp)

   763 apply simp

   764 done

   765

   766 lemma complete_real:

   767   fixes S :: "real set"

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

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

   770 proof -

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

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

   773

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

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

   776   proof

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

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

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

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

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

   782       unfolding P_def of_rat_of_int_eq using x by fast

   783   qed

   784   obtain b where b: "P b"

   785   proof

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

   787     unfolding P_def of_rat_of_int_eq

   788     proof

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

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

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

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

   793     qed

   794   qed

   795

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

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

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

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

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

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

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

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

   804     unfolding A_def B_def C_def bisect_def split_def by simp

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

   806     unfolding A_def B_def C_def bisect_def split_def by simp

   807

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

   809     apply (simp add: eq_divide_eq)

   810     apply (induct_tac n, simp)

   811     apply (simp add: C_def avg_def algebra_simps)

   812     done

   813

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

   815     apply (simp add: divide_less_eq)

   816     apply (subst mult_commute)

   817     apply (frule_tac y=y in ex_less_of_nat_mult)

   818     apply clarify

   819     apply (rule_tac x=n in exI)

   820     apply (erule less_trans)

   821     apply (rule mult_strict_right_mono)

   822     apply (rule le_less_trans [OF _ of_nat_less_two_power])

   823     apply simp

   824     apply assumption

   825     done

   826

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

   828     by (induct_tac n, simp_all add: a)

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

   830     by (induct_tac n, simp_all add: b)

   831   have ab: "a < b"

   832     using a b unfolding P_def

   833     apply (clarsimp simp add: not_le)

   834     apply (drule (1) bspec)

   835     apply (drule (1) less_le_trans)

   836     apply (simp add: of_rat_less)

   837     done

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

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

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

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

   842     apply (erule less_Suc_induct)

   843     apply (clarsimp simp add: C_def avg_def)

   844     apply (simp add: add_divide_distrib [symmetric])

   845     apply (rule AB [THEN less_imp_le])

   846     apply simp

   847     done

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

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

   850     apply (erule less_Suc_induct)

   851     apply (clarsimp simp add: C_def avg_def)

   852     apply (simp add: add_divide_distrib [symmetric])

   853     apply (rule AB [THEN less_imp_le])

   854     apply simp

   855     done

   856   have cauchy_lemma:

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

   858     apply (rule cauchyI)

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

   860     apply (erule exE)

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

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

   863     apply (simp add: width)

   864     apply (drule_tac x=n in spec)

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

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

   867     apply (frule A_mono, drule B_mono)

   868     apply (frule A_mono, drule B_mono)

   869     apply arith

   870     done

   871   have "cauchy A"

   872     apply (rule cauchy_lemma [rule_format])

   873     apply (simp add: A_mono)

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

   875     done

   876   have "cauchy B"

   877     apply (rule cauchy_lemma [rule_format])

   878     apply (simp add: B_mono)

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

   880     done

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

   882   proof

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

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

   885       using PB [unfolded P_def] cauchy B

   886       by (simp add: le_RealI)

   887   qed

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

   889     apply clarify

   890     apply (erule contrapos_pp)

   891     apply (simp add: not_le)

   892     apply (drule less_RealD [OF cauchy A], clarify)

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

   894     apply (simp add: P_def not_le, clarify)

   895     apply (erule rev_bexI)

   896     apply (erule (1) less_trans)

   897     apply (simp add: PA)

   898     done

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

   900   proof (rule vanishesI)

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

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

   903       using twos by fast

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

   905     proof (clarify)

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

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

   908         by simp

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

   910         using n by (simp add: divide_left_mono mult_pos_pos)

   911       also note k

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

   913     qed

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

   915   qed

   916   hence 3: "Real B = Real A"

   917     by (simp add: eq_Real cauchy A cauchy B width)

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

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

   920 qed

   921

   922 instantiation real :: linear_continuum

   923 begin

   924

   925 subsection{*Supremum of a set of reals*}

   926

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

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

   929

   930 instance

   931 proof

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

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

   934     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"

   935       using complete_real[of X] unfolding bdd_above_def by blast

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

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

   938   note Sup_upper = this

   939

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

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

   942     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"

   943       using complete_real[of X] by blast

   944     then have "Sup X = s"

   945       unfolding Sup_real_def by (best intro: Least_equality)

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

   947       by blast

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

   949   note Sup_least = this

   950

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

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

   953   { 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"

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

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

   956     using zero_neq_one by blast

   957 qed

   958 end

   959

   960

   961 subsection {* Hiding implementation details *}

   962

   963 hide_const (open) vanishes cauchy positive Real

   964

   965 declare Real_induct [induct del]

   966 declare Abs_real_induct [induct del]

   967 declare Abs_real_cases [cases del]

   968

   969 lifting_update real.lifting

   970 lifting_forget real.lifting

   971

   972 subsection{*More Lemmas*}

   973

   974 text {* BH: These lemmas should not be necessary; they should be

   975 covered by existing simp rules and simplification procedures. *}

   976

   977 lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"

   978 by simp (* redundant with mult_cancel_left *)

   979

   980 lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"

   981 by simp (* redundant with mult_cancel_right *)

   982

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

   984 by simp (* solved by linordered_ring_less_cancel_factor simproc *)

   985

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

   987 by simp (* solved by linordered_ring_le_cancel_factor simproc *)

   988

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

   990 by simp (* solved by linordered_ring_le_cancel_factor simproc *)

   991

   992

   993 subsection {* Embedding numbers into the Reals *}

   994

   995 abbreviation

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

   997 where

   998   "real_of_nat \<equiv> of_nat"

   999

  1000 abbreviation

  1001   real_of_int :: "int \<Rightarrow> real"

  1002 where

  1003   "real_of_int \<equiv> of_int"

  1004

  1005 abbreviation

  1006   real_of_rat :: "rat \<Rightarrow> real"

  1007 where

  1008   "real_of_rat \<equiv> of_rat"

  1009

  1010 consts

  1011   (*overloaded constant for injecting other types into "real"*)

  1012   real :: "'a => real"

  1013

  1014 defs (overloaded)

  1015   real_of_nat_def [code_unfold]: "real == real_of_nat"

  1016   real_of_int_def [code_unfold]: "real == real_of_int"

  1017

  1018 declare [[coercion_enabled]]

  1019 declare [[coercion "real::nat\<Rightarrow>real"]]

  1020 declare [[coercion "real::int\<Rightarrow>real"]]

  1021 declare [[coercion "int"]]

  1022

  1023 declare [[coercion_map map]]

  1024 declare [[coercion_map "% f g h x. g (h (f x))"]]

  1025 declare [[coercion_map "% f g (x,y) . (f x, g y)"]]

  1026

  1027 lemma real_eq_of_nat: "real = of_nat"

  1028   unfolding real_of_nat_def ..

  1029

  1030 lemma real_eq_of_int: "real = of_int"

  1031   unfolding real_of_int_def ..

  1032

  1033 lemma real_of_int_zero [simp]: "real (0::int) = 0"

  1034 by (simp add: real_of_int_def)

  1035

  1036 lemma real_of_one [simp]: "real (1::int) = (1::real)"

  1037 by (simp add: real_of_int_def)

  1038

  1039 lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"

  1040 by (simp add: real_of_int_def)

  1041

  1042 lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"

  1043 by (simp add: real_of_int_def)

  1044

  1045 lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"

  1046 by (simp add: real_of_int_def)

  1047

  1048 lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"

  1049 by (simp add: real_of_int_def)

  1050

  1051 lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"

  1052 by (simp add: real_of_int_def of_int_power)

  1053

  1054 lemmas power_real_of_int = real_of_int_power [symmetric]

  1055

  1056 lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"

  1057   apply (subst real_eq_of_int)+

  1058   apply (rule of_int_setsum)

  1059 done

  1060

  1061 lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =

  1062     (PROD x:A. real(f x))"

  1063   apply (subst real_eq_of_int)+

  1064   apply (rule of_int_setprod)

  1065 done

  1066

  1067 lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"

  1068 by (simp add: real_of_int_def)

  1069

  1070 lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"

  1071 by (simp add: real_of_int_def)

  1072

  1073 lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"

  1074 by (simp add: real_of_int_def)

  1075

  1076 lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"

  1077 by (simp add: real_of_int_def)

  1078

  1079 lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"

  1080 by (simp add: real_of_int_def)

  1081

  1082 lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"

  1083 by (simp add: real_of_int_def)

  1084

  1085 lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"

  1086 by (simp add: real_of_int_def)

  1087

  1088 lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"

  1089 by (simp add: real_of_int_def)

  1090

  1091 lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"

  1092   unfolding real_of_one[symmetric] real_of_int_less_iff ..

  1093

  1094 lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"

  1095   unfolding real_of_one[symmetric] real_of_int_le_iff ..

  1096

  1097 lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"

  1098   unfolding real_of_one[symmetric] real_of_int_less_iff ..

  1099

  1100 lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"

  1101   unfolding real_of_one[symmetric] real_of_int_le_iff ..

  1102

  1103 lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"

  1104 by (auto simp add: abs_if)

  1105

  1106 lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"

  1107   apply (subgoal_tac "real n + 1 = real (n + 1)")

  1108   apply (simp del: real_of_int_add)

  1109   apply auto

  1110 done

  1111

  1112 lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"

  1113   apply (subgoal_tac "real m + 1 = real (m + 1)")

  1114   apply (simp del: real_of_int_add)

  1115   apply simp

  1116 done

  1117

  1118 lemma real_of_int_div_aux: "(real (x::int)) / (real d) =

  1119     real (x div d) + (real (x mod d)) / (real d)"

  1120 proof -

  1121   have "x = (x div d) * d + x mod d"

  1122     by auto

  1123   then have "real x = real (x div d) * real d + real(x mod d)"

  1124     by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])

  1125   then have "real x / real d = ... / real d"

  1126     by simp

  1127   then show ?thesis

  1128     by (auto simp add: add_divide_distrib algebra_simps)

  1129 qed

  1130

  1131 lemma real_of_int_div: "(d :: int) dvd n ==>

  1132     real(n div d) = real n / real d"

  1133   apply (subst real_of_int_div_aux)

  1134   apply simp

  1135   apply (simp add: dvd_eq_mod_eq_0)

  1136 done

  1137

  1138 lemma real_of_int_div2:

  1139   "0 <= real (n::int) / real (x) - real (n div x)"

  1140   apply (case_tac "x = 0")

  1141   apply simp

  1142   apply (case_tac "0 < x")

  1143   apply (simp add: algebra_simps)

  1144   apply (subst real_of_int_div_aux)

  1145   apply simp

  1146   apply (subst zero_le_divide_iff)

  1147   apply auto

  1148   apply (simp add: algebra_simps)

  1149   apply (subst real_of_int_div_aux)

  1150   apply simp

  1151   apply (subst zero_le_divide_iff)

  1152   apply auto

  1153 done

  1154

  1155 lemma real_of_int_div3:

  1156   "real (n::int) / real (x) - real (n div x) <= 1"

  1157   apply (simp add: algebra_simps)

  1158   apply (subst real_of_int_div_aux)

  1159   apply (auto simp add: divide_le_eq intro: order_less_imp_le)

  1160 done

  1161

  1162 lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"

  1163 by (insert real_of_int_div2 [of n x], simp)

  1164

  1165 lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"

  1166 unfolding real_of_int_def by (rule Ints_of_int)

  1167

  1168

  1169 subsection{*Embedding the Naturals into the Reals*}

  1170

  1171 lemma real_of_nat_zero [simp]: "real (0::nat) = 0"

  1172 by (simp add: real_of_nat_def)

  1173

  1174 lemma real_of_nat_1 [simp]: "real (1::nat) = 1"

  1175 by (simp add: real_of_nat_def)

  1176

  1177 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"

  1178 by (simp add: real_of_nat_def)

  1179

  1180 lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"

  1181 by (simp add: real_of_nat_def)

  1182

  1183 (*Not for addsimps: often the LHS is used to represent a positive natural*)

  1184 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"

  1185 by (simp add: real_of_nat_def)

  1186

  1187 lemma real_of_nat_less_iff [iff]:

  1188      "(real (n::nat) < real m) = (n < m)"

  1189 by (simp add: real_of_nat_def)

  1190

  1191 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"

  1192 by (simp add: real_of_nat_def)

  1193

  1194 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"

  1195 by (simp add: real_of_nat_def)

  1196

  1197 lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"

  1198 by (simp add: real_of_nat_def del: of_nat_Suc)

  1199

  1200 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"

  1201 by (simp add: real_of_nat_def of_nat_mult)

  1202

  1203 lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"

  1204 by (simp add: real_of_nat_def of_nat_power)

  1205

  1206 lemmas power_real_of_nat = real_of_nat_power [symmetric]

  1207

  1208 lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =

  1209     (SUM x:A. real(f x))"

  1210   apply (subst real_eq_of_nat)+

  1211   apply (rule of_nat_setsum)

  1212 done

  1213

  1214 lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =

  1215     (PROD x:A. real(f x))"

  1216   apply (subst real_eq_of_nat)+

  1217   apply (rule of_nat_setprod)

  1218 done

  1219

  1220 lemma real_of_card: "real (card A) = setsum (%x.1) A"

  1221   apply (subst card_eq_setsum)

  1222   apply (subst real_of_nat_setsum)

  1223   apply simp

  1224 done

  1225

  1226 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"

  1227 by (simp add: real_of_nat_def)

  1228

  1229 lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"

  1230 by (simp add: real_of_nat_def)

  1231

  1232 lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"

  1233 by (simp add: add: real_of_nat_def of_nat_diff)

  1234

  1235 lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"

  1236 by (auto simp: real_of_nat_def)

  1237

  1238 lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"

  1239 by (simp add: add: real_of_nat_def)

  1240

  1241 lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"

  1242 by (simp add: add: real_of_nat_def)

  1243

  1244 lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"

  1245   apply (subgoal_tac "real n + 1 = real (Suc n)")

  1246   apply simp

  1247   apply (auto simp add: real_of_nat_Suc)

  1248 done

  1249

  1250 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"

  1251   apply (subgoal_tac "real m + 1 = real (Suc m)")

  1252   apply (simp add: less_Suc_eq_le)

  1253   apply (simp add: real_of_nat_Suc)

  1254 done

  1255

  1256 lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) =

  1257     real (x div d) + (real (x mod d)) / (real d)"

  1258 proof -

  1259   have "x = (x div d) * d + x mod d"

  1260     by auto

  1261   then have "real x = real (x div d) * real d + real(x mod d)"

  1262     by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])

  1263   then have "real x / real d = \<dots> / real d"

  1264     by simp

  1265   then show ?thesis

  1266     by (auto simp add: add_divide_distrib algebra_simps)

  1267 qed

  1268

  1269 lemma real_of_nat_div: "(d :: nat) dvd n ==>

  1270     real(n div d) = real n / real d"

  1271   by (subst real_of_nat_div_aux)

  1272     (auto simp add: dvd_eq_mod_eq_0 [symmetric])

  1273

  1274 lemma real_of_nat_div2:

  1275   "0 <= real (n::nat) / real (x) - real (n div x)"

  1276 apply (simp add: algebra_simps)

  1277 apply (subst real_of_nat_div_aux)

  1278 apply simp

  1279 apply (subst zero_le_divide_iff)

  1280 apply simp

  1281 done

  1282

  1283 lemma real_of_nat_div3:

  1284   "real (n::nat) / real (x) - real (n div x) <= 1"

  1285 apply(case_tac "x = 0")

  1286 apply (simp)

  1287 apply (simp add: algebra_simps)

  1288 apply (subst real_of_nat_div_aux)

  1289 apply simp

  1290 done

  1291

  1292 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"

  1293 by (insert real_of_nat_div2 [of n x], simp)

  1294

  1295 lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"

  1296 by (simp add: real_of_int_def real_of_nat_def)

  1297

  1298 lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"

  1299   apply (subgoal_tac "real(int(nat x)) = real(nat x)")

  1300   apply force

  1301   apply (simp only: real_of_int_of_nat_eq)

  1302 done

  1303

  1304 lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"

  1305 unfolding real_of_nat_def by (rule of_nat_in_Nats)

  1306

  1307 lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"

  1308 unfolding real_of_nat_def by (rule Ints_of_nat)

  1309

  1310 subsection {* The Archimedean Property of the Reals *}

  1311

  1312 theorem reals_Archimedean:

  1313   assumes x_pos: "0 < x"

  1314   shows "\<exists>n. inverse (real (Suc n)) < x"

  1315   unfolding real_of_nat_def using x_pos

  1316   by (rule ex_inverse_of_nat_Suc_less)

  1317

  1318 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"

  1319   unfolding real_of_nat_def by (rule ex_less_of_nat)

  1320

  1321 lemma reals_Archimedean3:

  1322   assumes x_greater_zero: "0 < x"

  1323   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"

  1324   unfolding real_of_nat_def using 0 < x

  1325   by (auto intro: ex_less_of_nat_mult)

  1326

  1327

  1328 subsection{* Rationals *}

  1329

  1330 lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"

  1331 by (simp add: real_eq_of_nat)

  1332

  1333

  1334 lemma Rats_eq_int_div_int:

  1335   "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")

  1336 proof

  1337   show "\<rat> \<subseteq> ?S"

  1338   proof

  1339     fix x::real assume "x : \<rat>"

  1340     then obtain r where "x = of_rat r" unfolding Rats_def ..

  1341     have "of_rat r : ?S"

  1342       by (cases r)(auto simp add:of_rat_rat real_eq_of_int)

  1343     thus "x : ?S" using x = of_rat r by simp

  1344   qed

  1345 next

  1346   show "?S \<subseteq> \<rat>"

  1347   proof(auto simp:Rats_def)

  1348     fix i j :: int assume "j \<noteq> 0"

  1349     hence "real i / real j = of_rat(Fract i j)"

  1350       by (simp add:of_rat_rat real_eq_of_int)

  1351     thus "real i / real j \<in> range of_rat" by blast

  1352   qed

  1353 qed

  1354

  1355 lemma Rats_eq_int_div_nat:

  1356   "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"

  1357 proof(auto simp:Rats_eq_int_div_int)

  1358   fix i j::int assume "j \<noteq> 0"

  1359   show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"

  1360   proof cases

  1361     assume "j>0"

  1362     hence "real i/real j = real i/real(nat j) \<and> 0<nat j"

  1363       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)

  1364     thus ?thesis by blast

  1365   next

  1366     assume "~ j>0"

  1367     hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using j\<noteq>0

  1368       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)

  1369     thus ?thesis by blast

  1370   qed

  1371 next

  1372   fix i::int and n::nat assume "0 < n"

  1373   hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp

  1374   thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast

  1375 qed

  1376

  1377 lemma Rats_abs_nat_div_natE:

  1378   assumes "x \<in> \<rat>"

  1379   obtains m n :: nat

  1380   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"

  1381 proof -

  1382   from x \<in> \<rat> obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"

  1383     by(auto simp add: Rats_eq_int_div_nat)

  1384   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp

  1385   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast

  1386   let ?gcd = "gcd m n"

  1387   from n\<noteq>0 have gcd: "?gcd \<noteq> 0" by simp

  1388   let ?k = "m div ?gcd"

  1389   let ?l = "n div ?gcd"

  1390   let ?gcd' = "gcd ?k ?l"

  1391   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"

  1392     by (rule dvd_mult_div_cancel)

  1393   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"

  1394     by (rule dvd_mult_div_cancel)

  1395   from n\<noteq>0 and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)

  1396   moreover

  1397   have "\<bar>x\<bar> = real ?k / real ?l"

  1398   proof -

  1399     from gcd have "real ?k / real ?l =

  1400         real (?gcd * ?k) / real (?gcd * ?l)" by simp

  1401     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp

  1402     also from x_rat have "\<dots> = \<bar>x\<bar>" ..

  1403     finally show ?thesis ..

  1404   qed

  1405   moreover

  1406   have "?gcd' = 1"

  1407   proof -

  1408     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"

  1409       by (rule gcd_mult_distrib_nat)

  1410     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp

  1411     with gcd show ?thesis by auto

  1412   qed

  1413   ultimately show ?thesis ..

  1414 qed

  1415

  1416 subsection{*Density of the Rational Reals in the Reals*}

  1417

  1418 text{* This density proof is due to Stefan Richter and was ported by TN.  The

  1419 original source is \emph{Real Analysis} by H.L. Royden.

  1420 It employs the Archimedean property of the reals. *}

  1421

  1422 lemma Rats_dense_in_real:

  1423   fixes x :: real

  1424   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"

  1425 proof -

  1426   from x<y have "0 < y-x" by simp

  1427   with reals_Archimedean obtain q::nat

  1428     where q: "inverse (real q) < y-x" and "0 < q" by auto

  1429   def p \<equiv> "ceiling (y * real q) - 1"

  1430   def r \<equiv> "of_int p / real q"

  1431   from q have "x < y - inverse (real q)" by simp

  1432   also have "y - inverse (real q) \<le> r"

  1433     unfolding r_def p_def

  1434     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling 0 < q)

  1435   finally have "x < r" .

  1436   moreover have "r < y"

  1437     unfolding r_def p_def

  1438     by (simp add: divide_less_eq diff_less_eq 0 < q

  1439       less_ceiling_iff [symmetric])

  1440   moreover from r_def have "r \<in> \<rat>" by simp

  1441   ultimately show ?thesis by fast

  1442 qed

  1443

  1444

  1445

  1446 subsection{*Numerals and Arithmetic*}

  1447

  1448 lemma [code_abbrev]:

  1449   "real_of_int (numeral k) = numeral k"

  1450   "real_of_int (neg_numeral k) = neg_numeral k"

  1451   by simp_all

  1452

  1453 text{*Collapse applications of @{term real} to @{term number_of}*}

  1454 lemma real_numeral [simp]:

  1455   "real (numeral v :: int) = numeral v"

  1456   "real (neg_numeral v :: int) = neg_numeral v"

  1457 by (simp_all add: real_of_int_def)

  1458

  1459 lemma real_of_nat_numeral [simp]:

  1460   "real (numeral v :: nat) = numeral v"

  1461 by (simp add: real_of_nat_def)

  1462

  1463 declaration {*

  1464   K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]

  1465     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)

  1466   #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]

  1467     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)

  1468   #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},

  1469       @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},

  1470       @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},

  1471       @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},

  1472       @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)}]

  1473   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})

  1474   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"}))

  1475 *}

  1476

  1477

  1478 subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}

  1479

  1480 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"

  1481 by arith

  1482

  1483 text {* FIXME: redundant with @{text add_eq_0_iff} below *}

  1484 lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"

  1485 by auto

  1486

  1487 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"

  1488 by auto

  1489

  1490 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"

  1491 by auto

  1492

  1493 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"

  1494 by auto

  1495

  1496 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"

  1497 by auto

  1498

  1499 subsection {* Lemmas about powers *}

  1500

  1501 text {* FIXME: declare this in Rings.thy or not at all *}

  1502 declare abs_mult_self [simp]

  1503

  1504 (* used by Import/HOL/real.imp *)

  1505 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"

  1506 by simp

  1507

  1508 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"

  1509 apply (induct "n")

  1510 apply (auto simp add: real_of_nat_Suc)

  1511 apply (subst mult_2)

  1512 apply (erule add_less_le_mono)

  1513 apply (rule two_realpow_ge_one)

  1514 done

  1515

  1516 text {* TODO: no longer real-specific; rename and move elsewhere *}

  1517 lemma realpow_Suc_le_self:

  1518   fixes r :: "'a::linordered_semidom"

  1519   shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"

  1520 by (insert power_decreasing [of 1 "Suc n" r], simp)

  1521

  1522 text {* TODO: no longer real-specific; rename and move elsewhere *}

  1523 lemma realpow_minus_mult:

  1524   fixes x :: "'a::monoid_mult"

  1525   shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"

  1526 by (simp add: power_commutes split add: nat_diff_split)

  1527

  1528 text {* FIXME: declare this [simp] for all types, or not at all *}

  1529 lemma real_two_squares_add_zero_iff [simp]:

  1530   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"

  1531 by (rule sum_squares_eq_zero_iff)

  1532

  1533 text {* FIXME: declare this [simp] for all types, or not at all *}

  1534 lemma realpow_two_sum_zero_iff [simp]:

  1535      "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"

  1536 by (rule sum_power2_eq_zero_iff)

  1537

  1538 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"

  1539 by (rule_tac y = 0 in order_trans, auto)

  1540

  1541 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"

  1542 by (auto simp add: power2_eq_square)

  1543

  1544

  1545 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:

  1546   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"

  1547   unfolding real_of_nat_le_iff[symmetric] by simp

  1548

  1549 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:

  1550   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"

  1551   unfolding real_of_nat_le_iff[symmetric] by simp

  1552

  1553 lemma numeral_power_le_real_of_int_cancel_iff[simp]:

  1554   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"

  1555   unfolding real_of_int_le_iff[symmetric] by simp

  1556

  1557 lemma real_of_int_le_numeral_power_cancel_iff[simp]:

  1558   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"

  1559   unfolding real_of_int_le_iff[symmetric] by simp

  1560

  1561 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:

  1562   "(neg_numeral x::real) ^ n \<le> real a \<longleftrightarrow> (neg_numeral x::int) ^ n \<le> a"

  1563   unfolding real_of_int_le_iff[symmetric] by simp

  1564

  1565 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:

  1566   "real a \<le> (neg_numeral x::real) ^ n \<longleftrightarrow> a \<le> (neg_numeral x::int) ^ n"

  1567   unfolding real_of_int_le_iff[symmetric] by simp

  1568

  1569 subsection{*Density of the Reals*}

  1570

  1571 lemma real_lbound_gt_zero:

  1572      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"

  1573 apply (rule_tac x = " (min d1 d2) /2" in exI)

  1574 apply (simp add: min_def)

  1575 done

  1576

  1577

  1578 text{*Similar results are proved in @{text Fields}*}

  1579 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"

  1580   by auto

  1581

  1582 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"

  1583   by auto

  1584

  1585 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

  1586   by simp

  1587

  1588 subsection{*Absolute Value Function for the Reals*}

  1589

  1590 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"

  1591 by (simp add: abs_if)

  1592

  1593 (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)

  1594 lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"

  1595 by (force simp add: abs_le_iff)

  1596

  1597 lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"

  1598 by (simp add: abs_if)

  1599

  1600 lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"

  1601 by (rule abs_of_nonneg [OF real_of_nat_ge_zero])

  1602

  1603 lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"

  1604 by simp

  1605

  1606 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"

  1607 by simp

  1608

  1609

  1610 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}

  1611

  1612 (* FIXME: theorems for negative numerals *)

  1613 lemma numeral_less_real_of_int_iff [simp]:

  1614      "((numeral n) < real (m::int)) = (numeral n < m)"

  1615 apply auto

  1616 apply (rule real_of_int_less_iff [THEN iffD1])

  1617 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

  1618 done

  1619

  1620 lemma numeral_less_real_of_int_iff2 [simp]:

  1621      "(real (m::int) < (numeral n)) = (m < numeral n)"

  1622 apply auto

  1623 apply (rule real_of_int_less_iff [THEN iffD1])

  1624 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

  1625 done

  1626

  1627 lemma numeral_le_real_of_int_iff [simp]:

  1628      "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"

  1629 by (simp add: linorder_not_less [symmetric])

  1630

  1631 lemma numeral_le_real_of_int_iff2 [simp]:

  1632      "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"

  1633 by (simp add: linorder_not_less [symmetric])

  1634

  1635 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"

  1636 unfolding real_of_nat_def by simp

  1637

  1638 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"

  1639 unfolding real_of_nat_def by (simp add: floor_minus)

  1640

  1641 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"

  1642 unfolding real_of_int_def by simp

  1643

  1644 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"

  1645 unfolding real_of_int_def by (simp add: floor_minus)

  1646

  1647 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"

  1648 unfolding real_of_int_def by (rule floor_exists)

  1649

  1650 lemma lemma_floor:

  1651   assumes a1: "real m \<le> r" and a2: "r < real n + 1"

  1652   shows "m \<le> (n::int)"

  1653 proof -

  1654   have "real m < real n + 1" using a1 a2 by (rule order_le_less_trans)

  1655   also have "... = real (n + 1)" by simp

  1656   finally have "m < n + 1" by (simp only: real_of_int_less_iff)

  1657   thus ?thesis by arith

  1658 qed

  1659

  1660 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"

  1661 unfolding real_of_int_def by (rule of_int_floor_le)

  1662

  1663 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"

  1664 by (auto intro: lemma_floor)

  1665

  1666 lemma real_of_int_floor_cancel [simp]:

  1667     "(real (floor x) = x) = (\<exists>n::int. x = real n)"

  1668   using floor_real_of_int by metis

  1669

  1670 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"

  1671   unfolding real_of_int_def using floor_unique [of n x] by simp

  1672

  1673 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"

  1674   unfolding real_of_int_def by (rule floor_unique)

  1675

  1676 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"

  1677 apply (rule inj_int [THEN injD])

  1678 apply (simp add: real_of_nat_Suc)

  1679 apply (simp add: real_of_nat_Suc floor_eq floor_eq [where n = "int n"])

  1680 done

  1681

  1682 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"

  1683 apply (drule order_le_imp_less_or_eq)

  1684 apply (auto intro: floor_eq3)

  1685 done

  1686

  1687 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"

  1688   unfolding real_of_int_def using floor_correct [of r] by simp

  1689

  1690 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"

  1691   unfolding real_of_int_def using floor_correct [of r] by simp

  1692

  1693 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"

  1694   unfolding real_of_int_def using floor_correct [of r] by simp

  1695

  1696 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"

  1697   unfolding real_of_int_def using floor_correct [of r] by simp

  1698

  1699 lemma le_floor: "real a <= x ==> a <= floor x"

  1700   unfolding real_of_int_def by (simp add: le_floor_iff)

  1701

  1702 lemma real_le_floor: "a <= floor x ==> real a <= x"

  1703   unfolding real_of_int_def by (simp add: le_floor_iff)

  1704

  1705 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"

  1706   unfolding real_of_int_def by (rule le_floor_iff)

  1707

  1708 lemma floor_less_eq: "(floor x < a) = (x < real a)"

  1709   unfolding real_of_int_def by (rule floor_less_iff)

  1710

  1711 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"

  1712   unfolding real_of_int_def by (rule less_floor_iff)

  1713

  1714 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"

  1715   unfolding real_of_int_def by (rule floor_le_iff)

  1716

  1717 lemma floor_add [simp]: "floor (x + real a) = floor x + a"

  1718   unfolding real_of_int_def by (rule floor_add_of_int)

  1719

  1720 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"

  1721   unfolding real_of_int_def by (rule floor_diff_of_int)

  1722

  1723 lemma le_mult_floor:

  1724   assumes "0 \<le> (a :: real)" and "0 \<le> b"

  1725   shows "floor a * floor b \<le> floor (a * b)"

  1726 proof -

  1727   have "real (floor a) \<le> a"

  1728     and "real (floor b) \<le> b" by auto

  1729   hence "real (floor a * floor b) \<le> a * b"

  1730     using assms by (auto intro!: mult_mono)

  1731   also have "a * b < real (floor (a * b) + 1)" by auto

  1732   finally show ?thesis unfolding real_of_int_less_iff by simp

  1733 qed

  1734

  1735 lemma floor_divide_eq_div:

  1736   "floor (real a / real b) = a div b"

  1737 proof cases

  1738   assume "b \<noteq> 0 \<or> b dvd a"

  1739   with real_of_int_div3[of a b] show ?thesis

  1740     by (auto simp: real_of_int_div[symmetric] intro!: floor_eq2 real_of_int_div4 neq_le_trans)

  1741        (metis add_left_cancel zero_neq_one real_of_int_div_aux real_of_int_inject

  1742               real_of_int_zero_cancel right_inverse_eq div_self mod_div_trivial)

  1743 qed (auto simp: real_of_int_div)

  1744

  1745 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"

  1746   unfolding real_of_nat_def by simp

  1747

  1748 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"

  1749   unfolding real_of_int_def by (rule le_of_int_ceiling)

  1750

  1751 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"

  1752   unfolding real_of_int_def by simp

  1753

  1754 lemma real_of_int_ceiling_cancel [simp]:

  1755      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"

  1756   using ceiling_real_of_int by metis

  1757

  1758 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"

  1759   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

  1760

  1761 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"

  1762   unfolding real_of_int_def using ceiling_unique [of "n + 1" x] by simp

  1763

  1764 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"

  1765   unfolding real_of_int_def using ceiling_unique [of n x] by simp

  1766

  1767 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"

  1768   unfolding real_of_int_def using ceiling_correct [of r] by simp

  1769

  1770 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"

  1771   unfolding real_of_int_def using ceiling_correct [of r] by simp

  1772

  1773 lemma ceiling_le: "x <= real a ==> ceiling x <= a"

  1774   unfolding real_of_int_def by (simp add: ceiling_le_iff)

  1775

  1776 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"

  1777   unfolding real_of_int_def by (simp add: ceiling_le_iff)

  1778

  1779 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"

  1780   unfolding real_of_int_def by (rule ceiling_le_iff)

  1781

  1782 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"

  1783   unfolding real_of_int_def by (rule less_ceiling_iff)

  1784

  1785 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"

  1786   unfolding real_of_int_def by (rule ceiling_less_iff)

  1787

  1788 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"

  1789   unfolding real_of_int_def by (rule le_ceiling_iff)

  1790

  1791 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"

  1792   unfolding real_of_int_def by (rule ceiling_add_of_int)

  1793

  1794 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"

  1795   unfolding real_of_int_def by (rule ceiling_diff_of_int)

  1796

  1797

  1798 subsubsection {* Versions for the natural numbers *}

  1799

  1800 definition

  1801   natfloor :: "real => nat" where

  1802   "natfloor x = nat(floor x)"

  1803

  1804 definition

  1805   natceiling :: "real => nat" where

  1806   "natceiling x = nat(ceiling x)"

  1807

  1808 lemma natfloor_zero [simp]: "natfloor 0 = 0"

  1809   by (unfold natfloor_def, simp)

  1810

  1811 lemma natfloor_one [simp]: "natfloor 1 = 1"

  1812   by (unfold natfloor_def, simp)

  1813

  1814 lemma zero_le_natfloor [simp]: "0 <= natfloor x"

  1815   by (unfold natfloor_def, simp)

  1816

  1817 lemma natfloor_numeral_eq [simp]: "natfloor (numeral n) = numeral n"

  1818   by (unfold natfloor_def, simp)

  1819

  1820 lemma natfloor_real_of_nat [simp]: "natfloor(real n) = n"

  1821   by (unfold natfloor_def, simp)

  1822

  1823 lemma real_natfloor_le: "0 <= x ==> real(natfloor x) <= x"

  1824   by (unfold natfloor_def, simp)

  1825

  1826 lemma natfloor_neg: "x <= 0 ==> natfloor x = 0"

  1827   unfolding natfloor_def by simp

  1828

  1829 lemma natfloor_mono: "x <= y ==> natfloor x <= natfloor y"

  1830   unfolding natfloor_def by (intro nat_mono floor_mono)

  1831

  1832 lemma le_natfloor: "real x <= a ==> x <= natfloor a"

  1833   apply (unfold natfloor_def)

  1834   apply (subst nat_int [THEN sym])

  1835   apply (rule nat_mono)

  1836   apply (rule le_floor)

  1837   apply simp

  1838 done

  1839

  1840 lemma natfloor_less_iff: "0 \<le> x \<Longrightarrow> natfloor x < n \<longleftrightarrow> x < real n"

  1841   unfolding natfloor_def real_of_nat_def

  1842   by (simp add: nat_less_iff floor_less_iff)

  1843

  1844 lemma less_natfloor:

  1845   assumes "0 \<le> x" and "x < real (n :: nat)"

  1846   shows "natfloor x < n"

  1847   using assms by (simp add: natfloor_less_iff)

  1848

  1849 lemma le_natfloor_eq: "0 <= x ==> (a <= natfloor x) = (real a <= x)"

  1850   apply (rule iffI)

  1851   apply (rule order_trans)

  1852   prefer 2

  1853   apply (erule real_natfloor_le)

  1854   apply (subst real_of_nat_le_iff)

  1855   apply assumption

  1856   apply (erule le_natfloor)

  1857 done

  1858

  1859 lemma le_natfloor_eq_numeral [simp]:

  1860     "~ neg((numeral n)::int) ==> 0 <= x ==>

  1861       (numeral n <= natfloor x) = (numeral n <= x)"

  1862   apply (subst le_natfloor_eq, assumption)

  1863   apply simp

  1864 done

  1865

  1866 lemma le_natfloor_eq_one [simp]: "(1 <= natfloor x) = (1 <= x)"

  1867   apply (case_tac "0 <= x")

  1868   apply (subst le_natfloor_eq, assumption, simp)

  1869   apply (rule iffI)

  1870   apply (subgoal_tac "natfloor x <= natfloor 0")

  1871   apply simp

  1872   apply (rule natfloor_mono)

  1873   apply simp

  1874   apply simp

  1875 done

  1876

  1877 lemma natfloor_eq: "real n <= x ==> x < real n + 1 ==> natfloor x = n"

  1878   unfolding natfloor_def by (simp add: floor_eq2 [where n="int n"])

  1879

  1880 lemma real_natfloor_add_one_gt: "x < real(natfloor x) + 1"

  1881   apply (case_tac "0 <= x")

  1882   apply (unfold natfloor_def)

  1883   apply simp

  1884   apply simp_all

  1885 done

  1886

  1887 lemma real_natfloor_gt_diff_one: "x - 1 < real(natfloor x)"

  1888 using real_natfloor_add_one_gt by (simp add: algebra_simps)

  1889

  1890 lemma ge_natfloor_plus_one_imp_gt: "natfloor z + 1 <= n ==> z < real n"

  1891   apply (subgoal_tac "z < real(natfloor z) + 1")

  1892   apply arith

  1893   apply (rule real_natfloor_add_one_gt)

  1894 done

  1895

  1896 lemma natfloor_add [simp]: "0 <= x ==> natfloor (x + real a) = natfloor x + a"

  1897   unfolding natfloor_def

  1898   unfolding real_of_int_of_nat_eq [symmetric] floor_add

  1899   by (simp add: nat_add_distrib)

  1900

  1901 lemma natfloor_add_numeral [simp]:

  1902     "~neg ((numeral n)::int) ==> 0 <= x ==>

  1903       natfloor (x + numeral n) = natfloor x + numeral n"

  1904   by (simp add: natfloor_add [symmetric])

  1905

  1906 lemma natfloor_add_one: "0 <= x ==> natfloor(x + 1) = natfloor x + 1"

  1907   by (simp add: natfloor_add [symmetric] del: One_nat_def)

  1908

  1909 lemma natfloor_subtract [simp]:

  1910     "natfloor(x - real a) = natfloor x - a"

  1911   unfolding natfloor_def real_of_int_of_nat_eq [symmetric] floor_subtract

  1912   by simp

  1913

  1914 lemma natfloor_div_nat:

  1915   assumes "1 <= x" and "y > 0"

  1916   shows "natfloor (x / real y) = natfloor x div y"

  1917 proof (rule natfloor_eq)

  1918   have "(natfloor x) div y * y \<le> natfloor x"

  1919     by (rule add_leD1 [where k="natfloor x mod y"], simp)

  1920   thus "real (natfloor x div y) \<le> x / real y"

  1921     using assms by (simp add: le_divide_eq le_natfloor_eq)

  1922   have "natfloor x < (natfloor x) div y * y + y"

  1923     apply (subst mod_div_equality [symmetric])

  1924     apply (rule add_strict_left_mono)

  1925     apply (rule mod_less_divisor)

  1926     apply fact

  1927     done

  1928   thus "x / real y < real (natfloor x div y) + 1"

  1929     using assms

  1930     by (simp add: divide_less_eq natfloor_less_iff distrib_right)

  1931 qed

  1932

  1933 lemma le_mult_natfloor:

  1934   shows "natfloor a * natfloor b \<le> natfloor (a * b)"

  1935   by (cases "0 <= a & 0 <= b")

  1936     (auto simp add: le_natfloor_eq mult_nonneg_nonneg mult_mono' real_natfloor_le natfloor_neg)

  1937

  1938 lemma natceiling_zero [simp]: "natceiling 0 = 0"

  1939   by (unfold natceiling_def, simp)

  1940

  1941 lemma natceiling_one [simp]: "natceiling 1 = 1"

  1942   by (unfold natceiling_def, simp)

  1943

  1944 lemma zero_le_natceiling [simp]: "0 <= natceiling x"

  1945   by (unfold natceiling_def, simp)

  1946

  1947 lemma natceiling_numeral_eq [simp]: "natceiling (numeral n) = numeral n"

  1948   by (unfold natceiling_def, simp)

  1949

  1950 lemma natceiling_real_of_nat [simp]: "natceiling(real n) = n"

  1951   by (unfold natceiling_def, simp)

  1952

  1953 lemma real_natceiling_ge: "x <= real(natceiling x)"

  1954   unfolding natceiling_def by (cases "x < 0", simp_all)

  1955

  1956 lemma natceiling_neg: "x <= 0 ==> natceiling x = 0"

  1957   unfolding natceiling_def by simp

  1958

  1959 lemma natceiling_mono: "x <= y ==> natceiling x <= natceiling y"

  1960   unfolding natceiling_def by (intro nat_mono ceiling_mono)

  1961

  1962 lemma natceiling_le: "x <= real a ==> natceiling x <= a"

  1963   unfolding natceiling_def real_of_nat_def

  1964   by (simp add: nat_le_iff ceiling_le_iff)

  1965

  1966 lemma natceiling_le_eq: "(natceiling x <= a) = (x <= real a)"

  1967   unfolding natceiling_def real_of_nat_def

  1968   by (simp add: nat_le_iff ceiling_le_iff)

  1969

  1970 lemma natceiling_le_eq_numeral [simp]:

  1971     "~ neg((numeral n)::int) ==>

  1972       (natceiling x <= numeral n) = (x <= numeral n)"

  1973   by (simp add: natceiling_le_eq)

  1974

  1975 lemma natceiling_le_eq_one: "(natceiling x <= 1) = (x <= 1)"

  1976   unfolding natceiling_def

  1977   by (simp add: nat_le_iff ceiling_le_iff)

  1978

  1979 lemma natceiling_eq: "real n < x ==> x <= real n + 1 ==> natceiling x = n + 1"

  1980   unfolding natceiling_def

  1981   by (simp add: ceiling_eq2 [where n="int n"])

  1982

  1983 lemma natceiling_add [simp]: "0 <= x ==>

  1984     natceiling (x + real a) = natceiling x + a"

  1985   unfolding natceiling_def

  1986   unfolding real_of_int_of_nat_eq [symmetric] ceiling_add

  1987   by (simp add: nat_add_distrib)

  1988

  1989 lemma natceiling_add_numeral [simp]:

  1990     "~ neg ((numeral n)::int) ==> 0 <= x ==>

  1991       natceiling (x + numeral n) = natceiling x + numeral n"

  1992   by (simp add: natceiling_add [symmetric])

  1993

  1994 lemma natceiling_add_one: "0 <= x ==> natceiling(x + 1) = natceiling x + 1"

  1995   by (simp add: natceiling_add [symmetric] del: One_nat_def)

  1996

  1997 lemma natceiling_subtract [simp]: "natceiling(x - real a) = natceiling x - a"

  1998   unfolding natceiling_def real_of_int_of_nat_eq [symmetric] ceiling_subtract

  1999   by simp

  2000

  2001 subsection {* Exponentiation with floor *}

  2002

  2003 lemma floor_power:

  2004   assumes "x = real (floor x)"

  2005   shows "floor (x ^ n) = floor x ^ n"

  2006 proof -

  2007   have *: "x ^ n = real (floor x ^ n)"

  2008     using assms by (induct n arbitrary: x) simp_all

  2009   show ?thesis unfolding real_of_int_inject[symmetric]

  2010     unfolding * floor_real_of_int ..

  2011 qed

  2012

  2013 lemma natfloor_power:

  2014   assumes "x = real (natfloor x)"

  2015   shows "natfloor (x ^ n) = natfloor x ^ n"

  2016 proof -

  2017   from assms have "0 \<le> floor x" by auto

  2018   note assms[unfolded natfloor_def real_nat_eq_real[OF 0 \<le> floor x]]

  2019   from floor_power[OF this]

  2020   show ?thesis unfolding natfloor_def nat_power_eq[OF 0 \<le> floor x, symmetric]

  2021     by simp

  2022 qed

  2023

  2024

  2025 subsection {* Implementation of rational real numbers *}

  2026

  2027 text {* Formal constructor *}

  2028

  2029 definition Ratreal :: "rat \<Rightarrow> real" where

  2030   [code_abbrev, simp]: "Ratreal = of_rat"

  2031

  2032 code_datatype Ratreal

  2033

  2034

  2035 text {* Numerals *}

  2036

  2037 lemma [code_abbrev]:

  2038   "(of_rat (of_int a) :: real) = of_int a"

  2039   by simp

  2040

  2041 lemma [code_abbrev]:

  2042   "(of_rat 0 :: real) = 0"

  2043   by simp

  2044

  2045 lemma [code_abbrev]:

  2046   "(of_rat 1 :: real) = 1"

  2047   by simp

  2048

  2049 lemma [code_abbrev]:

  2050   "(of_rat (numeral k) :: real) = numeral k"

  2051   by simp

  2052

  2053 lemma [code_abbrev]:

  2054   "(of_rat (neg_numeral k) :: real) = neg_numeral k"

  2055   by simp

  2056

  2057 lemma [code_post]:

  2058   "(of_rat (0 / r)  :: real) = 0"

  2059   "(of_rat (r / 0)  :: real) = 0"

  2060   "(of_rat (1 / 1)  :: real) = 1"

  2061   "(of_rat (numeral k / 1) :: real) = numeral k"

  2062   "(of_rat (neg_numeral k / 1) :: real) = neg_numeral k"

  2063   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"

  2064   "(of_rat (1 / neg_numeral k) :: real) = 1 / neg_numeral k"

  2065   "(of_rat (numeral k / numeral l)  :: real) = numeral k / numeral l"

  2066   "(of_rat (numeral k / neg_numeral l)  :: real) = numeral k / neg_numeral l"

  2067   "(of_rat (neg_numeral k / numeral l)  :: real) = neg_numeral k / numeral l"

  2068   "(of_rat (neg_numeral k / neg_numeral l)  :: real) = neg_numeral k / neg_numeral l"

  2069   by (simp_all add: of_rat_divide)

  2070

  2071

  2072 text {* Operations *}

  2073

  2074 lemma zero_real_code [code]:

  2075   "0 = Ratreal 0"

  2076 by simp

  2077

  2078 lemma one_real_code [code]:

  2079   "1 = Ratreal 1"

  2080 by simp

  2081

  2082 instantiation real :: equal

  2083 begin

  2084

  2085 definition "HOL.equal (x\<Colon>real) y \<longleftrightarrow> x - y = 0"

  2086

  2087 instance proof

  2088 qed (simp add: equal_real_def)

  2089

  2090 lemma real_equal_code [code]:

  2091   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"

  2092   by (simp add: equal_real_def equal)

  2093

  2094 lemma [code nbe]:

  2095   "HOL.equal (x::real) x \<longleftrightarrow> True"

  2096   by (rule equal_refl)

  2097

  2098 end

  2099

  2100 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"

  2101   by (simp add: of_rat_less_eq)

  2102

  2103 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"

  2104   by (simp add: of_rat_less)

  2105

  2106 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"

  2107   by (simp add: of_rat_add)

  2108

  2109 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"

  2110   by (simp add: of_rat_mult)

  2111

  2112 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"

  2113   by (simp add: of_rat_minus)

  2114

  2115 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"

  2116   by (simp add: of_rat_diff)

  2117

  2118 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"

  2119   by (simp add: of_rat_inverse)

  2120

  2121 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"

  2122   by (simp add: of_rat_divide)

  2123

  2124 lemma real_floor_code [code]: "floor (Ratreal x) = floor x"

  2125   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)

  2126

  2127

  2128 text {* Quickcheck *}

  2129

  2130 definition (in term_syntax)

  2131   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

  2132   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"

  2133

  2134 notation fcomp (infixl "\<circ>>" 60)

  2135 notation scomp (infixl "\<circ>\<rightarrow>" 60)

  2136

  2137 instantiation real :: random

  2138 begin

  2139

  2140 definition

  2141   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"

  2142

  2143 instance ..

  2144

  2145 end

  2146

  2147 no_notation fcomp (infixl "\<circ>>" 60)

  2148 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

  2149

  2150 instantiation real :: exhaustive

  2151 begin

  2152

  2153 definition

  2154   "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"

  2155

  2156 instance ..

  2157

  2158 end

  2159

  2160 instantiation real :: full_exhaustive

  2161 begin

  2162

  2163 definition

  2164   "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"

  2165

  2166 instance ..

  2167

  2168 end

  2169

  2170 instantiation real :: narrowing

  2171 begin

  2172

  2173 definition

  2174   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"

  2175

  2176 instance ..

  2177

  2178 end

  2179

  2180

  2181 subsection {* Setup for Nitpick *}

  2182

  2183 declaration {*

  2184   Nitpick_HOL.register_frac_type @{type_name real}

  2185    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),

  2186     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),

  2187     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),

  2188     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),

  2189     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),

  2190     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),

  2191     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),

  2192     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]

  2193 *}

  2194

  2195 lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real

  2196     ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real

  2197     times_real_inst.times_real uminus_real_inst.uminus_real

  2198     zero_real_inst.zero_real

  2199

  2200 ML_file "Tools/SMT/smt_real.ML"

  2201 setup SMT_Real.setup

  2202

  2203 end
`