src/HOL/Real.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60429 d3d1e185cd63 child 60758 d8d85a8172b5 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
     1 (*  Title:      HOL/Real.thy

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

     3     Author:     Larry Paulson, University of Cambridge

     4     Author:     Jeremy Avigad, Carnegie Mellon University

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

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

     7     Construction of Cauchy Reals by Brian Huffman, 2010

     8 *)

     9

    10 section {* 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

   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 simp

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

   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" by (simp add: 0 < r b)

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

   287       using b by simp

   288   qed

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

   290     using cauchyD [OF X s] ..

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

   292   proof (clarsimp)

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

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

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

   296       by (simp add: inverse_diff_inverse nz * abs_mult)

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

   298       by (simp add: mult_strict_mono less_imp_inverse_less

   299                     i j b * s)

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

   301   qed

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

   303 qed

   304

   305 lemma vanishes_diff_inverse:

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

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

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

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

   310 proof (rule vanishesI)

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

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

   313     using cauchy_not_vanishes [OF X] by fast

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

   315     using cauchy_not_vanishes [OF Y] by fast

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

   317   proof

   318     show "0 < a * r * b"

   319       using a r b by simp

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

   321       using a r b by simp

   322   qed

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

   324     using vanishesD [OF XY s] ..

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

   326   proof (clarsimp)

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

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

   329       using i j a b n by auto

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

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

   332       by (simp add: inverse_diff_inverse abs_mult)

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

   334       apply (intro mult_strict_mono' less_imp_inverse_less)

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

   336       done

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

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

   339   qed

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

   341 qed

   342

   343 subsection {* Equivalence relation on Cauchy sequences *}

   344

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

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

   347

   348 lemma realrelI [intro?]:

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

   350   shows "realrel X Y"

   351   using assms unfolding realrel_def by simp

   352

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

   354   unfolding realrel_def by simp

   355

   356 lemma symp_realrel: "symp realrel"

   357   unfolding realrel_def

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

   359

   360 lemma transp_realrel: "transp realrel"

   361   unfolding realrel_def

   362   apply (rule transpI, clarify)

   363   apply (drule (1) vanishes_add)

   364   apply (simp add: algebra_simps)

   365   done

   366

   367 lemma part_equivp_realrel: "part_equivp realrel"

   368   by (fast intro: part_equivpI symp_realrel transp_realrel

   369     realrel_refl cauchy_const)

   370

   371 subsection {* The field of real numbers *}

   372

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

   374   morphisms rep_real Real

   375   by (rule part_equivp_realrel)

   376

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

   378   unfolding real.pcr_cr_eq cr_real_def realrel_def by auto

   379

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

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

   382 proof (induct x)

   383   case (1 X)

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

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

   386 qed

   387

   388 lemma eq_Real:

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

   390   using real.rel_eq_transfer

   391   unfolding real.pcr_cr_eq cr_real_def rel_fun_def realrel_def by simp

   392

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

   394 by (simp add: real.domain_eq realrel_def)

   395

   396 instantiation real :: field

   397 begin

   398

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

   400   by (simp add: realrel_refl)

   401

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

   403   by (simp add: realrel_refl)

   404

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

   406   unfolding realrel_def add_diff_add

   407   by (simp only: cauchy_add vanishes_add simp_thms)

   408

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

   410   unfolding realrel_def minus_diff_minus

   411   by (simp only: cauchy_minus vanishes_minus simp_thms)

   412

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

   414   unfolding realrel_def mult_diff_mult

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

   416     vanishes_mult_bounded cauchy_imp_bounded simp_thms)

   417

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

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

   420 proof -

   421   fix X Y assume "realrel X Y"

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

   423     unfolding realrel_def by simp_all

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

   425   proof

   426     assume "vanishes X"

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

   428   next

   429     assume "vanishes Y"

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

   431   qed

   432   thus "?thesis X Y"

   433     unfolding realrel_def

   434     by (simp add: vanishes_diff_inverse X Y XY)

   435 qed

   436

   437 definition

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

   439

   440 definition

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

   442

   443 lemma add_Real:

   444   assumes X: "cauchy X" and Y: "cauchy Y"

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

   446   using assms plus_real.transfer

   447   unfolding cr_real_eq rel_fun_def by simp

   448

   449 lemma minus_Real:

   450   assumes X: "cauchy X"

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

   452   using assms uminus_real.transfer

   453   unfolding cr_real_eq rel_fun_def by simp

   454

   455 lemma diff_Real:

   456   assumes X: "cauchy X" and Y: "cauchy Y"

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

   458   unfolding minus_real_def

   459   by (simp add: minus_Real add_Real X Y)

   460

   461 lemma mult_Real:

   462   assumes X: "cauchy X" and Y: "cauchy Y"

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

   464   using assms times_real.transfer

   465   unfolding cr_real_eq rel_fun_def by simp

   466

   467 lemma inverse_Real:

   468   assumes X: "cauchy X"

   469   shows "inverse (Real X) =

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

   471   using assms inverse_real.transfer zero_real.transfer

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

   473

   474 instance proof

   475   fix a b c :: real

   476   show "a + b = b + a"

   477     by transfer (simp add: ac_simps realrel_def)

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

   479     by transfer (simp add: ac_simps realrel_def)

   480   show "0 + a = a"

   481     by transfer (simp add: realrel_def)

   482   show "- a + a = 0"

   483     by transfer (simp add: realrel_def)

   484   show "a - b = a + - b"

   485     by (rule minus_real_def)

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

   487     by transfer (simp add: ac_simps realrel_def)

   488   show "a * b = b * a"

   489     by transfer (simp add: ac_simps realrel_def)

   490   show "1 * a = a"

   491     by transfer (simp add: ac_simps realrel_def)

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

   493     by transfer (simp add: distrib_right realrel_def)

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

   495     by transfer (simp add: realrel_def)

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

   497     apply transfer

   498     apply (simp add: realrel_def)

   499     apply (rule vanishesI)

   500     apply (frule (1) cauchy_not_vanishes, clarify)

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

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

   503     done

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

   505     by (rule divide_real_def)

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

   507     by transfer (simp add: realrel_def)

   508 qed

   509

   510 end

   511

   512 subsection {* Positive reals *}

   513

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

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

   516 proof -

   517   { fix X Y

   518     assume "realrel X Y"

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

   520       unfolding realrel_def by simp_all

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

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

   523       by fast

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

   525       using 0 < r by (rule obtain_pos_sum)

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

   527       using vanishesD [OF XY s] ..

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

   529     proof (clarsimp)

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

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

   532         using i j n by simp_all

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

   534     qed

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

   536   } note 1 = this

   537   fix X Y assume "realrel X Y"

   538   hence "realrel X Y" and "realrel Y X"

   539     using symp_realrel unfolding symp_def by auto

   540   thus "?thesis X Y"

   541     by (safe elim!: 1)

   542 qed

   543

   544 lemma positive_Real:

   545   assumes X: "cauchy X"

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

   547   using assms positive.transfer

   548   unfolding cr_real_eq rel_fun_def by simp

   549

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

   551   by transfer auto

   552

   553 lemma positive_add:

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

   555 apply transfer

   556 apply (clarify, rename_tac a b i j)

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

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

   559 apply (simp add: add_strict_mono)

   560 done

   561

   562 lemma positive_mult:

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

   564 apply transfer

   565 apply (clarify, rename_tac a b i j)

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

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

   568 apply (rule mult_strict_mono, auto)

   569 done

   570

   571 lemma positive_minus:

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

   573 apply transfer

   574 apply (simp add: realrel_def)

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

   576 done

   577

   578 instantiation real :: linordered_field

   579 begin

   580

   581 definition

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

   583

   584 definition

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

   586

   587 definition

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

   589

   590 definition

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

   592

   593 instance proof

   594   fix a b c :: real

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

   596     by (rule abs_real_def)

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

   598     unfolding less_eq_real_def less_real_def

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

   600   show "a \<le> a"

   601     unfolding less_eq_real_def by simp

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

   603     unfolding less_eq_real_def less_real_def

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

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

   606     unfolding less_eq_real_def less_real_def

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

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

   609     unfolding less_eq_real_def less_real_def by auto

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

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

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

   613     by (rule sgn_real_def)

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

   615     unfolding less_eq_real_def less_real_def

   616     by (auto dest!: positive_minus)

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

   618     unfolding less_real_def

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

   620 qed

   621

   622 end

   623

   624 instantiation real :: distrib_lattice

   625 begin

   626

   627 definition

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

   629

   630 definition

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

   632

   633 instance proof

   634 qed (auto simp add: inf_real_def sup_real_def max_min_distrib2)

   635

   636 end

   637

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

   639 apply (induct x)

   640 apply (simp add: zero_real_def)

   641 apply (simp add: one_real_def add_Real)

   642 done

   643

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

   645 apply (cases x rule: int_diff_cases)

   646 apply (simp add: of_nat_Real diff_Real)

   647 done

   648

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

   650 apply (induct x)

   651 apply (simp add: Fract_of_int_quotient of_rat_divide)

   652 apply (simp add: of_int_Real divide_inverse)

   653 apply (simp add: inverse_Real mult_Real)

   654 done

   655

   656 instance real :: archimedean_field

   657 proof

   658   fix x :: real

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

   660     apply (induct x)

   661     apply (frule cauchy_imp_bounded, clarify)

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

   663     apply (rule less_imp_le)

   664     apply (simp add: of_int_Real less_real_def diff_Real positive_Real)

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

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

   667     apply (rule le_less_trans [OF abs_ge_self])

   668     apply (rule less_le_trans [OF _ le_of_int_ceiling])

   669     apply simp

   670     done

   671 qed

   672

   673 instantiation real :: floor_ceiling

   674 begin

   675

   676 definition [code del]:

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

   678

   679 instance proof

   680   fix x :: real

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

   682     unfolding floor_real_def using floor_exists1 by (rule theI')

   683 qed

   684

   685 end

   686

   687 subsection {* Completeness *}

   688

   689 lemma not_positive_Real:

   690   assumes X: "cauchy X"

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

   692 unfolding positive_Real [OF X]

   693 apply (auto, unfold not_less)

   694 apply (erule obtain_pos_sum)

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

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

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

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

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

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

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

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

   703 done

   704

   705 lemma le_Real:

   706   assumes X: "cauchy X" and Y: "cauchy Y"

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

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

   709 apply (simp add: diff_Real not_positive_Real X Y)

   710 apply (simp add: diff_le_eq ac_simps)

   711 done

   712

   713 lemma le_RealI:

   714   assumes Y: "cauchy Y"

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

   716 proof (induct x)

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

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

   719     by (simp add: of_rat_Real le_Real)

   720   {

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

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

   723       by (rule obtain_pos_sum)

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

   725       using cauchyD [OF Y s] ..

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

   727       using le [OF t] ..

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

   729     proof (clarsimp)

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

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

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

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

   734     qed

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

   736   }

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

   738     by (simp add: of_rat_Real le_Real X Y)

   739 qed

   740

   741 lemma Real_leI:

   742   assumes X: "cauchy X"

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

   744   shows "Real X \<le> y"

   745 proof -

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

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

   748   thus ?thesis by simp

   749 qed

   750

   751 lemma less_RealD:

   752   assumes Y: "cauchy Y"

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

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

   755

   756 lemma of_nat_less_two_power:

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

   758 apply (induct n)

   759 apply simp

   760 by (metis add_le_less_mono mult_2 of_nat_Suc one_le_numeral one_le_power power_Suc)

   761

   762 lemma complete_real:

   763   fixes S :: "real set"

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

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

   766 proof -

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

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

   769

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

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

   772   proof

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

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

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

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

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

   778       unfolding P_def of_rat_of_int_eq using x by fast

   779   qed

   780   obtain b where b: "P b"

   781   proof

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

   783     unfolding P_def of_rat_of_int_eq

   784     proof

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

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

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

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

   789     qed

   790   qed

   791

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

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

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

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

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

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

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

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

   800     unfolding A_def B_def C_def bisect_def split_def by simp

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

   802     unfolding A_def B_def C_def bisect_def split_def by simp

   803

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

   805     apply (simp add: eq_divide_eq)

   806     apply (induct_tac n, simp)

   807     apply (simp add: C_def avg_def power_Suc algebra_simps)

   808     done

   809

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

   811     apply (simp add: divide_less_eq)

   812     apply (subst mult.commute)

   813     apply (frule_tac y=y in ex_less_of_nat_mult)

   814     apply clarify

   815     apply (rule_tac x=n in exI)

   816     apply (erule less_trans)

   817     apply (rule mult_strict_right_mono)

   818     apply (rule le_less_trans [OF _ of_nat_less_two_power])

   819     apply simp

   820     apply assumption

   821     done

   822

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

   824     by (induct_tac n, simp_all add: a)

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

   826     by (induct_tac n, simp_all add: b)

   827   have ab: "a < b"

   828     using a b unfolding P_def

   829     apply (clarsimp simp add: not_le)

   830     apply (drule (1) bspec)

   831     apply (drule (1) less_le_trans)

   832     apply (simp add: of_rat_less)

   833     done

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

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

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

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

   838     apply (erule less_Suc_induct)

   839     apply (clarsimp simp add: C_def avg_def)

   840     apply (simp add: add_divide_distrib [symmetric])

   841     apply (rule AB [THEN less_imp_le])

   842     apply simp

   843     done

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

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

   846     apply (erule less_Suc_induct)

   847     apply (clarsimp simp add: C_def avg_def)

   848     apply (simp add: add_divide_distrib [symmetric])

   849     apply (rule AB [THEN less_imp_le])

   850     apply simp

   851     done

   852   have cauchy_lemma:

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

   854     apply (rule cauchyI)

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

   856     apply (erule exE)

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

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

   859     apply (simp add: width)

   860     apply (drule_tac x=n in spec)

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

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

   863     apply (frule A_mono, drule B_mono)

   864     apply (frule A_mono, drule B_mono)

   865     apply arith

   866     done

   867   have "cauchy A"

   868     apply (rule cauchy_lemma [rule_format])

   869     apply (simp add: A_mono)

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

   871     done

   872   have "cauchy B"

   873     apply (rule cauchy_lemma [rule_format])

   874     apply (simp add: B_mono)

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

   876     done

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

   878   proof

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

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

   881       using PB [unfolded P_def] cauchy B

   882       by (simp add: le_RealI)

   883   qed

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

   885     apply clarify

   886     apply (erule contrapos_pp)

   887     apply (simp add: not_le)

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

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

   890     apply (simp add: P_def not_le, clarify)

   891     apply (erule rev_bexI)

   892     apply (erule (1) less_trans)

   893     apply (simp add: PA)

   894     done

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

   896   proof (rule vanishesI)

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

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

   899       using twos by fast

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

   901     proof (clarify)

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

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

   904         by simp

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

   906         using n by (simp add: divide_left_mono)

   907       also note k

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

   909     qed

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

   911   qed

   912   hence 3: "Real B = Real A"

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

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

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

   916 qed

   917

   918 instantiation real :: linear_continuum

   919 begin

   920

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

   922

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

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

   925

   926 instance

   927 proof

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

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

   930     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"

   931       using complete_real[of X] unfolding bdd_above_def by blast

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

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

   934   note Sup_upper = this

   935

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

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

   938     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"

   939       using complete_real[of X] by blast

   940     then have "Sup X = s"

   941       unfolding Sup_real_def by (best intro: Least_equality)

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

   943       by blast

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

   945   note Sup_least = this

   946

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

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

   949   { 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"

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

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

   952     using zero_neq_one by blast

   953 qed

   954 end

   955

   956

   957 subsection {* Hiding implementation details *}

   958

   959 hide_const (open) vanishes cauchy positive Real

   960

   961 declare Real_induct [induct del]

   962 declare Abs_real_induct [induct del]

   963 declare Abs_real_cases [cases del]

   964

   965 lifting_update real.lifting

   966 lifting_forget real.lifting

   967

   968 subsection{*More Lemmas*}

   969

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

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

   972

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

   974 by simp (* solved by linordered_ring_less_cancel_factor simproc *)

   975

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

   977 by simp (* solved by linordered_ring_le_cancel_factor simproc *)

   978

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

   980 by simp (* solved by linordered_ring_le_cancel_factor simproc *)

   981

   982

   983 subsection {* Embedding numbers into the Reals *}

   984

   985 abbreviation

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

   987 where

   988   "real_of_nat \<equiv> of_nat"

   989

   990 abbreviation

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

   992 where

   993   "real_of_int \<equiv> of_int"

   994

   995 abbreviation

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

   997 where

   998   "real_of_rat \<equiv> of_rat"

   999

  1000 class real_of =

  1001   fixes real :: "'a \<Rightarrow> real"

  1002

  1003 instantiation nat :: real_of

  1004 begin

  1005

  1006 definition real_nat :: "nat \<Rightarrow> real" where real_of_nat_def [code_unfold]: "real \<equiv> of_nat"

  1007

  1008 instance ..

  1009 end

  1010

  1011 instantiation int :: real_of

  1012 begin

  1013

  1014 definition real_int :: "int \<Rightarrow> real" where real_of_int_def [code_unfold]: "real \<equiv> of_int"

  1015

  1016 instance ..

  1017 end

  1018

  1019 declare [[coercion_enabled]]

  1020

  1021 declare [[coercion "of_nat :: nat \<Rightarrow> int"]]

  1022 declare [[coercion "real   :: nat \<Rightarrow> real"]]

  1023 declare [[coercion "real   :: int \<Rightarrow> real"]]

  1024

  1025 (* We do not add rat to the coerced types, this has often unpleasant side effects when writing

  1026 inverse (Suc n) which sometimes gets two coercions: of_rat (inverse (of_nat (Suc n))) *)

  1027

  1028 declare [[coercion_map map]]

  1029 declare [[coercion_map "\<lambda>f g h x. g (h (f x))"]]

  1030 declare [[coercion_map "\<lambda>f g (x,y). (f x, g y)"]]

  1031

  1032 lemma real_eq_of_nat: "real = of_nat"

  1033   unfolding real_of_nat_def ..

  1034

  1035 lemma real_eq_of_int: "real = of_int"

  1036   unfolding real_of_int_def ..

  1037

  1038 lemma real_of_int_zero [simp]: "real (0::int) = 0"

  1039 by (simp add: real_of_int_def)

  1040

  1041 lemma real_of_one [simp]: "real (1::int) = (1::real)"

  1042 by (simp add: real_of_int_def)

  1043

  1044 lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"

  1045 by (simp add: real_of_int_def)

  1046

  1047 lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"

  1048 by (simp add: real_of_int_def)

  1049

  1050 lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"

  1051 by (simp add: real_of_int_def)

  1052

  1053 lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"

  1054 by (simp add: real_of_int_def)

  1055

  1056 lemma real_of_int_power [simp]: "real (x ^ n) = real (x::int) ^ n"

  1057 by (simp add: real_of_int_def of_int_power)

  1058

  1059 lemmas power_real_of_int = real_of_int_power [symmetric]

  1060

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

  1062   apply (subst real_eq_of_int)+

  1063   apply (rule of_int_setsum)

  1064 done

  1065

  1066 lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) =

  1067     (PROD x:A. real(f x))"

  1068   apply (subst real_eq_of_int)+

  1069   apply (rule of_int_setprod)

  1070 done

  1071

  1072 lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"

  1073 by (simp add: real_of_int_def)

  1074

  1075 lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"

  1076 by (simp add: real_of_int_def)

  1077

  1078 lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"

  1079 by (simp add: real_of_int_def)

  1080

  1081 lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"

  1082 by (simp add: real_of_int_def)

  1083

  1084 lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"

  1085 by (simp add: real_of_int_def)

  1086

  1087 lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"

  1088 by (simp add: real_of_int_def)

  1089

  1090 lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)"

  1091 by (simp add: real_of_int_def)

  1092

  1093 lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"

  1094 by (simp add: real_of_int_def)

  1095

  1096 lemma one_less_real_of_int_cancel_iff: "1 < real (i :: int) \<longleftrightarrow> 1 < i"

  1097   unfolding real_of_one[symmetric] real_of_int_less_iff ..

  1098

  1099 lemma one_le_real_of_int_cancel_iff: "1 \<le> real (i :: int) \<longleftrightarrow> 1 \<le> i"

  1100   unfolding real_of_one[symmetric] real_of_int_le_iff ..

  1101

  1102 lemma real_of_int_less_one_cancel_iff: "real (i :: int) < 1 \<longleftrightarrow> i < 1"

  1103   unfolding real_of_one[symmetric] real_of_int_less_iff ..

  1104

  1105 lemma real_of_int_le_one_cancel_iff: "real (i :: int) \<le> 1 \<longleftrightarrow> i \<le> 1"

  1106   unfolding real_of_one[symmetric] real_of_int_le_iff ..

  1107

  1108 lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"

  1109 by (auto simp add: abs_if)

  1110

  1111 lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"

  1112   apply (subgoal_tac "real n + 1 = real (n + 1)")

  1113   apply (simp del: real_of_int_add)

  1114   apply auto

  1115 done

  1116

  1117 lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"

  1118   apply (subgoal_tac "real m + 1 = real (m + 1)")

  1119   apply (simp del: real_of_int_add)

  1120   apply simp

  1121 done

  1122

  1123 lemma real_of_int_div_aux: "(real (x::int)) / (real d) =

  1124     real (x div d) + (real (x mod d)) / (real d)"

  1125 proof -

  1126   have "x = (x div d) * d + x mod d"

  1127     by auto

  1128   then have "real x = real (x div d) * real d + real(x mod d)"

  1129     by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])

  1130   then have "real x / real d = ... / real d"

  1131     by simp

  1132   then show ?thesis

  1133     by (auto simp add: add_divide_distrib algebra_simps)

  1134 qed

  1135

  1136 lemma real_of_int_div:

  1137   fixes d n :: int

  1138   shows "d dvd n \<Longrightarrow> real (n div d) = real n / real d"

  1139   by (simp add: real_of_int_div_aux)

  1140

  1141 lemma real_of_int_div2:

  1142   "0 <= real (n::int) / real (x) - real (n div x)"

  1143   apply (case_tac "x = 0")

  1144   apply simp

  1145   apply (case_tac "0 < x")

  1146   apply (simp add: algebra_simps)

  1147   apply (subst real_of_int_div_aux)

  1148   apply simp

  1149   apply (simp add: algebra_simps)

  1150   apply (subst real_of_int_div_aux)

  1151   apply simp

  1152   apply (subst zero_le_divide_iff)

  1153   apply auto

  1154 done

  1155

  1156 lemma real_of_int_div3:

  1157   "real (n::int) / real (x) - real (n div x) <= 1"

  1158   apply (simp add: algebra_simps)

  1159   apply (subst real_of_int_div_aux)

  1160   apply (auto simp add: divide_le_eq intro: order_less_imp_le)

  1161 done

  1162

  1163 lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x"

  1164 by (insert real_of_int_div2 [of n x], simp)

  1165

  1166 lemma Ints_real_of_int [simp]: "real (x::int) \<in> Ints"

  1167 unfolding real_of_int_def by (rule Ints_of_int)

  1168

  1169

  1170 subsection{*Embedding the Naturals into the Reals*}

  1171

  1172 lemma real_of_nat_zero [simp]: "real (0::nat) = 0"

  1173 by (simp add: real_of_nat_def)

  1174

  1175 lemma real_of_nat_1 [simp]: "real (1::nat) = 1"

  1176 by (simp add: real_of_nat_def)

  1177

  1178 lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"

  1179 by (simp add: real_of_nat_def)

  1180

  1181 lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"

  1182 by (simp add: real_of_nat_def)

  1183

  1184 (*Not for addsimps: often the LHS is used to represent a positive natural*)

  1185 lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"

  1186 by (simp add: real_of_nat_def)

  1187

  1188 lemma real_of_nat_less_iff [iff]:

  1189      "(real (n::nat) < real m) = (n < m)"

  1190 by (simp add: real_of_nat_def)

  1191

  1192 lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"

  1193 by (simp add: real_of_nat_def)

  1194

  1195 lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"

  1196 by (simp add: real_of_nat_def)

  1197

  1198 lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"

  1199 by (simp add: real_of_nat_def del: of_nat_Suc)

  1200

  1201 lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"

  1202 by (simp add: real_of_nat_def of_nat_mult)

  1203

  1204 lemma real_of_nat_power [simp]: "real (m ^ n) = real (m::nat) ^ n"

  1205 by (simp add: real_of_nat_def of_nat_power)

  1206

  1207 lemmas power_real_of_nat = real_of_nat_power [symmetric]

  1208

  1209 lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) =

  1210     (SUM x:A. real(f x))"

  1211   apply (subst real_eq_of_nat)+

  1212   apply (rule of_nat_setsum)

  1213 done

  1214

  1215 lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) =

  1216     (PROD x:A. real(f x))"

  1217   apply (subst real_eq_of_nat)+

  1218   apply (rule of_nat_setprod)

  1219 done

  1220

  1221 lemma real_of_card: "real (card A) = setsum (%x.1) A"

  1222   apply (subst card_eq_setsum)

  1223   apply (subst real_of_nat_setsum)

  1224   apply simp

  1225 done

  1226

  1227 lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"

  1228 by (simp add: real_of_nat_def)

  1229

  1230 lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"

  1231 by (simp add: real_of_nat_def)

  1232

  1233 lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"

  1234 by (simp add: add: real_of_nat_def of_nat_diff)

  1235

  1236 lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"

  1237 by (auto simp: real_of_nat_def)

  1238

  1239 lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"

  1240 by (simp add: add: real_of_nat_def)

  1241

  1242 lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"

  1243 by (simp add: add: real_of_nat_def)

  1244

  1245 lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"

  1246   apply (subgoal_tac "real n + 1 = real (Suc n)")

  1247   apply simp

  1248   apply (auto simp add: real_of_nat_Suc)

  1249 done

  1250

  1251 lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"

  1252   apply (subgoal_tac "real m + 1 = real (Suc m)")

  1253   apply (simp add: less_Suc_eq_le)

  1254   apply (simp add: real_of_nat_Suc)

  1255 done

  1256

  1257 lemma real_of_nat_div_aux: "(real (x::nat)) / (real d) =

  1258     real (x div d) + (real (x mod d)) / (real d)"

  1259 proof -

  1260   have "x = (x div d) * d + x mod d"

  1261     by auto

  1262   then have "real x = real (x div d) * real d + real(x mod d)"

  1263     by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])

  1264   then have "real x / real d = \<dots> / real d"

  1265     by simp

  1266   then show ?thesis

  1267     by (auto simp add: add_divide_distrib algebra_simps)

  1268 qed

  1269

  1270 lemma real_of_nat_div: "(d :: nat) dvd n ==>

  1271     real(n div d) = real n / real d"

  1272   by (subst real_of_nat_div_aux)

  1273     (auto simp add: dvd_eq_mod_eq_0 [symmetric])

  1274

  1275 lemma real_of_nat_div2:

  1276   "0 <= real (n::nat) / real (x) - real (n div x)"

  1277 apply (simp add: algebra_simps)

  1278 apply (subst real_of_nat_div_aux)

  1279 apply simp

  1280 done

  1281

  1282 lemma real_of_nat_div3:

  1283   "real (n::nat) / real (x) - real (n div x) <= 1"

  1284 apply(case_tac "x = 0")

  1285 apply (simp)

  1286 apply (simp add: algebra_simps)

  1287 apply (subst real_of_nat_div_aux)

  1288 apply simp

  1289 done

  1290

  1291 lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x"

  1292 by (insert real_of_nat_div2 [of n x], simp)

  1293

  1294 lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"

  1295 by (simp add: real_of_int_def real_of_nat_def)

  1296

  1297 lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"

  1298   apply (subgoal_tac "real(int(nat x)) = real(nat x)")

  1299   apply force

  1300   apply (simp only: real_of_int_of_nat_eq)

  1301 done

  1302

  1303 lemma Nats_real_of_nat [simp]: "real (n::nat) \<in> Nats"

  1304 unfolding real_of_nat_def by (rule of_nat_in_Nats)

  1305

  1306 lemma Ints_real_of_nat [simp]: "real (n::nat) \<in> Ints"

  1307 unfolding real_of_nat_def by (rule Ints_of_nat)

  1308

  1309 subsection {* The Archimedean Property of the Reals *}

  1310

  1311 theorem reals_Archimedean:

  1312   assumes x_pos: "0 < x"

  1313   shows "\<exists>n. inverse (real (Suc n)) < x"

  1314   unfolding real_of_nat_def using x_pos

  1315   by (rule ex_inverse_of_nat_Suc_less)

  1316

  1317 lemma reals_Archimedean2: "\<exists>n. (x::real) < real (n::nat)"

  1318   unfolding real_of_nat_def by (rule ex_less_of_nat)

  1319

  1320 lemma reals_Archimedean3:

  1321   assumes x_greater_zero: "0 < x"

  1322   shows "\<forall>(y::real). \<exists>(n::nat). y < real n * x"

  1323   unfolding real_of_nat_def using 0 < x

  1324   by (auto intro: ex_less_of_nat_mult)

  1325

  1326

  1327 subsection{* Rationals *}

  1328

  1329 lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"

  1330 by (simp add: real_eq_of_nat)

  1331

  1332 lemma Rats_eq_int_div_int:

  1333   "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")

  1334 proof

  1335   show "\<rat> \<subseteq> ?S"

  1336   proof

  1337     fix x::real assume "x : \<rat>"

  1338     then obtain r where "x = of_rat r" unfolding Rats_def ..

  1339     have "of_rat r : ?S"

  1340       by (cases r)(auto simp add:of_rat_rat real_eq_of_int)

  1341     thus "x : ?S" using x = of_rat r by simp

  1342   qed

  1343 next

  1344   show "?S \<subseteq> \<rat>"

  1345   proof(auto simp:Rats_def)

  1346     fix i j :: int assume "j \<noteq> 0"

  1347     hence "real i / real j = of_rat(Fract i j)"

  1348       by (simp add:of_rat_rat real_eq_of_int)

  1349     thus "real i / real j \<in> range of_rat" by blast

  1350   qed

  1351 qed

  1352

  1353 lemma Rats_eq_int_div_nat:

  1354   "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"

  1355 proof(auto simp:Rats_eq_int_div_int)

  1356   fix i j::int assume "j \<noteq> 0"

  1357   show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"

  1358   proof cases

  1359     assume "j>0"

  1360     hence "real i/real j = real i/real(nat j) \<and> 0<nat j"

  1361       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)

  1362     thus ?thesis by blast

  1363   next

  1364     assume "~ j>0"

  1365     hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using j\<noteq>0

  1366       by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)

  1367     thus ?thesis by blast

  1368   qed

  1369 next

  1370   fix i::int and n::nat assume "0 < n"

  1371   hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp

  1372   thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast

  1373 qed

  1374

  1375 lemma Rats_abs_nat_div_natE:

  1376   assumes "x \<in> \<rat>"

  1377   obtains m n :: nat

  1378   where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"

  1379 proof -

  1380   from x \<in> \<rat> obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"

  1381     by(auto simp add: Rats_eq_int_div_nat)

  1382   hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp

  1383   then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast

  1384   let ?gcd = "gcd m n"

  1385   from n\<noteq>0 have gcd: "?gcd \<noteq> 0" by simp

  1386   let ?k = "m div ?gcd"

  1387   let ?l = "n div ?gcd"

  1388   let ?gcd' = "gcd ?k ?l"

  1389   have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"

  1390     by (rule dvd_mult_div_cancel)

  1391   have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"

  1392     by (rule dvd_mult_div_cancel)

  1393   from n \<noteq> 0 and gcd_l

  1394   have "?gcd * ?l \<noteq> 0" by simp

  1395   then have "?l \<noteq> 0" by (blast dest!: mult_not_zero)

  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)"

  1401       by (simp only: real_of_nat_mult) simp

  1402     also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp

  1403     also from x_rat have "\<dots> = \<bar>x\<bar>" ..

  1404     finally show ?thesis ..

  1405   qed

  1406   moreover

  1407   have "?gcd' = 1"

  1408   proof -

  1409     have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"

  1410       by (rule gcd_mult_distrib_nat)

  1411     with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp

  1412     with gcd show ?thesis by auto

  1413   qed

  1414   ultimately show ?thesis ..

  1415 qed

  1416

  1417 subsection{*Density of the Rational Reals in the Reals*}

  1418

  1419 text{* This density proof is due to Stefan Richter and was ported by TN.  The

  1420 original source is \emph{Real Analysis} by H.L. Royden.

  1421 It employs the Archimedean property of the reals. *}

  1422

  1423 lemma Rats_dense_in_real:

  1424   fixes x :: real

  1425   assumes "x < y" shows "\<exists>r\<in>\<rat>. x < r \<and> r < y"

  1426 proof -

  1427   from x<y have "0 < y-x" by simp

  1428   with reals_Archimedean obtain q::nat

  1429     where q: "inverse (real q) < y-x" and "0 < q" by auto

  1430   def p \<equiv> "ceiling (y * real q) - 1"

  1431   def r \<equiv> "of_int p / real q"

  1432   from q have "x < y - inverse (real q)" by simp

  1433   also have "y - inverse (real q) \<le> r"

  1434     unfolding r_def p_def

  1435     by (simp add: le_divide_eq left_diff_distrib le_of_int_ceiling 0 < q)

  1436   finally have "x < r" .

  1437   moreover have "r < y"

  1438     unfolding r_def p_def

  1439     by (simp add: divide_less_eq diff_less_eq 0 < q

  1440       less_ceiling_iff [symmetric])

  1441   moreover from r_def have "r \<in> \<rat>" by simp

  1442   ultimately show ?thesis by fast

  1443 qed

  1444

  1445 lemma of_rat_dense:

  1446   fixes x y :: real

  1447   assumes "x < y"

  1448   shows "\<exists>q :: rat. x < of_rat q \<and> of_rat q < y"

  1449 using Rats_dense_in_real [OF x < y]

  1450 by (auto elim: Rats_cases)

  1451

  1452

  1453 subsection{*Numerals and Arithmetic*}

  1454

  1455 lemma [code_abbrev]:

  1456   "real_of_int (numeral k) = numeral k"

  1457   "real_of_int (- numeral k) = - numeral k"

  1458   by simp_all

  1459

  1460 text{*Collapse applications of @{const real} to @{const numeral}*}

  1461 lemma real_numeral [simp]:

  1462   "real (numeral v :: int) = numeral v"

  1463   "real (- numeral v :: int) = - numeral v"

  1464 by (simp_all add: real_of_int_def)

  1465

  1466 lemma  real_of_nat_numeral [simp]:

  1467   "real (numeral v :: nat) = numeral v"

  1468 by (simp add: real_of_nat_def)

  1469

  1470 declaration {*

  1471   K (Lin_Arith.add_inj_thms [@{thm real_of_nat_le_iff} RS iffD2, @{thm real_of_nat_inject} RS iffD2]

  1472     (* not needed because x < (y::nat) can be rewritten as Suc x <= y: real_of_nat_less_iff RS iffD2 *)

  1473   #> Lin_Arith.add_inj_thms [@{thm real_of_int_le_iff} RS iffD2, @{thm real_of_int_inject} RS iffD2]

  1474     (* not needed because x < (y::int) can be rewritten as x + 1 <= y: real_of_int_less_iff RS iffD2 *)

  1475   #> Lin_Arith.add_simps [@{thm real_of_nat_zero}, @{thm real_of_nat_Suc}, @{thm real_of_nat_add},

  1476       @{thm real_of_nat_mult}, @{thm real_of_int_zero}, @{thm real_of_one},

  1477       @{thm real_of_int_add}, @{thm real_of_int_minus}, @{thm real_of_int_diff},

  1478       @{thm real_of_int_mult}, @{thm real_of_int_of_nat_eq},

  1479       @{thm real_of_nat_numeral}, @{thm real_numeral(1)}, @{thm real_numeral(2)},

  1480       @{thm real_of_int_def[symmetric]}, @{thm real_of_nat_def[symmetric]}]

  1481   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "nat \<Rightarrow> real"})

  1482   #> Lin_Arith.add_inj_const (@{const_name real}, @{typ "int \<Rightarrow> real"})

  1483   #> Lin_Arith.add_inj_const (@{const_name of_nat}, @{typ "nat \<Rightarrow> real"})

  1484   #> Lin_Arith.add_inj_const (@{const_name of_int}, @{typ "int \<Rightarrow> real"}))

  1485 *}

  1486

  1487 subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}

  1488

  1489 lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)"

  1490 by arith

  1491

  1492 text {* FIXME: redundant with @{text add_eq_0_iff} below *}

  1493 lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"

  1494 by auto

  1495

  1496 lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"

  1497 by auto

  1498

  1499 lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"

  1500 by auto

  1501

  1502 lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"

  1503 by auto

  1504

  1505 lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"

  1506 by auto

  1507

  1508 subsection {* Lemmas about powers *}

  1509

  1510 text {* FIXME: declare this in Rings.thy or not at all *}

  1511 declare abs_mult_self [simp]

  1512

  1513 (* used by Import/HOL/real.imp *)

  1514 lemma two_realpow_ge_one: "(1::real) \<le> 2 ^ n"

  1515 by simp

  1516

  1517 lemma two_realpow_gt [simp]: "real (n::nat) < 2 ^ n"

  1518   by (simp add: of_nat_less_two_power real_of_nat_def)

  1519

  1520 text {* TODO: no longer real-specific; rename and move elsewhere *}

  1521 lemma realpow_Suc_le_self:

  1522   fixes r :: "'a::linordered_semidom"

  1523   shows "[| 0 \<le> r; r \<le> 1 |] ==> r ^ Suc n \<le> r"

  1524 by (insert power_decreasing [of 1 "Suc n" r], simp)

  1525

  1526 text {* TODO: no longer real-specific; rename and move elsewhere *}

  1527 lemma realpow_minus_mult:

  1528   fixes x :: "'a::monoid_mult"

  1529   shows "0 < n \<Longrightarrow> x ^ (n - 1) * x = x ^ n"

  1530 by (simp add: power_Suc power_commutes split add: nat_diff_split)

  1531

  1532 text {* FIXME: declare this [simp] for all types, or not at all *}

  1533 lemma real_two_squares_add_zero_iff [simp]:

  1534   "(x * x + y * y = 0) = ((x::real) = 0 \<and> y = 0)"

  1535 by (rule sum_squares_eq_zero_iff)

  1536

  1537 text {* FIXME: declare this [simp] for all types, or not at all *}

  1538 lemma realpow_two_sum_zero_iff [simp]:

  1539      "(x\<^sup>2 + y\<^sup>2 = (0::real)) = (x = 0 & y = 0)"

  1540 by (rule sum_power2_eq_zero_iff)

  1541

  1542 lemma real_minus_mult_self_le [simp]: "-(u * u) \<le> (x * (x::real))"

  1543 by (rule_tac y = 0 in order_trans, auto)

  1544

  1545 lemma realpow_square_minus_le [simp]: "- u\<^sup>2 \<le> (x::real)\<^sup>2"

  1546 by (auto simp add: power2_eq_square)

  1547

  1548

  1549 lemma numeral_power_eq_real_of_int_cancel_iff[simp]:

  1550   "numeral x ^ n = real (y::int) \<longleftrightarrow> numeral x ^ n = y"

  1551   by (metis real_numeral(1) real_of_int_inject real_of_int_power)

  1552

  1553 lemma real_of_int_eq_numeral_power_cancel_iff[simp]:

  1554   "real (y::int) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"

  1555   using numeral_power_eq_real_of_int_cancel_iff[of x n y]

  1556   by metis

  1557

  1558 lemma numeral_power_eq_real_of_nat_cancel_iff[simp]:

  1559   "numeral x ^ n = real (y::nat) \<longleftrightarrow> numeral x ^ n = y"

  1560   by (metis of_nat_eq_iff of_nat_numeral real_of_int_eq_numeral_power_cancel_iff

  1561     real_of_int_of_nat_eq zpower_int)

  1562

  1563 lemma real_of_nat_eq_numeral_power_cancel_iff[simp]:

  1564   "real (y::nat) = numeral x ^ n \<longleftrightarrow> y = numeral x ^ n"

  1565   using numeral_power_eq_real_of_nat_cancel_iff[of x n y]

  1566   by metis

  1567

  1568 lemma numeral_power_le_real_of_nat_cancel_iff[simp]:

  1569   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::nat) ^ n \<le> a"

  1570   unfolding real_of_nat_le_iff[symmetric] by simp

  1571

  1572 lemma real_of_nat_le_numeral_power_cancel_iff[simp]:

  1573   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::nat) ^ n"

  1574   unfolding real_of_nat_le_iff[symmetric] by simp

  1575

  1576 lemma numeral_power_le_real_of_int_cancel_iff[simp]:

  1577   "(numeral x::real) ^ n \<le> real a \<longleftrightarrow> (numeral x::int) ^ n \<le> a"

  1578   unfolding real_of_int_le_iff[symmetric] by simp

  1579

  1580 lemma real_of_int_le_numeral_power_cancel_iff[simp]:

  1581   "real a \<le> (numeral x::real) ^ n \<longleftrightarrow> a \<le> (numeral x::int) ^ n"

  1582   unfolding real_of_int_le_iff[symmetric] by simp

  1583

  1584 lemma numeral_power_less_real_of_nat_cancel_iff[simp]:

  1585   "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::nat) ^ n < a"

  1586   unfolding real_of_nat_less_iff[symmetric] by simp

  1587

  1588 lemma real_of_nat_less_numeral_power_cancel_iff[simp]:

  1589   "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::nat) ^ n"

  1590   unfolding real_of_nat_less_iff[symmetric] by simp

  1591

  1592 lemma numeral_power_less_real_of_int_cancel_iff[simp]:

  1593   "(numeral x::real) ^ n < real a \<longleftrightarrow> (numeral x::int) ^ n < a"

  1594   unfolding real_of_int_less_iff[symmetric] by simp

  1595

  1596 lemma real_of_int_less_numeral_power_cancel_iff[simp]:

  1597   "real a < (numeral x::real) ^ n \<longleftrightarrow> a < (numeral x::int) ^ n"

  1598   unfolding real_of_int_less_iff[symmetric] by simp

  1599

  1600 lemma neg_numeral_power_le_real_of_int_cancel_iff[simp]:

  1601   "(- numeral x::real) ^ n \<le> real a \<longleftrightarrow> (- numeral x::int) ^ n \<le> a"

  1602   unfolding real_of_int_le_iff[symmetric] by simp

  1603

  1604 lemma real_of_int_le_neg_numeral_power_cancel_iff[simp]:

  1605   "real a \<le> (- numeral x::real) ^ n \<longleftrightarrow> a \<le> (- numeral x::int) ^ n"

  1606   unfolding real_of_int_le_iff[symmetric] by simp

  1607

  1608

  1609 subsection{*Density of the Reals*}

  1610

  1611 lemma real_lbound_gt_zero:

  1612      "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"

  1613 apply (rule_tac x = " (min d1 d2) /2" in exI)

  1614 apply (simp add: min_def)

  1615 done

  1616

  1617

  1618 text{*Similar results are proved in @{text Fields}*}

  1619 lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"

  1620   by auto

  1621

  1622 lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"

  1623   by auto

  1624

  1625 lemma real_sum_of_halves: "x/2 + x/2 = (x::real)"

  1626   by simp

  1627

  1628 subsection{*Absolute Value Function for the Reals*}

  1629

  1630 lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"

  1631 by (simp add: abs_if)

  1632

  1633 (* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)

  1634 lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"

  1635 by (force simp add: abs_le_iff)

  1636

  1637 lemma abs_add_one_gt_zero: "(0::real) < 1 + abs(x)"

  1638 by (simp add: abs_if)

  1639

  1640 lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"

  1641 by (rule abs_of_nonneg [OF real_of_nat_ge_zero])

  1642

  1643 lemma abs_add_one_not_less_self: "~ abs(x) + (1::real) < x"

  1644 by simp

  1645

  1646 lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"

  1647 by simp

  1648

  1649

  1650 subsection{*Floor and Ceiling Functions from the Reals to the Integers*}

  1651

  1652 (* FIXME: theorems for negative numerals *)

  1653 lemma numeral_less_real_of_int_iff [simp]:

  1654      "((numeral n) < real (m::int)) = (numeral n < m)"

  1655 apply auto

  1656 apply (rule real_of_int_less_iff [THEN iffD1])

  1657 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

  1658 done

  1659

  1660 lemma numeral_less_real_of_int_iff2 [simp]:

  1661      "(real (m::int) < (numeral n)) = (m < numeral n)"

  1662 apply auto

  1663 apply (rule real_of_int_less_iff [THEN iffD1])

  1664 apply (drule_tac [2] real_of_int_less_iff [THEN iffD2], auto)

  1665 done

  1666

  1667 lemma real_of_nat_less_numeral_iff [simp]:

  1668   "real (n::nat) < numeral w \<longleftrightarrow> n < numeral w"

  1669   using real_of_nat_less_iff[of n "numeral w"] by simp

  1670

  1671 lemma numeral_less_real_of_nat_iff [simp]:

  1672   "numeral w < real (n::nat) \<longleftrightarrow> numeral w < n"

  1673   using real_of_nat_less_iff[of "numeral w" n] by simp

  1674

  1675 lemma numeral_le_real_of_nat_iff[simp]:

  1676   "(numeral n \<le> real(m::nat)) = (numeral n \<le> m)"

  1677 by (metis not_le real_of_nat_less_numeral_iff)

  1678

  1679 lemma numeral_le_real_of_int_iff [simp]:

  1680      "((numeral n) \<le> real (m::int)) = (numeral n \<le> m)"

  1681 by (simp add: linorder_not_less [symmetric])

  1682

  1683 lemma numeral_le_real_of_int_iff2 [simp]:

  1684      "(real (m::int) \<le> (numeral n)) = (m \<le> numeral n)"

  1685 by (simp add: linorder_not_less [symmetric])

  1686

  1687 lemma floor_real_of_nat [simp]: "floor (real (n::nat)) = int n"

  1688 unfolding real_of_nat_def by simp

  1689

  1690 lemma floor_minus_real_of_nat [simp]: "floor (- real (n::nat)) = - int n"

  1691 unfolding real_of_nat_def by (simp add: floor_minus)

  1692

  1693 lemma floor_real_of_int [simp]: "floor (real (n::int)) = n"

  1694 unfolding real_of_int_def by simp

  1695

  1696 lemma floor_minus_real_of_int [simp]: "floor (- real (n::int)) = - n"

  1697 unfolding real_of_int_def by (simp add: floor_minus)

  1698

  1699 lemma real_lb_ub_int: " \<exists>n::int. real n \<le> r & r < real (n+1)"

  1700 unfolding real_of_int_def by (rule floor_exists)

  1701

  1702 lemma lemma_floor: "real m \<le> r \<Longrightarrow> r < real n + 1 \<Longrightarrow> m \<le> (n::int)"

  1703   by simp

  1704

  1705 lemma real_of_int_floor_le [simp]: "real (floor r) \<le> r"

  1706 unfolding real_of_int_def by (rule of_int_floor_le)

  1707

  1708 lemma lemma_floor2: "real n < real (x::int) + 1 ==> n \<le> x"

  1709   by simp

  1710

  1711 lemma real_of_int_floor_cancel [simp]:

  1712     "(real (floor x) = x) = (\<exists>n::int. x = real n)"

  1713   using floor_real_of_int by metis

  1714

  1715 lemma floor_eq: "[| real n < x; x < real n + 1 |] ==> floor x = n"

  1716   by linarith

  1717

  1718 lemma floor_eq2: "[| real n \<le> x; x < real n + 1 |] ==> floor x = n"

  1719   by linarith

  1720

  1721 lemma floor_eq3: "[| real n < x; x < real (Suc n) |] ==> nat(floor x) = n"

  1722   by linarith

  1723

  1724 lemma floor_eq4: "[| real n \<le> x; x < real (Suc n) |] ==> nat(floor x) = n"

  1725   by linarith

  1726

  1727 lemma real_of_int_floor_ge_diff_one [simp]: "r - 1 \<le> real(floor r)"

  1728   by linarith

  1729

  1730 lemma real_of_int_floor_gt_diff_one [simp]: "r - 1 < real(floor r)"

  1731   by linarith

  1732

  1733 lemma real_of_int_floor_add_one_ge [simp]: "r \<le> real(floor r) + 1"

  1734   by linarith

  1735

  1736 lemma real_of_int_floor_add_one_gt [simp]: "r < real(floor r) + 1"

  1737   by linarith

  1738

  1739 lemma le_floor: "real a <= x ==> a <= floor x"

  1740   by linarith

  1741

  1742 lemma real_le_floor: "a <= floor x ==> real a <= x"

  1743   by linarith

  1744

  1745 lemma le_floor_eq: "(a <= floor x) = (real a <= x)"

  1746   by linarith

  1747

  1748 lemma floor_less_eq: "(floor x < a) = (x < real a)"

  1749   by linarith

  1750

  1751 lemma less_floor_eq: "(a < floor x) = (real a + 1 <= x)"

  1752   by linarith

  1753

  1754 lemma floor_le_eq: "(floor x <= a) = (x < real a + 1)"

  1755   by linarith

  1756

  1757 lemma floor_eq_iff: "floor x = b \<longleftrightarrow> real b \<le> x \<and> x < real (b + 1)"

  1758   by linarith

  1759

  1760 lemma floor_add [simp]: "floor (x + real a) = floor x + a"

  1761   by linarith

  1762

  1763 lemma floor_add2[simp]: "floor (real a + x) = a + floor x"

  1764   by linarith

  1765

  1766 lemma floor_subtract [simp]: "floor (x - real a) = floor x - a"

  1767   by linarith

  1768

  1769 lemma floor_divide_real_eq_div: "0 \<le> b \<Longrightarrow> floor (a / real b) = floor a div b"

  1770 proof cases

  1771   assume "0 < b"

  1772   { fix i j :: int assume "real i \<le> a" "a < 1 + real i"

  1773       "real j * real b \<le> a" "a < real b + real j * real b"

  1774     then have "i < b + j * b" "j * b < 1 + i"

  1775       unfolding real_of_int_less_iff[symmetric] by auto

  1776     then have "(j - i div b) * b \<le> i mod b" "i mod b < ((j - i div b) + 1) * b"

  1777       by (auto simp: field_simps)

  1778     then have "(j - i div b) * b < 1 * b" "0 * b < ((j - i div b) + 1) * b"

  1779       using pos_mod_bound[OF 0<b, of i] pos_mod_sign[OF 0<b, of i] by linarith+

  1780     then have "j = i div b"

  1781       using 0 < b unfolding mult_less_cancel_right by auto }

  1782   with 0 < b show ?thesis

  1783     by (auto split: floor_split simp: field_simps)

  1784 qed auto

  1785

  1786 lemma floor_divide_eq_div:

  1787   "floor (real a / real b) = a div b"

  1788   using floor_divide_of_int_eq [of a b] real_eq_of_int by simp

  1789

  1790 lemma floor_divide_eq_div_numeral[simp]: "\<lfloor>numeral a / numeral b::real\<rfloor> = numeral a div numeral b"

  1791   using floor_divide_eq_div[of "numeral a" "numeral b"] by simp

  1792

  1793 lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"

  1794   using floor_divide_eq_div[of "- numeral a" "numeral b"] by simp

  1795

  1796 lemma ceiling_real_of_nat [simp]: "ceiling (real (n::nat)) = int n"

  1797   by linarith

  1798

  1799 lemma real_of_int_ceiling_ge [simp]: "r \<le> real (ceiling r)"

  1800   by linarith

  1801

  1802 lemma ceiling_real_of_int [simp]: "ceiling (real (n::int)) = n"

  1803   by linarith

  1804

  1805 lemma real_of_int_ceiling_cancel [simp]:

  1806      "(real (ceiling x) = x) = (\<exists>n::int. x = real n)"

  1807   using ceiling_real_of_int by metis

  1808

  1809 lemma ceiling_eq: "[| real n < x; x < real n + 1 |] ==> ceiling x = n + 1"

  1810   by linarith

  1811

  1812 lemma ceiling_eq2: "[| real n < x; x \<le> real n + 1 |] ==> ceiling x = n + 1"

  1813   by linarith

  1814

  1815 lemma ceiling_eq3: "[| real n - 1 < x; x \<le> real n  |] ==> ceiling x = n"

  1816   by linarith

  1817

  1818 lemma real_of_int_ceiling_diff_one_le [simp]: "real (ceiling r) - 1 \<le> r"

  1819   by linarith

  1820

  1821 lemma real_of_int_ceiling_le_add_one [simp]: "real (ceiling r) \<le> r + 1"

  1822   by linarith

  1823

  1824 lemma ceiling_le: "x <= real a ==> ceiling x <= a"

  1825   by linarith

  1826

  1827 lemma ceiling_le_real: "ceiling x <= a ==> x <= real a"

  1828   by linarith

  1829

  1830 lemma ceiling_le_eq: "(ceiling x <= a) = (x <= real a)"

  1831   by linarith

  1832

  1833 lemma less_ceiling_eq: "(a < ceiling x) = (real a < x)"

  1834   by linarith

  1835

  1836 lemma ceiling_less_eq: "(ceiling x < a) = (x <= real a - 1)"

  1837   by linarith

  1838

  1839 lemma le_ceiling_eq: "(a <= ceiling x) = (real a - 1 < x)"

  1840   by linarith

  1841

  1842 lemma ceiling_add [simp]: "ceiling (x + real a) = ceiling x + a"

  1843   by linarith

  1844

  1845 lemma ceiling_subtract [simp]: "ceiling (x - real a) = ceiling x - a"

  1846   by linarith

  1847

  1848 lemma ceiling_divide_eq_div: "\<lceil>real a / real b\<rceil> = - (- a div b)"

  1849   unfolding ceiling_def minus_divide_left real_of_int_minus[symmetric] floor_divide_eq_div by simp_all

  1850

  1851 lemma ceiling_divide_eq_div_numeral [simp]:

  1852   "\<lceil>numeral a / numeral b :: real\<rceil> = - (- numeral a div numeral b)"

  1853   using ceiling_divide_eq_div[of "numeral a" "numeral b"] by simp

  1854

  1855 lemma ceiling_minus_divide_eq_div_numeral [simp]:

  1856   "\<lceil>- (numeral a / numeral b :: real)\<rceil> = - (numeral a div numeral b)"

  1857   using ceiling_divide_eq_div[of "- numeral a" "numeral b"] by simp

  1858

  1859 text{* The following lemmas are remnants of the erstwhile functions natfloor

  1860 and natceiling. *}

  1861

  1862 lemma nat_floor_neg: "(x::real) <= 0 ==> nat(floor x) = 0"

  1863   by linarith

  1864

  1865 lemma le_nat_floor: "real x <= a ==> x <= nat(floor a)"

  1866   by linarith

  1867

  1868 lemma le_mult_nat_floor:

  1869   shows "nat(floor a) * nat(floor b) \<le> nat(floor (a * b))"

  1870   by (cases "0 <= a & 0 <= b")

  1871      (auto simp add: nat_mult_distrib[symmetric] nat_mono le_mult_floor)

  1872

  1873 lemma nat_ceiling_le_eq: "(nat(ceiling x) <= a) = (x <= real a)"

  1874   by linarith

  1875

  1876 lemma real_nat_ceiling_ge: "x <= real(nat(ceiling x))"

  1877   by linarith

  1878

  1879

  1880 lemma Rats_no_top_le: "\<exists> q \<in> \<rat>. (x :: real) \<le> q"

  1881   by (auto intro!: bexI[of _ "of_nat (nat(ceiling x))"]) linarith

  1882

  1883 lemma Rats_no_bot_less: "\<exists> q \<in> \<rat>. q < (x :: real)"

  1884   apply (auto intro!: bexI[of _ "of_int (floor x - 1)"])

  1885   apply (rule less_le_trans[OF _ of_int_floor_le])

  1886   apply simp

  1887   done

  1888

  1889 subsection {* Exponentiation with floor *}

  1890

  1891 lemma floor_power:

  1892   assumes "x = real (floor x)"

  1893   shows "floor (x ^ n) = floor x ^ n"

  1894 proof -

  1895   have *: "x ^ n = real (floor x ^ n)"

  1896     using assms by (induct n arbitrary: x) simp_all

  1897   show ?thesis unfolding real_of_int_inject[symmetric]

  1898     unfolding * floor_real_of_int ..

  1899 qed

  1900 (*

  1901 lemma natfloor_power:

  1902   assumes "x = real (natfloor x)"

  1903   shows "natfloor (x ^ n) = natfloor x ^ n"

  1904 proof -

  1905   from assms have "0 \<le> floor x" by auto

  1906   note assms[unfolded natfloor_def real_nat_eq_real[OF 0 \<le> floor x]]

  1907   from floor_power[OF this]

  1908   show ?thesis unfolding natfloor_def nat_power_eq[OF 0 \<le> floor x, symmetric]

  1909     by simp

  1910 qed

  1911 *)

  1912 lemma floor_numeral_power[simp]:

  1913   "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"

  1914   by (metis floor_of_int of_int_numeral of_int_power)

  1915

  1916 lemma ceiling_numeral_power[simp]:

  1917   "\<lceil>numeral x ^ n\<rceil> = numeral x ^ n"

  1918   by (metis ceiling_of_int of_int_numeral of_int_power)

  1919

  1920

  1921 subsection {* Implementation of rational real numbers *}

  1922

  1923 text {* Formal constructor *}

  1924

  1925 definition Ratreal :: "rat \<Rightarrow> real" where

  1926   [code_abbrev, simp]: "Ratreal = of_rat"

  1927

  1928 code_datatype Ratreal

  1929

  1930

  1931 text {* Numerals *}

  1932

  1933 lemma [code_abbrev]:

  1934   "(of_rat (of_int a) :: real) = of_int a"

  1935   by simp

  1936

  1937 lemma [code_abbrev]:

  1938   "(of_rat 0 :: real) = 0"

  1939   by simp

  1940

  1941 lemma [code_abbrev]:

  1942   "(of_rat 1 :: real) = 1"

  1943   by simp

  1944

  1945 lemma [code_abbrev]:

  1946   "(of_rat (- 1) :: real) = - 1"

  1947   by simp

  1948

  1949 lemma [code_abbrev]:

  1950   "(of_rat (numeral k) :: real) = numeral k"

  1951   by simp

  1952

  1953 lemma [code_abbrev]:

  1954   "(of_rat (- numeral k) :: real) = - numeral k"

  1955   by simp

  1956

  1957 lemma [code_post]:

  1958   "(of_rat (1 / numeral k) :: real) = 1 / numeral k"

  1959   "(of_rat (numeral k / numeral l) :: real) = numeral k / numeral l"

  1960   "(of_rat (- (1 / numeral k)) :: real) = - (1 / numeral k)"

  1961   "(of_rat (- (numeral k / numeral l)) :: real) = - (numeral k / numeral l)"

  1962   by (simp_all add: of_rat_divide of_rat_minus)

  1963

  1964

  1965 text {* Operations *}

  1966

  1967 lemma zero_real_code [code]:

  1968   "0 = Ratreal 0"

  1969 by simp

  1970

  1971 lemma one_real_code [code]:

  1972   "1 = Ratreal 1"

  1973 by simp

  1974

  1975 instantiation real :: equal

  1976 begin

  1977

  1978 definition "HOL.equal (x\<Colon>real) y \<longleftrightarrow> x - y = 0"

  1979

  1980 instance proof

  1981 qed (simp add: equal_real_def)

  1982

  1983 lemma real_equal_code [code]:

  1984   "HOL.equal (Ratreal x) (Ratreal y) \<longleftrightarrow> HOL.equal x y"

  1985   by (simp add: equal_real_def equal)

  1986

  1987 lemma [code nbe]:

  1988   "HOL.equal (x::real) x \<longleftrightarrow> True"

  1989   by (rule equal_refl)

  1990

  1991 end

  1992

  1993 lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"

  1994   by (simp add: of_rat_less_eq)

  1995

  1996 lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"

  1997   by (simp add: of_rat_less)

  1998

  1999 lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"

  2000   by (simp add: of_rat_add)

  2001

  2002 lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"

  2003   by (simp add: of_rat_mult)

  2004

  2005 lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"

  2006   by (simp add: of_rat_minus)

  2007

  2008 lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"

  2009   by (simp add: of_rat_diff)

  2010

  2011 lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"

  2012   by (simp add: of_rat_inverse)

  2013

  2014 lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"

  2015   by (simp add: of_rat_divide)

  2016

  2017 lemma real_floor_code [code]: "floor (Ratreal x) = floor x"

  2018   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)

  2019

  2020

  2021 text {* Quickcheck *}

  2022

  2023 definition (in term_syntax)

  2024   valterm_ratreal :: "rat \<times> (unit \<Rightarrow> Code_Evaluation.term) \<Rightarrow> real \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

  2025   [code_unfold]: "valterm_ratreal k = Code_Evaluation.valtermify Ratreal {\<cdot>} k"

  2026

  2027 notation fcomp (infixl "\<circ>>" 60)

  2028 notation scomp (infixl "\<circ>\<rightarrow>" 60)

  2029

  2030 instantiation real :: random

  2031 begin

  2032

  2033 definition

  2034   "Quickcheck_Random.random i = Quickcheck_Random.random i \<circ>\<rightarrow> (\<lambda>r. Pair (valterm_ratreal r))"

  2035

  2036 instance ..

  2037

  2038 end

  2039

  2040 no_notation fcomp (infixl "\<circ>>" 60)

  2041 no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

  2042

  2043 instantiation real :: exhaustive

  2044 begin

  2045

  2046 definition

  2047   "exhaustive_real f d = Quickcheck_Exhaustive.exhaustive (%r. f (Ratreal r)) d"

  2048

  2049 instance ..

  2050

  2051 end

  2052

  2053 instantiation real :: full_exhaustive

  2054 begin

  2055

  2056 definition

  2057   "full_exhaustive_real f d = Quickcheck_Exhaustive.full_exhaustive (%r. f (valterm_ratreal r)) d"

  2058

  2059 instance ..

  2060

  2061 end

  2062

  2063 instantiation real :: narrowing

  2064 begin

  2065

  2066 definition

  2067   "narrowing = Quickcheck_Narrowing.apply (Quickcheck_Narrowing.cons Ratreal) narrowing"

  2068

  2069 instance ..

  2070

  2071 end

  2072

  2073

  2074 subsection {* Setup for Nitpick *}

  2075

  2076 declaration {*

  2077   Nitpick_HOL.register_frac_type @{type_name real}

  2078    [(@{const_name zero_real_inst.zero_real}, @{const_name Nitpick.zero_frac}),

  2079     (@{const_name one_real_inst.one_real}, @{const_name Nitpick.one_frac}),

  2080     (@{const_name plus_real_inst.plus_real}, @{const_name Nitpick.plus_frac}),

  2081     (@{const_name times_real_inst.times_real}, @{const_name Nitpick.times_frac}),

  2082     (@{const_name uminus_real_inst.uminus_real}, @{const_name Nitpick.uminus_frac}),

  2083     (@{const_name inverse_real_inst.inverse_real}, @{const_name Nitpick.inverse_frac}),

  2084     (@{const_name ord_real_inst.less_real}, @{const_name Nitpick.less_frac}),

  2085     (@{const_name ord_real_inst.less_eq_real}, @{const_name Nitpick.less_eq_frac})]

  2086 *}

  2087

  2088 lemmas [nitpick_unfold] = inverse_real_inst.inverse_real one_real_inst.one_real

  2089     ord_real_inst.less_real ord_real_inst.less_eq_real plus_real_inst.plus_real

  2090     times_real_inst.times_real uminus_real_inst.uminus_real

  2091     zero_real_inst.zero_real

  2092

  2093

  2094 subsection {* Setup for SMT *}

  2095

  2096 ML_file "Tools/SMT/smt_real.ML"

  2097 ML_file "Tools/SMT/z3_real.ML"

  2098

  2099 lemma [z3_rule]:

  2100   "0 + (x::real) = x"

  2101   "x + 0 = x"

  2102   "0 * x = 0"

  2103   "1 * x = x"

  2104   "x + y = y + x"

  2105   by auto

  2106

  2107 end
`