src/HOL/RealDef.thy
author haftmann
Mon Apr 27 10:11:44 2009 +0200 (2009-04-27)
changeset 31001 7e6ffd8f51a9
parent 30242 aea5d7fa7ef5
child 31100 6a2e67fe4488
permissions -rw-r--r--
cleaned up theory power further
haftmann@28952
     1
(*  Title       : HOL/RealDef.thy
paulson@5588
     2
    Author      : Jacques D. Fleuriot
paulson@5588
     3
    Copyright   : 1998  University of Cambridge
paulson@14387
     4
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
avigad@16819
     5
    Additional contributions by Jeremy Avigad
paulson@14269
     6
*)
paulson@14269
     7
paulson@14387
     8
header{*Defining the Reals from the Positive Reals*}
paulson@14387
     9
nipkow@15131
    10
theory RealDef
nipkow@15140
    11
imports PReal
haftmann@28952
    12
uses ("Tools/real_arith.ML")
nipkow@15131
    13
begin
paulson@5588
    14
wenzelm@19765
    15
definition
wenzelm@21404
    16
  realrel   ::  "((preal * preal) * (preal * preal)) set" where
haftmann@28562
    17
  [code del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
paulson@14269
    18
paulson@14484
    19
typedef (Real)  real = "UNIV//realrel"
paulson@14269
    20
  by (auto simp add: quotient_def)
paulson@5588
    21
wenzelm@19765
    22
definition
paulson@14484
    23
  (** these don't use the overloaded "real" function: users don't see them **)
wenzelm@21404
    24
  real_of_preal :: "preal => real" where
haftmann@28562
    25
  [code del]: "real_of_preal m = Abs_Real (realrel `` {(m + 1, 1)})"
paulson@14484
    26
haftmann@25762
    27
instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
haftmann@25571
    28
begin
paulson@5588
    29
haftmann@25571
    30
definition
haftmann@28562
    31
  real_zero_def [code del]: "0 = Abs_Real(realrel``{(1, 1)})"
haftmann@25571
    32
haftmann@25571
    33
definition
haftmann@28562
    34
  real_one_def [code del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
paulson@5588
    35
haftmann@25571
    36
definition
haftmann@28562
    37
  real_add_def [code del]: "z + w =
paulson@14484
    38
       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
nipkow@27964
    39
                 { Abs_Real(realrel``{(x+u, y+v)}) })"
bauerg@10606
    40
haftmann@25571
    41
definition
haftmann@28562
    42
  real_minus_def [code del]: "- r =  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
haftmann@25571
    43
haftmann@25571
    44
definition
haftmann@28562
    45
  real_diff_def [code del]: "r - (s::real) = r + - s"
paulson@14484
    46
haftmann@25571
    47
definition
haftmann@28562
    48
  real_mult_def [code del]:
haftmann@25571
    49
    "z * w =
paulson@14484
    50
       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
nipkow@27964
    51
                 { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
paulson@5588
    52
haftmann@25571
    53
definition
haftmann@28562
    54
  real_inverse_def [code del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
haftmann@25571
    55
haftmann@25571
    56
definition
haftmann@28562
    57
  real_divide_def [code del]: "R / (S::real) = R * inverse S"
paulson@14269
    58
haftmann@25571
    59
definition
haftmann@28562
    60
  real_le_def [code del]: "z \<le> (w::real) \<longleftrightarrow>
haftmann@25571
    61
    (\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)"
haftmann@25571
    62
haftmann@25571
    63
definition
haftmann@28562
    64
  real_less_def [code del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
paulson@5588
    65
haftmann@25571
    66
definition
haftmann@25571
    67
  real_abs_def:  "abs (r::real) = (if r < 0 then - r else r)"
paulson@14334
    68
haftmann@25571
    69
definition
haftmann@25571
    70
  real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"
haftmann@25571
    71
haftmann@25571
    72
instance ..
haftmann@25571
    73
haftmann@25571
    74
end
paulson@14334
    75
huffman@23287
    76
subsection {* Equivalence relation over positive reals *}
paulson@14269
    77
paulson@14270
    78
lemma preal_trans_lemma:
paulson@14365
    79
  assumes "x + y1 = x1 + y"
paulson@14365
    80
      and "x + y2 = x2 + y"
paulson@14365
    81
  shows "x1 + y2 = x2 + (y1::preal)"
paulson@14365
    82
proof -
huffman@23287
    83
  have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
paulson@14365
    84
  also have "... = (x2 + y) + x1"  by (simp add: prems)
huffman@23287
    85
  also have "... = x2 + (x1 + y)"  by (simp add: add_ac)
paulson@14365
    86
  also have "... = x2 + (x + y1)"  by (simp add: prems)
huffman@23287
    87
  also have "... = (x2 + y1) + x"  by (simp add: add_ac)
paulson@14365
    88
  finally have "(x1 + y2) + x = (x2 + y1) + x" .
huffman@23287
    89
  thus ?thesis by (rule add_right_imp_eq)
paulson@14365
    90
qed
paulson@14365
    91
paulson@14269
    92
paulson@14484
    93
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
paulson@14484
    94
by (simp add: realrel_def)
paulson@14269
    95
paulson@14269
    96
lemma equiv_realrel: "equiv UNIV realrel"
nipkow@30198
    97
apply (auto simp add: equiv_def refl_on_def sym_def trans_def realrel_def)
paulson@14365
    98
apply (blast dest: preal_trans_lemma) 
paulson@14269
    99
done
paulson@14269
   100
paulson@14497
   101
text{*Reduces equality of equivalence classes to the @{term realrel} relation:
paulson@14497
   102
  @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
paulson@14269
   103
lemmas equiv_realrel_iff = 
paulson@14269
   104
       eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
paulson@14269
   105
paulson@14269
   106
declare equiv_realrel_iff [simp]
paulson@14269
   107
paulson@14497
   108
paulson@14484
   109
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
paulson@14484
   110
by (simp add: Real_def realrel_def quotient_def, blast)
paulson@14269
   111
huffman@22958
   112
declare Abs_Real_inject [simp]
paulson@14484
   113
declare Abs_Real_inverse [simp]
paulson@14269
   114
paulson@14269
   115
paulson@14484
   116
text{*Case analysis on the representation of a real number as an equivalence
paulson@14484
   117
      class of pairs of positive reals.*}
paulson@14484
   118
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 
paulson@14484
   119
     "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
paulson@14484
   120
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
paulson@14484
   121
apply (drule arg_cong [where f=Abs_Real])
paulson@14484
   122
apply (auto simp add: Rep_Real_inverse)
paulson@14269
   123
done
paulson@14269
   124
paulson@14269
   125
huffman@23287
   126
subsection {* Addition and Subtraction *}
paulson@14269
   127
paulson@14269
   128
lemma real_add_congruent2_lemma:
paulson@14269
   129
     "[|a + ba = aa + b; ab + bc = ac + bb|]
paulson@14269
   130
      ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
huffman@23287
   131
apply (simp add: add_assoc)
huffman@23287
   132
apply (rule add_left_commute [of ab, THEN ssubst])
huffman@23287
   133
apply (simp add: add_assoc [symmetric])
huffman@23287
   134
apply (simp add: add_ac)
paulson@14269
   135
done
paulson@14269
   136
paulson@14269
   137
lemma real_add:
paulson@14497
   138
     "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
paulson@14497
   139
      Abs_Real (realrel``{(x+u, y+v)})"
paulson@14497
   140
proof -
paulson@15169
   141
  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
paulson@15169
   142
        respects2 realrel"
paulson@14497
   143
    by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
paulson@14497
   144
  thus ?thesis
paulson@14497
   145
    by (simp add: real_add_def UN_UN_split_split_eq
paulson@14658
   146
                  UN_equiv_class2 [OF equiv_realrel equiv_realrel])
paulson@14497
   147
qed
paulson@14269
   148
paulson@14484
   149
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
paulson@14484
   150
proof -
paulson@15169
   151
  have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
huffman@23288
   152
    by (simp add: congruent_def add_commute) 
paulson@14484
   153
  thus ?thesis
paulson@14484
   154
    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
paulson@14484
   155
qed
paulson@14334
   156
huffman@23287
   157
instance real :: ab_group_add
huffman@23287
   158
proof
huffman@23287
   159
  fix x y z :: real
huffman@23287
   160
  show "(x + y) + z = x + (y + z)"
huffman@23287
   161
    by (cases x, cases y, cases z, simp add: real_add add_assoc)
huffman@23287
   162
  show "x + y = y + x"
huffman@23287
   163
    by (cases x, cases y, simp add: real_add add_commute)
huffman@23287
   164
  show "0 + x = x"
huffman@23287
   165
    by (cases x, simp add: real_add real_zero_def add_ac)
huffman@23287
   166
  show "- x + x = 0"
huffman@23287
   167
    by (cases x, simp add: real_minus real_add real_zero_def add_commute)
huffman@23287
   168
  show "x - y = x + - y"
huffman@23287
   169
    by (simp add: real_diff_def)
huffman@23287
   170
qed
paulson@14269
   171
paulson@14269
   172
huffman@23287
   173
subsection {* Multiplication *}
paulson@14269
   174
paulson@14329
   175
lemma real_mult_congruent2_lemma:
paulson@14329
   176
     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
paulson@14484
   177
          x * x1 + y * y1 + (x * y2 + y * x2) =
paulson@14484
   178
          x * x2 + y * y2 + (x * y1 + y * x1)"
huffman@23287
   179
apply (simp add: add_left_commute add_assoc [symmetric])
huffman@23288
   180
apply (simp add: add_assoc right_distrib [symmetric])
huffman@23288
   181
apply (simp add: add_commute)
paulson@14269
   182
done
paulson@14269
   183
paulson@14269
   184
lemma real_mult_congruent2:
paulson@15169
   185
    "(%p1 p2.
paulson@14484
   186
        (%(x1,y1). (%(x2,y2). 
paulson@15169
   187
          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
paulson@15169
   188
     respects2 realrel"
paulson@14658
   189
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
huffman@23288
   190
apply (simp add: mult_commute add_commute)
paulson@14269
   191
apply (auto simp add: real_mult_congruent2_lemma)
paulson@14269
   192
done
paulson@14269
   193
paulson@14269
   194
lemma real_mult:
paulson@14484
   195
      "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
paulson@14484
   196
       Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
paulson@14484
   197
by (simp add: real_mult_def UN_UN_split_split_eq
paulson@14658
   198
         UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
paulson@14269
   199
paulson@14269
   200
lemma real_mult_commute: "(z::real) * w = w * z"
huffman@23288
   201
by (cases z, cases w, simp add: real_mult add_ac mult_ac)
paulson@14269
   202
paulson@14269
   203
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
paulson@14484
   204
apply (cases z1, cases z2, cases z3)
nipkow@29667
   205
apply (simp add: real_mult algebra_simps)
paulson@14269
   206
done
paulson@14269
   207
paulson@14269
   208
lemma real_mult_1: "(1::real) * z = z"
paulson@14484
   209
apply (cases z)
nipkow@29667
   210
apply (simp add: real_mult real_one_def algebra_simps)
paulson@14269
   211
done
paulson@14269
   212
paulson@14269
   213
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14484
   214
apply (cases z1, cases z2, cases w)
nipkow@29667
   215
apply (simp add: real_add real_mult algebra_simps)
paulson@14269
   216
done
paulson@14269
   217
paulson@14329
   218
text{*one and zero are distinct*}
paulson@14365
   219
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
paulson@14484
   220
proof -
huffman@23287
   221
  have "(1::preal) < 1 + 1"
huffman@23287
   222
    by (simp add: preal_self_less_add_left)
paulson@14484
   223
  thus ?thesis
huffman@23288
   224
    by (simp add: real_zero_def real_one_def)
paulson@14484
   225
qed
paulson@14269
   226
huffman@23287
   227
instance real :: comm_ring_1
huffman@23287
   228
proof
huffman@23287
   229
  fix x y z :: real
huffman@23287
   230
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
huffman@23287
   231
  show "x * y = y * x" by (rule real_mult_commute)
huffman@23287
   232
  show "1 * x = x" by (rule real_mult_1)
huffman@23287
   233
  show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
huffman@23287
   234
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
huffman@23287
   235
qed
huffman@23287
   236
huffman@23287
   237
subsection {* Inverse and Division *}
paulson@14365
   238
paulson@14484
   239
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
huffman@23288
   240
by (simp add: real_zero_def add_commute)
paulson@14269
   241
paulson@14365
   242
text{*Instead of using an existential quantifier and constructing the inverse
paulson@14365
   243
within the proof, we could define the inverse explicitly.*}
paulson@14365
   244
paulson@14365
   245
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
paulson@14484
   246
apply (simp add: real_zero_def real_one_def, cases x)
paulson@14269
   247
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@14365
   248
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
paulson@14334
   249
apply (rule_tac
huffman@23287
   250
        x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
paulson@14334
   251
       in exI)
paulson@14334
   252
apply (rule_tac [2]
huffman@23287
   253
        x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 
paulson@14334
   254
       in exI)
nipkow@29667
   255
apply (auto simp add: real_mult preal_mult_inverse_right algebra_simps)
paulson@14269
   256
done
paulson@14269
   257
paulson@14365
   258
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
paulson@14484
   259
apply (simp add: real_inverse_def)
huffman@23287
   260
apply (drule real_mult_inverse_left_ex, safe)
huffman@23287
   261
apply (rule theI, assumption, rename_tac z)
huffman@23287
   262
apply (subgoal_tac "(z * x) * y = z * (x * y)")
huffman@23287
   263
apply (simp add: mult_commute)
huffman@23287
   264
apply (rule mult_assoc)
paulson@14269
   265
done
paulson@14334
   266
paulson@14341
   267
paulson@14341
   268
subsection{*The Real Numbers form a Field*}
paulson@14341
   269
paulson@14334
   270
instance real :: field
paulson@14334
   271
proof
paulson@14334
   272
  fix x y z :: real
paulson@14365
   273
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
paulson@14430
   274
  show "x / y = x * inverse y" by (simp add: real_divide_def)
paulson@14334
   275
qed
paulson@14334
   276
paulson@14334
   277
paulson@14341
   278
text{*Inverse of zero!  Useful to simplify certain equations*}
paulson@14269
   279
paulson@14334
   280
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
paulson@14484
   281
by (simp add: real_inverse_def)
paulson@14334
   282
paulson@14334
   283
instance real :: division_by_zero
paulson@14334
   284
proof
paulson@14334
   285
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
paulson@14334
   286
qed
paulson@14334
   287
paulson@14269
   288
paulson@14365
   289
subsection{*The @{text "\<le>"} Ordering*}
paulson@14269
   290
paulson@14365
   291
lemma real_le_refl: "w \<le> (w::real)"
paulson@14484
   292
by (cases w, force simp add: real_le_def)
paulson@14269
   293
paulson@14378
   294
text{*The arithmetic decision procedure is not set up for type preal.
paulson@14378
   295
  This lemma is currently unused, but it could simplify the proofs of the
paulson@14378
   296
  following two lemmas.*}
paulson@14378
   297
lemma preal_eq_le_imp_le:
paulson@14378
   298
  assumes eq: "a+b = c+d" and le: "c \<le> a"
paulson@14378
   299
  shows "b \<le> (d::preal)"
paulson@14378
   300
proof -
huffman@23288
   301
  have "c+d \<le> a+d" by (simp add: prems)
paulson@14378
   302
  hence "a+b \<le> a+d" by (simp add: prems)
huffman@23288
   303
  thus "b \<le> d" by simp
paulson@14378
   304
qed
paulson@14378
   305
paulson@14378
   306
lemma real_le_lemma:
paulson@14378
   307
  assumes l: "u1 + v2 \<le> u2 + v1"
paulson@14378
   308
      and "x1 + v1 = u1 + y1"
paulson@14378
   309
      and "x2 + v2 = u2 + y2"
paulson@14378
   310
  shows "x1 + y2 \<le> x2 + (y1::preal)"
paulson@14365
   311
proof -
paulson@14378
   312
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
huffman@23288
   313
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
huffman@23288
   314
  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
huffman@23288
   315
  finally show ?thesis by simp
huffman@23288
   316
qed
paulson@14378
   317
paulson@14378
   318
lemma real_le: 
paulson@14484
   319
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
paulson@14484
   320
      (x1 + y2 \<le> x2 + y1)"
huffman@23288
   321
apply (simp add: real_le_def)
paulson@14387
   322
apply (auto intro: real_le_lemma)
paulson@14378
   323
done
paulson@14378
   324
paulson@14378
   325
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
nipkow@15542
   326
by (cases z, cases w, simp add: real_le)
paulson@14378
   327
paulson@14378
   328
lemma real_trans_lemma:
paulson@14378
   329
  assumes "x + v \<le> u + y"
paulson@14378
   330
      and "u + v' \<le> u' + v"
paulson@14378
   331
      and "x2 + v2 = u2 + y2"
paulson@14378
   332
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   333
proof -
huffman@23288
   334
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
huffman@23288
   335
  also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
huffman@23288
   336
  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
huffman@23288
   337
  also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
huffman@23288
   338
  finally show ?thesis by simp
nipkow@15542
   339
qed
paulson@14269
   340
paulson@14365
   341
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14484
   342
apply (cases i, cases j, cases k)
paulson@14484
   343
apply (simp add: real_le)
huffman@23288
   344
apply (blast intro: real_trans_lemma)
paulson@14334
   345
done
paulson@14334
   346
paulson@14365
   347
instance real :: order
haftmann@27682
   348
proof
haftmann@27682
   349
  fix u v :: real
haftmann@27682
   350
  show "u < v \<longleftrightarrow> u \<le> v \<and> \<not> v \<le> u" 
haftmann@27682
   351
    by (auto simp add: real_less_def intro: real_le_anti_sym)
haftmann@27682
   352
qed (assumption | rule real_le_refl real_le_trans real_le_anti_sym)+
paulson@14365
   353
paulson@14378
   354
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   355
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
huffman@23288
   356
apply (cases z, cases w)
huffman@23288
   357
apply (auto simp add: real_le real_zero_def add_ac)
paulson@14334
   358
done
paulson@14334
   359
paulson@14334
   360
instance real :: linorder
paulson@14334
   361
  by (intro_classes, rule real_le_linear)
paulson@14334
   362
paulson@14334
   363
paulson@14378
   364
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14484
   365
apply (cases x, cases y) 
paulson@14378
   366
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
huffman@23288
   367
                      add_ac)
huffman@23288
   368
apply (simp_all add: add_assoc [symmetric])
nipkow@15542
   369
done
paulson@14378
   370
paulson@14484
   371
lemma real_add_left_mono: 
paulson@14484
   372
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
paulson@14484
   373
proof -
chaieb@27668
   374
  have "z + x - (z + y) = (z + -z) + (x - y)" 
nipkow@29667
   375
    by (simp add: algebra_simps) 
paulson@14484
   376
  with le show ?thesis 
obua@14754
   377
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
paulson@14484
   378
qed
paulson@14334
   379
paulson@14365
   380
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   381
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   382
paulson@14365
   383
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   384
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   385
paulson@14334
   386
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14484
   387
apply (cases x, cases y)
paulson@14378
   388
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   389
                 linorder_not_le [where 'a = preal] 
paulson@14378
   390
                  real_zero_def real_le real_mult)
paulson@14365
   391
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
wenzelm@16973
   392
apply (auto dest!: less_add_left_Ex
nipkow@29667
   393
     simp add: algebra_simps preal_self_less_add_left)
paulson@14334
   394
done
paulson@14334
   395
paulson@14334
   396
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   397
apply (rule real_sum_gt_zero_less)
paulson@14334
   398
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   399
apply (drule real_mult_order, assumption)
paulson@14334
   400
apply (simp add: right_distrib)
paulson@14334
   401
done
paulson@14334
   402
haftmann@25571
   403
instantiation real :: distrib_lattice
haftmann@25571
   404
begin
haftmann@25571
   405
haftmann@25571
   406
definition
haftmann@25571
   407
  "(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
haftmann@25571
   408
haftmann@25571
   409
definition
haftmann@25571
   410
  "(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
haftmann@25571
   411
haftmann@25571
   412
instance
haftmann@22456
   413
  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
haftmann@22456
   414
haftmann@25571
   415
end
haftmann@25571
   416
paulson@14378
   417
paulson@14334
   418
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   419
paulson@14334
   420
instance real :: ordered_field
paulson@14334
   421
proof
paulson@14334
   422
  fix x y z :: real
paulson@14334
   423
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
huffman@22962
   424
  show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
huffman@22962
   425
  show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
nipkow@24506
   426
  show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
nipkow@24506
   427
    by (simp only: real_sgn_def)
paulson@14334
   428
qed
paulson@14334
   429
haftmann@25303
   430
instance real :: lordered_ab_group_add ..
haftmann@25303
   431
paulson@14365
   432
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   433
to be essential for proving completeness of the reals from that of the
paulson@14365
   434
positive reals.*}
paulson@14365
   435
paulson@14365
   436
lemma real_of_preal_add:
paulson@14365
   437
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
nipkow@29667
   438
by (simp add: real_of_preal_def real_add algebra_simps)
paulson@14365
   439
paulson@14365
   440
lemma real_of_preal_mult:
paulson@14365
   441
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
nipkow@29667
   442
by (simp add: real_of_preal_def real_mult algebra_simps)
paulson@14365
   443
paulson@14365
   444
paulson@14365
   445
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   446
lemma real_of_preal_trichotomy:
paulson@14365
   447
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14484
   448
apply (simp add: real_of_preal_def real_zero_def, cases x)
huffman@23288
   449
apply (auto simp add: real_minus add_ac)
paulson@14365
   450
apply (cut_tac x = x and y = y in linorder_less_linear)
huffman@23288
   451
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
paulson@14365
   452
done
paulson@14365
   453
paulson@14365
   454
lemma real_of_preal_leD:
paulson@14365
   455
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
huffman@23288
   456
by (simp add: real_of_preal_def real_le)
paulson@14365
   457
paulson@14365
   458
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   459
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   460
paulson@14365
   461
lemma real_of_preal_lessD:
paulson@14365
   462
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
huffman@23288
   463
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
paulson@14365
   464
paulson@14365
   465
lemma real_of_preal_less_iff [simp]:
paulson@14365
   466
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   467
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   468
paulson@14365
   469
lemma real_of_preal_le_iff:
paulson@14365
   470
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
huffman@23288
   471
by (simp add: linorder_not_less [symmetric])
paulson@14365
   472
paulson@14365
   473
lemma real_of_preal_zero_less: "0 < real_of_preal m"
huffman@23288
   474
apply (insert preal_self_less_add_left [of 1 m])
huffman@23288
   475
apply (auto simp add: real_zero_def real_of_preal_def
huffman@23288
   476
                      real_less_def real_le_def add_ac)
huffman@23288
   477
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
huffman@23288
   478
apply (simp add: add_ac)
paulson@14365
   479
done
paulson@14365
   480
paulson@14365
   481
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   482
by (simp add: real_of_preal_zero_less)
paulson@14365
   483
paulson@14365
   484
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14484
   485
proof -
paulson@14484
   486
  from real_of_preal_minus_less_zero
paulson@14484
   487
  show ?thesis by (blast dest: order_less_trans)
paulson@14484
   488
qed
paulson@14365
   489
paulson@14365
   490
paulson@14365
   491
subsection{*Theorems About the Ordering*}
paulson@14365
   492
paulson@14365
   493
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   494
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   495
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   496
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   497
done
paulson@14365
   498
paulson@14365
   499
lemma real_gt_preal_preal_Ex:
paulson@14365
   500
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   501
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   502
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   503
paulson@14365
   504
lemma real_ge_preal_preal_Ex:
paulson@14365
   505
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   506
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   507
paulson@14365
   508
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   509
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   510
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   511
            simp add: real_of_preal_zero_less)
paulson@14365
   512
paulson@14365
   513
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   514
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   515
paulson@14334
   516
paulson@14334
   517
subsection{*More Lemmas*}
paulson@14334
   518
paulson@14334
   519
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   520
by auto
paulson@14334
   521
paulson@14334
   522
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   523
by auto
paulson@14334
   524
paulson@14334
   525
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   526
  by (force elim: order_less_asym
paulson@14334
   527
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   528
paulson@14334
   529
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   530
apply (simp add: mult_le_cancel_right)
huffman@23289
   531
apply (blast intro: elim: order_less_asym)
paulson@14365
   532
done
paulson@14334
   533
paulson@14334
   534
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
nipkow@15923
   535
by(simp add:mult_commute)
paulson@14334
   536
paulson@14365
   537
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
huffman@23289
   538
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
paulson@14334
   539
paulson@14334
   540
haftmann@24198
   541
subsection {* Embedding numbers into the Reals *}
haftmann@24198
   542
haftmann@24198
   543
abbreviation
haftmann@24198
   544
  real_of_nat :: "nat \<Rightarrow> real"
haftmann@24198
   545
where
haftmann@24198
   546
  "real_of_nat \<equiv> of_nat"
haftmann@24198
   547
haftmann@24198
   548
abbreviation
haftmann@24198
   549
  real_of_int :: "int \<Rightarrow> real"
haftmann@24198
   550
where
haftmann@24198
   551
  "real_of_int \<equiv> of_int"
haftmann@24198
   552
haftmann@24198
   553
abbreviation
haftmann@24198
   554
  real_of_rat :: "rat \<Rightarrow> real"
haftmann@24198
   555
where
haftmann@24198
   556
  "real_of_rat \<equiv> of_rat"
haftmann@24198
   557
haftmann@24198
   558
consts
haftmann@24198
   559
  (*overloaded constant for injecting other types into "real"*)
haftmann@24198
   560
  real :: "'a => real"
paulson@14365
   561
paulson@14378
   562
defs (overloaded)
haftmann@28520
   563
  real_of_nat_def [code unfold]: "real == real_of_nat"
haftmann@28520
   564
  real_of_int_def [code unfold]: "real == real_of_int"
paulson@14365
   565
avigad@16819
   566
lemma real_eq_of_nat: "real = of_nat"
haftmann@24198
   567
  unfolding real_of_nat_def ..
avigad@16819
   568
avigad@16819
   569
lemma real_eq_of_int: "real = of_int"
haftmann@24198
   570
  unfolding real_of_int_def ..
avigad@16819
   571
paulson@14365
   572
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   573
by (simp add: real_of_int_def) 
paulson@14365
   574
paulson@14365
   575
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   576
by (simp add: real_of_int_def) 
paulson@14334
   577
avigad@16819
   578
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378
   579
by (simp add: real_of_int_def) 
paulson@14365
   580
avigad@16819
   581
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378
   582
by (simp add: real_of_int_def) 
avigad@16819
   583
avigad@16819
   584
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819
   585
by (simp add: real_of_int_def) 
paulson@14365
   586
avigad@16819
   587
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378
   588
by (simp add: real_of_int_def) 
paulson@14334
   589
avigad@16819
   590
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819
   591
  apply (subst real_eq_of_int)+
avigad@16819
   592
  apply (rule of_int_setsum)
avigad@16819
   593
done
avigad@16819
   594
avigad@16819
   595
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819
   596
    (PROD x:A. real(f x))"
avigad@16819
   597
  apply (subst real_eq_of_int)+
avigad@16819
   598
  apply (rule of_int_setprod)
avigad@16819
   599
done
paulson@14365
   600
chaieb@27668
   601
lemma real_of_int_zero_cancel [simp, algebra, presburger]: "(real x = 0) = (x = (0::int))"
paulson@14378
   602
by (simp add: real_of_int_def) 
paulson@14365
   603
chaieb@27668
   604
lemma real_of_int_inject [iff, algebra, presburger]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   605
by (simp add: real_of_int_def) 
paulson@14365
   606
chaieb@27668
   607
lemma real_of_int_less_iff [iff, presburger]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   608
by (simp add: real_of_int_def) 
paulson@14365
   609
chaieb@27668
   610
lemma real_of_int_le_iff [simp, presburger]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   611
by (simp add: real_of_int_def) 
paulson@14365
   612
chaieb@27668
   613
lemma real_of_int_gt_zero_cancel_iff [simp, presburger]: "(0 < real (n::int)) = (0 < n)"
avigad@16819
   614
by (simp add: real_of_int_def) 
avigad@16819
   615
chaieb@27668
   616
lemma real_of_int_ge_zero_cancel_iff [simp, presburger]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819
   617
by (simp add: real_of_int_def) 
avigad@16819
   618
chaieb@27668
   619
lemma real_of_int_lt_zero_cancel_iff [simp, presburger]: "(real (n::int) < 0) = (n < 0)" 
avigad@16819
   620
by (simp add: real_of_int_def)
avigad@16819
   621
chaieb@27668
   622
lemma real_of_int_le_zero_cancel_iff [simp, presburger]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819
   623
by (simp add: real_of_int_def)
avigad@16819
   624
avigad@16888
   625
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888
   626
by (auto simp add: abs_if)
avigad@16888
   627
avigad@16819
   628
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819
   629
  apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819
   630
  apply (simp del: real_of_int_add)
avigad@16819
   631
  apply auto
avigad@16819
   632
done
avigad@16819
   633
avigad@16819
   634
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819
   635
  apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819
   636
  apply (simp del: real_of_int_add)
avigad@16819
   637
  apply simp
avigad@16819
   638
done
avigad@16819
   639
avigad@16819
   640
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
avigad@16819
   641
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   642
proof -
avigad@16819
   643
  assume "d ~= 0"
avigad@16819
   644
  have "x = (x div d) * d + x mod d"
avigad@16819
   645
    by auto
avigad@16819
   646
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   647
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819
   648
  then have "real x / real d = ... / real d"
avigad@16819
   649
    by simp
avigad@16819
   650
  then show ?thesis
nipkow@29667
   651
    by (auto simp add: add_divide_distrib algebra_simps prems)
avigad@16819
   652
qed
avigad@16819
   653
avigad@16819
   654
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
avigad@16819
   655
    real(n div d) = real n / real d"
avigad@16819
   656
  apply (frule real_of_int_div_aux [of d n])
avigad@16819
   657
  apply simp
nipkow@30042
   658
  apply (simp add: dvd_eq_mod_eq_0)
avigad@16819
   659
done
avigad@16819
   660
avigad@16819
   661
lemma real_of_int_div2:
avigad@16819
   662
  "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819
   663
  apply (case_tac "x = 0")
avigad@16819
   664
  apply simp
avigad@16819
   665
  apply (case_tac "0 < x")
nipkow@29667
   666
  apply (simp add: algebra_simps)
avigad@16819
   667
  apply (subst real_of_int_div_aux)
avigad@16819
   668
  apply simp
avigad@16819
   669
  apply simp
avigad@16819
   670
  apply (subst zero_le_divide_iff)
avigad@16819
   671
  apply auto
nipkow@29667
   672
  apply (simp add: algebra_simps)
avigad@16819
   673
  apply (subst real_of_int_div_aux)
avigad@16819
   674
  apply simp
avigad@16819
   675
  apply simp
avigad@16819
   676
  apply (subst zero_le_divide_iff)
avigad@16819
   677
  apply auto
avigad@16819
   678
done
avigad@16819
   679
avigad@16819
   680
lemma real_of_int_div3:
avigad@16819
   681
  "real (n::int) / real (x) - real (n div x) <= 1"
avigad@16819
   682
  apply(case_tac "x = 0")
avigad@16819
   683
  apply simp
nipkow@29667
   684
  apply (simp add: algebra_simps)
avigad@16819
   685
  apply (subst real_of_int_div_aux)
avigad@16819
   686
  apply assumption
avigad@16819
   687
  apply simp
avigad@16819
   688
  apply (subst divide_le_eq)
avigad@16819
   689
  apply clarsimp
avigad@16819
   690
  apply (rule conjI)
avigad@16819
   691
  apply (rule impI)
avigad@16819
   692
  apply (rule order_less_imp_le)
avigad@16819
   693
  apply simp
avigad@16819
   694
  apply (rule impI)
avigad@16819
   695
  apply (rule order_less_imp_le)
avigad@16819
   696
  apply simp
avigad@16819
   697
done
avigad@16819
   698
avigad@16819
   699
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
nipkow@27964
   700
by (insert real_of_int_div2 [of n x], simp)
nipkow@27964
   701
nipkow@27964
   702
paulson@14365
   703
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   704
paulson@14334
   705
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   706
by (simp add: real_of_nat_def)
paulson@14334
   707
huffman@30082
   708
lemma real_of_nat_1 [simp]: "real (1::nat) = 1"
huffman@30082
   709
by (simp add: real_of_nat_def)
huffman@30082
   710
paulson@14334
   711
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   712
by (simp add: real_of_nat_def)
paulson@14334
   713
paulson@14365
   714
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   715
by (simp add: real_of_nat_def)
paulson@14334
   716
paulson@14334
   717
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   718
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   719
by (simp add: real_of_nat_def)
paulson@14334
   720
paulson@14334
   721
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   722
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   723
by (simp add: real_of_nat_def)
paulson@14334
   724
paulson@14334
   725
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   726
by (simp add: real_of_nat_def)
paulson@14334
   727
paulson@14334
   728
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   729
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   730
paulson@14365
   731
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   732
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   733
paulson@14334
   734
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
huffman@23431
   735
by (simp add: real_of_nat_def of_nat_mult)
paulson@14334
   736
avigad@16819
   737
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819
   738
    (SUM x:A. real(f x))"
avigad@16819
   739
  apply (subst real_eq_of_nat)+
avigad@16819
   740
  apply (rule of_nat_setsum)
avigad@16819
   741
done
avigad@16819
   742
avigad@16819
   743
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819
   744
    (PROD x:A. real(f x))"
avigad@16819
   745
  apply (subst real_eq_of_nat)+
avigad@16819
   746
  apply (rule of_nat_setprod)
avigad@16819
   747
done
avigad@16819
   748
avigad@16819
   749
lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819
   750
  apply (subst card_eq_setsum)
avigad@16819
   751
  apply (subst real_of_nat_setsum)
avigad@16819
   752
  apply simp
avigad@16819
   753
done
avigad@16819
   754
paulson@14334
   755
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   756
by (simp add: real_of_nat_def)
paulson@14334
   757
paulson@14387
   758
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   759
by (simp add: real_of_nat_def)
paulson@14334
   760
paulson@14365
   761
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
huffman@23438
   762
by (simp add: add: real_of_nat_def of_nat_diff)
paulson@14334
   763
nipkow@25162
   764
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
nipkow@25140
   765
by (auto simp: real_of_nat_def)
paulson@14365
   766
paulson@14365
   767
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   768
by (simp add: add: real_of_nat_def)
paulson@14334
   769
paulson@14365
   770
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   771
by (simp add: add: real_of_nat_def)
paulson@14334
   772
nipkow@25140
   773
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))"
paulson@14378
   774
by (simp add: add: real_of_nat_def)
paulson@14334
   775
avigad@16819
   776
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819
   777
  apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819
   778
  apply simp
avigad@16819
   779
  apply (auto simp add: real_of_nat_Suc)
avigad@16819
   780
done
avigad@16819
   781
avigad@16819
   782
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819
   783
  apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819
   784
  apply (simp add: less_Suc_eq_le)
avigad@16819
   785
  apply (simp add: real_of_nat_Suc)
avigad@16819
   786
done
avigad@16819
   787
avigad@16819
   788
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
avigad@16819
   789
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   790
proof -
avigad@16819
   791
  assume "0 < d"
avigad@16819
   792
  have "x = (x div d) * d + x mod d"
avigad@16819
   793
    by auto
avigad@16819
   794
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   795
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819
   796
  then have "real x / real d = \<dots> / real d"
avigad@16819
   797
    by simp
avigad@16819
   798
  then show ?thesis
nipkow@29667
   799
    by (auto simp add: add_divide_distrib algebra_simps prems)
avigad@16819
   800
qed
avigad@16819
   801
avigad@16819
   802
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
avigad@16819
   803
    real(n div d) = real n / real d"
avigad@16819
   804
  apply (frule real_of_nat_div_aux [of d n])
avigad@16819
   805
  apply simp
avigad@16819
   806
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
avigad@16819
   807
  apply assumption
avigad@16819
   808
done
avigad@16819
   809
avigad@16819
   810
lemma real_of_nat_div2:
avigad@16819
   811
  "0 <= real (n::nat) / real (x) - real (n div x)"
nipkow@25134
   812
apply(case_tac "x = 0")
nipkow@25134
   813
 apply (simp)
nipkow@29667
   814
apply (simp add: algebra_simps)
nipkow@25134
   815
apply (subst real_of_nat_div_aux)
nipkow@25134
   816
 apply simp
nipkow@25134
   817
apply simp
nipkow@25134
   818
apply (subst zero_le_divide_iff)
nipkow@25134
   819
apply simp
avigad@16819
   820
done
avigad@16819
   821
avigad@16819
   822
lemma real_of_nat_div3:
avigad@16819
   823
  "real (n::nat) / real (x) - real (n div x) <= 1"
nipkow@25134
   824
apply(case_tac "x = 0")
nipkow@25134
   825
apply (simp)
nipkow@29667
   826
apply (simp add: algebra_simps)
nipkow@25134
   827
apply (subst real_of_nat_div_aux)
nipkow@25134
   828
 apply simp
nipkow@25134
   829
apply simp
avigad@16819
   830
done
avigad@16819
   831
avigad@16819
   832
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
nipkow@29667
   833
by (insert real_of_nat_div2 [of n x], simp)
avigad@16819
   834
paulson@14365
   835
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   836
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   837
paulson@14426
   838
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426
   839
by (simp add: real_of_int_def real_of_nat_def)
paulson@14334
   840
avigad@16819
   841
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819
   842
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819
   843
  apply force
avigad@16819
   844
  apply (simp only: real_of_int_real_of_nat)
avigad@16819
   845
done
paulson@14387
   846
nipkow@28001
   847
nipkow@28001
   848
subsection{* Rationals *}
nipkow@28001
   849
nipkow@28091
   850
lemma Rats_real_nat[simp]: "real(n::nat) \<in> \<rat>"
nipkow@28091
   851
by (simp add: real_eq_of_nat)
nipkow@28091
   852
nipkow@28091
   853
nipkow@28001
   854
lemma Rats_eq_int_div_int:
nipkow@28091
   855
  "\<rat> = { real(i::int)/real(j::int) |i j. j \<noteq> 0}" (is "_ = ?S")
nipkow@28001
   856
proof
nipkow@28091
   857
  show "\<rat> \<subseteq> ?S"
nipkow@28001
   858
  proof
nipkow@28091
   859
    fix x::real assume "x : \<rat>"
nipkow@28001
   860
    then obtain r where "x = of_rat r" unfolding Rats_def ..
nipkow@28001
   861
    have "of_rat r : ?S"
nipkow@28001
   862
      by (cases r)(auto simp add:of_rat_rat real_eq_of_int)
nipkow@28001
   863
    thus "x : ?S" using `x = of_rat r` by simp
nipkow@28001
   864
  qed
nipkow@28001
   865
next
nipkow@28091
   866
  show "?S \<subseteq> \<rat>"
nipkow@28001
   867
  proof(auto simp:Rats_def)
nipkow@28001
   868
    fix i j :: int assume "j \<noteq> 0"
nipkow@28001
   869
    hence "real i / real j = of_rat(Fract i j)"
nipkow@28001
   870
      by (simp add:of_rat_rat real_eq_of_int)
nipkow@28001
   871
    thus "real i / real j \<in> range of_rat" by blast
nipkow@28001
   872
  qed
nipkow@28001
   873
qed
nipkow@28001
   874
nipkow@28001
   875
lemma Rats_eq_int_div_nat:
nipkow@28091
   876
  "\<rat> = { real(i::int)/real(n::nat) |i n. n \<noteq> 0}"
nipkow@28001
   877
proof(auto simp:Rats_eq_int_div_int)
nipkow@28001
   878
  fix i j::int assume "j \<noteq> 0"
nipkow@28001
   879
  show "EX (i'::int) (n::nat). real i/real j = real i'/real n \<and> 0<n"
nipkow@28001
   880
  proof cases
nipkow@28001
   881
    assume "j>0"
nipkow@28001
   882
    hence "real i/real j = real i/real(nat j) \<and> 0<nat j"
nipkow@28001
   883
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
nipkow@28001
   884
    thus ?thesis by blast
nipkow@28001
   885
  next
nipkow@28001
   886
    assume "~ j>0"
nipkow@28001
   887
    hence "real i/real j = real(-i)/real(nat(-j)) \<and> 0<nat(-j)" using `j\<noteq>0`
nipkow@28001
   888
      by (simp add: real_eq_of_int real_eq_of_nat of_nat_nat)
nipkow@28001
   889
    thus ?thesis by blast
nipkow@28001
   890
  qed
nipkow@28001
   891
next
nipkow@28001
   892
  fix i::int and n::nat assume "0 < n"
nipkow@28001
   893
  hence "real i/real n = real i/real(int n) \<and> int n \<noteq> 0" by simp
nipkow@28001
   894
  thus "\<exists>(i'::int) j::int. real i/real n = real i'/real j \<and> j \<noteq> 0" by blast
nipkow@28001
   895
qed
nipkow@28001
   896
nipkow@28001
   897
lemma Rats_abs_nat_div_natE:
nipkow@28001
   898
  assumes "x \<in> \<rat>"
nipkow@28001
   899
  obtains m n where "n \<noteq> 0" and "\<bar>x\<bar> = real m / real n" and "gcd m n = 1"
nipkow@28001
   900
proof -
nipkow@28001
   901
  from `x \<in> \<rat>` obtain i::int and n::nat where "n \<noteq> 0" and "x = real i / real n"
nipkow@28001
   902
    by(auto simp add: Rats_eq_int_div_nat)
nipkow@28001
   903
  hence "\<bar>x\<bar> = real(nat(abs i)) / real n" by simp
nipkow@28001
   904
  then obtain m :: nat where x_rat: "\<bar>x\<bar> = real m / real n" by blast
nipkow@28001
   905
  let ?gcd = "gcd m n"
nipkow@28001
   906
  from `n\<noteq>0` have gcd: "?gcd \<noteq> 0" by (simp add: gcd_zero)
nipkow@28001
   907
  let ?k = "m div ?gcd"
nipkow@28001
   908
  let ?l = "n div ?gcd"
nipkow@28001
   909
  let ?gcd' = "gcd ?k ?l"
nipkow@28001
   910
  have "?gcd dvd m" .. then have gcd_k: "?gcd * ?k = m"
nipkow@28001
   911
    by (rule dvd_mult_div_cancel)
nipkow@28001
   912
  have "?gcd dvd n" .. then have gcd_l: "?gcd * ?l = n"
nipkow@28001
   913
    by (rule dvd_mult_div_cancel)
nipkow@28001
   914
  from `n\<noteq>0` and gcd_l have "?l \<noteq> 0" by (auto iff del: neq0_conv)
nipkow@28001
   915
  moreover
nipkow@28001
   916
  have "\<bar>x\<bar> = real ?k / real ?l"
nipkow@28001
   917
  proof -
nipkow@28001
   918
    from gcd have "real ?k / real ?l =
nipkow@28001
   919
        real (?gcd * ?k) / real (?gcd * ?l)" by simp
nipkow@28001
   920
    also from gcd_k and gcd_l have "\<dots> = real m / real n" by simp
nipkow@28001
   921
    also from x_rat have "\<dots> = \<bar>x\<bar>" ..
nipkow@28001
   922
    finally show ?thesis ..
nipkow@28001
   923
  qed
nipkow@28001
   924
  moreover
nipkow@28001
   925
  have "?gcd' = 1"
nipkow@28001
   926
  proof -
nipkow@28001
   927
    have "?gcd * ?gcd' = gcd (?gcd * ?k) (?gcd * ?l)"
nipkow@28001
   928
      by (rule gcd_mult_distrib2)
nipkow@28001
   929
    with gcd_k gcd_l have "?gcd * ?gcd' = ?gcd" by simp
nipkow@28001
   930
    with gcd show ?thesis by simp
nipkow@28001
   931
  qed
nipkow@28001
   932
  ultimately show ?thesis ..
nipkow@28001
   933
qed
nipkow@28001
   934
nipkow@28001
   935
paulson@14387
   936
subsection{*Numerals and Arithmetic*}
paulson@14387
   937
haftmann@25571
   938
instantiation real :: number_ring
haftmann@25571
   939
begin
haftmann@25571
   940
haftmann@25571
   941
definition
haftmann@28562
   942
  real_number_of_def [code del]: "number_of w = real_of_int w"
haftmann@25571
   943
haftmann@25571
   944
instance
haftmann@24198
   945
  by intro_classes (simp add: real_number_of_def)
paulson@14387
   946
haftmann@25571
   947
end
haftmann@25571
   948
haftmann@25965
   949
lemma [code unfold, symmetric, code post]:
haftmann@24198
   950
  "number_of k = real_of_int (number_of k)"
haftmann@24198
   951
  unfolding number_of_is_id real_number_of_def ..
paulson@14387
   952
paulson@14387
   953
paulson@14387
   954
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   955
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   956
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   957
paulson@14387
   958
lemma real_of_nat_number_of [simp]:
paulson@14387
   959
     "real (number_of v :: nat) =  
paulson@14387
   960
        (if neg (number_of v :: int) then 0  
paulson@14387
   961
         else (number_of v :: real))"
paulson@14387
   962
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   963
 
paulson@14387
   964
haftmann@28952
   965
use "Tools/real_arith.ML"
wenzelm@24075
   966
declaration {* K real_arith_setup *}
paulson@14387
   967
kleing@19023
   968
paulson@14387
   969
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   970
paulson@14387
   971
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   972
lemma real_0_le_divide_iff:
paulson@14387
   973
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   974
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   975
paulson@14387
   976
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   977
by arith
paulson@14387
   978
paulson@15085
   979
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   980
by auto
paulson@14387
   981
paulson@15085
   982
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   983
by auto
paulson@14387
   984
paulson@15085
   985
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387
   986
by auto
paulson@14387
   987
paulson@15085
   988
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   989
by auto
paulson@14387
   990
paulson@15085
   991
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   992
by auto
paulson@14387
   993
paulson@14387
   994
paulson@14387
   995
(*
paulson@14387
   996
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   997
It replaces x+-y by x-y.
paulson@15086
   998
declare real_diff_def [symmetric, simp]
paulson@14387
   999
*)
paulson@14387
  1000
paulson@14387
  1001
subsubsection{*Density of the Reals*}
paulson@14387
  1002
paulson@14387
  1003
lemma real_lbound_gt_zero:
paulson@14387
  1004
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
  1005
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
  1006
apply (simp add: min_def)
paulson@14387
  1007
done
paulson@14387
  1008
paulson@14387
  1009
paulson@14387
  1010
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
  1011
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
  1012
  by auto
paulson@14387
  1013
paulson@14387
  1014
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
  1015
  by auto
paulson@14387
  1016
paulson@14387
  1017
paulson@14387
  1018
subsection{*Absolute Value Function for the Reals*}
paulson@14387
  1019
paulson@14387
  1020
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003
  1021
by (simp add: abs_if)
paulson@14387
  1022
huffman@23289
  1023
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
paulson@14387
  1024
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
obua@14738
  1025
by (force simp add: OrderedGroup.abs_le_iff)
paulson@14387
  1026
paulson@14387
  1027
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@15003
  1028
by (simp add: abs_if)
paulson@14387
  1029
paulson@14387
  1030
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
huffman@22958
  1031
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
paulson@14387
  1032
paulson@14387
  1033
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
webertj@20217
  1034
by simp
paulson@14387
  1035
 
paulson@14387
  1036
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217
  1037
by simp
paulson@14387
  1038
haftmann@26732
  1039
instance real :: lordered_ring
haftmann@26732
  1040
proof
haftmann@26732
  1041
  fix a::real
haftmann@26732
  1042
  show "abs a = sup a (-a)"
haftmann@26732
  1043
    by (auto simp add: real_abs_def sup_real_def)
haftmann@26732
  1044
qed
haftmann@26732
  1045
berghofe@24534
  1046
haftmann@27544
  1047
subsection {* Implementation of rational real numbers *}
berghofe@24534
  1048
haftmann@27544
  1049
definition Ratreal :: "rat \<Rightarrow> real" where
haftmann@27544
  1050
  [simp]: "Ratreal = of_rat"
berghofe@24534
  1051
haftmann@24623
  1052
code_datatype Ratreal
berghofe@24534
  1053
haftmann@27544
  1054
lemma Ratreal_number_collapse [code post]:
haftmann@27544
  1055
  "Ratreal 0 = 0"
haftmann@27544
  1056
  "Ratreal 1 = 1"
haftmann@27544
  1057
  "Ratreal (number_of k) = number_of k"
haftmann@27544
  1058
by simp_all
berghofe@24534
  1059
berghofe@24534
  1060
lemma zero_real_code [code, code unfold]:
haftmann@27544
  1061
  "0 = Ratreal 0"
haftmann@27544
  1062
by simp
berghofe@24534
  1063
berghofe@24534
  1064
lemma one_real_code [code, code unfold]:
haftmann@27544
  1065
  "1 = Ratreal 1"
haftmann@27544
  1066
by simp
haftmann@27544
  1067
haftmann@27544
  1068
lemma number_of_real_code [code unfold]:
haftmann@27544
  1069
  "number_of k = Ratreal (number_of k)"
haftmann@27544
  1070
by simp
haftmann@27544
  1071
haftmann@27544
  1072
lemma Ratreal_number_of_quotient [code post]:
haftmann@27544
  1073
  "Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s"
haftmann@27544
  1074
by simp
haftmann@27544
  1075
haftmann@27544
  1076
lemma Ratreal_number_of_quotient2 [code post]:
haftmann@27544
  1077
  "Ratreal (number_of r / number_of s) = number_of r / number_of s"
haftmann@27544
  1078
unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide ..
berghofe@24534
  1079
haftmann@26513
  1080
instantiation real :: eq
haftmann@26513
  1081
begin
haftmann@26513
  1082
haftmann@27544
  1083
definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
haftmann@26513
  1084
haftmann@26513
  1085
instance by default (simp add: eq_real_def)
berghofe@24534
  1086
haftmann@27544
  1087
lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y"
haftmann@27544
  1088
  by (simp add: eq_real_def eq)
haftmann@26513
  1089
haftmann@28351
  1090
lemma real_eq_refl [code nbe]:
haftmann@28351
  1091
  "eq_class.eq (x::real) x \<longleftrightarrow> True"
haftmann@28351
  1092
  by (rule HOL.eq_refl)
haftmann@28351
  1093
haftmann@26513
  1094
end
berghofe@24534
  1095
haftmann@27544
  1096
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
haftmann@27652
  1097
  by (simp add: of_rat_less_eq)
berghofe@24534
  1098
haftmann@27544
  1099
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
haftmann@27652
  1100
  by (simp add: of_rat_less)
berghofe@24534
  1101
haftmann@27544
  1102
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
haftmann@27544
  1103
  by (simp add: of_rat_add)
berghofe@24534
  1104
haftmann@27544
  1105
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
haftmann@27544
  1106
  by (simp add: of_rat_mult)
haftmann@27544
  1107
haftmann@27544
  1108
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
haftmann@27544
  1109
  by (simp add: of_rat_minus)
berghofe@24534
  1110
haftmann@27544
  1111
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
haftmann@27544
  1112
  by (simp add: of_rat_diff)
berghofe@24534
  1113
haftmann@27544
  1114
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
haftmann@27544
  1115
  by (simp add: of_rat_inverse)
haftmann@27544
  1116
 
haftmann@27544
  1117
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
haftmann@27544
  1118
  by (simp add: of_rat_divide)
berghofe@24534
  1119
haftmann@24623
  1120
text {* Setup for SML code generator *}
nipkow@23031
  1121
nipkow@23031
  1122
types_code
berghofe@24534
  1123
  real ("(int */ int)")
nipkow@23031
  1124
attach (term_of) {*
berghofe@24534
  1125
fun term_of_real (p, q) =
haftmann@24623
  1126
  let
haftmann@24623
  1127
    val rT = HOLogic.realT
berghofe@24534
  1128
  in
berghofe@24534
  1129
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
haftmann@24623
  1130
    else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
berghofe@24534
  1131
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24534
  1132
  end;
nipkow@23031
  1133
*}
nipkow@23031
  1134
attach (test) {*
nipkow@23031
  1135
fun gen_real i =
berghofe@24534
  1136
  let
berghofe@24534
  1137
    val p = random_range 0 i;
berghofe@24534
  1138
    val q = random_range 1 (i + 1);
berghofe@24534
  1139
    val g = Integer.gcd p q;
wenzelm@24630
  1140
    val p' = p div g;
wenzelm@24630
  1141
    val q' = q div g;
berghofe@25885
  1142
    val r = (if one_of [true, false] then p' else ~ p',
berghofe@25885
  1143
      if p' = 0 then 0 else q')
berghofe@24534
  1144
  in
berghofe@25885
  1145
    (r, fn () => term_of_real r)
berghofe@24534
  1146
  end;
nipkow@23031
  1147
*}
nipkow@23031
  1148
nipkow@23031
  1149
consts_code
haftmann@24623
  1150
  Ratreal ("(_)")
berghofe@24534
  1151
berghofe@24534
  1152
consts_code
berghofe@24534
  1153
  "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
berghofe@24534
  1154
attach {*
berghofe@24534
  1155
fun real_of_int 0 = (0, 0)
berghofe@24534
  1156
  | real_of_int i = (i, 1);
berghofe@24534
  1157
*}
berghofe@24534
  1158
paulson@5588
  1159
end