src/HOL/Real/RealDef.thy
author haftmann
Fri Jul 18 18:25:56 2008 +0200 (2008-07-18)
changeset 27652 818666de6c24
parent 27544 5b293a8cf476
child 27668 6eb20b2cecf8
permissions -rw-r--r--
refined code generator setup for rational numbers; more simplification rules for rational numbers
paulson@5588
     1
(*  Title       : Real/RealDef.thy
paulson@7219
     2
    ID          : $Id$
paulson@5588
     3
    Author      : Jacques D. Fleuriot
paulson@5588
     4
    Copyright   : 1998  University of Cambridge
paulson@14387
     5
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
avigad@16819
     6
    Additional contributions by Jeremy Avigad
paulson@14269
     7
*)
paulson@14269
     8
paulson@14387
     9
header{*Defining the Reals from the Positive Reals*}
paulson@14387
    10
nipkow@15131
    11
theory RealDef
nipkow@15140
    12
imports PReal
haftmann@16417
    13
uses ("real_arith.ML")
nipkow@15131
    14
begin
paulson@5588
    15
wenzelm@19765
    16
definition
wenzelm@21404
    17
  realrel   ::  "((preal * preal) * (preal * preal)) set" where
haftmann@27106
    18
  [code func del]: "realrel = {p. \<exists>x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}"
paulson@14269
    19
paulson@14484
    20
typedef (Real)  real = "UNIV//realrel"
paulson@14269
    21
  by (auto simp add: quotient_def)
paulson@5588
    22
wenzelm@19765
    23
definition
paulson@14484
    24
  (** these don't use the overloaded "real" function: users don't see them **)
wenzelm@21404
    25
  real_of_preal :: "preal => real" where
huffman@23288
    26
  "real_of_preal m = Abs_Real(realrel``{(m + 1, 1)})"
paulson@14484
    27
haftmann@25762
    28
instantiation real :: "{zero, one, plus, minus, uminus, times, inverse, ord, abs, sgn}"
haftmann@25571
    29
begin
paulson@5588
    30
haftmann@25571
    31
definition
haftmann@25571
    32
  real_zero_def [code func del]: "0 = Abs_Real(realrel``{(1, 1)})"
haftmann@25571
    33
haftmann@25571
    34
definition
haftmann@25571
    35
  real_one_def [code func del]: "1 = Abs_Real(realrel``{(1 + 1, 1)})"
paulson@5588
    36
haftmann@25571
    37
definition
haftmann@25571
    38
  real_add_def [code func del]: "z + w =
paulson@14484
    39
       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
haftmann@25571
    40
		 { Abs_Real(realrel``{(x+u, y+v)}) })"
bauerg@10606
    41
haftmann@25571
    42
definition
haftmann@25571
    43
  real_minus_def [code func del]: "- r =  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
haftmann@25571
    44
haftmann@25571
    45
definition
haftmann@25571
    46
  real_diff_def [code func del]: "r - (s::real) = r + - s"
paulson@14484
    47
haftmann@25571
    48
definition
haftmann@25571
    49
  real_mult_def [code func del]:
haftmann@25571
    50
    "z * w =
paulson@14484
    51
       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
haftmann@25571
    52
		 { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })"
paulson@5588
    53
haftmann@25571
    54
definition
haftmann@25571
    55
  real_inverse_def [code func del]: "inverse (R::real) = (THE S. (R = 0 & S = 0) | S * R = 1)"
haftmann@25571
    56
haftmann@25571
    57
definition
haftmann@25571
    58
  real_divide_def [code func del]: "R / (S::real) = R * inverse S"
paulson@14269
    59
haftmann@25571
    60
definition
haftmann@25571
    61
  real_le_def [code func del]: "z \<le> (w::real) \<longleftrightarrow>
haftmann@25571
    62
    (\<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w)"
haftmann@25571
    63
haftmann@25571
    64
definition
haftmann@25571
    65
  real_less_def [code func del]: "x < (y\<Colon>real) \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
paulson@5588
    66
haftmann@25571
    67
definition
haftmann@25571
    68
  real_abs_def:  "abs (r::real) = (if r < 0 then - r else r)"
paulson@14334
    69
haftmann@25571
    70
definition
haftmann@25571
    71
  real_sgn_def: "sgn (x::real) = (if x=0 then 0 else if 0<x then 1 else - 1)"
haftmann@25571
    72
haftmann@25571
    73
instance ..
haftmann@25571
    74
haftmann@25571
    75
end
paulson@14334
    76
huffman@23287
    77
subsection {* Equivalence relation over positive reals *}
paulson@14269
    78
paulson@14270
    79
lemma preal_trans_lemma:
paulson@14365
    80
  assumes "x + y1 = x1 + y"
paulson@14365
    81
      and "x + y2 = x2 + y"
paulson@14365
    82
  shows "x1 + y2 = x2 + (y1::preal)"
paulson@14365
    83
proof -
huffman@23287
    84
  have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
paulson@14365
    85
  also have "... = (x2 + y) + x1"  by (simp add: prems)
huffman@23287
    86
  also have "... = x2 + (x1 + y)"  by (simp add: add_ac)
paulson@14365
    87
  also have "... = x2 + (x + y1)"  by (simp add: prems)
huffman@23287
    88
  also have "... = (x2 + y1) + x"  by (simp add: add_ac)
paulson@14365
    89
  finally have "(x1 + y2) + x = (x2 + y1) + x" .
huffman@23287
    90
  thus ?thesis by (rule add_right_imp_eq)
paulson@14365
    91
qed
paulson@14365
    92
paulson@14269
    93
paulson@14484
    94
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
paulson@14484
    95
by (simp add: realrel_def)
paulson@14269
    96
paulson@14269
    97
lemma equiv_realrel: "equiv UNIV realrel"
paulson@14365
    98
apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def)
paulson@14365
    99
apply (blast dest: preal_trans_lemma) 
paulson@14269
   100
done
paulson@14269
   101
paulson@14497
   102
text{*Reduces equality of equivalence classes to the @{term realrel} relation:
paulson@14497
   103
  @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
paulson@14269
   104
lemmas equiv_realrel_iff = 
paulson@14269
   105
       eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
paulson@14269
   106
paulson@14269
   107
declare equiv_realrel_iff [simp]
paulson@14269
   108
paulson@14497
   109
paulson@14484
   110
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
paulson@14484
   111
by (simp add: Real_def realrel_def quotient_def, blast)
paulson@14269
   112
huffman@22958
   113
declare Abs_Real_inject [simp]
paulson@14484
   114
declare Abs_Real_inverse [simp]
paulson@14269
   115
paulson@14269
   116
paulson@14484
   117
text{*Case analysis on the representation of a real number as an equivalence
paulson@14484
   118
      class of pairs of positive reals.*}
paulson@14484
   119
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 
paulson@14484
   120
     "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
paulson@14484
   121
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
paulson@14484
   122
apply (drule arg_cong [where f=Abs_Real])
paulson@14484
   123
apply (auto simp add: Rep_Real_inverse)
paulson@14269
   124
done
paulson@14269
   125
paulson@14269
   126
huffman@23287
   127
subsection {* Addition and Subtraction *}
paulson@14269
   128
paulson@14269
   129
lemma real_add_congruent2_lemma:
paulson@14269
   130
     "[|a + ba = aa + b; ab + bc = ac + bb|]
paulson@14269
   131
      ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
huffman@23287
   132
apply (simp add: add_assoc)
huffman@23287
   133
apply (rule add_left_commute [of ab, THEN ssubst])
huffman@23287
   134
apply (simp add: add_assoc [symmetric])
huffman@23287
   135
apply (simp add: add_ac)
paulson@14269
   136
done
paulson@14269
   137
paulson@14269
   138
lemma real_add:
paulson@14497
   139
     "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
paulson@14497
   140
      Abs_Real (realrel``{(x+u, y+v)})"
paulson@14497
   141
proof -
paulson@15169
   142
  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
paulson@15169
   143
        respects2 realrel"
paulson@14497
   144
    by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
paulson@14497
   145
  thus ?thesis
paulson@14497
   146
    by (simp add: real_add_def UN_UN_split_split_eq
paulson@14658
   147
                  UN_equiv_class2 [OF equiv_realrel equiv_realrel])
paulson@14497
   148
qed
paulson@14269
   149
paulson@14484
   150
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
paulson@14484
   151
proof -
paulson@15169
   152
  have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
huffman@23288
   153
    by (simp add: congruent_def add_commute) 
paulson@14484
   154
  thus ?thesis
paulson@14484
   155
    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
paulson@14484
   156
qed
paulson@14334
   157
huffman@23287
   158
instance real :: ab_group_add
huffman@23287
   159
proof
huffman@23287
   160
  fix x y z :: real
huffman@23287
   161
  show "(x + y) + z = x + (y + z)"
huffman@23287
   162
    by (cases x, cases y, cases z, simp add: real_add add_assoc)
huffman@23287
   163
  show "x + y = y + x"
huffman@23287
   164
    by (cases x, cases y, simp add: real_add add_commute)
huffman@23287
   165
  show "0 + x = x"
huffman@23287
   166
    by (cases x, simp add: real_add real_zero_def add_ac)
huffman@23287
   167
  show "- x + x = 0"
huffman@23287
   168
    by (cases x, simp add: real_minus real_add real_zero_def add_commute)
huffman@23287
   169
  show "x - y = x + - y"
huffman@23287
   170
    by (simp add: real_diff_def)
huffman@23287
   171
qed
paulson@14269
   172
paulson@14269
   173
huffman@23287
   174
subsection {* Multiplication *}
paulson@14269
   175
paulson@14329
   176
lemma real_mult_congruent2_lemma:
paulson@14329
   177
     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
paulson@14484
   178
          x * x1 + y * y1 + (x * y2 + y * x2) =
paulson@14484
   179
          x * x2 + y * y2 + (x * y1 + y * x1)"
huffman@23287
   180
apply (simp add: add_left_commute add_assoc [symmetric])
huffman@23288
   181
apply (simp add: add_assoc right_distrib [symmetric])
huffman@23288
   182
apply (simp add: add_commute)
paulson@14269
   183
done
paulson@14269
   184
paulson@14269
   185
lemma real_mult_congruent2:
paulson@15169
   186
    "(%p1 p2.
paulson@14484
   187
        (%(x1,y1). (%(x2,y2). 
paulson@15169
   188
          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
paulson@15169
   189
     respects2 realrel"
paulson@14658
   190
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
huffman@23288
   191
apply (simp add: mult_commute add_commute)
paulson@14269
   192
apply (auto simp add: real_mult_congruent2_lemma)
paulson@14269
   193
done
paulson@14269
   194
paulson@14269
   195
lemma real_mult:
paulson@14484
   196
      "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
paulson@14484
   197
       Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
paulson@14484
   198
by (simp add: real_mult_def UN_UN_split_split_eq
paulson@14658
   199
         UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
paulson@14269
   200
paulson@14269
   201
lemma real_mult_commute: "(z::real) * w = w * z"
huffman@23288
   202
by (cases z, cases w, simp add: real_mult add_ac mult_ac)
paulson@14269
   203
paulson@14269
   204
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
paulson@14484
   205
apply (cases z1, cases z2, cases z3)
huffman@23288
   206
apply (simp add: real_mult right_distrib add_ac mult_ac)
paulson@14269
   207
done
paulson@14269
   208
paulson@14269
   209
lemma real_mult_1: "(1::real) * z = z"
paulson@14484
   210
apply (cases z)
huffman@23288
   211
apply (simp add: real_mult real_one_def right_distrib
huffman@23288
   212
                  mult_1_right mult_ac add_ac)
paulson@14269
   213
done
paulson@14269
   214
paulson@14269
   215
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14484
   216
apply (cases z1, cases z2, cases w)
huffman@23288
   217
apply (simp add: real_add real_mult right_distrib add_ac mult_ac)
paulson@14269
   218
done
paulson@14269
   219
paulson@14329
   220
text{*one and zero are distinct*}
paulson@14365
   221
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
paulson@14484
   222
proof -
huffman@23287
   223
  have "(1::preal) < 1 + 1"
huffman@23287
   224
    by (simp add: preal_self_less_add_left)
paulson@14484
   225
  thus ?thesis
huffman@23288
   226
    by (simp add: real_zero_def real_one_def)
paulson@14484
   227
qed
paulson@14269
   228
huffman@23287
   229
instance real :: comm_ring_1
huffman@23287
   230
proof
huffman@23287
   231
  fix x y z :: real
huffman@23287
   232
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
huffman@23287
   233
  show "x * y = y * x" by (rule real_mult_commute)
huffman@23287
   234
  show "1 * x = x" by (rule real_mult_1)
huffman@23287
   235
  show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
huffman@23287
   236
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
huffman@23287
   237
qed
huffman@23287
   238
huffman@23287
   239
subsection {* Inverse and Division *}
paulson@14365
   240
paulson@14484
   241
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
huffman@23288
   242
by (simp add: real_zero_def add_commute)
paulson@14269
   243
paulson@14365
   244
text{*Instead of using an existential quantifier and constructing the inverse
paulson@14365
   245
within the proof, we could define the inverse explicitly.*}
paulson@14365
   246
paulson@14365
   247
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
paulson@14484
   248
apply (simp add: real_zero_def real_one_def, cases x)
paulson@14269
   249
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@14365
   250
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
paulson@14334
   251
apply (rule_tac
huffman@23287
   252
        x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
paulson@14334
   253
       in exI)
paulson@14334
   254
apply (rule_tac [2]
huffman@23287
   255
        x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 
paulson@14334
   256
       in exI)
nipkow@23477
   257
apply (auto simp add: real_mult preal_mult_inverse_right ring_simps)
paulson@14269
   258
done
paulson@14269
   259
paulson@14365
   260
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
paulson@14484
   261
apply (simp add: real_inverse_def)
huffman@23287
   262
apply (drule real_mult_inverse_left_ex, safe)
huffman@23287
   263
apply (rule theI, assumption, rename_tac z)
huffman@23287
   264
apply (subgoal_tac "(z * x) * y = z * (x * y)")
huffman@23287
   265
apply (simp add: mult_commute)
huffman@23287
   266
apply (rule mult_assoc)
paulson@14269
   267
done
paulson@14334
   268
paulson@14341
   269
paulson@14341
   270
subsection{*The Real Numbers form a Field*}
paulson@14341
   271
paulson@14334
   272
instance real :: field
paulson@14334
   273
proof
paulson@14334
   274
  fix x y z :: real
paulson@14365
   275
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
paulson@14430
   276
  show "x / y = x * inverse y" by (simp add: real_divide_def)
paulson@14334
   277
qed
paulson@14334
   278
paulson@14334
   279
paulson@14341
   280
text{*Inverse of zero!  Useful to simplify certain equations*}
paulson@14269
   281
paulson@14334
   282
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
paulson@14484
   283
by (simp add: real_inverse_def)
paulson@14334
   284
paulson@14334
   285
instance real :: division_by_zero
paulson@14334
   286
proof
paulson@14334
   287
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
paulson@14334
   288
qed
paulson@14334
   289
paulson@14269
   290
paulson@14365
   291
subsection{*The @{text "\<le>"} Ordering*}
paulson@14269
   292
paulson@14365
   293
lemma real_le_refl: "w \<le> (w::real)"
paulson@14484
   294
by (cases w, force simp add: real_le_def)
paulson@14269
   295
paulson@14378
   296
text{*The arithmetic decision procedure is not set up for type preal.
paulson@14378
   297
  This lemma is currently unused, but it could simplify the proofs of the
paulson@14378
   298
  following two lemmas.*}
paulson@14378
   299
lemma preal_eq_le_imp_le:
paulson@14378
   300
  assumes eq: "a+b = c+d" and le: "c \<le> a"
paulson@14378
   301
  shows "b \<le> (d::preal)"
paulson@14378
   302
proof -
huffman@23288
   303
  have "c+d \<le> a+d" by (simp add: prems)
paulson@14378
   304
  hence "a+b \<le> a+d" by (simp add: prems)
huffman@23288
   305
  thus "b \<le> d" by simp
paulson@14378
   306
qed
paulson@14378
   307
paulson@14378
   308
lemma real_le_lemma:
paulson@14378
   309
  assumes l: "u1 + v2 \<le> u2 + v1"
paulson@14378
   310
      and "x1 + v1 = u1 + y1"
paulson@14378
   311
      and "x2 + v2 = u2 + y2"
paulson@14378
   312
  shows "x1 + y2 \<le> x2 + (y1::preal)"
paulson@14365
   313
proof -
paulson@14378
   314
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
huffman@23288
   315
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
huffman@23288
   316
  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
huffman@23288
   317
  finally show ?thesis by simp
huffman@23288
   318
qed
paulson@14378
   319
paulson@14378
   320
lemma real_le: 
paulson@14484
   321
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
paulson@14484
   322
      (x1 + y2 \<le> x2 + y1)"
huffman@23288
   323
apply (simp add: real_le_def)
paulson@14387
   324
apply (auto intro: real_le_lemma)
paulson@14378
   325
done
paulson@14378
   326
paulson@14378
   327
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
nipkow@15542
   328
by (cases z, cases w, simp add: real_le)
paulson@14378
   329
paulson@14378
   330
lemma real_trans_lemma:
paulson@14378
   331
  assumes "x + v \<le> u + y"
paulson@14378
   332
      and "u + v' \<le> u' + v"
paulson@14378
   333
      and "x2 + v2 = u2 + y2"
paulson@14378
   334
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   335
proof -
huffman@23288
   336
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
huffman@23288
   337
  also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
huffman@23288
   338
  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
huffman@23288
   339
  also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
huffman@23288
   340
  finally show ?thesis by simp
nipkow@15542
   341
qed
paulson@14269
   342
paulson@14365
   343
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14484
   344
apply (cases i, cases j, cases k)
paulson@14484
   345
apply (simp add: real_le)
huffman@23288
   346
apply (blast intro: real_trans_lemma)
paulson@14334
   347
done
paulson@14334
   348
paulson@14334
   349
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   350
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   351
by (simp add: real_less_def)
paulson@14365
   352
paulson@14365
   353
instance real :: order
paulson@14365
   354
proof qed
paulson@14365
   355
 (assumption |
paulson@14365
   356
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   357
paulson@14378
   358
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   359
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
huffman@23288
   360
apply (cases z, cases w)
huffman@23288
   361
apply (auto simp add: real_le real_zero_def add_ac)
paulson@14334
   362
done
paulson@14334
   363
paulson@14334
   364
paulson@14334
   365
instance real :: linorder
paulson@14334
   366
  by (intro_classes, rule real_le_linear)
paulson@14334
   367
paulson@14334
   368
paulson@14378
   369
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14484
   370
apply (cases x, cases y) 
paulson@14378
   371
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
huffman@23288
   372
                      add_ac)
huffman@23288
   373
apply (simp_all add: add_assoc [symmetric])
nipkow@15542
   374
done
paulson@14378
   375
paulson@14484
   376
lemma real_add_left_mono: 
paulson@14484
   377
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
paulson@14484
   378
proof -
paulson@14484
   379
  have "z + x - (z + y) = (z + -z) + (x - y)"
paulson@14484
   380
    by (simp add: diff_minus add_ac) 
paulson@14484
   381
  with le show ?thesis 
obua@14754
   382
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
paulson@14484
   383
qed
paulson@14334
   384
paulson@14365
   385
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   386
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   387
paulson@14365
   388
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   389
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   390
paulson@14334
   391
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14484
   392
apply (cases x, cases y)
paulson@14378
   393
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   394
                 linorder_not_le [where 'a = preal] 
paulson@14378
   395
                  real_zero_def real_le real_mult)
paulson@14365
   396
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
wenzelm@16973
   397
apply (auto dest!: less_add_left_Ex
huffman@23288
   398
     simp add: add_ac mult_ac
huffman@23288
   399
          right_distrib preal_self_less_add_left)
paulson@14334
   400
done
paulson@14334
   401
paulson@14334
   402
lemma real_mult_less_mono2: "[| (0::real) < z; x < y |] ==> z * x < z * y"
paulson@14334
   403
apply (rule real_sum_gt_zero_less)
paulson@14334
   404
apply (drule real_less_sum_gt_zero [of x y])
paulson@14334
   405
apply (drule real_mult_order, assumption)
paulson@14334
   406
apply (simp add: right_distrib)
paulson@14334
   407
done
paulson@14334
   408
haftmann@25571
   409
instantiation real :: distrib_lattice
haftmann@25571
   410
begin
haftmann@25571
   411
haftmann@25571
   412
definition
haftmann@25571
   413
  "(inf \<Colon> real \<Rightarrow> real \<Rightarrow> real) = min"
haftmann@25571
   414
haftmann@25571
   415
definition
haftmann@25571
   416
  "(sup \<Colon> real \<Rightarrow> real \<Rightarrow> real) = max"
haftmann@25571
   417
haftmann@25571
   418
instance
haftmann@22456
   419
  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
haftmann@22456
   420
haftmann@25571
   421
end
haftmann@25571
   422
paulson@14378
   423
paulson@14334
   424
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   425
paulson@14334
   426
instance real :: ordered_field
paulson@14334
   427
proof
paulson@14334
   428
  fix x y z :: real
paulson@14334
   429
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
huffman@22962
   430
  show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
huffman@22962
   431
  show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
nipkow@24506
   432
  show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
nipkow@24506
   433
    by (simp only: real_sgn_def)
paulson@14334
   434
qed
paulson@14334
   435
haftmann@25303
   436
instance real :: lordered_ab_group_add ..
haftmann@25303
   437
paulson@14365
   438
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   439
to be essential for proving completeness of the reals from that of the
paulson@14365
   440
positive reals.*}
paulson@14365
   441
paulson@14365
   442
lemma real_of_preal_add:
paulson@14365
   443
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
huffman@23288
   444
by (simp add: real_of_preal_def real_add left_distrib add_ac)
paulson@14365
   445
paulson@14365
   446
lemma real_of_preal_mult:
paulson@14365
   447
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
huffman@23288
   448
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
paulson@14365
   449
paulson@14365
   450
paulson@14365
   451
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   452
lemma real_of_preal_trichotomy:
paulson@14365
   453
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14484
   454
apply (simp add: real_of_preal_def real_zero_def, cases x)
huffman@23288
   455
apply (auto simp add: real_minus add_ac)
paulson@14365
   456
apply (cut_tac x = x and y = y in linorder_less_linear)
huffman@23288
   457
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
paulson@14365
   458
done
paulson@14365
   459
paulson@14365
   460
lemma real_of_preal_leD:
paulson@14365
   461
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
huffman@23288
   462
by (simp add: real_of_preal_def real_le)
paulson@14365
   463
paulson@14365
   464
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   465
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   466
paulson@14365
   467
lemma real_of_preal_lessD:
paulson@14365
   468
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
huffman@23288
   469
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
paulson@14365
   470
paulson@14365
   471
lemma real_of_preal_less_iff [simp]:
paulson@14365
   472
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   473
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   474
paulson@14365
   475
lemma real_of_preal_le_iff:
paulson@14365
   476
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
huffman@23288
   477
by (simp add: linorder_not_less [symmetric])
paulson@14365
   478
paulson@14365
   479
lemma real_of_preal_zero_less: "0 < real_of_preal m"
huffman@23288
   480
apply (insert preal_self_less_add_left [of 1 m])
huffman@23288
   481
apply (auto simp add: real_zero_def real_of_preal_def
huffman@23288
   482
                      real_less_def real_le_def add_ac)
huffman@23288
   483
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
huffman@23288
   484
apply (simp add: add_ac)
paulson@14365
   485
done
paulson@14365
   486
paulson@14365
   487
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   488
by (simp add: real_of_preal_zero_less)
paulson@14365
   489
paulson@14365
   490
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14484
   491
proof -
paulson@14484
   492
  from real_of_preal_minus_less_zero
paulson@14484
   493
  show ?thesis by (blast dest: order_less_trans)
paulson@14484
   494
qed
paulson@14365
   495
paulson@14365
   496
paulson@14365
   497
subsection{*Theorems About the Ordering*}
paulson@14365
   498
paulson@14365
   499
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   500
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   501
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   502
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   503
done
paulson@14365
   504
paulson@14365
   505
lemma real_gt_preal_preal_Ex:
paulson@14365
   506
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   507
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   508
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   509
paulson@14365
   510
lemma real_ge_preal_preal_Ex:
paulson@14365
   511
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   512
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   513
paulson@14365
   514
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   515
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   516
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   517
            simp add: real_of_preal_zero_less)
paulson@14365
   518
paulson@14365
   519
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   520
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   521
paulson@14334
   522
paulson@14334
   523
subsection{*More Lemmas*}
paulson@14334
   524
paulson@14334
   525
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   526
by auto
paulson@14334
   527
paulson@14334
   528
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   529
by auto
paulson@14334
   530
paulson@14334
   531
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   532
  by (force elim: order_less_asym
paulson@14334
   533
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   534
paulson@14334
   535
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   536
apply (simp add: mult_le_cancel_right)
huffman@23289
   537
apply (blast intro: elim: order_less_asym)
paulson@14365
   538
done
paulson@14334
   539
paulson@14334
   540
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
nipkow@15923
   541
by(simp add:mult_commute)
paulson@14334
   542
paulson@14365
   543
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
huffman@23289
   544
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
paulson@14334
   545
paulson@14334
   546
haftmann@24198
   547
subsection {* Embedding numbers into the Reals *}
haftmann@24198
   548
haftmann@24198
   549
abbreviation
haftmann@24198
   550
  real_of_nat :: "nat \<Rightarrow> real"
haftmann@24198
   551
where
haftmann@24198
   552
  "real_of_nat \<equiv> of_nat"
haftmann@24198
   553
haftmann@24198
   554
abbreviation
haftmann@24198
   555
  real_of_int :: "int \<Rightarrow> real"
haftmann@24198
   556
where
haftmann@24198
   557
  "real_of_int \<equiv> of_int"
haftmann@24198
   558
haftmann@24198
   559
abbreviation
haftmann@24198
   560
  real_of_rat :: "rat \<Rightarrow> real"
haftmann@24198
   561
where
haftmann@24198
   562
  "real_of_rat \<equiv> of_rat"
haftmann@24198
   563
haftmann@24198
   564
consts
haftmann@24198
   565
  (*overloaded constant for injecting other types into "real"*)
haftmann@24198
   566
  real :: "'a => real"
paulson@14365
   567
paulson@14378
   568
defs (overloaded)
berghofe@24534
   569
  real_of_nat_def [code inline]: "real == real_of_nat"
berghofe@24534
   570
  real_of_int_def [code inline]: "real == real_of_int"
paulson@14365
   571
avigad@16819
   572
lemma real_eq_of_nat: "real = of_nat"
haftmann@24198
   573
  unfolding real_of_nat_def ..
avigad@16819
   574
avigad@16819
   575
lemma real_eq_of_int: "real = of_int"
haftmann@24198
   576
  unfolding real_of_int_def ..
avigad@16819
   577
paulson@14365
   578
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   579
by (simp add: real_of_int_def) 
paulson@14365
   580
paulson@14365
   581
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   582
by (simp add: real_of_int_def) 
paulson@14334
   583
avigad@16819
   584
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378
   585
by (simp add: real_of_int_def) 
paulson@14365
   586
avigad@16819
   587
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378
   588
by (simp add: real_of_int_def) 
avigad@16819
   589
avigad@16819
   590
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819
   591
by (simp add: real_of_int_def) 
paulson@14365
   592
avigad@16819
   593
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378
   594
by (simp add: real_of_int_def) 
paulson@14334
   595
avigad@16819
   596
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819
   597
  apply (subst real_eq_of_int)+
avigad@16819
   598
  apply (rule of_int_setsum)
avigad@16819
   599
done
avigad@16819
   600
avigad@16819
   601
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819
   602
    (PROD x:A. real(f x))"
avigad@16819
   603
  apply (subst real_eq_of_int)+
avigad@16819
   604
  apply (rule of_int_setprod)
avigad@16819
   605
done
paulson@14365
   606
paulson@14365
   607
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   608
by (simp add: real_of_int_def) 
paulson@14365
   609
paulson@14365
   610
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   611
by (simp add: real_of_int_def) 
paulson@14365
   612
paulson@14365
   613
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   614
by (simp add: real_of_int_def) 
paulson@14365
   615
paulson@14365
   616
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   617
by (simp add: real_of_int_def) 
paulson@14365
   618
avigad@16819
   619
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
avigad@16819
   620
by (simp add: real_of_int_def) 
avigad@16819
   621
avigad@16819
   622
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819
   623
by (simp add: real_of_int_def) 
avigad@16819
   624
avigad@16819
   625
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
avigad@16819
   626
by (simp add: real_of_int_def)
avigad@16819
   627
avigad@16819
   628
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819
   629
by (simp add: real_of_int_def)
avigad@16819
   630
avigad@16888
   631
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888
   632
by (auto simp add: abs_if)
avigad@16888
   633
avigad@16819
   634
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819
   635
  apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819
   636
  apply (simp del: real_of_int_add)
avigad@16819
   637
  apply auto
avigad@16819
   638
done
avigad@16819
   639
avigad@16819
   640
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819
   641
  apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819
   642
  apply (simp del: real_of_int_add)
avigad@16819
   643
  apply simp
avigad@16819
   644
done
avigad@16819
   645
avigad@16819
   646
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
avigad@16819
   647
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   648
proof -
avigad@16819
   649
  assume "d ~= 0"
avigad@16819
   650
  have "x = (x div d) * d + x mod d"
avigad@16819
   651
    by auto
avigad@16819
   652
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   653
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819
   654
  then have "real x / real d = ... / real d"
avigad@16819
   655
    by simp
avigad@16819
   656
  then show ?thesis
nipkow@23477
   657
    by (auto simp add: add_divide_distrib ring_simps prems)
avigad@16819
   658
qed
avigad@16819
   659
avigad@16819
   660
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
avigad@16819
   661
    real(n div d) = real n / real d"
avigad@16819
   662
  apply (frule real_of_int_div_aux [of d n])
avigad@16819
   663
  apply simp
avigad@16819
   664
  apply (simp add: zdvd_iff_zmod_eq_0)
avigad@16819
   665
done
avigad@16819
   666
avigad@16819
   667
lemma real_of_int_div2:
avigad@16819
   668
  "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819
   669
  apply (case_tac "x = 0")
avigad@16819
   670
  apply simp
avigad@16819
   671
  apply (case_tac "0 < x")
avigad@16819
   672
  apply (simp add: compare_rls)
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
  apply (simp add: compare_rls)
avigad@16819
   679
  apply (subst real_of_int_div_aux)
avigad@16819
   680
  apply simp
avigad@16819
   681
  apply simp
avigad@16819
   682
  apply (subst zero_le_divide_iff)
avigad@16819
   683
  apply auto
avigad@16819
   684
done
avigad@16819
   685
avigad@16819
   686
lemma real_of_int_div3:
avigad@16819
   687
  "real (n::int) / real (x) - real (n div x) <= 1"
avigad@16819
   688
  apply(case_tac "x = 0")
avigad@16819
   689
  apply simp
avigad@16819
   690
  apply (simp add: compare_rls)
avigad@16819
   691
  apply (subst real_of_int_div_aux)
avigad@16819
   692
  apply assumption
avigad@16819
   693
  apply simp
avigad@16819
   694
  apply (subst divide_le_eq)
avigad@16819
   695
  apply clarsimp
avigad@16819
   696
  apply (rule conjI)
avigad@16819
   697
  apply (rule impI)
avigad@16819
   698
  apply (rule order_less_imp_le)
avigad@16819
   699
  apply simp
avigad@16819
   700
  apply (rule impI)
avigad@16819
   701
  apply (rule order_less_imp_le)
avigad@16819
   702
  apply simp
avigad@16819
   703
done
avigad@16819
   704
avigad@16819
   705
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
avigad@16819
   706
  by (insert real_of_int_div2 [of n x], simp)
paulson@14365
   707
paulson@14365
   708
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   709
paulson@14334
   710
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   711
by (simp add: real_of_nat_def)
paulson@14334
   712
paulson@14334
   713
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   714
by (simp add: real_of_nat_def)
paulson@14334
   715
paulson@14365
   716
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   717
by (simp add: real_of_nat_def)
paulson@14334
   718
paulson@14334
   719
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   720
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   721
by (simp add: real_of_nat_def)
paulson@14334
   722
paulson@14334
   723
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   724
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   725
by (simp add: real_of_nat_def)
paulson@14334
   726
paulson@14334
   727
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   728
by (simp add: real_of_nat_def)
paulson@14334
   729
paulson@14334
   730
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   731
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   732
paulson@14365
   733
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   734
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   735
paulson@14334
   736
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
huffman@23431
   737
by (simp add: real_of_nat_def of_nat_mult)
paulson@14334
   738
avigad@16819
   739
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819
   740
    (SUM x:A. real(f x))"
avigad@16819
   741
  apply (subst real_eq_of_nat)+
avigad@16819
   742
  apply (rule of_nat_setsum)
avigad@16819
   743
done
avigad@16819
   744
avigad@16819
   745
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819
   746
    (PROD x:A. real(f x))"
avigad@16819
   747
  apply (subst real_eq_of_nat)+
avigad@16819
   748
  apply (rule of_nat_setprod)
avigad@16819
   749
done
avigad@16819
   750
avigad@16819
   751
lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819
   752
  apply (subst card_eq_setsum)
avigad@16819
   753
  apply (subst real_of_nat_setsum)
avigad@16819
   754
  apply simp
avigad@16819
   755
done
avigad@16819
   756
paulson@14334
   757
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   758
by (simp add: real_of_nat_def)
paulson@14334
   759
paulson@14387
   760
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   761
by (simp add: real_of_nat_def)
paulson@14334
   762
paulson@14365
   763
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
huffman@23438
   764
by (simp add: add: real_of_nat_def of_nat_diff)
paulson@14334
   765
nipkow@25162
   766
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (0 < n)"
nipkow@25140
   767
by (auto simp: real_of_nat_def)
paulson@14365
   768
paulson@14365
   769
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   770
by (simp add: add: real_of_nat_def)
paulson@14334
   771
paulson@14365
   772
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   773
by (simp add: add: real_of_nat_def)
paulson@14334
   774
nipkow@25140
   775
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))"
paulson@14378
   776
by (simp add: add: real_of_nat_def)
paulson@14334
   777
avigad@16819
   778
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819
   779
  apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819
   780
  apply simp
avigad@16819
   781
  apply (auto simp add: real_of_nat_Suc)
avigad@16819
   782
done
avigad@16819
   783
avigad@16819
   784
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819
   785
  apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819
   786
  apply (simp add: less_Suc_eq_le)
avigad@16819
   787
  apply (simp add: real_of_nat_Suc)
avigad@16819
   788
done
avigad@16819
   789
avigad@16819
   790
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
avigad@16819
   791
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   792
proof -
avigad@16819
   793
  assume "0 < d"
avigad@16819
   794
  have "x = (x div d) * d + x mod d"
avigad@16819
   795
    by auto
avigad@16819
   796
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   797
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819
   798
  then have "real x / real d = \<dots> / real d"
avigad@16819
   799
    by simp
avigad@16819
   800
  then show ?thesis
nipkow@23477
   801
    by (auto simp add: add_divide_distrib ring_simps prems)
avigad@16819
   802
qed
avigad@16819
   803
avigad@16819
   804
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
avigad@16819
   805
    real(n div d) = real n / real d"
avigad@16819
   806
  apply (frule real_of_nat_div_aux [of d n])
avigad@16819
   807
  apply simp
avigad@16819
   808
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
avigad@16819
   809
  apply assumption
avigad@16819
   810
done
avigad@16819
   811
avigad@16819
   812
lemma real_of_nat_div2:
avigad@16819
   813
  "0 <= real (n::nat) / real (x) - real (n div x)"
nipkow@25134
   814
apply(case_tac "x = 0")
nipkow@25134
   815
 apply (simp)
nipkow@25134
   816
apply (simp add: compare_rls)
nipkow@25134
   817
apply (subst real_of_nat_div_aux)
nipkow@25134
   818
 apply simp
nipkow@25134
   819
apply simp
nipkow@25134
   820
apply (subst zero_le_divide_iff)
nipkow@25134
   821
apply simp
avigad@16819
   822
done
avigad@16819
   823
avigad@16819
   824
lemma real_of_nat_div3:
avigad@16819
   825
  "real (n::nat) / real (x) - real (n div x) <= 1"
nipkow@25134
   826
apply(case_tac "x = 0")
nipkow@25134
   827
apply (simp)
nipkow@25134
   828
apply (simp add: compare_rls)
nipkow@25134
   829
apply (subst real_of_nat_div_aux)
nipkow@25134
   830
 apply simp
nipkow@25134
   831
apply simp
avigad@16819
   832
done
avigad@16819
   833
avigad@16819
   834
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
avigad@16819
   835
  by (insert real_of_nat_div2 [of n x], simp)
avigad@16819
   836
paulson@14365
   837
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   838
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   839
paulson@14426
   840
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426
   841
by (simp add: real_of_int_def real_of_nat_def)
paulson@14334
   842
avigad@16819
   843
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819
   844
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819
   845
  apply force
avigad@16819
   846
  apply (simp only: real_of_int_real_of_nat)
avigad@16819
   847
done
paulson@14387
   848
paulson@14387
   849
subsection{*Numerals and Arithmetic*}
paulson@14387
   850
haftmann@25571
   851
instantiation real :: number_ring
haftmann@25571
   852
begin
haftmann@25571
   853
haftmann@25571
   854
definition
haftmann@25965
   855
  real_number_of_def [code func del]: "number_of w = real_of_int w"
haftmann@25571
   856
haftmann@25571
   857
instance
haftmann@24198
   858
  by intro_classes (simp add: real_number_of_def)
paulson@14387
   859
haftmann@25571
   860
end
haftmann@25571
   861
haftmann@25965
   862
lemma [code unfold, symmetric, code post]:
haftmann@24198
   863
  "number_of k = real_of_int (number_of k)"
haftmann@24198
   864
  unfolding number_of_is_id real_number_of_def ..
paulson@14387
   865
paulson@14387
   866
paulson@14387
   867
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   868
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   869
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   870
paulson@14387
   871
lemma real_of_nat_number_of [simp]:
paulson@14387
   872
     "real (number_of v :: nat) =  
paulson@14387
   873
        (if neg (number_of v :: int) then 0  
paulson@14387
   874
         else (number_of v :: real))"
paulson@14387
   875
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   876
 
paulson@14387
   877
paulson@14387
   878
use "real_arith.ML"
wenzelm@24075
   879
declaration {* K real_arith_setup *}
paulson@14387
   880
kleing@19023
   881
paulson@14387
   882
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   883
paulson@14387
   884
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   885
lemma real_0_le_divide_iff:
paulson@14387
   886
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   887
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   888
paulson@14387
   889
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   890
by arith
paulson@14387
   891
paulson@15085
   892
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   893
by auto
paulson@14387
   894
paulson@15085
   895
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   896
by auto
paulson@14387
   897
paulson@15085
   898
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387
   899
by auto
paulson@14387
   900
paulson@15085
   901
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   902
by auto
paulson@14387
   903
paulson@15085
   904
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   905
by auto
paulson@14387
   906
paulson@14387
   907
paulson@14387
   908
(*
paulson@14387
   909
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   910
It replaces x+-y by x-y.
paulson@15086
   911
declare real_diff_def [symmetric, simp]
paulson@14387
   912
*)
paulson@14387
   913
paulson@14387
   914
paulson@14387
   915
subsubsection{*Density of the Reals*}
paulson@14387
   916
paulson@14387
   917
lemma real_lbound_gt_zero:
paulson@14387
   918
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
   919
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
   920
apply (simp add: min_def)
paulson@14387
   921
done
paulson@14387
   922
paulson@14387
   923
paulson@14387
   924
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
   925
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
   926
  by auto
paulson@14387
   927
paulson@14387
   928
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
   929
  by auto
paulson@14387
   930
paulson@14387
   931
paulson@14387
   932
subsection{*Absolute Value Function for the Reals*}
paulson@14387
   933
paulson@14387
   934
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003
   935
by (simp add: abs_if)
paulson@14387
   936
huffman@23289
   937
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
paulson@14387
   938
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
obua@14738
   939
by (force simp add: OrderedGroup.abs_le_iff)
paulson@14387
   940
paulson@14387
   941
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@15003
   942
by (simp add: abs_if)
paulson@14387
   943
paulson@14387
   944
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
huffman@22958
   945
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
paulson@14387
   946
paulson@14387
   947
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
webertj@20217
   948
by simp
paulson@14387
   949
 
paulson@14387
   950
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217
   951
by simp
paulson@14387
   952
haftmann@26732
   953
instance real :: lordered_ring
haftmann@26732
   954
proof
haftmann@26732
   955
  fix a::real
haftmann@26732
   956
  show "abs a = sup a (-a)"
haftmann@26732
   957
    by (auto simp add: real_abs_def sup_real_def)
haftmann@26732
   958
qed
haftmann@26732
   959
berghofe@24534
   960
haftmann@27544
   961
subsection {* Implementation of rational real numbers *}
berghofe@24534
   962
haftmann@27544
   963
definition Ratreal :: "rat \<Rightarrow> real" where
haftmann@27544
   964
  [simp]: "Ratreal = of_rat"
berghofe@24534
   965
haftmann@24623
   966
code_datatype Ratreal
berghofe@24534
   967
haftmann@27544
   968
lemma Ratreal_number_collapse [code post]:
haftmann@27544
   969
  "Ratreal 0 = 0"
haftmann@27544
   970
  "Ratreal 1 = 1"
haftmann@27544
   971
  "Ratreal (number_of k) = number_of k"
haftmann@27544
   972
by simp_all
berghofe@24534
   973
berghofe@24534
   974
lemma zero_real_code [code, code unfold]:
haftmann@27544
   975
  "0 = Ratreal 0"
haftmann@27544
   976
by simp
berghofe@24534
   977
berghofe@24534
   978
lemma one_real_code [code, code unfold]:
haftmann@27544
   979
  "1 = Ratreal 1"
haftmann@27544
   980
by simp
haftmann@27544
   981
haftmann@27544
   982
lemma number_of_real_code [code unfold]:
haftmann@27544
   983
  "number_of k = Ratreal (number_of k)"
haftmann@27544
   984
by simp
haftmann@27544
   985
haftmann@27544
   986
lemma Ratreal_number_of_quotient [code post]:
haftmann@27544
   987
  "Ratreal (number_of r) / Ratreal (number_of s) = number_of r / number_of s"
haftmann@27544
   988
by simp
haftmann@27544
   989
haftmann@27544
   990
lemma Ratreal_number_of_quotient2 [code post]:
haftmann@27544
   991
  "Ratreal (number_of r / number_of s) = number_of r / number_of s"
haftmann@27544
   992
unfolding Ratreal_number_of_quotient [symmetric] Ratreal_def of_rat_divide ..
berghofe@24534
   993
haftmann@26513
   994
instantiation real :: eq
haftmann@26513
   995
begin
haftmann@26513
   996
haftmann@27544
   997
definition "eq_class.eq (x\<Colon>real) y \<longleftrightarrow> x - y = 0"
haftmann@26513
   998
haftmann@26513
   999
instance by default (simp add: eq_real_def)
berghofe@24534
  1000
haftmann@27544
  1001
lemma real_eq_code [code]: "eq_class.eq (Ratreal x) (Ratreal y) \<longleftrightarrow> eq_class.eq x y"
haftmann@27544
  1002
  by (simp add: eq_real_def eq)
haftmann@26513
  1003
haftmann@26513
  1004
end
berghofe@24534
  1005
haftmann@27544
  1006
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> x \<le> y"
haftmann@27652
  1007
  by (simp add: of_rat_less_eq)
berghofe@24534
  1008
haftmann@27544
  1009
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> x < y"
haftmann@27652
  1010
  by (simp add: of_rat_less)
berghofe@24534
  1011
haftmann@27544
  1012
lemma real_plus_code [code]: "Ratreal x + Ratreal y = Ratreal (x + y)"
haftmann@27544
  1013
  by (simp add: of_rat_add)
berghofe@24534
  1014
haftmann@27544
  1015
lemma real_times_code [code]: "Ratreal x * Ratreal y = Ratreal (x * y)"
haftmann@27544
  1016
  by (simp add: of_rat_mult)
haftmann@27544
  1017
haftmann@27544
  1018
lemma real_uminus_code [code]: "- Ratreal x = Ratreal (- x)"
haftmann@27544
  1019
  by (simp add: of_rat_minus)
berghofe@24534
  1020
haftmann@27544
  1021
lemma real_minus_code [code]: "Ratreal x - Ratreal y = Ratreal (x - y)"
haftmann@27544
  1022
  by (simp add: of_rat_diff)
berghofe@24534
  1023
haftmann@27544
  1024
lemma real_inverse_code [code]: "inverse (Ratreal x) = Ratreal (inverse x)"
haftmann@27544
  1025
  by (simp add: of_rat_inverse)
haftmann@27544
  1026
 
haftmann@27544
  1027
lemma real_divide_code [code]: "Ratreal x / Ratreal y = Ratreal (x / y)"
haftmann@27544
  1028
  by (simp add: of_rat_divide)
berghofe@24534
  1029
haftmann@24623
  1030
text {* Setup for SML code generator *}
nipkow@23031
  1031
nipkow@23031
  1032
types_code
berghofe@24534
  1033
  real ("(int */ int)")
nipkow@23031
  1034
attach (term_of) {*
berghofe@24534
  1035
fun term_of_real (p, q) =
haftmann@24623
  1036
  let
haftmann@24623
  1037
    val rT = HOLogic.realT
berghofe@24534
  1038
  in
berghofe@24534
  1039
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
haftmann@24623
  1040
    else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
berghofe@24534
  1041
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24534
  1042
  end;
nipkow@23031
  1043
*}
nipkow@23031
  1044
attach (test) {*
nipkow@23031
  1045
fun gen_real i =
berghofe@24534
  1046
  let
berghofe@24534
  1047
    val p = random_range 0 i;
berghofe@24534
  1048
    val q = random_range 1 (i + 1);
berghofe@24534
  1049
    val g = Integer.gcd p q;
wenzelm@24630
  1050
    val p' = p div g;
wenzelm@24630
  1051
    val q' = q div g;
berghofe@25885
  1052
    val r = (if one_of [true, false] then p' else ~ p',
berghofe@25885
  1053
      if p' = 0 then 0 else q')
berghofe@24534
  1054
  in
berghofe@25885
  1055
    (r, fn () => term_of_real r)
berghofe@24534
  1056
  end;
nipkow@23031
  1057
*}
nipkow@23031
  1058
nipkow@23031
  1059
consts_code
haftmann@24623
  1060
  Ratreal ("(_)")
berghofe@24534
  1061
berghofe@24534
  1062
consts_code
berghofe@24534
  1063
  "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
berghofe@24534
  1064
attach {*
berghofe@24534
  1065
fun real_of_int 0 = (0, 0)
berghofe@24534
  1066
  | real_of_int i = (i, 1);
berghofe@24534
  1067
*}
berghofe@24534
  1068
berghofe@24534
  1069
declare real_of_int_of_nat_eq [symmetric, code]
nipkow@23031
  1070
paulson@5588
  1071
end