src/HOL/Real/RealDef.thy
author nipkow
Sun Oct 21 22:33:35 2007 +0200 (2007-10-21)
changeset 25140 273772abbea2
parent 25134 3d4953e88449
child 25162 ad4d5365d9d8
permissions -rw-r--r--
More changes from >0 to ~=0::nat
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
wenzelm@19765
    18
  "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@23879
    28
instance real :: zero
haftmann@23879
    29
  real_zero_def: "0 == Abs_Real(realrel``{(1, 1)})" ..
haftmann@24198
    30
lemmas [code func del] = real_zero_def
paulson@5588
    31
haftmann@23879
    32
instance real :: one
haftmann@23879
    33
  real_one_def: "1 == Abs_Real(realrel``{(1 + 1, 1)})" ..
haftmann@24198
    34
lemmas [code func del] = real_one_def
paulson@5588
    35
haftmann@23879
    36
instance real :: plus
haftmann@23879
    37
  real_add_def: "z + w ==
paulson@14484
    38
       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
haftmann@23879
    39
		 { Abs_Real(realrel``{(x+u, y+v)}) })" ..
haftmann@24198
    40
lemmas [code func del] = real_add_def
bauerg@10606
    41
haftmann@23879
    42
instance real :: minus
haftmann@23879
    43
  real_minus_def: "- r ==  contents (\<Union>(x,y) \<in> Rep_Real(r). { Abs_Real(realrel``{(y,x)}) })"
haftmann@23879
    44
  real_diff_def: "r - (s::real) == r + - s" ..
haftmann@24198
    45
lemmas [code func del] = real_minus_def real_diff_def
paulson@14484
    46
haftmann@23879
    47
instance real :: times
paulson@14484
    48
  real_mult_def:
paulson@14484
    49
    "z * w ==
paulson@14484
    50
       contents (\<Union>(x,y) \<in> Rep_Real(z). \<Union>(u,v) \<in> Rep_Real(w).
haftmann@23879
    51
		 { Abs_Real(realrel``{(x*u + y*v, x*v + y*u)}) })" ..
haftmann@24198
    52
lemmas [code func del] = real_mult_def
paulson@5588
    53
haftmann@23879
    54
instance real :: inverse
haftmann@23879
    55
  real_inverse_def: "inverse (R::real) == (THE S. (R = 0 & S = 0) | S * R = 1)"
haftmann@23879
    56
  real_divide_def: "R / (S::real) == R * inverse S" ..
haftmann@24198
    57
lemmas [code func del] = real_inverse_def real_divide_def
paulson@14269
    58
haftmann@23879
    59
instance real :: ord
haftmann@23879
    60
  real_le_def: "z \<le> (w::real) == 
paulson@14484
    61
    \<exists>x y u v. x+v \<le> u+y & (x,y) \<in> Rep_Real z & (u,v) \<in> Rep_Real w"
haftmann@23879
    62
  real_less_def: "(x < (y::real)) == (x \<le> y & x \<noteq> y)" ..
haftmann@24198
    63
lemmas [code func del] = real_le_def real_less_def
paulson@5588
    64
haftmann@23879
    65
instance real :: abs
haftmann@23879
    66
  real_abs_def:  "abs (r::real) == (if r < 0 then - r else r)" ..
paulson@14334
    67
nipkow@24506
    68
instance real :: sgn
nipkow@24506
    69
  real_sgn_def: "sgn x == (if x=0 then 0 else if 0<x then 1 else - 1)" ..
paulson@14334
    70
huffman@23287
    71
subsection {* Equivalence relation over positive reals *}
paulson@14269
    72
paulson@14270
    73
lemma preal_trans_lemma:
paulson@14365
    74
  assumes "x + y1 = x1 + y"
paulson@14365
    75
      and "x + y2 = x2 + y"
paulson@14365
    76
  shows "x1 + y2 = x2 + (y1::preal)"
paulson@14365
    77
proof -
huffman@23287
    78
  have "(x1 + y2) + x = (x + y2) + x1" by (simp add: add_ac)
paulson@14365
    79
  also have "... = (x2 + y) + x1"  by (simp add: prems)
huffman@23287
    80
  also have "... = x2 + (x1 + y)"  by (simp add: add_ac)
paulson@14365
    81
  also have "... = x2 + (x + y1)"  by (simp add: prems)
huffman@23287
    82
  also have "... = (x2 + y1) + x"  by (simp add: add_ac)
paulson@14365
    83
  finally have "(x1 + y2) + x = (x2 + y1) + x" .
huffman@23287
    84
  thus ?thesis by (rule add_right_imp_eq)
paulson@14365
    85
qed
paulson@14365
    86
paulson@14269
    87
paulson@14484
    88
lemma realrel_iff [simp]: "(((x1,y1),(x2,y2)) \<in> realrel) = (x1 + y2 = x2 + y1)"
paulson@14484
    89
by (simp add: realrel_def)
paulson@14269
    90
paulson@14269
    91
lemma equiv_realrel: "equiv UNIV realrel"
paulson@14365
    92
apply (auto simp add: equiv_def refl_def sym_def trans_def realrel_def)
paulson@14365
    93
apply (blast dest: preal_trans_lemma) 
paulson@14269
    94
done
paulson@14269
    95
paulson@14497
    96
text{*Reduces equality of equivalence classes to the @{term realrel} relation:
paulson@14497
    97
  @{term "(realrel `` {x} = realrel `` {y}) = ((x,y) \<in> realrel)"} *}
paulson@14269
    98
lemmas equiv_realrel_iff = 
paulson@14269
    99
       eq_equiv_class_iff [OF equiv_realrel UNIV_I UNIV_I]
paulson@14269
   100
paulson@14269
   101
declare equiv_realrel_iff [simp]
paulson@14269
   102
paulson@14497
   103
paulson@14484
   104
lemma realrel_in_real [simp]: "realrel``{(x,y)}: Real"
paulson@14484
   105
by (simp add: Real_def realrel_def quotient_def, blast)
paulson@14269
   106
huffman@22958
   107
declare Abs_Real_inject [simp]
paulson@14484
   108
declare Abs_Real_inverse [simp]
paulson@14269
   109
paulson@14269
   110
paulson@14484
   111
text{*Case analysis on the representation of a real number as an equivalence
paulson@14484
   112
      class of pairs of positive reals.*}
paulson@14484
   113
lemma eq_Abs_Real [case_names Abs_Real, cases type: real]: 
paulson@14484
   114
     "(!!x y. z = Abs_Real(realrel``{(x,y)}) ==> P) ==> P"
paulson@14484
   115
apply (rule Rep_Real [of z, unfolded Real_def, THEN quotientE])
paulson@14484
   116
apply (drule arg_cong [where f=Abs_Real])
paulson@14484
   117
apply (auto simp add: Rep_Real_inverse)
paulson@14269
   118
done
paulson@14269
   119
paulson@14269
   120
huffman@23287
   121
subsection {* Addition and Subtraction *}
paulson@14269
   122
paulson@14269
   123
lemma real_add_congruent2_lemma:
paulson@14269
   124
     "[|a + ba = aa + b; ab + bc = ac + bb|]
paulson@14269
   125
      ==> a + ab + (ba + bc) = aa + ac + (b + (bb::preal))"
huffman@23287
   126
apply (simp add: add_assoc)
huffman@23287
   127
apply (rule add_left_commute [of ab, THEN ssubst])
huffman@23287
   128
apply (simp add: add_assoc [symmetric])
huffman@23287
   129
apply (simp add: add_ac)
paulson@14269
   130
done
paulson@14269
   131
paulson@14269
   132
lemma real_add:
paulson@14497
   133
     "Abs_Real (realrel``{(x,y)}) + Abs_Real (realrel``{(u,v)}) =
paulson@14497
   134
      Abs_Real (realrel``{(x+u, y+v)})"
paulson@14497
   135
proof -
paulson@15169
   136
  have "(\<lambda>z w. (\<lambda>(x,y). (\<lambda>(u,v). {Abs_Real (realrel `` {(x+u, y+v)})}) w) z)
paulson@15169
   137
        respects2 realrel"
paulson@14497
   138
    by (simp add: congruent2_def, blast intro: real_add_congruent2_lemma) 
paulson@14497
   139
  thus ?thesis
paulson@14497
   140
    by (simp add: real_add_def UN_UN_split_split_eq
paulson@14658
   141
                  UN_equiv_class2 [OF equiv_realrel equiv_realrel])
paulson@14497
   142
qed
paulson@14269
   143
paulson@14484
   144
lemma real_minus: "- Abs_Real(realrel``{(x,y)}) = Abs_Real(realrel `` {(y,x)})"
paulson@14484
   145
proof -
paulson@15169
   146
  have "(\<lambda>(x,y). {Abs_Real (realrel``{(y,x)})}) respects realrel"
huffman@23288
   147
    by (simp add: congruent_def add_commute) 
paulson@14484
   148
  thus ?thesis
paulson@14484
   149
    by (simp add: real_minus_def UN_equiv_class [OF equiv_realrel])
paulson@14484
   150
qed
paulson@14334
   151
huffman@23287
   152
instance real :: ab_group_add
huffman@23287
   153
proof
huffman@23287
   154
  fix x y z :: real
huffman@23287
   155
  show "(x + y) + z = x + (y + z)"
huffman@23287
   156
    by (cases x, cases y, cases z, simp add: real_add add_assoc)
huffman@23287
   157
  show "x + y = y + x"
huffman@23287
   158
    by (cases x, cases y, simp add: real_add add_commute)
huffman@23287
   159
  show "0 + x = x"
huffman@23287
   160
    by (cases x, simp add: real_add real_zero_def add_ac)
huffman@23287
   161
  show "- x + x = 0"
huffman@23287
   162
    by (cases x, simp add: real_minus real_add real_zero_def add_commute)
huffman@23287
   163
  show "x - y = x + - y"
huffman@23287
   164
    by (simp add: real_diff_def)
huffman@23287
   165
qed
paulson@14269
   166
paulson@14269
   167
huffman@23287
   168
subsection {* Multiplication *}
paulson@14269
   169
paulson@14329
   170
lemma real_mult_congruent2_lemma:
paulson@14329
   171
     "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==>
paulson@14484
   172
          x * x1 + y * y1 + (x * y2 + y * x2) =
paulson@14484
   173
          x * x2 + y * y2 + (x * y1 + y * x1)"
huffman@23287
   174
apply (simp add: add_left_commute add_assoc [symmetric])
huffman@23288
   175
apply (simp add: add_assoc right_distrib [symmetric])
huffman@23288
   176
apply (simp add: add_commute)
paulson@14269
   177
done
paulson@14269
   178
paulson@14269
   179
lemma real_mult_congruent2:
paulson@15169
   180
    "(%p1 p2.
paulson@14484
   181
        (%(x1,y1). (%(x2,y2). 
paulson@15169
   182
          { Abs_Real (realrel``{(x1*x2 + y1*y2, x1*y2+y1*x2)}) }) p2) p1)
paulson@15169
   183
     respects2 realrel"
paulson@14658
   184
apply (rule congruent2_commuteI [OF equiv_realrel], clarify)
huffman@23288
   185
apply (simp add: mult_commute add_commute)
paulson@14269
   186
apply (auto simp add: real_mult_congruent2_lemma)
paulson@14269
   187
done
paulson@14269
   188
paulson@14269
   189
lemma real_mult:
paulson@14484
   190
      "Abs_Real((realrel``{(x1,y1)})) * Abs_Real((realrel``{(x2,y2)})) =
paulson@14484
   191
       Abs_Real(realrel `` {(x1*x2+y1*y2,x1*y2+y1*x2)})"
paulson@14484
   192
by (simp add: real_mult_def UN_UN_split_split_eq
paulson@14658
   193
         UN_equiv_class2 [OF equiv_realrel equiv_realrel real_mult_congruent2])
paulson@14269
   194
paulson@14269
   195
lemma real_mult_commute: "(z::real) * w = w * z"
huffman@23288
   196
by (cases z, cases w, simp add: real_mult add_ac mult_ac)
paulson@14269
   197
paulson@14269
   198
lemma real_mult_assoc: "((z1::real) * z2) * z3 = z1 * (z2 * z3)"
paulson@14484
   199
apply (cases z1, cases z2, cases z3)
huffman@23288
   200
apply (simp add: real_mult right_distrib add_ac mult_ac)
paulson@14269
   201
done
paulson@14269
   202
paulson@14269
   203
lemma real_mult_1: "(1::real) * z = z"
paulson@14484
   204
apply (cases z)
huffman@23288
   205
apply (simp add: real_mult real_one_def right_distrib
huffman@23288
   206
                  mult_1_right mult_ac add_ac)
paulson@14269
   207
done
paulson@14269
   208
paulson@14269
   209
lemma real_add_mult_distrib: "((z1::real) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14484
   210
apply (cases z1, cases z2, cases w)
huffman@23288
   211
apply (simp add: real_add real_mult right_distrib add_ac mult_ac)
paulson@14269
   212
done
paulson@14269
   213
paulson@14329
   214
text{*one and zero are distinct*}
paulson@14365
   215
lemma real_zero_not_eq_one: "0 \<noteq> (1::real)"
paulson@14484
   216
proof -
huffman@23287
   217
  have "(1::preal) < 1 + 1"
huffman@23287
   218
    by (simp add: preal_self_less_add_left)
paulson@14484
   219
  thus ?thesis
huffman@23288
   220
    by (simp add: real_zero_def real_one_def)
paulson@14484
   221
qed
paulson@14269
   222
huffman@23287
   223
instance real :: comm_ring_1
huffman@23287
   224
proof
huffman@23287
   225
  fix x y z :: real
huffman@23287
   226
  show "(x * y) * z = x * (y * z)" by (rule real_mult_assoc)
huffman@23287
   227
  show "x * y = y * x" by (rule real_mult_commute)
huffman@23287
   228
  show "1 * x = x" by (rule real_mult_1)
huffman@23287
   229
  show "(x + y) * z = x * z + y * z" by (rule real_add_mult_distrib)
huffman@23287
   230
  show "0 \<noteq> (1::real)" by (rule real_zero_not_eq_one)
huffman@23287
   231
qed
huffman@23287
   232
huffman@23287
   233
subsection {* Inverse and Division *}
paulson@14365
   234
paulson@14484
   235
lemma real_zero_iff: "Abs_Real (realrel `` {(x, x)}) = 0"
huffman@23288
   236
by (simp add: real_zero_def add_commute)
paulson@14269
   237
paulson@14365
   238
text{*Instead of using an existential quantifier and constructing the inverse
paulson@14365
   239
within the proof, we could define the inverse explicitly.*}
paulson@14365
   240
paulson@14365
   241
lemma real_mult_inverse_left_ex: "x \<noteq> 0 ==> \<exists>y. y*x = (1::real)"
paulson@14484
   242
apply (simp add: real_zero_def real_one_def, cases x)
paulson@14269
   243
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@14365
   244
apply (auto dest!: less_add_left_Ex simp add: real_zero_iff)
paulson@14334
   245
apply (rule_tac
huffman@23287
   246
        x = "Abs_Real (realrel``{(1, inverse (D) + 1)})"
paulson@14334
   247
       in exI)
paulson@14334
   248
apply (rule_tac [2]
huffman@23287
   249
        x = "Abs_Real (realrel``{(inverse (D) + 1, 1)})" 
paulson@14334
   250
       in exI)
nipkow@23477
   251
apply (auto simp add: real_mult preal_mult_inverse_right ring_simps)
paulson@14269
   252
done
paulson@14269
   253
paulson@14365
   254
lemma real_mult_inverse_left: "x \<noteq> 0 ==> inverse(x)*x = (1::real)"
paulson@14484
   255
apply (simp add: real_inverse_def)
huffman@23287
   256
apply (drule real_mult_inverse_left_ex, safe)
huffman@23287
   257
apply (rule theI, assumption, rename_tac z)
huffman@23287
   258
apply (subgoal_tac "(z * x) * y = z * (x * y)")
huffman@23287
   259
apply (simp add: mult_commute)
huffman@23287
   260
apply (rule mult_assoc)
paulson@14269
   261
done
paulson@14334
   262
paulson@14341
   263
paulson@14341
   264
subsection{*The Real Numbers form a Field*}
paulson@14341
   265
paulson@14334
   266
instance real :: field
paulson@14334
   267
proof
paulson@14334
   268
  fix x y z :: real
paulson@14365
   269
  show "x \<noteq> 0 ==> inverse x * x = 1" by (rule real_mult_inverse_left)
paulson@14430
   270
  show "x / y = x * inverse y" by (simp add: real_divide_def)
paulson@14334
   271
qed
paulson@14334
   272
paulson@14334
   273
paulson@14341
   274
text{*Inverse of zero!  Useful to simplify certain equations*}
paulson@14269
   275
paulson@14334
   276
lemma INVERSE_ZERO: "inverse 0 = (0::real)"
paulson@14484
   277
by (simp add: real_inverse_def)
paulson@14334
   278
paulson@14334
   279
instance real :: division_by_zero
paulson@14334
   280
proof
paulson@14334
   281
  show "inverse 0 = (0::real)" by (rule INVERSE_ZERO)
paulson@14334
   282
qed
paulson@14334
   283
paulson@14269
   284
paulson@14365
   285
subsection{*The @{text "\<le>"} Ordering*}
paulson@14269
   286
paulson@14365
   287
lemma real_le_refl: "w \<le> (w::real)"
paulson@14484
   288
by (cases w, force simp add: real_le_def)
paulson@14269
   289
paulson@14378
   290
text{*The arithmetic decision procedure is not set up for type preal.
paulson@14378
   291
  This lemma is currently unused, but it could simplify the proofs of the
paulson@14378
   292
  following two lemmas.*}
paulson@14378
   293
lemma preal_eq_le_imp_le:
paulson@14378
   294
  assumes eq: "a+b = c+d" and le: "c \<le> a"
paulson@14378
   295
  shows "b \<le> (d::preal)"
paulson@14378
   296
proof -
huffman@23288
   297
  have "c+d \<le> a+d" by (simp add: prems)
paulson@14378
   298
  hence "a+b \<le> a+d" by (simp add: prems)
huffman@23288
   299
  thus "b \<le> d" by simp
paulson@14378
   300
qed
paulson@14378
   301
paulson@14378
   302
lemma real_le_lemma:
paulson@14378
   303
  assumes l: "u1 + v2 \<le> u2 + v1"
paulson@14378
   304
      and "x1 + v1 = u1 + y1"
paulson@14378
   305
      and "x2 + v2 = u2 + y2"
paulson@14378
   306
  shows "x1 + y2 \<le> x2 + (y1::preal)"
paulson@14365
   307
proof -
paulson@14378
   308
  have "(x1+v1) + (u2+y2) = (u1+y1) + (x2+v2)" by (simp add: prems)
huffman@23288
   309
  hence "(x1+y2) + (u2+v1) = (x2+y1) + (u1+v2)" by (simp add: add_ac)
huffman@23288
   310
  also have "... \<le> (x2+y1) + (u2+v1)" by (simp add: prems)
huffman@23288
   311
  finally show ?thesis by simp
huffman@23288
   312
qed
paulson@14378
   313
paulson@14378
   314
lemma real_le: 
paulson@14484
   315
     "(Abs_Real(realrel``{(x1,y1)}) \<le> Abs_Real(realrel``{(x2,y2)})) =  
paulson@14484
   316
      (x1 + y2 \<le> x2 + y1)"
huffman@23288
   317
apply (simp add: real_le_def)
paulson@14387
   318
apply (auto intro: real_le_lemma)
paulson@14378
   319
done
paulson@14378
   320
paulson@14378
   321
lemma real_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::real)"
nipkow@15542
   322
by (cases z, cases w, simp add: real_le)
paulson@14378
   323
paulson@14378
   324
lemma real_trans_lemma:
paulson@14378
   325
  assumes "x + v \<le> u + y"
paulson@14378
   326
      and "u + v' \<le> u' + v"
paulson@14378
   327
      and "x2 + v2 = u2 + y2"
paulson@14378
   328
  shows "x + v' \<le> u' + (y::preal)"
paulson@14378
   329
proof -
huffman@23288
   330
  have "(x+v') + (u+v) = (x+v) + (u+v')" by (simp add: add_ac)
huffman@23288
   331
  also have "... \<le> (u+y) + (u+v')" by (simp add: prems)
huffman@23288
   332
  also have "... \<le> (u+y) + (u'+v)" by (simp add: prems)
huffman@23288
   333
  also have "... = (u'+y) + (u+v)"  by (simp add: add_ac)
huffman@23288
   334
  finally show ?thesis by simp
nipkow@15542
   335
qed
paulson@14269
   336
paulson@14365
   337
lemma real_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::real)"
paulson@14484
   338
apply (cases i, cases j, cases k)
paulson@14484
   339
apply (simp add: real_le)
huffman@23288
   340
apply (blast intro: real_trans_lemma)
paulson@14334
   341
done
paulson@14334
   342
paulson@14334
   343
(* Axiom 'order_less_le' of class 'order': *)
paulson@14334
   344
lemma real_less_le: "((w::real) < z) = (w \<le> z & w \<noteq> z)"
paulson@14365
   345
by (simp add: real_less_def)
paulson@14365
   346
paulson@14365
   347
instance real :: order
paulson@14365
   348
proof qed
paulson@14365
   349
 (assumption |
paulson@14365
   350
  rule real_le_refl real_le_trans real_le_anti_sym real_less_le)+
paulson@14365
   351
paulson@14378
   352
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14378
   353
lemma real_le_linear: "(z::real) \<le> w | w \<le> z"
huffman@23288
   354
apply (cases z, cases w)
huffman@23288
   355
apply (auto simp add: real_le real_zero_def add_ac)
paulson@14334
   356
done
paulson@14334
   357
paulson@14334
   358
paulson@14334
   359
instance real :: linorder
paulson@14334
   360
  by (intro_classes, rule real_le_linear)
paulson@14334
   361
paulson@14334
   362
paulson@14378
   363
lemma real_le_eq_diff: "(x \<le> y) = (x-y \<le> (0::real))"
paulson@14484
   364
apply (cases x, cases y) 
paulson@14378
   365
apply (auto simp add: real_le real_zero_def real_diff_def real_add real_minus
huffman@23288
   366
                      add_ac)
huffman@23288
   367
apply (simp_all add: add_assoc [symmetric])
nipkow@15542
   368
done
paulson@14378
   369
paulson@14484
   370
lemma real_add_left_mono: 
paulson@14484
   371
  assumes le: "x \<le> y" shows "z + x \<le> z + (y::real)"
paulson@14484
   372
proof -
paulson@14484
   373
  have "z + x - (z + y) = (z + -z) + (x - y)"
paulson@14484
   374
    by (simp add: diff_minus add_ac) 
paulson@14484
   375
  with le show ?thesis 
obua@14754
   376
    by (simp add: real_le_eq_diff[of x] real_le_eq_diff[of "z+x"] diff_minus)
paulson@14484
   377
qed
paulson@14334
   378
paulson@14365
   379
lemma real_sum_gt_zero_less: "(0 < S + (-W::real)) ==> (W < S)"
paulson@14365
   380
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14365
   381
paulson@14365
   382
lemma real_less_sum_gt_zero: "(W < S) ==> (0 < S + (-W::real))"
paulson@14365
   383
by (simp add: linorder_not_le [symmetric] real_le_eq_diff [of S] diff_minus)
paulson@14334
   384
paulson@14334
   385
lemma real_mult_order: "[| 0 < x; 0 < y |] ==> (0::real) < x * y"
paulson@14484
   386
apply (cases x, cases y)
paulson@14378
   387
apply (simp add: linorder_not_le [where 'a = real, symmetric] 
paulson@14378
   388
                 linorder_not_le [where 'a = preal] 
paulson@14378
   389
                  real_zero_def real_le real_mult)
paulson@14365
   390
  --{*Reduce to the (simpler) @{text "\<le>"} relation *}
wenzelm@16973
   391
apply (auto dest!: less_add_left_Ex
huffman@23288
   392
     simp add: add_ac mult_ac
huffman@23288
   393
          right_distrib 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@22456
   403
instance real :: distrib_lattice
haftmann@22456
   404
  "inf x y \<equiv> min x y"
haftmann@22456
   405
  "sup x y \<equiv> max x y"
haftmann@22456
   406
  by default (auto simp add: inf_real_def sup_real_def min_max.sup_inf_distrib1)
haftmann@22456
   407
paulson@14378
   408
paulson@14334
   409
subsection{*The Reals Form an Ordered Field*}
paulson@14334
   410
paulson@14334
   411
instance real :: ordered_field
paulson@14334
   412
proof
paulson@14334
   413
  fix x y z :: real
paulson@14334
   414
  show "x \<le> y ==> z + x \<le> z + y" by (rule real_add_left_mono)
huffman@22962
   415
  show "x < y ==> 0 < z ==> z * x < z * y" by (rule real_mult_less_mono2)
huffman@22962
   416
  show "\<bar>x\<bar> = (if x < 0 then -x else x)" by (simp only: real_abs_def)
nipkow@24506
   417
  show "sgn x = (if x=0 then 0 else if 0<x then 1 else - 1)"
nipkow@24506
   418
    by (simp only: real_sgn_def)
paulson@14334
   419
qed
paulson@14334
   420
paulson@14365
   421
text{*The function @{term real_of_preal} requires many proofs, but it seems
paulson@14365
   422
to be essential for proving completeness of the reals from that of the
paulson@14365
   423
positive reals.*}
paulson@14365
   424
paulson@14365
   425
lemma real_of_preal_add:
paulson@14365
   426
     "real_of_preal ((x::preal) + y) = real_of_preal x + real_of_preal y"
huffman@23288
   427
by (simp add: real_of_preal_def real_add left_distrib add_ac)
paulson@14365
   428
paulson@14365
   429
lemma real_of_preal_mult:
paulson@14365
   430
     "real_of_preal ((x::preal) * y) = real_of_preal x* real_of_preal y"
huffman@23288
   431
by (simp add: real_of_preal_def real_mult right_distrib add_ac mult_ac)
paulson@14365
   432
paulson@14365
   433
paulson@14365
   434
text{*Gleason prop 9-4.4 p 127*}
paulson@14365
   435
lemma real_of_preal_trichotomy:
paulson@14365
   436
      "\<exists>m. (x::real) = real_of_preal m | x = 0 | x = -(real_of_preal m)"
paulson@14484
   437
apply (simp add: real_of_preal_def real_zero_def, cases x)
huffman@23288
   438
apply (auto simp add: real_minus add_ac)
paulson@14365
   439
apply (cut_tac x = x and y = y in linorder_less_linear)
huffman@23288
   440
apply (auto dest!: less_add_left_Ex simp add: add_assoc [symmetric])
paulson@14365
   441
done
paulson@14365
   442
paulson@14365
   443
lemma real_of_preal_leD:
paulson@14365
   444
      "real_of_preal m1 \<le> real_of_preal m2 ==> m1 \<le> m2"
huffman@23288
   445
by (simp add: real_of_preal_def real_le)
paulson@14365
   446
paulson@14365
   447
lemma real_of_preal_lessI: "m1 < m2 ==> real_of_preal m1 < real_of_preal m2"
paulson@14365
   448
by (auto simp add: real_of_preal_leD linorder_not_le [symmetric])
paulson@14365
   449
paulson@14365
   450
lemma real_of_preal_lessD:
paulson@14365
   451
      "real_of_preal m1 < real_of_preal m2 ==> m1 < m2"
huffman@23288
   452
by (simp add: real_of_preal_def real_le linorder_not_le [symmetric])
paulson@14365
   453
paulson@14365
   454
lemma real_of_preal_less_iff [simp]:
paulson@14365
   455
     "(real_of_preal m1 < real_of_preal m2) = (m1 < m2)"
paulson@14365
   456
by (blast intro: real_of_preal_lessI real_of_preal_lessD)
paulson@14365
   457
paulson@14365
   458
lemma real_of_preal_le_iff:
paulson@14365
   459
     "(real_of_preal m1 \<le> real_of_preal m2) = (m1 \<le> m2)"
huffman@23288
   460
by (simp add: linorder_not_less [symmetric])
paulson@14365
   461
paulson@14365
   462
lemma real_of_preal_zero_less: "0 < real_of_preal m"
huffman@23288
   463
apply (insert preal_self_less_add_left [of 1 m])
huffman@23288
   464
apply (auto simp add: real_zero_def real_of_preal_def
huffman@23288
   465
                      real_less_def real_le_def add_ac)
huffman@23288
   466
apply (rule_tac x="m + 1" in exI, rule_tac x="1" in exI)
huffman@23288
   467
apply (simp add: add_ac)
paulson@14365
   468
done
paulson@14365
   469
paulson@14365
   470
lemma real_of_preal_minus_less_zero: "- real_of_preal m < 0"
paulson@14365
   471
by (simp add: real_of_preal_zero_less)
paulson@14365
   472
paulson@14365
   473
lemma real_of_preal_not_minus_gt_zero: "~ 0 < - real_of_preal m"
paulson@14484
   474
proof -
paulson@14484
   475
  from real_of_preal_minus_less_zero
paulson@14484
   476
  show ?thesis by (blast dest: order_less_trans)
paulson@14484
   477
qed
paulson@14365
   478
paulson@14365
   479
paulson@14365
   480
subsection{*Theorems About the Ordering*}
paulson@14365
   481
paulson@14365
   482
lemma real_gt_zero_preal_Ex: "(0 < x) = (\<exists>y. x = real_of_preal y)"
paulson@14365
   483
apply (auto simp add: real_of_preal_zero_less)
paulson@14365
   484
apply (cut_tac x = x in real_of_preal_trichotomy)
paulson@14365
   485
apply (blast elim!: real_of_preal_not_minus_gt_zero [THEN notE])
paulson@14365
   486
done
paulson@14365
   487
paulson@14365
   488
lemma real_gt_preal_preal_Ex:
paulson@14365
   489
     "real_of_preal z < x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   490
by (blast dest!: real_of_preal_zero_less [THEN order_less_trans]
paulson@14365
   491
             intro: real_gt_zero_preal_Ex [THEN iffD1])
paulson@14365
   492
paulson@14365
   493
lemma real_ge_preal_preal_Ex:
paulson@14365
   494
     "real_of_preal z \<le> x ==> \<exists>y. x = real_of_preal y"
paulson@14365
   495
by (blast dest: order_le_imp_less_or_eq real_gt_preal_preal_Ex)
paulson@14365
   496
paulson@14365
   497
lemma real_less_all_preal: "y \<le> 0 ==> \<forall>x. y < real_of_preal x"
paulson@14365
   498
by (auto elim: order_le_imp_less_or_eq [THEN disjE] 
paulson@14365
   499
            intro: real_of_preal_zero_less [THEN [2] order_less_trans] 
paulson@14365
   500
            simp add: real_of_preal_zero_less)
paulson@14365
   501
paulson@14365
   502
lemma real_less_all_real2: "~ 0 < y ==> \<forall>x. y < real_of_preal x"
paulson@14365
   503
by (blast intro!: real_less_all_preal linorder_not_less [THEN iffD1])
paulson@14365
   504
paulson@14334
   505
paulson@14334
   506
subsection{*More Lemmas*}
paulson@14334
   507
paulson@14334
   508
lemma real_mult_left_cancel: "(c::real) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14334
   509
by auto
paulson@14334
   510
paulson@14334
   511
lemma real_mult_right_cancel: "(c::real) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14334
   512
by auto
paulson@14334
   513
paulson@14334
   514
lemma real_mult_less_iff1 [simp]: "(0::real) < z ==> (x*z < y*z) = (x < y)"
paulson@14334
   515
  by (force elim: order_less_asym
paulson@14334
   516
            simp add: Ring_and_Field.mult_less_cancel_right)
paulson@14334
   517
paulson@14334
   518
lemma real_mult_le_cancel_iff1 [simp]: "(0::real) < z ==> (x*z \<le> y*z) = (x\<le>y)"
paulson@14365
   519
apply (simp add: mult_le_cancel_right)
huffman@23289
   520
apply (blast intro: elim: order_less_asym)
paulson@14365
   521
done
paulson@14334
   522
paulson@14334
   523
lemma real_mult_le_cancel_iff2 [simp]: "(0::real) < z ==> (z*x \<le> z*y) = (x\<le>y)"
nipkow@15923
   524
by(simp add:mult_commute)
paulson@14334
   525
paulson@14365
   526
lemma real_inverse_gt_one: "[| (0::real) < x; x < 1 |] ==> 1 < inverse x"
huffman@23289
   527
by (simp add: one_less_inverse_iff) (* TODO: generalize/move *)
paulson@14334
   528
paulson@14334
   529
haftmann@24198
   530
subsection {* Embedding numbers into the Reals *}
haftmann@24198
   531
haftmann@24198
   532
abbreviation
haftmann@24198
   533
  real_of_nat :: "nat \<Rightarrow> real"
haftmann@24198
   534
where
haftmann@24198
   535
  "real_of_nat \<equiv> of_nat"
haftmann@24198
   536
haftmann@24198
   537
abbreviation
haftmann@24198
   538
  real_of_int :: "int \<Rightarrow> real"
haftmann@24198
   539
where
haftmann@24198
   540
  "real_of_int \<equiv> of_int"
haftmann@24198
   541
haftmann@24198
   542
abbreviation
haftmann@24198
   543
  real_of_rat :: "rat \<Rightarrow> real"
haftmann@24198
   544
where
haftmann@24198
   545
  "real_of_rat \<equiv> of_rat"
haftmann@24198
   546
haftmann@24198
   547
consts
haftmann@24198
   548
  (*overloaded constant for injecting other types into "real"*)
haftmann@24198
   549
  real :: "'a => real"
paulson@14365
   550
paulson@14378
   551
defs (overloaded)
berghofe@24534
   552
  real_of_nat_def [code inline]: "real == real_of_nat"
berghofe@24534
   553
  real_of_int_def [code inline]: "real == real_of_int"
paulson@14365
   554
avigad@16819
   555
lemma real_eq_of_nat: "real = of_nat"
haftmann@24198
   556
  unfolding real_of_nat_def ..
avigad@16819
   557
avigad@16819
   558
lemma real_eq_of_int: "real = of_int"
haftmann@24198
   559
  unfolding real_of_int_def ..
avigad@16819
   560
paulson@14365
   561
lemma real_of_int_zero [simp]: "real (0::int) = 0"  
paulson@14378
   562
by (simp add: real_of_int_def) 
paulson@14365
   563
paulson@14365
   564
lemma real_of_one [simp]: "real (1::int) = (1::real)"
paulson@14378
   565
by (simp add: real_of_int_def) 
paulson@14334
   566
avigad@16819
   567
lemma real_of_int_add [simp]: "real(x + y) = real (x::int) + real y"
paulson@14378
   568
by (simp add: real_of_int_def) 
paulson@14365
   569
avigad@16819
   570
lemma real_of_int_minus [simp]: "real(-x) = -real (x::int)"
paulson@14378
   571
by (simp add: real_of_int_def) 
avigad@16819
   572
avigad@16819
   573
lemma real_of_int_diff [simp]: "real(x - y) = real (x::int) - real y"
avigad@16819
   574
by (simp add: real_of_int_def) 
paulson@14365
   575
avigad@16819
   576
lemma real_of_int_mult [simp]: "real(x * y) = real (x::int) * real y"
paulson@14378
   577
by (simp add: real_of_int_def) 
paulson@14334
   578
avigad@16819
   579
lemma real_of_int_setsum [simp]: "real ((SUM x:A. f x)::int) = (SUM x:A. real(f x))"
avigad@16819
   580
  apply (subst real_eq_of_int)+
avigad@16819
   581
  apply (rule of_int_setsum)
avigad@16819
   582
done
avigad@16819
   583
avigad@16819
   584
lemma real_of_int_setprod [simp]: "real ((PROD x:A. f x)::int) = 
avigad@16819
   585
    (PROD x:A. real(f x))"
avigad@16819
   586
  apply (subst real_eq_of_int)+
avigad@16819
   587
  apply (rule of_int_setprod)
avigad@16819
   588
done
paulson@14365
   589
paulson@14365
   590
lemma real_of_int_zero_cancel [simp]: "(real x = 0) = (x = (0::int))"
paulson@14378
   591
by (simp add: real_of_int_def) 
paulson@14365
   592
paulson@14365
   593
lemma real_of_int_inject [iff]: "(real (x::int) = real y) = (x = y)"
paulson@14378
   594
by (simp add: real_of_int_def) 
paulson@14365
   595
paulson@14365
   596
lemma real_of_int_less_iff [iff]: "(real (x::int) < real y) = (x < y)"
paulson@14378
   597
by (simp add: real_of_int_def) 
paulson@14365
   598
paulson@14365
   599
lemma real_of_int_le_iff [simp]: "(real (x::int) \<le> real y) = (x \<le> y)"
paulson@14378
   600
by (simp add: real_of_int_def) 
paulson@14365
   601
avigad@16819
   602
lemma real_of_int_gt_zero_cancel_iff [simp]: "(0 < real (n::int)) = (0 < n)"
avigad@16819
   603
by (simp add: real_of_int_def) 
avigad@16819
   604
avigad@16819
   605
lemma real_of_int_ge_zero_cancel_iff [simp]: "(0 <= real (n::int)) = (0 <= n)"
avigad@16819
   606
by (simp add: real_of_int_def) 
avigad@16819
   607
avigad@16819
   608
lemma real_of_int_lt_zero_cancel_iff [simp]: "(real (n::int) < 0) = (n < 0)"
avigad@16819
   609
by (simp add: real_of_int_def)
avigad@16819
   610
avigad@16819
   611
lemma real_of_int_le_zero_cancel_iff [simp]: "(real (n::int) <= 0) = (n <= 0)"
avigad@16819
   612
by (simp add: real_of_int_def)
avigad@16819
   613
avigad@16888
   614
lemma real_of_int_abs [simp]: "real (abs x) = abs(real (x::int))"
avigad@16888
   615
by (auto simp add: abs_if)
avigad@16888
   616
avigad@16819
   617
lemma int_less_real_le: "((n::int) < m) = (real n + 1 <= real m)"
avigad@16819
   618
  apply (subgoal_tac "real n + 1 = real (n + 1)")
avigad@16819
   619
  apply (simp del: real_of_int_add)
avigad@16819
   620
  apply auto
avigad@16819
   621
done
avigad@16819
   622
avigad@16819
   623
lemma int_le_real_less: "((n::int) <= m) = (real n < real m + 1)"
avigad@16819
   624
  apply (subgoal_tac "real m + 1 = real (m + 1)")
avigad@16819
   625
  apply (simp del: real_of_int_add)
avigad@16819
   626
  apply simp
avigad@16819
   627
done
avigad@16819
   628
avigad@16819
   629
lemma real_of_int_div_aux: "d ~= 0 ==> (real (x::int)) / (real d) = 
avigad@16819
   630
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   631
proof -
avigad@16819
   632
  assume "d ~= 0"
avigad@16819
   633
  have "x = (x div d) * d + x mod d"
avigad@16819
   634
    by auto
avigad@16819
   635
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   636
    by (simp only: real_of_int_mult [THEN sym] real_of_int_add [THEN sym])
avigad@16819
   637
  then have "real x / real d = ... / real d"
avigad@16819
   638
    by simp
avigad@16819
   639
  then show ?thesis
nipkow@23477
   640
    by (auto simp add: add_divide_distrib ring_simps prems)
avigad@16819
   641
qed
avigad@16819
   642
avigad@16819
   643
lemma real_of_int_div: "(d::int) ~= 0 ==> d dvd n ==>
avigad@16819
   644
    real(n div d) = real n / real d"
avigad@16819
   645
  apply (frule real_of_int_div_aux [of d n])
avigad@16819
   646
  apply simp
avigad@16819
   647
  apply (simp add: zdvd_iff_zmod_eq_0)
avigad@16819
   648
done
avigad@16819
   649
avigad@16819
   650
lemma real_of_int_div2:
avigad@16819
   651
  "0 <= real (n::int) / real (x) - real (n div x)"
avigad@16819
   652
  apply (case_tac "x = 0")
avigad@16819
   653
  apply simp
avigad@16819
   654
  apply (case_tac "0 < x")
avigad@16819
   655
  apply (simp add: compare_rls)
avigad@16819
   656
  apply (subst real_of_int_div_aux)
avigad@16819
   657
  apply simp
avigad@16819
   658
  apply simp
avigad@16819
   659
  apply (subst zero_le_divide_iff)
avigad@16819
   660
  apply auto
avigad@16819
   661
  apply (simp add: compare_rls)
avigad@16819
   662
  apply (subst real_of_int_div_aux)
avigad@16819
   663
  apply simp
avigad@16819
   664
  apply simp
avigad@16819
   665
  apply (subst zero_le_divide_iff)
avigad@16819
   666
  apply auto
avigad@16819
   667
done
avigad@16819
   668
avigad@16819
   669
lemma real_of_int_div3:
avigad@16819
   670
  "real (n::int) / real (x) - real (n div x) <= 1"
avigad@16819
   671
  apply(case_tac "x = 0")
avigad@16819
   672
  apply simp
avigad@16819
   673
  apply (simp add: compare_rls)
avigad@16819
   674
  apply (subst real_of_int_div_aux)
avigad@16819
   675
  apply assumption
avigad@16819
   676
  apply simp
avigad@16819
   677
  apply (subst divide_le_eq)
avigad@16819
   678
  apply clarsimp
avigad@16819
   679
  apply (rule conjI)
avigad@16819
   680
  apply (rule impI)
avigad@16819
   681
  apply (rule order_less_imp_le)
avigad@16819
   682
  apply simp
avigad@16819
   683
  apply (rule impI)
avigad@16819
   684
  apply (rule order_less_imp_le)
avigad@16819
   685
  apply simp
avigad@16819
   686
done
avigad@16819
   687
avigad@16819
   688
lemma real_of_int_div4: "real (n div x) <= real (n::int) / real x" 
avigad@16819
   689
  by (insert real_of_int_div2 [of n x], simp)
paulson@14365
   690
paulson@14365
   691
subsection{*Embedding the Naturals into the Reals*}
paulson@14365
   692
paulson@14334
   693
lemma real_of_nat_zero [simp]: "real (0::nat) = 0"
paulson@14365
   694
by (simp add: real_of_nat_def)
paulson@14334
   695
paulson@14334
   696
lemma real_of_nat_one [simp]: "real (Suc 0) = (1::real)"
paulson@14365
   697
by (simp add: real_of_nat_def)
paulson@14334
   698
paulson@14365
   699
lemma real_of_nat_add [simp]: "real (m + n) = real (m::nat) + real n"
paulson@14378
   700
by (simp add: real_of_nat_def)
paulson@14334
   701
paulson@14334
   702
(*Not for addsimps: often the LHS is used to represent a positive natural*)
paulson@14334
   703
lemma real_of_nat_Suc: "real (Suc n) = real n + (1::real)"
paulson@14378
   704
by (simp add: real_of_nat_def)
paulson@14334
   705
paulson@14334
   706
lemma real_of_nat_less_iff [iff]: 
paulson@14334
   707
     "(real (n::nat) < real m) = (n < m)"
paulson@14365
   708
by (simp add: real_of_nat_def)
paulson@14334
   709
paulson@14334
   710
lemma real_of_nat_le_iff [iff]: "(real (n::nat) \<le> real m) = (n \<le> m)"
paulson@14378
   711
by (simp add: real_of_nat_def)
paulson@14334
   712
paulson@14334
   713
lemma real_of_nat_ge_zero [iff]: "0 \<le> real (n::nat)"
paulson@14378
   714
by (simp add: real_of_nat_def zero_le_imp_of_nat)
paulson@14334
   715
paulson@14365
   716
lemma real_of_nat_Suc_gt_zero: "0 < real (Suc n)"
paulson@14378
   717
by (simp add: real_of_nat_def del: of_nat_Suc)
paulson@14365
   718
paulson@14334
   719
lemma real_of_nat_mult [simp]: "real (m * n) = real (m::nat) * real n"
huffman@23431
   720
by (simp add: real_of_nat_def of_nat_mult)
paulson@14334
   721
avigad@16819
   722
lemma real_of_nat_setsum [simp]: "real ((SUM x:A. f x)::nat) = 
avigad@16819
   723
    (SUM x:A. real(f x))"
avigad@16819
   724
  apply (subst real_eq_of_nat)+
avigad@16819
   725
  apply (rule of_nat_setsum)
avigad@16819
   726
done
avigad@16819
   727
avigad@16819
   728
lemma real_of_nat_setprod [simp]: "real ((PROD x:A. f x)::nat) = 
avigad@16819
   729
    (PROD x:A. real(f x))"
avigad@16819
   730
  apply (subst real_eq_of_nat)+
avigad@16819
   731
  apply (rule of_nat_setprod)
avigad@16819
   732
done
avigad@16819
   733
avigad@16819
   734
lemma real_of_card: "real (card A) = setsum (%x.1) A"
avigad@16819
   735
  apply (subst card_eq_setsum)
avigad@16819
   736
  apply (subst real_of_nat_setsum)
avigad@16819
   737
  apply simp
avigad@16819
   738
done
avigad@16819
   739
paulson@14334
   740
lemma real_of_nat_inject [iff]: "(real (n::nat) = real m) = (n = m)"
paulson@14378
   741
by (simp add: real_of_nat_def)
paulson@14334
   742
paulson@14387
   743
lemma real_of_nat_zero_iff [iff]: "(real (n::nat) = 0) = (n = 0)"
paulson@14378
   744
by (simp add: real_of_nat_def)
paulson@14334
   745
paulson@14365
   746
lemma real_of_nat_diff: "n \<le> m ==> real (m - n) = real (m::nat) - real n"
huffman@23438
   747
by (simp add: add: real_of_nat_def of_nat_diff)
paulson@14334
   748
nipkow@25140
   749
lemma real_of_nat_gt_zero_cancel_iff [simp]: "(0 < real (n::nat)) = (n \<noteq> 0)"
nipkow@25140
   750
by (auto simp: real_of_nat_def)
paulson@14365
   751
paulson@14365
   752
lemma real_of_nat_le_zero_cancel_iff [simp]: "(real (n::nat) \<le> 0) = (n = 0)"
paulson@14378
   753
by (simp add: add: real_of_nat_def)
paulson@14334
   754
paulson@14365
   755
lemma not_real_of_nat_less_zero [simp]: "~ real (n::nat) < 0"
paulson@14378
   756
by (simp add: add: real_of_nat_def)
paulson@14334
   757
nipkow@25140
   758
lemma real_of_nat_ge_zero_cancel_iff [simp]: "(0 \<le> real (n::nat))"
paulson@14378
   759
by (simp add: add: real_of_nat_def)
paulson@14334
   760
avigad@16819
   761
lemma nat_less_real_le: "((n::nat) < m) = (real n + 1 <= real m)"
avigad@16819
   762
  apply (subgoal_tac "real n + 1 = real (Suc n)")
avigad@16819
   763
  apply simp
avigad@16819
   764
  apply (auto simp add: real_of_nat_Suc)
avigad@16819
   765
done
avigad@16819
   766
avigad@16819
   767
lemma nat_le_real_less: "((n::nat) <= m) = (real n < real m + 1)"
avigad@16819
   768
  apply (subgoal_tac "real m + 1 = real (Suc m)")
avigad@16819
   769
  apply (simp add: less_Suc_eq_le)
avigad@16819
   770
  apply (simp add: real_of_nat_Suc)
avigad@16819
   771
done
avigad@16819
   772
avigad@16819
   773
lemma real_of_nat_div_aux: "0 < d ==> (real (x::nat)) / (real d) = 
avigad@16819
   774
    real (x div d) + (real (x mod d)) / (real d)"
avigad@16819
   775
proof -
avigad@16819
   776
  assume "0 < d"
avigad@16819
   777
  have "x = (x div d) * d + x mod d"
avigad@16819
   778
    by auto
avigad@16819
   779
  then have "real x = real (x div d) * real d + real(x mod d)"
avigad@16819
   780
    by (simp only: real_of_nat_mult [THEN sym] real_of_nat_add [THEN sym])
avigad@16819
   781
  then have "real x / real d = \<dots> / real d"
avigad@16819
   782
    by simp
avigad@16819
   783
  then show ?thesis
nipkow@23477
   784
    by (auto simp add: add_divide_distrib ring_simps prems)
avigad@16819
   785
qed
avigad@16819
   786
avigad@16819
   787
lemma real_of_nat_div: "0 < (d::nat) ==> d dvd n ==>
avigad@16819
   788
    real(n div d) = real n / real d"
avigad@16819
   789
  apply (frule real_of_nat_div_aux [of d n])
avigad@16819
   790
  apply simp
avigad@16819
   791
  apply (subst dvd_eq_mod_eq_0 [THEN sym])
avigad@16819
   792
  apply assumption
avigad@16819
   793
done
avigad@16819
   794
avigad@16819
   795
lemma real_of_nat_div2:
avigad@16819
   796
  "0 <= real (n::nat) / real (x) - real (n div x)"
nipkow@25134
   797
apply(case_tac "x = 0")
nipkow@25134
   798
 apply (simp)
nipkow@25134
   799
apply (simp add: compare_rls)
nipkow@25134
   800
apply (subst real_of_nat_div_aux)
nipkow@25134
   801
 apply simp
nipkow@25134
   802
apply simp
nipkow@25134
   803
apply (subst zero_le_divide_iff)
nipkow@25134
   804
apply simp
avigad@16819
   805
done
avigad@16819
   806
avigad@16819
   807
lemma real_of_nat_div3:
avigad@16819
   808
  "real (n::nat) / real (x) - real (n div x) <= 1"
nipkow@25134
   809
apply(case_tac "x = 0")
nipkow@25134
   810
apply (simp)
nipkow@25134
   811
apply (simp add: compare_rls)
nipkow@25134
   812
apply (subst real_of_nat_div_aux)
nipkow@25134
   813
 apply simp
nipkow@25134
   814
apply simp
avigad@16819
   815
done
avigad@16819
   816
avigad@16819
   817
lemma real_of_nat_div4: "real (n div x) <= real (n::nat) / real x" 
avigad@16819
   818
  by (insert real_of_nat_div2 [of n x], simp)
avigad@16819
   819
paulson@14365
   820
lemma real_of_int_real_of_nat: "real (int n) = real n"
paulson@14378
   821
by (simp add: real_of_nat_def real_of_int_def int_eq_of_nat)
paulson@14378
   822
paulson@14426
   823
lemma real_of_int_of_nat_eq [simp]: "real (of_nat n :: int) = real n"
paulson@14426
   824
by (simp add: real_of_int_def real_of_nat_def)
paulson@14334
   825
avigad@16819
   826
lemma real_nat_eq_real [simp]: "0 <= x ==> real(nat x) = real x"
avigad@16819
   827
  apply (subgoal_tac "real(int(nat x)) = real(nat x)")
avigad@16819
   828
  apply force
avigad@16819
   829
  apply (simp only: real_of_int_real_of_nat)
avigad@16819
   830
done
paulson@14387
   831
paulson@14387
   832
subsection{*Numerals and Arithmetic*}
paulson@14387
   833
haftmann@24198
   834
instance real :: number_ring
haftmann@24198
   835
  real_number_of_def: "number_of w \<equiv> real_of_int w"
haftmann@24198
   836
  by intro_classes (simp add: real_number_of_def)
paulson@14387
   837
haftmann@24198
   838
lemma [code, code unfold]:
haftmann@24198
   839
  "number_of k = real_of_int (number_of k)"
haftmann@24198
   840
  unfolding number_of_is_id real_number_of_def ..
paulson@14387
   841
paulson@14387
   842
paulson@14387
   843
text{*Collapse applications of @{term real} to @{term number_of}*}
paulson@14387
   844
lemma real_number_of [simp]: "real (number_of v :: int) = number_of v"
paulson@14387
   845
by (simp add:  real_of_int_def of_int_number_of_eq)
paulson@14387
   846
paulson@14387
   847
lemma real_of_nat_number_of [simp]:
paulson@14387
   848
     "real (number_of v :: nat) =  
paulson@14387
   849
        (if neg (number_of v :: int) then 0  
paulson@14387
   850
         else (number_of v :: real))"
paulson@14387
   851
by (simp add: real_of_int_real_of_nat [symmetric] int_nat_number_of)
paulson@14387
   852
 
paulson@14387
   853
paulson@14387
   854
use "real_arith.ML"
wenzelm@24075
   855
declaration {* K real_arith_setup *}
paulson@14387
   856
kleing@19023
   857
paulson@14387
   858
subsection{* Simprules combining x+y and 0: ARE THEY NEEDED?*}
paulson@14387
   859
paulson@14387
   860
text{*Needed in this non-standard form by Hyperreal/Transcendental*}
paulson@14387
   861
lemma real_0_le_divide_iff:
paulson@14387
   862
     "((0::real) \<le> x/y) = ((x \<le> 0 | 0 \<le> y) & (0 \<le> x | y \<le> 0))"
paulson@14387
   863
by (simp add: real_divide_def zero_le_mult_iff, auto)
paulson@14387
   864
paulson@14387
   865
lemma real_add_minus_iff [simp]: "(x + - a = (0::real)) = (x=a)" 
paulson@14387
   866
by arith
paulson@14387
   867
paulson@15085
   868
lemma real_add_eq_0_iff: "(x+y = (0::real)) = (y = -x)"
paulson@14387
   869
by auto
paulson@14387
   870
paulson@15085
   871
lemma real_add_less_0_iff: "(x+y < (0::real)) = (y < -x)"
paulson@14387
   872
by auto
paulson@14387
   873
paulson@15085
   874
lemma real_0_less_add_iff: "((0::real) < x+y) = (-x < y)"
paulson@14387
   875
by auto
paulson@14387
   876
paulson@15085
   877
lemma real_add_le_0_iff: "(x+y \<le> (0::real)) = (y \<le> -x)"
paulson@14387
   878
by auto
paulson@14387
   879
paulson@15085
   880
lemma real_0_le_add_iff: "((0::real) \<le> x+y) = (-x \<le> y)"
paulson@14387
   881
by auto
paulson@14387
   882
paulson@14387
   883
paulson@14387
   884
(*
paulson@14387
   885
FIXME: we should have this, as for type int, but many proofs would break.
paulson@14387
   886
It replaces x+-y by x-y.
paulson@15086
   887
declare real_diff_def [symmetric, simp]
paulson@14387
   888
*)
paulson@14387
   889
paulson@14387
   890
paulson@14387
   891
subsubsection{*Density of the Reals*}
paulson@14387
   892
paulson@14387
   893
lemma real_lbound_gt_zero:
paulson@14387
   894
     "[| (0::real) < d1; 0 < d2 |] ==> \<exists>e. 0 < e & e < d1 & e < d2"
paulson@14387
   895
apply (rule_tac x = " (min d1 d2) /2" in exI)
paulson@14387
   896
apply (simp add: min_def)
paulson@14387
   897
done
paulson@14387
   898
paulson@14387
   899
paulson@14387
   900
text{*Similar results are proved in @{text Ring_and_Field}*}
paulson@14387
   901
lemma real_less_half_sum: "x < y ==> x < (x+y) / (2::real)"
paulson@14387
   902
  by auto
paulson@14387
   903
paulson@14387
   904
lemma real_gt_half_sum: "x < y ==> (x+y)/(2::real) < y"
paulson@14387
   905
  by auto
paulson@14387
   906
paulson@14387
   907
paulson@14387
   908
subsection{*Absolute Value Function for the Reals*}
paulson@14387
   909
paulson@14387
   910
lemma abs_minus_add_cancel: "abs(x + (-y)) = abs (y + (-(x::real)))"
paulson@15003
   911
by (simp add: abs_if)
paulson@14387
   912
huffman@23289
   913
(* FIXME: redundant, but used by Integration/RealRandVar.thy in AFP *)
paulson@14387
   914
lemma abs_le_interval_iff: "(abs x \<le> r) = (-r\<le>x & x\<le>(r::real))"
obua@14738
   915
by (force simp add: OrderedGroup.abs_le_iff)
paulson@14387
   916
paulson@14387
   917
lemma abs_add_one_gt_zero [simp]: "(0::real) < 1 + abs(x)"
paulson@15003
   918
by (simp add: abs_if)
paulson@14387
   919
paulson@14387
   920
lemma abs_real_of_nat_cancel [simp]: "abs (real x) = real (x::nat)"
huffman@22958
   921
by (rule abs_of_nonneg [OF real_of_nat_ge_zero])
paulson@14387
   922
paulson@14387
   923
lemma abs_add_one_not_less_self [simp]: "~ abs(x) + (1::real) < x"
webertj@20217
   924
by simp
paulson@14387
   925
 
paulson@14387
   926
lemma abs_sum_triangle_ineq: "abs ((x::real) + y + (-l + -m)) \<le> abs(x + -l) + abs(y + -m)"
webertj@20217
   927
by simp
paulson@14387
   928
berghofe@24534
   929
berghofe@24534
   930
subsection {* Implementation of rational real numbers as pairs of integers *}
berghofe@24534
   931
berghofe@24534
   932
definition
haftmann@24623
   933
  Ratreal :: "int \<times> int \<Rightarrow> real"
berghofe@24534
   934
where
haftmann@24623
   935
  "Ratreal = INum"
berghofe@24534
   936
haftmann@24623
   937
code_datatype Ratreal
berghofe@24534
   938
haftmann@24623
   939
lemma Ratreal_simp:
haftmann@24623
   940
  "Ratreal (k, l) = real_of_int k / real_of_int l"
haftmann@24623
   941
  unfolding Ratreal_def INum_def by simp
berghofe@24534
   942
haftmann@24623
   943
lemma Ratreal_zero [simp]: "Ratreal 0\<^sub>N = 0"
haftmann@24623
   944
  by (simp add: Ratreal_simp)
berghofe@24534
   945
haftmann@24623
   946
lemma Ratreal_lit [simp]: "Ratreal i\<^sub>N = real_of_int i"
haftmann@24623
   947
  by (simp add: Ratreal_simp)
berghofe@24534
   948
berghofe@24534
   949
lemma zero_real_code [code, code unfold]:
haftmann@24623
   950
  "0 = Ratreal 0\<^sub>N" by simp
berghofe@24534
   951
berghofe@24534
   952
lemma one_real_code [code, code unfold]:
haftmann@24623
   953
  "1 = Ratreal 1\<^sub>N" by simp
berghofe@24534
   954
berghofe@24534
   955
instance real :: eq ..
berghofe@24534
   956
haftmann@24623
   957
lemma real_eq_code [code]: "Ratreal x = Ratreal y \<longleftrightarrow> normNum x = normNum y"
haftmann@24623
   958
  unfolding Ratreal_def INum_normNum_iff ..
berghofe@24534
   959
haftmann@24623
   960
lemma real_less_eq_code [code]: "Ratreal x \<le> Ratreal y \<longleftrightarrow> normNum x \<le>\<^sub>N normNum y"
berghofe@24534
   961
proof -
haftmann@24623
   962
  have "normNum x \<le>\<^sub>N normNum y \<longleftrightarrow> Ratreal (normNum x) \<le> Ratreal (normNum y)" 
haftmann@24623
   963
    by (simp add: Ratreal_def del: normNum)
haftmann@24623
   964
  also have "\<dots> = (Ratreal x \<le> Ratreal y)" by (simp add: Ratreal_def)
berghofe@24534
   965
  finally show ?thesis by simp
berghofe@24534
   966
qed
berghofe@24534
   967
haftmann@24623
   968
lemma real_less_code [code]: "Ratreal x < Ratreal y \<longleftrightarrow> normNum x <\<^sub>N normNum y"
berghofe@24534
   969
proof -
haftmann@24623
   970
  have "normNum x <\<^sub>N normNum y \<longleftrightarrow> Ratreal (normNum x) < Ratreal (normNum y)" 
haftmann@24623
   971
    by (simp add: Ratreal_def del: normNum)
haftmann@24623
   972
  also have "\<dots> = (Ratreal x < Ratreal y)" by (simp add: Ratreal_def)
berghofe@24534
   973
  finally show ?thesis by simp
berghofe@24534
   974
qed
berghofe@24534
   975
haftmann@24623
   976
lemma real_add_code [code]: "Ratreal x + Ratreal y = Ratreal (x +\<^sub>N y)"
haftmann@24623
   977
  unfolding Ratreal_def by simp
berghofe@24534
   978
haftmann@24623
   979
lemma real_mul_code [code]: "Ratreal x * Ratreal y = Ratreal (x *\<^sub>N y)"
haftmann@24623
   980
  unfolding Ratreal_def by simp
berghofe@24534
   981
haftmann@24623
   982
lemma real_neg_code [code]: "- Ratreal x = Ratreal (~\<^sub>N x)"
haftmann@24623
   983
  unfolding Ratreal_def by simp
berghofe@24534
   984
haftmann@24623
   985
lemma real_sub_code [code]: "Ratreal x - Ratreal y = Ratreal (x -\<^sub>N y)"
haftmann@24623
   986
  unfolding Ratreal_def by simp
berghofe@24534
   987
haftmann@24623
   988
lemma real_inv_code [code]: "inverse (Ratreal x) = Ratreal (Ninv x)"
haftmann@24623
   989
  unfolding Ratreal_def Ninv real_divide_def by simp
berghofe@24534
   990
haftmann@24623
   991
lemma real_div_code [code]: "Ratreal x / Ratreal y = Ratreal (x \<div>\<^sub>N y)"
haftmann@24623
   992
  unfolding Ratreal_def by simp
berghofe@24534
   993
haftmann@24623
   994
text {* Setup for SML code generator *}
nipkow@23031
   995
nipkow@23031
   996
types_code
berghofe@24534
   997
  real ("(int */ int)")
nipkow@23031
   998
attach (term_of) {*
berghofe@24534
   999
fun term_of_real (p, q) =
haftmann@24623
  1000
  let
haftmann@24623
  1001
    val rT = HOLogic.realT
berghofe@24534
  1002
  in
berghofe@24534
  1003
    if q = 1 orelse p = 0 then HOLogic.mk_number rT p
haftmann@24623
  1004
    else @{term "op / \<Colon> real \<Rightarrow> real \<Rightarrow> real"} $
berghofe@24534
  1005
      HOLogic.mk_number rT p $ HOLogic.mk_number rT q
berghofe@24534
  1006
  end;
nipkow@23031
  1007
*}
nipkow@23031
  1008
attach (test) {*
nipkow@23031
  1009
fun gen_real i =
berghofe@24534
  1010
  let
berghofe@24534
  1011
    val p = random_range 0 i;
berghofe@24534
  1012
    val q = random_range 1 (i + 1);
berghofe@24534
  1013
    val g = Integer.gcd p q;
wenzelm@24630
  1014
    val p' = p div g;
wenzelm@24630
  1015
    val q' = q div g;
berghofe@24534
  1016
  in
berghofe@24534
  1017
    (if one_of [true, false] then p' else ~ p',
berghofe@24534
  1018
     if p' = 0 then 0 else q')
berghofe@24534
  1019
  end;
nipkow@23031
  1020
*}
nipkow@23031
  1021
nipkow@23031
  1022
consts_code
haftmann@24623
  1023
  Ratreal ("(_)")
berghofe@24534
  1024
berghofe@24534
  1025
consts_code
berghofe@24534
  1026
  "of_int :: int \<Rightarrow> real" ("\<module>real'_of'_int")
berghofe@24534
  1027
attach {*
berghofe@24534
  1028
fun real_of_int 0 = (0, 0)
berghofe@24534
  1029
  | real_of_int i = (i, 1);
berghofe@24534
  1030
*}
berghofe@24534
  1031
berghofe@24534
  1032
declare real_of_int_of_nat_eq [symmetric, code]
nipkow@23031
  1033
paulson@5588
  1034
end