src/HOL/Real/RealVector.thy
author huffman
Fri Jul 11 16:56:20 2008 +0200 (2008-07-11)
changeset 27553 d315a513a150
parent 27552 15cf4ed9c2a1
child 28009 e93b121074fb
permissions -rw-r--r--
instance real_field < field_char_0;
instance star :: (field_char_0) field_char_0
haftmann@27552
     1
(*  Title:      RealVector.thy
huffman@20504
     2
    ID:         $Id$
haftmann@27552
     3
    Author:     Brian Huffman
huffman@20504
     4
*)
huffman@20504
     5
huffman@20504
     6
header {* Vector Spaces and Algebras over the Reals *}
huffman@20504
     7
huffman@20504
     8
theory RealVector
huffman@20684
     9
imports RealPow
huffman@20504
    10
begin
huffman@20504
    11
huffman@20504
    12
subsection {* Locale for additive functions *}
huffman@20504
    13
huffman@20504
    14
locale additive =
huffman@20504
    15
  fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
huffman@20504
    16
  assumes add: "f (x + y) = f x + f y"
huffman@27443
    17
begin
huffman@20504
    18
huffman@27443
    19
lemma zero: "f 0 = 0"
huffman@20504
    20
proof -
huffman@20504
    21
  have "f 0 = f (0 + 0)" by simp
huffman@20504
    22
  also have "\<dots> = f 0 + f 0" by (rule add)
huffman@20504
    23
  finally show "f 0 = 0" by simp
huffman@20504
    24
qed
huffman@20504
    25
huffman@27443
    26
lemma minus: "f (- x) = - f x"
huffman@20504
    27
proof -
huffman@20504
    28
  have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
huffman@20504
    29
  also have "\<dots> = - f x + f x" by (simp add: zero)
huffman@20504
    30
  finally show "f (- x) = - f x" by (rule add_right_imp_eq)
huffman@20504
    31
qed
huffman@20504
    32
huffman@27443
    33
lemma diff: "f (x - y) = f x - f y"
huffman@20504
    34
by (simp add: diff_def add minus)
huffman@20504
    35
huffman@27443
    36
lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
huffman@22942
    37
apply (cases "finite A")
huffman@22942
    38
apply (induct set: finite)
huffman@22942
    39
apply (simp add: zero)
huffman@22942
    40
apply (simp add: add)
huffman@22942
    41
apply (simp add: zero)
huffman@22942
    42
done
huffman@22942
    43
huffman@27443
    44
end
huffman@20504
    45
huffman@20504
    46
subsection {* Real vector spaces *}
huffman@20504
    47
huffman@22636
    48
class scaleR = type +
haftmann@25062
    49
  fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
haftmann@24748
    50
begin
huffman@20504
    51
huffman@20763
    52
abbreviation
haftmann@25062
    53
  divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
haftmann@24748
    54
where
haftmann@25062
    55
  "x /\<^sub>R r == scaleR (inverse r) x"
haftmann@24748
    56
haftmann@24748
    57
end
haftmann@24748
    58
haftmann@25571
    59
instantiation real :: scaleR
haftmann@25571
    60
begin
haftmann@25571
    61
haftmann@25571
    62
definition
haftmann@25571
    63
  real_scaleR_def [simp]: "scaleR a x = a * x"
haftmann@25571
    64
haftmann@25571
    65
instance ..
haftmann@25571
    66
haftmann@25571
    67
end
huffman@20554
    68
haftmann@24588
    69
class real_vector = scaleR + ab_group_add +
haftmann@25062
    70
  assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
haftmann@25062
    71
  and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
haftmann@24588
    72
  and scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
haftmann@24588
    73
  and scaleR_one [simp]: "scaleR 1 x = x"
huffman@20504
    74
haftmann@24588
    75
class real_algebra = real_vector + ring +
haftmann@25062
    76
  assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
haftmann@25062
    77
  and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
huffman@20504
    78
haftmann@24588
    79
class real_algebra_1 = real_algebra + ring_1
huffman@20554
    80
haftmann@24588
    81
class real_div_algebra = real_algebra_1 + division_ring
huffman@20584
    82
haftmann@24588
    83
class real_field = real_div_algebra + field
huffman@20584
    84
huffman@20584
    85
instance real :: real_field
huffman@20554
    86
apply (intro_classes, unfold real_scaleR_def)
huffman@20554
    87
apply (rule right_distrib)
huffman@20554
    88
apply (rule left_distrib)
huffman@20763
    89
apply (rule mult_assoc [symmetric])
huffman@20554
    90
apply (rule mult_1_left)
huffman@20554
    91
apply (rule mult_assoc)
huffman@20554
    92
apply (rule mult_left_commute)
huffman@20554
    93
done
huffman@20554
    94
huffman@20504
    95
lemma scaleR_left_commute:
huffman@20504
    96
  fixes x :: "'a::real_vector"
huffman@21809
    97
  shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)"
huffman@20763
    98
by (simp add: mult_commute)
huffman@20504
    99
huffman@23127
   100
interpretation scaleR_left: additive ["(\<lambda>a. scaleR a x::'a::real_vector)"]
huffman@23127
   101
by unfold_locales (rule scaleR_left_distrib)
huffman@20504
   102
huffman@23127
   103
interpretation scaleR_right: additive ["(\<lambda>x. scaleR a x::'a::real_vector)"]
huffman@23127
   104
by unfold_locales (rule scaleR_right_distrib)
huffman@20504
   105
huffman@23127
   106
lemmas scaleR_zero_left [simp] = scaleR_left.zero
huffman@20504
   107
huffman@23127
   108
lemmas scaleR_zero_right [simp] = scaleR_right.zero
huffman@20504
   109
huffman@23127
   110
lemmas scaleR_minus_left [simp] = scaleR_left.minus
huffman@23113
   111
huffman@23127
   112
lemmas scaleR_minus_right [simp] = scaleR_right.minus
huffman@20504
   113
huffman@23127
   114
lemmas scaleR_left_diff_distrib = scaleR_left.diff
huffman@20504
   115
huffman@23127
   116
lemmas scaleR_right_diff_distrib = scaleR_right.diff
huffman@20504
   117
huffman@22973
   118
lemma scaleR_eq_0_iff [simp]:
huffman@20554
   119
  fixes x :: "'a::real_vector"
huffman@21809
   120
  shows "(scaleR a x = 0) = (a = 0 \<or> x = 0)"
huffman@20554
   121
proof cases
huffman@20554
   122
  assume "a = 0" thus ?thesis by simp
huffman@20554
   123
next
huffman@20554
   124
  assume anz [simp]: "a \<noteq> 0"
huffman@21809
   125
  { assume "scaleR a x = 0"
huffman@21809
   126
    hence "scaleR (inverse a) (scaleR a x) = 0" by simp
huffman@20763
   127
    hence "x = 0" by simp }
huffman@20554
   128
  thus ?thesis by force
huffman@20554
   129
qed
huffman@20554
   130
huffman@20554
   131
lemma scaleR_left_imp_eq:
huffman@20554
   132
  fixes x y :: "'a::real_vector"
huffman@21809
   133
  shows "\<lbrakk>a \<noteq> 0; scaleR a x = scaleR a y\<rbrakk> \<Longrightarrow> x = y"
huffman@20554
   134
proof -
huffman@20554
   135
  assume nonzero: "a \<noteq> 0"
huffman@21809
   136
  assume "scaleR a x = scaleR a y"
huffman@21809
   137
  hence "scaleR a (x - y) = 0"
huffman@20554
   138
     by (simp add: scaleR_right_diff_distrib)
huffman@22973
   139
  hence "x - y = 0" by (simp add: nonzero)
huffman@20554
   140
  thus "x = y" by simp
huffman@20554
   141
qed
huffman@20554
   142
huffman@20554
   143
lemma scaleR_right_imp_eq:
huffman@20554
   144
  fixes x y :: "'a::real_vector"
huffman@21809
   145
  shows "\<lbrakk>x \<noteq> 0; scaleR a x = scaleR b x\<rbrakk> \<Longrightarrow> a = b"
huffman@20554
   146
proof -
huffman@20554
   147
  assume nonzero: "x \<noteq> 0"
huffman@21809
   148
  assume "scaleR a x = scaleR b x"
huffman@21809
   149
  hence "scaleR (a - b) x = 0"
huffman@20554
   150
     by (simp add: scaleR_left_diff_distrib)
huffman@22973
   151
  hence "a - b = 0" by (simp add: nonzero)
huffman@20554
   152
  thus "a = b" by simp
huffman@20554
   153
qed
huffman@20554
   154
huffman@20554
   155
lemma scaleR_cancel_left:
huffman@20554
   156
  fixes x y :: "'a::real_vector"
huffman@21809
   157
  shows "(scaleR a x = scaleR a y) = (x = y \<or> a = 0)"
huffman@20554
   158
by (auto intro: scaleR_left_imp_eq)
huffman@20554
   159
huffman@20554
   160
lemma scaleR_cancel_right:
huffman@20554
   161
  fixes x y :: "'a::real_vector"
huffman@21809
   162
  shows "(scaleR a x = scaleR b x) = (a = b \<or> x = 0)"
huffman@20554
   163
by (auto intro: scaleR_right_imp_eq)
huffman@20554
   164
huffman@20584
   165
lemma nonzero_inverse_scaleR_distrib:
huffman@21809
   166
  fixes x :: "'a::real_div_algebra" shows
huffman@21809
   167
  "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
huffman@20763
   168
by (rule inverse_unique, simp)
huffman@20584
   169
huffman@20584
   170
lemma inverse_scaleR_distrib:
huffman@20584
   171
  fixes x :: "'a::{real_div_algebra,division_by_zero}"
huffman@21809
   172
  shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
huffman@20584
   173
apply (case_tac "a = 0", simp)
huffman@20584
   174
apply (case_tac "x = 0", simp)
huffman@20584
   175
apply (erule (1) nonzero_inverse_scaleR_distrib)
huffman@20584
   176
done
huffman@20584
   177
huffman@20554
   178
huffman@20554
   179
subsection {* Embedding of the Reals into any @{text real_algebra_1}:
huffman@20554
   180
@{term of_real} *}
huffman@20554
   181
huffman@20554
   182
definition
wenzelm@21404
   183
  of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
huffman@21809
   184
  "of_real r = scaleR r 1"
huffman@20554
   185
huffman@21809
   186
lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
huffman@20763
   187
by (simp add: of_real_def)
huffman@20763
   188
huffman@20554
   189
lemma of_real_0 [simp]: "of_real 0 = 0"
huffman@20554
   190
by (simp add: of_real_def)
huffman@20554
   191
huffman@20554
   192
lemma of_real_1 [simp]: "of_real 1 = 1"
huffman@20554
   193
by (simp add: of_real_def)
huffman@20554
   194
huffman@20554
   195
lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
huffman@20554
   196
by (simp add: of_real_def scaleR_left_distrib)
huffman@20554
   197
huffman@20554
   198
lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
huffman@20554
   199
by (simp add: of_real_def)
huffman@20554
   200
huffman@20554
   201
lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
huffman@20554
   202
by (simp add: of_real_def scaleR_left_diff_distrib)
huffman@20554
   203
huffman@20554
   204
lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
huffman@20763
   205
by (simp add: of_real_def mult_commute)
huffman@20554
   206
huffman@20584
   207
lemma nonzero_of_real_inverse:
huffman@20584
   208
  "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
huffman@20584
   209
   inverse (of_real x :: 'a::real_div_algebra)"
huffman@20584
   210
by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
huffman@20584
   211
huffman@20584
   212
lemma of_real_inverse [simp]:
huffman@20584
   213
  "of_real (inverse x) =
huffman@20584
   214
   inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
huffman@20584
   215
by (simp add: of_real_def inverse_scaleR_distrib)
huffman@20584
   216
huffman@20584
   217
lemma nonzero_of_real_divide:
huffman@20584
   218
  "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
huffman@20584
   219
   (of_real x / of_real y :: 'a::real_field)"
huffman@20584
   220
by (simp add: divide_inverse nonzero_of_real_inverse)
huffman@20722
   221
huffman@20722
   222
lemma of_real_divide [simp]:
huffman@20584
   223
  "of_real (x / y) =
huffman@20584
   224
   (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
huffman@20584
   225
by (simp add: divide_inverse)
huffman@20584
   226
huffman@20722
   227
lemma of_real_power [simp]:
huffman@20722
   228
  "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
wenzelm@20772
   229
by (induct n) (simp_all add: power_Suc)
huffman@20722
   230
huffman@20554
   231
lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
huffman@20554
   232
by (simp add: of_real_def scaleR_cancel_right)
huffman@20554
   233
huffman@20584
   234
lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
huffman@20554
   235
huffman@20554
   236
lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
huffman@20554
   237
proof
huffman@20554
   238
  fix r
huffman@20554
   239
  show "of_real r = id r"
huffman@22973
   240
    by (simp add: of_real_def)
huffman@20554
   241
qed
huffman@20554
   242
huffman@20554
   243
text{*Collapse nested embeddings*}
huffman@20554
   244
lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
wenzelm@20772
   245
by (induct n) auto
huffman@20554
   246
huffman@20554
   247
lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
huffman@20554
   248
by (cases z rule: int_diff_cases, simp)
huffman@20554
   249
huffman@20554
   250
lemma of_real_number_of_eq:
huffman@20554
   251
  "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
huffman@20554
   252
by (simp add: number_of_eq)
huffman@20554
   253
huffman@22912
   254
text{*Every real algebra has characteristic zero*}
huffman@22912
   255
instance real_algebra_1 < ring_char_0
huffman@22912
   256
proof
huffman@23282
   257
  fix m n :: nat
huffman@23282
   258
  have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
huffman@23282
   259
    by (simp only: of_real_eq_iff of_nat_eq_iff)
huffman@23282
   260
  thus "(of_nat m = (of_nat n::'a)) = (m = n)"
huffman@23282
   261
    by (simp only: of_real_of_nat_eq)
huffman@22912
   262
qed
huffman@22912
   263
huffman@27553
   264
instance real_field < field_char_0 ..
huffman@27553
   265
huffman@20554
   266
huffman@20554
   267
subsection {* The Set of Real Numbers *}
huffman@20554
   268
wenzelm@20772
   269
definition
wenzelm@21404
   270
  Reals :: "'a::real_algebra_1 set" where
haftmann@27435
   271
  [code func del]: "Reals \<equiv> range of_real"
huffman@20554
   272
wenzelm@21210
   273
notation (xsymbols)
huffman@20554
   274
  Reals  ("\<real>")
huffman@20554
   275
huffman@21809
   276
lemma Reals_of_real [simp]: "of_real r \<in> Reals"
huffman@20554
   277
by (simp add: Reals_def)
huffman@20554
   278
huffman@21809
   279
lemma Reals_of_int [simp]: "of_int z \<in> Reals"
huffman@21809
   280
by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
huffman@20718
   281
huffman@21809
   282
lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
huffman@21809
   283
by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
huffman@21809
   284
huffman@21809
   285
lemma Reals_number_of [simp]:
huffman@21809
   286
  "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
huffman@21809
   287
by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
huffman@20718
   288
huffman@20554
   289
lemma Reals_0 [simp]: "0 \<in> Reals"
huffman@20554
   290
apply (unfold Reals_def)
huffman@20554
   291
apply (rule range_eqI)
huffman@20554
   292
apply (rule of_real_0 [symmetric])
huffman@20554
   293
done
huffman@20554
   294
huffman@20554
   295
lemma Reals_1 [simp]: "1 \<in> Reals"
huffman@20554
   296
apply (unfold Reals_def)
huffman@20554
   297
apply (rule range_eqI)
huffman@20554
   298
apply (rule of_real_1 [symmetric])
huffman@20554
   299
done
huffman@20554
   300
huffman@20584
   301
lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
huffman@20554
   302
apply (auto simp add: Reals_def)
huffman@20554
   303
apply (rule range_eqI)
huffman@20554
   304
apply (rule of_real_add [symmetric])
huffman@20554
   305
done
huffman@20554
   306
huffman@20584
   307
lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
huffman@20584
   308
apply (auto simp add: Reals_def)
huffman@20584
   309
apply (rule range_eqI)
huffman@20584
   310
apply (rule of_real_minus [symmetric])
huffman@20584
   311
done
huffman@20584
   312
huffman@20584
   313
lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
huffman@20584
   314
apply (auto simp add: Reals_def)
huffman@20584
   315
apply (rule range_eqI)
huffman@20584
   316
apply (rule of_real_diff [symmetric])
huffman@20584
   317
done
huffman@20584
   318
huffman@20584
   319
lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
huffman@20554
   320
apply (auto simp add: Reals_def)
huffman@20554
   321
apply (rule range_eqI)
huffman@20554
   322
apply (rule of_real_mult [symmetric])
huffman@20554
   323
done
huffman@20554
   324
huffman@20584
   325
lemma nonzero_Reals_inverse:
huffman@20584
   326
  fixes a :: "'a::real_div_algebra"
huffman@20584
   327
  shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   328
apply (auto simp add: Reals_def)
huffman@20584
   329
apply (rule range_eqI)
huffman@20584
   330
apply (erule nonzero_of_real_inverse [symmetric])
huffman@20584
   331
done
huffman@20584
   332
huffman@20584
   333
lemma Reals_inverse [simp]:
huffman@20584
   334
  fixes a :: "'a::{real_div_algebra,division_by_zero}"
huffman@20584
   335
  shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
huffman@20584
   336
apply (auto simp add: Reals_def)
huffman@20584
   337
apply (rule range_eqI)
huffman@20584
   338
apply (rule of_real_inverse [symmetric])
huffman@20584
   339
done
huffman@20584
   340
huffman@20584
   341
lemma nonzero_Reals_divide:
huffman@20584
   342
  fixes a b :: "'a::real_field"
huffman@20584
   343
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   344
apply (auto simp add: Reals_def)
huffman@20584
   345
apply (rule range_eqI)
huffman@20584
   346
apply (erule nonzero_of_real_divide [symmetric])
huffman@20584
   347
done
huffman@20584
   348
huffman@20584
   349
lemma Reals_divide [simp]:
huffman@20584
   350
  fixes a b :: "'a::{real_field,division_by_zero}"
huffman@20584
   351
  shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
huffman@20584
   352
apply (auto simp add: Reals_def)
huffman@20584
   353
apply (rule range_eqI)
huffman@20584
   354
apply (rule of_real_divide [symmetric])
huffman@20584
   355
done
huffman@20584
   356
huffman@20722
   357
lemma Reals_power [simp]:
huffman@20722
   358
  fixes a :: "'a::{real_algebra_1,recpower}"
huffman@20722
   359
  shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
huffman@20722
   360
apply (auto simp add: Reals_def)
huffman@20722
   361
apply (rule range_eqI)
huffman@20722
   362
apply (rule of_real_power [symmetric])
huffman@20722
   363
done
huffman@20722
   364
huffman@20554
   365
lemma Reals_cases [cases set: Reals]:
huffman@20554
   366
  assumes "q \<in> \<real>"
huffman@20554
   367
  obtains (of_real) r where "q = of_real r"
huffman@20554
   368
  unfolding Reals_def
huffman@20554
   369
proof -
huffman@20554
   370
  from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
huffman@20554
   371
  then obtain r where "q = of_real r" ..
huffman@20554
   372
  then show thesis ..
huffman@20554
   373
qed
huffman@20554
   374
huffman@20554
   375
lemma Reals_induct [case_names of_real, induct set: Reals]:
huffman@20554
   376
  "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
huffman@20554
   377
  by (rule Reals_cases) auto
huffman@20554
   378
huffman@20504
   379
huffman@20504
   380
subsection {* Real normed vector spaces *}
huffman@20504
   381
huffman@22636
   382
class norm = type +
huffman@22636
   383
  fixes norm :: "'a \<Rightarrow> real"
huffman@20504
   384
haftmann@25571
   385
instantiation real :: norm
haftmann@25571
   386
begin
haftmann@25571
   387
haftmann@25571
   388
definition
haftmann@25571
   389
  real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>"
haftmann@25571
   390
haftmann@25571
   391
instance ..
haftmann@25571
   392
haftmann@25571
   393
end
huffman@20554
   394
huffman@24520
   395
class sgn_div_norm = scaleR + norm + sgn +
haftmann@25062
   396
  assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
nipkow@24506
   397
haftmann@24588
   398
class real_normed_vector = real_vector + sgn_div_norm +
haftmann@24588
   399
  assumes norm_ge_zero [simp]: "0 \<le> norm x"
haftmann@25062
   400
  and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
haftmann@25062
   401
  and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
haftmann@24588
   402
  and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
huffman@20504
   403
haftmann@24588
   404
class real_normed_algebra = real_algebra + real_normed_vector +
haftmann@25062
   405
  assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
huffman@20504
   406
haftmann@24588
   407
class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
haftmann@25062
   408
  assumes norm_one [simp]: "norm 1 = 1"
huffman@22852
   409
haftmann@24588
   410
class real_normed_div_algebra = real_div_algebra + real_normed_vector +
haftmann@25062
   411
  assumes norm_mult: "norm (x * y) = norm x * norm y"
huffman@20504
   412
haftmann@24588
   413
class real_normed_field = real_field + real_normed_div_algebra
huffman@20584
   414
huffman@22852
   415
instance real_normed_div_algebra < real_normed_algebra_1
huffman@20554
   416
proof
huffman@20554
   417
  fix x y :: 'a
huffman@20554
   418
  show "norm (x * y) \<le> norm x * norm y"
huffman@20554
   419
    by (simp add: norm_mult)
huffman@22852
   420
next
huffman@22852
   421
  have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
huffman@22852
   422
    by (rule norm_mult)
huffman@22852
   423
  thus "norm (1::'a) = 1" by simp
huffman@20554
   424
qed
huffman@20554
   425
huffman@20584
   426
instance real :: real_normed_field
huffman@22852
   427
apply (intro_classes, unfold real_norm_def real_scaleR_def)
nipkow@24506
   428
apply (simp add: real_sgn_def)
huffman@20554
   429
apply (rule abs_ge_zero)
huffman@20554
   430
apply (rule abs_eq_0)
huffman@20554
   431
apply (rule abs_triangle_ineq)
huffman@22852
   432
apply (rule abs_mult)
huffman@20554
   433
apply (rule abs_mult)
huffman@20554
   434
done
huffman@20504
   435
huffman@22852
   436
lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
huffman@20504
   437
by simp
huffman@20504
   438
huffman@22852
   439
lemma zero_less_norm_iff [simp]:
huffman@22852
   440
  fixes x :: "'a::real_normed_vector"
huffman@22852
   441
  shows "(0 < norm x) = (x \<noteq> 0)"
huffman@20504
   442
by (simp add: order_less_le)
huffman@20504
   443
huffman@22852
   444
lemma norm_not_less_zero [simp]:
huffman@22852
   445
  fixes x :: "'a::real_normed_vector"
huffman@22852
   446
  shows "\<not> norm x < 0"
huffman@20828
   447
by (simp add: linorder_not_less)
huffman@20828
   448
huffman@22852
   449
lemma norm_le_zero_iff [simp]:
huffman@22852
   450
  fixes x :: "'a::real_normed_vector"
huffman@22852
   451
  shows "(norm x \<le> 0) = (x = 0)"
huffman@20828
   452
by (simp add: order_le_less)
huffman@20828
   453
huffman@20504
   454
lemma norm_minus_cancel [simp]:
huffman@20584
   455
  fixes x :: "'a::real_normed_vector"
huffman@20584
   456
  shows "norm (- x) = norm x"
huffman@20504
   457
proof -
huffman@21809
   458
  have "norm (- x) = norm (scaleR (- 1) x)"
huffman@20504
   459
    by (simp only: scaleR_minus_left scaleR_one)
huffman@20533
   460
  also have "\<dots> = \<bar>- 1\<bar> * norm x"
huffman@20504
   461
    by (rule norm_scaleR)
huffman@20504
   462
  finally show ?thesis by simp
huffman@20504
   463
qed
huffman@20504
   464
huffman@20504
   465
lemma norm_minus_commute:
huffman@20584
   466
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   467
  shows "norm (a - b) = norm (b - a)"
huffman@20504
   468
proof -
huffman@22898
   469
  have "norm (- (b - a)) = norm (b - a)"
huffman@22898
   470
    by (rule norm_minus_cancel)
huffman@22898
   471
  thus ?thesis by simp
huffman@20504
   472
qed
huffman@20504
   473
huffman@20504
   474
lemma norm_triangle_ineq2:
huffman@20584
   475
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   476
  shows "norm a - norm b \<le> norm (a - b)"
huffman@20504
   477
proof -
huffman@20533
   478
  have "norm (a - b + b) \<le> norm (a - b) + norm b"
huffman@20504
   479
    by (rule norm_triangle_ineq)
huffman@22898
   480
  thus ?thesis by simp
huffman@20504
   481
qed
huffman@20504
   482
huffman@20584
   483
lemma norm_triangle_ineq3:
huffman@20584
   484
  fixes a b :: "'a::real_normed_vector"
huffman@20584
   485
  shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
huffman@20584
   486
apply (subst abs_le_iff)
huffman@20584
   487
apply auto
huffman@20584
   488
apply (rule norm_triangle_ineq2)
huffman@20584
   489
apply (subst norm_minus_commute)
huffman@20584
   490
apply (rule norm_triangle_ineq2)
huffman@20584
   491
done
huffman@20584
   492
huffman@20504
   493
lemma norm_triangle_ineq4:
huffman@20584
   494
  fixes a b :: "'a::real_normed_vector"
huffman@20533
   495
  shows "norm (a - b) \<le> norm a + norm b"
huffman@20504
   496
proof -
huffman@22898
   497
  have "norm (a + - b) \<le> norm a + norm (- b)"
huffman@20504
   498
    by (rule norm_triangle_ineq)
huffman@22898
   499
  thus ?thesis
huffman@22898
   500
    by (simp only: diff_minus norm_minus_cancel)
huffman@22898
   501
qed
huffman@22898
   502
huffman@22898
   503
lemma norm_diff_ineq:
huffman@22898
   504
  fixes a b :: "'a::real_normed_vector"
huffman@22898
   505
  shows "norm a - norm b \<le> norm (a + b)"
huffman@22898
   506
proof -
huffman@22898
   507
  have "norm a - norm (- b) \<le> norm (a - - b)"
huffman@22898
   508
    by (rule norm_triangle_ineq2)
huffman@22898
   509
  thus ?thesis by simp
huffman@20504
   510
qed
huffman@20504
   511
huffman@20551
   512
lemma norm_diff_triangle_ineq:
huffman@20551
   513
  fixes a b c d :: "'a::real_normed_vector"
huffman@20551
   514
  shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
huffman@20551
   515
proof -
huffman@20551
   516
  have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
huffman@20551
   517
    by (simp add: diff_minus add_ac)
huffman@20551
   518
  also have "\<dots> \<le> norm (a - c) + norm (b - d)"
huffman@20551
   519
    by (rule norm_triangle_ineq)
huffman@20551
   520
  finally show ?thesis .
huffman@20551
   521
qed
huffman@20551
   522
huffman@22857
   523
lemma abs_norm_cancel [simp]:
huffman@22857
   524
  fixes a :: "'a::real_normed_vector"
huffman@22857
   525
  shows "\<bar>norm a\<bar> = norm a"
huffman@22857
   526
by (rule abs_of_nonneg [OF norm_ge_zero])
huffman@22857
   527
huffman@22880
   528
lemma norm_add_less:
huffman@22880
   529
  fixes x y :: "'a::real_normed_vector"
huffman@22880
   530
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
huffman@22880
   531
by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
huffman@22880
   532
huffman@22880
   533
lemma norm_mult_less:
huffman@22880
   534
  fixes x y :: "'a::real_normed_algebra"
huffman@22880
   535
  shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
huffman@22880
   536
apply (rule order_le_less_trans [OF norm_mult_ineq])
huffman@22880
   537
apply (simp add: mult_strict_mono')
huffman@22880
   538
done
huffman@22880
   539
huffman@22857
   540
lemma norm_of_real [simp]:
huffman@22857
   541
  "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
huffman@22852
   542
unfolding of_real_def by (simp add: norm_scaleR)
huffman@20560
   543
huffman@22876
   544
lemma norm_number_of [simp]:
huffman@22876
   545
  "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
huffman@22876
   546
    = \<bar>number_of w\<bar>"
huffman@22876
   547
by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
huffman@22876
   548
huffman@22876
   549
lemma norm_of_int [simp]:
huffman@22876
   550
  "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
huffman@22876
   551
by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
huffman@22876
   552
huffman@22876
   553
lemma norm_of_nat [simp]:
huffman@22876
   554
  "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
huffman@22876
   555
apply (subst of_real_of_nat_eq [symmetric])
huffman@22876
   556
apply (subst norm_of_real, simp)
huffman@22876
   557
done
huffman@22876
   558
huffman@20504
   559
lemma nonzero_norm_inverse:
huffman@20504
   560
  fixes a :: "'a::real_normed_div_algebra"
huffman@20533
   561
  shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
huffman@20504
   562
apply (rule inverse_unique [symmetric])
huffman@20504
   563
apply (simp add: norm_mult [symmetric])
huffman@20504
   564
done
huffman@20504
   565
huffman@20504
   566
lemma norm_inverse:
huffman@20504
   567
  fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
huffman@20533
   568
  shows "norm (inverse a) = inverse (norm a)"
huffman@20504
   569
apply (case_tac "a = 0", simp)
huffman@20504
   570
apply (erule nonzero_norm_inverse)
huffman@20504
   571
done
huffman@20504
   572
huffman@20584
   573
lemma nonzero_norm_divide:
huffman@20584
   574
  fixes a b :: "'a::real_normed_field"
huffman@20584
   575
  shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
huffman@20584
   576
by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
huffman@20584
   577
huffman@20584
   578
lemma norm_divide:
huffman@20584
   579
  fixes a b :: "'a::{real_normed_field,division_by_zero}"
huffman@20584
   580
  shows "norm (a / b) = norm a / norm b"
huffman@20584
   581
by (simp add: divide_inverse norm_mult norm_inverse)
huffman@20584
   582
huffman@22852
   583
lemma norm_power_ineq:
huffman@22852
   584
  fixes x :: "'a::{real_normed_algebra_1,recpower}"
huffman@22852
   585
  shows "norm (x ^ n) \<le> norm x ^ n"
huffman@22852
   586
proof (induct n)
huffman@22852
   587
  case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
huffman@22852
   588
next
huffman@22852
   589
  case (Suc n)
huffman@22852
   590
  have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
huffman@22852
   591
    by (rule norm_mult_ineq)
huffman@22852
   592
  also from Suc have "\<dots> \<le> norm x * norm x ^ n"
huffman@22852
   593
    using norm_ge_zero by (rule mult_left_mono)
huffman@22852
   594
  finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
huffman@22852
   595
    by (simp add: power_Suc)
huffman@22852
   596
qed
huffman@22852
   597
huffman@20684
   598
lemma norm_power:
huffman@20684
   599
  fixes x :: "'a::{real_normed_div_algebra,recpower}"
huffman@20684
   600
  shows "norm (x ^ n) = norm x ^ n"
wenzelm@20772
   601
by (induct n) (simp_all add: power_Suc norm_mult)
huffman@20684
   602
huffman@22442
   603
huffman@22972
   604
subsection {* Sign function *}
huffman@22972
   605
nipkow@24506
   606
lemma norm_sgn:
nipkow@24506
   607
  "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
nipkow@24506
   608
by (simp add: sgn_div_norm norm_scaleR)
huffman@22972
   609
nipkow@24506
   610
lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
nipkow@24506
   611
by (simp add: sgn_div_norm)
huffman@22972
   612
nipkow@24506
   613
lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
nipkow@24506
   614
by (simp add: sgn_div_norm)
huffman@22972
   615
nipkow@24506
   616
lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
nipkow@24506
   617
by (simp add: sgn_div_norm)
huffman@22972
   618
nipkow@24506
   619
lemma sgn_scaleR:
nipkow@24506
   620
  "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
nipkow@24506
   621
by (simp add: sgn_div_norm norm_scaleR mult_ac)
huffman@22973
   622
huffman@22972
   623
lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
nipkow@24506
   624
by (simp add: sgn_div_norm)
huffman@22972
   625
huffman@22972
   626
lemma sgn_of_real:
huffman@22972
   627
  "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
huffman@22972
   628
unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
huffman@22972
   629
huffman@22973
   630
lemma sgn_mult:
huffman@22973
   631
  fixes x y :: "'a::real_normed_div_algebra"
huffman@22973
   632
  shows "sgn (x * y) = sgn x * sgn y"
nipkow@24506
   633
by (simp add: sgn_div_norm norm_mult mult_commute)
huffman@22973
   634
huffman@22972
   635
lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
nipkow@24506
   636
by (simp add: sgn_div_norm divide_inverse)
huffman@22972
   637
huffman@22972
   638
lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
huffman@22972
   639
unfolding real_sgn_eq by simp
huffman@22972
   640
huffman@22972
   641
lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
huffman@22972
   642
unfolding real_sgn_eq by simp
huffman@22972
   643
huffman@22972
   644
huffman@22442
   645
subsection {* Bounded Linear and Bilinear Operators *}
huffman@22442
   646
huffman@22442
   647
locale bounded_linear = additive +
huffman@22442
   648
  constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@22442
   649
  assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
huffman@22442
   650
  assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
huffman@27443
   651
begin
huffman@22442
   652
huffman@27443
   653
lemma pos_bounded:
huffman@22442
   654
  "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   655
proof -
huffman@22442
   656
  obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
huffman@22442
   657
    using bounded by fast
huffman@22442
   658
  show ?thesis
huffman@22442
   659
  proof (intro exI impI conjI allI)
huffman@22442
   660
    show "0 < max 1 K"
huffman@22442
   661
      by (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   662
  next
huffman@22442
   663
    fix x
huffman@22442
   664
    have "norm (f x) \<le> norm x * K" using K .
huffman@22442
   665
    also have "\<dots> \<le> norm x * max 1 K"
huffman@22442
   666
      by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
huffman@22442
   667
    finally show "norm (f x) \<le> norm x * max 1 K" .
huffman@22442
   668
  qed
huffman@22442
   669
qed
huffman@22442
   670
huffman@27443
   671
lemma nonneg_bounded:
huffman@22442
   672
  "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
huffman@22442
   673
proof -
huffman@22442
   674
  from pos_bounded
huffman@22442
   675
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   676
qed
huffman@22442
   677
huffman@27443
   678
end
huffman@27443
   679
huffman@22442
   680
locale bounded_bilinear =
huffman@22442
   681
  fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
huffman@22442
   682
                 \<Rightarrow> 'c::real_normed_vector"
huffman@22442
   683
    (infixl "**" 70)
huffman@22442
   684
  assumes add_left: "prod (a + a') b = prod a b + prod a' b"
huffman@22442
   685
  assumes add_right: "prod a (b + b') = prod a b + prod a b'"
huffman@22442
   686
  assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
huffman@22442
   687
  assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
huffman@22442
   688
  assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
huffman@27443
   689
begin
huffman@22442
   690
huffman@27443
   691
lemma pos_bounded:
huffman@22442
   692
  "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   693
apply (cut_tac bounded, erule exE)
huffman@22442
   694
apply (rule_tac x="max 1 K" in exI, safe)
huffman@22442
   695
apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
huffman@22442
   696
apply (drule spec, drule spec, erule order_trans)
huffman@22442
   697
apply (rule mult_left_mono [OF le_maxI2])
huffman@22442
   698
apply (intro mult_nonneg_nonneg norm_ge_zero)
huffman@22442
   699
done
huffman@22442
   700
huffman@27443
   701
lemma nonneg_bounded:
huffman@22442
   702
  "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
huffman@22442
   703
proof -
huffman@22442
   704
  from pos_bounded
huffman@22442
   705
  show ?thesis by (auto intro: order_less_imp_le)
huffman@22442
   706
qed
huffman@22442
   707
huffman@27443
   708
lemma additive_right: "additive (\<lambda>b. prod a b)"
huffman@22442
   709
by (rule additive.intro, rule add_right)
huffman@22442
   710
huffman@27443
   711
lemma additive_left: "additive (\<lambda>a. prod a b)"
huffman@22442
   712
by (rule additive.intro, rule add_left)
huffman@22442
   713
huffman@27443
   714
lemma zero_left: "prod 0 b = 0"
huffman@22442
   715
by (rule additive.zero [OF additive_left])
huffman@22442
   716
huffman@27443
   717
lemma zero_right: "prod a 0 = 0"
huffman@22442
   718
by (rule additive.zero [OF additive_right])
huffman@22442
   719
huffman@27443
   720
lemma minus_left: "prod (- a) b = - prod a b"
huffman@22442
   721
by (rule additive.minus [OF additive_left])
huffman@22442
   722
huffman@27443
   723
lemma minus_right: "prod a (- b) = - prod a b"
huffman@22442
   724
by (rule additive.minus [OF additive_right])
huffman@22442
   725
huffman@27443
   726
lemma diff_left:
huffman@22442
   727
  "prod (a - a') b = prod a b - prod a' b"
huffman@22442
   728
by (rule additive.diff [OF additive_left])
huffman@22442
   729
huffman@27443
   730
lemma diff_right:
huffman@22442
   731
  "prod a (b - b') = prod a b - prod a b'"
huffman@22442
   732
by (rule additive.diff [OF additive_right])
huffman@22442
   733
huffman@27443
   734
lemma bounded_linear_left:
huffman@22442
   735
  "bounded_linear (\<lambda>a. a ** b)"
huffman@22442
   736
apply (unfold_locales)
huffman@22442
   737
apply (rule add_left)
huffman@22442
   738
apply (rule scaleR_left)
huffman@22442
   739
apply (cut_tac bounded, safe)
huffman@22442
   740
apply (rule_tac x="norm b * K" in exI)
huffman@22442
   741
apply (simp add: mult_ac)
huffman@22442
   742
done
huffman@22442
   743
huffman@27443
   744
lemma bounded_linear_right:
huffman@22442
   745
  "bounded_linear (\<lambda>b. a ** b)"
huffman@22442
   746
apply (unfold_locales)
huffman@22442
   747
apply (rule add_right)
huffman@22442
   748
apply (rule scaleR_right)
huffman@22442
   749
apply (cut_tac bounded, safe)
huffman@22442
   750
apply (rule_tac x="norm a * K" in exI)
huffman@22442
   751
apply (simp add: mult_ac)
huffman@22442
   752
done
huffman@22442
   753
huffman@27443
   754
lemma prod_diff_prod:
huffman@22442
   755
  "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
huffman@22442
   756
by (simp add: diff_left diff_right)
huffman@22442
   757
huffman@27443
   758
end
huffman@27443
   759
huffman@23127
   760
interpretation mult:
huffman@22442
   761
  bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
huffman@22442
   762
apply (rule bounded_bilinear.intro)
huffman@22442
   763
apply (rule left_distrib)
huffman@22442
   764
apply (rule right_distrib)
huffman@22442
   765
apply (rule mult_scaleR_left)
huffman@22442
   766
apply (rule mult_scaleR_right)
huffman@22442
   767
apply (rule_tac x="1" in exI)
huffman@22442
   768
apply (simp add: norm_mult_ineq)
huffman@22442
   769
done
huffman@22442
   770
huffman@23127
   771
interpretation mult_left:
huffman@22442
   772
  bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
huffman@23127
   773
by (rule mult.bounded_linear_left)
huffman@22442
   774
huffman@23127
   775
interpretation mult_right:
huffman@23127
   776
  bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
huffman@23127
   777
by (rule mult.bounded_linear_right)
huffman@23127
   778
huffman@23127
   779
interpretation divide:
huffman@23120
   780
  bounded_linear ["(\<lambda>x::'a::real_normed_field. x / y)"]
huffman@23127
   781
unfolding divide_inverse by (rule mult.bounded_linear_left)
huffman@23120
   782
huffman@23127
   783
interpretation scaleR: bounded_bilinear ["scaleR"]
huffman@22442
   784
apply (rule bounded_bilinear.intro)
huffman@22442
   785
apply (rule scaleR_left_distrib)
huffman@22442
   786
apply (rule scaleR_right_distrib)
huffman@22973
   787
apply simp
huffman@22442
   788
apply (rule scaleR_left_commute)
huffman@22442
   789
apply (rule_tac x="1" in exI)
huffman@22442
   790
apply (simp add: norm_scaleR)
huffman@22442
   791
done
huffman@22442
   792
huffman@23127
   793
interpretation scaleR_left: bounded_linear ["\<lambda>r. scaleR r x"]
huffman@23127
   794
by (rule scaleR.bounded_linear_left)
huffman@23127
   795
huffman@23127
   796
interpretation scaleR_right: bounded_linear ["\<lambda>x. scaleR r x"]
huffman@23127
   797
by (rule scaleR.bounded_linear_right)
huffman@23127
   798
huffman@23127
   799
interpretation of_real: bounded_linear ["\<lambda>r. of_real r"]
huffman@23127
   800
unfolding of_real_def by (rule scaleR.bounded_linear_left)
huffman@22625
   801
huffman@20504
   802
end