src/HOL/Real/RealVector.thy
 author huffman Tue Apr 10 21:51:08 2007 +0200 (2007-04-10) changeset 22625 a2967023d674 parent 22442 15d9ed9b5051 child 22636 c40465deaf20 permissions -rw-r--r--
interpretation bounded_linear_of_real
```     1 (*  Title       : RealVector.thy
```
```     2     ID:         \$Id\$
```
```     3     Author      : Brian Huffman
```
```     4 *)
```
```     5
```
```     6 header {* Vector Spaces and Algebras over the Reals *}
```
```     7
```
```     8 theory RealVector
```
```     9 imports RealPow
```
```    10 begin
```
```    11
```
```    12 subsection {* Locale for additive functions *}
```
```    13
```
```    14 locale additive =
```
```    15   fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
```
```    16   assumes add: "f (x + y) = f x + f y"
```
```    17
```
```    18 lemma (in additive) zero: "f 0 = 0"
```
```    19 proof -
```
```    20   have "f 0 = f (0 + 0)" by simp
```
```    21   also have "\<dots> = f 0 + f 0" by (rule add)
```
```    22   finally show "f 0 = 0" by simp
```
```    23 qed
```
```    24
```
```    25 lemma (in additive) minus: "f (- x) = - f x"
```
```    26 proof -
```
```    27   have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
```
```    28   also have "\<dots> = - f x + f x" by (simp add: zero)
```
```    29   finally show "f (- x) = - f x" by (rule add_right_imp_eq)
```
```    30 qed
```
```    31
```
```    32 lemma (in additive) diff: "f (x - y) = f x - f y"
```
```    33 by (simp add: diff_def add minus)
```
```    34
```
```    35
```
```    36 subsection {* Real vector spaces *}
```
```    37
```
```    38 axclass scaleR < type
```
```    39
```
```    40 consts
```
```    41   scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a::scaleR" (infixr "*#" 75)
```
```    42
```
```    43 abbreviation
```
```    44   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a::scaleR" (infixl "'/#" 70) where
```
```    45   "x /# r == scaleR (inverse r) x"
```
```    46
```
```    47 notation (xsymbols)
```
```    48   scaleR (infixr "*\<^sub>R" 75) and
```
```    49   divideR (infixl "'/\<^sub>R" 70)
```
```    50
```
```    51 instance real :: scaleR ..
```
```    52
```
```    53 defs (overloaded)
```
```    54   real_scaleR_def: "scaleR a x \<equiv> a * x"
```
```    55
```
```    56 axclass real_vector < scaleR, ab_group_add
```
```    57   scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```    58   scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```    59   scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```    60   scaleR_one [simp]: "scaleR 1 x = x"
```
```    61
```
```    62 axclass real_algebra < real_vector, ring
```
```    63   mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```    64   mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```    65
```
```    66 axclass real_algebra_1 < real_algebra, ring_1
```
```    67
```
```    68 axclass real_div_algebra < real_algebra_1, division_ring
```
```    69
```
```    70 axclass real_field < real_div_algebra, field
```
```    71
```
```    72 instance real :: real_field
```
```    73 apply (intro_classes, unfold real_scaleR_def)
```
```    74 apply (rule right_distrib)
```
```    75 apply (rule left_distrib)
```
```    76 apply (rule mult_assoc [symmetric])
```
```    77 apply (rule mult_1_left)
```
```    78 apply (rule mult_assoc)
```
```    79 apply (rule mult_left_commute)
```
```    80 done
```
```    81
```
```    82 lemma scaleR_left_commute:
```
```    83   fixes x :: "'a::real_vector"
```
```    84   shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)"
```
```    85 by (simp add: mult_commute)
```
```    86
```
```    87 lemma additive_scaleR_right: "additive (\<lambda>x. scaleR a x::'a::real_vector)"
```
```    88 by (rule additive.intro, rule scaleR_right_distrib)
```
```    89
```
```    90 lemma additive_scaleR_left: "additive (\<lambda>a. scaleR a x::'a::real_vector)"
```
```    91 by (rule additive.intro, rule scaleR_left_distrib)
```
```    92
```
```    93 lemmas scaleR_zero_left [simp] =
```
```    94   additive.zero [OF additive_scaleR_left, standard]
```
```    95
```
```    96 lemmas scaleR_zero_right [simp] =
```
```    97   additive.zero [OF additive_scaleR_right, standard]
```
```    98
```
```    99 lemmas scaleR_minus_left [simp] =
```
```   100   additive.minus [OF additive_scaleR_left, standard]
```
```   101
```
```   102 lemmas scaleR_minus_right [simp] =
```
```   103   additive.minus [OF additive_scaleR_right, standard]
```
```   104
```
```   105 lemmas scaleR_left_diff_distrib =
```
```   106   additive.diff [OF additive_scaleR_left, standard]
```
```   107
```
```   108 lemmas scaleR_right_diff_distrib =
```
```   109   additive.diff [OF additive_scaleR_right, standard]
```
```   110
```
```   111 lemma scaleR_eq_0_iff:
```
```   112   fixes x :: "'a::real_vector"
```
```   113   shows "(scaleR a x = 0) = (a = 0 \<or> x = 0)"
```
```   114 proof cases
```
```   115   assume "a = 0" thus ?thesis by simp
```
```   116 next
```
```   117   assume anz [simp]: "a \<noteq> 0"
```
```   118   { assume "scaleR a x = 0"
```
```   119     hence "scaleR (inverse a) (scaleR a x) = 0" by simp
```
```   120     hence "x = 0" by simp }
```
```   121   thus ?thesis by force
```
```   122 qed
```
```   123
```
```   124 lemma scaleR_left_imp_eq:
```
```   125   fixes x y :: "'a::real_vector"
```
```   126   shows "\<lbrakk>a \<noteq> 0; scaleR a x = scaleR a y\<rbrakk> \<Longrightarrow> x = y"
```
```   127 proof -
```
```   128   assume nonzero: "a \<noteq> 0"
```
```   129   assume "scaleR a x = scaleR a y"
```
```   130   hence "scaleR a (x - y) = 0"
```
```   131      by (simp add: scaleR_right_diff_distrib)
```
```   132   hence "x - y = 0"
```
```   133      by (simp add: scaleR_eq_0_iff nonzero)
```
```   134   thus "x = y" by simp
```
```   135 qed
```
```   136
```
```   137 lemma scaleR_right_imp_eq:
```
```   138   fixes x y :: "'a::real_vector"
```
```   139   shows "\<lbrakk>x \<noteq> 0; scaleR a x = scaleR b x\<rbrakk> \<Longrightarrow> a = b"
```
```   140 proof -
```
```   141   assume nonzero: "x \<noteq> 0"
```
```   142   assume "scaleR a x = scaleR b x"
```
```   143   hence "scaleR (a - b) x = 0"
```
```   144      by (simp add: scaleR_left_diff_distrib)
```
```   145   hence "a - b = 0"
```
```   146      by (simp add: scaleR_eq_0_iff nonzero)
```
```   147   thus "a = b" by simp
```
```   148 qed
```
```   149
```
```   150 lemma scaleR_cancel_left:
```
```   151   fixes x y :: "'a::real_vector"
```
```   152   shows "(scaleR a x = scaleR a y) = (x = y \<or> a = 0)"
```
```   153 by (auto intro: scaleR_left_imp_eq)
```
```   154
```
```   155 lemma scaleR_cancel_right:
```
```   156   fixes x y :: "'a::real_vector"
```
```   157   shows "(scaleR a x = scaleR b x) = (a = b \<or> x = 0)"
```
```   158 by (auto intro: scaleR_right_imp_eq)
```
```   159
```
```   160 lemma nonzero_inverse_scaleR_distrib:
```
```   161   fixes x :: "'a::real_div_algebra" shows
```
```   162   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   163 by (rule inverse_unique, simp)
```
```   164
```
```   165 lemma inverse_scaleR_distrib:
```
```   166   fixes x :: "'a::{real_div_algebra,division_by_zero}"
```
```   167   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   168 apply (case_tac "a = 0", simp)
```
```   169 apply (case_tac "x = 0", simp)
```
```   170 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   171 done
```
```   172
```
```   173
```
```   174 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   175 @{term of_real} *}
```
```   176
```
```   177 definition
```
```   178   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   179   "of_real r = scaleR r 1"
```
```   180
```
```   181 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   182 by (simp add: of_real_def)
```
```   183
```
```   184 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   185 by (simp add: of_real_def)
```
```   186
```
```   187 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   188 by (simp add: of_real_def)
```
```   189
```
```   190 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   191 by (simp add: of_real_def scaleR_left_distrib)
```
```   192
```
```   193 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   194 by (simp add: of_real_def)
```
```   195
```
```   196 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   197 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   198
```
```   199 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   200 by (simp add: of_real_def mult_commute)
```
```   201
```
```   202 lemma nonzero_of_real_inverse:
```
```   203   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   204    inverse (of_real x :: 'a::real_div_algebra)"
```
```   205 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   206
```
```   207 lemma of_real_inverse [simp]:
```
```   208   "of_real (inverse x) =
```
```   209    inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
```
```   210 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   211
```
```   212 lemma nonzero_of_real_divide:
```
```   213   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   214    (of_real x / of_real y :: 'a::real_field)"
```
```   215 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   216
```
```   217 lemma of_real_divide [simp]:
```
```   218   "of_real (x / y) =
```
```   219    (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
```
```   220 by (simp add: divide_inverse)
```
```   221
```
```   222 lemma of_real_power [simp]:
```
```   223   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
```
```   224 by (induct n) (simp_all add: power_Suc)
```
```   225
```
```   226 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   227 by (simp add: of_real_def scaleR_cancel_right)
```
```   228
```
```   229 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   230
```
```   231 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   232 proof
```
```   233   fix r
```
```   234   show "of_real r = id r"
```
```   235     by (simp add: of_real_def real_scaleR_def)
```
```   236 qed
```
```   237
```
```   238 text{*Collapse nested embeddings*}
```
```   239 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   240 by (induct n) auto
```
```   241
```
```   242 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   243 by (cases z rule: int_diff_cases, simp)
```
```   244
```
```   245 lemma of_real_number_of_eq:
```
```   246   "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
```
```   247 by (simp add: number_of_eq)
```
```   248
```
```   249
```
```   250 subsection {* The Set of Real Numbers *}
```
```   251
```
```   252 definition
```
```   253   Reals :: "'a::real_algebra_1 set" where
```
```   254   "Reals \<equiv> range of_real"
```
```   255
```
```   256 notation (xsymbols)
```
```   257   Reals  ("\<real>")
```
```   258
```
```   259 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   260 by (simp add: Reals_def)
```
```   261
```
```   262 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   263 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   264
```
```   265 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   266 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   267
```
```   268 lemma Reals_number_of [simp]:
```
```   269   "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
```
```   270 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
```
```   271
```
```   272 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   273 apply (unfold Reals_def)
```
```   274 apply (rule range_eqI)
```
```   275 apply (rule of_real_0 [symmetric])
```
```   276 done
```
```   277
```
```   278 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   279 apply (unfold Reals_def)
```
```   280 apply (rule range_eqI)
```
```   281 apply (rule of_real_1 [symmetric])
```
```   282 done
```
```   283
```
```   284 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   285 apply (auto simp add: Reals_def)
```
```   286 apply (rule range_eqI)
```
```   287 apply (rule of_real_add [symmetric])
```
```   288 done
```
```   289
```
```   290 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   291 apply (auto simp add: Reals_def)
```
```   292 apply (rule range_eqI)
```
```   293 apply (rule of_real_minus [symmetric])
```
```   294 done
```
```   295
```
```   296 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   297 apply (auto simp add: Reals_def)
```
```   298 apply (rule range_eqI)
```
```   299 apply (rule of_real_diff [symmetric])
```
```   300 done
```
```   301
```
```   302 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   303 apply (auto simp add: Reals_def)
```
```   304 apply (rule range_eqI)
```
```   305 apply (rule of_real_mult [symmetric])
```
```   306 done
```
```   307
```
```   308 lemma nonzero_Reals_inverse:
```
```   309   fixes a :: "'a::real_div_algebra"
```
```   310   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   311 apply (auto simp add: Reals_def)
```
```   312 apply (rule range_eqI)
```
```   313 apply (erule nonzero_of_real_inverse [symmetric])
```
```   314 done
```
```   315
```
```   316 lemma Reals_inverse [simp]:
```
```   317   fixes a :: "'a::{real_div_algebra,division_by_zero}"
```
```   318   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   319 apply (auto simp add: Reals_def)
```
```   320 apply (rule range_eqI)
```
```   321 apply (rule of_real_inverse [symmetric])
```
```   322 done
```
```   323
```
```   324 lemma nonzero_Reals_divide:
```
```   325   fixes a b :: "'a::real_field"
```
```   326   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   327 apply (auto simp add: Reals_def)
```
```   328 apply (rule range_eqI)
```
```   329 apply (erule nonzero_of_real_divide [symmetric])
```
```   330 done
```
```   331
```
```   332 lemma Reals_divide [simp]:
```
```   333   fixes a b :: "'a::{real_field,division_by_zero}"
```
```   334   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   335 apply (auto simp add: Reals_def)
```
```   336 apply (rule range_eqI)
```
```   337 apply (rule of_real_divide [symmetric])
```
```   338 done
```
```   339
```
```   340 lemma Reals_power [simp]:
```
```   341   fixes a :: "'a::{real_algebra_1,recpower}"
```
```   342   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   343 apply (auto simp add: Reals_def)
```
```   344 apply (rule range_eqI)
```
```   345 apply (rule of_real_power [symmetric])
```
```   346 done
```
```   347
```
```   348 lemma Reals_cases [cases set: Reals]:
```
```   349   assumes "q \<in> \<real>"
```
```   350   obtains (of_real) r where "q = of_real r"
```
```   351   unfolding Reals_def
```
```   352 proof -
```
```   353   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   354   then obtain r where "q = of_real r" ..
```
```   355   then show thesis ..
```
```   356 qed
```
```   357
```
```   358 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   359   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   360   by (rule Reals_cases) auto
```
```   361
```
```   362
```
```   363 subsection {* Real normed vector spaces *}
```
```   364
```
```   365 axclass norm < type
```
```   366 consts norm :: "'a::norm \<Rightarrow> real"
```
```   367
```
```   368 instance real :: norm ..
```
```   369
```
```   370 defs (overloaded)
```
```   371   real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>"
```
```   372
```
```   373 axclass normed < plus, zero, norm
```
```   374   norm_ge_zero [simp]: "0 \<le> norm x"
```
```   375   norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
```
```   376   norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   377
```
```   378 axclass real_normed_vector < real_vector, normed
```
```   379   norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   380
```
```   381 axclass real_normed_algebra < real_algebra, real_normed_vector
```
```   382   norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   383
```
```   384 axclass real_normed_div_algebra < real_div_algebra, normed
```
```   385   norm_of_real: "norm (of_real r) = abs r"
```
```   386   norm_mult: "norm (x * y) = norm x * norm y"
```
```   387
```
```   388 axclass real_normed_field < real_field, real_normed_div_algebra
```
```   389
```
```   390 instance real_normed_div_algebra < real_normed_algebra
```
```   391 proof
```
```   392   fix a :: real and x :: 'a
```
```   393   have "norm (scaleR a x) = norm (of_real a * x)"
```
```   394     by (simp add: of_real_def)
```
```   395   also have "\<dots> = abs a * norm x"
```
```   396     by (simp add: norm_mult norm_of_real)
```
```   397   finally show "norm (scaleR a x) = abs a * norm x" .
```
```   398 next
```
```   399   fix x y :: 'a
```
```   400   show "norm (x * y) \<le> norm x * norm y"
```
```   401     by (simp add: norm_mult)
```
```   402 qed
```
```   403
```
```   404 instance real :: real_normed_field
```
```   405 apply (intro_classes, unfold real_norm_def)
```
```   406 apply (rule abs_ge_zero)
```
```   407 apply (rule abs_eq_0)
```
```   408 apply (rule abs_triangle_ineq)
```
```   409 apply simp
```
```   410 apply (rule abs_mult)
```
```   411 done
```
```   412
```
```   413 lemma norm_zero [simp]: "norm (0::'a::normed) = 0"
```
```   414 by simp
```
```   415
```
```   416 lemma zero_less_norm_iff [simp]: "(0 < norm x) = (x \<noteq> (0::'a::normed))"
```
```   417 by (simp add: order_less_le)
```
```   418
```
```   419 lemma norm_not_less_zero [simp]: "\<not> norm (x::'a::normed) < 0"
```
```   420 by (simp add: linorder_not_less)
```
```   421
```
```   422 lemma norm_le_zero_iff [simp]: "(norm x \<le> 0) = (x = (0::'a::normed))"
```
```   423 by (simp add: order_le_less)
```
```   424
```
```   425 lemma norm_minus_cancel [simp]:
```
```   426   fixes x :: "'a::real_normed_vector"
```
```   427   shows "norm (- x) = norm x"
```
```   428 proof -
```
```   429   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   430     by (simp only: scaleR_minus_left scaleR_one)
```
```   431   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   432     by (rule norm_scaleR)
```
```   433   finally show ?thesis by simp
```
```   434 qed
```
```   435
```
```   436 lemma norm_minus_commute:
```
```   437   fixes a b :: "'a::real_normed_vector"
```
```   438   shows "norm (a - b) = norm (b - a)"
```
```   439 proof -
```
```   440   have "norm (a - b) = norm (- (a - b))"
```
```   441     by (simp only: norm_minus_cancel)
```
```   442   also have "\<dots> = norm (b - a)" by simp
```
```   443   finally show ?thesis .
```
```   444 qed
```
```   445
```
```   446 lemma norm_triangle_ineq2:
```
```   447   fixes a b :: "'a::real_normed_vector"
```
```   448   shows "norm a - norm b \<le> norm (a - b)"
```
```   449 proof -
```
```   450   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   451     by (rule norm_triangle_ineq)
```
```   452   also have "(a - b + b) = a"
```
```   453     by simp
```
```   454   finally show ?thesis
```
```   455     by (simp add: compare_rls)
```
```   456 qed
```
```   457
```
```   458 lemma norm_triangle_ineq3:
```
```   459   fixes a b :: "'a::real_normed_vector"
```
```   460   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   461 apply (subst abs_le_iff)
```
```   462 apply auto
```
```   463 apply (rule norm_triangle_ineq2)
```
```   464 apply (subst norm_minus_commute)
```
```   465 apply (rule norm_triangle_ineq2)
```
```   466 done
```
```   467
```
```   468 lemma norm_triangle_ineq4:
```
```   469   fixes a b :: "'a::real_normed_vector"
```
```   470   shows "norm (a - b) \<le> norm a + norm b"
```
```   471 proof -
```
```   472   have "norm (a - b) = norm (a + - b)"
```
```   473     by (simp only: diff_minus)
```
```   474   also have "\<dots> \<le> norm a + norm (- b)"
```
```   475     by (rule norm_triangle_ineq)
```
```   476   finally show ?thesis
```
```   477     by simp
```
```   478 qed
```
```   479
```
```   480 lemma norm_diff_triangle_ineq:
```
```   481   fixes a b c d :: "'a::real_normed_vector"
```
```   482   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   483 proof -
```
```   484   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   485     by (simp add: diff_minus add_ac)
```
```   486   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   487     by (rule norm_triangle_ineq)
```
```   488   finally show ?thesis .
```
```   489 qed
```
```   490
```
```   491 lemma norm_one [simp]: "norm (1::'a::real_normed_div_algebra) = 1"
```
```   492 proof -
```
```   493   have "norm (of_real 1 :: 'a) = abs 1"
```
```   494     by (rule norm_of_real)
```
```   495   thus ?thesis by simp
```
```   496 qed
```
```   497
```
```   498 lemma nonzero_norm_inverse:
```
```   499   fixes a :: "'a::real_normed_div_algebra"
```
```   500   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   501 apply (rule inverse_unique [symmetric])
```
```   502 apply (simp add: norm_mult [symmetric])
```
```   503 done
```
```   504
```
```   505 lemma norm_inverse:
```
```   506   fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
```
```   507   shows "norm (inverse a) = inverse (norm a)"
```
```   508 apply (case_tac "a = 0", simp)
```
```   509 apply (erule nonzero_norm_inverse)
```
```   510 done
```
```   511
```
```   512 lemma nonzero_norm_divide:
```
```   513   fixes a b :: "'a::real_normed_field"
```
```   514   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   515 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   516
```
```   517 lemma norm_divide:
```
```   518   fixes a b :: "'a::{real_normed_field,division_by_zero}"
```
```   519   shows "norm (a / b) = norm a / norm b"
```
```   520 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   521
```
```   522 lemma norm_power:
```
```   523   fixes x :: "'a::{real_normed_div_algebra,recpower}"
```
```   524   shows "norm (x ^ n) = norm x ^ n"
```
```   525 by (induct n) (simp_all add: power_Suc norm_mult)
```
```   526
```
```   527
```
```   528 subsection {* Bounded Linear and Bilinear Operators *}
```
```   529
```
```   530 locale bounded_linear = additive +
```
```   531   constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   532   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```   533   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   534
```
```   535 lemma (in bounded_linear) pos_bounded:
```
```   536   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   537 proof -
```
```   538   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```   539     using bounded by fast
```
```   540   show ?thesis
```
```   541   proof (intro exI impI conjI allI)
```
```   542     show "0 < max 1 K"
```
```   543       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   544   next
```
```   545     fix x
```
```   546     have "norm (f x) \<le> norm x * K" using K .
```
```   547     also have "\<dots> \<le> norm x * max 1 K"
```
```   548       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```   549     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```   550   qed
```
```   551 qed
```
```   552
```
```   553 lemma (in bounded_linear) nonneg_bounded:
```
```   554   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   555 proof -
```
```   556   from pos_bounded
```
```   557   show ?thesis by (auto intro: order_less_imp_le)
```
```   558 qed
```
```   559
```
```   560 locale bounded_bilinear =
```
```   561   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```   562                  \<Rightarrow> 'c::real_normed_vector"
```
```   563     (infixl "**" 70)
```
```   564   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```   565   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```   566   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```   567   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```   568   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```   569
```
```   570 lemma (in bounded_bilinear) pos_bounded:
```
```   571   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   572 apply (cut_tac bounded, erule exE)
```
```   573 apply (rule_tac x="max 1 K" in exI, safe)
```
```   574 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   575 apply (drule spec, drule spec, erule order_trans)
```
```   576 apply (rule mult_left_mono [OF le_maxI2])
```
```   577 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```   578 done
```
```   579
```
```   580 lemma (in bounded_bilinear) nonneg_bounded:
```
```   581   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   582 proof -
```
```   583   from pos_bounded
```
```   584   show ?thesis by (auto intro: order_less_imp_le)
```
```   585 qed
```
```   586
```
```   587 lemma (in bounded_bilinear) additive_right: "additive (\<lambda>b. prod a b)"
```
```   588 by (rule additive.intro, rule add_right)
```
```   589
```
```   590 lemma (in bounded_bilinear) additive_left: "additive (\<lambda>a. prod a b)"
```
```   591 by (rule additive.intro, rule add_left)
```
```   592
```
```   593 lemma (in bounded_bilinear) zero_left: "prod 0 b = 0"
```
```   594 by (rule additive.zero [OF additive_left])
```
```   595
```
```   596 lemma (in bounded_bilinear) zero_right: "prod a 0 = 0"
```
```   597 by (rule additive.zero [OF additive_right])
```
```   598
```
```   599 lemma (in bounded_bilinear) minus_left: "prod (- a) b = - prod a b"
```
```   600 by (rule additive.minus [OF additive_left])
```
```   601
```
```   602 lemma (in bounded_bilinear) minus_right: "prod a (- b) = - prod a b"
```
```   603 by (rule additive.minus [OF additive_right])
```
```   604
```
```   605 lemma (in bounded_bilinear) diff_left:
```
```   606   "prod (a - a') b = prod a b - prod a' b"
```
```   607 by (rule additive.diff [OF additive_left])
```
```   608
```
```   609 lemma (in bounded_bilinear) diff_right:
```
```   610   "prod a (b - b') = prod a b - prod a b'"
```
```   611 by (rule additive.diff [OF additive_right])
```
```   612
```
```   613 lemma (in bounded_bilinear) bounded_linear_left:
```
```   614   "bounded_linear (\<lambda>a. a ** b)"
```
```   615 apply (unfold_locales)
```
```   616 apply (rule add_left)
```
```   617 apply (rule scaleR_left)
```
```   618 apply (cut_tac bounded, safe)
```
```   619 apply (rule_tac x="norm b * K" in exI)
```
```   620 apply (simp add: mult_ac)
```
```   621 done
```
```   622
```
```   623 lemma (in bounded_bilinear) bounded_linear_right:
```
```   624   "bounded_linear (\<lambda>b. a ** b)"
```
```   625 apply (unfold_locales)
```
```   626 apply (rule add_right)
```
```   627 apply (rule scaleR_right)
```
```   628 apply (cut_tac bounded, safe)
```
```   629 apply (rule_tac x="norm a * K" in exI)
```
```   630 apply (simp add: mult_ac)
```
```   631 done
```
```   632
```
```   633 lemma (in bounded_bilinear) prod_diff_prod:
```
```   634   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```   635 by (simp add: diff_left diff_right)
```
```   636
```
```   637 interpretation bounded_bilinear_mult:
```
```   638   bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
```
```   639 apply (rule bounded_bilinear.intro)
```
```   640 apply (rule left_distrib)
```
```   641 apply (rule right_distrib)
```
```   642 apply (rule mult_scaleR_left)
```
```   643 apply (rule mult_scaleR_right)
```
```   644 apply (rule_tac x="1" in exI)
```
```   645 apply (simp add: norm_mult_ineq)
```
```   646 done
```
```   647
```
```   648 interpretation bounded_linear_mult_left:
```
```   649   bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
```
```   650 by (rule bounded_bilinear_mult.bounded_linear_left)
```
```   651
```
```   652 interpretation bounded_linear_mult_right:
```
```   653   bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
```
```   654 by (rule bounded_bilinear_mult.bounded_linear_right)
```
```   655
```
```   656 interpretation bounded_bilinear_scaleR:
```
```   657   bounded_bilinear ["scaleR"]
```
```   658 apply (rule bounded_bilinear.intro)
```
```   659 apply (rule scaleR_left_distrib)
```
```   660 apply (rule scaleR_right_distrib)
```
```   661 apply (simp add: real_scaleR_def)
```
```   662 apply (rule scaleR_left_commute)
```
```   663 apply (rule_tac x="1" in exI)
```
```   664 apply (simp add: norm_scaleR)
```
```   665 done
```
```   666
```
```   667 interpretation bounded_linear_of_real:
```
```   668   bounded_linear ["\<lambda>r. of_real r"]
```
```   669 apply (unfold of_real_def)
```
```   670 apply (rule bounded_bilinear_scaleR.bounded_linear_left)
```
```   671 done
```
```   672
```
```   673 end
```