src/HOL/Real/RealVector.thy
 author huffman Mon May 28 04:22:44 2007 +0200 (2007-05-28) changeset 23113 d5cdaa3b7517 parent 22973 64d300e16370 child 23120 a34f204e9c88 permissions -rw-r--r--
```     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 lemma (in additive) setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
```
```    36 apply (cases "finite A")
```
```    37 apply (induct set: finite)
```
```    38 apply (simp add: zero)
```
```    39 apply (simp add: add)
```
```    40 apply (simp add: zero)
```
```    41 done
```
```    42
```
```    43
```
```    44 subsection {* Real vector spaces *}
```
```    45
```
```    46 class scaleR = type +
```
```    47   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"
```
```    48
```
```    49 notation
```
```    50   scaleR (infixr "*#" 75)
```
```    51
```
```    52 abbreviation
```
```    53   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a::scaleR" (infixl "'/#" 70) where
```
```    54   "x /# r == scaleR (inverse r) x"
```
```    55
```
```    56 notation (xsymbols)
```
```    57   scaleR (infixr "*\<^sub>R" 75) and
```
```    58   divideR (infixl "'/\<^sub>R" 70)
```
```    59
```
```    60 instance real :: scaleR
```
```    61   real_scaleR_def [simp]: "scaleR a x \<equiv> a * x" ..
```
```    62
```
```    63 axclass real_vector < scaleR, ab_group_add
```
```    64   scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```    65   scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```    66   scaleR_scaleR [simp]: "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```    67   scaleR_one [simp]: "scaleR 1 x = x"
```
```    68
```
```    69 axclass real_algebra < real_vector, ring
```
```    70   mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```    71   mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```    72
```
```    73 axclass real_algebra_1 < real_algebra, ring_1
```
```    74
```
```    75 axclass real_div_algebra < real_algebra_1, division_ring
```
```    76
```
```    77 axclass real_field < real_div_algebra, field
```
```    78
```
```    79 instance real :: real_field
```
```    80 apply (intro_classes, unfold real_scaleR_def)
```
```    81 apply (rule right_distrib)
```
```    82 apply (rule left_distrib)
```
```    83 apply (rule mult_assoc [symmetric])
```
```    84 apply (rule mult_1_left)
```
```    85 apply (rule mult_assoc)
```
```    86 apply (rule mult_left_commute)
```
```    87 done
```
```    88
```
```    89 lemma scaleR_left_commute:
```
```    90   fixes x :: "'a::real_vector"
```
```    91   shows "scaleR a (scaleR b x) = scaleR b (scaleR a x)"
```
```    92 by (simp add: mult_commute)
```
```    93
```
```    94 interpretation additive_scaleR_left:
```
```    95   additive ["(\<lambda>a. scaleR a x::'a::real_vector)"]
```
```    96 by (rule additive.intro, rule scaleR_left_distrib)
```
```    97
```
```    98 interpretation additive_scaleR_right:
```
```    99   additive ["(\<lambda>x. scaleR a x::'a::real_vector)"]
```
```   100 by (rule additive.intro, rule scaleR_right_distrib)
```
```   101
```
```   102 lemmas scaleR_zero_left [simp] = additive_scaleR_left.zero
```
```   103
```
```   104 lemmas scaleR_zero_right [simp] = additive_scaleR_right.zero
```
```   105
```
```   106 lemmas scaleR_minus_left [simp] = additive_scaleR_left.minus
```
```   107
```
```   108 lemmas scaleR_minus_right [simp] = additive_scaleR_right.minus
```
```   109
```
```   110 lemmas scaleR_left_diff_distrib = additive_scaleR_left.diff
```
```   111
```
```   112 lemmas scaleR_right_diff_distrib = additive_scaleR_right.diff
```
```   113
```
```   114 lemma scaleR_eq_0_iff [simp]:
```
```   115   fixes x :: "'a::real_vector"
```
```   116   shows "(scaleR a x = 0) = (a = 0 \<or> x = 0)"
```
```   117 proof cases
```
```   118   assume "a = 0" thus ?thesis by simp
```
```   119 next
```
```   120   assume anz [simp]: "a \<noteq> 0"
```
```   121   { assume "scaleR a x = 0"
```
```   122     hence "scaleR (inverse a) (scaleR a x) = 0" by simp
```
```   123     hence "x = 0" by simp }
```
```   124   thus ?thesis by force
```
```   125 qed
```
```   126
```
```   127 lemma scaleR_left_imp_eq:
```
```   128   fixes x y :: "'a::real_vector"
```
```   129   shows "\<lbrakk>a \<noteq> 0; scaleR a x = scaleR a y\<rbrakk> \<Longrightarrow> x = y"
```
```   130 proof -
```
```   131   assume nonzero: "a \<noteq> 0"
```
```   132   assume "scaleR a x = scaleR a y"
```
```   133   hence "scaleR a (x - y) = 0"
```
```   134      by (simp add: scaleR_right_diff_distrib)
```
```   135   hence "x - y = 0" by (simp add: nonzero)
```
```   136   thus "x = y" by simp
```
```   137 qed
```
```   138
```
```   139 lemma scaleR_right_imp_eq:
```
```   140   fixes x y :: "'a::real_vector"
```
```   141   shows "\<lbrakk>x \<noteq> 0; scaleR a x = scaleR b x\<rbrakk> \<Longrightarrow> a = b"
```
```   142 proof -
```
```   143   assume nonzero: "x \<noteq> 0"
```
```   144   assume "scaleR a x = scaleR b x"
```
```   145   hence "scaleR (a - b) x = 0"
```
```   146      by (simp add: scaleR_left_diff_distrib)
```
```   147   hence "a - b = 0" by (simp add: nonzero)
```
```   148   thus "a = b" by simp
```
```   149 qed
```
```   150
```
```   151 lemma scaleR_cancel_left:
```
```   152   fixes x y :: "'a::real_vector"
```
```   153   shows "(scaleR a x = scaleR a y) = (x = y \<or> a = 0)"
```
```   154 by (auto intro: scaleR_left_imp_eq)
```
```   155
```
```   156 lemma scaleR_cancel_right:
```
```   157   fixes x y :: "'a::real_vector"
```
```   158   shows "(scaleR a x = scaleR b x) = (a = b \<or> x = 0)"
```
```   159 by (auto intro: scaleR_right_imp_eq)
```
```   160
```
```   161 lemma nonzero_inverse_scaleR_distrib:
```
```   162   fixes x :: "'a::real_div_algebra" shows
```
```   163   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   164 by (rule inverse_unique, simp)
```
```   165
```
```   166 lemma inverse_scaleR_distrib:
```
```   167   fixes x :: "'a::{real_div_algebra,division_by_zero}"
```
```   168   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   169 apply (case_tac "a = 0", simp)
```
```   170 apply (case_tac "x = 0", simp)
```
```   171 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   172 done
```
```   173
```
```   174
```
```   175 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   176 @{term of_real} *}
```
```   177
```
```   178 definition
```
```   179   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   180   "of_real r = scaleR r 1"
```
```   181
```
```   182 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   183 by (simp add: of_real_def)
```
```   184
```
```   185 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   186 by (simp add: of_real_def)
```
```   187
```
```   188 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   189 by (simp add: of_real_def)
```
```   190
```
```   191 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   192 by (simp add: of_real_def scaleR_left_distrib)
```
```   193
```
```   194 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   195 by (simp add: of_real_def)
```
```   196
```
```   197 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   198 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   199
```
```   200 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   201 by (simp add: of_real_def mult_commute)
```
```   202
```
```   203 lemma nonzero_of_real_inverse:
```
```   204   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   205    inverse (of_real x :: 'a::real_div_algebra)"
```
```   206 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   207
```
```   208 lemma of_real_inverse [simp]:
```
```   209   "of_real (inverse x) =
```
```   210    inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
```
```   211 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   212
```
```   213 lemma nonzero_of_real_divide:
```
```   214   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   215    (of_real x / of_real y :: 'a::real_field)"
```
```   216 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   217
```
```   218 lemma of_real_divide [simp]:
```
```   219   "of_real (x / y) =
```
```   220    (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
```
```   221 by (simp add: divide_inverse)
```
```   222
```
```   223 lemma of_real_power [simp]:
```
```   224   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
```
```   225 by (induct n) (simp_all add: power_Suc)
```
```   226
```
```   227 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   228 by (simp add: of_real_def scaleR_cancel_right)
```
```   229
```
```   230 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   231
```
```   232 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   233 proof
```
```   234   fix r
```
```   235   show "of_real r = id r"
```
```   236     by (simp add: of_real_def)
```
```   237 qed
```
```   238
```
```   239 text{*Collapse nested embeddings*}
```
```   240 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   241 by (induct n) auto
```
```   242
```
```   243 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   244 by (cases z rule: int_diff_cases, simp)
```
```   245
```
```   246 lemma of_real_number_of_eq:
```
```   247   "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
```
```   248 by (simp add: number_of_eq)
```
```   249
```
```   250 text{*Every real algebra has characteristic zero*}
```
```   251 instance real_algebra_1 < ring_char_0
```
```   252 proof
```
```   253   fix w z :: int
```
```   254   assume "of_int w = (of_int z::'a)"
```
```   255   hence "of_real (of_int w) = (of_real (of_int z)::'a)"
```
```   256     by (simp only: of_real_of_int_eq)
```
```   257   thus "w = z"
```
```   258     by (simp only: of_real_eq_iff of_int_eq_iff)
```
```   259 qed
```
```   260
```
```   261
```
```   262 subsection {* The Set of Real Numbers *}
```
```   263
```
```   264 definition
```
```   265   Reals :: "'a::real_algebra_1 set" where
```
```   266   "Reals \<equiv> range of_real"
```
```   267
```
```   268 notation (xsymbols)
```
```   269   Reals  ("\<real>")
```
```   270
```
```   271 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   272 by (simp add: Reals_def)
```
```   273
```
```   274 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   275 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   276
```
```   277 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   278 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   279
```
```   280 lemma Reals_number_of [simp]:
```
```   281   "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
```
```   282 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
```
```   283
```
```   284 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   285 apply (unfold Reals_def)
```
```   286 apply (rule range_eqI)
```
```   287 apply (rule of_real_0 [symmetric])
```
```   288 done
```
```   289
```
```   290 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   291 apply (unfold Reals_def)
```
```   292 apply (rule range_eqI)
```
```   293 apply (rule of_real_1 [symmetric])
```
```   294 done
```
```   295
```
```   296 lemma Reals_add [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_add [symmetric])
```
```   300 done
```
```   301
```
```   302 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   303 apply (auto simp add: Reals_def)
```
```   304 apply (rule range_eqI)
```
```   305 apply (rule of_real_minus [symmetric])
```
```   306 done
```
```   307
```
```   308 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   309 apply (auto simp add: Reals_def)
```
```   310 apply (rule range_eqI)
```
```   311 apply (rule of_real_diff [symmetric])
```
```   312 done
```
```   313
```
```   314 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   315 apply (auto simp add: Reals_def)
```
```   316 apply (rule range_eqI)
```
```   317 apply (rule of_real_mult [symmetric])
```
```   318 done
```
```   319
```
```   320 lemma nonzero_Reals_inverse:
```
```   321   fixes a :: "'a::real_div_algebra"
```
```   322   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   323 apply (auto simp add: Reals_def)
```
```   324 apply (rule range_eqI)
```
```   325 apply (erule nonzero_of_real_inverse [symmetric])
```
```   326 done
```
```   327
```
```   328 lemma Reals_inverse [simp]:
```
```   329   fixes a :: "'a::{real_div_algebra,division_by_zero}"
```
```   330   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   331 apply (auto simp add: Reals_def)
```
```   332 apply (rule range_eqI)
```
```   333 apply (rule of_real_inverse [symmetric])
```
```   334 done
```
```   335
```
```   336 lemma nonzero_Reals_divide:
```
```   337   fixes a b :: "'a::real_field"
```
```   338   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   339 apply (auto simp add: Reals_def)
```
```   340 apply (rule range_eqI)
```
```   341 apply (erule nonzero_of_real_divide [symmetric])
```
```   342 done
```
```   343
```
```   344 lemma Reals_divide [simp]:
```
```   345   fixes a b :: "'a::{real_field,division_by_zero}"
```
```   346   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   347 apply (auto simp add: Reals_def)
```
```   348 apply (rule range_eqI)
```
```   349 apply (rule of_real_divide [symmetric])
```
```   350 done
```
```   351
```
```   352 lemma Reals_power [simp]:
```
```   353   fixes a :: "'a::{real_algebra_1,recpower}"
```
```   354   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   355 apply (auto simp add: Reals_def)
```
```   356 apply (rule range_eqI)
```
```   357 apply (rule of_real_power [symmetric])
```
```   358 done
```
```   359
```
```   360 lemma Reals_cases [cases set: Reals]:
```
```   361   assumes "q \<in> \<real>"
```
```   362   obtains (of_real) r where "q = of_real r"
```
```   363   unfolding Reals_def
```
```   364 proof -
```
```   365   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   366   then obtain r where "q = of_real r" ..
```
```   367   then show thesis ..
```
```   368 qed
```
```   369
```
```   370 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   371   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   372   by (rule Reals_cases) auto
```
```   373
```
```   374
```
```   375 subsection {* Real normed vector spaces *}
```
```   376
```
```   377 class norm = type +
```
```   378   fixes norm :: "'a \<Rightarrow> real"
```
```   379
```
```   380 instance real :: norm
```
```   381   real_norm_def [simp]: "norm r \<equiv> \<bar>r\<bar>" ..
```
```   382
```
```   383 axclass real_normed_vector < real_vector, norm
```
```   384   norm_ge_zero [simp]: "0 \<le> norm x"
```
```   385   norm_eq_zero [simp]: "(norm x = 0) = (x = 0)"
```
```   386   norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   387   norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   388
```
```   389 axclass real_normed_algebra < real_algebra, real_normed_vector
```
```   390   norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   391
```
```   392 axclass real_normed_algebra_1 < real_algebra_1, real_normed_algebra
```
```   393   norm_one [simp]: "norm 1 = 1"
```
```   394
```
```   395 axclass real_normed_div_algebra < real_div_algebra, real_normed_vector
```
```   396   norm_mult: "norm (x * y) = norm x * norm y"
```
```   397
```
```   398 axclass real_normed_field < real_field, real_normed_div_algebra
```
```   399
```
```   400 instance real_normed_div_algebra < real_normed_algebra_1
```
```   401 proof
```
```   402   fix x y :: 'a
```
```   403   show "norm (x * y) \<le> norm x * norm y"
```
```   404     by (simp add: norm_mult)
```
```   405 next
```
```   406   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   407     by (rule norm_mult)
```
```   408   thus "norm (1::'a) = 1" by simp
```
```   409 qed
```
```   410
```
```   411 instance real :: real_normed_field
```
```   412 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```   413 apply (rule abs_ge_zero)
```
```   414 apply (rule abs_eq_0)
```
```   415 apply (rule abs_triangle_ineq)
```
```   416 apply (rule abs_mult)
```
```   417 apply (rule abs_mult)
```
```   418 done
```
```   419
```
```   420 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   421 by simp
```
```   422
```
```   423 lemma zero_less_norm_iff [simp]:
```
```   424   fixes x :: "'a::real_normed_vector"
```
```   425   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   426 by (simp add: order_less_le)
```
```   427
```
```   428 lemma norm_not_less_zero [simp]:
```
```   429   fixes x :: "'a::real_normed_vector"
```
```   430   shows "\<not> norm x < 0"
```
```   431 by (simp add: linorder_not_less)
```
```   432
```
```   433 lemma norm_le_zero_iff [simp]:
```
```   434   fixes x :: "'a::real_normed_vector"
```
```   435   shows "(norm x \<le> 0) = (x = 0)"
```
```   436 by (simp add: order_le_less)
```
```   437
```
```   438 lemma norm_minus_cancel [simp]:
```
```   439   fixes x :: "'a::real_normed_vector"
```
```   440   shows "norm (- x) = norm x"
```
```   441 proof -
```
```   442   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   443     by (simp only: scaleR_minus_left scaleR_one)
```
```   444   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   445     by (rule norm_scaleR)
```
```   446   finally show ?thesis by simp
```
```   447 qed
```
```   448
```
```   449 lemma norm_minus_commute:
```
```   450   fixes a b :: "'a::real_normed_vector"
```
```   451   shows "norm (a - b) = norm (b - a)"
```
```   452 proof -
```
```   453   have "norm (- (b - a)) = norm (b - a)"
```
```   454     by (rule norm_minus_cancel)
```
```   455   thus ?thesis by simp
```
```   456 qed
```
```   457
```
```   458 lemma norm_triangle_ineq2:
```
```   459   fixes a b :: "'a::real_normed_vector"
```
```   460   shows "norm a - norm b \<le> norm (a - b)"
```
```   461 proof -
```
```   462   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   463     by (rule norm_triangle_ineq)
```
```   464   thus ?thesis by simp
```
```   465 qed
```
```   466
```
```   467 lemma norm_triangle_ineq3:
```
```   468   fixes a b :: "'a::real_normed_vector"
```
```   469   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   470 apply (subst abs_le_iff)
```
```   471 apply auto
```
```   472 apply (rule norm_triangle_ineq2)
```
```   473 apply (subst norm_minus_commute)
```
```   474 apply (rule norm_triangle_ineq2)
```
```   475 done
```
```   476
```
```   477 lemma norm_triangle_ineq4:
```
```   478   fixes a b :: "'a::real_normed_vector"
```
```   479   shows "norm (a - b) \<le> norm a + norm b"
```
```   480 proof -
```
```   481   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   482     by (rule norm_triangle_ineq)
```
```   483   thus ?thesis
```
```   484     by (simp only: diff_minus norm_minus_cancel)
```
```   485 qed
```
```   486
```
```   487 lemma norm_diff_ineq:
```
```   488   fixes a b :: "'a::real_normed_vector"
```
```   489   shows "norm a - norm b \<le> norm (a + b)"
```
```   490 proof -
```
```   491   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   492     by (rule norm_triangle_ineq2)
```
```   493   thus ?thesis by simp
```
```   494 qed
```
```   495
```
```   496 lemma norm_diff_triangle_ineq:
```
```   497   fixes a b c d :: "'a::real_normed_vector"
```
```   498   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   499 proof -
```
```   500   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   501     by (simp add: diff_minus add_ac)
```
```   502   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   503     by (rule norm_triangle_ineq)
```
```   504   finally show ?thesis .
```
```   505 qed
```
```   506
```
```   507 lemma abs_norm_cancel [simp]:
```
```   508   fixes a :: "'a::real_normed_vector"
```
```   509   shows "\<bar>norm a\<bar> = norm a"
```
```   510 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   511
```
```   512 lemma norm_add_less:
```
```   513   fixes x y :: "'a::real_normed_vector"
```
```   514   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   515 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   516
```
```   517 lemma norm_mult_less:
```
```   518   fixes x y :: "'a::real_normed_algebra"
```
```   519   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   520 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   521 apply (simp add: mult_strict_mono')
```
```   522 done
```
```   523
```
```   524 lemma norm_of_real [simp]:
```
```   525   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   526 unfolding of_real_def by (simp add: norm_scaleR)
```
```   527
```
```   528 lemma norm_number_of [simp]:
```
```   529   "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
```
```   530     = \<bar>number_of w\<bar>"
```
```   531 by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
```
```   532
```
```   533 lemma norm_of_int [simp]:
```
```   534   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   535 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   536
```
```   537 lemma norm_of_nat [simp]:
```
```   538   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   539 apply (subst of_real_of_nat_eq [symmetric])
```
```   540 apply (subst norm_of_real, simp)
```
```   541 done
```
```   542
```
```   543 lemma nonzero_norm_inverse:
```
```   544   fixes a :: "'a::real_normed_div_algebra"
```
```   545   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   546 apply (rule inverse_unique [symmetric])
```
```   547 apply (simp add: norm_mult [symmetric])
```
```   548 done
```
```   549
```
```   550 lemma norm_inverse:
```
```   551   fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
```
```   552   shows "norm (inverse a) = inverse (norm a)"
```
```   553 apply (case_tac "a = 0", simp)
```
```   554 apply (erule nonzero_norm_inverse)
```
```   555 done
```
```   556
```
```   557 lemma nonzero_norm_divide:
```
```   558   fixes a b :: "'a::real_normed_field"
```
```   559   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   560 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   561
```
```   562 lemma norm_divide:
```
```   563   fixes a b :: "'a::{real_normed_field,division_by_zero}"
```
```   564   shows "norm (a / b) = norm a / norm b"
```
```   565 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   566
```
```   567 lemma norm_power_ineq:
```
```   568   fixes x :: "'a::{real_normed_algebra_1,recpower}"
```
```   569   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   570 proof (induct n)
```
```   571   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   572 next
```
```   573   case (Suc n)
```
```   574   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   575     by (rule norm_mult_ineq)
```
```   576   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   577     using norm_ge_zero by (rule mult_left_mono)
```
```   578   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   579     by (simp add: power_Suc)
```
```   580 qed
```
```   581
```
```   582 lemma norm_power:
```
```   583   fixes x :: "'a::{real_normed_div_algebra,recpower}"
```
```   584   shows "norm (x ^ n) = norm x ^ n"
```
```   585 by (induct n) (simp_all add: power_Suc norm_mult)
```
```   586
```
```   587
```
```   588 subsection {* Sign function *}
```
```   589
```
```   590 definition
```
```   591   sgn :: "'a::real_normed_vector \<Rightarrow> 'a" where
```
```   592   "sgn x = scaleR (inverse (norm x)) x"
```
```   593
```
```   594 lemma norm_sgn: "norm (sgn x) = (if x = 0 then 0 else 1)"
```
```   595 unfolding sgn_def by (simp add: norm_scaleR)
```
```   596
```
```   597 lemma sgn_zero [simp]: "sgn 0 = 0"
```
```   598 unfolding sgn_def by simp
```
```   599
```
```   600 lemma sgn_zero_iff: "(sgn x = 0) = (x = 0)"
```
```   601 unfolding sgn_def by simp
```
```   602
```
```   603 lemma sgn_minus: "sgn (- x) = - sgn x"
```
```   604 unfolding sgn_def by simp
```
```   605
```
```   606 lemma sgn_scaleR: "sgn (scaleR r x) = scaleR (sgn r) (sgn x)"
```
```   607 unfolding sgn_def by (simp add: norm_scaleR mult_ac)
```
```   608
```
```   609 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```   610 unfolding sgn_def by simp
```
```   611
```
```   612 lemma sgn_of_real:
```
```   613   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```   614 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```   615
```
```   616 lemma sgn_mult:
```
```   617   fixes x y :: "'a::real_normed_div_algebra"
```
```   618   shows "sgn (x * y) = sgn x * sgn y"
```
```   619 unfolding sgn_def by (simp add: norm_mult mult_commute)
```
```   620
```
```   621 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```   622 unfolding sgn_def by (simp add: divide_inverse)
```
```   623
```
```   624 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```   625 unfolding real_sgn_eq by simp
```
```   626
```
```   627 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```   628 unfolding real_sgn_eq by simp
```
```   629
```
```   630
```
```   631 subsection {* Bounded Linear and Bilinear Operators *}
```
```   632
```
```   633 locale bounded_linear = additive +
```
```   634   constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   635   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```   636   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   637
```
```   638 lemma (in bounded_linear) pos_bounded:
```
```   639   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   640 proof -
```
```   641   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```   642     using bounded by fast
```
```   643   show ?thesis
```
```   644   proof (intro exI impI conjI allI)
```
```   645     show "0 < max 1 K"
```
```   646       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   647   next
```
```   648     fix x
```
```   649     have "norm (f x) \<le> norm x * K" using K .
```
```   650     also have "\<dots> \<le> norm x * max 1 K"
```
```   651       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```   652     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```   653   qed
```
```   654 qed
```
```   655
```
```   656 lemma (in bounded_linear) nonneg_bounded:
```
```   657   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   658 proof -
```
```   659   from pos_bounded
```
```   660   show ?thesis by (auto intro: order_less_imp_le)
```
```   661 qed
```
```   662
```
```   663 locale bounded_bilinear =
```
```   664   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```   665                  \<Rightarrow> 'c::real_normed_vector"
```
```   666     (infixl "**" 70)
```
```   667   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```   668   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```   669   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```   670   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```   671   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```   672
```
```   673 lemma (in bounded_bilinear) pos_bounded:
```
```   674   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   675 apply (cut_tac bounded, erule exE)
```
```   676 apply (rule_tac x="max 1 K" in exI, safe)
```
```   677 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   678 apply (drule spec, drule spec, erule order_trans)
```
```   679 apply (rule mult_left_mono [OF le_maxI2])
```
```   680 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```   681 done
```
```   682
```
```   683 lemma (in bounded_bilinear) nonneg_bounded:
```
```   684   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   685 proof -
```
```   686   from pos_bounded
```
```   687   show ?thesis by (auto intro: order_less_imp_le)
```
```   688 qed
```
```   689
```
```   690 lemma (in bounded_bilinear) additive_right: "additive (\<lambda>b. prod a b)"
```
```   691 by (rule additive.intro, rule add_right)
```
```   692
```
```   693 lemma (in bounded_bilinear) additive_left: "additive (\<lambda>a. prod a b)"
```
```   694 by (rule additive.intro, rule add_left)
```
```   695
```
```   696 lemma (in bounded_bilinear) zero_left: "prod 0 b = 0"
```
```   697 by (rule additive.zero [OF additive_left])
```
```   698
```
```   699 lemma (in bounded_bilinear) zero_right: "prod a 0 = 0"
```
```   700 by (rule additive.zero [OF additive_right])
```
```   701
```
```   702 lemma (in bounded_bilinear) minus_left: "prod (- a) b = - prod a b"
```
```   703 by (rule additive.minus [OF additive_left])
```
```   704
```
```   705 lemma (in bounded_bilinear) minus_right: "prod a (- b) = - prod a b"
```
```   706 by (rule additive.minus [OF additive_right])
```
```   707
```
```   708 lemma (in bounded_bilinear) diff_left:
```
```   709   "prod (a - a') b = prod a b - prod a' b"
```
```   710 by (rule additive.diff [OF additive_left])
```
```   711
```
```   712 lemma (in bounded_bilinear) diff_right:
```
```   713   "prod a (b - b') = prod a b - prod a b'"
```
```   714 by (rule additive.diff [OF additive_right])
```
```   715
```
```   716 lemma (in bounded_bilinear) bounded_linear_left:
```
```   717   "bounded_linear (\<lambda>a. a ** b)"
```
```   718 apply (unfold_locales)
```
```   719 apply (rule add_left)
```
```   720 apply (rule scaleR_left)
```
```   721 apply (cut_tac bounded, safe)
```
```   722 apply (rule_tac x="norm b * K" in exI)
```
```   723 apply (simp add: mult_ac)
```
```   724 done
```
```   725
```
```   726 lemma (in bounded_bilinear) bounded_linear_right:
```
```   727   "bounded_linear (\<lambda>b. a ** b)"
```
```   728 apply (unfold_locales)
```
```   729 apply (rule add_right)
```
```   730 apply (rule scaleR_right)
```
```   731 apply (cut_tac bounded, safe)
```
```   732 apply (rule_tac x="norm a * K" in exI)
```
```   733 apply (simp add: mult_ac)
```
```   734 done
```
```   735
```
```   736 lemma (in bounded_bilinear) prod_diff_prod:
```
```   737   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```   738 by (simp add: diff_left diff_right)
```
```   739
```
```   740 interpretation bounded_bilinear_mult:
```
```   741   bounded_bilinear ["op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"]
```
```   742 apply (rule bounded_bilinear.intro)
```
```   743 apply (rule left_distrib)
```
```   744 apply (rule right_distrib)
```
```   745 apply (rule mult_scaleR_left)
```
```   746 apply (rule mult_scaleR_right)
```
```   747 apply (rule_tac x="1" in exI)
```
```   748 apply (simp add: norm_mult_ineq)
```
```   749 done
```
```   750
```
```   751 interpretation bounded_linear_mult_left:
```
```   752   bounded_linear ["(\<lambda>x::'a::real_normed_algebra. x * y)"]
```
```   753 by (rule bounded_bilinear_mult.bounded_linear_left)
```
```   754
```
```   755 interpretation bounded_linear_mult_right:
```
```   756   bounded_linear ["(\<lambda>y::'a::real_normed_algebra. x * y)"]
```
```   757 by (rule bounded_bilinear_mult.bounded_linear_right)
```
```   758
```
```   759 interpretation bounded_bilinear_scaleR:
```
```   760   bounded_bilinear ["scaleR"]
```
```   761 apply (rule bounded_bilinear.intro)
```
```   762 apply (rule scaleR_left_distrib)
```
```   763 apply (rule scaleR_right_distrib)
```
```   764 apply simp
```
```   765 apply (rule scaleR_left_commute)
```
```   766 apply (rule_tac x="1" in exI)
```
```   767 apply (simp add: norm_scaleR)
```
```   768 done
```
```   769
```
```   770 interpretation bounded_linear_of_real:
```
```   771   bounded_linear ["\<lambda>r. of_real r"]
```
```   772 apply (unfold of_real_def)
```
```   773 apply (rule bounded_bilinear_scaleR.bounded_linear_left)
```
```   774 done
```
```   775
```
```   776 end
```