src/HOL/RealVector.thy
 author nipkow Fri Mar 06 17:38:47 2009 +0100 (2009-03-06) changeset 30313 b2441b0c8d38 parent 30273 ecd6f0ca62ea child 30729 461ee3e49ad3 permissions -rw-r--r--
```     1 (*  Title:      HOL/RealVector.thy
```
```     2     Author:     Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Vector Spaces and Algebras over the Reals *}
```
```     6
```
```     7 theory RealVector
```
```     8 imports RealPow
```
```     9 begin
```
```    10
```
```    11 subsection {* Locale for additive functions *}
```
```    12
```
```    13 locale additive =
```
```    14   fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
```
```    15   assumes add: "f (x + y) = f x + f y"
```
```    16 begin
```
```    17
```
```    18 lemma 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 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 diff: "f (x - y) = f x - f y"
```
```    33 by (simp add: diff_def add minus)
```
```    34
```
```    35 lemma 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 end
```
```    44
```
```    45 subsection {* Vector spaces *}
```
```    46
```
```    47 locale vector_space =
```
```    48   fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
```
```    49   assumes scale_right_distrib [algebra_simps]:
```
```    50     "scale a (x + y) = scale a x + scale a y"
```
```    51   and scale_left_distrib [algebra_simps]:
```
```    52     "scale (a + b) x = scale a x + scale b x"
```
```    53   and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
```
```    54   and scale_one [simp]: "scale 1 x = x"
```
```    55 begin
```
```    56
```
```    57 lemma scale_left_commute:
```
```    58   "scale a (scale b x) = scale b (scale a x)"
```
```    59 by (simp add: mult_commute)
```
```    60
```
```    61 lemma scale_zero_left [simp]: "scale 0 x = 0"
```
```    62   and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
```
```    63   and scale_left_diff_distrib [algebra_simps]:
```
```    64         "scale (a - b) x = scale a x - scale b x"
```
```    65 proof -
```
```    66   interpret s: additive "\<lambda>a. scale a x"
```
```    67     proof qed (rule scale_left_distrib)
```
```    68   show "scale 0 x = 0" by (rule s.zero)
```
```    69   show "scale (- a) x = - (scale a x)" by (rule s.minus)
```
```    70   show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
```
```    71 qed
```
```    72
```
```    73 lemma scale_zero_right [simp]: "scale a 0 = 0"
```
```    74   and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
```
```    75   and scale_right_diff_distrib [algebra_simps]:
```
```    76         "scale a (x - y) = scale a x - scale a y"
```
```    77 proof -
```
```    78   interpret s: additive "\<lambda>x. scale a x"
```
```    79     proof qed (rule scale_right_distrib)
```
```    80   show "scale a 0 = 0" by (rule s.zero)
```
```    81   show "scale a (- x) = - (scale a x)" by (rule s.minus)
```
```    82   show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
```
```    83 qed
```
```    84
```
```    85 lemma scale_eq_0_iff [simp]:
```
```    86   "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
```
```    87 proof cases
```
```    88   assume "a = 0" thus ?thesis by simp
```
```    89 next
```
```    90   assume anz [simp]: "a \<noteq> 0"
```
```    91   { assume "scale a x = 0"
```
```    92     hence "scale (inverse a) (scale a x) = 0" by simp
```
```    93     hence "x = 0" by simp }
```
```    94   thus ?thesis by force
```
```    95 qed
```
```    96
```
```    97 lemma scale_left_imp_eq:
```
```    98   "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
```
```    99 proof -
```
```   100   assume nonzero: "a \<noteq> 0"
```
```   101   assume "scale a x = scale a y"
```
```   102   hence "scale a (x - y) = 0"
```
```   103      by (simp add: scale_right_diff_distrib)
```
```   104   hence "x - y = 0" by (simp add: nonzero)
```
```   105   thus "x = y" by (simp only: right_minus_eq)
```
```   106 qed
```
```   107
```
```   108 lemma scale_right_imp_eq:
```
```   109   "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
```
```   110 proof -
```
```   111   assume nonzero: "x \<noteq> 0"
```
```   112   assume "scale a x = scale b x"
```
```   113   hence "scale (a - b) x = 0"
```
```   114      by (simp add: scale_left_diff_distrib)
```
```   115   hence "a - b = 0" by (simp add: nonzero)
```
```   116   thus "a = b" by (simp only: right_minus_eq)
```
```   117 qed
```
```   118
```
```   119 lemma scale_cancel_left:
```
```   120   "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
```
```   121 by (auto intro: scale_left_imp_eq)
```
```   122
```
```   123 lemma scale_cancel_right:
```
```   124   "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
```
```   125 by (auto intro: scale_right_imp_eq)
```
```   126
```
```   127 end
```
```   128
```
```   129 subsection {* Real vector spaces *}
```
```   130
```
```   131 class scaleR =
```
```   132   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
```
```   133 begin
```
```   134
```
```   135 abbreviation
```
```   136   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
```
```   137 where
```
```   138   "x /\<^sub>R r == scaleR (inverse r) x"
```
```   139
```
```   140 end
```
```   141
```
```   142 class real_vector = scaleR + ab_group_add +
```
```   143   assumes scaleR_right_distrib: "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```   144   and scaleR_left_distrib: "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```   145   and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```   146   and scaleR_one: "scaleR 1 x = x"
```
```   147
```
```   148 interpretation real_vector!:
```
```   149   vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
```
```   150 apply unfold_locales
```
```   151 apply (rule scaleR_right_distrib)
```
```   152 apply (rule scaleR_left_distrib)
```
```   153 apply (rule scaleR_scaleR)
```
```   154 apply (rule scaleR_one)
```
```   155 done
```
```   156
```
```   157 text {* Recover original theorem names *}
```
```   158
```
```   159 lemmas scaleR_left_commute = real_vector.scale_left_commute
```
```   160 lemmas scaleR_zero_left = real_vector.scale_zero_left
```
```   161 lemmas scaleR_minus_left = real_vector.scale_minus_left
```
```   162 lemmas scaleR_left_diff_distrib = real_vector.scale_left_diff_distrib
```
```   163 lemmas scaleR_zero_right = real_vector.scale_zero_right
```
```   164 lemmas scaleR_minus_right = real_vector.scale_minus_right
```
```   165 lemmas scaleR_right_diff_distrib = real_vector.scale_right_diff_distrib
```
```   166 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
```
```   167 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
```
```   168 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
```
```   169 lemmas scaleR_cancel_left = real_vector.scale_cancel_left
```
```   170 lemmas scaleR_cancel_right = real_vector.scale_cancel_right
```
```   171
```
```   172 class real_algebra = real_vector + ring +
```
```   173   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```   174   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```   175
```
```   176 class real_algebra_1 = real_algebra + ring_1
```
```   177
```
```   178 class real_div_algebra = real_algebra_1 + division_ring
```
```   179
```
```   180 class real_field = real_div_algebra + field
```
```   181
```
```   182 instantiation real :: real_field
```
```   183 begin
```
```   184
```
```   185 definition
```
```   186   real_scaleR_def [simp]: "scaleR a x = a * x"
```
```   187
```
```   188 instance proof
```
```   189 qed (simp_all add: algebra_simps)
```
```   190
```
```   191 end
```
```   192
```
```   193 interpretation scaleR_left!: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
```
```   194 proof qed (rule scaleR_left_distrib)
```
```   195
```
```   196 interpretation scaleR_right!: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
```
```   197 proof qed (rule scaleR_right_distrib)
```
```   198
```
```   199 lemma nonzero_inverse_scaleR_distrib:
```
```   200   fixes x :: "'a::real_div_algebra" shows
```
```   201   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   202 by (rule inverse_unique, simp)
```
```   203
```
```   204 lemma inverse_scaleR_distrib:
```
```   205   fixes x :: "'a::{real_div_algebra,division_by_zero}"
```
```   206   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   207 apply (case_tac "a = 0", simp)
```
```   208 apply (case_tac "x = 0", simp)
```
```   209 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   210 done
```
```   211
```
```   212
```
```   213 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   214 @{term of_real} *}
```
```   215
```
```   216 definition
```
```   217   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   218   "of_real r = scaleR r 1"
```
```   219
```
```   220 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   221 by (simp add: of_real_def)
```
```   222
```
```   223 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   224 by (simp add: of_real_def)
```
```   225
```
```   226 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   227 by (simp add: of_real_def)
```
```   228
```
```   229 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   230 by (simp add: of_real_def scaleR_left_distrib)
```
```   231
```
```   232 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   233 by (simp add: of_real_def)
```
```   234
```
```   235 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   236 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   237
```
```   238 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   239 by (simp add: of_real_def mult_commute)
```
```   240
```
```   241 lemma nonzero_of_real_inverse:
```
```   242   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   243    inverse (of_real x :: 'a::real_div_algebra)"
```
```   244 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   245
```
```   246 lemma of_real_inverse [simp]:
```
```   247   "of_real (inverse x) =
```
```   248    inverse (of_real x :: 'a::{real_div_algebra,division_by_zero})"
```
```   249 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   250
```
```   251 lemma nonzero_of_real_divide:
```
```   252   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   253    (of_real x / of_real y :: 'a::real_field)"
```
```   254 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   255
```
```   256 lemma of_real_divide [simp]:
```
```   257   "of_real (x / y) =
```
```   258    (of_real x / of_real y :: 'a::{real_field,division_by_zero})"
```
```   259 by (simp add: divide_inverse)
```
```   260
```
```   261 lemma of_real_power [simp]:
```
```   262   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1,recpower}) ^ n"
```
```   263 by (induct n) simp_all
```
```   264
```
```   265 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   266 by (simp add: of_real_def scaleR_cancel_right)
```
```   267
```
```   268 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   269
```
```   270 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   271 proof
```
```   272   fix r
```
```   273   show "of_real r = id r"
```
```   274     by (simp add: of_real_def)
```
```   275 qed
```
```   276
```
```   277 text{*Collapse nested embeddings*}
```
```   278 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   279 by (induct n) auto
```
```   280
```
```   281 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   282 by (cases z rule: int_diff_cases, simp)
```
```   283
```
```   284 lemma of_real_number_of_eq:
```
```   285   "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
```
```   286 by (simp add: number_of_eq)
```
```   287
```
```   288 text{*Every real algebra has characteristic zero*}
```
```   289 instance real_algebra_1 < ring_char_0
```
```   290 proof
```
```   291   fix m n :: nat
```
```   292   have "(of_real (of_nat m) = (of_real (of_nat n)::'a)) = (m = n)"
```
```   293     by (simp only: of_real_eq_iff of_nat_eq_iff)
```
```   294   thus "(of_nat m = (of_nat n::'a)) = (m = n)"
```
```   295     by (simp only: of_real_of_nat_eq)
```
```   296 qed
```
```   297
```
```   298 instance real_field < field_char_0 ..
```
```   299
```
```   300
```
```   301 subsection {* The Set of Real Numbers *}
```
```   302
```
```   303 definition
```
```   304   Reals :: "'a::real_algebra_1 set" where
```
```   305   [code del]: "Reals = range of_real"
```
```   306
```
```   307 notation (xsymbols)
```
```   308   Reals  ("\<real>")
```
```   309
```
```   310 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   311 by (simp add: Reals_def)
```
```   312
```
```   313 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   314 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   315
```
```   316 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   317 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   318
```
```   319 lemma Reals_number_of [simp]:
```
```   320   "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
```
```   321 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
```
```   322
```
```   323 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   324 apply (unfold Reals_def)
```
```   325 apply (rule range_eqI)
```
```   326 apply (rule of_real_0 [symmetric])
```
```   327 done
```
```   328
```
```   329 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   330 apply (unfold Reals_def)
```
```   331 apply (rule range_eqI)
```
```   332 apply (rule of_real_1 [symmetric])
```
```   333 done
```
```   334
```
```   335 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   336 apply (auto simp add: Reals_def)
```
```   337 apply (rule range_eqI)
```
```   338 apply (rule of_real_add [symmetric])
```
```   339 done
```
```   340
```
```   341 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   342 apply (auto simp add: Reals_def)
```
```   343 apply (rule range_eqI)
```
```   344 apply (rule of_real_minus [symmetric])
```
```   345 done
```
```   346
```
```   347 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   348 apply (auto simp add: Reals_def)
```
```   349 apply (rule range_eqI)
```
```   350 apply (rule of_real_diff [symmetric])
```
```   351 done
```
```   352
```
```   353 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   354 apply (auto simp add: Reals_def)
```
```   355 apply (rule range_eqI)
```
```   356 apply (rule of_real_mult [symmetric])
```
```   357 done
```
```   358
```
```   359 lemma nonzero_Reals_inverse:
```
```   360   fixes a :: "'a::real_div_algebra"
```
```   361   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   362 apply (auto simp add: Reals_def)
```
```   363 apply (rule range_eqI)
```
```   364 apply (erule nonzero_of_real_inverse [symmetric])
```
```   365 done
```
```   366
```
```   367 lemma Reals_inverse [simp]:
```
```   368   fixes a :: "'a::{real_div_algebra,division_by_zero}"
```
```   369   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   370 apply (auto simp add: Reals_def)
```
```   371 apply (rule range_eqI)
```
```   372 apply (rule of_real_inverse [symmetric])
```
```   373 done
```
```   374
```
```   375 lemma nonzero_Reals_divide:
```
```   376   fixes a b :: "'a::real_field"
```
```   377   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   378 apply (auto simp add: Reals_def)
```
```   379 apply (rule range_eqI)
```
```   380 apply (erule nonzero_of_real_divide [symmetric])
```
```   381 done
```
```   382
```
```   383 lemma Reals_divide [simp]:
```
```   384   fixes a b :: "'a::{real_field,division_by_zero}"
```
```   385   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   386 apply (auto simp add: Reals_def)
```
```   387 apply (rule range_eqI)
```
```   388 apply (rule of_real_divide [symmetric])
```
```   389 done
```
```   390
```
```   391 lemma Reals_power [simp]:
```
```   392   fixes a :: "'a::{real_algebra_1,recpower}"
```
```   393   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   394 apply (auto simp add: Reals_def)
```
```   395 apply (rule range_eqI)
```
```   396 apply (rule of_real_power [symmetric])
```
```   397 done
```
```   398
```
```   399 lemma Reals_cases [cases set: Reals]:
```
```   400   assumes "q \<in> \<real>"
```
```   401   obtains (of_real) r where "q = of_real r"
```
```   402   unfolding Reals_def
```
```   403 proof -
```
```   404   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   405   then obtain r where "q = of_real r" ..
```
```   406   then show thesis ..
```
```   407 qed
```
```   408
```
```   409 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   410   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   411   by (rule Reals_cases) auto
```
```   412
```
```   413
```
```   414 subsection {* Real normed vector spaces *}
```
```   415
```
```   416 class norm =
```
```   417   fixes norm :: "'a \<Rightarrow> real"
```
```   418
```
```   419 class sgn_div_norm = scaleR + norm + sgn +
```
```   420   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   421
```
```   422 class real_normed_vector = real_vector + sgn_div_norm +
```
```   423   assumes norm_ge_zero [simp]: "0 \<le> norm x"
```
```   424   and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   425   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   426   and norm_scaleR: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   427
```
```   428 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   429   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   430
```
```   431 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   432   assumes norm_one [simp]: "norm 1 = 1"
```
```   433
```
```   434 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   435   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   436
```
```   437 class real_normed_field = real_field + real_normed_div_algebra
```
```   438
```
```   439 instance real_normed_div_algebra < real_normed_algebra_1
```
```   440 proof
```
```   441   fix x y :: 'a
```
```   442   show "norm (x * y) \<le> norm x * norm y"
```
```   443     by (simp add: norm_mult)
```
```   444 next
```
```   445   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   446     by (rule norm_mult)
```
```   447   thus "norm (1::'a) = 1" by simp
```
```   448 qed
```
```   449
```
```   450 instantiation real :: real_normed_field
```
```   451 begin
```
```   452
```
```   453 definition
```
```   454   real_norm_def [simp]: "norm r = \<bar>r\<bar>"
```
```   455
```
```   456 instance
```
```   457 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```   458 apply (simp add: real_sgn_def)
```
```   459 apply (rule abs_ge_zero)
```
```   460 apply (rule abs_eq_0)
```
```   461 apply (rule abs_triangle_ineq)
```
```   462 apply (rule abs_mult)
```
```   463 apply (rule abs_mult)
```
```   464 done
```
```   465
```
```   466 end
```
```   467
```
```   468 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   469 by simp
```
```   470
```
```   471 lemma zero_less_norm_iff [simp]:
```
```   472   fixes x :: "'a::real_normed_vector"
```
```   473   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   474 by (simp add: order_less_le)
```
```   475
```
```   476 lemma norm_not_less_zero [simp]:
```
```   477   fixes x :: "'a::real_normed_vector"
```
```   478   shows "\<not> norm x < 0"
```
```   479 by (simp add: linorder_not_less)
```
```   480
```
```   481 lemma norm_le_zero_iff [simp]:
```
```   482   fixes x :: "'a::real_normed_vector"
```
```   483   shows "(norm x \<le> 0) = (x = 0)"
```
```   484 by (simp add: order_le_less)
```
```   485
```
```   486 lemma norm_minus_cancel [simp]:
```
```   487   fixes x :: "'a::real_normed_vector"
```
```   488   shows "norm (- x) = norm x"
```
```   489 proof -
```
```   490   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   491     by (simp only: scaleR_minus_left scaleR_one)
```
```   492   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   493     by (rule norm_scaleR)
```
```   494   finally show ?thesis by simp
```
```   495 qed
```
```   496
```
```   497 lemma norm_minus_commute:
```
```   498   fixes a b :: "'a::real_normed_vector"
```
```   499   shows "norm (a - b) = norm (b - a)"
```
```   500 proof -
```
```   501   have "norm (- (b - a)) = norm (b - a)"
```
```   502     by (rule norm_minus_cancel)
```
```   503   thus ?thesis by simp
```
```   504 qed
```
```   505
```
```   506 lemma norm_triangle_ineq2:
```
```   507   fixes a b :: "'a::real_normed_vector"
```
```   508   shows "norm a - norm b \<le> norm (a - b)"
```
```   509 proof -
```
```   510   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   511     by (rule norm_triangle_ineq)
```
```   512   thus ?thesis by simp
```
```   513 qed
```
```   514
```
```   515 lemma norm_triangle_ineq3:
```
```   516   fixes a b :: "'a::real_normed_vector"
```
```   517   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   518 apply (subst abs_le_iff)
```
```   519 apply auto
```
```   520 apply (rule norm_triangle_ineq2)
```
```   521 apply (subst norm_minus_commute)
```
```   522 apply (rule norm_triangle_ineq2)
```
```   523 done
```
```   524
```
```   525 lemma norm_triangle_ineq4:
```
```   526   fixes a b :: "'a::real_normed_vector"
```
```   527   shows "norm (a - b) \<le> norm a + norm b"
```
```   528 proof -
```
```   529   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   530     by (rule norm_triangle_ineq)
```
```   531   thus ?thesis
```
```   532     by (simp only: diff_minus norm_minus_cancel)
```
```   533 qed
```
```   534
```
```   535 lemma norm_diff_ineq:
```
```   536   fixes a b :: "'a::real_normed_vector"
```
```   537   shows "norm a - norm b \<le> norm (a + b)"
```
```   538 proof -
```
```   539   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   540     by (rule norm_triangle_ineq2)
```
```   541   thus ?thesis by simp
```
```   542 qed
```
```   543
```
```   544 lemma norm_diff_triangle_ineq:
```
```   545   fixes a b c d :: "'a::real_normed_vector"
```
```   546   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   547 proof -
```
```   548   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   549     by (simp add: diff_minus add_ac)
```
```   550   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   551     by (rule norm_triangle_ineq)
```
```   552   finally show ?thesis .
```
```   553 qed
```
```   554
```
```   555 lemma abs_norm_cancel [simp]:
```
```   556   fixes a :: "'a::real_normed_vector"
```
```   557   shows "\<bar>norm a\<bar> = norm a"
```
```   558 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   559
```
```   560 lemma norm_add_less:
```
```   561   fixes x y :: "'a::real_normed_vector"
```
```   562   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   563 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   564
```
```   565 lemma norm_mult_less:
```
```   566   fixes x y :: "'a::real_normed_algebra"
```
```   567   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   568 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   569 apply (simp add: mult_strict_mono')
```
```   570 done
```
```   571
```
```   572 lemma norm_of_real [simp]:
```
```   573   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   574 unfolding of_real_def by (simp add: norm_scaleR)
```
```   575
```
```   576 lemma norm_number_of [simp]:
```
```   577   "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
```
```   578     = \<bar>number_of w\<bar>"
```
```   579 by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
```
```   580
```
```   581 lemma norm_of_int [simp]:
```
```   582   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   583 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   584
```
```   585 lemma norm_of_nat [simp]:
```
```   586   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   587 apply (subst of_real_of_nat_eq [symmetric])
```
```   588 apply (subst norm_of_real, simp)
```
```   589 done
```
```   590
```
```   591 lemma nonzero_norm_inverse:
```
```   592   fixes a :: "'a::real_normed_div_algebra"
```
```   593   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   594 apply (rule inverse_unique [symmetric])
```
```   595 apply (simp add: norm_mult [symmetric])
```
```   596 done
```
```   597
```
```   598 lemma norm_inverse:
```
```   599   fixes a :: "'a::{real_normed_div_algebra,division_by_zero}"
```
```   600   shows "norm (inverse a) = inverse (norm a)"
```
```   601 apply (case_tac "a = 0", simp)
```
```   602 apply (erule nonzero_norm_inverse)
```
```   603 done
```
```   604
```
```   605 lemma nonzero_norm_divide:
```
```   606   fixes a b :: "'a::real_normed_field"
```
```   607   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   608 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   609
```
```   610 lemma norm_divide:
```
```   611   fixes a b :: "'a::{real_normed_field,division_by_zero}"
```
```   612   shows "norm (a / b) = norm a / norm b"
```
```   613 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   614
```
```   615 lemma norm_power_ineq:
```
```   616   fixes x :: "'a::{real_normed_algebra_1,recpower}"
```
```   617   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   618 proof (induct n)
```
```   619   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   620 next
```
```   621   case (Suc n)
```
```   622   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   623     by (rule norm_mult_ineq)
```
```   624   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   625     using norm_ge_zero by (rule mult_left_mono)
```
```   626   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   627     by simp
```
```   628 qed
```
```   629
```
```   630 lemma norm_power:
```
```   631   fixes x :: "'a::{real_normed_div_algebra,recpower}"
```
```   632   shows "norm (x ^ n) = norm x ^ n"
```
```   633 by (induct n) (simp_all add: norm_mult)
```
```   634
```
```   635
```
```   636 subsection {* Sign function *}
```
```   637
```
```   638 lemma norm_sgn:
```
```   639   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```   640 by (simp add: sgn_div_norm norm_scaleR)
```
```   641
```
```   642 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```   643 by (simp add: sgn_div_norm)
```
```   644
```
```   645 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```   646 by (simp add: sgn_div_norm)
```
```   647
```
```   648 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```   649 by (simp add: sgn_div_norm)
```
```   650
```
```   651 lemma sgn_scaleR:
```
```   652   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```   653 by (simp add: sgn_div_norm norm_scaleR mult_ac)
```
```   654
```
```   655 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```   656 by (simp add: sgn_div_norm)
```
```   657
```
```   658 lemma sgn_of_real:
```
```   659   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```   660 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```   661
```
```   662 lemma sgn_mult:
```
```   663   fixes x y :: "'a::real_normed_div_algebra"
```
```   664   shows "sgn (x * y) = sgn x * sgn y"
```
```   665 by (simp add: sgn_div_norm norm_mult mult_commute)
```
```   666
```
```   667 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```   668 by (simp add: sgn_div_norm divide_inverse)
```
```   669
```
```   670 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```   671 unfolding real_sgn_eq by simp
```
```   672
```
```   673 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```   674 unfolding real_sgn_eq by simp
```
```   675
```
```   676
```
```   677 subsection {* Bounded Linear and Bilinear Operators *}
```
```   678
```
```   679 locale bounded_linear = additive +
```
```   680   constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   681   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```   682   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   683 begin
```
```   684
```
```   685 lemma pos_bounded:
```
```   686   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   687 proof -
```
```   688   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```   689     using bounded by fast
```
```   690   show ?thesis
```
```   691   proof (intro exI impI conjI allI)
```
```   692     show "0 < max 1 K"
```
```   693       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   694   next
```
```   695     fix x
```
```   696     have "norm (f x) \<le> norm x * K" using K .
```
```   697     also have "\<dots> \<le> norm x * max 1 K"
```
```   698       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```   699     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```   700   qed
```
```   701 qed
```
```   702
```
```   703 lemma nonneg_bounded:
```
```   704   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   705 proof -
```
```   706   from pos_bounded
```
```   707   show ?thesis by (auto intro: order_less_imp_le)
```
```   708 qed
```
```   709
```
```   710 end
```
```   711
```
```   712 locale bounded_bilinear =
```
```   713   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```   714                  \<Rightarrow> 'c::real_normed_vector"
```
```   715     (infixl "**" 70)
```
```   716   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```   717   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```   718   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```   719   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```   720   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```   721 begin
```
```   722
```
```   723 lemma pos_bounded:
```
```   724   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   725 apply (cut_tac bounded, erule exE)
```
```   726 apply (rule_tac x="max 1 K" in exI, safe)
```
```   727 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   728 apply (drule spec, drule spec, erule order_trans)
```
```   729 apply (rule mult_left_mono [OF le_maxI2])
```
```   730 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```   731 done
```
```   732
```
```   733 lemma nonneg_bounded:
```
```   734   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   735 proof -
```
```   736   from pos_bounded
```
```   737   show ?thesis by (auto intro: order_less_imp_le)
```
```   738 qed
```
```   739
```
```   740 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```   741 by (rule additive.intro, rule add_right)
```
```   742
```
```   743 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```   744 by (rule additive.intro, rule add_left)
```
```   745
```
```   746 lemma zero_left: "prod 0 b = 0"
```
```   747 by (rule additive.zero [OF additive_left])
```
```   748
```
```   749 lemma zero_right: "prod a 0 = 0"
```
```   750 by (rule additive.zero [OF additive_right])
```
```   751
```
```   752 lemma minus_left: "prod (- a) b = - prod a b"
```
```   753 by (rule additive.minus [OF additive_left])
```
```   754
```
```   755 lemma minus_right: "prod a (- b) = - prod a b"
```
```   756 by (rule additive.minus [OF additive_right])
```
```   757
```
```   758 lemma diff_left:
```
```   759   "prod (a - a') b = prod a b - prod a' b"
```
```   760 by (rule additive.diff [OF additive_left])
```
```   761
```
```   762 lemma diff_right:
```
```   763   "prod a (b - b') = prod a b - prod a b'"
```
```   764 by (rule additive.diff [OF additive_right])
```
```   765
```
```   766 lemma bounded_linear_left:
```
```   767   "bounded_linear (\<lambda>a. a ** b)"
```
```   768 apply (unfold_locales)
```
```   769 apply (rule add_left)
```
```   770 apply (rule scaleR_left)
```
```   771 apply (cut_tac bounded, safe)
```
```   772 apply (rule_tac x="norm b * K" in exI)
```
```   773 apply (simp add: mult_ac)
```
```   774 done
```
```   775
```
```   776 lemma bounded_linear_right:
```
```   777   "bounded_linear (\<lambda>b. a ** b)"
```
```   778 apply (unfold_locales)
```
```   779 apply (rule add_right)
```
```   780 apply (rule scaleR_right)
```
```   781 apply (cut_tac bounded, safe)
```
```   782 apply (rule_tac x="norm a * K" in exI)
```
```   783 apply (simp add: mult_ac)
```
```   784 done
```
```   785
```
```   786 lemma prod_diff_prod:
```
```   787   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```   788 by (simp add: diff_left diff_right)
```
```   789
```
```   790 end
```
```   791
```
```   792 interpretation mult!:
```
```   793   bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"
```
```   794 apply (rule bounded_bilinear.intro)
```
```   795 apply (rule left_distrib)
```
```   796 apply (rule right_distrib)
```
```   797 apply (rule mult_scaleR_left)
```
```   798 apply (rule mult_scaleR_right)
```
```   799 apply (rule_tac x="1" in exI)
```
```   800 apply (simp add: norm_mult_ineq)
```
```   801 done
```
```   802
```
```   803 interpretation mult_left!:
```
```   804   bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)"
```
```   805 by (rule mult.bounded_linear_left)
```
```   806
```
```   807 interpretation mult_right!:
```
```   808   bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)"
```
```   809 by (rule mult.bounded_linear_right)
```
```   810
```
```   811 interpretation divide!:
```
```   812   bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)"
```
```   813 unfolding divide_inverse by (rule mult.bounded_linear_left)
```
```   814
```
```   815 interpretation scaleR!: bounded_bilinear "scaleR"
```
```   816 apply (rule bounded_bilinear.intro)
```
```   817 apply (rule scaleR_left_distrib)
```
```   818 apply (rule scaleR_right_distrib)
```
```   819 apply simp
```
```   820 apply (rule scaleR_left_commute)
```
```   821 apply (rule_tac x="1" in exI)
```
```   822 apply (simp add: norm_scaleR)
```
```   823 done
```
```   824
```
```   825 interpretation scaleR_left!: bounded_linear "\<lambda>r. scaleR r x"
```
```   826 by (rule scaleR.bounded_linear_left)
```
```   827
```
```   828 interpretation scaleR_right!: bounded_linear "\<lambda>x. scaleR r x"
```
```   829 by (rule scaleR.bounded_linear_right)
```
```   830
```
```   831 interpretation of_real!: bounded_linear "\<lambda>r. of_real r"
```
```   832 unfolding of_real_def by (rule scaleR.bounded_linear_left)
```
```   833
```
```   834 end
```