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