src/HOL/RealVector.thy
 author wenzelm Fri Dec 17 17:43:54 2010 +0100 (2010-12-17) changeset 41229 d797baa3d57c parent 38621 d6cb7e625d75 child 41969 1cf3e4107a2a permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
```     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 RComplete
```
```     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: add minus diff_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 [simp]:
```
```   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 [simp]:
```
```   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 lemma scaleR_minus1_left [simp]:
```
```   173   fixes x :: "'a::real_vector"
```
```   174   shows "scaleR (-1) x = - x"
```
```   175   using scaleR_minus_left [of 1 x] by simp
```
```   176
```
```   177 class real_algebra = real_vector + ring +
```
```   178   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```   179   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```   180
```
```   181 class real_algebra_1 = real_algebra + ring_1
```
```   182
```
```   183 class real_div_algebra = real_algebra_1 + division_ring
```
```   184
```
```   185 class real_field = real_div_algebra + field
```
```   186
```
```   187 instantiation real :: real_field
```
```   188 begin
```
```   189
```
```   190 definition
```
```   191   real_scaleR_def [simp]: "scaleR a x = a * x"
```
```   192
```
```   193 instance proof
```
```   194 qed (simp_all add: algebra_simps)
```
```   195
```
```   196 end
```
```   197
```
```   198 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
```
```   199 proof qed (rule scaleR_left_distrib)
```
```   200
```
```   201 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
```
```   202 proof qed (rule scaleR_right_distrib)
```
```   203
```
```   204 lemma nonzero_inverse_scaleR_distrib:
```
```   205   fixes x :: "'a::real_div_algebra" shows
```
```   206   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   207 by (rule inverse_unique, simp)
```
```   208
```
```   209 lemma inverse_scaleR_distrib:
```
```   210   fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   211   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   212 apply (case_tac "a = 0", simp)
```
```   213 apply (case_tac "x = 0", simp)
```
```   214 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   215 done
```
```   216
```
```   217
```
```   218 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   219 @{term of_real} *}
```
```   220
```
```   221 definition
```
```   222   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   223   "of_real r = scaleR r 1"
```
```   224
```
```   225 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   226 by (simp add: of_real_def)
```
```   227
```
```   228 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   229 by (simp add: of_real_def)
```
```   230
```
```   231 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   232 by (simp add: of_real_def)
```
```   233
```
```   234 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   235 by (simp add: of_real_def scaleR_left_distrib)
```
```   236
```
```   237 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   238 by (simp add: of_real_def)
```
```   239
```
```   240 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   241 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   242
```
```   243 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   244 by (simp add: of_real_def mult_commute)
```
```   245
```
```   246 lemma nonzero_of_real_inverse:
```
```   247   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   248    inverse (of_real x :: 'a::real_div_algebra)"
```
```   249 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   250
```
```   251 lemma of_real_inverse [simp]:
```
```   252   "of_real (inverse x) =
```
```   253    inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
```
```   254 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   255
```
```   256 lemma nonzero_of_real_divide:
```
```   257   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   258    (of_real x / of_real y :: 'a::real_field)"
```
```   259 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   260
```
```   261 lemma of_real_divide [simp]:
```
```   262   "of_real (x / y) =
```
```   263    (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
```
```   264 by (simp add: divide_inverse)
```
```   265
```
```   266 lemma of_real_power [simp]:
```
```   267   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
```
```   268 by (induct n) simp_all
```
```   269
```
```   270 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   271 by (simp add: of_real_def)
```
```   272
```
```   273 lemma inj_of_real:
```
```   274   "inj of_real"
```
```   275   by (auto intro: injI)
```
```   276
```
```   277 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   278
```
```   279 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   280 proof
```
```   281   fix r
```
```   282   show "of_real r = id r"
```
```   283     by (simp add: of_real_def)
```
```   284 qed
```
```   285
```
```   286 text{*Collapse nested embeddings*}
```
```   287 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   288 by (induct n) auto
```
```   289
```
```   290 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   291 by (cases z rule: int_diff_cases, simp)
```
```   292
```
```   293 lemma of_real_number_of_eq:
```
```   294   "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
```
```   295 by (simp add: number_of_eq)
```
```   296
```
```   297 text{*Every real algebra has characteristic zero*}
```
```   298
```
```   299 instance real_algebra_1 < ring_char_0
```
```   300 proof
```
```   301   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
```
```   302   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
```
```   303 qed
```
```   304
```
```   305 instance real_field < field_char_0 ..
```
```   306
```
```   307
```
```   308 subsection {* The Set of Real Numbers *}
```
```   309
```
```   310 definition Reals :: "'a::real_algebra_1 set" where
```
```   311   "Reals = range of_real"
```
```   312
```
```   313 notation (xsymbols)
```
```   314   Reals  ("\<real>")
```
```   315
```
```   316 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   317 by (simp add: Reals_def)
```
```   318
```
```   319 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   320 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   321
```
```   322 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   323 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   324
```
```   325 lemma Reals_number_of [simp]:
```
```   326   "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
```
```   327 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
```
```   328
```
```   329 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   330 apply (unfold Reals_def)
```
```   331 apply (rule range_eqI)
```
```   332 apply (rule of_real_0 [symmetric])
```
```   333 done
```
```   334
```
```   335 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   336 apply (unfold Reals_def)
```
```   337 apply (rule range_eqI)
```
```   338 apply (rule of_real_1 [symmetric])
```
```   339 done
```
```   340
```
```   341 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   342 apply (auto simp add: Reals_def)
```
```   343 apply (rule range_eqI)
```
```   344 apply (rule of_real_add [symmetric])
```
```   345 done
```
```   346
```
```   347 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   348 apply (auto simp add: Reals_def)
```
```   349 apply (rule range_eqI)
```
```   350 apply (rule of_real_minus [symmetric])
```
```   351 done
```
```   352
```
```   353 lemma Reals_diff [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_diff [symmetric])
```
```   357 done
```
```   358
```
```   359 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   360 apply (auto simp add: Reals_def)
```
```   361 apply (rule range_eqI)
```
```   362 apply (rule of_real_mult [symmetric])
```
```   363 done
```
```   364
```
```   365 lemma nonzero_Reals_inverse:
```
```   366   fixes a :: "'a::real_div_algebra"
```
```   367   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   368 apply (auto simp add: Reals_def)
```
```   369 apply (rule range_eqI)
```
```   370 apply (erule nonzero_of_real_inverse [symmetric])
```
```   371 done
```
```   372
```
```   373 lemma Reals_inverse [simp]:
```
```   374   fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   375   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   376 apply (auto simp add: Reals_def)
```
```   377 apply (rule range_eqI)
```
```   378 apply (rule of_real_inverse [symmetric])
```
```   379 done
```
```   380
```
```   381 lemma nonzero_Reals_divide:
```
```   382   fixes a b :: "'a::real_field"
```
```   383   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   384 apply (auto simp add: Reals_def)
```
```   385 apply (rule range_eqI)
```
```   386 apply (erule nonzero_of_real_divide [symmetric])
```
```   387 done
```
```   388
```
```   389 lemma Reals_divide [simp]:
```
```   390   fixes a b :: "'a::{real_field, field_inverse_zero}"
```
```   391   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   392 apply (auto simp add: Reals_def)
```
```   393 apply (rule range_eqI)
```
```   394 apply (rule of_real_divide [symmetric])
```
```   395 done
```
```   396
```
```   397 lemma Reals_power [simp]:
```
```   398   fixes a :: "'a::{real_algebra_1}"
```
```   399   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   400 apply (auto simp add: Reals_def)
```
```   401 apply (rule range_eqI)
```
```   402 apply (rule of_real_power [symmetric])
```
```   403 done
```
```   404
```
```   405 lemma Reals_cases [cases set: Reals]:
```
```   406   assumes "q \<in> \<real>"
```
```   407   obtains (of_real) r where "q = of_real r"
```
```   408   unfolding Reals_def
```
```   409 proof -
```
```   410   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   411   then obtain r where "q = of_real r" ..
```
```   412   then show thesis ..
```
```   413 qed
```
```   414
```
```   415 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   416   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   417   by (rule Reals_cases) auto
```
```   418
```
```   419
```
```   420 subsection {* Topological spaces *}
```
```   421
```
```   422 class "open" =
```
```   423   fixes "open" :: "'a set \<Rightarrow> bool"
```
```   424
```
```   425 class topological_space = "open" +
```
```   426   assumes open_UNIV [simp, intro]: "open UNIV"
```
```   427   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
```
```   428   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
```
```   429 begin
```
```   430
```
```   431 definition
```
```   432   closed :: "'a set \<Rightarrow> bool" where
```
```   433   "closed S \<longleftrightarrow> open (- S)"
```
```   434
```
```   435 lemma open_empty [intro, simp]: "open {}"
```
```   436   using open_Union [of "{}"] by simp
```
```   437
```
```   438 lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
```
```   439   using open_Union [of "{S, T}"] by simp
```
```   440
```
```   441 lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
```
```   442   unfolding UN_eq by (rule open_Union) auto
```
```   443
```
```   444 lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
```
```   445   by (induct set: finite) auto
```
```   446
```
```   447 lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
```
```   448   unfolding Inter_def by (rule open_INT)
```
```   449
```
```   450 lemma closed_empty [intro, simp]:  "closed {}"
```
```   451   unfolding closed_def by simp
```
```   452
```
```   453 lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
```
```   454   unfolding closed_def by auto
```
```   455
```
```   456 lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
```
```   457   unfolding closed_def Inter_def by auto
```
```   458
```
```   459 lemma closed_UNIV [intro, simp]: "closed UNIV"
```
```   460   unfolding closed_def by simp
```
```   461
```
```   462 lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
```
```   463   unfolding closed_def by auto
```
```   464
```
```   465 lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
```
```   466   unfolding closed_def by auto
```
```   467
```
```   468 lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
```
```   469   by (induct set: finite) auto
```
```   470
```
```   471 lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
```
```   472   unfolding Union_def by (rule closed_UN)
```
```   473
```
```   474 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
```
```   475   unfolding closed_def by simp
```
```   476
```
```   477 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
```
```   478   unfolding closed_def by simp
```
```   479
```
```   480 lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
```
```   481   unfolding closed_open Diff_eq by (rule open_Int)
```
```   482
```
```   483 lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
```
```   484   unfolding open_closed Diff_eq by (rule closed_Int)
```
```   485
```
```   486 lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
```
```   487   unfolding closed_open .
```
```   488
```
```   489 lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
```
```   490   unfolding open_closed .
```
```   491
```
```   492 end
```
```   493
```
```   494
```
```   495 subsection {* Metric spaces *}
```
```   496
```
```   497 class dist =
```
```   498   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```   499
```
```   500 class open_dist = "open" + dist +
```
```   501   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   502
```
```   503 class metric_space = open_dist +
```
```   504   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```   505   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```   506 begin
```
```   507
```
```   508 lemma dist_self [simp]: "dist x x = 0"
```
```   509 by simp
```
```   510
```
```   511 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```   512 using dist_triangle2 [of x x y] by simp
```
```   513
```
```   514 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```   515 by (simp add: less_le)
```
```   516
```
```   517 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```   518 by (simp add: not_less)
```
```   519
```
```   520 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```   521 by (simp add: le_less)
```
```   522
```
```   523 lemma dist_commute: "dist x y = dist y x"
```
```   524 proof (rule order_antisym)
```
```   525   show "dist x y \<le> dist y x"
```
```   526     using dist_triangle2 [of x y x] by simp
```
```   527   show "dist y x \<le> dist x y"
```
```   528     using dist_triangle2 [of y x y] by simp
```
```   529 qed
```
```   530
```
```   531 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```   532 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```   533
```
```   534 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```   535 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```   536
```
```   537 subclass topological_space
```
```   538 proof
```
```   539   have "\<exists>e::real. 0 < e"
```
```   540     by (fast intro: zero_less_one)
```
```   541   then show "open UNIV"
```
```   542     unfolding open_dist by simp
```
```   543 next
```
```   544   fix S T assume "open S" "open T"
```
```   545   then show "open (S \<inter> T)"
```
```   546     unfolding open_dist
```
```   547     apply clarify
```
```   548     apply (drule (1) bspec)+
```
```   549     apply (clarify, rename_tac r s)
```
```   550     apply (rule_tac x="min r s" in exI, simp)
```
```   551     done
```
```   552 next
```
```   553   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```   554     unfolding open_dist by fast
```
```   555 qed
```
```   556
```
```   557 end
```
```   558
```
```   559
```
```   560 subsection {* Real normed vector spaces *}
```
```   561
```
```   562 class norm =
```
```   563   fixes norm :: "'a \<Rightarrow> real"
```
```   564
```
```   565 class sgn_div_norm = scaleR + norm + sgn +
```
```   566   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   567
```
```   568 class dist_norm = dist + norm + minus +
```
```   569   assumes dist_norm: "dist x y = norm (x - y)"
```
```   570
```
```   571 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
```
```   572   assumes norm_ge_zero [simp]: "0 \<le> norm x"
```
```   573   and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   574   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   575   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   576
```
```   577 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   578   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   579
```
```   580 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   581   assumes norm_one [simp]: "norm 1 = 1"
```
```   582
```
```   583 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   584   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   585
```
```   586 class real_normed_field = real_field + real_normed_div_algebra
```
```   587
```
```   588 instance real_normed_div_algebra < real_normed_algebra_1
```
```   589 proof
```
```   590   fix x y :: 'a
```
```   591   show "norm (x * y) \<le> norm x * norm y"
```
```   592     by (simp add: norm_mult)
```
```   593 next
```
```   594   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   595     by (rule norm_mult)
```
```   596   thus "norm (1::'a) = 1" by simp
```
```   597 qed
```
```   598
```
```   599 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   600 by simp
```
```   601
```
```   602 lemma zero_less_norm_iff [simp]:
```
```   603   fixes x :: "'a::real_normed_vector"
```
```   604   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   605 by (simp add: order_less_le)
```
```   606
```
```   607 lemma norm_not_less_zero [simp]:
```
```   608   fixes x :: "'a::real_normed_vector"
```
```   609   shows "\<not> norm x < 0"
```
```   610 by (simp add: linorder_not_less)
```
```   611
```
```   612 lemma norm_le_zero_iff [simp]:
```
```   613   fixes x :: "'a::real_normed_vector"
```
```   614   shows "(norm x \<le> 0) = (x = 0)"
```
```   615 by (simp add: order_le_less)
```
```   616
```
```   617 lemma norm_minus_cancel [simp]:
```
```   618   fixes x :: "'a::real_normed_vector"
```
```   619   shows "norm (- x) = norm x"
```
```   620 proof -
```
```   621   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   622     by (simp only: scaleR_minus_left scaleR_one)
```
```   623   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   624     by (rule norm_scaleR)
```
```   625   finally show ?thesis by simp
```
```   626 qed
```
```   627
```
```   628 lemma norm_minus_commute:
```
```   629   fixes a b :: "'a::real_normed_vector"
```
```   630   shows "norm (a - b) = norm (b - a)"
```
```   631 proof -
```
```   632   have "norm (- (b - a)) = norm (b - a)"
```
```   633     by (rule norm_minus_cancel)
```
```   634   thus ?thesis by simp
```
```   635 qed
```
```   636
```
```   637 lemma norm_triangle_ineq2:
```
```   638   fixes a b :: "'a::real_normed_vector"
```
```   639   shows "norm a - norm b \<le> norm (a - b)"
```
```   640 proof -
```
```   641   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   642     by (rule norm_triangle_ineq)
```
```   643   thus ?thesis by simp
```
```   644 qed
```
```   645
```
```   646 lemma norm_triangle_ineq3:
```
```   647   fixes a b :: "'a::real_normed_vector"
```
```   648   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   649 apply (subst abs_le_iff)
```
```   650 apply auto
```
```   651 apply (rule norm_triangle_ineq2)
```
```   652 apply (subst norm_minus_commute)
```
```   653 apply (rule norm_triangle_ineq2)
```
```   654 done
```
```   655
```
```   656 lemma norm_triangle_ineq4:
```
```   657   fixes a b :: "'a::real_normed_vector"
```
```   658   shows "norm (a - b) \<le> norm a + norm b"
```
```   659 proof -
```
```   660   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   661     by (rule norm_triangle_ineq)
```
```   662   thus ?thesis
```
```   663     by (simp only: diff_minus norm_minus_cancel)
```
```   664 qed
```
```   665
```
```   666 lemma norm_diff_ineq:
```
```   667   fixes a b :: "'a::real_normed_vector"
```
```   668   shows "norm a - norm b \<le> norm (a + b)"
```
```   669 proof -
```
```   670   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   671     by (rule norm_triangle_ineq2)
```
```   672   thus ?thesis by simp
```
```   673 qed
```
```   674
```
```   675 lemma norm_diff_triangle_ineq:
```
```   676   fixes a b c d :: "'a::real_normed_vector"
```
```   677   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   678 proof -
```
```   679   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   680     by (simp add: diff_minus add_ac)
```
```   681   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   682     by (rule norm_triangle_ineq)
```
```   683   finally show ?thesis .
```
```   684 qed
```
```   685
```
```   686 lemma abs_norm_cancel [simp]:
```
```   687   fixes a :: "'a::real_normed_vector"
```
```   688   shows "\<bar>norm a\<bar> = norm a"
```
```   689 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   690
```
```   691 lemma norm_add_less:
```
```   692   fixes x y :: "'a::real_normed_vector"
```
```   693   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   694 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   695
```
```   696 lemma norm_mult_less:
```
```   697   fixes x y :: "'a::real_normed_algebra"
```
```   698   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   699 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   700 apply (simp add: mult_strict_mono')
```
```   701 done
```
```   702
```
```   703 lemma norm_of_real [simp]:
```
```   704   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   705 unfolding of_real_def by simp
```
```   706
```
```   707 lemma norm_number_of [simp]:
```
```   708   "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
```
```   709     = \<bar>number_of w\<bar>"
```
```   710 by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
```
```   711
```
```   712 lemma norm_of_int [simp]:
```
```   713   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   714 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   715
```
```   716 lemma norm_of_nat [simp]:
```
```   717   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   718 apply (subst of_real_of_nat_eq [symmetric])
```
```   719 apply (subst norm_of_real, simp)
```
```   720 done
```
```   721
```
```   722 lemma nonzero_norm_inverse:
```
```   723   fixes a :: "'a::real_normed_div_algebra"
```
```   724   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   725 apply (rule inverse_unique [symmetric])
```
```   726 apply (simp add: norm_mult [symmetric])
```
```   727 done
```
```   728
```
```   729 lemma norm_inverse:
```
```   730   fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   731   shows "norm (inverse a) = inverse (norm a)"
```
```   732 apply (case_tac "a = 0", simp)
```
```   733 apply (erule nonzero_norm_inverse)
```
```   734 done
```
```   735
```
```   736 lemma nonzero_norm_divide:
```
```   737   fixes a b :: "'a::real_normed_field"
```
```   738   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   739 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   740
```
```   741 lemma norm_divide:
```
```   742   fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
```
```   743   shows "norm (a / b) = norm a / norm b"
```
```   744 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   745
```
```   746 lemma norm_power_ineq:
```
```   747   fixes x :: "'a::{real_normed_algebra_1}"
```
```   748   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   749 proof (induct n)
```
```   750   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   751 next
```
```   752   case (Suc n)
```
```   753   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   754     by (rule norm_mult_ineq)
```
```   755   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   756     using norm_ge_zero by (rule mult_left_mono)
```
```   757   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   758     by simp
```
```   759 qed
```
```   760
```
```   761 lemma norm_power:
```
```   762   fixes x :: "'a::{real_normed_div_algebra}"
```
```   763   shows "norm (x ^ n) = norm x ^ n"
```
```   764 by (induct n) (simp_all add: norm_mult)
```
```   765
```
```   766 text {* Every normed vector space is a metric space. *}
```
```   767
```
```   768 instance real_normed_vector < metric_space
```
```   769 proof
```
```   770   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   771     unfolding dist_norm by simp
```
```   772 next
```
```   773   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
```
```   774     unfolding dist_norm
```
```   775     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
```
```   776 qed
```
```   777
```
```   778
```
```   779 subsection {* Class instances for real numbers *}
```
```   780
```
```   781 instantiation real :: real_normed_field
```
```   782 begin
```
```   783
```
```   784 definition real_norm_def [simp]:
```
```   785   "norm r = \<bar>r\<bar>"
```
```   786
```
```   787 definition dist_real_def:
```
```   788   "dist x y = \<bar>x - y\<bar>"
```
```   789
```
```   790 definition open_real_def:
```
```   791   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   792
```
```   793 instance
```
```   794 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```   795 apply (rule dist_real_def)
```
```   796 apply (rule open_real_def)
```
```   797 apply (simp add: sgn_real_def)
```
```   798 apply (rule abs_ge_zero)
```
```   799 apply (rule abs_eq_0)
```
```   800 apply (rule abs_triangle_ineq)
```
```   801 apply (rule abs_mult)
```
```   802 apply (rule abs_mult)
```
```   803 done
```
```   804
```
```   805 end
```
```   806
```
```   807 lemma open_real_lessThan [simp]:
```
```   808   fixes a :: real shows "open {..<a}"
```
```   809 unfolding open_real_def dist_real_def
```
```   810 proof (clarify)
```
```   811   fix x assume "x < a"
```
```   812   hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```   813   thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```   814 qed
```
```   815
```
```   816 lemma open_real_greaterThan [simp]:
```
```   817   fixes a :: real shows "open {a<..}"
```
```   818 unfolding open_real_def dist_real_def
```
```   819 proof (clarify)
```
```   820   fix x assume "a < x"
```
```   821   hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```   822   thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```   823 qed
```
```   824
```
```   825 lemma open_real_greaterThanLessThan [simp]:
```
```   826   fixes a b :: real shows "open {a<..<b}"
```
```   827 proof -
```
```   828   have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
```
```   829   thus "open {a<..<b}" by (simp add: open_Int)
```
```   830 qed
```
```   831
```
```   832 lemma closed_real_atMost [simp]:
```
```   833   fixes a :: real shows "closed {..a}"
```
```   834 unfolding closed_open by simp
```
```   835
```
```   836 lemma closed_real_atLeast [simp]:
```
```   837   fixes a :: real shows "closed {a..}"
```
```   838 unfolding closed_open by simp
```
```   839
```
```   840 lemma closed_real_atLeastAtMost [simp]:
```
```   841   fixes a b :: real shows "closed {a..b}"
```
```   842 proof -
```
```   843   have "{a..b} = {a..} \<inter> {..b}" by auto
```
```   844   thus "closed {a..b}" by (simp add: closed_Int)
```
```   845 qed
```
```   846
```
```   847
```
```   848 subsection {* Extra type constraints *}
```
```   849
```
```   850 text {* Only allow @{term "open"} in class @{text topological_space}. *}
```
```   851
```
```   852 setup {* Sign.add_const_constraint
```
```   853   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
```
```   854
```
```   855 text {* Only allow @{term dist} in class @{text metric_space}. *}
```
```   856
```
```   857 setup {* Sign.add_const_constraint
```
```   858   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
```
```   859
```
```   860 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
```
```   861
```
```   862 setup {* Sign.add_const_constraint
```
```   863   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
```
```   864
```
```   865
```
```   866 subsection {* Sign function *}
```
```   867
```
```   868 lemma norm_sgn:
```
```   869   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```   870 by (simp add: sgn_div_norm)
```
```   871
```
```   872 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```   873 by (simp add: sgn_div_norm)
```
```   874
```
```   875 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```   876 by (simp add: sgn_div_norm)
```
```   877
```
```   878 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```   879 by (simp add: sgn_div_norm)
```
```   880
```
```   881 lemma sgn_scaleR:
```
```   882   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```   883 by (simp add: sgn_div_norm mult_ac)
```
```   884
```
```   885 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```   886 by (simp add: sgn_div_norm)
```
```   887
```
```   888 lemma sgn_of_real:
```
```   889   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```   890 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```   891
```
```   892 lemma sgn_mult:
```
```   893   fixes x y :: "'a::real_normed_div_algebra"
```
```   894   shows "sgn (x * y) = sgn x * sgn y"
```
```   895 by (simp add: sgn_div_norm norm_mult mult_commute)
```
```   896
```
```   897 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```   898 by (simp add: sgn_div_norm divide_inverse)
```
```   899
```
```   900 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```   901 unfolding real_sgn_eq by simp
```
```   902
```
```   903 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```   904 unfolding real_sgn_eq by simp
```
```   905
```
```   906
```
```   907 subsection {* Bounded Linear and Bilinear Operators *}
```
```   908
```
```   909 locale bounded_linear = additive +
```
```   910   constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   911   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```   912   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   913 begin
```
```   914
```
```   915 lemma pos_bounded:
```
```   916   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   917 proof -
```
```   918   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```   919     using bounded by fast
```
```   920   show ?thesis
```
```   921   proof (intro exI impI conjI allI)
```
```   922     show "0 < max 1 K"
```
```   923       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   924   next
```
```   925     fix x
```
```   926     have "norm (f x) \<le> norm x * K" using K .
```
```   927     also have "\<dots> \<le> norm x * max 1 K"
```
```   928       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```   929     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```   930   qed
```
```   931 qed
```
```   932
```
```   933 lemma nonneg_bounded:
```
```   934   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   935 proof -
```
```   936   from pos_bounded
```
```   937   show ?thesis by (auto intro: order_less_imp_le)
```
```   938 qed
```
```   939
```
```   940 end
```
```   941
```
```   942 locale bounded_bilinear =
```
```   943   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```   944                  \<Rightarrow> 'c::real_normed_vector"
```
```   945     (infixl "**" 70)
```
```   946   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```   947   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```   948   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```   949   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```   950   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```   951 begin
```
```   952
```
```   953 lemma pos_bounded:
```
```   954   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   955 apply (cut_tac bounded, erule exE)
```
```   956 apply (rule_tac x="max 1 K" in exI, safe)
```
```   957 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   958 apply (drule spec, drule spec, erule order_trans)
```
```   959 apply (rule mult_left_mono [OF le_maxI2])
```
```   960 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```   961 done
```
```   962
```
```   963 lemma nonneg_bounded:
```
```   964   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   965 proof -
```
```   966   from pos_bounded
```
```   967   show ?thesis by (auto intro: order_less_imp_le)
```
```   968 qed
```
```   969
```
```   970 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```   971 by (rule additive.intro, rule add_right)
```
```   972
```
```   973 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```   974 by (rule additive.intro, rule add_left)
```
```   975
```
```   976 lemma zero_left: "prod 0 b = 0"
```
```   977 by (rule additive.zero [OF additive_left])
```
```   978
```
```   979 lemma zero_right: "prod a 0 = 0"
```
```   980 by (rule additive.zero [OF additive_right])
```
```   981
```
```   982 lemma minus_left: "prod (- a) b = - prod a b"
```
```   983 by (rule additive.minus [OF additive_left])
```
```   984
```
```   985 lemma minus_right: "prod a (- b) = - prod a b"
```
```   986 by (rule additive.minus [OF additive_right])
```
```   987
```
```   988 lemma diff_left:
```
```   989   "prod (a - a') b = prod a b - prod a' b"
```
```   990 by (rule additive.diff [OF additive_left])
```
```   991
```
```   992 lemma diff_right:
```
```   993   "prod a (b - b') = prod a b - prod a b'"
```
```   994 by (rule additive.diff [OF additive_right])
```
```   995
```
```   996 lemma bounded_linear_left:
```
```   997   "bounded_linear (\<lambda>a. a ** b)"
```
```   998 apply (unfold_locales)
```
```   999 apply (rule add_left)
```
```  1000 apply (rule scaleR_left)
```
```  1001 apply (cut_tac bounded, safe)
```
```  1002 apply (rule_tac x="norm b * K" in exI)
```
```  1003 apply (simp add: mult_ac)
```
```  1004 done
```
```  1005
```
```  1006 lemma bounded_linear_right:
```
```  1007   "bounded_linear (\<lambda>b. a ** b)"
```
```  1008 apply (unfold_locales)
```
```  1009 apply (rule add_right)
```
```  1010 apply (rule scaleR_right)
```
```  1011 apply (cut_tac bounded, safe)
```
```  1012 apply (rule_tac x="norm a * K" in exI)
```
```  1013 apply (simp add: mult_ac)
```
```  1014 done
```
```  1015
```
```  1016 lemma prod_diff_prod:
```
```  1017   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```  1018 by (simp add: diff_left diff_right)
```
```  1019
```
```  1020 end
```
```  1021
```
```  1022 interpretation mult:
```
```  1023   bounded_bilinear "op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra"
```
```  1024 apply (rule bounded_bilinear.intro)
```
```  1025 apply (rule left_distrib)
```
```  1026 apply (rule right_distrib)
```
```  1027 apply (rule mult_scaleR_left)
```
```  1028 apply (rule mult_scaleR_right)
```
```  1029 apply (rule_tac x="1" in exI)
```
```  1030 apply (simp add: norm_mult_ineq)
```
```  1031 done
```
```  1032
```
```  1033 interpretation mult_left:
```
```  1034   bounded_linear "(\<lambda>x::'a::real_normed_algebra. x * y)"
```
```  1035 by (rule mult.bounded_linear_left)
```
```  1036
```
```  1037 interpretation mult_right:
```
```  1038   bounded_linear "(\<lambda>y::'a::real_normed_algebra. x * y)"
```
```  1039 by (rule mult.bounded_linear_right)
```
```  1040
```
```  1041 interpretation divide:
```
```  1042   bounded_linear "(\<lambda>x::'a::real_normed_field. x / y)"
```
```  1043 unfolding divide_inverse by (rule mult.bounded_linear_left)
```
```  1044
```
```  1045 interpretation scaleR: bounded_bilinear "scaleR"
```
```  1046 apply (rule bounded_bilinear.intro)
```
```  1047 apply (rule scaleR_left_distrib)
```
```  1048 apply (rule scaleR_right_distrib)
```
```  1049 apply simp
```
```  1050 apply (rule scaleR_left_commute)
```
```  1051 apply (rule_tac x="1" in exI, simp)
```
```  1052 done
```
```  1053
```
```  1054 interpretation scaleR_left: bounded_linear "\<lambda>r. scaleR r x"
```
```  1055 by (rule scaleR.bounded_linear_left)
```
```  1056
```
```  1057 interpretation scaleR_right: bounded_linear "\<lambda>x. scaleR r x"
```
```  1058 by (rule scaleR.bounded_linear_right)
```
```  1059
```
```  1060 interpretation of_real: bounded_linear "\<lambda>r. of_real r"
```
```  1061 unfolding of_real_def by (rule scaleR.bounded_linear_left)
```
```  1062
```
```  1063 end
```