src/HOL/RealVector.thy
 author bulwahn Fri Oct 21 11:17:14 2011 +0200 (2011-10-21) changeset 45231 d85a2fdc586c parent 44937 22c0857b8aab child 46868 6c250adbe101 permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
```     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   and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
```
```    66 proof -
```
```    67   interpret s: additive "\<lambda>a. scale a x"
```
```    68     proof qed (rule scale_left_distrib)
```
```    69   show "scale 0 x = 0" by (rule s.zero)
```
```    70   show "scale (- a) x = - (scale a x)" by (rule s.minus)
```
```    71   show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
```
```    72   show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum)
```
```    73 qed
```
```    74
```
```    75 lemma scale_zero_right [simp]: "scale a 0 = 0"
```
```    76   and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
```
```    77   and scale_right_diff_distrib [algebra_simps]:
```
```    78         "scale a (x - y) = scale a x - scale a y"
```
```    79   and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"
```
```    80 proof -
```
```    81   interpret s: additive "\<lambda>x. scale a x"
```
```    82     proof qed (rule scale_right_distrib)
```
```    83   show "scale a 0 = 0" by (rule s.zero)
```
```    84   show "scale a (- x) = - (scale a x)" by (rule s.minus)
```
```    85   show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
```
```    86   show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)
```
```    87 qed
```
```    88
```
```    89 lemma scale_eq_0_iff [simp]:
```
```    90   "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
```
```    91 proof cases
```
```    92   assume "a = 0" thus ?thesis by simp
```
```    93 next
```
```    94   assume anz [simp]: "a \<noteq> 0"
```
```    95   { assume "scale a x = 0"
```
```    96     hence "scale (inverse a) (scale a x) = 0" by simp
```
```    97     hence "x = 0" by simp }
```
```    98   thus ?thesis by force
```
```    99 qed
```
```   100
```
```   101 lemma scale_left_imp_eq:
```
```   102   "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
```
```   103 proof -
```
```   104   assume nonzero: "a \<noteq> 0"
```
```   105   assume "scale a x = scale a y"
```
```   106   hence "scale a (x - y) = 0"
```
```   107      by (simp add: scale_right_diff_distrib)
```
```   108   hence "x - y = 0" by (simp add: nonzero)
```
```   109   thus "x = y" by (simp only: right_minus_eq)
```
```   110 qed
```
```   111
```
```   112 lemma scale_right_imp_eq:
```
```   113   "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
```
```   114 proof -
```
```   115   assume nonzero: "x \<noteq> 0"
```
```   116   assume "scale a x = scale b x"
```
```   117   hence "scale (a - b) x = 0"
```
```   118      by (simp add: scale_left_diff_distrib)
```
```   119   hence "a - b = 0" by (simp add: nonzero)
```
```   120   thus "a = b" by (simp only: right_minus_eq)
```
```   121 qed
```
```   122
```
```   123 lemma scale_cancel_left [simp]:
```
```   124   "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
```
```   125 by (auto intro: scale_left_imp_eq)
```
```   126
```
```   127 lemma scale_cancel_right [simp]:
```
```   128   "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
```
```   129 by (auto intro: scale_right_imp_eq)
```
```   130
```
```   131 end
```
```   132
```
```   133 subsection {* Real vector spaces *}
```
```   134
```
```   135 class scaleR =
```
```   136   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
```
```   137 begin
```
```   138
```
```   139 abbreviation
```
```   140   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
```
```   141 where
```
```   142   "x /\<^sub>R r == scaleR (inverse r) x"
```
```   143
```
```   144 end
```
```   145
```
```   146 class real_vector = scaleR + ab_group_add +
```
```   147   assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```   148   and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```   149   and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```   150   and scaleR_one: "scaleR 1 x = x"
```
```   151
```
```   152 interpretation real_vector:
```
```   153   vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
```
```   154 apply unfold_locales
```
```   155 apply (rule scaleR_add_right)
```
```   156 apply (rule scaleR_add_left)
```
```   157 apply (rule scaleR_scaleR)
```
```   158 apply (rule scaleR_one)
```
```   159 done
```
```   160
```
```   161 text {* Recover original theorem names *}
```
```   162
```
```   163 lemmas scaleR_left_commute = real_vector.scale_left_commute
```
```   164 lemmas scaleR_zero_left = real_vector.scale_zero_left
```
```   165 lemmas scaleR_minus_left = real_vector.scale_minus_left
```
```   166 lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib
```
```   167 lemmas scaleR_setsum_left = real_vector.scale_setsum_left
```
```   168 lemmas scaleR_zero_right = real_vector.scale_zero_right
```
```   169 lemmas scaleR_minus_right = real_vector.scale_minus_right
```
```   170 lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib
```
```   171 lemmas scaleR_setsum_right = real_vector.scale_setsum_right
```
```   172 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
```
```   173 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
```
```   174 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
```
```   175 lemmas scaleR_cancel_left = real_vector.scale_cancel_left
```
```   176 lemmas scaleR_cancel_right = real_vector.scale_cancel_right
```
```   177
```
```   178 text {* Legacy names *}
```
```   179
```
```   180 lemmas scaleR_left_distrib = scaleR_add_left
```
```   181 lemmas scaleR_right_distrib = scaleR_add_right
```
```   182 lemmas scaleR_left_diff_distrib = scaleR_diff_left
```
```   183 lemmas scaleR_right_diff_distrib = scaleR_diff_right
```
```   184
```
```   185 lemma scaleR_minus1_left [simp]:
```
```   186   fixes x :: "'a::real_vector"
```
```   187   shows "scaleR (-1) x = - x"
```
```   188   using scaleR_minus_left [of 1 x] by simp
```
```   189
```
```   190 class real_algebra = real_vector + ring +
```
```   191   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```   192   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```   193
```
```   194 class real_algebra_1 = real_algebra + ring_1
```
```   195
```
```   196 class real_div_algebra = real_algebra_1 + division_ring
```
```   197
```
```   198 class real_field = real_div_algebra + field
```
```   199
```
```   200 instantiation real :: real_field
```
```   201 begin
```
```   202
```
```   203 definition
```
```   204   real_scaleR_def [simp]: "scaleR a x = a * x"
```
```   205
```
```   206 instance proof
```
```   207 qed (simp_all add: algebra_simps)
```
```   208
```
```   209 end
```
```   210
```
```   211 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
```
```   212 proof qed (rule scaleR_left_distrib)
```
```   213
```
```   214 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
```
```   215 proof qed (rule scaleR_right_distrib)
```
```   216
```
```   217 lemma nonzero_inverse_scaleR_distrib:
```
```   218   fixes x :: "'a::real_div_algebra" shows
```
```   219   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   220 by (rule inverse_unique, simp)
```
```   221
```
```   222 lemma inverse_scaleR_distrib:
```
```   223   fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   224   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   225 apply (case_tac "a = 0", simp)
```
```   226 apply (case_tac "x = 0", simp)
```
```   227 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   228 done
```
```   229
```
```   230
```
```   231 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   232 @{term of_real} *}
```
```   233
```
```   234 definition
```
```   235   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   236   "of_real r = scaleR r 1"
```
```   237
```
```   238 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   239 by (simp add: of_real_def)
```
```   240
```
```   241 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   242 by (simp add: of_real_def)
```
```   243
```
```   244 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   245 by (simp add: of_real_def)
```
```   246
```
```   247 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   248 by (simp add: of_real_def scaleR_left_distrib)
```
```   249
```
```   250 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   251 by (simp add: of_real_def)
```
```   252
```
```   253 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   254 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   255
```
```   256 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   257 by (simp add: of_real_def mult_commute)
```
```   258
```
```   259 lemma nonzero_of_real_inverse:
```
```   260   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   261    inverse (of_real x :: 'a::real_div_algebra)"
```
```   262 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   263
```
```   264 lemma of_real_inverse [simp]:
```
```   265   "of_real (inverse x) =
```
```   266    inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
```
```   267 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   268
```
```   269 lemma nonzero_of_real_divide:
```
```   270   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   271    (of_real x / of_real y :: 'a::real_field)"
```
```   272 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   273
```
```   274 lemma of_real_divide [simp]:
```
```   275   "of_real (x / y) =
```
```   276    (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
```
```   277 by (simp add: divide_inverse)
```
```   278
```
```   279 lemma of_real_power [simp]:
```
```   280   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
```
```   281 by (induct n) simp_all
```
```   282
```
```   283 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   284 by (simp add: of_real_def)
```
```   285
```
```   286 lemma inj_of_real:
```
```   287   "inj of_real"
```
```   288   by (auto intro: injI)
```
```   289
```
```   290 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   291
```
```   292 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   293 proof
```
```   294   fix r
```
```   295   show "of_real r = id r"
```
```   296     by (simp add: of_real_def)
```
```   297 qed
```
```   298
```
```   299 text{*Collapse nested embeddings*}
```
```   300 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   301 by (induct n) auto
```
```   302
```
```   303 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   304 by (cases z rule: int_diff_cases, simp)
```
```   305
```
```   306 lemma of_real_number_of_eq:
```
```   307   "of_real (number_of w) = (number_of w :: 'a::{number_ring,real_algebra_1})"
```
```   308 by (simp add: number_of_eq)
```
```   309
```
```   310 text{*Every real algebra has characteristic zero*}
```
```   311
```
```   312 instance real_algebra_1 < ring_char_0
```
```   313 proof
```
```   314   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
```
```   315   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
```
```   316 qed
```
```   317
```
```   318 instance real_field < field_char_0 ..
```
```   319
```
```   320
```
```   321 subsection {* The Set of Real Numbers *}
```
```   322
```
```   323 definition Reals :: "'a::real_algebra_1 set" where
```
```   324   "Reals = range of_real"
```
```   325
```
```   326 notation (xsymbols)
```
```   327   Reals  ("\<real>")
```
```   328
```
```   329 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   330 by (simp add: Reals_def)
```
```   331
```
```   332 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   333 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   334
```
```   335 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   336 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   337
```
```   338 lemma Reals_number_of [simp]:
```
```   339   "(number_of w::'a::{number_ring,real_algebra_1}) \<in> Reals"
```
```   340 by (subst of_real_number_of_eq [symmetric], rule Reals_of_real)
```
```   341
```
```   342 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   343 apply (unfold Reals_def)
```
```   344 apply (rule range_eqI)
```
```   345 apply (rule of_real_0 [symmetric])
```
```   346 done
```
```   347
```
```   348 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   349 apply (unfold Reals_def)
```
```   350 apply (rule range_eqI)
```
```   351 apply (rule of_real_1 [symmetric])
```
```   352 done
```
```   353
```
```   354 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   355 apply (auto simp add: Reals_def)
```
```   356 apply (rule range_eqI)
```
```   357 apply (rule of_real_add [symmetric])
```
```   358 done
```
```   359
```
```   360 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   361 apply (auto simp add: Reals_def)
```
```   362 apply (rule range_eqI)
```
```   363 apply (rule of_real_minus [symmetric])
```
```   364 done
```
```   365
```
```   366 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   367 apply (auto simp add: Reals_def)
```
```   368 apply (rule range_eqI)
```
```   369 apply (rule of_real_diff [symmetric])
```
```   370 done
```
```   371
```
```   372 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   373 apply (auto simp add: Reals_def)
```
```   374 apply (rule range_eqI)
```
```   375 apply (rule of_real_mult [symmetric])
```
```   376 done
```
```   377
```
```   378 lemma nonzero_Reals_inverse:
```
```   379   fixes a :: "'a::real_div_algebra"
```
```   380   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   381 apply (auto simp add: Reals_def)
```
```   382 apply (rule range_eqI)
```
```   383 apply (erule nonzero_of_real_inverse [symmetric])
```
```   384 done
```
```   385
```
```   386 lemma Reals_inverse [simp]:
```
```   387   fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   388   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   389 apply (auto simp add: Reals_def)
```
```   390 apply (rule range_eqI)
```
```   391 apply (rule of_real_inverse [symmetric])
```
```   392 done
```
```   393
```
```   394 lemma nonzero_Reals_divide:
```
```   395   fixes a b :: "'a::real_field"
```
```   396   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   397 apply (auto simp add: Reals_def)
```
```   398 apply (rule range_eqI)
```
```   399 apply (erule nonzero_of_real_divide [symmetric])
```
```   400 done
```
```   401
```
```   402 lemma Reals_divide [simp]:
```
```   403   fixes a b :: "'a::{real_field, field_inverse_zero}"
```
```   404   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   405 apply (auto simp add: Reals_def)
```
```   406 apply (rule range_eqI)
```
```   407 apply (rule of_real_divide [symmetric])
```
```   408 done
```
```   409
```
```   410 lemma Reals_power [simp]:
```
```   411   fixes a :: "'a::{real_algebra_1}"
```
```   412   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   413 apply (auto simp add: Reals_def)
```
```   414 apply (rule range_eqI)
```
```   415 apply (rule of_real_power [symmetric])
```
```   416 done
```
```   417
```
```   418 lemma Reals_cases [cases set: Reals]:
```
```   419   assumes "q \<in> \<real>"
```
```   420   obtains (of_real) r where "q = of_real r"
```
```   421   unfolding Reals_def
```
```   422 proof -
```
```   423   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   424   then obtain r where "q = of_real r" ..
```
```   425   then show thesis ..
```
```   426 qed
```
```   427
```
```   428 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   429   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   430   by (rule Reals_cases) auto
```
```   431
```
```   432
```
```   433 subsection {* Topological spaces *}
```
```   434
```
```   435 class "open" =
```
```   436   fixes "open" :: "'a set \<Rightarrow> bool"
```
```   437
```
```   438 class topological_space = "open" +
```
```   439   assumes open_UNIV [simp, intro]: "open UNIV"
```
```   440   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
```
```   441   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
```
```   442 begin
```
```   443
```
```   444 definition
```
```   445   closed :: "'a set \<Rightarrow> bool" where
```
```   446   "closed S \<longleftrightarrow> open (- S)"
```
```   447
```
```   448 lemma open_empty [intro, simp]: "open {}"
```
```   449   using open_Union [of "{}"] by simp
```
```   450
```
```   451 lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
```
```   452   using open_Union [of "{S, T}"] by simp
```
```   453
```
```   454 lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
```
```   455   unfolding SUP_def by (rule open_Union) auto
```
```   456
```
```   457 lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
```
```   458   by (induct set: finite) auto
```
```   459
```
```   460 lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
```
```   461   unfolding INF_def by (rule open_Inter) auto
```
```   462
```
```   463 lemma closed_empty [intro, simp]:  "closed {}"
```
```   464   unfolding closed_def by simp
```
```   465
```
```   466 lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
```
```   467   unfolding closed_def by auto
```
```   468
```
```   469 lemma closed_UNIV [intro, simp]: "closed UNIV"
```
```   470   unfolding closed_def by simp
```
```   471
```
```   472 lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
```
```   473   unfolding closed_def by auto
```
```   474
```
```   475 lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
```
```   476   unfolding closed_def by auto
```
```   477
```
```   478 lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
```
```   479   unfolding closed_def uminus_Inf by auto
```
```   480
```
```   481 lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
```
```   482   by (induct set: finite) auto
```
```   483
```
```   484 lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
```
```   485   unfolding SUP_def by (rule closed_Union) auto
```
```   486
```
```   487 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
```
```   488   unfolding closed_def by simp
```
```   489
```
```   490 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
```
```   491   unfolding closed_def by simp
```
```   492
```
```   493 lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
```
```   494   unfolding closed_open Diff_eq by (rule open_Int)
```
```   495
```
```   496 lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
```
```   497   unfolding open_closed Diff_eq by (rule closed_Int)
```
```   498
```
```   499 lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
```
```   500   unfolding closed_open .
```
```   501
```
```   502 lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
```
```   503   unfolding open_closed .
```
```   504
```
```   505 end
```
```   506
```
```   507
```
```   508 subsection {* Metric spaces *}
```
```   509
```
```   510 class dist =
```
```   511   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```   512
```
```   513 class open_dist = "open" + dist +
```
```   514   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   515
```
```   516 class metric_space = open_dist +
```
```   517   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```   518   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```   519 begin
```
```   520
```
```   521 lemma dist_self [simp]: "dist x x = 0"
```
```   522 by simp
```
```   523
```
```   524 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```   525 using dist_triangle2 [of x x y] by simp
```
```   526
```
```   527 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```   528 by (simp add: less_le)
```
```   529
```
```   530 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```   531 by (simp add: not_less)
```
```   532
```
```   533 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```   534 by (simp add: le_less)
```
```   535
```
```   536 lemma dist_commute: "dist x y = dist y x"
```
```   537 proof (rule order_antisym)
```
```   538   show "dist x y \<le> dist y x"
```
```   539     using dist_triangle2 [of x y x] by simp
```
```   540   show "dist y x \<le> dist x y"
```
```   541     using dist_triangle2 [of y x y] by simp
```
```   542 qed
```
```   543
```
```   544 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```   545 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```   546
```
```   547 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```   548 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```   549
```
```   550 lemma dist_triangle_alt:
```
```   551   shows "dist y z <= dist x y + dist x z"
```
```   552 by (rule dist_triangle3)
```
```   553
```
```   554 lemma dist_pos_lt:
```
```   555   shows "x \<noteq> y ==> 0 < dist x y"
```
```   556 by (simp add: zero_less_dist_iff)
```
```   557
```
```   558 lemma dist_nz:
```
```   559   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
```
```   560 by (simp add: zero_less_dist_iff)
```
```   561
```
```   562 lemma dist_triangle_le:
```
```   563   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
```
```   564 by (rule order_trans [OF dist_triangle2])
```
```   565
```
```   566 lemma dist_triangle_lt:
```
```   567   shows "dist x z + dist y z < e ==> dist x y < e"
```
```   568 by (rule le_less_trans [OF dist_triangle2])
```
```   569
```
```   570 lemma dist_triangle_half_l:
```
```   571   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   572 by (rule dist_triangle_lt [where z=y], simp)
```
```   573
```
```   574 lemma dist_triangle_half_r:
```
```   575   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   576 by (rule dist_triangle_half_l, simp_all add: dist_commute)
```
```   577
```
```   578 subclass topological_space
```
```   579 proof
```
```   580   have "\<exists>e::real. 0 < e"
```
```   581     by (fast intro: zero_less_one)
```
```   582   then show "open UNIV"
```
```   583     unfolding open_dist by simp
```
```   584 next
```
```   585   fix S T assume "open S" "open T"
```
```   586   then show "open (S \<inter> T)"
```
```   587     unfolding open_dist
```
```   588     apply clarify
```
```   589     apply (drule (1) bspec)+
```
```   590     apply (clarify, rename_tac r s)
```
```   591     apply (rule_tac x="min r s" in exI, simp)
```
```   592     done
```
```   593 next
```
```   594   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```   595     unfolding open_dist by fast
```
```   596 qed
```
```   597
```
```   598 lemma (in metric_space) open_ball: "open {y. dist x y < d}"
```
```   599 proof (unfold open_dist, intro ballI)
```
```   600   fix y assume *: "y \<in> {y. dist x y < d}"
```
```   601   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
```
```   602     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
```
```   603 qed
```
```   604
```
```   605 end
```
```   606
```
```   607
```
```   608 subsection {* Real normed vector spaces *}
```
```   609
```
```   610 class norm =
```
```   611   fixes norm :: "'a \<Rightarrow> real"
```
```   612
```
```   613 class sgn_div_norm = scaleR + norm + sgn +
```
```   614   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   615
```
```   616 class dist_norm = dist + norm + minus +
```
```   617   assumes dist_norm: "dist x y = norm (x - y)"
```
```   618
```
```   619 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
```
```   620   assumes norm_ge_zero [simp]: "0 \<le> norm x"
```
```   621   and norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   622   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   623   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   624
```
```   625 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   626   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   627
```
```   628 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   629   assumes norm_one [simp]: "norm 1 = 1"
```
```   630
```
```   631 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   632   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   633
```
```   634 class real_normed_field = real_field + real_normed_div_algebra
```
```   635
```
```   636 instance real_normed_div_algebra < real_normed_algebra_1
```
```   637 proof
```
```   638   fix x y :: 'a
```
```   639   show "norm (x * y) \<le> norm x * norm y"
```
```   640     by (simp add: norm_mult)
```
```   641 next
```
```   642   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   643     by (rule norm_mult)
```
```   644   thus "norm (1::'a) = 1" by simp
```
```   645 qed
```
```   646
```
```   647 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   648 by simp
```
```   649
```
```   650 lemma zero_less_norm_iff [simp]:
```
```   651   fixes x :: "'a::real_normed_vector"
```
```   652   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   653 by (simp add: order_less_le)
```
```   654
```
```   655 lemma norm_not_less_zero [simp]:
```
```   656   fixes x :: "'a::real_normed_vector"
```
```   657   shows "\<not> norm x < 0"
```
```   658 by (simp add: linorder_not_less)
```
```   659
```
```   660 lemma norm_le_zero_iff [simp]:
```
```   661   fixes x :: "'a::real_normed_vector"
```
```   662   shows "(norm x \<le> 0) = (x = 0)"
```
```   663 by (simp add: order_le_less)
```
```   664
```
```   665 lemma norm_minus_cancel [simp]:
```
```   666   fixes x :: "'a::real_normed_vector"
```
```   667   shows "norm (- x) = norm x"
```
```   668 proof -
```
```   669   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   670     by (simp only: scaleR_minus_left scaleR_one)
```
```   671   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   672     by (rule norm_scaleR)
```
```   673   finally show ?thesis by simp
```
```   674 qed
```
```   675
```
```   676 lemma norm_minus_commute:
```
```   677   fixes a b :: "'a::real_normed_vector"
```
```   678   shows "norm (a - b) = norm (b - a)"
```
```   679 proof -
```
```   680   have "norm (- (b - a)) = norm (b - a)"
```
```   681     by (rule norm_minus_cancel)
```
```   682   thus ?thesis by simp
```
```   683 qed
```
```   684
```
```   685 lemma norm_triangle_ineq2:
```
```   686   fixes a b :: "'a::real_normed_vector"
```
```   687   shows "norm a - norm b \<le> norm (a - b)"
```
```   688 proof -
```
```   689   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   690     by (rule norm_triangle_ineq)
```
```   691   thus ?thesis by simp
```
```   692 qed
```
```   693
```
```   694 lemma norm_triangle_ineq3:
```
```   695   fixes a b :: "'a::real_normed_vector"
```
```   696   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   697 apply (subst abs_le_iff)
```
```   698 apply auto
```
```   699 apply (rule norm_triangle_ineq2)
```
```   700 apply (subst norm_minus_commute)
```
```   701 apply (rule norm_triangle_ineq2)
```
```   702 done
```
```   703
```
```   704 lemma norm_triangle_ineq4:
```
```   705   fixes a b :: "'a::real_normed_vector"
```
```   706   shows "norm (a - b) \<le> norm a + norm b"
```
```   707 proof -
```
```   708   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   709     by (rule norm_triangle_ineq)
```
```   710   thus ?thesis
```
```   711     by (simp only: diff_minus norm_minus_cancel)
```
```   712 qed
```
```   713
```
```   714 lemma norm_diff_ineq:
```
```   715   fixes a b :: "'a::real_normed_vector"
```
```   716   shows "norm a - norm b \<le> norm (a + b)"
```
```   717 proof -
```
```   718   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   719     by (rule norm_triangle_ineq2)
```
```   720   thus ?thesis by simp
```
```   721 qed
```
```   722
```
```   723 lemma norm_diff_triangle_ineq:
```
```   724   fixes a b c d :: "'a::real_normed_vector"
```
```   725   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   726 proof -
```
```   727   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   728     by (simp add: diff_minus add_ac)
```
```   729   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   730     by (rule norm_triangle_ineq)
```
```   731   finally show ?thesis .
```
```   732 qed
```
```   733
```
```   734 lemma abs_norm_cancel [simp]:
```
```   735   fixes a :: "'a::real_normed_vector"
```
```   736   shows "\<bar>norm a\<bar> = norm a"
```
```   737 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   738
```
```   739 lemma norm_add_less:
```
```   740   fixes x y :: "'a::real_normed_vector"
```
```   741   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   742 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   743
```
```   744 lemma norm_mult_less:
```
```   745   fixes x y :: "'a::real_normed_algebra"
```
```   746   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   747 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   748 apply (simp add: mult_strict_mono')
```
```   749 done
```
```   750
```
```   751 lemma norm_of_real [simp]:
```
```   752   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   753 unfolding of_real_def by simp
```
```   754
```
```   755 lemma norm_number_of [simp]:
```
```   756   "norm (number_of w::'a::{number_ring,real_normed_algebra_1})
```
```   757     = \<bar>number_of w\<bar>"
```
```   758 by (subst of_real_number_of_eq [symmetric], rule norm_of_real)
```
```   759
```
```   760 lemma norm_of_int [simp]:
```
```   761   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   762 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   763
```
```   764 lemma norm_of_nat [simp]:
```
```   765   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   766 apply (subst of_real_of_nat_eq [symmetric])
```
```   767 apply (subst norm_of_real, simp)
```
```   768 done
```
```   769
```
```   770 lemma nonzero_norm_inverse:
```
```   771   fixes a :: "'a::real_normed_div_algebra"
```
```   772   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   773 apply (rule inverse_unique [symmetric])
```
```   774 apply (simp add: norm_mult [symmetric])
```
```   775 done
```
```   776
```
```   777 lemma norm_inverse:
```
```   778   fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   779   shows "norm (inverse a) = inverse (norm a)"
```
```   780 apply (case_tac "a = 0", simp)
```
```   781 apply (erule nonzero_norm_inverse)
```
```   782 done
```
```   783
```
```   784 lemma nonzero_norm_divide:
```
```   785   fixes a b :: "'a::real_normed_field"
```
```   786   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   787 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   788
```
```   789 lemma norm_divide:
```
```   790   fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
```
```   791   shows "norm (a / b) = norm a / norm b"
```
```   792 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   793
```
```   794 lemma norm_power_ineq:
```
```   795   fixes x :: "'a::{real_normed_algebra_1}"
```
```   796   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   797 proof (induct n)
```
```   798   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   799 next
```
```   800   case (Suc n)
```
```   801   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   802     by (rule norm_mult_ineq)
```
```   803   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   804     using norm_ge_zero by (rule mult_left_mono)
```
```   805   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   806     by simp
```
```   807 qed
```
```   808
```
```   809 lemma norm_power:
```
```   810   fixes x :: "'a::{real_normed_div_algebra}"
```
```   811   shows "norm (x ^ n) = norm x ^ n"
```
```   812 by (induct n) (simp_all add: norm_mult)
```
```   813
```
```   814 text {* Every normed vector space is a metric space. *}
```
```   815
```
```   816 instance real_normed_vector < metric_space
```
```   817 proof
```
```   818   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   819     unfolding dist_norm by simp
```
```   820 next
```
```   821   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
```
```   822     unfolding dist_norm
```
```   823     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
```
```   824 qed
```
```   825
```
```   826
```
```   827 subsection {* Class instances for real numbers *}
```
```   828
```
```   829 instantiation real :: real_normed_field
```
```   830 begin
```
```   831
```
```   832 definition real_norm_def [simp]:
```
```   833   "norm r = \<bar>r\<bar>"
```
```   834
```
```   835 definition dist_real_def:
```
```   836   "dist x y = \<bar>x - y\<bar>"
```
```   837
```
```   838 definition open_real_def:
```
```   839   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   840
```
```   841 instance
```
```   842 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```   843 apply (rule dist_real_def)
```
```   844 apply (rule open_real_def)
```
```   845 apply (simp add: sgn_real_def)
```
```   846 apply (rule abs_ge_zero)
```
```   847 apply (rule abs_eq_0)
```
```   848 apply (rule abs_triangle_ineq)
```
```   849 apply (rule abs_mult)
```
```   850 apply (rule abs_mult)
```
```   851 done
```
```   852
```
```   853 end
```
```   854
```
```   855 lemma open_real_lessThan [simp]:
```
```   856   fixes a :: real shows "open {..<a}"
```
```   857 unfolding open_real_def dist_real_def
```
```   858 proof (clarify)
```
```   859   fix x assume "x < a"
```
```   860   hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```   861   thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```   862 qed
```
```   863
```
```   864 lemma open_real_greaterThan [simp]:
```
```   865   fixes a :: real shows "open {a<..}"
```
```   866 unfolding open_real_def dist_real_def
```
```   867 proof (clarify)
```
```   868   fix x assume "a < x"
```
```   869   hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```   870   thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```   871 qed
```
```   872
```
```   873 lemma open_real_greaterThanLessThan [simp]:
```
```   874   fixes a b :: real shows "open {a<..<b}"
```
```   875 proof -
```
```   876   have "{a<..<b} = {a<..} \<inter> {..<b}" by auto
```
```   877   thus "open {a<..<b}" by (simp add: open_Int)
```
```   878 qed
```
```   879
```
```   880 lemma closed_real_atMost [simp]:
```
```   881   fixes a :: real shows "closed {..a}"
```
```   882 unfolding closed_open by simp
```
```   883
```
```   884 lemma closed_real_atLeast [simp]:
```
```   885   fixes a :: real shows "closed {a..}"
```
```   886 unfolding closed_open by simp
```
```   887
```
```   888 lemma closed_real_atLeastAtMost [simp]:
```
```   889   fixes a b :: real shows "closed {a..b}"
```
```   890 proof -
```
```   891   have "{a..b} = {a..} \<inter> {..b}" by auto
```
```   892   thus "closed {a..b}" by (simp add: closed_Int)
```
```   893 qed
```
```   894
```
```   895
```
```   896 subsection {* Extra type constraints *}
```
```   897
```
```   898 text {* Only allow @{term "open"} in class @{text topological_space}. *}
```
```   899
```
```   900 setup {* Sign.add_const_constraint
```
```   901   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
```
```   902
```
```   903 text {* Only allow @{term dist} in class @{text metric_space}. *}
```
```   904
```
```   905 setup {* Sign.add_const_constraint
```
```   906   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
```
```   907
```
```   908 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
```
```   909
```
```   910 setup {* Sign.add_const_constraint
```
```   911   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
```
```   912
```
```   913
```
```   914 subsection {* Sign function *}
```
```   915
```
```   916 lemma norm_sgn:
```
```   917   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```   918 by (simp add: sgn_div_norm)
```
```   919
```
```   920 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```   921 by (simp add: sgn_div_norm)
```
```   922
```
```   923 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```   924 by (simp add: sgn_div_norm)
```
```   925
```
```   926 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```   927 by (simp add: sgn_div_norm)
```
```   928
```
```   929 lemma sgn_scaleR:
```
```   930   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```   931 by (simp add: sgn_div_norm mult_ac)
```
```   932
```
```   933 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```   934 by (simp add: sgn_div_norm)
```
```   935
```
```   936 lemma sgn_of_real:
```
```   937   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```   938 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```   939
```
```   940 lemma sgn_mult:
```
```   941   fixes x y :: "'a::real_normed_div_algebra"
```
```   942   shows "sgn (x * y) = sgn x * sgn y"
```
```   943 by (simp add: sgn_div_norm norm_mult mult_commute)
```
```   944
```
```   945 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```   946 by (simp add: sgn_div_norm divide_inverse)
```
```   947
```
```   948 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```   949 unfolding real_sgn_eq by simp
```
```   950
```
```   951 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```   952 unfolding real_sgn_eq by simp
```
```   953
```
```   954
```
```   955 subsection {* Bounded Linear and Bilinear Operators *}
```
```   956
```
```   957 locale bounded_linear = additive +
```
```   958   constrains f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```   959   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```   960   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   961 begin
```
```   962
```
```   963 lemma pos_bounded:
```
```   964   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   965 proof -
```
```   966   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```   967     using bounded by fast
```
```   968   show ?thesis
```
```   969   proof (intro exI impI conjI allI)
```
```   970     show "0 < max 1 K"
```
```   971       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   972   next
```
```   973     fix x
```
```   974     have "norm (f x) \<le> norm x * K" using K .
```
```   975     also have "\<dots> \<le> norm x * max 1 K"
```
```   976       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```   977     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```   978   qed
```
```   979 qed
```
```   980
```
```   981 lemma nonneg_bounded:
```
```   982   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   983 proof -
```
```   984   from pos_bounded
```
```   985   show ?thesis by (auto intro: order_less_imp_le)
```
```   986 qed
```
```   987
```
```   988 end
```
```   989
```
```   990 lemma bounded_linear_intro:
```
```   991   assumes "\<And>x y. f (x + y) = f x + f y"
```
```   992   assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
```
```   993   assumes "\<And>x. norm (f x) \<le> norm x * K"
```
```   994   shows "bounded_linear f"
```
```   995   by default (fast intro: assms)+
```
```   996
```
```   997 locale bounded_bilinear =
```
```   998   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```   999                  \<Rightarrow> 'c::real_normed_vector"
```
```  1000     (infixl "**" 70)
```
```  1001   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```  1002   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```  1003   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```  1004   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```  1005   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```  1006 begin
```
```  1007
```
```  1008 lemma pos_bounded:
```
```  1009   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1010 apply (cut_tac bounded, erule exE)
```
```  1011 apply (rule_tac x="max 1 K" in exI, safe)
```
```  1012 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```  1013 apply (drule spec, drule spec, erule order_trans)
```
```  1014 apply (rule mult_left_mono [OF le_maxI2])
```
```  1015 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```  1016 done
```
```  1017
```
```  1018 lemma nonneg_bounded:
```
```  1019   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1020 proof -
```
```  1021   from pos_bounded
```
```  1022   show ?thesis by (auto intro: order_less_imp_le)
```
```  1023 qed
```
```  1024
```
```  1025 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```  1026 by (rule additive.intro, rule add_right)
```
```  1027
```
```  1028 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```  1029 by (rule additive.intro, rule add_left)
```
```  1030
```
```  1031 lemma zero_left: "prod 0 b = 0"
```
```  1032 by (rule additive.zero [OF additive_left])
```
```  1033
```
```  1034 lemma zero_right: "prod a 0 = 0"
```
```  1035 by (rule additive.zero [OF additive_right])
```
```  1036
```
```  1037 lemma minus_left: "prod (- a) b = - prod a b"
```
```  1038 by (rule additive.minus [OF additive_left])
```
```  1039
```
```  1040 lemma minus_right: "prod a (- b) = - prod a b"
```
```  1041 by (rule additive.minus [OF additive_right])
```
```  1042
```
```  1043 lemma diff_left:
```
```  1044   "prod (a - a') b = prod a b - prod a' b"
```
```  1045 by (rule additive.diff [OF additive_left])
```
```  1046
```
```  1047 lemma diff_right:
```
```  1048   "prod a (b - b') = prod a b - prod a b'"
```
```  1049 by (rule additive.diff [OF additive_right])
```
```  1050
```
```  1051 lemma bounded_linear_left:
```
```  1052   "bounded_linear (\<lambda>a. a ** b)"
```
```  1053 apply (cut_tac bounded, safe)
```
```  1054 apply (rule_tac K="norm b * K" in bounded_linear_intro)
```
```  1055 apply (rule add_left)
```
```  1056 apply (rule scaleR_left)
```
```  1057 apply (simp add: mult_ac)
```
```  1058 done
```
```  1059
```
```  1060 lemma bounded_linear_right:
```
```  1061   "bounded_linear (\<lambda>b. a ** b)"
```
```  1062 apply (cut_tac bounded, safe)
```
```  1063 apply (rule_tac K="norm a * K" in bounded_linear_intro)
```
```  1064 apply (rule add_right)
```
```  1065 apply (rule scaleR_right)
```
```  1066 apply (simp add: mult_ac)
```
```  1067 done
```
```  1068
```
```  1069 lemma prod_diff_prod:
```
```  1070   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```  1071 by (simp add: diff_left diff_right)
```
```  1072
```
```  1073 end
```
```  1074
```
```  1075 lemma bounded_bilinear_mult:
```
```  1076   "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
```
```  1077 apply (rule bounded_bilinear.intro)
```
```  1078 apply (rule left_distrib)
```
```  1079 apply (rule right_distrib)
```
```  1080 apply (rule mult_scaleR_left)
```
```  1081 apply (rule mult_scaleR_right)
```
```  1082 apply (rule_tac x="1" in exI)
```
```  1083 apply (simp add: norm_mult_ineq)
```
```  1084 done
```
```  1085
```
```  1086 lemma bounded_linear_mult_left:
```
```  1087   "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
```
```  1088   using bounded_bilinear_mult
```
```  1089   by (rule bounded_bilinear.bounded_linear_left)
```
```  1090
```
```  1091 lemma bounded_linear_mult_right:
```
```  1092   "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
```
```  1093   using bounded_bilinear_mult
```
```  1094   by (rule bounded_bilinear.bounded_linear_right)
```
```  1095
```
```  1096 lemma bounded_linear_divide:
```
```  1097   "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
```
```  1098   unfolding divide_inverse by (rule bounded_linear_mult_left)
```
```  1099
```
```  1100 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
```
```  1101 apply (rule bounded_bilinear.intro)
```
```  1102 apply (rule scaleR_left_distrib)
```
```  1103 apply (rule scaleR_right_distrib)
```
```  1104 apply simp
```
```  1105 apply (rule scaleR_left_commute)
```
```  1106 apply (rule_tac x="1" in exI, simp)
```
```  1107 done
```
```  1108
```
```  1109 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
```
```  1110   using bounded_bilinear_scaleR
```
```  1111   by (rule bounded_bilinear.bounded_linear_left)
```
```  1112
```
```  1113 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
```
```  1114   using bounded_bilinear_scaleR
```
```  1115   by (rule bounded_bilinear.bounded_linear_right)
```
```  1116
```
```  1117 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
```
```  1118   unfolding of_real_def by (rule bounded_linear_scaleR_left)
```
```  1119
```
```  1120 subsection{* Hausdorff and other separation properties *}
```
```  1121
```
```  1122 class t0_space = topological_space +
```
```  1123   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
```
```  1124
```
```  1125 class t1_space = topological_space +
```
```  1126   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
```
```  1127
```
```  1128 instance t1_space \<subseteq> t0_space
```
```  1129 proof qed (fast dest: t1_space)
```
```  1130
```
```  1131 lemma separation_t1:
```
```  1132   fixes x y :: "'a::t1_space"
```
```  1133   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
```
```  1134   using t1_space[of x y] by blast
```
```  1135
```
```  1136 lemma closed_singleton:
```
```  1137   fixes a :: "'a::t1_space"
```
```  1138   shows "closed {a}"
```
```  1139 proof -
```
```  1140   let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
```
```  1141   have "open ?T" by (simp add: open_Union)
```
```  1142   also have "?T = - {a}"
```
```  1143     by (simp add: set_eq_iff separation_t1, auto)
```
```  1144   finally show "closed {a}" unfolding closed_def .
```
```  1145 qed
```
```  1146
```
```  1147 lemma closed_insert [simp]:
```
```  1148   fixes a :: "'a::t1_space"
```
```  1149   assumes "closed S" shows "closed (insert a S)"
```
```  1150 proof -
```
```  1151   from closed_singleton assms
```
```  1152   have "closed ({a} \<union> S)" by (rule closed_Un)
```
```  1153   thus "closed (insert a S)" by simp
```
```  1154 qed
```
```  1155
```
```  1156 lemma finite_imp_closed:
```
```  1157   fixes S :: "'a::t1_space set"
```
```  1158   shows "finite S \<Longrightarrow> closed S"
```
```  1159 by (induct set: finite, simp_all)
```
```  1160
```
```  1161 text {* T2 spaces are also known as Hausdorff spaces. *}
```
```  1162
```
```  1163 class t2_space = topological_space +
```
```  1164   assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  1165
```
```  1166 instance t2_space \<subseteq> t1_space
```
```  1167 proof qed (fast dest: hausdorff)
```
```  1168
```
```  1169 instance metric_space \<subseteq> t2_space
```
```  1170 proof
```
```  1171   fix x y :: "'a::metric_space"
```
```  1172   assume xy: "x \<noteq> y"
```
```  1173   let ?U = "{y'. dist x y' < dist x y / 2}"
```
```  1174   let ?V = "{x'. dist y x' < dist x y / 2}"
```
```  1175   have th0: "\<And>d x y z. (d x z :: real) \<le> d x y + d y z \<Longrightarrow> d y z = d z y
```
```  1176                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```  1177   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```  1178     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
```
```  1179     using open_ball[of _ "dist x y / 2"] by auto
```
```  1180   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  1181     by blast
```
```  1182 qed
```
```  1183
```
```  1184 lemma separation_t2:
```
```  1185   fixes x y :: "'a::t2_space"
```
```  1186   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
```
```  1187   using hausdorff[of x y] by blast
```
```  1188
```
```  1189 lemma separation_t0:
```
```  1190   fixes x y :: "'a::t0_space"
```
```  1191   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
```
```  1192   using t0_space[of x y] by blast
```
```  1193
```
```  1194 text {* A perfect space is a topological space with no isolated points. *}
```
```  1195
```
```  1196 class perfect_space = topological_space +
```
```  1197   assumes not_open_singleton: "\<not> open {x}"
```
```  1198
```
```  1199 instance real_normed_algebra_1 \<subseteq> perfect_space
```
```  1200 proof
```
```  1201   fix x::'a
```
```  1202   show "\<not> open {x}"
```
```  1203     unfolding open_dist dist_norm
```
```  1204     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
```
```  1205 qed
```
```  1206
```
```  1207 end
```