src/HOL/RealVector.thy
 author hoelzl Thu Jan 31 17:42:12 2013 +0100 (2013-01-31) changeset 51002 496013a6eb38 parent 50999 3de230ed0547 child 51022 78de6c7e8a58 permissions -rw-r--r--
remove unnecessary assumption from real_normed_vector
```     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_numeral: "of_real (numeral w) = numeral w"
```
```   307 using of_real_of_int_eq [of "numeral w"] by simp
```
```   308
```
```   309 lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w"
```
```   310 using of_real_of_int_eq [of "neg_numeral w"] by simp
```
```   311
```
```   312 text{*Every real algebra has characteristic zero*}
```
```   313
```
```   314 instance real_algebra_1 < ring_char_0
```
```   315 proof
```
```   316   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
```
```   317   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
```
```   318 qed
```
```   319
```
```   320 instance real_field < field_char_0 ..
```
```   321
```
```   322
```
```   323 subsection {* The Set of Real Numbers *}
```
```   324
```
```   325 definition Reals :: "'a::real_algebra_1 set" where
```
```   326   "Reals = range of_real"
```
```   327
```
```   328 notation (xsymbols)
```
```   329   Reals  ("\<real>")
```
```   330
```
```   331 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   332 by (simp add: Reals_def)
```
```   333
```
```   334 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   335 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   336
```
```   337 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   338 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   339
```
```   340 lemma Reals_numeral [simp]: "numeral w \<in> Reals"
```
```   341 by (subst of_real_numeral [symmetric], rule Reals_of_real)
```
```   342
```
```   343 lemma Reals_neg_numeral [simp]: "neg_numeral w \<in> Reals"
```
```   344 by (subst of_real_neg_numeral [symmetric], rule Reals_of_real)
```
```   345
```
```   346 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   347 apply (unfold Reals_def)
```
```   348 apply (rule range_eqI)
```
```   349 apply (rule of_real_0 [symmetric])
```
```   350 done
```
```   351
```
```   352 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   353 apply (unfold Reals_def)
```
```   354 apply (rule range_eqI)
```
```   355 apply (rule of_real_1 [symmetric])
```
```   356 done
```
```   357
```
```   358 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   359 apply (auto simp add: Reals_def)
```
```   360 apply (rule range_eqI)
```
```   361 apply (rule of_real_add [symmetric])
```
```   362 done
```
```   363
```
```   364 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   365 apply (auto simp add: Reals_def)
```
```   366 apply (rule range_eqI)
```
```   367 apply (rule of_real_minus [symmetric])
```
```   368 done
```
```   369
```
```   370 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   371 apply (auto simp add: Reals_def)
```
```   372 apply (rule range_eqI)
```
```   373 apply (rule of_real_diff [symmetric])
```
```   374 done
```
```   375
```
```   376 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   377 apply (auto simp add: Reals_def)
```
```   378 apply (rule range_eqI)
```
```   379 apply (rule of_real_mult [symmetric])
```
```   380 done
```
```   381
```
```   382 lemma nonzero_Reals_inverse:
```
```   383   fixes a :: "'a::real_div_algebra"
```
```   384   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   385 apply (auto simp add: Reals_def)
```
```   386 apply (rule range_eqI)
```
```   387 apply (erule nonzero_of_real_inverse [symmetric])
```
```   388 done
```
```   389
```
```   390 lemma Reals_inverse [simp]:
```
```   391   fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   392   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   393 apply (auto simp add: Reals_def)
```
```   394 apply (rule range_eqI)
```
```   395 apply (rule of_real_inverse [symmetric])
```
```   396 done
```
```   397
```
```   398 lemma nonzero_Reals_divide:
```
```   399   fixes a b :: "'a::real_field"
```
```   400   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   401 apply (auto simp add: Reals_def)
```
```   402 apply (rule range_eqI)
```
```   403 apply (erule nonzero_of_real_divide [symmetric])
```
```   404 done
```
```   405
```
```   406 lemma Reals_divide [simp]:
```
```   407   fixes a b :: "'a::{real_field, field_inverse_zero}"
```
```   408   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   409 apply (auto simp add: Reals_def)
```
```   410 apply (rule range_eqI)
```
```   411 apply (rule of_real_divide [symmetric])
```
```   412 done
```
```   413
```
```   414 lemma Reals_power [simp]:
```
```   415   fixes a :: "'a::{real_algebra_1}"
```
```   416   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   417 apply (auto simp add: Reals_def)
```
```   418 apply (rule range_eqI)
```
```   419 apply (rule of_real_power [symmetric])
```
```   420 done
```
```   421
```
```   422 lemma Reals_cases [cases set: Reals]:
```
```   423   assumes "q \<in> \<real>"
```
```   424   obtains (of_real) r where "q = of_real r"
```
```   425   unfolding Reals_def
```
```   426 proof -
```
```   427   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   428   then obtain r where "q = of_real r" ..
```
```   429   then show thesis ..
```
```   430 qed
```
```   431
```
```   432 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   433   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   434   by (rule Reals_cases) auto
```
```   435
```
```   436
```
```   437 subsection {* Topological spaces *}
```
```   438
```
```   439 class "open" =
```
```   440   fixes "open" :: "'a set \<Rightarrow> bool"
```
```   441
```
```   442 class topological_space = "open" +
```
```   443   assumes open_UNIV [simp, intro]: "open UNIV"
```
```   444   assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
```
```   445   assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
```
```   446 begin
```
```   447
```
```   448 definition
```
```   449   closed :: "'a set \<Rightarrow> bool" where
```
```   450   "closed S \<longleftrightarrow> open (- S)"
```
```   451
```
```   452 lemma open_empty [intro, simp]: "open {}"
```
```   453   using open_Union [of "{}"] by simp
```
```   454
```
```   455 lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
```
```   456   using open_Union [of "{S, T}"] by simp
```
```   457
```
```   458 lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
```
```   459   unfolding SUP_def by (rule open_Union) auto
```
```   460
```
```   461 lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
```
```   462   by (induct set: finite) auto
```
```   463
```
```   464 lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
```
```   465   unfolding INF_def by (rule open_Inter) auto
```
```   466
```
```   467 lemma closed_empty [intro, simp]:  "closed {}"
```
```   468   unfolding closed_def by simp
```
```   469
```
```   470 lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
```
```   471   unfolding closed_def by auto
```
```   472
```
```   473 lemma closed_UNIV [intro, simp]: "closed UNIV"
```
```   474   unfolding closed_def by simp
```
```   475
```
```   476 lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
```
```   477   unfolding closed_def by auto
```
```   478
```
```   479 lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
```
```   480   unfolding closed_def by auto
```
```   481
```
```   482 lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
```
```   483   unfolding closed_def uminus_Inf by auto
```
```   484
```
```   485 lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
```
```   486   by (induct set: finite) auto
```
```   487
```
```   488 lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
```
```   489   unfolding SUP_def by (rule closed_Union) auto
```
```   490
```
```   491 lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
```
```   492   unfolding closed_def by simp
```
```   493
```
```   494 lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
```
```   495   unfolding closed_def by simp
```
```   496
```
```   497 lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
```
```   498   unfolding closed_open Diff_eq by (rule open_Int)
```
```   499
```
```   500 lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
```
```   501   unfolding open_closed Diff_eq by (rule closed_Int)
```
```   502
```
```   503 lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
```
```   504   unfolding closed_open .
```
```   505
```
```   506 lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
```
```   507   unfolding open_closed .
```
```   508
```
```   509 end
```
```   510
```
```   511 inductive generate_topology for S where
```
```   512   UNIV: "generate_topology S UNIV"
```
```   513 | Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"
```
```   514 | UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"
```
```   515 | Basis: "s \<in> S \<Longrightarrow> generate_topology S s"
```
```   516
```
```   517 hide_fact (open) UNIV Int UN Basis
```
```   518
```
```   519 lemma generate_topology_Union:
```
```   520   "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
```
```   521   unfolding SUP_def by (intro generate_topology.UN) auto
```
```   522
```
```   523 lemma topological_space_generate_topology:
```
```   524   "class.topological_space (generate_topology S)"
```
```   525   by default (auto intro: generate_topology.intros)
```
```   526
```
```   527 class order_topology = order + "open" +
```
```   528   assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
```
```   529 begin
```
```   530
```
```   531 subclass topological_space
```
```   532   unfolding open_generated_order
```
```   533   by (rule topological_space_generate_topology)
```
```   534
```
```   535 lemma open_greaterThan [simp]: "open {a <..}"
```
```   536   unfolding open_generated_order by (auto intro: generate_topology.Basis)
```
```   537
```
```   538 lemma open_lessThan [simp]: "open {..< a}"
```
```   539   unfolding open_generated_order by (auto intro: generate_topology.Basis)
```
```   540
```
```   541 lemma open_greaterThanLessThan [simp]: "open {a <..< b}"
```
```   542    unfolding greaterThanLessThan_eq by (simp add: open_Int)
```
```   543
```
```   544 end
```
```   545
```
```   546 class linorder_topology = linorder + order_topology
```
```   547
```
```   548 lemma closed_atMost [simp]: "closed {.. a::'a::linorder_topology}"
```
```   549   by (simp add: closed_open)
```
```   550
```
```   551 lemma closed_atLeast [simp]: "closed {a::'a::linorder_topology ..}"
```
```   552   by (simp add: closed_open)
```
```   553
```
```   554 lemma closed_atLeastAtMost [simp]: "closed {a::'a::linorder_topology .. b}"
```
```   555 proof -
```
```   556   have "{a .. b} = {a ..} \<inter> {.. b}"
```
```   557     by auto
```
```   558   then show ?thesis
```
```   559     by (simp add: closed_Int)
```
```   560 qed
```
```   561
```
```   562 inductive open_interval :: "'a::order set \<Rightarrow> bool" where
```
```   563   empty[intro]: "open_interval {}" |
```
```   564   UNIV[intro]: "open_interval UNIV" |
```
```   565   greaterThan[intro]: "open_interval {a <..}" |
```
```   566   lessThan[intro]: "open_interval {..< b}" |
```
```   567   greaterThanLessThan[intro]: "open_interval {a <..< b}"
```
```   568 hide_fact (open) empty UNIV greaterThan lessThan greaterThanLessThan
```
```   569
```
```   570 lemma open_intervalD:
```
```   571   "open_interval S \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> S"
```
```   572   by (cases rule: open_interval.cases) auto
```
```   573
```
```   574 lemma open_interval_Int[intro]:
```
```   575   fixes S T :: "'a :: linorder set"
```
```   576   assumes S: "open_interval S" and T: "open_interval T"
```
```   577   shows "open_interval (S \<inter> T)"
```
```   578 proof -
```
```   579   { fix a b :: 'a have "{..<b} \<inter> {a<..} = { a <..} \<inter> {..< b }" by auto } note this[simp]
```
```   580   { fix a b :: 'a and A have "{a <..} \<inter> ({b <..} \<inter> A) = {max a b <..} \<inter> A" by auto } note this[simp]
```
```   581   { fix a b :: 'a and A have "{..<b} \<inter> (A \<inter> {..<a}) = A \<inter> {..<min a b}" by auto } note this[simp]
```
```   582   { fix a b :: 'a have "open_interval ({ a <..} \<inter> {..< b})"
```
```   583       unfolding greaterThanLessThan_eq[symmetric] by auto } note this[simp]
```
```   584   show ?thesis
```
```   585     by (cases rule: open_interval.cases[OF S, case_product open_interval.cases[OF T]])
```
```   586        (auto simp: greaterThanLessThan_eq lessThan_Int_lessThan greaterThan_Int_greaterThan Int_assoc)
```
```   587 qed
```
```   588
```
```   589 lemma open_interval_imp_open: "open_interval S \<Longrightarrow> open (S::'a::order_topology set)"
```
```   590   by (cases S rule: open_interval.cases) auto
```
```   591
```
```   592 lemma open_orderD:
```
```   593   "open (S::'a::linorder_topology set) \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>T. open_interval T \<and> T \<subseteq> S \<and> x \<in> T"
```
```   594   unfolding open_generated_order
```
```   595 proof (induct rule: generate_topology.induct)
```
```   596   case (UN K) then obtain k where "k \<in> K" "x \<in> k" by auto
```
```   597   with UN(2)[of k] show ?case by auto
```
```   598 qed auto
```
```   599
```
```   600 lemma open_order_induct[consumes 2, case_names subset UNIV lessThan greaterThan greaterThanLessThan]:
```
```   601   fixes S :: "'a::linorder_topology set"
```
```   602   assumes S: "open S" "x \<in> S"
```
```   603   assumes subset: "\<And>S T. P S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> P T"
```
```   604   assumes univ: "P UNIV"
```
```   605   assumes lt: "\<And>a. x < a \<Longrightarrow> P {..< a}"
```
```   606   assumes gt: "\<And>a. a < x \<Longrightarrow> P {a <..}"
```
```   607   assumes lgt: "\<And>a b. a < x \<Longrightarrow> x < b \<Longrightarrow> P {a <..< b}"
```
```   608   shows "P S"
```
```   609 proof -
```
```   610   from open_orderD[OF S] obtain T where "open_interval T" "T \<subseteq> S" "x \<in> T"
```
```   611     by auto
```
```   612   then show "P S"
```
```   613     by induct (auto intro: univ subset lt gt lgt)
```
```   614 qed
```
```   615
```
```   616 subsection {* Metric spaces *}
```
```   617
```
```   618 class dist =
```
```   619   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```   620
```
```   621 class open_dist = "open" + dist +
```
```   622   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   623
```
```   624 class metric_space = open_dist +
```
```   625   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```   626   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```   627 begin
```
```   628
```
```   629 lemma dist_self [simp]: "dist x x = 0"
```
```   630 by simp
```
```   631
```
```   632 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```   633 using dist_triangle2 [of x x y] by simp
```
```   634
```
```   635 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```   636 by (simp add: less_le)
```
```   637
```
```   638 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```   639 by (simp add: not_less)
```
```   640
```
```   641 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```   642 by (simp add: le_less)
```
```   643
```
```   644 lemma dist_commute: "dist x y = dist y x"
```
```   645 proof (rule order_antisym)
```
```   646   show "dist x y \<le> dist y x"
```
```   647     using dist_triangle2 [of x y x] by simp
```
```   648   show "dist y x \<le> dist x y"
```
```   649     using dist_triangle2 [of y x y] by simp
```
```   650 qed
```
```   651
```
```   652 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```   653 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```   654
```
```   655 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```   656 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```   657
```
```   658 lemma dist_triangle_alt:
```
```   659   shows "dist y z <= dist x y + dist x z"
```
```   660 by (rule dist_triangle3)
```
```   661
```
```   662 lemma dist_pos_lt:
```
```   663   shows "x \<noteq> y ==> 0 < dist x y"
```
```   664 by (simp add: zero_less_dist_iff)
```
```   665
```
```   666 lemma dist_nz:
```
```   667   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
```
```   668 by (simp add: zero_less_dist_iff)
```
```   669
```
```   670 lemma dist_triangle_le:
```
```   671   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
```
```   672 by (rule order_trans [OF dist_triangle2])
```
```   673
```
```   674 lemma dist_triangle_lt:
```
```   675   shows "dist x z + dist y z < e ==> dist x y < e"
```
```   676 by (rule le_less_trans [OF dist_triangle2])
```
```   677
```
```   678 lemma dist_triangle_half_l:
```
```   679   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   680 by (rule dist_triangle_lt [where z=y], simp)
```
```   681
```
```   682 lemma dist_triangle_half_r:
```
```   683   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   684 by (rule dist_triangle_half_l, simp_all add: dist_commute)
```
```   685
```
```   686 subclass topological_space
```
```   687 proof
```
```   688   have "\<exists>e::real. 0 < e"
```
```   689     by (fast intro: zero_less_one)
```
```   690   then show "open UNIV"
```
```   691     unfolding open_dist by simp
```
```   692 next
```
```   693   fix S T assume "open S" "open T"
```
```   694   then show "open (S \<inter> T)"
```
```   695     unfolding open_dist
```
```   696     apply clarify
```
```   697     apply (drule (1) bspec)+
```
```   698     apply (clarify, rename_tac r s)
```
```   699     apply (rule_tac x="min r s" in exI, simp)
```
```   700     done
```
```   701 next
```
```   702   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```   703     unfolding open_dist by fast
```
```   704 qed
```
```   705
```
```   706 lemma (in metric_space) open_ball: "open {y. dist x y < d}"
```
```   707 proof (unfold open_dist, intro ballI)
```
```   708   fix y assume *: "y \<in> {y. dist x y < d}"
```
```   709   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
```
```   710     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
```
```   711 qed
```
```   712
```
```   713 end
```
```   714
```
```   715
```
```   716 subsection {* Real normed vector spaces *}
```
```   717
```
```   718 class norm =
```
```   719   fixes norm :: "'a \<Rightarrow> real"
```
```   720
```
```   721 class sgn_div_norm = scaleR + norm + sgn +
```
```   722   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   723
```
```   724 class dist_norm = dist + norm + minus +
```
```   725   assumes dist_norm: "dist x y = norm (x - y)"
```
```   726
```
```   727 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
```
```   728   assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   729   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   730   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   731 begin
```
```   732
```
```   733 lemma norm_ge_zero [simp]: "0 \<le> norm x"
```
```   734 proof -
```
```   735   have "0 = norm (x + -1 *\<^sub>R x)"
```
```   736     using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
```
```   737   also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
```
```   738   finally show ?thesis by simp
```
```   739 qed
```
```   740
```
```   741 end
```
```   742
```
```   743 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   744   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   745
```
```   746 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   747   assumes norm_one [simp]: "norm 1 = 1"
```
```   748
```
```   749 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   750   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   751
```
```   752 class real_normed_field = real_field + real_normed_div_algebra
```
```   753
```
```   754 instance real_normed_div_algebra < real_normed_algebra_1
```
```   755 proof
```
```   756   fix x y :: 'a
```
```   757   show "norm (x * y) \<le> norm x * norm y"
```
```   758     by (simp add: norm_mult)
```
```   759 next
```
```   760   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   761     by (rule norm_mult)
```
```   762   thus "norm (1::'a) = 1" by simp
```
```   763 qed
```
```   764
```
```   765 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   766 by simp
```
```   767
```
```   768 lemma zero_less_norm_iff [simp]:
```
```   769   fixes x :: "'a::real_normed_vector"
```
```   770   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   771 by (simp add: order_less_le)
```
```   772
```
```   773 lemma norm_not_less_zero [simp]:
```
```   774   fixes x :: "'a::real_normed_vector"
```
```   775   shows "\<not> norm x < 0"
```
```   776 by (simp add: linorder_not_less)
```
```   777
```
```   778 lemma norm_le_zero_iff [simp]:
```
```   779   fixes x :: "'a::real_normed_vector"
```
```   780   shows "(norm x \<le> 0) = (x = 0)"
```
```   781 by (simp add: order_le_less)
```
```   782
```
```   783 lemma norm_minus_cancel [simp]:
```
```   784   fixes x :: "'a::real_normed_vector"
```
```   785   shows "norm (- x) = norm x"
```
```   786 proof -
```
```   787   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   788     by (simp only: scaleR_minus_left scaleR_one)
```
```   789   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   790     by (rule norm_scaleR)
```
```   791   finally show ?thesis by simp
```
```   792 qed
```
```   793
```
```   794 lemma norm_minus_commute:
```
```   795   fixes a b :: "'a::real_normed_vector"
```
```   796   shows "norm (a - b) = norm (b - a)"
```
```   797 proof -
```
```   798   have "norm (- (b - a)) = norm (b - a)"
```
```   799     by (rule norm_minus_cancel)
```
```   800   thus ?thesis by simp
```
```   801 qed
```
```   802
```
```   803 lemma norm_triangle_ineq2:
```
```   804   fixes a b :: "'a::real_normed_vector"
```
```   805   shows "norm a - norm b \<le> norm (a - b)"
```
```   806 proof -
```
```   807   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   808     by (rule norm_triangle_ineq)
```
```   809   thus ?thesis by simp
```
```   810 qed
```
```   811
```
```   812 lemma norm_triangle_ineq3:
```
```   813   fixes a b :: "'a::real_normed_vector"
```
```   814   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   815 apply (subst abs_le_iff)
```
```   816 apply auto
```
```   817 apply (rule norm_triangle_ineq2)
```
```   818 apply (subst norm_minus_commute)
```
```   819 apply (rule norm_triangle_ineq2)
```
```   820 done
```
```   821
```
```   822 lemma norm_triangle_ineq4:
```
```   823   fixes a b :: "'a::real_normed_vector"
```
```   824   shows "norm (a - b) \<le> norm a + norm b"
```
```   825 proof -
```
```   826   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   827     by (rule norm_triangle_ineq)
```
```   828   thus ?thesis
```
```   829     by (simp only: diff_minus norm_minus_cancel)
```
```   830 qed
```
```   831
```
```   832 lemma norm_diff_ineq:
```
```   833   fixes a b :: "'a::real_normed_vector"
```
```   834   shows "norm a - norm b \<le> norm (a + b)"
```
```   835 proof -
```
```   836   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   837     by (rule norm_triangle_ineq2)
```
```   838   thus ?thesis by simp
```
```   839 qed
```
```   840
```
```   841 lemma norm_diff_triangle_ineq:
```
```   842   fixes a b c d :: "'a::real_normed_vector"
```
```   843   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   844 proof -
```
```   845   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   846     by (simp add: diff_minus add_ac)
```
```   847   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   848     by (rule norm_triangle_ineq)
```
```   849   finally show ?thesis .
```
```   850 qed
```
```   851
```
```   852 lemma abs_norm_cancel [simp]:
```
```   853   fixes a :: "'a::real_normed_vector"
```
```   854   shows "\<bar>norm a\<bar> = norm a"
```
```   855 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   856
```
```   857 lemma norm_add_less:
```
```   858   fixes x y :: "'a::real_normed_vector"
```
```   859   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   860 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   861
```
```   862 lemma norm_mult_less:
```
```   863   fixes x y :: "'a::real_normed_algebra"
```
```   864   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   865 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   866 apply (simp add: mult_strict_mono')
```
```   867 done
```
```   868
```
```   869 lemma norm_of_real [simp]:
```
```   870   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   871 unfolding of_real_def by simp
```
```   872
```
```   873 lemma norm_numeral [simp]:
```
```   874   "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   875 by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
```
```   876
```
```   877 lemma norm_neg_numeral [simp]:
```
```   878   "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   879 by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
```
```   880
```
```   881 lemma norm_of_int [simp]:
```
```   882   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   883 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   884
```
```   885 lemma norm_of_nat [simp]:
```
```   886   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   887 apply (subst of_real_of_nat_eq [symmetric])
```
```   888 apply (subst norm_of_real, simp)
```
```   889 done
```
```   890
```
```   891 lemma nonzero_norm_inverse:
```
```   892   fixes a :: "'a::real_normed_div_algebra"
```
```   893   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   894 apply (rule inverse_unique [symmetric])
```
```   895 apply (simp add: norm_mult [symmetric])
```
```   896 done
```
```   897
```
```   898 lemma norm_inverse:
```
```   899   fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   900   shows "norm (inverse a) = inverse (norm a)"
```
```   901 apply (case_tac "a = 0", simp)
```
```   902 apply (erule nonzero_norm_inverse)
```
```   903 done
```
```   904
```
```   905 lemma nonzero_norm_divide:
```
```   906   fixes a b :: "'a::real_normed_field"
```
```   907   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   908 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   909
```
```   910 lemma norm_divide:
```
```   911   fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
```
```   912   shows "norm (a / b) = norm a / norm b"
```
```   913 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   914
```
```   915 lemma norm_power_ineq:
```
```   916   fixes x :: "'a::{real_normed_algebra_1}"
```
```   917   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   918 proof (induct n)
```
```   919   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   920 next
```
```   921   case (Suc n)
```
```   922   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   923     by (rule norm_mult_ineq)
```
```   924   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   925     using norm_ge_zero by (rule mult_left_mono)
```
```   926   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   927     by simp
```
```   928 qed
```
```   929
```
```   930 lemma norm_power:
```
```   931   fixes x :: "'a::{real_normed_div_algebra}"
```
```   932   shows "norm (x ^ n) = norm x ^ n"
```
```   933 by (induct n) (simp_all add: norm_mult)
```
```   934
```
```   935 text {* Every normed vector space is a metric space. *}
```
```   936
```
```   937 instance real_normed_vector < metric_space
```
```   938 proof
```
```   939   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   940     unfolding dist_norm by simp
```
```   941 next
```
```   942   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
```
```   943     unfolding dist_norm
```
```   944     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
```
```   945 qed
```
```   946
```
```   947
```
```   948 subsection {* Class instances for real numbers *}
```
```   949
```
```   950 instantiation real :: real_normed_field
```
```   951 begin
```
```   952
```
```   953 definition real_norm_def [simp]:
```
```   954   "norm r = \<bar>r\<bar>"
```
```   955
```
```   956 definition dist_real_def:
```
```   957   "dist x y = \<bar>x - y\<bar>"
```
```   958
```
```   959 definition open_real_def:
```
```   960   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   961
```
```   962 instance
```
```   963 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```   964 apply (rule dist_real_def)
```
```   965 apply (rule open_real_def)
```
```   966 apply (simp add: sgn_real_def)
```
```   967 apply (rule abs_eq_0)
```
```   968 apply (rule abs_triangle_ineq)
```
```   969 apply (rule abs_mult)
```
```   970 apply (rule abs_mult)
```
```   971 done
```
```   972
```
```   973 end
```
```   974
```
```   975 instance real :: linorder_topology
```
```   976 proof
```
```   977   show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
```
```   978   proof (rule ext, safe)
```
```   979     fix S :: "real set" assume "open S"
```
```   980     then guess f unfolding open_real_def bchoice_iff ..
```
```   981     then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
```
```   982       by (fastforce simp: dist_real_def)
```
```   983     show "generate_topology (range lessThan \<union> range greaterThan) S"
```
```   984       apply (subst *)
```
```   985       apply (intro generate_topology_Union generate_topology.Int)
```
```   986       apply (auto intro: generate_topology.Basis)
```
```   987       done
```
```   988   next
```
```   989     fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
```
```   990     moreover have "\<And>a::real. open {..<a}"
```
```   991       unfolding open_real_def dist_real_def
```
```   992     proof clarify
```
```   993       fix x a :: real assume "x < a"
```
```   994       hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```   995       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```   996     qed
```
```   997     moreover have "\<And>a::real. open {a <..}"
```
```   998       unfolding open_real_def dist_real_def
```
```   999     proof clarify
```
```  1000       fix x a :: real assume "a < x"
```
```  1001       hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```  1002       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```  1003     qed
```
```  1004     ultimately show "open S"
```
```  1005       by induct auto
```
```  1006   qed
```
```  1007 qed
```
```  1008
```
```  1009 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
```
```  1010 lemmas open_real_lessThan = open_lessThan[where 'a=real]
```
```  1011 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
```
```  1012 lemmas closed_real_atMost = closed_atMost[where 'a=real]
```
```  1013 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
```
```  1014 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
```
```  1015
```
```  1016 subsection {* Extra type constraints *}
```
```  1017
```
```  1018 text {* Only allow @{term "open"} in class @{text topological_space}. *}
```
```  1019
```
```  1020 setup {* Sign.add_const_constraint
```
```  1021   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
```
```  1022
```
```  1023 text {* Only allow @{term dist} in class @{text metric_space}. *}
```
```  1024
```
```  1025 setup {* Sign.add_const_constraint
```
```  1026   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
```
```  1027
```
```  1028 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
```
```  1029
```
```  1030 setup {* Sign.add_const_constraint
```
```  1031   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
```
```  1032
```
```  1033
```
```  1034 subsection {* Sign function *}
```
```  1035
```
```  1036 lemma norm_sgn:
```
```  1037   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```  1038 by (simp add: sgn_div_norm)
```
```  1039
```
```  1040 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```  1041 by (simp add: sgn_div_norm)
```
```  1042
```
```  1043 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```  1044 by (simp add: sgn_div_norm)
```
```  1045
```
```  1046 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```  1047 by (simp add: sgn_div_norm)
```
```  1048
```
```  1049 lemma sgn_scaleR:
```
```  1050   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```  1051 by (simp add: sgn_div_norm mult_ac)
```
```  1052
```
```  1053 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```  1054 by (simp add: sgn_div_norm)
```
```  1055
```
```  1056 lemma sgn_of_real:
```
```  1057   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```  1058 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```  1059
```
```  1060 lemma sgn_mult:
```
```  1061   fixes x y :: "'a::real_normed_div_algebra"
```
```  1062   shows "sgn (x * y) = sgn x * sgn y"
```
```  1063 by (simp add: sgn_div_norm norm_mult mult_commute)
```
```  1064
```
```  1065 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```  1066 by (simp add: sgn_div_norm divide_inverse)
```
```  1067
```
```  1068 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```  1069 unfolding real_sgn_eq by simp
```
```  1070
```
```  1071 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```  1072 unfolding real_sgn_eq by simp
```
```  1073
```
```  1074
```
```  1075 subsection {* Bounded Linear and Bilinear Operators *}
```
```  1076
```
```  1077 locale bounded_linear = additive f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
```
```  1078   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```  1079   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1080 begin
```
```  1081
```
```  1082 lemma pos_bounded:
```
```  1083   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1084 proof -
```
```  1085   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```  1086     using bounded by fast
```
```  1087   show ?thesis
```
```  1088   proof (intro exI impI conjI allI)
```
```  1089     show "0 < max 1 K"
```
```  1090       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```  1091   next
```
```  1092     fix x
```
```  1093     have "norm (f x) \<le> norm x * K" using K .
```
```  1094     also have "\<dots> \<le> norm x * max 1 K"
```
```  1095       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```  1096     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```  1097   qed
```
```  1098 qed
```
```  1099
```
```  1100 lemma nonneg_bounded:
```
```  1101   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1102 proof -
```
```  1103   from pos_bounded
```
```  1104   show ?thesis by (auto intro: order_less_imp_le)
```
```  1105 qed
```
```  1106
```
```  1107 end
```
```  1108
```
```  1109 lemma bounded_linear_intro:
```
```  1110   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1111   assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
```
```  1112   assumes "\<And>x. norm (f x) \<le> norm x * K"
```
```  1113   shows "bounded_linear f"
```
```  1114   by default (fast intro: assms)+
```
```  1115
```
```  1116 locale bounded_bilinear =
```
```  1117   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```  1118                  \<Rightarrow> 'c::real_normed_vector"
```
```  1119     (infixl "**" 70)
```
```  1120   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```  1121   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```  1122   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```  1123   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```  1124   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```  1125 begin
```
```  1126
```
```  1127 lemma pos_bounded:
```
```  1128   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1129 apply (cut_tac bounded, erule exE)
```
```  1130 apply (rule_tac x="max 1 K" in exI, safe)
```
```  1131 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```  1132 apply (drule spec, drule spec, erule order_trans)
```
```  1133 apply (rule mult_left_mono [OF le_maxI2])
```
```  1134 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```  1135 done
```
```  1136
```
```  1137 lemma nonneg_bounded:
```
```  1138   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1139 proof -
```
```  1140   from pos_bounded
```
```  1141   show ?thesis by (auto intro: order_less_imp_le)
```
```  1142 qed
```
```  1143
```
```  1144 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```  1145 by (rule additive.intro, rule add_right)
```
```  1146
```
```  1147 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```  1148 by (rule additive.intro, rule add_left)
```
```  1149
```
```  1150 lemma zero_left: "prod 0 b = 0"
```
```  1151 by (rule additive.zero [OF additive_left])
```
```  1152
```
```  1153 lemma zero_right: "prod a 0 = 0"
```
```  1154 by (rule additive.zero [OF additive_right])
```
```  1155
```
```  1156 lemma minus_left: "prod (- a) b = - prod a b"
```
```  1157 by (rule additive.minus [OF additive_left])
```
```  1158
```
```  1159 lemma minus_right: "prod a (- b) = - prod a b"
```
```  1160 by (rule additive.minus [OF additive_right])
```
```  1161
```
```  1162 lemma diff_left:
```
```  1163   "prod (a - a') b = prod a b - prod a' b"
```
```  1164 by (rule additive.diff [OF additive_left])
```
```  1165
```
```  1166 lemma diff_right:
```
```  1167   "prod a (b - b') = prod a b - prod a b'"
```
```  1168 by (rule additive.diff [OF additive_right])
```
```  1169
```
```  1170 lemma bounded_linear_left:
```
```  1171   "bounded_linear (\<lambda>a. a ** b)"
```
```  1172 apply (cut_tac bounded, safe)
```
```  1173 apply (rule_tac K="norm b * K" in bounded_linear_intro)
```
```  1174 apply (rule add_left)
```
```  1175 apply (rule scaleR_left)
```
```  1176 apply (simp add: mult_ac)
```
```  1177 done
```
```  1178
```
```  1179 lemma bounded_linear_right:
```
```  1180   "bounded_linear (\<lambda>b. a ** b)"
```
```  1181 apply (cut_tac bounded, safe)
```
```  1182 apply (rule_tac K="norm a * K" in bounded_linear_intro)
```
```  1183 apply (rule add_right)
```
```  1184 apply (rule scaleR_right)
```
```  1185 apply (simp add: mult_ac)
```
```  1186 done
```
```  1187
```
```  1188 lemma prod_diff_prod:
```
```  1189   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```  1190 by (simp add: diff_left diff_right)
```
```  1191
```
```  1192 end
```
```  1193
```
```  1194 lemma bounded_bilinear_mult:
```
```  1195   "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
```
```  1196 apply (rule bounded_bilinear.intro)
```
```  1197 apply (rule distrib_right)
```
```  1198 apply (rule distrib_left)
```
```  1199 apply (rule mult_scaleR_left)
```
```  1200 apply (rule mult_scaleR_right)
```
```  1201 apply (rule_tac x="1" in exI)
```
```  1202 apply (simp add: norm_mult_ineq)
```
```  1203 done
```
```  1204
```
```  1205 lemma bounded_linear_mult_left:
```
```  1206   "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
```
```  1207   using bounded_bilinear_mult
```
```  1208   by (rule bounded_bilinear.bounded_linear_left)
```
```  1209
```
```  1210 lemma bounded_linear_mult_right:
```
```  1211   "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
```
```  1212   using bounded_bilinear_mult
```
```  1213   by (rule bounded_bilinear.bounded_linear_right)
```
```  1214
```
```  1215 lemma bounded_linear_divide:
```
```  1216   "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
```
```  1217   unfolding divide_inverse by (rule bounded_linear_mult_left)
```
```  1218
```
```  1219 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
```
```  1220 apply (rule bounded_bilinear.intro)
```
```  1221 apply (rule scaleR_left_distrib)
```
```  1222 apply (rule scaleR_right_distrib)
```
```  1223 apply simp
```
```  1224 apply (rule scaleR_left_commute)
```
```  1225 apply (rule_tac x="1" in exI, simp)
```
```  1226 done
```
```  1227
```
```  1228 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
```
```  1229   using bounded_bilinear_scaleR
```
```  1230   by (rule bounded_bilinear.bounded_linear_left)
```
```  1231
```
```  1232 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
```
```  1233   using bounded_bilinear_scaleR
```
```  1234   by (rule bounded_bilinear.bounded_linear_right)
```
```  1235
```
```  1236 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
```
```  1237   unfolding of_real_def by (rule bounded_linear_scaleR_left)
```
```  1238
```
```  1239 subsection{* Hausdorff and other separation properties *}
```
```  1240
```
```  1241 class t0_space = topological_space +
```
```  1242   assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
```
```  1243
```
```  1244 class t1_space = topological_space +
```
```  1245   assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
```
```  1246
```
```  1247 instance t1_space \<subseteq> t0_space
```
```  1248 proof qed (fast dest: t1_space)
```
```  1249
```
```  1250 lemma separation_t1:
```
```  1251   fixes x y :: "'a::t1_space"
```
```  1252   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
```
```  1253   using t1_space[of x y] by blast
```
```  1254
```
```  1255 lemma closed_singleton:
```
```  1256   fixes a :: "'a::t1_space"
```
```  1257   shows "closed {a}"
```
```  1258 proof -
```
```  1259   let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
```
```  1260   have "open ?T" by (simp add: open_Union)
```
```  1261   also have "?T = - {a}"
```
```  1262     by (simp add: set_eq_iff separation_t1, auto)
```
```  1263   finally show "closed {a}" unfolding closed_def .
```
```  1264 qed
```
```  1265
```
```  1266 lemma closed_insert [simp]:
```
```  1267   fixes a :: "'a::t1_space"
```
```  1268   assumes "closed S" shows "closed (insert a S)"
```
```  1269 proof -
```
```  1270   from closed_singleton assms
```
```  1271   have "closed ({a} \<union> S)" by (rule closed_Un)
```
```  1272   thus "closed (insert a S)" by simp
```
```  1273 qed
```
```  1274
```
```  1275 lemma finite_imp_closed:
```
```  1276   fixes S :: "'a::t1_space set"
```
```  1277   shows "finite S \<Longrightarrow> closed S"
```
```  1278 by (induct set: finite, simp_all)
```
```  1279
```
```  1280 text {* T2 spaces are also known as Hausdorff spaces. *}
```
```  1281
```
```  1282 class t2_space = topological_space +
```
```  1283   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 = {}"
```
```  1284
```
```  1285 instance t2_space \<subseteq> t1_space
```
```  1286 proof qed (fast dest: hausdorff)
```
```  1287
```
```  1288 lemma (in linorder) less_separate:
```
```  1289   assumes "x < y"
```
```  1290   shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"
```
```  1291 proof cases
```
```  1292   assume "\<exists>z. x < z \<and> z < y"
```
```  1293   then guess z ..
```
```  1294   then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
```
```  1295     by auto
```
```  1296   then show ?thesis by blast
```
```  1297 next
```
```  1298   assume "\<not> (\<exists>z. x < z \<and> z < y)"
```
```  1299   with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"
```
```  1300     by auto
```
```  1301   then show ?thesis by blast
```
```  1302 qed
```
```  1303
```
```  1304 instance linorder_topology \<subseteq> t2_space
```
```  1305 proof
```
```  1306   fix x y :: 'a
```
```  1307   from less_separate[of x y] less_separate[of y x]
```
```  1308   show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  1309     by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+
```
```  1310 qed
```
```  1311
```
```  1312 instance metric_space \<subseteq> t2_space
```
```  1313 proof
```
```  1314   fix x y :: "'a::metric_space"
```
```  1315   assume xy: "x \<noteq> y"
```
```  1316   let ?U = "{y'. dist x y' < dist x y / 2}"
```
```  1317   let ?V = "{x'. dist y x' < dist x y / 2}"
```
```  1318   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
```
```  1319                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```  1320   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```  1321     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
```
```  1322     using open_ball[of _ "dist x y / 2"] by auto
```
```  1323   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  1324     by blast
```
```  1325 qed
```
```  1326
```
```  1327 lemma separation_t2:
```
```  1328   fixes x y :: "'a::t2_space"
```
```  1329   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 = {})"
```
```  1330   using hausdorff[of x y] by blast
```
```  1331
```
```  1332 lemma separation_t0:
```
```  1333   fixes x y :: "'a::t0_space"
```
```  1334   shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
```
```  1335   using t0_space[of x y] by blast
```
```  1336
```
```  1337 text {* A perfect space is a topological space with no isolated points. *}
```
```  1338
```
```  1339 class perfect_space = topological_space +
```
```  1340   assumes not_open_singleton: "\<not> open {x}"
```
```  1341
```
```  1342 instance real_normed_algebra_1 \<subseteq> perfect_space
```
```  1343 proof
```
```  1344   fix x::'a
```
```  1345   show "\<not> open {x}"
```
```  1346     unfolding open_dist dist_norm
```
```  1347     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
```
```  1348 qed
```
```  1349
```
```  1350 end
```