src/HOL/Real_Vector_Spaces.thy
 author wenzelm Tue Sep 03 01:12:40 2013 +0200 (2013-09-03) changeset 53374 a14d2a854c02 parent 52381 63eec9cea2c7 child 53381 355a4cac5440 permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
```     1 (*  Title:      HOL/Real_Vector_Spaces.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 header {* Vector Spaces and Algebras over the Reals *}
```
```     7
```
```     8 theory Real_Vector_Spaces
```
```     9 imports Real Topological_Spaces
```
```    10 begin
```
```    11
```
```    12 subsection {* Locale for additive functions *}
```
```    13
```
```    14 locale additive =
```
```    15   fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
```
```    16   assumes add: "f (x + y) = f x + f y"
```
```    17 begin
```
```    18
```
```    19 lemma zero: "f 0 = 0"
```
```    20 proof -
```
```    21   have "f 0 = f (0 + 0)" by simp
```
```    22   also have "\<dots> = f 0 + f 0" by (rule add)
```
```    23   finally show "f 0 = 0" by simp
```
```    24 qed
```
```    25
```
```    26 lemma minus: "f (- x) = - f x"
```
```    27 proof -
```
```    28   have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
```
```    29   also have "\<dots> = - f x + f x" by (simp add: zero)
```
```    30   finally show "f (- x) = - f x" by (rule add_right_imp_eq)
```
```    31 qed
```
```    32
```
```    33 lemma diff: "f (x - y) = f x - f y"
```
```    34 by (simp add: add minus diff_minus)
```
```    35
```
```    36 lemma setsum: "f (setsum g A) = (\<Sum>x\<in>A. f (g x))"
```
```    37 apply (cases "finite A")
```
```    38 apply (induct set: finite)
```
```    39 apply (simp add: zero)
```
```    40 apply (simp add: add)
```
```    41 apply (simp add: zero)
```
```    42 done
```
```    43
```
```    44 end
```
```    45
```
```    46 subsection {* Vector spaces *}
```
```    47
```
```    48 locale vector_space =
```
```    49   fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
```
```    50   assumes scale_right_distrib [algebra_simps]:
```
```    51     "scale a (x + y) = scale a x + scale a y"
```
```    52   and scale_left_distrib [algebra_simps]:
```
```    53     "scale (a + b) x = scale a x + scale b x"
```
```    54   and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
```
```    55   and scale_one [simp]: "scale 1 x = x"
```
```    56 begin
```
```    57
```
```    58 lemma scale_left_commute:
```
```    59   "scale a (scale b x) = scale b (scale a x)"
```
```    60 by (simp add: mult_commute)
```
```    61
```
```    62 lemma scale_zero_left [simp]: "scale 0 x = 0"
```
```    63   and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
```
```    64   and scale_left_diff_distrib [algebra_simps]:
```
```    65         "scale (a - b) x = scale a x - scale b x"
```
```    66   and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
```
```    67 proof -
```
```    68   interpret s: additive "\<lambda>a. scale a x"
```
```    69     proof qed (rule scale_left_distrib)
```
```    70   show "scale 0 x = 0" by (rule s.zero)
```
```    71   show "scale (- a) x = - (scale a x)" by (rule s.minus)
```
```    72   show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
```
```    73   show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum)
```
```    74 qed
```
```    75
```
```    76 lemma scale_zero_right [simp]: "scale a 0 = 0"
```
```    77   and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
```
```    78   and scale_right_diff_distrib [algebra_simps]:
```
```    79         "scale a (x - y) = scale a x - scale a y"
```
```    80   and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"
```
```    81 proof -
```
```    82   interpret s: additive "\<lambda>x. scale a x"
```
```    83     proof qed (rule scale_right_distrib)
```
```    84   show "scale a 0 = 0" by (rule s.zero)
```
```    85   show "scale a (- x) = - (scale a x)" by (rule s.minus)
```
```    86   show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
```
```    87   show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)
```
```    88 qed
```
```    89
```
```    90 lemma scale_eq_0_iff [simp]:
```
```    91   "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
```
```    92 proof cases
```
```    93   assume "a = 0" thus ?thesis by simp
```
```    94 next
```
```    95   assume anz [simp]: "a \<noteq> 0"
```
```    96   { assume "scale a x = 0"
```
```    97     hence "scale (inverse a) (scale a x) = 0" by simp
```
```    98     hence "x = 0" by simp }
```
```    99   thus ?thesis by force
```
```   100 qed
```
```   101
```
```   102 lemma scale_left_imp_eq:
```
```   103   "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
```
```   104 proof -
```
```   105   assume nonzero: "a \<noteq> 0"
```
```   106   assume "scale a x = scale a y"
```
```   107   hence "scale a (x - y) = 0"
```
```   108      by (simp add: scale_right_diff_distrib)
```
```   109   hence "x - y = 0" by (simp add: nonzero)
```
```   110   thus "x = y" by (simp only: right_minus_eq)
```
```   111 qed
```
```   112
```
```   113 lemma scale_right_imp_eq:
```
```   114   "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
```
```   115 proof -
```
```   116   assume nonzero: "x \<noteq> 0"
```
```   117   assume "scale a x = scale b x"
```
```   118   hence "scale (a - b) x = 0"
```
```   119      by (simp add: scale_left_diff_distrib)
```
```   120   hence "a - b = 0" by (simp add: nonzero)
```
```   121   thus "a = b" by (simp only: right_minus_eq)
```
```   122 qed
```
```   123
```
```   124 lemma scale_cancel_left [simp]:
```
```   125   "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
```
```   126 by (auto intro: scale_left_imp_eq)
```
```   127
```
```   128 lemma scale_cancel_right [simp]:
```
```   129   "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
```
```   130 by (auto intro: scale_right_imp_eq)
```
```   131
```
```   132 end
```
```   133
```
```   134 subsection {* Real vector spaces *}
```
```   135
```
```   136 class scaleR =
```
```   137   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
```
```   138 begin
```
```   139
```
```   140 abbreviation
```
```   141   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
```
```   142 where
```
```   143   "x /\<^sub>R r == scaleR (inverse r) x"
```
```   144
```
```   145 end
```
```   146
```
```   147 class real_vector = scaleR + ab_group_add +
```
```   148   assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```   149   and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```   150   and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```   151   and scaleR_one: "scaleR 1 x = x"
```
```   152
```
```   153 interpretation real_vector:
```
```   154   vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
```
```   155 apply unfold_locales
```
```   156 apply (rule scaleR_add_right)
```
```   157 apply (rule scaleR_add_left)
```
```   158 apply (rule scaleR_scaleR)
```
```   159 apply (rule scaleR_one)
```
```   160 done
```
```   161
```
```   162 text {* Recover original theorem names *}
```
```   163
```
```   164 lemmas scaleR_left_commute = real_vector.scale_left_commute
```
```   165 lemmas scaleR_zero_left = real_vector.scale_zero_left
```
```   166 lemmas scaleR_minus_left = real_vector.scale_minus_left
```
```   167 lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib
```
```   168 lemmas scaleR_setsum_left = real_vector.scale_setsum_left
```
```   169 lemmas scaleR_zero_right = real_vector.scale_zero_right
```
```   170 lemmas scaleR_minus_right = real_vector.scale_minus_right
```
```   171 lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib
```
```   172 lemmas scaleR_setsum_right = real_vector.scale_setsum_right
```
```   173 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
```
```   174 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
```
```   175 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
```
```   176 lemmas scaleR_cancel_left = real_vector.scale_cancel_left
```
```   177 lemmas scaleR_cancel_right = real_vector.scale_cancel_right
```
```   178
```
```   179 text {* Legacy names *}
```
```   180
```
```   181 lemmas scaleR_left_distrib = scaleR_add_left
```
```   182 lemmas scaleR_right_distrib = scaleR_add_right
```
```   183 lemmas scaleR_left_diff_distrib = scaleR_diff_left
```
```   184 lemmas scaleR_right_diff_distrib = scaleR_diff_right
```
```   185
```
```   186 lemma scaleR_minus1_left [simp]:
```
```   187   fixes x :: "'a::real_vector"
```
```   188   shows "scaleR (-1) x = - x"
```
```   189   using scaleR_minus_left [of 1 x] by simp
```
```   190
```
```   191 class real_algebra = real_vector + ring +
```
```   192   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```   193   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```   194
```
```   195 class real_algebra_1 = real_algebra + ring_1
```
```   196
```
```   197 class real_div_algebra = real_algebra_1 + division_ring
```
```   198
```
```   199 class real_field = real_div_algebra + field
```
```   200
```
```   201 instantiation real :: real_field
```
```   202 begin
```
```   203
```
```   204 definition
```
```   205   real_scaleR_def [simp]: "scaleR a x = a * x"
```
```   206
```
```   207 instance proof
```
```   208 qed (simp_all add: algebra_simps)
```
```   209
```
```   210 end
```
```   211
```
```   212 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
```
```   213 proof qed (rule scaleR_left_distrib)
```
```   214
```
```   215 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
```
```   216 proof qed (rule scaleR_right_distrib)
```
```   217
```
```   218 lemma nonzero_inverse_scaleR_distrib:
```
```   219   fixes x :: "'a::real_div_algebra" shows
```
```   220   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   221 by (rule inverse_unique, simp)
```
```   222
```
```   223 lemma inverse_scaleR_distrib:
```
```   224   fixes x :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   225   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   226 apply (case_tac "a = 0", simp)
```
```   227 apply (case_tac "x = 0", simp)
```
```   228 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   229 done
```
```   230
```
```   231
```
```   232 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   233 @{term of_real} *}
```
```   234
```
```   235 definition
```
```   236   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   237   "of_real r = scaleR r 1"
```
```   238
```
```   239 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   240 by (simp add: of_real_def)
```
```   241
```
```   242 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   243 by (simp add: of_real_def)
```
```   244
```
```   245 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   246 by (simp add: of_real_def)
```
```   247
```
```   248 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   249 by (simp add: of_real_def scaleR_left_distrib)
```
```   250
```
```   251 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   252 by (simp add: of_real_def)
```
```   253
```
```   254 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   255 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   256
```
```   257 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   258 by (simp add: of_real_def mult_commute)
```
```   259
```
```   260 lemma nonzero_of_real_inverse:
```
```   261   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   262    inverse (of_real x :: 'a::real_div_algebra)"
```
```   263 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   264
```
```   265 lemma of_real_inverse [simp]:
```
```   266   "of_real (inverse x) =
```
```   267    inverse (of_real x :: 'a::{real_div_algebra, division_ring_inverse_zero})"
```
```   268 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   269
```
```   270 lemma nonzero_of_real_divide:
```
```   271   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   272    (of_real x / of_real y :: 'a::real_field)"
```
```   273 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   274
```
```   275 lemma of_real_divide [simp]:
```
```   276   "of_real (x / y) =
```
```   277    (of_real x / of_real y :: 'a::{real_field, field_inverse_zero})"
```
```   278 by (simp add: divide_inverse)
```
```   279
```
```   280 lemma of_real_power [simp]:
```
```   281   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
```
```   282 by (induct n) simp_all
```
```   283
```
```   284 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   285 by (simp add: of_real_def)
```
```   286
```
```   287 lemma inj_of_real:
```
```   288   "inj of_real"
```
```   289   by (auto intro: injI)
```
```   290
```
```   291 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   292
```
```   293 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   294 proof
```
```   295   fix r
```
```   296   show "of_real r = id r"
```
```   297     by (simp add: of_real_def)
```
```   298 qed
```
```   299
```
```   300 text{*Collapse nested embeddings*}
```
```   301 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   302 by (induct n) auto
```
```   303
```
```   304 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   305 by (cases z rule: int_diff_cases, simp)
```
```   306
```
```   307 lemma of_real_numeral: "of_real (numeral w) = numeral w"
```
```   308 using of_real_of_int_eq [of "numeral w"] by simp
```
```   309
```
```   310 lemma of_real_neg_numeral: "of_real (neg_numeral w) = neg_numeral w"
```
```   311 using of_real_of_int_eq [of "neg_numeral w"] by simp
```
```   312
```
```   313 text{*Every real algebra has characteristic zero*}
```
```   314
```
```   315 instance real_algebra_1 < ring_char_0
```
```   316 proof
```
```   317   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
```
```   318   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
```
```   319 qed
```
```   320
```
```   321 instance real_field < field_char_0 ..
```
```   322
```
```   323
```
```   324 subsection {* The Set of Real Numbers *}
```
```   325
```
```   326 definition Reals :: "'a::real_algebra_1 set" where
```
```   327   "Reals = range of_real"
```
```   328
```
```   329 notation (xsymbols)
```
```   330   Reals  ("\<real>")
```
```   331
```
```   332 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   333 by (simp add: Reals_def)
```
```   334
```
```   335 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   336 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   337
```
```   338 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   339 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   340
```
```   341 lemma Reals_numeral [simp]: "numeral w \<in> Reals"
```
```   342 by (subst of_real_numeral [symmetric], rule Reals_of_real)
```
```   343
```
```   344 lemma Reals_neg_numeral [simp]: "neg_numeral w \<in> Reals"
```
```   345 by (subst of_real_neg_numeral [symmetric], rule Reals_of_real)
```
```   346
```
```   347 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   348 apply (unfold Reals_def)
```
```   349 apply (rule range_eqI)
```
```   350 apply (rule of_real_0 [symmetric])
```
```   351 done
```
```   352
```
```   353 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   354 apply (unfold Reals_def)
```
```   355 apply (rule range_eqI)
```
```   356 apply (rule of_real_1 [symmetric])
```
```   357 done
```
```   358
```
```   359 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   360 apply (auto simp add: Reals_def)
```
```   361 apply (rule range_eqI)
```
```   362 apply (rule of_real_add [symmetric])
```
```   363 done
```
```   364
```
```   365 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   366 apply (auto simp add: Reals_def)
```
```   367 apply (rule range_eqI)
```
```   368 apply (rule of_real_minus [symmetric])
```
```   369 done
```
```   370
```
```   371 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   372 apply (auto simp add: Reals_def)
```
```   373 apply (rule range_eqI)
```
```   374 apply (rule of_real_diff [symmetric])
```
```   375 done
```
```   376
```
```   377 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   378 apply (auto simp add: Reals_def)
```
```   379 apply (rule range_eqI)
```
```   380 apply (rule of_real_mult [symmetric])
```
```   381 done
```
```   382
```
```   383 lemma nonzero_Reals_inverse:
```
```   384   fixes a :: "'a::real_div_algebra"
```
```   385   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   386 apply (auto simp add: Reals_def)
```
```   387 apply (rule range_eqI)
```
```   388 apply (erule nonzero_of_real_inverse [symmetric])
```
```   389 done
```
```   390
```
```   391 lemma Reals_inverse [simp]:
```
```   392   fixes a :: "'a::{real_div_algebra, division_ring_inverse_zero}"
```
```   393   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   394 apply (auto simp add: Reals_def)
```
```   395 apply (rule range_eqI)
```
```   396 apply (rule of_real_inverse [symmetric])
```
```   397 done
```
```   398
```
```   399 lemma nonzero_Reals_divide:
```
```   400   fixes a b :: "'a::real_field"
```
```   401   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   402 apply (auto simp add: Reals_def)
```
```   403 apply (rule range_eqI)
```
```   404 apply (erule nonzero_of_real_divide [symmetric])
```
```   405 done
```
```   406
```
```   407 lemma Reals_divide [simp]:
```
```   408   fixes a b :: "'a::{real_field, field_inverse_zero}"
```
```   409   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   410 apply (auto simp add: Reals_def)
```
```   411 apply (rule range_eqI)
```
```   412 apply (rule of_real_divide [symmetric])
```
```   413 done
```
```   414
```
```   415 lemma Reals_power [simp]:
```
```   416   fixes a :: "'a::{real_algebra_1}"
```
```   417   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   418 apply (auto simp add: Reals_def)
```
```   419 apply (rule range_eqI)
```
```   420 apply (rule of_real_power [symmetric])
```
```   421 done
```
```   422
```
```   423 lemma Reals_cases [cases set: Reals]:
```
```   424   assumes "q \<in> \<real>"
```
```   425   obtains (of_real) r where "q = of_real r"
```
```   426   unfolding Reals_def
```
```   427 proof -
```
```   428   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   429   then obtain r where "q = of_real r" ..
```
```   430   then show thesis ..
```
```   431 qed
```
```   432
```
```   433 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   434   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   435   by (rule Reals_cases) auto
```
```   436
```
```   437
```
```   438 subsection {* Real normed vector spaces *}
```
```   439
```
```   440 class dist =
```
```   441   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```   442
```
```   443 class norm =
```
```   444   fixes norm :: "'a \<Rightarrow> real"
```
```   445
```
```   446 class sgn_div_norm = scaleR + norm + sgn +
```
```   447   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   448
```
```   449 class dist_norm = dist + norm + minus +
```
```   450   assumes dist_norm: "dist x y = norm (x - y)"
```
```   451
```
```   452 class open_dist = "open" + dist +
```
```   453   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   454
```
```   455 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
```
```   456   assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   457   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   458   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   459 begin
```
```   460
```
```   461 lemma norm_ge_zero [simp]: "0 \<le> norm x"
```
```   462 proof -
```
```   463   have "0 = norm (x + -1 *\<^sub>R x)"
```
```   464     using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
```
```   465   also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
```
```   466   finally show ?thesis by simp
```
```   467 qed
```
```   468
```
```   469 end
```
```   470
```
```   471 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   472   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   473
```
```   474 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   475   assumes norm_one [simp]: "norm 1 = 1"
```
```   476
```
```   477 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   478   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   479
```
```   480 class real_normed_field = real_field + real_normed_div_algebra
```
```   481
```
```   482 instance real_normed_div_algebra < real_normed_algebra_1
```
```   483 proof
```
```   484   fix x y :: 'a
```
```   485   show "norm (x * y) \<le> norm x * norm y"
```
```   486     by (simp add: norm_mult)
```
```   487 next
```
```   488   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   489     by (rule norm_mult)
```
```   490   thus "norm (1::'a) = 1" by simp
```
```   491 qed
```
```   492
```
```   493 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   494 by simp
```
```   495
```
```   496 lemma zero_less_norm_iff [simp]:
```
```   497   fixes x :: "'a::real_normed_vector"
```
```   498   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   499 by (simp add: order_less_le)
```
```   500
```
```   501 lemma norm_not_less_zero [simp]:
```
```   502   fixes x :: "'a::real_normed_vector"
```
```   503   shows "\<not> norm x < 0"
```
```   504 by (simp add: linorder_not_less)
```
```   505
```
```   506 lemma norm_le_zero_iff [simp]:
```
```   507   fixes x :: "'a::real_normed_vector"
```
```   508   shows "(norm x \<le> 0) = (x = 0)"
```
```   509 by (simp add: order_le_less)
```
```   510
```
```   511 lemma norm_minus_cancel [simp]:
```
```   512   fixes x :: "'a::real_normed_vector"
```
```   513   shows "norm (- x) = norm x"
```
```   514 proof -
```
```   515   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   516     by (simp only: scaleR_minus_left scaleR_one)
```
```   517   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   518     by (rule norm_scaleR)
```
```   519   finally show ?thesis by simp
```
```   520 qed
```
```   521
```
```   522 lemma norm_minus_commute:
```
```   523   fixes a b :: "'a::real_normed_vector"
```
```   524   shows "norm (a - b) = norm (b - a)"
```
```   525 proof -
```
```   526   have "norm (- (b - a)) = norm (b - a)"
```
```   527     by (rule norm_minus_cancel)
```
```   528   thus ?thesis by simp
```
```   529 qed
```
```   530
```
```   531 lemma norm_triangle_ineq2:
```
```   532   fixes a b :: "'a::real_normed_vector"
```
```   533   shows "norm a - norm b \<le> norm (a - b)"
```
```   534 proof -
```
```   535   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   536     by (rule norm_triangle_ineq)
```
```   537   thus ?thesis by simp
```
```   538 qed
```
```   539
```
```   540 lemma norm_triangle_ineq3:
```
```   541   fixes a b :: "'a::real_normed_vector"
```
```   542   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   543 apply (subst abs_le_iff)
```
```   544 apply auto
```
```   545 apply (rule norm_triangle_ineq2)
```
```   546 apply (subst norm_minus_commute)
```
```   547 apply (rule norm_triangle_ineq2)
```
```   548 done
```
```   549
```
```   550 lemma norm_triangle_ineq4:
```
```   551   fixes a b :: "'a::real_normed_vector"
```
```   552   shows "norm (a - b) \<le> norm a + norm b"
```
```   553 proof -
```
```   554   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   555     by (rule norm_triangle_ineq)
```
```   556   thus ?thesis
```
```   557     by (simp only: diff_minus norm_minus_cancel)
```
```   558 qed
```
```   559
```
```   560 lemma norm_diff_ineq:
```
```   561   fixes a b :: "'a::real_normed_vector"
```
```   562   shows "norm a - norm b \<le> norm (a + b)"
```
```   563 proof -
```
```   564   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   565     by (rule norm_triangle_ineq2)
```
```   566   thus ?thesis by simp
```
```   567 qed
```
```   568
```
```   569 lemma norm_diff_triangle_ineq:
```
```   570   fixes a b c d :: "'a::real_normed_vector"
```
```   571   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   572 proof -
```
```   573   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   574     by (simp add: diff_minus add_ac)
```
```   575   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   576     by (rule norm_triangle_ineq)
```
```   577   finally show ?thesis .
```
```   578 qed
```
```   579
```
```   580 lemma abs_norm_cancel [simp]:
```
```   581   fixes a :: "'a::real_normed_vector"
```
```   582   shows "\<bar>norm a\<bar> = norm a"
```
```   583 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   584
```
```   585 lemma norm_add_less:
```
```   586   fixes x y :: "'a::real_normed_vector"
```
```   587   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   588 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   589
```
```   590 lemma norm_mult_less:
```
```   591   fixes x y :: "'a::real_normed_algebra"
```
```   592   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   593 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   594 apply (simp add: mult_strict_mono')
```
```   595 done
```
```   596
```
```   597 lemma norm_of_real [simp]:
```
```   598   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   599 unfolding of_real_def by simp
```
```   600
```
```   601 lemma norm_numeral [simp]:
```
```   602   "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   603 by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
```
```   604
```
```   605 lemma norm_neg_numeral [simp]:
```
```   606   "norm (neg_numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   607 by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
```
```   608
```
```   609 lemma norm_of_int [simp]:
```
```   610   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   611 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   612
```
```   613 lemma norm_of_nat [simp]:
```
```   614   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   615 apply (subst of_real_of_nat_eq [symmetric])
```
```   616 apply (subst norm_of_real, simp)
```
```   617 done
```
```   618
```
```   619 lemma nonzero_norm_inverse:
```
```   620   fixes a :: "'a::real_normed_div_algebra"
```
```   621   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   622 apply (rule inverse_unique [symmetric])
```
```   623 apply (simp add: norm_mult [symmetric])
```
```   624 done
```
```   625
```
```   626 lemma norm_inverse:
```
```   627   fixes a :: "'a::{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   628   shows "norm (inverse a) = inverse (norm a)"
```
```   629 apply (case_tac "a = 0", simp)
```
```   630 apply (erule nonzero_norm_inverse)
```
```   631 done
```
```   632
```
```   633 lemma nonzero_norm_divide:
```
```   634   fixes a b :: "'a::real_normed_field"
```
```   635   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   636 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   637
```
```   638 lemma norm_divide:
```
```   639   fixes a b :: "'a::{real_normed_field, field_inverse_zero}"
```
```   640   shows "norm (a / b) = norm a / norm b"
```
```   641 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   642
```
```   643 lemma norm_power_ineq:
```
```   644   fixes x :: "'a::{real_normed_algebra_1}"
```
```   645   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   646 proof (induct n)
```
```   647   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   648 next
```
```   649   case (Suc n)
```
```   650   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   651     by (rule norm_mult_ineq)
```
```   652   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   653     using norm_ge_zero by (rule mult_left_mono)
```
```   654   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   655     by simp
```
```   656 qed
```
```   657
```
```   658 lemma norm_power:
```
```   659   fixes x :: "'a::{real_normed_div_algebra}"
```
```   660   shows "norm (x ^ n) = norm x ^ n"
```
```   661 by (induct n) (simp_all add: norm_mult)
```
```   662
```
```   663
```
```   664 subsection {* Metric spaces *}
```
```   665
```
```   666 class metric_space = open_dist +
```
```   667   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```   668   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```   669 begin
```
```   670
```
```   671 lemma dist_self [simp]: "dist x x = 0"
```
```   672 by simp
```
```   673
```
```   674 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```   675 using dist_triangle2 [of x x y] by simp
```
```   676
```
```   677 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```   678 by (simp add: less_le)
```
```   679
```
```   680 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```   681 by (simp add: not_less)
```
```   682
```
```   683 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```   684 by (simp add: le_less)
```
```   685
```
```   686 lemma dist_commute: "dist x y = dist y x"
```
```   687 proof (rule order_antisym)
```
```   688   show "dist x y \<le> dist y x"
```
```   689     using dist_triangle2 [of x y x] by simp
```
```   690   show "dist y x \<le> dist x y"
```
```   691     using dist_triangle2 [of y x y] by simp
```
```   692 qed
```
```   693
```
```   694 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```   695 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```   696
```
```   697 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```   698 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```   699
```
```   700 lemma dist_triangle_alt:
```
```   701   shows "dist y z <= dist x y + dist x z"
```
```   702 by (rule dist_triangle3)
```
```   703
```
```   704 lemma dist_pos_lt:
```
```   705   shows "x \<noteq> y ==> 0 < dist x y"
```
```   706 by (simp add: zero_less_dist_iff)
```
```   707
```
```   708 lemma dist_nz:
```
```   709   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
```
```   710 by (simp add: zero_less_dist_iff)
```
```   711
```
```   712 lemma dist_triangle_le:
```
```   713   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
```
```   714 by (rule order_trans [OF dist_triangle2])
```
```   715
```
```   716 lemma dist_triangle_lt:
```
```   717   shows "dist x z + dist y z < e ==> dist x y < e"
```
```   718 by (rule le_less_trans [OF dist_triangle2])
```
```   719
```
```   720 lemma dist_triangle_half_l:
```
```   721   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   722 by (rule dist_triangle_lt [where z=y], simp)
```
```   723
```
```   724 lemma dist_triangle_half_r:
```
```   725   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   726 by (rule dist_triangle_half_l, simp_all add: dist_commute)
```
```   727
```
```   728 subclass topological_space
```
```   729 proof
```
```   730   have "\<exists>e::real. 0 < e"
```
```   731     by (fast intro: zero_less_one)
```
```   732   then show "open UNIV"
```
```   733     unfolding open_dist by simp
```
```   734 next
```
```   735   fix S T assume "open S" "open T"
```
```   736   then show "open (S \<inter> T)"
```
```   737     unfolding open_dist
```
```   738     apply clarify
```
```   739     apply (drule (1) bspec)+
```
```   740     apply (clarify, rename_tac r s)
```
```   741     apply (rule_tac x="min r s" in exI, simp)
```
```   742     done
```
```   743 next
```
```   744   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```   745     unfolding open_dist by fast
```
```   746 qed
```
```   747
```
```   748 lemma open_ball: "open {y. dist x y < d}"
```
```   749 proof (unfold open_dist, intro ballI)
```
```   750   fix y assume *: "y \<in> {y. dist x y < d}"
```
```   751   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
```
```   752     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
```
```   753 qed
```
```   754
```
```   755 subclass first_countable_topology
```
```   756 proof
```
```   757   fix x
```
```   758   show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
```
```   759   proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
```
```   760     fix S assume "open S" "x \<in> S"
```
```   761     then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
```
```   762       by (auto simp: open_dist subset_eq dist_commute)
```
```   763     moreover
```
```   764     from e obtain i where "inverse (Suc i) < e"
```
```   765       by (auto dest!: reals_Archimedean)
```
```   766     then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
```
```   767       by auto
```
```   768     ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
```
```   769       by blast
```
```   770   qed (auto intro: open_ball)
```
```   771 qed
```
```   772
```
```   773 end
```
```   774
```
```   775 instance metric_space \<subseteq> t2_space
```
```   776 proof
```
```   777   fix x y :: "'a::metric_space"
```
```   778   assume xy: "x \<noteq> y"
```
```   779   let ?U = "{y'. dist x y' < dist x y / 2}"
```
```   780   let ?V = "{x'. dist y x' < dist x y / 2}"
```
```   781   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
```
```   782                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```   783   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```   784     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
```
```   785     using open_ball[of _ "dist x y / 2"] by auto
```
```   786   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```   787     by blast
```
```   788 qed
```
```   789
```
```   790 text {* Every normed vector space is a metric space. *}
```
```   791
```
```   792 instance real_normed_vector < metric_space
```
```   793 proof
```
```   794   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
```
```   795     unfolding dist_norm by simp
```
```   796 next
```
```   797   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
```
```   798     unfolding dist_norm
```
```   799     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
```
```   800 qed
```
```   801
```
```   802 subsection {* Class instances for real numbers *}
```
```   803
```
```   804 instantiation real :: real_normed_field
```
```   805 begin
```
```   806
```
```   807 definition dist_real_def:
```
```   808   "dist x y = \<bar>x - y\<bar>"
```
```   809
```
```   810 definition open_real_def [code del]:
```
```   811   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   812
```
```   813 definition real_norm_def [simp]:
```
```   814   "norm r = \<bar>r\<bar>"
```
```   815
```
```   816 instance
```
```   817 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```   818 apply (rule dist_real_def)
```
```   819 apply (rule open_real_def)
```
```   820 apply (simp add: sgn_real_def)
```
```   821 apply (rule abs_eq_0)
```
```   822 apply (rule abs_triangle_ineq)
```
```   823 apply (rule abs_mult)
```
```   824 apply (rule abs_mult)
```
```   825 done
```
```   826
```
```   827 end
```
```   828
```
```   829 code_abort "open :: real set \<Rightarrow> bool"
```
```   830
```
```   831 instance real :: linorder_topology
```
```   832 proof
```
```   833   show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
```
```   834   proof (rule ext, safe)
```
```   835     fix S :: "real set" assume "open S"
```
```   836     then guess f unfolding open_real_def bchoice_iff ..
```
```   837     then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
```
```   838       by (fastforce simp: dist_real_def)
```
```   839     show "generate_topology (range lessThan \<union> range greaterThan) S"
```
```   840       apply (subst *)
```
```   841       apply (intro generate_topology_Union generate_topology.Int)
```
```   842       apply (auto intro: generate_topology.Basis)
```
```   843       done
```
```   844   next
```
```   845     fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
```
```   846     moreover have "\<And>a::real. open {..<a}"
```
```   847       unfolding open_real_def dist_real_def
```
```   848     proof clarify
```
```   849       fix x a :: real assume "x < a"
```
```   850       hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```   851       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```   852     qed
```
```   853     moreover have "\<And>a::real. open {a <..}"
```
```   854       unfolding open_real_def dist_real_def
```
```   855     proof clarify
```
```   856       fix x a :: real assume "a < x"
```
```   857       hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```   858       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```   859     qed
```
```   860     ultimately show "open S"
```
```   861       by induct auto
```
```   862   qed
```
```   863 qed
```
```   864
```
```   865 instance real :: linear_continuum_topology ..
```
```   866
```
```   867 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
```
```   868 lemmas open_real_lessThan = open_lessThan[where 'a=real]
```
```   869 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
```
```   870 lemmas closed_real_atMost = closed_atMost[where 'a=real]
```
```   871 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
```
```   872 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
```
```   873
```
```   874 subsection {* Extra type constraints *}
```
```   875
```
```   876 text {* Only allow @{term "open"} in class @{text topological_space}. *}
```
```   877
```
```   878 setup {* Sign.add_const_constraint
```
```   879   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
```
```   880
```
```   881 text {* Only allow @{term dist} in class @{text metric_space}. *}
```
```   882
```
```   883 setup {* Sign.add_const_constraint
```
```   884   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
```
```   885
```
```   886 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
```
```   887
```
```   888 setup {* Sign.add_const_constraint
```
```   889   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
```
```   890
```
```   891 subsection {* Sign function *}
```
```   892
```
```   893 lemma norm_sgn:
```
```   894   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```   895 by (simp add: sgn_div_norm)
```
```   896
```
```   897 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```   898 by (simp add: sgn_div_norm)
```
```   899
```
```   900 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```   901 by (simp add: sgn_div_norm)
```
```   902
```
```   903 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```   904 by (simp add: sgn_div_norm)
```
```   905
```
```   906 lemma sgn_scaleR:
```
```   907   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```   908 by (simp add: sgn_div_norm mult_ac)
```
```   909
```
```   910 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```   911 by (simp add: sgn_div_norm)
```
```   912
```
```   913 lemma sgn_of_real:
```
```   914   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```   915 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```   916
```
```   917 lemma sgn_mult:
```
```   918   fixes x y :: "'a::real_normed_div_algebra"
```
```   919   shows "sgn (x * y) = sgn x * sgn y"
```
```   920 by (simp add: sgn_div_norm norm_mult mult_commute)
```
```   921
```
```   922 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```   923 by (simp add: sgn_div_norm divide_inverse)
```
```   924
```
```   925 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```   926 unfolding real_sgn_eq by simp
```
```   927
```
```   928 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```   929 unfolding real_sgn_eq by simp
```
```   930
```
```   931 lemma norm_conv_dist: "norm x = dist x 0"
```
```   932   unfolding dist_norm by simp
```
```   933
```
```   934 subsection {* Bounded Linear and Bilinear Operators *}
```
```   935
```
```   936 locale bounded_linear = additive f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
```
```   937   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```   938   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```   939 begin
```
```   940
```
```   941 lemma pos_bounded:
```
```   942   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   943 proof -
```
```   944   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```   945     using bounded by fast
```
```   946   show ?thesis
```
```   947   proof (intro exI impI conjI allI)
```
```   948     show "0 < max 1 K"
```
```   949       by (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   950   next
```
```   951     fix x
```
```   952     have "norm (f x) \<le> norm x * K" using K .
```
```   953     also have "\<dots> \<le> norm x * max 1 K"
```
```   954       by (rule mult_left_mono [OF le_maxI2 norm_ge_zero])
```
```   955     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```   956   qed
```
```   957 qed
```
```   958
```
```   959 lemma nonneg_bounded:
```
```   960   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```   961 proof -
```
```   962   from pos_bounded
```
```   963   show ?thesis by (auto intro: order_less_imp_le)
```
```   964 qed
```
```   965
```
```   966 end
```
```   967
```
```   968 lemma bounded_linear_intro:
```
```   969   assumes "\<And>x y. f (x + y) = f x + f y"
```
```   970   assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
```
```   971   assumes "\<And>x. norm (f x) \<le> norm x * K"
```
```   972   shows "bounded_linear f"
```
```   973   by default (fast intro: assms)+
```
```   974
```
```   975 locale bounded_bilinear =
```
```   976   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```   977                  \<Rightarrow> 'c::real_normed_vector"
```
```   978     (infixl "**" 70)
```
```   979   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```   980   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```   981   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```   982   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```   983   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```   984 begin
```
```   985
```
```   986 lemma pos_bounded:
```
```   987   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   988 apply (cut_tac bounded, erule exE)
```
```   989 apply (rule_tac x="max 1 K" in exI, safe)
```
```   990 apply (rule order_less_le_trans [OF zero_less_one le_maxI1])
```
```   991 apply (drule spec, drule spec, erule order_trans)
```
```   992 apply (rule mult_left_mono [OF le_maxI2])
```
```   993 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```   994 done
```
```   995
```
```   996 lemma nonneg_bounded:
```
```   997   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```   998 proof -
```
```   999   from pos_bounded
```
```  1000   show ?thesis by (auto intro: order_less_imp_le)
```
```  1001 qed
```
```  1002
```
```  1003 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```  1004 by (rule additive.intro, rule add_right)
```
```  1005
```
```  1006 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```  1007 by (rule additive.intro, rule add_left)
```
```  1008
```
```  1009 lemma zero_left: "prod 0 b = 0"
```
```  1010 by (rule additive.zero [OF additive_left])
```
```  1011
```
```  1012 lemma zero_right: "prod a 0 = 0"
```
```  1013 by (rule additive.zero [OF additive_right])
```
```  1014
```
```  1015 lemma minus_left: "prod (- a) b = - prod a b"
```
```  1016 by (rule additive.minus [OF additive_left])
```
```  1017
```
```  1018 lemma minus_right: "prod a (- b) = - prod a b"
```
```  1019 by (rule additive.minus [OF additive_right])
```
```  1020
```
```  1021 lemma diff_left:
```
```  1022   "prod (a - a') b = prod a b - prod a' b"
```
```  1023 by (rule additive.diff [OF additive_left])
```
```  1024
```
```  1025 lemma diff_right:
```
```  1026   "prod a (b - b') = prod a b - prod a b'"
```
```  1027 by (rule additive.diff [OF additive_right])
```
```  1028
```
```  1029 lemma bounded_linear_left:
```
```  1030   "bounded_linear (\<lambda>a. a ** b)"
```
```  1031 apply (cut_tac bounded, safe)
```
```  1032 apply (rule_tac K="norm b * K" in bounded_linear_intro)
```
```  1033 apply (rule add_left)
```
```  1034 apply (rule scaleR_left)
```
```  1035 apply (simp add: mult_ac)
```
```  1036 done
```
```  1037
```
```  1038 lemma bounded_linear_right:
```
```  1039   "bounded_linear (\<lambda>b. a ** b)"
```
```  1040 apply (cut_tac bounded, safe)
```
```  1041 apply (rule_tac K="norm a * K" in bounded_linear_intro)
```
```  1042 apply (rule add_right)
```
```  1043 apply (rule scaleR_right)
```
```  1044 apply (simp add: mult_ac)
```
```  1045 done
```
```  1046
```
```  1047 lemma prod_diff_prod:
```
```  1048   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```  1049 by (simp add: diff_left diff_right)
```
```  1050
```
```  1051 end
```
```  1052
```
```  1053 lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
```
```  1054   by default (auto intro!: exI[of _ 1])
```
```  1055
```
```  1056 lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
```
```  1057   by default (auto intro!: exI[of _ 1])
```
```  1058
```
```  1059 lemma bounded_linear_add:
```
```  1060   assumes "bounded_linear f"
```
```  1061   assumes "bounded_linear g"
```
```  1062   shows "bounded_linear (\<lambda>x. f x + g x)"
```
```  1063 proof -
```
```  1064   interpret f: bounded_linear f by fact
```
```  1065   interpret g: bounded_linear g by fact
```
```  1066   show ?thesis
```
```  1067   proof
```
```  1068     from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
```
```  1069     from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
```
```  1070     show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
```
```  1071       using add_mono[OF Kf Kg]
```
```  1072       by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
```
```  1073   qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
```
```  1074 qed
```
```  1075
```
```  1076 lemma bounded_linear_minus:
```
```  1077   assumes "bounded_linear f"
```
```  1078   shows "bounded_linear (\<lambda>x. - f x)"
```
```  1079 proof -
```
```  1080   interpret f: bounded_linear f by fact
```
```  1081   show ?thesis apply (unfold_locales)
```
```  1082     apply (simp add: f.add)
```
```  1083     apply (simp add: f.scaleR)
```
```  1084     apply (simp add: f.bounded)
```
```  1085     done
```
```  1086 qed
```
```  1087
```
```  1088 lemma bounded_linear_compose:
```
```  1089   assumes "bounded_linear f"
```
```  1090   assumes "bounded_linear g"
```
```  1091   shows "bounded_linear (\<lambda>x. f (g x))"
```
```  1092 proof -
```
```  1093   interpret f: bounded_linear f by fact
```
```  1094   interpret g: bounded_linear g by fact
```
```  1095   show ?thesis proof (unfold_locales)
```
```  1096     fix x y show "f (g (x + y)) = f (g x) + f (g y)"
```
```  1097       by (simp only: f.add g.add)
```
```  1098   next
```
```  1099     fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
```
```  1100       by (simp only: f.scaleR g.scaleR)
```
```  1101   next
```
```  1102     from f.pos_bounded
```
```  1103     obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
```
```  1104     from g.pos_bounded
```
```  1105     obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
```
```  1106     show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
```
```  1107     proof (intro exI allI)
```
```  1108       fix x
```
```  1109       have "norm (f (g x)) \<le> norm (g x) * Kf"
```
```  1110         using f .
```
```  1111       also have "\<dots> \<le> (norm x * Kg) * Kf"
```
```  1112         using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
```
```  1113       also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
```
```  1114         by (rule mult_assoc)
```
```  1115       finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
```
```  1116     qed
```
```  1117   qed
```
```  1118 qed
```
```  1119
```
```  1120 lemma bounded_bilinear_mult:
```
```  1121   "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
```
```  1122 apply (rule bounded_bilinear.intro)
```
```  1123 apply (rule distrib_right)
```
```  1124 apply (rule distrib_left)
```
```  1125 apply (rule mult_scaleR_left)
```
```  1126 apply (rule mult_scaleR_right)
```
```  1127 apply (rule_tac x="1" in exI)
```
```  1128 apply (simp add: norm_mult_ineq)
```
```  1129 done
```
```  1130
```
```  1131 lemma bounded_linear_mult_left:
```
```  1132   "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
```
```  1133   using bounded_bilinear_mult
```
```  1134   by (rule bounded_bilinear.bounded_linear_left)
```
```  1135
```
```  1136 lemma bounded_linear_mult_right:
```
```  1137   "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
```
```  1138   using bounded_bilinear_mult
```
```  1139   by (rule bounded_bilinear.bounded_linear_right)
```
```  1140
```
```  1141 lemmas bounded_linear_mult_const =
```
```  1142   bounded_linear_mult_left [THEN bounded_linear_compose]
```
```  1143
```
```  1144 lemmas bounded_linear_const_mult =
```
```  1145   bounded_linear_mult_right [THEN bounded_linear_compose]
```
```  1146
```
```  1147 lemma bounded_linear_divide:
```
```  1148   "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
```
```  1149   unfolding divide_inverse by (rule bounded_linear_mult_left)
```
```  1150
```
```  1151 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
```
```  1152 apply (rule bounded_bilinear.intro)
```
```  1153 apply (rule scaleR_left_distrib)
```
```  1154 apply (rule scaleR_right_distrib)
```
```  1155 apply simp
```
```  1156 apply (rule scaleR_left_commute)
```
```  1157 apply (rule_tac x="1" in exI, simp)
```
```  1158 done
```
```  1159
```
```  1160 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
```
```  1161   using bounded_bilinear_scaleR
```
```  1162   by (rule bounded_bilinear.bounded_linear_left)
```
```  1163
```
```  1164 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
```
```  1165   using bounded_bilinear_scaleR
```
```  1166   by (rule bounded_bilinear.bounded_linear_right)
```
```  1167
```
```  1168 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
```
```  1169   unfolding of_real_def by (rule bounded_linear_scaleR_left)
```
```  1170
```
```  1171 lemma real_bounded_linear:
```
```  1172   fixes f :: "real \<Rightarrow> real"
```
```  1173   shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
```
```  1174 proof -
```
```  1175   { fix x assume "bounded_linear f"
```
```  1176     then interpret bounded_linear f .
```
```  1177     from scaleR[of x 1] have "f x = x * f 1"
```
```  1178       by simp }
```
```  1179   then show ?thesis
```
```  1180     by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
```
```  1181 qed
```
```  1182
```
```  1183 instance real_normed_algebra_1 \<subseteq> perfect_space
```
```  1184 proof
```
```  1185   fix x::'a
```
```  1186   show "\<not> open {x}"
```
```  1187     unfolding open_dist dist_norm
```
```  1188     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
```
```  1189 qed
```
```  1190
```
```  1191 subsection {* Filters and Limits on Metric Space *}
```
```  1192
```
```  1193 lemma eventually_nhds_metric:
```
```  1194   fixes a :: "'a :: metric_space"
```
```  1195   shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```  1196 unfolding eventually_nhds open_dist
```
```  1197 apply safe
```
```  1198 apply fast
```
```  1199 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```  1200 apply clarsimp
```
```  1201 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```  1202 apply (simp only: less_diff_eq)
```
```  1203 apply (erule le_less_trans [OF dist_triangle])
```
```  1204 done
```
```  1205
```
```  1206 lemma eventually_at:
```
```  1207   fixes a :: "'a :: metric_space"
```
```  1208   shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```  1209   unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)
```
```  1210
```
```  1211 lemma eventually_at_le:
```
```  1212   fixes a :: "'a::metric_space"
```
```  1213   shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<and> dist x a \<le> d \<longrightarrow> P x)"
```
```  1214   unfolding eventually_at_filter eventually_nhds_metric
```
```  1215   apply auto
```
```  1216   apply (rule_tac x="d / 2" in exI)
```
```  1217   apply auto
```
```  1218   done
```
```  1219
```
```  1220 lemma tendstoI:
```
```  1221   fixes l :: "'a :: metric_space"
```
```  1222   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```  1223   shows "(f ---> l) F"
```
```  1224   apply (rule topological_tendstoI)
```
```  1225   apply (simp add: open_dist)
```
```  1226   apply (drule (1) bspec, clarify)
```
```  1227   apply (drule assms)
```
```  1228   apply (erule eventually_elim1, simp)
```
```  1229   done
```
```  1230
```
```  1231 lemma tendstoD:
```
```  1232   fixes l :: "'a :: metric_space"
```
```  1233   shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```  1234   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```  1235   apply (clarsimp simp add: open_dist)
```
```  1236   apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```  1237   apply (simp only: less_diff_eq)
```
```  1238   apply (erule le_less_trans [OF dist_triangle])
```
```  1239   apply simp
```
```  1240   apply simp
```
```  1241   done
```
```  1242
```
```  1243 lemma tendsto_iff:
```
```  1244   fixes l :: "'a :: metric_space"
```
```  1245   shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
```
```  1246   using tendstoI tendstoD by fast
```
```  1247
```
```  1248 lemma metric_tendsto_imp_tendsto:
```
```  1249   fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
```
```  1250   assumes f: "(f ---> a) F"
```
```  1251   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
```
```  1252   shows "(g ---> b) F"
```
```  1253 proof (rule tendstoI)
```
```  1254   fix e :: real assume "0 < e"
```
```  1255   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
```
```  1256   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
```
```  1257     using le_less_trans by (rule eventually_elim2)
```
```  1258 qed
```
```  1259
```
```  1260 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
```
```  1261   unfolding filterlim_at_top
```
```  1262   apply (intro allI)
```
```  1263   apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
```
```  1264   apply (auto simp: natceiling_le_eq)
```
```  1265   done
```
```  1266
```
```  1267 subsubsection {* Limits of Sequences *}
```
```  1268
```
```  1269 lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```  1270   unfolding tendsto_iff eventually_sequentially ..
```
```  1271
```
```  1272 lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
```
```  1273   unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
```
```  1274
```
```  1275 lemma metric_LIMSEQ_I:
```
```  1276   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
```
```  1277 by (simp add: LIMSEQ_def)
```
```  1278
```
```  1279 lemma metric_LIMSEQ_D:
```
```  1280   "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```  1281 by (simp add: LIMSEQ_def)
```
```  1282
```
```  1283
```
```  1284 subsubsection {* Limits of Functions *}
```
```  1285
```
```  1286 lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
```
```  1287      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
```
```  1288         --> dist (f x) L < r)"
```
```  1289   unfolding tendsto_iff eventually_at by simp
```
```  1290
```
```  1291 lemma metric_LIM_I:
```
```  1292   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
```
```  1293     \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
```
```  1294 by (simp add: LIM_def)
```
```  1295
```
```  1296 lemma metric_LIM_D:
```
```  1297   "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
```
```  1298     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
```
```  1299 by (simp add: LIM_def)
```
```  1300
```
```  1301 lemma metric_LIM_imp_LIM:
```
```  1302   assumes f: "f -- a --> (l::'a::metric_space)"
```
```  1303   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
```
```  1304   shows "g -- a --> (m::'b::metric_space)"
```
```  1305   by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
```
```  1306
```
```  1307 lemma metric_LIM_equal2:
```
```  1308   assumes 1: "0 < R"
```
```  1309   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1310   shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
```
```  1311 apply (rule topological_tendstoI)
```
```  1312 apply (drule (2) topological_tendstoD)
```
```  1313 apply (simp add: eventually_at, safe)
```
```  1314 apply (rule_tac x="min d R" in exI, safe)
```
```  1315 apply (simp add: 1)
```
```  1316 apply (simp add: 2)
```
```  1317 done
```
```  1318
```
```  1319 lemma metric_LIM_compose2:
```
```  1320   assumes f: "f -- (a::'a::metric_space) --> b"
```
```  1321   assumes g: "g -- b --> c"
```
```  1322   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```  1323   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  1324   using inj
```
```  1325   by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
```
```  1326
```
```  1327 lemma metric_isCont_LIM_compose2:
```
```  1328   fixes f :: "'a :: metric_space \<Rightarrow> _"
```
```  1329   assumes f [unfolded isCont_def]: "isCont f a"
```
```  1330   assumes g: "g -- f a --> l"
```
```  1331   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```  1332   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```  1333 by (rule metric_LIM_compose2 [OF f g inj])
```
```  1334
```
```  1335 subsection {* Complete metric spaces *}
```
```  1336
```
```  1337 subsection {* Cauchy sequences *}
```
```  1338
```
```  1339 definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```  1340   "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```  1341
```
```  1342 subsection {* Cauchy Sequences *}
```
```  1343
```
```  1344 lemma metric_CauchyI:
```
```  1345   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```  1346   by (simp add: Cauchy_def)
```
```  1347
```
```  1348 lemma metric_CauchyD:
```
```  1349   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```  1350   by (simp add: Cauchy_def)
```
```  1351
```
```  1352 lemma metric_Cauchy_iff2:
```
```  1353   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
```
```  1354 apply (simp add: Cauchy_def, auto)
```
```  1355 apply (drule reals_Archimedean, safe)
```
```  1356 apply (drule_tac x = n in spec, auto)
```
```  1357 apply (rule_tac x = M in exI, auto)
```
```  1358 apply (drule_tac x = m in spec, simp)
```
```  1359 apply (drule_tac x = na in spec, auto)
```
```  1360 done
```
```  1361
```
```  1362 lemma Cauchy_iff2:
```
```  1363   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
```
```  1364   unfolding metric_Cauchy_iff2 dist_real_def ..
```
```  1365
```
```  1366 lemma Cauchy_subseq_Cauchy:
```
```  1367   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```  1368 apply (auto simp add: Cauchy_def)
```
```  1369 apply (drule_tac x=e in spec, clarify)
```
```  1370 apply (rule_tac x=M in exI, clarify)
```
```  1371 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```  1372 done
```
```  1373
```
```  1374 theorem LIMSEQ_imp_Cauchy:
```
```  1375   assumes X: "X ----> a" shows "Cauchy X"
```
```  1376 proof (rule metric_CauchyI)
```
```  1377   fix e::real assume "0 < e"
```
```  1378   hence "0 < e/2" by simp
```
```  1379   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```  1380   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```  1381   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```  1382   proof (intro exI allI impI)
```
```  1383     fix m assume "N \<le> m"
```
```  1384     hence m: "dist (X m) a < e/2" using N by fast
```
```  1385     fix n assume "N \<le> n"
```
```  1386     hence n: "dist (X n) a < e/2" using N by fast
```
```  1387     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```  1388       by (rule dist_triangle2)
```
```  1389     also from m n have "\<dots> < e" by simp
```
```  1390     finally show "dist (X m) (X n) < e" .
```
```  1391   qed
```
```  1392 qed
```
```  1393
```
```  1394 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1395 unfolding convergent_def
```
```  1396 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1397
```
```  1398 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1399
```
```  1400 class complete_space = metric_space +
```
```  1401   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1402
```
```  1403 lemma Cauchy_convergent_iff:
```
```  1404   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1405   shows "Cauchy X = convergent X"
```
```  1406 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1407
```
```  1408 subsection {* The set of real numbers is a complete metric space *}
```
```  1409
```
```  1410 text {*
```
```  1411 Proof that Cauchy sequences converge based on the one from
```
```  1412 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```  1413 *}
```
```  1414
```
```  1415 text {*
```
```  1416   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1417   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1418 *}
```
```  1419
```
```  1420 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```  1421 by (simp add: isUbI setleI)
```
```  1422
```
```  1423 lemma increasing_LIMSEQ:
```
```  1424   fixes f :: "nat \<Rightarrow> real"
```
```  1425   assumes inc: "\<And>n. f n \<le> f (Suc n)"
```
```  1426       and bdd: "\<And>n. f n \<le> l"
```
```  1427       and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
```
```  1428   shows "f ----> l"
```
```  1429 proof (rule increasing_tendsto)
```
```  1430   fix x assume "x < l"
```
```  1431   with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
```
```  1432     by auto
```
```  1433   from en[OF `0 < e`] obtain n where "l - e \<le> f n"
```
```  1434     by (auto simp: field_simps)
```
```  1435   with `e < l - x` `0 < e` have "x < f n" by simp
```
```  1436   with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
```
```  1437     by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
```
```  1438 qed (insert bdd, auto)
```
```  1439
```
```  1440 lemma real_Cauchy_convergent:
```
```  1441   fixes X :: "nat \<Rightarrow> real"
```
```  1442   assumes X: "Cauchy X"
```
```  1443   shows "convergent X"
```
```  1444 proof -
```
```  1445   def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1446   then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
```
```  1447
```
```  1448   { fix N x assume N: "\<forall>n\<ge>N. X n < x"
```
```  1449   have "isUb UNIV S x"
```
```  1450   proof (rule isUb_UNIV_I)
```
```  1451   fix y::real assume "y \<in> S"
```
```  1452   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1453     by (simp add: S_def)
```
```  1454   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1455   hence "y < X (max M N)" by simp
```
```  1456   also have "\<dots> < x" using N by simp
```
```  1457   finally show "y \<le> x"
```
```  1458     by (rule order_less_imp_le)
```
```  1459   qed }
```
```  1460   note bound_isUb = this
```
```  1461
```
```  1462   have "\<exists>u. isLub UNIV S u"
```
```  1463   proof (rule reals_complete)
```
```  1464   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
```
```  1465     using X[THEN metric_CauchyD, OF zero_less_one] by auto
```
```  1466   hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
```
```  1467   show "\<exists>x. x \<in> S"
```
```  1468   proof
```
```  1469     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1470       by (simp add: abs_diff_less_iff dist_real_def)
```
```  1471     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1472   qed
```
```  1473   show "\<exists>u. isUb UNIV S u"
```
```  1474   proof
```
```  1475     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1476       by (simp add: abs_diff_less_iff dist_real_def)
```
```  1477     thus "isUb UNIV S (X N + 1)"
```
```  1478       by (rule bound_isUb)
```
```  1479   qed
```
```  1480   qed
```
```  1481   then obtain x where x: "isLub UNIV S x" ..
```
```  1482   have "X ----> x"
```
```  1483   proof (rule metric_LIMSEQ_I)
```
```  1484   fix r::real assume "0 < r"
```
```  1485   hence r: "0 < r/2" by simp
```
```  1486   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
```
```  1487     using metric_CauchyD [OF X r] by auto
```
```  1488   hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
```
```  1489   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1490     by (simp only: dist_real_def abs_diff_less_iff)
```
```  1491
```
```  1492   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1493   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1494   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```  1495
```
```  1496   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1497   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```  1498   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```  1499
```
```  1500   show "\<exists>N. \<forall>n\<ge>N. dist (X n) x < r"
```
```  1501   proof (intro exI allI impI)
```
```  1502     fix n assume n: "N \<le> n"
```
```  1503     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1504     thus "dist (X n) x < r" using 1 2
```
```  1505       by (simp add: abs_diff_less_iff dist_real_def)
```
```  1506   qed
```
```  1507   qed
```
```  1508   then show ?thesis unfolding convergent_def by auto
```
```  1509 qed
```
```  1510
```
```  1511 instance real :: complete_space
```
```  1512   by intro_classes (rule real_Cauchy_convergent)
```
```  1513
```
```  1514 class banach = real_normed_vector + complete_space
```
```  1515
```
```  1516 instance real :: banach by default
```
```  1517
```
```  1518 lemma tendsto_at_topI_sequentially:
```
```  1519   fixes f :: "real \<Rightarrow> real"
```
```  1520   assumes mono: "mono f"
```
```  1521   assumes limseq: "(\<lambda>n. f (real n)) ----> y"
```
```  1522   shows "(f ---> y) at_top"
```
```  1523 proof (rule tendstoI)
```
```  1524   fix e :: real assume "0 < e"
```
```  1525   with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
```
```  1526     by (auto simp: LIMSEQ_def dist_real_def)
```
```  1527   { fix x :: real
```
```  1528     from ex_le_of_nat[of x] guess n ..
```
```  1529     note monoD[OF mono this]
```
```  1530     also have "f (real_of_nat n) \<le> y"
```
```  1531       by (rule LIMSEQ_le_const[OF limseq])
```
```  1532          (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
```
```  1533     finally have "f x \<le> y" . }
```
```  1534   note le = this
```
```  1535   have "eventually (\<lambda>x. real N \<le> x) at_top"
```
```  1536     by (rule eventually_ge_at_top)
```
```  1537   then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
```
```  1538   proof eventually_elim
```
```  1539     fix x assume N': "real N \<le> x"
```
```  1540     with N[of N] le have "y - f (real N) < e" by auto
```
```  1541     moreover note monoD[OF mono N']
```
```  1542     ultimately show "dist (f x) y < e"
```
```  1543       using le[of x] by (auto simp: dist_real_def field_simps)
```
```  1544   qed
```
```  1545 qed
```
```  1546
```
```  1547 end
```
```  1548
```