src/HOL/Real_Vector_Spaces.thy
 author paulson Mon Feb 22 14:37:56 2016 +0000 (2016-02-22) changeset 62379 340738057c8c parent 62368 106569399cd6 child 62397 5ae24f33d343 permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
```     1 (*  Title:      HOL/Real_Vector_Spaces.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 section \<open>Vector Spaces and Algebras over the Reals\<close>
```
```     7
```
```     8 theory Real_Vector_Spaces
```
```     9 imports Real Topological_Spaces
```
```    10 begin
```
```    11
```
```    12 subsection \<open>Locale for additive functions\<close>
```
```    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   using add [of x "- y"] by (simp add: 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 \<open>Vector spaces\<close>
```
```    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 \<open>Real vector spaces\<close>
```
```   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 \<open>Recover original theorem names\<close>
```
```   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 \<open>Legacy names\<close>
```
```   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}"
```
```   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 lemma setsum_constant_scaleR:
```
```   232   fixes y :: "'a::real_vector"
```
```   233   shows "(\<Sum>x\<in>A. y) = of_nat (card A) *\<^sub>R y"
```
```   234   apply (cases "finite A")
```
```   235   apply (induct set: finite)
```
```   236   apply (simp_all add: algebra_simps)
```
```   237   done
```
```   238
```
```   239 lemma real_vector_affinity_eq:
```
```   240   fixes x :: "'a :: real_vector"
```
```   241   assumes m0: "m \<noteq> 0"
```
```   242   shows "m *\<^sub>R x + c = y \<longleftrightarrow> x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
```
```   243 proof
```
```   244   assume h: "m *\<^sub>R x + c = y"
```
```   245   hence "m *\<^sub>R x = y - c" by (simp add: field_simps)
```
```   246   hence "inverse m *\<^sub>R (m *\<^sub>R x) = inverse m *\<^sub>R (y - c)" by simp
```
```   247   then show "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
```
```   248     using m0
```
```   249   by (simp add: real_vector.scale_right_diff_distrib)
```
```   250 next
```
```   251   assume h: "x = inverse m *\<^sub>R y - (inverse m *\<^sub>R c)"
```
```   252   show "m *\<^sub>R x + c = y" unfolding h
```
```   253     using m0  by (simp add: real_vector.scale_right_diff_distrib)
```
```   254 qed
```
```   255
```
```   256 lemma real_vector_eq_affinity:
```
```   257   fixes x :: "'a :: real_vector"
```
```   258   shows "m \<noteq> 0 ==> (y = m *\<^sub>R x + c \<longleftrightarrow> inverse m *\<^sub>R y - (inverse m *\<^sub>R c) = x)"
```
```   259   using real_vector_affinity_eq[where m=m and x=x and y=y and c=c]
```
```   260   by metis
```
```   261
```
```   262
```
```   263 subsection \<open>Embedding of the Reals into any \<open>real_algebra_1\<close>:
```
```   264 @{term of_real}\<close>
```
```   265
```
```   266 definition
```
```   267   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   268   "of_real r = scaleR r 1"
```
```   269
```
```   270 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   271 by (simp add: of_real_def)
```
```   272
```
```   273 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   274 by (simp add: of_real_def)
```
```   275
```
```   276 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   277 by (simp add: of_real_def)
```
```   278
```
```   279 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   280 by (simp add: of_real_def scaleR_left_distrib)
```
```   281
```
```   282 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   283 by (simp add: of_real_def)
```
```   284
```
```   285 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   286 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   287
```
```   288 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   289 by (simp add: of_real_def mult.commute)
```
```   290
```
```   291 lemma of_real_setsum[simp]: "of_real (setsum f s) = (\<Sum>x\<in>s. of_real (f x))"
```
```   292   by (induct s rule: infinite_finite_induct) auto
```
```   293
```
```   294 lemma of_real_setprod[simp]: "of_real (setprod f s) = (\<Prod>x\<in>s. of_real (f x))"
```
```   295   by (induct s rule: infinite_finite_induct) auto
```
```   296
```
```   297 lemma nonzero_of_real_inverse:
```
```   298   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   299    inverse (of_real x :: 'a::real_div_algebra)"
```
```   300 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   301
```
```   302 lemma of_real_inverse [simp]:
```
```   303   "of_real (inverse x) =
```
```   304    inverse (of_real x :: 'a::{real_div_algebra, division_ring})"
```
```   305 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   306
```
```   307 lemma nonzero_of_real_divide:
```
```   308   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   309    (of_real x / of_real y :: 'a::real_field)"
```
```   310 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   311
```
```   312 lemma of_real_divide [simp]:
```
```   313   "of_real (x / y) = (of_real x / of_real y :: 'a::real_div_algebra)"
```
```   314 by (simp add: divide_inverse)
```
```   315
```
```   316 lemma of_real_power [simp]:
```
```   317   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
```
```   318 by (induct n) simp_all
```
```   319
```
```   320 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   321 by (simp add: of_real_def)
```
```   322
```
```   323 lemma inj_of_real:
```
```   324   "inj of_real"
```
```   325   by (auto intro: injI)
```
```   326
```
```   327 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   328
```
```   329 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   330 proof
```
```   331   fix r
```
```   332   show "of_real r = id r"
```
```   333     by (simp add: of_real_def)
```
```   334 qed
```
```   335
```
```   336 text\<open>Collapse nested embeddings\<close>
```
```   337 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   338 by (induct n) auto
```
```   339
```
```   340 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   341 by (cases z rule: int_diff_cases, simp)
```
```   342
```
```   343 lemma of_real_numeral [simp]: "of_real (numeral w) = numeral w"
```
```   344 using of_real_of_int_eq [of "numeral w"] by simp
```
```   345
```
```   346 lemma of_real_neg_numeral [simp]: "of_real (- numeral w) = - numeral w"
```
```   347 using of_real_of_int_eq [of "- numeral w"] by simp
```
```   348
```
```   349 text\<open>Every real algebra has characteristic zero\<close>
```
```   350
```
```   351 instance real_algebra_1 < ring_char_0
```
```   352 proof
```
```   353   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
```
```   354   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
```
```   355 qed
```
```   356
```
```   357 instance real_field < field_char_0 ..
```
```   358
```
```   359
```
```   360 subsection \<open>The Set of Real Numbers\<close>
```
```   361
```
```   362 definition Reals :: "'a::real_algebra_1 set"  ("\<real>")
```
```   363   where "\<real> = range of_real"
```
```   364
```
```   365 lemma Reals_of_real [simp]: "of_real r \<in> \<real>"
```
```   366 by (simp add: Reals_def)
```
```   367
```
```   368 lemma Reals_of_int [simp]: "of_int z \<in> \<real>"
```
```   369 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   370
```
```   371 lemma Reals_of_nat [simp]: "of_nat n \<in> \<real>"
```
```   372 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   373
```
```   374 lemma Reals_numeral [simp]: "numeral w \<in> \<real>"
```
```   375 by (subst of_real_numeral [symmetric], rule Reals_of_real)
```
```   376
```
```   377 lemma Reals_0 [simp]: "0 \<in> \<real>"
```
```   378 apply (unfold Reals_def)
```
```   379 apply (rule range_eqI)
```
```   380 apply (rule of_real_0 [symmetric])
```
```   381 done
```
```   382
```
```   383 lemma Reals_1 [simp]: "1 \<in> \<real>"
```
```   384 apply (unfold Reals_def)
```
```   385 apply (rule range_eqI)
```
```   386 apply (rule of_real_1 [symmetric])
```
```   387 done
```
```   388
```
```   389 lemma Reals_add [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a + b \<in> \<real>"
```
```   390 apply (auto simp add: Reals_def)
```
```   391 apply (rule range_eqI)
```
```   392 apply (rule of_real_add [symmetric])
```
```   393 done
```
```   394
```
```   395 lemma Reals_minus [simp]: "a \<in> \<real> \<Longrightarrow> - a \<in> \<real>"
```
```   396 apply (auto simp add: Reals_def)
```
```   397 apply (rule range_eqI)
```
```   398 apply (rule of_real_minus [symmetric])
```
```   399 done
```
```   400
```
```   401 lemma Reals_diff [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a - b \<in> \<real>"
```
```   402 apply (auto simp add: Reals_def)
```
```   403 apply (rule range_eqI)
```
```   404 apply (rule of_real_diff [symmetric])
```
```   405 done
```
```   406
```
```   407 lemma Reals_mult [simp]: "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a * b \<in> \<real>"
```
```   408 apply (auto simp add: Reals_def)
```
```   409 apply (rule range_eqI)
```
```   410 apply (rule of_real_mult [symmetric])
```
```   411 done
```
```   412
```
```   413 lemma nonzero_Reals_inverse:
```
```   414   fixes a :: "'a::real_div_algebra"
```
```   415   shows "\<lbrakk>a \<in> \<real>; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> \<real>"
```
```   416 apply (auto simp add: Reals_def)
```
```   417 apply (rule range_eqI)
```
```   418 apply (erule nonzero_of_real_inverse [symmetric])
```
```   419 done
```
```   420
```
```   421 lemma Reals_inverse:
```
```   422   fixes a :: "'a::{real_div_algebra, division_ring}"
```
```   423   shows "a \<in> \<real> \<Longrightarrow> inverse a \<in> \<real>"
```
```   424 apply (auto simp add: Reals_def)
```
```   425 apply (rule range_eqI)
```
```   426 apply (rule of_real_inverse [symmetric])
```
```   427 done
```
```   428
```
```   429 lemma Reals_inverse_iff [simp]:
```
```   430   fixes x:: "'a :: {real_div_algebra, division_ring}"
```
```   431   shows "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
```
```   432 by (metis Reals_inverse inverse_inverse_eq)
```
```   433
```
```   434 lemma nonzero_Reals_divide:
```
```   435   fixes a b :: "'a::real_field"
```
```   436   shows "\<lbrakk>a \<in> \<real>; b \<in> \<real>; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> \<real>"
```
```   437 apply (auto simp add: Reals_def)
```
```   438 apply (rule range_eqI)
```
```   439 apply (erule nonzero_of_real_divide [symmetric])
```
```   440 done
```
```   441
```
```   442 lemma Reals_divide [simp]:
```
```   443   fixes a b :: "'a::{real_field, field}"
```
```   444   shows "\<lbrakk>a \<in> \<real>; b \<in> \<real>\<rbrakk> \<Longrightarrow> a / b \<in> \<real>"
```
```   445 apply (auto simp add: Reals_def)
```
```   446 apply (rule range_eqI)
```
```   447 apply (rule of_real_divide [symmetric])
```
```   448 done
```
```   449
```
```   450 lemma Reals_power [simp]:
```
```   451   fixes a :: "'a::{real_algebra_1}"
```
```   452   shows "a \<in> \<real> \<Longrightarrow> a ^ n \<in> \<real>"
```
```   453 apply (auto simp add: Reals_def)
```
```   454 apply (rule range_eqI)
```
```   455 apply (rule of_real_power [symmetric])
```
```   456 done
```
```   457
```
```   458 lemma Reals_cases [cases set: Reals]:
```
```   459   assumes "q \<in> \<real>"
```
```   460   obtains (of_real) r where "q = of_real r"
```
```   461   unfolding Reals_def
```
```   462 proof -
```
```   463   from \<open>q \<in> \<real>\<close> have "q \<in> range of_real" unfolding Reals_def .
```
```   464   then obtain r where "q = of_real r" ..
```
```   465   then show thesis ..
```
```   466 qed
```
```   467
```
```   468 lemma setsum_in_Reals [intro,simp]:
```
```   469   assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setsum f s \<in> \<real>"
```
```   470 proof (cases "finite s")
```
```   471   case True then show ?thesis using assms
```
```   472     by (induct s rule: finite_induct) auto
```
```   473 next
```
```   474   case False then show ?thesis using assms
```
```   475     by (metis Reals_0 setsum.infinite)
```
```   476 qed
```
```   477
```
```   478 lemma setprod_in_Reals [intro,simp]:
```
```   479   assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setprod f s \<in> \<real>"
```
```   480 proof (cases "finite s")
```
```   481   case True then show ?thesis using assms
```
```   482     by (induct s rule: finite_induct) auto
```
```   483 next
```
```   484   case False then show ?thesis using assms
```
```   485     by (metis Reals_1 setprod.infinite)
```
```   486 qed
```
```   487
```
```   488 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   489   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   490   by (rule Reals_cases) auto
```
```   491
```
```   492 subsection \<open>Ordered real vector spaces\<close>
```
```   493
```
```   494 class ordered_real_vector = real_vector + ordered_ab_group_add +
```
```   495   assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y"
```
```   496   assumes scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"
```
```   497 begin
```
```   498
```
```   499 lemma scaleR_mono:
```
```   500   "a \<le> b \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R y"
```
```   501 apply (erule scaleR_right_mono [THEN order_trans], assumption)
```
```   502 apply (erule scaleR_left_mono, assumption)
```
```   503 done
```
```   504
```
```   505 lemma scaleR_mono':
```
```   506   "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R d"
```
```   507   by (rule scaleR_mono) (auto intro: order.trans)
```
```   508
```
```   509 lemma pos_le_divideRI:
```
```   510   assumes "0 < c"
```
```   511   assumes "c *\<^sub>R a \<le> b"
```
```   512   shows "a \<le> b /\<^sub>R c"
```
```   513 proof -
```
```   514   from scaleR_left_mono[OF assms(2)] assms(1)
```
```   515   have "c *\<^sub>R a /\<^sub>R c \<le> b /\<^sub>R c"
```
```   516     by simp
```
```   517   with assms show ?thesis
```
```   518     by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
```
```   519 qed
```
```   520
```
```   521 lemma pos_le_divideR_eq:
```
```   522   assumes "0 < c"
```
```   523   shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b"
```
```   524 proof rule
```
```   525   assume "a \<le> b /\<^sub>R c"
```
```   526   from scaleR_left_mono[OF this] assms
```
```   527   have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"
```
```   528     by simp
```
```   529   with assms show "c *\<^sub>R a \<le> b"
```
```   530     by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
```
```   531 qed (rule pos_le_divideRI[OF assms])
```
```   532
```
```   533 lemma scaleR_image_atLeastAtMost:
```
```   534   "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"
```
```   535   apply (auto intro!: scaleR_left_mono)
```
```   536   apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI)
```
```   537   apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)
```
```   538   done
```
```   539
```
```   540 end
```
```   541
```
```   542 lemma neg_le_divideR_eq:
```
```   543   fixes a :: "'a :: ordered_real_vector"
```
```   544   assumes "c < 0"
```
```   545   shows "a \<le> b /\<^sub>R c \<longleftrightarrow> b \<le> c *\<^sub>R a"
```
```   546   using pos_le_divideR_eq [of "-c" a "-b"] assms
```
```   547   by simp
```
```   548
```
```   549 lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> 0 \<le> a *\<^sub>R x"
```
```   550   using scaleR_left_mono [of 0 x a]
```
```   551   by simp
```
```   552
```
```   553 lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> (x::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"
```
```   554   using scaleR_left_mono [of x 0 a] by simp
```
```   555
```
```   556 lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> a *\<^sub>R x \<le> 0"
```
```   557   using scaleR_right_mono [of a 0 x] by simp
```
```   558
```
```   559 lemma split_scaleR_neg_le: "(0 \<le> a & x \<le> 0) | (a \<le> 0 & 0 \<le> x) \<Longrightarrow>
```
```   560   a *\<^sub>R (x::'a::ordered_real_vector) \<le> 0"
```
```   561   by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)
```
```   562
```
```   563 lemma le_add_iff1:
```
```   564   fixes c d e::"'a::ordered_real_vector"
```
```   565   shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"
```
```   566   by (simp add: algebra_simps)
```
```   567
```
```   568 lemma le_add_iff2:
```
```   569   fixes c d e::"'a::ordered_real_vector"
```
```   570   shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"
```
```   571   by (simp add: algebra_simps)
```
```   572
```
```   573 lemma scaleR_left_mono_neg:
```
```   574   fixes a b::"'a::ordered_real_vector"
```
```   575   shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"
```
```   576   apply (drule scaleR_left_mono [of _ _ "- c"])
```
```   577   apply simp_all
```
```   578   done
```
```   579
```
```   580 lemma scaleR_right_mono_neg:
```
```   581   fixes c::"'a::ordered_real_vector"
```
```   582   shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"
```
```   583   apply (drule scaleR_right_mono [of _ _ "- c"])
```
```   584   apply simp_all
```
```   585   done
```
```   586
```
```   587 lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> (b::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"
```
```   588 using scaleR_right_mono_neg [of a 0 b] by simp
```
```   589
```
```   590 lemma split_scaleR_pos_le:
```
```   591   fixes b::"'a::ordered_real_vector"
```
```   592   shows "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"
```
```   593   by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)
```
```   594
```
```   595 lemma zero_le_scaleR_iff:
```
```   596   fixes b::"'a::ordered_real_vector"
```
```   597   shows "0 \<le> a *\<^sub>R b \<longleftrightarrow> 0 < a \<and> 0 \<le> b \<or> a < 0 \<and> b \<le> 0 \<or> a = 0" (is "?lhs = ?rhs")
```
```   598 proof cases
```
```   599   assume "a \<noteq> 0"
```
```   600   show ?thesis
```
```   601   proof
```
```   602     assume lhs: ?lhs
```
```   603     {
```
```   604       assume "0 < a"
```
```   605       with lhs have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"
```
```   606         by (intro scaleR_mono) auto
```
```   607       hence ?rhs using \<open>0 < a\<close>
```
```   608         by simp
```
```   609     } moreover {
```
```   610       assume "0 > a"
```
```   611       with lhs have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"
```
```   612         by (intro scaleR_mono) auto
```
```   613       hence ?rhs using \<open>0 > a\<close>
```
```   614         by simp
```
```   615     } ultimately show ?rhs using \<open>a \<noteq> 0\<close> by arith
```
```   616   qed (auto simp: not_le \<open>a \<noteq> 0\<close> intro!: split_scaleR_pos_le)
```
```   617 qed simp
```
```   618
```
```   619 lemma scaleR_le_0_iff:
```
```   620   fixes b::"'a::ordered_real_vector"
```
```   621   shows "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
```
```   622   by (insert zero_le_scaleR_iff [of "-a" b]) force
```
```   623
```
```   624 lemma scaleR_le_cancel_left:
```
```   625   fixes b::"'a::ordered_real_vector"
```
```   626   shows "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```   627   by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg
```
```   628     dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])
```
```   629
```
```   630 lemma scaleR_le_cancel_left_pos:
```
```   631   fixes b::"'a::ordered_real_vector"
```
```   632   shows "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"
```
```   633   by (auto simp: scaleR_le_cancel_left)
```
```   634
```
```   635 lemma scaleR_le_cancel_left_neg:
```
```   636   fixes b::"'a::ordered_real_vector"
```
```   637   shows "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"
```
```   638   by (auto simp: scaleR_le_cancel_left)
```
```   639
```
```   640 lemma scaleR_left_le_one_le:
```
```   641   fixes x::"'a::ordered_real_vector" and a::real
```
```   642   shows "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"
```
```   643   using scaleR_right_mono[of a 1 x] by simp
```
```   644
```
```   645
```
```   646 subsection \<open>Real normed vector spaces\<close>
```
```   647
```
```   648 class dist =
```
```   649   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```   650
```
```   651 class norm =
```
```   652   fixes norm :: "'a \<Rightarrow> real"
```
```   653
```
```   654 class sgn_div_norm = scaleR + norm + sgn +
```
```   655   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   656
```
```   657 class dist_norm = dist + norm + minus +
```
```   658   assumes dist_norm: "dist x y = norm (x - y)"
```
```   659
```
```   660 class uniformity_dist = dist + uniformity +
```
```   661   assumes uniformity_dist: "uniformity = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
```
```   662 begin
```
```   663
```
```   664 lemma eventually_uniformity_metric:
```
```   665   "eventually P uniformity \<longleftrightarrow> (\<exists>e>0. \<forall>x y. dist x y < e \<longrightarrow> P (x, y))"
```
```   666   unfolding uniformity_dist
```
```   667   by (subst eventually_INF_base)
```
```   668      (auto simp: eventually_principal subset_eq intro: bexI[of _ "min _ _"])
```
```   669
```
```   670 end
```
```   671
```
```   672 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + uniformity_dist + open_uniformity +
```
```   673   assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   674   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   675   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   676 begin
```
```   677
```
```   678 lemma norm_ge_zero [simp]: "0 \<le> norm x"
```
```   679 proof -
```
```   680   have "0 = norm (x + -1 *\<^sub>R x)"
```
```   681     using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
```
```   682   also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
```
```   683   finally show ?thesis by simp
```
```   684 qed
```
```   685
```
```   686 end
```
```   687
```
```   688 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   689   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   690
```
```   691 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   692   assumes norm_one [simp]: "norm 1 = 1"
```
```   693
```
```   694 lemma (in real_normed_algebra_1) scaleR_power [simp]:
```
```   695   "(scaleR x y) ^ n = scaleR (x^n) (y^n)"
```
```   696   by (induction n) (simp_all add: scaleR_one scaleR_scaleR mult_ac)
```
```   697
```
```   698 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   699   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   700
```
```   701 class real_normed_field = real_field + real_normed_div_algebra
```
```   702
```
```   703 instance real_normed_div_algebra < real_normed_algebra_1
```
```   704 proof
```
```   705   fix x y :: 'a
```
```   706   show "norm (x * y) \<le> norm x * norm y"
```
```   707     by (simp add: norm_mult)
```
```   708 next
```
```   709   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   710     by (rule norm_mult)
```
```   711   thus "norm (1::'a) = 1" by simp
```
```   712 qed
```
```   713
```
```   714 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   715 by simp
```
```   716
```
```   717 lemma zero_less_norm_iff [simp]:
```
```   718   fixes x :: "'a::real_normed_vector"
```
```   719   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   720 by (simp add: order_less_le)
```
```   721
```
```   722 lemma norm_not_less_zero [simp]:
```
```   723   fixes x :: "'a::real_normed_vector"
```
```   724   shows "\<not> norm x < 0"
```
```   725 by (simp add: linorder_not_less)
```
```   726
```
```   727 lemma norm_le_zero_iff [simp]:
```
```   728   fixes x :: "'a::real_normed_vector"
```
```   729   shows "(norm x \<le> 0) = (x = 0)"
```
```   730 by (simp add: order_le_less)
```
```   731
```
```   732 lemma norm_minus_cancel [simp]:
```
```   733   fixes x :: "'a::real_normed_vector"
```
```   734   shows "norm (- x) = norm x"
```
```   735 proof -
```
```   736   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   737     by (simp only: scaleR_minus_left scaleR_one)
```
```   738   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   739     by (rule norm_scaleR)
```
```   740   finally show ?thesis by simp
```
```   741 qed
```
```   742
```
```   743 lemma norm_minus_commute:
```
```   744   fixes a b :: "'a::real_normed_vector"
```
```   745   shows "norm (a - b) = norm (b - a)"
```
```   746 proof -
```
```   747   have "norm (- (b - a)) = norm (b - a)"
```
```   748     by (rule norm_minus_cancel)
```
```   749   thus ?thesis by simp
```
```   750 qed
```
```   751
```
```   752 lemma norm_uminus_minus: "norm (-x - y :: 'a :: real_normed_vector) = norm (x + y)"
```
```   753   by (subst (2) norm_minus_cancel[symmetric], subst minus_add_distrib) simp
```
```   754
```
```   755 lemma norm_triangle_ineq2:
```
```   756   fixes a b :: "'a::real_normed_vector"
```
```   757   shows "norm a - norm b \<le> norm (a - b)"
```
```   758 proof -
```
```   759   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   760     by (rule norm_triangle_ineq)
```
```   761   thus ?thesis by simp
```
```   762 qed
```
```   763
```
```   764 lemma norm_triangle_ineq3:
```
```   765   fixes a b :: "'a::real_normed_vector"
```
```   766   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   767 apply (subst abs_le_iff)
```
```   768 apply auto
```
```   769 apply (rule norm_triangle_ineq2)
```
```   770 apply (subst norm_minus_commute)
```
```   771 apply (rule norm_triangle_ineq2)
```
```   772 done
```
```   773
```
```   774 lemma norm_triangle_ineq4:
```
```   775   fixes a b :: "'a::real_normed_vector"
```
```   776   shows "norm (a - b) \<le> norm a + norm b"
```
```   777 proof -
```
```   778   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   779     by (rule norm_triangle_ineq)
```
```   780   then show ?thesis by simp
```
```   781 qed
```
```   782
```
```   783 lemma norm_diff_ineq:
```
```   784   fixes a b :: "'a::real_normed_vector"
```
```   785   shows "norm a - norm b \<le> norm (a + b)"
```
```   786 proof -
```
```   787   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   788     by (rule norm_triangle_ineq2)
```
```   789   thus ?thesis by simp
```
```   790 qed
```
```   791
```
```   792 lemma norm_add_leD:
```
```   793   fixes a b :: "'a::real_normed_vector"
```
```   794   shows "norm (a + b) \<le> c \<Longrightarrow> norm b \<le> norm a + c"
```
```   795     by (metis add.commute diff_le_eq norm_diff_ineq order.trans)
```
```   796
```
```   797 lemma norm_diff_triangle_ineq:
```
```   798   fixes a b c d :: "'a::real_normed_vector"
```
```   799   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   800 proof -
```
```   801   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   802     by (simp add: algebra_simps)
```
```   803   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   804     by (rule norm_triangle_ineq)
```
```   805   finally show ?thesis .
```
```   806 qed
```
```   807
```
```   808 lemma norm_diff_triangle_le:
```
```   809   fixes x y z :: "'a::real_normed_vector"
```
```   810   assumes "norm (x - y) \<le> e1"  "norm (y - z) \<le> e2"
```
```   811     shows "norm (x - z) \<le> e1 + e2"
```
```   812   using norm_diff_triangle_ineq [of x y y z] assms by simp
```
```   813
```
```   814 lemma norm_diff_triangle_less:
```
```   815   fixes x y z :: "'a::real_normed_vector"
```
```   816   assumes "norm (x - y) < e1"  "norm (y - z) < e2"
```
```   817     shows "norm (x - z) < e1 + e2"
```
```   818   using norm_diff_triangle_ineq [of x y y z] assms by simp
```
```   819
```
```   820 lemma norm_triangle_mono:
```
```   821   fixes a b :: "'a::real_normed_vector"
```
```   822   shows "\<lbrakk>norm a \<le> r; norm b \<le> s\<rbrakk> \<Longrightarrow> norm (a + b) \<le> r + s"
```
```   823 by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)
```
```   824
```
```   825 lemma norm_setsum:
```
```   826   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   827   shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
```
```   828   by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono)
```
```   829
```
```   830 lemma setsum_norm_le:
```
```   831   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   832   assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
```
```   833   shows "norm (setsum f S) \<le> setsum g S"
```
```   834   by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
```
```   835
```
```   836 lemma abs_norm_cancel [simp]:
```
```   837   fixes a :: "'a::real_normed_vector"
```
```   838   shows "\<bar>norm a\<bar> = norm a"
```
```   839 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   840
```
```   841 lemma norm_add_less:
```
```   842   fixes x y :: "'a::real_normed_vector"
```
```   843   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   844 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   845
```
```   846 lemma norm_mult_less:
```
```   847   fixes x y :: "'a::real_normed_algebra"
```
```   848   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   849 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   850 apply (simp add: mult_strict_mono')
```
```   851 done
```
```   852
```
```   853 lemma norm_of_real [simp]:
```
```   854   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   855 unfolding of_real_def by simp
```
```   856
```
```   857 lemma norm_numeral [simp]:
```
```   858   "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   859 by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
```
```   860
```
```   861 lemma norm_neg_numeral [simp]:
```
```   862   "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   863 by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
```
```   864
```
```   865 lemma norm_of_real_add1 [simp]:
```
```   866      "norm (of_real x + 1 :: 'a :: real_normed_div_algebra) = abs (x + 1)"
```
```   867   by (metis norm_of_real of_real_1 of_real_add)
```
```   868
```
```   869 lemma norm_of_real_addn [simp]:
```
```   870      "norm (of_real x + numeral b :: 'a :: real_normed_div_algebra) = abs (x + numeral b)"
```
```   871   by (metis norm_of_real of_real_add of_real_numeral)
```
```   872
```
```   873 lemma norm_of_int [simp]:
```
```   874   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   875 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   876
```
```   877 lemma norm_of_nat [simp]:
```
```   878   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   879 apply (subst of_real_of_nat_eq [symmetric])
```
```   880 apply (subst norm_of_real, simp)
```
```   881 done
```
```   882
```
```   883 lemma nonzero_norm_inverse:
```
```   884   fixes a :: "'a::real_normed_div_algebra"
```
```   885   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   886 apply (rule inverse_unique [symmetric])
```
```   887 apply (simp add: norm_mult [symmetric])
```
```   888 done
```
```   889
```
```   890 lemma norm_inverse:
```
```   891   fixes a :: "'a::{real_normed_div_algebra, division_ring}"
```
```   892   shows "norm (inverse a) = inverse (norm a)"
```
```   893 apply (case_tac "a = 0", simp)
```
```   894 apply (erule nonzero_norm_inverse)
```
```   895 done
```
```   896
```
```   897 lemma nonzero_norm_divide:
```
```   898   fixes a b :: "'a::real_normed_field"
```
```   899   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   900 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   901
```
```   902 lemma norm_divide:
```
```   903   fixes a b :: "'a::{real_normed_field, field}"
```
```   904   shows "norm (a / b) = norm a / norm b"
```
```   905 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   906
```
```   907 lemma norm_power_ineq:
```
```   908   fixes x :: "'a::{real_normed_algebra_1}"
```
```   909   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   910 proof (induct n)
```
```   911   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   912 next
```
```   913   case (Suc n)
```
```   914   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   915     by (rule norm_mult_ineq)
```
```   916   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   917     using norm_ge_zero by (rule mult_left_mono)
```
```   918   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   919     by simp
```
```   920 qed
```
```   921
```
```   922 lemma norm_power:
```
```   923   fixes x :: "'a::{real_normed_div_algebra}"
```
```   924   shows "norm (x ^ n) = norm x ^ n"
```
```   925 by (induct n) (simp_all add: norm_mult)
```
```   926
```
```   927 lemma norm_mult_numeral1 [simp]:
```
```   928   fixes a b :: "'a::{real_normed_field, field}"
```
```   929   shows "norm (numeral w * a) = numeral w * norm a"
```
```   930 by (simp add: norm_mult)
```
```   931
```
```   932 lemma norm_mult_numeral2 [simp]:
```
```   933   fixes a b :: "'a::{real_normed_field, field}"
```
```   934   shows "norm (a * numeral w) = norm a * numeral w"
```
```   935 by (simp add: norm_mult)
```
```   936
```
```   937 lemma norm_divide_numeral [simp]:
```
```   938   fixes a b :: "'a::{real_normed_field, field}"
```
```   939   shows "norm (a / numeral w) = norm a / numeral w"
```
```   940 by (simp add: norm_divide)
```
```   941
```
```   942 lemma norm_of_real_diff [simp]:
```
```   943     "norm (of_real b - of_real a :: 'a::real_normed_algebra_1) \<le> \<bar>b - a\<bar>"
```
```   944   by (metis norm_of_real of_real_diff order_refl)
```
```   945
```
```   946 text\<open>Despite a superficial resemblance, \<open>norm_eq_1\<close> is not relevant.\<close>
```
```   947 lemma square_norm_one:
```
```   948   fixes x :: "'a::real_normed_div_algebra"
```
```   949   assumes "x^2 = 1" shows "norm x = 1"
```
```   950   by (metis assms norm_minus_cancel norm_one power2_eq_1_iff)
```
```   951
```
```   952 lemma norm_less_p1:
```
```   953   fixes x :: "'a::real_normed_algebra_1"
```
```   954   shows "norm x < norm (of_real (norm x) + 1 :: 'a)"
```
```   955 proof -
```
```   956   have "norm x < norm (of_real (norm x + 1) :: 'a)"
```
```   957     by (simp add: of_real_def)
```
```   958   then show ?thesis
```
```   959     by simp
```
```   960 qed
```
```   961
```
```   962 lemma setprod_norm:
```
```   963   fixes f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}"
```
```   964   shows "setprod (\<lambda>x. norm(f x)) A = norm (setprod f A)"
```
```   965   by (induct A rule: infinite_finite_induct) (auto simp: norm_mult)
```
```   966
```
```   967 lemma norm_setprod_le:
```
```   968   "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))"
```
```   969 proof (induction A rule: infinite_finite_induct)
```
```   970   case (insert a A)
```
```   971   then have "norm (setprod f (insert a A)) \<le> norm (f a) * norm (setprod f A)"
```
```   972     by (simp add: norm_mult_ineq)
```
```   973   also have "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a))"
```
```   974     by (rule insert)
```
```   975   finally show ?case
```
```   976     by (simp add: insert mult_left_mono)
```
```   977 qed simp_all
```
```   978
```
```   979 lemma norm_setprod_diff:
```
```   980   fixes z w :: "'i \<Rightarrow> 'a::{real_normed_algebra_1, comm_monoid_mult}"
```
```   981   shows "(\<And>i. i \<in> I \<Longrightarrow> norm (z i) \<le> 1) \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> norm (w i) \<le> 1) \<Longrightarrow>
```
```   982     norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
```
```   983 proof (induction I rule: infinite_finite_induct)
```
```   984   case (insert i I)
```
```   985   note insert.hyps[simp]
```
```   986
```
```   987   have "norm ((\<Prod>i\<in>insert i I. z i) - (\<Prod>i\<in>insert i I. w i)) =
```
```   988     norm ((\<Prod>i\<in>I. z i) * (z i - w i) + ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * w i)"
```
```   989     (is "_ = norm (?t1 + ?t2)")
```
```   990     by (auto simp add: field_simps)
```
```   991   also have "... \<le> norm ?t1 + norm ?t2"
```
```   992     by (rule norm_triangle_ineq)
```
```   993   also have "norm ?t1 \<le> norm (\<Prod>i\<in>I. z i) * norm (z i - w i)"
```
```   994     by (rule norm_mult_ineq)
```
```   995   also have "\<dots> \<le> (\<Prod>i\<in>I. norm (z i)) * norm(z i - w i)"
```
```   996     by (rule mult_right_mono) (auto intro: norm_setprod_le)
```
```   997   also have "(\<Prod>i\<in>I. norm (z i)) \<le> (\<Prod>i\<in>I. 1)"
```
```   998     by (intro setprod_mono) (auto intro!: insert)
```
```   999   also have "norm ?t2 \<le> norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * norm (w i)"
```
```  1000     by (rule norm_mult_ineq)
```
```  1001   also have "norm (w i) \<le> 1"
```
```  1002     by (auto intro: insert)
```
```  1003   also have "norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
```
```  1004     using insert by auto
```
```  1005   finally show ?case
```
```  1006     by (auto simp add: ac_simps mult_right_mono mult_left_mono)
```
```  1007 qed simp_all
```
```  1008
```
```  1009 lemma norm_power_diff:
```
```  1010   fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}"
```
```  1011   assumes "norm z \<le> 1" "norm w \<le> 1"
```
```  1012   shows "norm (z^m - w^m) \<le> m * norm (z - w)"
```
```  1013 proof -
```
```  1014   have "norm (z^m - w^m) = norm ((\<Prod> i < m. z) - (\<Prod> i < m. w))"
```
```  1015     by (simp add: setprod_constant)
```
```  1016   also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))"
```
```  1017     by (intro norm_setprod_diff) (auto simp add: assms)
```
```  1018   also have "\<dots> = m * norm (z - w)"
```
```  1019     by simp
```
```  1020   finally show ?thesis .
```
```  1021 qed
```
```  1022
```
```  1023 subsection \<open>Metric spaces\<close>
```
```  1024
```
```  1025 class metric_space = uniformity_dist + open_uniformity +
```
```  1026   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```  1027   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```  1028 begin
```
```  1029
```
```  1030 lemma dist_self [simp]: "dist x x = 0"
```
```  1031 by simp
```
```  1032
```
```  1033 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```  1034 using dist_triangle2 [of x x y] by simp
```
```  1035
```
```  1036 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```  1037 by (simp add: less_le)
```
```  1038
```
```  1039 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```  1040 by (simp add: not_less)
```
```  1041
```
```  1042 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```  1043 by (simp add: le_less)
```
```  1044
```
```  1045 lemma dist_commute: "dist x y = dist y x"
```
```  1046 proof (rule order_antisym)
```
```  1047   show "dist x y \<le> dist y x"
```
```  1048     using dist_triangle2 [of x y x] by simp
```
```  1049   show "dist y x \<le> dist x y"
```
```  1050     using dist_triangle2 [of y x y] by simp
```
```  1051 qed
```
```  1052
```
```  1053 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```  1054 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```  1055
```
```  1056 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```  1057 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```  1058
```
```  1059 lemma dist_triangle_alt:
```
```  1060   shows "dist y z <= dist x y + dist x z"
```
```  1061 by (rule dist_triangle3)
```
```  1062
```
```  1063 lemma dist_pos_lt:
```
```  1064   shows "x \<noteq> y ==> 0 < dist x y"
```
```  1065 by (simp add: zero_less_dist_iff)
```
```  1066
```
```  1067 lemma dist_nz:
```
```  1068   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
```
```  1069 by (simp add: zero_less_dist_iff)
```
```  1070
```
```  1071 declare dist_nz [symmetric, simp]
```
```  1072
```
```  1073 lemma dist_triangle_le:
```
```  1074   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
```
```  1075 by (rule order_trans [OF dist_triangle2])
```
```  1076
```
```  1077 lemma dist_triangle_lt:
```
```  1078   shows "dist x z + dist y z < e ==> dist x y < e"
```
```  1079 by (rule le_less_trans [OF dist_triangle2])
```
```  1080
```
```  1081 lemma dist_triangle_half_l:
```
```  1082   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```  1083 by (rule dist_triangle_lt [where z=y], simp)
```
```  1084
```
```  1085 lemma dist_triangle_half_r:
```
```  1086   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```  1087 by (rule dist_triangle_half_l, simp_all add: dist_commute)
```
```  1088
```
```  1089 subclass uniform_space
```
```  1090 proof
```
```  1091   fix E x assume "eventually E uniformity"
```
```  1092   then obtain e where E: "0 < e" "\<And>x y. dist x y < e \<Longrightarrow> E (x, y)"
```
```  1093     unfolding eventually_uniformity_metric by auto
```
```  1094   then show "E (x, x)" "\<forall>\<^sub>F (x, y) in uniformity. E (y, x)"
```
```  1095     unfolding eventually_uniformity_metric by (auto simp: dist_commute)
```
```  1096
```
```  1097   show "\<exists>D. eventually D uniformity \<and> (\<forall>x y z. D (x, y) \<longrightarrow> D (y, z) \<longrightarrow> E (x, z))"
```
```  1098     using E dist_triangle_half_l[where e=e] unfolding eventually_uniformity_metric
```
```  1099     by (intro exI[of _ "\<lambda>(x, y). dist x y < e / 2"] exI[of _ "e/2"] conjI)
```
```  1100        (auto simp: dist_commute)
```
```  1101 qed
```
```  1102
```
```  1103 lemma open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```  1104   unfolding open_uniformity eventually_uniformity_metric by (simp add: dist_commute)
```
```  1105
```
```  1106 lemma open_ball: "open {y. dist x y < d}"
```
```  1107 proof (unfold open_dist, intro ballI)
```
```  1108   fix y assume *: "y \<in> {y. dist x y < d}"
```
```  1109   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
```
```  1110     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
```
```  1111 qed
```
```  1112
```
```  1113 subclass first_countable_topology
```
```  1114 proof
```
```  1115   fix x
```
```  1116   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))"
```
```  1117   proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
```
```  1118     fix S assume "open S" "x \<in> S"
```
```  1119     then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
```
```  1120       by (auto simp: open_dist subset_eq dist_commute)
```
```  1121     moreover
```
```  1122     from e obtain i where "inverse (Suc i) < e"
```
```  1123       by (auto dest!: reals_Archimedean)
```
```  1124     then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
```
```  1125       by auto
```
```  1126     ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
```
```  1127       by blast
```
```  1128   qed (auto intro: open_ball)
```
```  1129 qed
```
```  1130
```
```  1131 end
```
```  1132
```
```  1133 instance metric_space \<subseteq> t2_space
```
```  1134 proof
```
```  1135   fix x y :: "'a::metric_space"
```
```  1136   assume xy: "x \<noteq> y"
```
```  1137   let ?U = "{y'. dist x y' < dist x y / 2}"
```
```  1138   let ?V = "{x'. dist y x' < dist x y / 2}"
```
```  1139   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
```
```  1140                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```  1141   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```  1142     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
```
```  1143     using open_ball[of _ "dist x y / 2"] by auto
```
```  1144   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  1145     by blast
```
```  1146 qed
```
```  1147
```
```  1148 text \<open>Every normed vector space is a metric space.\<close>
```
```  1149
```
```  1150 instance real_normed_vector < metric_space
```
```  1151 proof
```
```  1152   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
```
```  1153     unfolding dist_norm by simp
```
```  1154 next
```
```  1155   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
```
```  1156     unfolding dist_norm
```
```  1157     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
```
```  1158 qed
```
```  1159
```
```  1160 subsection \<open>Class instances for real numbers\<close>
```
```  1161
```
```  1162 instantiation real :: real_normed_field
```
```  1163 begin
```
```  1164
```
```  1165 definition dist_real_def:
```
```  1166   "dist x y = \<bar>x - y\<bar>"
```
```  1167
```
```  1168 definition uniformity_real_def [code del]:
```
```  1169   "(uniformity :: (real \<times> real) filter) = (INF e:{0 <..}. principal {(x, y). dist x y < e})"
```
```  1170
```
```  1171 definition open_real_def [code del]:
```
```  1172   "open (U :: real set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
```
```  1173
```
```  1174 definition real_norm_def [simp]:
```
```  1175   "norm r = \<bar>r\<bar>"
```
```  1176
```
```  1177 instance
```
```  1178 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```  1179 apply (rule dist_real_def)
```
```  1180 apply (simp add: sgn_real_def)
```
```  1181 apply (rule uniformity_real_def)
```
```  1182 apply (rule open_real_def)
```
```  1183 apply (rule abs_eq_0)
```
```  1184 apply (rule abs_triangle_ineq)
```
```  1185 apply (rule abs_mult)
```
```  1186 apply (rule abs_mult)
```
```  1187 done
```
```  1188
```
```  1189 end
```
```  1190
```
```  1191 declare uniformity_Abort[where 'a=real, code]
```
```  1192
```
```  1193 lemma dist_of_real [simp]:
```
```  1194   fixes a :: "'a::real_normed_div_algebra"
```
```  1195   shows "dist (of_real x :: 'a) (of_real y) = dist x y"
```
```  1196 by (metis dist_norm norm_of_real of_real_diff real_norm_def)
```
```  1197
```
```  1198 declare [[code abort: "open :: real set \<Rightarrow> bool"]]
```
```  1199
```
```  1200 instance real :: linorder_topology
```
```  1201 proof
```
```  1202   show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
```
```  1203   proof (rule ext, safe)
```
```  1204     fix S :: "real set" assume "open S"
```
```  1205     then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"
```
```  1206       unfolding open_dist bchoice_iff ..
```
```  1207     then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
```
```  1208       by (fastforce simp: dist_real_def)
```
```  1209     show "generate_topology (range lessThan \<union> range greaterThan) S"
```
```  1210       apply (subst *)
```
```  1211       apply (intro generate_topology_Union generate_topology.Int)
```
```  1212       apply (auto intro: generate_topology.Basis)
```
```  1213       done
```
```  1214   next
```
```  1215     fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
```
```  1216     moreover have "\<And>a::real. open {..<a}"
```
```  1217       unfolding open_dist dist_real_def
```
```  1218     proof clarify
```
```  1219       fix x a :: real assume "x < a"
```
```  1220       hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```  1221       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```  1222     qed
```
```  1223     moreover have "\<And>a::real. open {a <..}"
```
```  1224       unfolding open_dist dist_real_def
```
```  1225     proof clarify
```
```  1226       fix x a :: real assume "a < x"
```
```  1227       hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```  1228       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```  1229     qed
```
```  1230     ultimately show "open S"
```
```  1231       by induct auto
```
```  1232   qed
```
```  1233 qed
```
```  1234
```
```  1235 instance real :: linear_continuum_topology ..
```
```  1236
```
```  1237 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
```
```  1238 lemmas open_real_lessThan = open_lessThan[where 'a=real]
```
```  1239 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
```
```  1240 lemmas closed_real_atMost = closed_atMost[where 'a=real]
```
```  1241 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
```
```  1242 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
```
```  1243
```
```  1244 subsection \<open>Extra type constraints\<close>
```
```  1245
```
```  1246 text \<open>Only allow @{term "open"} in class \<open>topological_space\<close>.\<close>
```
```  1247
```
```  1248 setup \<open>Sign.add_const_constraint
```
```  1249   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"})\<close>
```
```  1250
```
```  1251 text \<open>Only allow @{term "uniformity"} in class \<open>uniform_space\<close>.\<close>
```
```  1252
```
```  1253 setup \<open>Sign.add_const_constraint
```
```  1254   (@{const_name "uniformity"}, SOME @{typ "('a::uniformity \<times> 'a) filter"})\<close>
```
```  1255
```
```  1256 text \<open>Only allow @{term dist} in class \<open>metric_space\<close>.\<close>
```
```  1257
```
```  1258 setup \<open>Sign.add_const_constraint
```
```  1259   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"})\<close>
```
```  1260
```
```  1261 text \<open>Only allow @{term norm} in class \<open>real_normed_vector\<close>.\<close>
```
```  1262
```
```  1263 setup \<open>Sign.add_const_constraint
```
```  1264   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"})\<close>
```
```  1265
```
```  1266 subsection \<open>Sign function\<close>
```
```  1267
```
```  1268 lemma norm_sgn:
```
```  1269   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```  1270 by (simp add: sgn_div_norm)
```
```  1271
```
```  1272 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```  1273 by (simp add: sgn_div_norm)
```
```  1274
```
```  1275 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```  1276 by (simp add: sgn_div_norm)
```
```  1277
```
```  1278 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```  1279 by (simp add: sgn_div_norm)
```
```  1280
```
```  1281 lemma sgn_scaleR:
```
```  1282   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```  1283 by (simp add: sgn_div_norm ac_simps)
```
```  1284
```
```  1285 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```  1286 by (simp add: sgn_div_norm)
```
```  1287
```
```  1288 lemma sgn_of_real:
```
```  1289   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```  1290 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```  1291
```
```  1292 lemma sgn_mult:
```
```  1293   fixes x y :: "'a::real_normed_div_algebra"
```
```  1294   shows "sgn (x * y) = sgn x * sgn y"
```
```  1295 by (simp add: sgn_div_norm norm_mult mult.commute)
```
```  1296
```
```  1297 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```  1298   by (simp add: sgn_div_norm divide_inverse)
```
```  1299
```
```  1300 lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> (x::real)"
```
```  1301   by (cases "0::real" x rule: linorder_cases) simp_all
```
```  1302
```
```  1303 lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> (x::real) \<le> 0"
```
```  1304   by (cases "0::real" x rule: linorder_cases) simp_all
```
```  1305
```
```  1306 lemma norm_conv_dist: "norm x = dist x 0"
```
```  1307   unfolding dist_norm by simp
```
```  1308
```
```  1309 declare norm_conv_dist [symmetric, simp]
```
```  1310
```
```  1311 lemma dist_diff [simp]: "dist a (a - b) = norm b"  "dist (a - b) a = norm b"
```
```  1312   by (simp_all add: dist_norm)
```
```  1313
```
```  1314 lemma dist_of_int: "dist (of_int m) (of_int n :: 'a :: real_normed_algebra_1) = of_int \<bar>m - n\<bar>"
```
```  1315 proof -
```
```  1316   have "dist (of_int m) (of_int n :: 'a) = dist (of_int m :: 'a) (of_int m - (of_int (m - n)))"
```
```  1317     by simp
```
```  1318   also have "\<dots> = of_int \<bar>m - n\<bar>" by (subst dist_diff, subst norm_of_int) simp
```
```  1319   finally show ?thesis .
```
```  1320 qed
```
```  1321
```
```  1322 lemma dist_of_nat:
```
```  1323   "dist (of_nat m) (of_nat n :: 'a :: real_normed_algebra_1) = of_int \<bar>int m - int n\<bar>"
```
```  1324   by (subst (1 2) of_int_of_nat_eq [symmetric]) (rule dist_of_int)
```
```  1325
```
```  1326 subsection \<open>Bounded Linear and Bilinear Operators\<close>
```
```  1327
```
```  1328 locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
```
```  1329   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```  1330
```
```  1331 lemma linear_imp_scaleR:
```
```  1332   assumes "linear D" obtains d where "D = (\<lambda>x. x *\<^sub>R d)"
```
```  1333   by (metis assms linear.scaleR mult.commute mult.left_neutral real_scaleR_def)
```
```  1334
```
```  1335 lemma linearI:
```
```  1336   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1337   assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```  1338   shows "linear f"
```
```  1339   by standard (rule assms)+
```
```  1340
```
```  1341 locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
```
```  1342   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1343 begin
```
```  1344
```
```  1345 lemma pos_bounded:
```
```  1346   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1347 proof -
```
```  1348   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```  1349     using bounded by blast
```
```  1350   show ?thesis
```
```  1351   proof (intro exI impI conjI allI)
```
```  1352     show "0 < max 1 K"
```
```  1353       by (rule order_less_le_trans [OF zero_less_one max.cobounded1])
```
```  1354   next
```
```  1355     fix x
```
```  1356     have "norm (f x) \<le> norm x * K" using K .
```
```  1357     also have "\<dots> \<le> norm x * max 1 K"
```
```  1358       by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero])
```
```  1359     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```  1360   qed
```
```  1361 qed
```
```  1362
```
```  1363 lemma nonneg_bounded:
```
```  1364   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1365 proof -
```
```  1366   from pos_bounded
```
```  1367   show ?thesis by (auto intro: order_less_imp_le)
```
```  1368 qed
```
```  1369
```
```  1370 lemma linear: "linear f" ..
```
```  1371
```
```  1372 end
```
```  1373
```
```  1374 lemma bounded_linear_intro:
```
```  1375   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1376   assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
```
```  1377   assumes "\<And>x. norm (f x) \<le> norm x * K"
```
```  1378   shows "bounded_linear f"
```
```  1379   by standard (blast intro: assms)+
```
```  1380
```
```  1381 locale bounded_bilinear =
```
```  1382   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```  1383                  \<Rightarrow> 'c::real_normed_vector"
```
```  1384     (infixl "**" 70)
```
```  1385   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```  1386   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```  1387   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```  1388   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```  1389   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```  1390 begin
```
```  1391
```
```  1392 lemma pos_bounded:
```
```  1393   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1394 apply (cut_tac bounded, erule exE)
```
```  1395 apply (rule_tac x="max 1 K" in exI, safe)
```
```  1396 apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])
```
```  1397 apply (drule spec, drule spec, erule order_trans)
```
```  1398 apply (rule mult_left_mono [OF max.cobounded2])
```
```  1399 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```  1400 done
```
```  1401
```
```  1402 lemma nonneg_bounded:
```
```  1403   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1404 proof -
```
```  1405   from pos_bounded
```
```  1406   show ?thesis by (auto intro: order_less_imp_le)
```
```  1407 qed
```
```  1408
```
```  1409 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```  1410 by (rule additive.intro, rule add_right)
```
```  1411
```
```  1412 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```  1413 by (rule additive.intro, rule add_left)
```
```  1414
```
```  1415 lemma zero_left: "prod 0 b = 0"
```
```  1416 by (rule additive.zero [OF additive_left])
```
```  1417
```
```  1418 lemma zero_right: "prod a 0 = 0"
```
```  1419 by (rule additive.zero [OF additive_right])
```
```  1420
```
```  1421 lemma minus_left: "prod (- a) b = - prod a b"
```
```  1422 by (rule additive.minus [OF additive_left])
```
```  1423
```
```  1424 lemma minus_right: "prod a (- b) = - prod a b"
```
```  1425 by (rule additive.minus [OF additive_right])
```
```  1426
```
```  1427 lemma diff_left:
```
```  1428   "prod (a - a') b = prod a b - prod a' b"
```
```  1429 by (rule additive.diff [OF additive_left])
```
```  1430
```
```  1431 lemma diff_right:
```
```  1432   "prod a (b - b') = prod a b - prod a b'"
```
```  1433 by (rule additive.diff [OF additive_right])
```
```  1434
```
```  1435 lemma setsum_left:
```
```  1436   "prod (setsum g S) x = setsum ((\<lambda>i. prod (g i) x)) S"
```
```  1437 by (rule additive.setsum [OF additive_left])
```
```  1438
```
```  1439 lemma setsum_right:
```
```  1440   "prod x (setsum g S) = setsum ((\<lambda>i. (prod x (g i)))) S"
```
```  1441 by (rule additive.setsum [OF additive_right])
```
```  1442
```
```  1443
```
```  1444 lemma bounded_linear_left:
```
```  1445   "bounded_linear (\<lambda>a. a ** b)"
```
```  1446 apply (cut_tac bounded, safe)
```
```  1447 apply (rule_tac K="norm b * K" in bounded_linear_intro)
```
```  1448 apply (rule add_left)
```
```  1449 apply (rule scaleR_left)
```
```  1450 apply (simp add: ac_simps)
```
```  1451 done
```
```  1452
```
```  1453 lemma bounded_linear_right:
```
```  1454   "bounded_linear (\<lambda>b. a ** b)"
```
```  1455 apply (cut_tac bounded, safe)
```
```  1456 apply (rule_tac K="norm a * K" in bounded_linear_intro)
```
```  1457 apply (rule add_right)
```
```  1458 apply (rule scaleR_right)
```
```  1459 apply (simp add: ac_simps)
```
```  1460 done
```
```  1461
```
```  1462 lemma prod_diff_prod:
```
```  1463   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```  1464 by (simp add: diff_left diff_right)
```
```  1465
```
```  1466 lemma flip: "bounded_bilinear (\<lambda>x y. y ** x)"
```
```  1467   apply standard
```
```  1468   apply (rule add_right)
```
```  1469   apply (rule add_left)
```
```  1470   apply (rule scaleR_right)
```
```  1471   apply (rule scaleR_left)
```
```  1472   apply (subst mult.commute)
```
```  1473   using bounded
```
```  1474   apply blast
```
```  1475   done
```
```  1476
```
```  1477 lemma comp1:
```
```  1478   assumes "bounded_linear g"
```
```  1479   shows "bounded_bilinear (\<lambda>x. op ** (g x))"
```
```  1480 proof unfold_locales
```
```  1481   interpret g: bounded_linear g by fact
```
```  1482   show "\<And>a a' b. g (a + a') ** b = g a ** b + g a' ** b"
```
```  1483     "\<And>a b b'. g a ** (b + b') = g a ** b + g a ** b'"
```
```  1484     "\<And>r a b. g (r *\<^sub>R a) ** b = r *\<^sub>R (g a ** b)"
```
```  1485     "\<And>a r b. g a ** (r *\<^sub>R b) = r *\<^sub>R (g a ** b)"
```
```  1486     by (auto simp: g.add add_left add_right g.scaleR scaleR_left scaleR_right)
```
```  1487   from g.nonneg_bounded nonneg_bounded
```
```  1488   obtain K L
```
```  1489   where nn: "0 \<le> K" "0 \<le> L"
```
```  1490     and K: "\<And>x. norm (g x) \<le> norm x * K"
```
```  1491     and L: "\<And>a b. norm (a ** b) \<le> norm a * norm b * L"
```
```  1492     by auto
```
```  1493   have "norm (g a ** b) \<le> norm a * K * norm b * L" for a b
```
```  1494     by (auto intro!:  order_trans[OF K] order_trans[OF L] mult_mono simp: nn)
```
```  1495   then show "\<exists>K. \<forall>a b. norm (g a ** b) \<le> norm a * norm b * K"
```
```  1496     by (auto intro!: exI[where x="K * L"] simp: ac_simps)
```
```  1497 qed
```
```  1498
```
```  1499 lemma comp:
```
```  1500   "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_bilinear (\<lambda>x y. f x ** g y)"
```
```  1501   by (rule bounded_bilinear.flip[OF bounded_bilinear.comp1[OF bounded_bilinear.flip[OF comp1]]])
```
```  1502
```
```  1503 end
```
```  1504
```
```  1505 lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
```
```  1506   by standard (auto intro!: exI[of _ 1])
```
```  1507
```
```  1508 lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
```
```  1509   by standard (auto intro!: exI[of _ 1])
```
```  1510
```
```  1511 lemma bounded_linear_add:
```
```  1512   assumes "bounded_linear f"
```
```  1513   assumes "bounded_linear g"
```
```  1514   shows "bounded_linear (\<lambda>x. f x + g x)"
```
```  1515 proof -
```
```  1516   interpret f: bounded_linear f by fact
```
```  1517   interpret g: bounded_linear g by fact
```
```  1518   show ?thesis
```
```  1519   proof
```
```  1520     from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
```
```  1521     from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
```
```  1522     show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
```
```  1523       using add_mono[OF Kf Kg]
```
```  1524       by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
```
```  1525   qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
```
```  1526 qed
```
```  1527
```
```  1528 lemma bounded_linear_minus:
```
```  1529   assumes "bounded_linear f"
```
```  1530   shows "bounded_linear (\<lambda>x. - f x)"
```
```  1531 proof -
```
```  1532   interpret f: bounded_linear f by fact
```
```  1533   show ?thesis apply (unfold_locales)
```
```  1534     apply (simp add: f.add)
```
```  1535     apply (simp add: f.scaleR)
```
```  1536     apply (simp add: f.bounded)
```
```  1537     done
```
```  1538 qed
```
```  1539
```
```  1540 lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g \<Longrightarrow> bounded_linear (\<lambda>x. f x - g x)"
```
```  1541   using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g]
```
```  1542   by (auto simp add: algebra_simps)
```
```  1543
```
```  1544 lemma bounded_linear_setsum:
```
```  1545   fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
```
```  1546   assumes "\<And>i. i \<in> I \<Longrightarrow> bounded_linear (f i)"
```
```  1547   shows "bounded_linear (\<lambda>x. \<Sum>i\<in>I. f i x)"
```
```  1548 proof cases
```
```  1549   assume "finite I"
```
```  1550   from this show ?thesis
```
```  1551     using assms
```
```  1552     by (induct I) (auto intro!: bounded_linear_add)
```
```  1553 qed simp
```
```  1554
```
```  1555 lemma bounded_linear_compose:
```
```  1556   assumes "bounded_linear f"
```
```  1557   assumes "bounded_linear g"
```
```  1558   shows "bounded_linear (\<lambda>x. f (g x))"
```
```  1559 proof -
```
```  1560   interpret f: bounded_linear f by fact
```
```  1561   interpret g: bounded_linear g by fact
```
```  1562   show ?thesis proof (unfold_locales)
```
```  1563     fix x y show "f (g (x + y)) = f (g x) + f (g y)"
```
```  1564       by (simp only: f.add g.add)
```
```  1565   next
```
```  1566     fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
```
```  1567       by (simp only: f.scaleR g.scaleR)
```
```  1568   next
```
```  1569     from f.pos_bounded
```
```  1570     obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by blast
```
```  1571     from g.pos_bounded
```
```  1572     obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
```
```  1573     show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
```
```  1574     proof (intro exI allI)
```
```  1575       fix x
```
```  1576       have "norm (f (g x)) \<le> norm (g x) * Kf"
```
```  1577         using f .
```
```  1578       also have "\<dots> \<le> (norm x * Kg) * Kf"
```
```  1579         using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
```
```  1580       also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
```
```  1581         by (rule mult.assoc)
```
```  1582       finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
```
```  1583     qed
```
```  1584   qed
```
```  1585 qed
```
```  1586
```
```  1587 lemma bounded_bilinear_mult:
```
```  1588   "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
```
```  1589 apply (rule bounded_bilinear.intro)
```
```  1590 apply (rule distrib_right)
```
```  1591 apply (rule distrib_left)
```
```  1592 apply (rule mult_scaleR_left)
```
```  1593 apply (rule mult_scaleR_right)
```
```  1594 apply (rule_tac x="1" in exI)
```
```  1595 apply (simp add: norm_mult_ineq)
```
```  1596 done
```
```  1597
```
```  1598 lemma bounded_linear_mult_left:
```
```  1599   "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
```
```  1600   using bounded_bilinear_mult
```
```  1601   by (rule bounded_bilinear.bounded_linear_left)
```
```  1602
```
```  1603 lemma bounded_linear_mult_right:
```
```  1604   "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
```
```  1605   using bounded_bilinear_mult
```
```  1606   by (rule bounded_bilinear.bounded_linear_right)
```
```  1607
```
```  1608 lemmas bounded_linear_mult_const =
```
```  1609   bounded_linear_mult_left [THEN bounded_linear_compose]
```
```  1610
```
```  1611 lemmas bounded_linear_const_mult =
```
```  1612   bounded_linear_mult_right [THEN bounded_linear_compose]
```
```  1613
```
```  1614 lemma bounded_linear_divide:
```
```  1615   "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
```
```  1616   unfolding divide_inverse by (rule bounded_linear_mult_left)
```
```  1617
```
```  1618 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
```
```  1619 apply (rule bounded_bilinear.intro)
```
```  1620 apply (rule scaleR_left_distrib)
```
```  1621 apply (rule scaleR_right_distrib)
```
```  1622 apply simp
```
```  1623 apply (rule scaleR_left_commute)
```
```  1624 apply (rule_tac x="1" in exI, simp)
```
```  1625 done
```
```  1626
```
```  1627 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
```
```  1628   using bounded_bilinear_scaleR
```
```  1629   by (rule bounded_bilinear.bounded_linear_left)
```
```  1630
```
```  1631 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
```
```  1632   using bounded_bilinear_scaleR
```
```  1633   by (rule bounded_bilinear.bounded_linear_right)
```
```  1634
```
```  1635 lemmas bounded_linear_scaleR_const =
```
```  1636   bounded_linear_scaleR_left[THEN bounded_linear_compose]
```
```  1637
```
```  1638 lemmas bounded_linear_const_scaleR =
```
```  1639   bounded_linear_scaleR_right[THEN bounded_linear_compose]
```
```  1640
```
```  1641 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
```
```  1642   unfolding of_real_def by (rule bounded_linear_scaleR_left)
```
```  1643
```
```  1644 lemma real_bounded_linear:
```
```  1645   fixes f :: "real \<Rightarrow> real"
```
```  1646   shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
```
```  1647 proof -
```
```  1648   { fix x assume "bounded_linear f"
```
```  1649     then interpret bounded_linear f .
```
```  1650     from scaleR[of x 1] have "f x = x * f 1"
```
```  1651       by simp }
```
```  1652   then show ?thesis
```
```  1653     by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
```
```  1654 qed
```
```  1655
```
```  1656 lemma bij_linear_imp_inv_linear:
```
```  1657   assumes "linear f" "bij f" shows "linear (inv f)"
```
```  1658   using assms unfolding linear_def linear_axioms_def additive_def
```
```  1659   by (auto simp: bij_is_surj bij_is_inj surj_f_inv_f intro!:  Hilbert_Choice.inv_f_eq)
```
```  1660
```
```  1661 instance real_normed_algebra_1 \<subseteq> perfect_space
```
```  1662 proof
```
```  1663   fix x::'a
```
```  1664   show "\<not> open {x}"
```
```  1665     unfolding open_dist dist_norm
```
```  1666     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
```
```  1667 qed
```
```  1668
```
```  1669 subsection \<open>Filters and Limits on Metric Space\<close>
```
```  1670
```
```  1671 lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})"
```
```  1672   unfolding nhds_def
```
```  1673 proof (safe intro!: INF_eq)
```
```  1674   fix S assume "open S" "x \<in> S"
```
```  1675   then obtain e where "{y. dist y x < e} \<subseteq> S" "0 < e"
```
```  1676     by (auto simp: open_dist subset_eq)
```
```  1677   then show "\<exists>e\<in>{0<..}. principal {y. dist y x < e} \<le> principal S"
```
```  1678     by auto
```
```  1679 qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute)
```
```  1680
```
```  1681 lemma (in metric_space) tendsto_iff:
```
```  1682   "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
```
```  1683   unfolding nhds_metric filterlim_INF filterlim_principal by auto
```
```  1684
```
```  1685 lemma (in metric_space) tendstoI: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
```
```  1686   by (auto simp: tendsto_iff)
```
```  1687
```
```  1688 lemma (in metric_space) tendstoD: "(f \<longlongrightarrow> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```  1689   by (auto simp: tendsto_iff)
```
```  1690
```
```  1691 lemma (in metric_space) eventually_nhds_metric:
```
```  1692   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```  1693   unfolding nhds_metric
```
```  1694   by (subst eventually_INF_base)
```
```  1695      (auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b])
```
```  1696
```
```  1697 lemma eventually_at:
```
```  1698   fixes a :: "'a :: metric_space"
```
```  1699   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)"
```
```  1700   unfolding eventually_at_filter eventually_nhds_metric by auto
```
```  1701
```
```  1702 lemma eventually_at_le:
```
```  1703   fixes a :: "'a::metric_space"
```
```  1704   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)"
```
```  1705   unfolding eventually_at_filter eventually_nhds_metric
```
```  1706   apply auto
```
```  1707   apply (rule_tac x="d / 2" in exI)
```
```  1708   apply auto
```
```  1709   done
```
```  1710
```
```  1711 lemma eventually_at_left_real: "a > (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {b<..<a}) (at_left a)"
```
```  1712   by (subst eventually_at, rule exI[of _ "a - b"]) (force simp: dist_real_def)
```
```  1713
```
```  1714 lemma eventually_at_right_real: "a < (b :: real) \<Longrightarrow> eventually (\<lambda>x. x \<in> {a<..<b}) (at_right a)"
```
```  1715   by (subst eventually_at, rule exI[of _ "b - a"]) (force simp: dist_real_def)
```
```  1716
```
```  1717 lemma metric_tendsto_imp_tendsto:
```
```  1718   fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
```
```  1719   assumes f: "(f \<longlongrightarrow> a) F"
```
```  1720   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
```
```  1721   shows "(g \<longlongrightarrow> b) F"
```
```  1722 proof (rule tendstoI)
```
```  1723   fix e :: real assume "0 < e"
```
```  1724   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
```
```  1725   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
```
```  1726     using le_less_trans by (rule eventually_elim2)
```
```  1727 qed
```
```  1728
```
```  1729 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
```
```  1730   unfolding filterlim_at_top
```
```  1731   apply (intro allI)
```
```  1732   apply (rule_tac c="nat \<lceil>Z + 1\<rceil>" in eventually_sequentiallyI)
```
```  1733   apply linarith
```
```  1734   done
```
```  1735
```
```  1736
```
```  1737 subsubsection \<open>Limits of Sequences\<close>
```
```  1738
```
```  1739 lemma lim_sequentially: "X \<longlonglongrightarrow> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```  1740   unfolding tendsto_iff eventually_sequentially ..
```
```  1741
```
```  1742 lemmas LIMSEQ_def = lim_sequentially  (*legacy binding*)
```
```  1743
```
```  1744 lemma LIMSEQ_iff_nz: "X \<longlonglongrightarrow> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
```
```  1745   unfolding lim_sequentially by (metis Suc_leD zero_less_Suc)
```
```  1746
```
```  1747 lemma metric_LIMSEQ_I:
```
```  1748   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X \<longlonglongrightarrow> (L::'a::metric_space)"
```
```  1749 by (simp add: lim_sequentially)
```
```  1750
```
```  1751 lemma metric_LIMSEQ_D:
```
```  1752   "\<lbrakk>X \<longlonglongrightarrow> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```  1753 by (simp add: lim_sequentially)
```
```  1754
```
```  1755
```
```  1756 subsubsection \<open>Limits of Functions\<close>
```
```  1757
```
```  1758 lemma LIM_def: "f \<midarrow>(a::'a::metric_space)\<rightarrow> (L::'b::metric_space) =
```
```  1759      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
```
```  1760         --> dist (f x) L < r)"
```
```  1761   unfolding tendsto_iff eventually_at by simp
```
```  1762
```
```  1763 lemma metric_LIM_I:
```
```  1764   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
```
```  1765     \<Longrightarrow> f \<midarrow>(a::'a::metric_space)\<rightarrow> (L::'b::metric_space)"
```
```  1766 by (simp add: LIM_def)
```
```  1767
```
```  1768 lemma metric_LIM_D:
```
```  1769   "\<lbrakk>f \<midarrow>(a::'a::metric_space)\<rightarrow> (L::'b::metric_space); 0 < r\<rbrakk>
```
```  1770     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
```
```  1771 by (simp add: LIM_def)
```
```  1772
```
```  1773 lemma metric_LIM_imp_LIM:
```
```  1774   assumes f: "f \<midarrow>a\<rightarrow> (l::'a::metric_space)"
```
```  1775   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
```
```  1776   shows "g \<midarrow>a\<rightarrow> (m::'b::metric_space)"
```
```  1777   by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
```
```  1778
```
```  1779 lemma metric_LIM_equal2:
```
```  1780   assumes 1: "0 < R"
```
```  1781   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1782   shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>(a::'a::metric_space)\<rightarrow> l"
```
```  1783 apply (rule topological_tendstoI)
```
```  1784 apply (drule (2) topological_tendstoD)
```
```  1785 apply (simp add: eventually_at, safe)
```
```  1786 apply (rule_tac x="min d R" in exI, safe)
```
```  1787 apply (simp add: 1)
```
```  1788 apply (simp add: 2)
```
```  1789 done
```
```  1790
```
```  1791 lemma metric_LIM_compose2:
```
```  1792   assumes f: "f \<midarrow>(a::'a::metric_space)\<rightarrow> b"
```
```  1793   assumes g: "g \<midarrow>b\<rightarrow> c"
```
```  1794   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```  1795   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
```
```  1796   using inj
```
```  1797   by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
```
```  1798
```
```  1799 lemma metric_isCont_LIM_compose2:
```
```  1800   fixes f :: "'a :: metric_space \<Rightarrow> _"
```
```  1801   assumes f [unfolded isCont_def]: "isCont f a"
```
```  1802   assumes g: "g \<midarrow>f a\<rightarrow> l"
```
```  1803   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```  1804   shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
```
```  1805 by (rule metric_LIM_compose2 [OF f g inj])
```
```  1806
```
```  1807 subsection \<open>Complete metric spaces\<close>
```
```  1808
```
```  1809 subsection \<open>Cauchy sequences\<close>
```
```  1810
```
```  1811 lemma (in metric_space) Cauchy_def: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
```
```  1812 proof -
```
```  1813   have *: "eventually P (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) =
```
```  1814     (\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. P (X m, X n))" for P
```
```  1815   proof (subst eventually_INF_base, goal_cases)
```
```  1816     case (2 a b) then show ?case
```
```  1817       by (intro bexI[of _ "max a b"]) (auto simp: eventually_principal subset_eq)
```
```  1818   qed (auto simp: eventually_principal, blast)
```
```  1819   have "Cauchy X \<longleftrightarrow> (INF M. principal {(X m, X n) | n m. m \<ge> M \<and> n \<ge> M}) \<le> uniformity"
```
```  1820     unfolding Cauchy_uniform_iff le_filter_def * ..
```
```  1821   also have "\<dots> = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
```
```  1822     unfolding uniformity_dist le_INF_iff by (auto simp: * le_principal)
```
```  1823   finally show ?thesis .
```
```  1824 qed
```
```  1825
```
```  1826 lemma (in metric_space) Cauchy_altdef:
```
```  1827   "Cauchy f = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e)"
```
```  1828 proof
```
```  1829   assume A: "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
```
```  1830   show "Cauchy f" unfolding Cauchy_def
```
```  1831   proof (intro allI impI)
```
```  1832     fix e :: real assume e: "e > 0"
```
```  1833     with A obtain M where M: "\<And>m n. m \<ge> M \<Longrightarrow> n > m \<Longrightarrow> dist (f m) (f n) < e" by blast
```
```  1834     have "dist (f m) (f n) < e" if "m \<ge> M" "n \<ge> M" for m n
```
```  1835       using M[of m n] M[of n m] e that by (cases m n rule: linorder_cases) (auto simp: dist_commute)
```
```  1836     thus "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m) (f n) < e" by blast
```
```  1837   qed
```
```  1838 next
```
```  1839   assume "Cauchy f"
```
```  1840   show "\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e"
```
```  1841   proof (intro allI impI)
```
```  1842     fix e :: real assume e: "e > 0"
```
```  1843     with \<open>Cauchy f\<close> obtain M where "\<And>m n. m \<ge> M \<Longrightarrow> n \<ge> M \<Longrightarrow> dist (f m) (f n) < e"
```
```  1844       unfolding Cauchy_def by blast
```
```  1845     thus "\<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (f m) (f n) < e" by (intro exI[of _ M]) force
```
```  1846   qed
```
```  1847 qed
```
```  1848
```
```  1849 lemma (in metric_space) metric_CauchyI:
```
```  1850   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```  1851   by (simp add: Cauchy_def)
```
```  1852
```
```  1853 lemma (in metric_space) CauchyI': "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n>m. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```  1854   unfolding Cauchy_altdef by blast
```
```  1855
```
```  1856 lemma (in metric_space) metric_CauchyD:
```
```  1857   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```  1858   by (simp add: Cauchy_def)
```
```  1859
```
```  1860 lemma (in metric_space) metric_Cauchy_iff2:
```
```  1861   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
```
```  1862 apply (simp add: Cauchy_def, auto)
```
```  1863 apply (drule reals_Archimedean, safe)
```
```  1864 apply (drule_tac x = n in spec, auto)
```
```  1865 apply (rule_tac x = M in exI, auto)
```
```  1866 apply (drule_tac x = m in spec, simp)
```
```  1867 apply (drule_tac x = na in spec, auto)
```
```  1868 done
```
```  1869
```
```  1870 lemma Cauchy_iff2:
```
```  1871   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
```
```  1872   unfolding metric_Cauchy_iff2 dist_real_def ..
```
```  1873
```
```  1874 lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
```
```  1875 proof (subst lim_sequentially, intro allI impI exI)
```
```  1876   fix e :: real assume e: "e > 0"
```
```  1877   fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
```
```  1878   have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
```
```  1879   also note n
```
```  1880   finally show "dist (1 / of_nat n :: 'a) 0 < e" using e
```
```  1881     by (simp add: divide_simps mult.commute norm_divide)
```
```  1882 qed
```
```  1883
```
```  1884 lemma (in metric_space) complete_def:
```
```  1885   shows "complete S = (\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l))"
```
```  1886   unfolding complete_uniform
```
```  1887 proof safe
```
```  1888   fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> S" "Cauchy f"
```
```  1889     and *: "\<forall>F\<le>principal S. F \<noteq> bot \<longrightarrow> cauchy_filter F \<longrightarrow> (\<exists>x\<in>S. F \<le> nhds x)"
```
```  1890   then show "\<exists>l\<in>S. f \<longlonglongrightarrow> l"
```
```  1891     unfolding filterlim_def using f
```
```  1892     by (intro *[rule_format])
```
```  1893        (auto simp: filtermap_sequentually_ne_bot le_principal eventually_filtermap Cauchy_uniform)
```
```  1894 next
```
```  1895   fix F :: "'a filter" assume "F \<le> principal S" "F \<noteq> bot" "cauchy_filter F"
```
```  1896   assume seq: "\<forall>f. (\<forall>n. f n \<in> S) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>S. f \<longlonglongrightarrow> l)"
```
```  1897
```
```  1898   from \<open>F \<le> principal S\<close> \<open>cauchy_filter F\<close> have FF_le: "F \<times>\<^sub>F F \<le> uniformity_on S"
```
```  1899     by (simp add: cauchy_filter_def principal_prod_principal[symmetric] prod_filter_mono)
```
```  1900
```
```  1901   let ?P = "\<lambda>P e. eventually P F \<and> (\<forall>x. P x \<longrightarrow> x \<in> S) \<and> (\<forall>x y. P x \<longrightarrow> P y \<longrightarrow> dist x y < e)"
```
```  1902
```
```  1903   { fix \<epsilon> :: real assume "0 < \<epsilon>"
```
```  1904     then have "eventually (\<lambda>(x, y). x \<in> S \<and> y \<in> S \<and> dist x y < \<epsilon>) (uniformity_on S)"
```
```  1905       unfolding eventually_inf_principal eventually_uniformity_metric by auto
```
```  1906     from filter_leD[OF FF_le this] have "\<exists>P. ?P P \<epsilon>"
```
```  1907       unfolding eventually_prod_same by auto }
```
```  1908   note P = this
```
```  1909
```
```  1910   have "\<exists>P. \<forall>n. ?P (P n) (1 / Suc n) \<and> P (Suc n) \<le> P n"
```
```  1911   proof (rule dependent_nat_choice)
```
```  1912     show "\<exists>P. ?P P (1 / Suc 0)"
```
```  1913       using P[of 1] by auto
```
```  1914   next
```
```  1915     fix P n assume "?P P (1/Suc n)"
```
```  1916     moreover obtain Q where "?P Q (1 / Suc (Suc n))"
```
```  1917       using P[of "1/Suc (Suc n)"] by auto
```
```  1918     ultimately show "\<exists>Q. ?P Q (1 / Suc (Suc n)) \<and> Q \<le> P"
```
```  1919       by (intro exI[of _ "\<lambda>x. P x \<and> Q x"]) (auto simp: eventually_conj_iff)
```
```  1920   qed
```
```  1921   then obtain P where P: "\<And>n. eventually (P n) F" "\<And>n x. P n x \<Longrightarrow> x \<in> S"
```
```  1922     "\<And>n x y. P n x \<Longrightarrow> P n y \<Longrightarrow> dist x y < 1 / Suc n" "\<And>n. P (Suc n) \<le> P n"
```
```  1923     by metis
```
```  1924   have "antimono P"
```
```  1925     using P(4) unfolding decseq_Suc_iff le_fun_def by blast
```
```  1926
```
```  1927   obtain X where X: "\<And>n. P n (X n)"
```
```  1928     using P(1)[THEN eventually_happens'[OF \<open>F \<noteq> bot\<close>]] by metis
```
```  1929   have "Cauchy X"
```
```  1930     unfolding metric_Cauchy_iff2 inverse_eq_divide
```
```  1931   proof (intro exI allI impI)
```
```  1932     fix j m n :: nat assume "j \<le> m" "j \<le> n"
```
```  1933     with \<open>antimono P\<close> X have "P j (X m)" "P j (X n)"
```
```  1934       by (auto simp: antimono_def)
```
```  1935     then show "dist (X m) (X n) < 1 / Suc j"
```
```  1936       by (rule P)
```
```  1937   qed
```
```  1938   moreover have "\<forall>n. X n \<in> S"
```
```  1939     using P(2) X by auto
```
```  1940   ultimately obtain x where "X \<longlonglongrightarrow> x" "x \<in> S"
```
```  1941     using seq by blast
```
```  1942
```
```  1943   show "\<exists>x\<in>S. F \<le> nhds x"
```
```  1944   proof (rule bexI)
```
```  1945     { fix e :: real assume "0 < e"
```
```  1946       then have "(\<lambda>n. 1 / Suc n :: real) \<longlonglongrightarrow> 0 \<and> 0 < e / 2"
```
```  1947         by (subst LIMSEQ_Suc_iff) (auto intro!: lim_1_over_n)
```
```  1948       then have "\<forall>\<^sub>F n in sequentially. dist (X n) x < e / 2 \<and> 1 / Suc n < e / 2"
```
```  1949         using \<open>X \<longlonglongrightarrow> x\<close> unfolding tendsto_iff order_tendsto_iff[where 'a=real] eventually_conj_iff by blast
```
```  1950       then obtain n where "dist x (X n) < e / 2" "1 / Suc n < e / 2"
```
```  1951         by (auto simp: eventually_sequentially dist_commute)
```
```  1952       have "eventually (\<lambda>y. dist y x < e) F"
```
```  1953         using \<open>eventually (P n) F\<close>
```
```  1954       proof eventually_elim
```
```  1955         fix y assume "P n y"
```
```  1956         then have "dist y (X n) < 1 / Suc n"
```
```  1957           by (intro X P)
```
```  1958         also have "\<dots> < e / 2" by fact
```
```  1959         finally show "dist y x < e"
```
```  1960           by (rule dist_triangle_half_l) fact
```
```  1961       qed }
```
```  1962     then show "F \<le> nhds x"
```
```  1963       unfolding nhds_metric le_INF_iff le_principal by auto
```
```  1964   qed fact
```
```  1965 qed
```
```  1966
```
```  1967 lemma (in metric_space) totally_bounded_metric:
```
```  1968   "totally_bounded S \<longleftrightarrow> (\<forall>e>0. \<exists>k. finite k \<and> S \<subseteq> (\<Union>x\<in>k. {y. dist x y < e}))"
```
```  1969   unfolding totally_bounded_def eventually_uniformity_metric imp_ex
```
```  1970   apply (subst all_comm)
```
```  1971   apply (intro arg_cong[where f=All] ext)
```
```  1972   apply safe
```
```  1973   subgoal for e
```
```  1974     apply (erule allE[of _ "\<lambda>(x, y). dist x y < e"])
```
```  1975     apply auto
```
```  1976     done
```
```  1977   subgoal for e P k
```
```  1978     apply (intro exI[of _ k])
```
```  1979     apply (force simp: subset_eq)
```
```  1980     done
```
```  1981   done
```
```  1982
```
```  1983 subsubsection \<open>Cauchy Sequences are Convergent\<close>
```
```  1984
```
```  1985 (* TODO: update to uniform_space *)
```
```  1986 class complete_space = metric_space +
```
```  1987   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1988
```
```  1989 lemma Cauchy_convergent_iff:
```
```  1990   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1991   shows "Cauchy X = convergent X"
```
```  1992 by (blast intro: Cauchy_convergent convergent_Cauchy)
```
```  1993
```
```  1994 subsection \<open>The set of real numbers is a complete metric space\<close>
```
```  1995
```
```  1996 text \<open>
```
```  1997 Proof that Cauchy sequences converge based on the one from
```
```  1998 @{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
```
```  1999 \<close>
```
```  2000
```
```  2001 text \<open>
```
```  2002   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  2003   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  2004 \<close>
```
```  2005
```
```  2006 lemma increasing_LIMSEQ:
```
```  2007   fixes f :: "nat \<Rightarrow> real"
```
```  2008   assumes inc: "\<And>n. f n \<le> f (Suc n)"
```
```  2009       and bdd: "\<And>n. f n \<le> l"
```
```  2010       and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
```
```  2011   shows "f \<longlonglongrightarrow> l"
```
```  2012 proof (rule increasing_tendsto)
```
```  2013   fix x assume "x < l"
```
```  2014   with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
```
```  2015     by auto
```
```  2016   from en[OF \<open>0 < e\<close>] obtain n where "l - e \<le> f n"
```
```  2017     by (auto simp: field_simps)
```
```  2018   with \<open>e < l - x\<close> \<open>0 < e\<close> have "x < f n" by simp
```
```  2019   with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
```
```  2020     by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
```
```  2021 qed (insert bdd, auto)
```
```  2022
```
```  2023 lemma real_Cauchy_convergent:
```
```  2024   fixes X :: "nat \<Rightarrow> real"
```
```  2025   assumes X: "Cauchy X"
```
```  2026   shows "convergent X"
```
```  2027 proof -
```
```  2028   def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  2029   then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
```
```  2030
```
```  2031   { fix N x assume N: "\<forall>n\<ge>N. X n < x"
```
```  2032   fix y::real assume "y \<in> S"
```
```  2033   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  2034     by (simp add: S_def)
```
```  2035   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  2036   hence "y < X (max M N)" by simp
```
```  2037   also have "\<dots> < x" using N by simp
```
```  2038   finally have "y \<le> x"
```
```  2039     by (rule order_less_imp_le) }
```
```  2040   note bound_isUb = this
```
```  2041
```
```  2042   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
```
```  2043     using X[THEN metric_CauchyD, OF zero_less_one] by auto
```
```  2044   hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
```
```  2045   have [simp]: "S \<noteq> {}"
```
```  2046   proof (intro exI ex_in_conv[THEN iffD1])
```
```  2047     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  2048       by (simp add: abs_diff_less_iff dist_real_def)
```
```  2049     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  2050   qed
```
```  2051   have [simp]: "bdd_above S"
```
```  2052   proof
```
```  2053     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  2054       by (simp add: abs_diff_less_iff dist_real_def)
```
```  2055     thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
```
```  2056       by (rule bound_isUb)
```
```  2057   qed
```
```  2058   have "X \<longlonglongrightarrow> Sup S"
```
```  2059   proof (rule metric_LIMSEQ_I)
```
```  2060   fix r::real assume "0 < r"
```
```  2061   hence r: "0 < r/2" by simp
```
```  2062   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
```
```  2063     using metric_CauchyD [OF X r] by auto
```
```  2064   hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
```
```  2065   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  2066     by (simp only: dist_real_def abs_diff_less_iff)
```
```  2067
```
```  2068   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by blast
```
```  2069   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  2070   hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
```
```  2071
```
```  2072   from N have "\<forall>n\<ge>N. X n < X N + r/2" by blast
```
```  2073   from bound_isUb[OF this]
```
```  2074   have 2: "Sup S \<le> X N + r/2"
```
```  2075     by (intro cSup_least) simp_all
```
```  2076
```
```  2077   show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
```
```  2078   proof (intro exI allI impI)
```
```  2079     fix n assume n: "N \<le> n"
```
```  2080     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  2081     thus "dist (X n) (Sup S) < r" using 1 2
```
```  2082       by (simp add: abs_diff_less_iff dist_real_def)
```
```  2083   qed
```
```  2084   qed
```
```  2085   then show ?thesis unfolding convergent_def by auto
```
```  2086 qed
```
```  2087
```
```  2088 instance real :: complete_space
```
```  2089   by intro_classes (rule real_Cauchy_convergent)
```
```  2090
```
```  2091 class banach = real_normed_vector + complete_space
```
```  2092
```
```  2093 instance real :: banach ..
```
```  2094
```
```  2095 lemma tendsto_at_topI_sequentially:
```
```  2096   fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
```
```  2097   assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) \<longlonglongrightarrow> y"
```
```  2098   shows "(f \<longlongrightarrow> y) at_top"
```
```  2099 proof -
```
```  2100   from nhds_countable[of y] guess A . note A = this
```
```  2101
```
```  2102   have "\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m"
```
```  2103   proof (rule ccontr)
```
```  2104     assume "\<not> (\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m)"
```
```  2105     then obtain m where "\<And>k. \<exists>x\<ge>k. f x \<notin> A m"
```
```  2106       by auto
```
```  2107     then have "\<exists>X. \<forall>n. (f (X n) \<notin> A m) \<and> max n (X n) + 1 \<le> X (Suc n)"
```
```  2108       by (intro dependent_nat_choice) (auto simp del: max.bounded_iff)
```
```  2109     then obtain X where X: "\<And>n. f (X n) \<notin> A m" "\<And>n. max n (X n) + 1 \<le> X (Suc n)"
```
```  2110       by auto
```
```  2111     { fix n have "1 \<le> n \<longrightarrow> real n \<le> X n"
```
```  2112         using X[of "n - 1"] by auto }
```
```  2113     then have "filterlim X at_top sequentially"
```
```  2114       by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially]
```
```  2115                 simp: eventually_sequentially)
```
```  2116     from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False
```
```  2117       by auto
```
```  2118   qed
```
```  2119   then obtain k where "\<And>m x. k m \<le> x \<Longrightarrow> f x \<in> A m"
```
```  2120     by metis
```
```  2121   then show ?thesis
```
```  2122     unfolding at_top_def A
```
```  2123     by (intro filterlim_base[where i=k]) auto
```
```  2124 qed
```
```  2125
```
```  2126 lemma tendsto_at_topI_sequentially_real:
```
```  2127   fixes f :: "real \<Rightarrow> real"
```
```  2128   assumes mono: "mono f"
```
```  2129   assumes limseq: "(\<lambda>n. f (real n)) \<longlonglongrightarrow> y"
```
```  2130   shows "(f \<longlongrightarrow> y) at_top"
```
```  2131 proof (rule tendstoI)
```
```  2132   fix e :: real assume "0 < e"
```
```  2133   with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
```
```  2134     by (auto simp: lim_sequentially dist_real_def)
```
```  2135   { fix x :: real
```
```  2136     obtain n where "x \<le> real_of_nat n"
```
```  2137       using ex_le_of_nat[of x] ..
```
```  2138     note monoD[OF mono this]
```
```  2139     also have "f (real_of_nat n) \<le> y"
```
```  2140       by (rule LIMSEQ_le_const[OF limseq]) (auto intro!: exI[of _ n] monoD[OF mono])
```
```  2141     finally have "f x \<le> y" . }
```
```  2142   note le = this
```
```  2143   have "eventually (\<lambda>x. real N \<le> x) at_top"
```
```  2144     by (rule eventually_ge_at_top)
```
```  2145   then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
```
```  2146   proof eventually_elim
```
```  2147     fix x assume N': "real N \<le> x"
```
```  2148     with N[of N] le have "y - f (real N) < e" by auto
```
```  2149     moreover note monoD[OF mono N']
```
```  2150     ultimately show "dist (f x) y < e"
```
```  2151       using le[of x] by (auto simp: dist_real_def field_simps)
```
```  2152   qed
```
```  2153 qed
```
```  2154
```
```  2155 end
```