src/HOL/Real_Vector_Spaces.thy
 author paulson Sat Apr 11 11:56:40 2015 +0100 (2015-04-11) changeset 60017 b785d6d06430 parent 59867 58043346ca64 child 60026 41d81b4a0a21 permissions -rw-r--r--
Overloading of ln and powr, but "approximation" no longer works for powr. Code generation also fails due to type ambiguity in scala.
```     1 (*  Title:      HOL/Real_Vector_Spaces.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 section {* Vector Spaces and Algebras over the Reals *}
```
```     7
```
```     8 theory Real_Vector_Spaces
```
```     9 imports Real Topological_Spaces
```
```    10 begin
```
```    11
```
```    12 subsection {* Locale for additive functions *}
```
```    13
```
```    14 locale additive =
```
```    15   fixes f :: "'a::ab_group_add \<Rightarrow> 'b::ab_group_add"
```
```    16   assumes add: "f (x + y) = f x + f y"
```
```    17 begin
```
```    18
```
```    19 lemma zero: "f 0 = 0"
```
```    20 proof -
```
```    21   have "f 0 = f (0 + 0)" by simp
```
```    22   also have "\<dots> = f 0 + f 0" by (rule add)
```
```    23   finally show "f 0 = 0" by simp
```
```    24 qed
```
```    25
```
```    26 lemma minus: "f (- x) = - f x"
```
```    27 proof -
```
```    28   have "f (- x) + f x = f (- x + x)" by (rule add [symmetric])
```
```    29   also have "\<dots> = - f x + f x" by (simp add: zero)
```
```    30   finally show "f (- x) = - f x" by (rule add_right_imp_eq)
```
```    31 qed
```
```    32
```
```    33 lemma diff: "f (x - y) = f x - f y"
```
```    34   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 {* Vector spaces *}
```
```    47
```
```    48 locale vector_space =
```
```    49   fixes scale :: "'a::field \<Rightarrow> 'b::ab_group_add \<Rightarrow> 'b"
```
```    50   assumes scale_right_distrib [algebra_simps]:
```
```    51     "scale a (x + y) = scale a x + scale a y"
```
```    52   and scale_left_distrib [algebra_simps]:
```
```    53     "scale (a + b) x = scale a x + scale b x"
```
```    54   and scale_scale [simp]: "scale a (scale b x) = scale (a * b) x"
```
```    55   and scale_one [simp]: "scale 1 x = x"
```
```    56 begin
```
```    57
```
```    58 lemma scale_left_commute:
```
```    59   "scale a (scale b x) = scale b (scale a x)"
```
```    60 by (simp add: mult.commute)
```
```    61
```
```    62 lemma scale_zero_left [simp]: "scale 0 x = 0"
```
```    63   and scale_minus_left [simp]: "scale (- a) x = - (scale a x)"
```
```    64   and scale_left_diff_distrib [algebra_simps]:
```
```    65         "scale (a - b) x = scale a x - scale b x"
```
```    66   and scale_setsum_left: "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)"
```
```    67 proof -
```
```    68   interpret s: additive "\<lambda>a. scale a x"
```
```    69     proof qed (rule scale_left_distrib)
```
```    70   show "scale 0 x = 0" by (rule s.zero)
```
```    71   show "scale (- a) x = - (scale a x)" by (rule s.minus)
```
```    72   show "scale (a - b) x = scale a x - scale b x" by (rule s.diff)
```
```    73   show "scale (setsum f A) x = (\<Sum>a\<in>A. scale (f a) x)" by (rule s.setsum)
```
```    74 qed
```
```    75
```
```    76 lemma scale_zero_right [simp]: "scale a 0 = 0"
```
```    77   and scale_minus_right [simp]: "scale a (- x) = - (scale a x)"
```
```    78   and scale_right_diff_distrib [algebra_simps]:
```
```    79         "scale a (x - y) = scale a x - scale a y"
```
```    80   and scale_setsum_right: "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))"
```
```    81 proof -
```
```    82   interpret s: additive "\<lambda>x. scale a x"
```
```    83     proof qed (rule scale_right_distrib)
```
```    84   show "scale a 0 = 0" by (rule s.zero)
```
```    85   show "scale a (- x) = - (scale a x)" by (rule s.minus)
```
```    86   show "scale a (x - y) = scale a x - scale a y" by (rule s.diff)
```
```    87   show "scale a (setsum f A) = (\<Sum>x\<in>A. scale a (f x))" by (rule s.setsum)
```
```    88 qed
```
```    89
```
```    90 lemma scale_eq_0_iff [simp]:
```
```    91   "scale a x = 0 \<longleftrightarrow> a = 0 \<or> x = 0"
```
```    92 proof cases
```
```    93   assume "a = 0" thus ?thesis by simp
```
```    94 next
```
```    95   assume anz [simp]: "a \<noteq> 0"
```
```    96   { assume "scale a x = 0"
```
```    97     hence "scale (inverse a) (scale a x) = 0" by simp
```
```    98     hence "x = 0" by simp }
```
```    99   thus ?thesis by force
```
```   100 qed
```
```   101
```
```   102 lemma scale_left_imp_eq:
```
```   103   "\<lbrakk>a \<noteq> 0; scale a x = scale a y\<rbrakk> \<Longrightarrow> x = y"
```
```   104 proof -
```
```   105   assume nonzero: "a \<noteq> 0"
```
```   106   assume "scale a x = scale a y"
```
```   107   hence "scale a (x - y) = 0"
```
```   108      by (simp add: scale_right_diff_distrib)
```
```   109   hence "x - y = 0" by (simp add: nonzero)
```
```   110   thus "x = y" by (simp only: right_minus_eq)
```
```   111 qed
```
```   112
```
```   113 lemma scale_right_imp_eq:
```
```   114   "\<lbrakk>x \<noteq> 0; scale a x = scale b x\<rbrakk> \<Longrightarrow> a = b"
```
```   115 proof -
```
```   116   assume nonzero: "x \<noteq> 0"
```
```   117   assume "scale a x = scale b x"
```
```   118   hence "scale (a - b) x = 0"
```
```   119      by (simp add: scale_left_diff_distrib)
```
```   120   hence "a - b = 0" by (simp add: nonzero)
```
```   121   thus "a = b" by (simp only: right_minus_eq)
```
```   122 qed
```
```   123
```
```   124 lemma scale_cancel_left [simp]:
```
```   125   "scale a x = scale a y \<longleftrightarrow> x = y \<or> a = 0"
```
```   126 by (auto intro: scale_left_imp_eq)
```
```   127
```
```   128 lemma scale_cancel_right [simp]:
```
```   129   "scale a x = scale b x \<longleftrightarrow> a = b \<or> x = 0"
```
```   130 by (auto intro: scale_right_imp_eq)
```
```   131
```
```   132 end
```
```   133
```
```   134 subsection {* Real vector spaces *}
```
```   135
```
```   136 class scaleR =
```
```   137   fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a" (infixr "*\<^sub>R" 75)
```
```   138 begin
```
```   139
```
```   140 abbreviation
```
```   141   divideR :: "'a \<Rightarrow> real \<Rightarrow> 'a" (infixl "'/\<^sub>R" 70)
```
```   142 where
```
```   143   "x /\<^sub>R r == scaleR (inverse r) x"
```
```   144
```
```   145 end
```
```   146
```
```   147 class real_vector = scaleR + ab_group_add +
```
```   148   assumes scaleR_add_right: "scaleR a (x + y) = scaleR a x + scaleR a y"
```
```   149   and scaleR_add_left: "scaleR (a + b) x = scaleR a x + scaleR b x"
```
```   150   and scaleR_scaleR: "scaleR a (scaleR b x) = scaleR (a * b) x"
```
```   151   and scaleR_one: "scaleR 1 x = x"
```
```   152
```
```   153 interpretation real_vector:
```
```   154   vector_space "scaleR :: real \<Rightarrow> 'a \<Rightarrow> 'a::real_vector"
```
```   155 apply unfold_locales
```
```   156 apply (rule scaleR_add_right)
```
```   157 apply (rule scaleR_add_left)
```
```   158 apply (rule scaleR_scaleR)
```
```   159 apply (rule scaleR_one)
```
```   160 done
```
```   161
```
```   162 text {* Recover original theorem names *}
```
```   163
```
```   164 lemmas scaleR_left_commute = real_vector.scale_left_commute
```
```   165 lemmas scaleR_zero_left = real_vector.scale_zero_left
```
```   166 lemmas scaleR_minus_left = real_vector.scale_minus_left
```
```   167 lemmas scaleR_diff_left = real_vector.scale_left_diff_distrib
```
```   168 lemmas scaleR_setsum_left = real_vector.scale_setsum_left
```
```   169 lemmas scaleR_zero_right = real_vector.scale_zero_right
```
```   170 lemmas scaleR_minus_right = real_vector.scale_minus_right
```
```   171 lemmas scaleR_diff_right = real_vector.scale_right_diff_distrib
```
```   172 lemmas scaleR_setsum_right = real_vector.scale_setsum_right
```
```   173 lemmas scaleR_eq_0_iff = real_vector.scale_eq_0_iff
```
```   174 lemmas scaleR_left_imp_eq = real_vector.scale_left_imp_eq
```
```   175 lemmas scaleR_right_imp_eq = real_vector.scale_right_imp_eq
```
```   176 lemmas scaleR_cancel_left = real_vector.scale_cancel_left
```
```   177 lemmas scaleR_cancel_right = real_vector.scale_cancel_right
```
```   178
```
```   179 text {* Legacy names *}
```
```   180
```
```   181 lemmas scaleR_left_distrib = scaleR_add_left
```
```   182 lemmas scaleR_right_distrib = scaleR_add_right
```
```   183 lemmas scaleR_left_diff_distrib = scaleR_diff_left
```
```   184 lemmas scaleR_right_diff_distrib = scaleR_diff_right
```
```   185
```
```   186 lemma scaleR_minus1_left [simp]:
```
```   187   fixes x :: "'a::real_vector"
```
```   188   shows "scaleR (-1) x = - x"
```
```   189   using scaleR_minus_left [of 1 x] by simp
```
```   190
```
```   191 class real_algebra = real_vector + ring +
```
```   192   assumes mult_scaleR_left [simp]: "scaleR a x * y = scaleR a (x * y)"
```
```   193   and mult_scaleR_right [simp]: "x * scaleR a y = scaleR a (x * y)"
```
```   194
```
```   195 class real_algebra_1 = real_algebra + ring_1
```
```   196
```
```   197 class real_div_algebra = real_algebra_1 + division_ring
```
```   198
```
```   199 class real_field = real_div_algebra + field
```
```   200
```
```   201 instantiation real :: real_field
```
```   202 begin
```
```   203
```
```   204 definition
```
```   205   real_scaleR_def [simp]: "scaleR a x = a * x"
```
```   206
```
```   207 instance proof
```
```   208 qed (simp_all add: algebra_simps)
```
```   209
```
```   210 end
```
```   211
```
```   212 interpretation scaleR_left: additive "(\<lambda>a. scaleR a x::'a::real_vector)"
```
```   213 proof qed (rule scaleR_left_distrib)
```
```   214
```
```   215 interpretation scaleR_right: additive "(\<lambda>x. scaleR a x::'a::real_vector)"
```
```   216 proof qed (rule scaleR_right_distrib)
```
```   217
```
```   218 lemma nonzero_inverse_scaleR_distrib:
```
```   219   fixes x :: "'a::real_div_algebra" shows
```
```   220   "\<lbrakk>a \<noteq> 0; x \<noteq> 0\<rbrakk> \<Longrightarrow> inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   221 by (rule inverse_unique, simp)
```
```   222
```
```   223 lemma inverse_scaleR_distrib:
```
```   224   fixes x :: "'a::{real_div_algebra, division_ring}"
```
```   225   shows "inverse (scaleR a x) = scaleR (inverse a) (inverse x)"
```
```   226 apply (case_tac "a = 0", simp)
```
```   227 apply (case_tac "x = 0", simp)
```
```   228 apply (erule (1) nonzero_inverse_scaleR_distrib)
```
```   229 done
```
```   230
```
```   231
```
```   232 subsection {* Embedding of the Reals into any @{text real_algebra_1}:
```
```   233 @{term of_real} *}
```
```   234
```
```   235 definition
```
```   236   of_real :: "real \<Rightarrow> 'a::real_algebra_1" where
```
```   237   "of_real r = scaleR r 1"
```
```   238
```
```   239 lemma scaleR_conv_of_real: "scaleR r x = of_real r * x"
```
```   240 by (simp add: of_real_def)
```
```   241
```
```   242 lemma of_real_0 [simp]: "of_real 0 = 0"
```
```   243 by (simp add: of_real_def)
```
```   244
```
```   245 lemma of_real_1 [simp]: "of_real 1 = 1"
```
```   246 by (simp add: of_real_def)
```
```   247
```
```   248 lemma of_real_add [simp]: "of_real (x + y) = of_real x + of_real y"
```
```   249 by (simp add: of_real_def scaleR_left_distrib)
```
```   250
```
```   251 lemma of_real_minus [simp]: "of_real (- x) = - of_real x"
```
```   252 by (simp add: of_real_def)
```
```   253
```
```   254 lemma of_real_diff [simp]: "of_real (x - y) = of_real x - of_real y"
```
```   255 by (simp add: of_real_def scaleR_left_diff_distrib)
```
```   256
```
```   257 lemma of_real_mult [simp]: "of_real (x * y) = of_real x * of_real y"
```
```   258 by (simp add: of_real_def mult.commute)
```
```   259
```
```   260 lemma of_real_setsum[simp]: "of_real (setsum f s) = (\<Sum>x\<in>s. of_real (f x))"
```
```   261   by (induct s rule: infinite_finite_induct) auto
```
```   262
```
```   263 lemma of_real_setprod[simp]: "of_real (setprod f s) = (\<Prod>x\<in>s. of_real (f x))"
```
```   264   by (induct s rule: infinite_finite_induct) auto
```
```   265
```
```   266 lemma nonzero_of_real_inverse:
```
```   267   "x \<noteq> 0 \<Longrightarrow> of_real (inverse x) =
```
```   268    inverse (of_real x :: 'a::real_div_algebra)"
```
```   269 by (simp add: of_real_def nonzero_inverse_scaleR_distrib)
```
```   270
```
```   271 lemma of_real_inverse [simp]:
```
```   272   "of_real (inverse x) =
```
```   273    inverse (of_real x :: 'a::{real_div_algebra, division_ring})"
```
```   274 by (simp add: of_real_def inverse_scaleR_distrib)
```
```   275
```
```   276 lemma nonzero_of_real_divide:
```
```   277   "y \<noteq> 0 \<Longrightarrow> of_real (x / y) =
```
```   278    (of_real x / of_real y :: 'a::real_field)"
```
```   279 by (simp add: divide_inverse nonzero_of_real_inverse)
```
```   280
```
```   281 lemma of_real_divide [simp]:
```
```   282   "of_real (x / y) =
```
```   283    (of_real x / of_real y :: 'a::{real_field, field})"
```
```   284 by (simp add: divide_inverse)
```
```   285
```
```   286 lemma of_real_power [simp]:
```
```   287   "of_real (x ^ n) = (of_real x :: 'a::{real_algebra_1}) ^ n"
```
```   288 by (induct n) simp_all
```
```   289
```
```   290 lemma of_real_eq_iff [simp]: "(of_real x = of_real y) = (x = y)"
```
```   291 by (simp add: of_real_def)
```
```   292
```
```   293 lemma inj_of_real:
```
```   294   "inj of_real"
```
```   295   by (auto intro: injI)
```
```   296
```
```   297 lemmas of_real_eq_0_iff [simp] = of_real_eq_iff [of _ 0, simplified]
```
```   298
```
```   299 lemma of_real_eq_id [simp]: "of_real = (id :: real \<Rightarrow> real)"
```
```   300 proof
```
```   301   fix r
```
```   302   show "of_real r = id r"
```
```   303     by (simp add: of_real_def)
```
```   304 qed
```
```   305
```
```   306 text{*Collapse nested embeddings*}
```
```   307 lemma of_real_of_nat_eq [simp]: "of_real (of_nat n) = of_nat n"
```
```   308 by (induct n) auto
```
```   309
```
```   310 lemma of_real_of_int_eq [simp]: "of_real (of_int z) = of_int z"
```
```   311 by (cases z rule: int_diff_cases, simp)
```
```   312
```
```   313 lemma of_real_real_of_nat_eq [simp]: "of_real (real n) = of_nat n"
```
```   314   by (simp add: real_of_nat_def)
```
```   315
```
```   316 lemma of_real_real_of_int_eq [simp]: "of_real (real z) = of_int z"
```
```   317   by (simp add: real_of_int_def)
```
```   318
```
```   319 lemma of_real_numeral: "of_real (numeral w) = numeral w"
```
```   320 using of_real_of_int_eq [of "numeral w"] by simp
```
```   321
```
```   322 lemma of_real_neg_numeral: "of_real (- numeral w) = - numeral w"
```
```   323 using of_real_of_int_eq [of "- numeral w"] by simp
```
```   324
```
```   325 text{*Every real algebra has characteristic zero*}
```
```   326
```
```   327 instance real_algebra_1 < ring_char_0
```
```   328 proof
```
```   329   from inj_of_real inj_of_nat have "inj (of_real \<circ> of_nat)" by (rule inj_comp)
```
```   330   then show "inj (of_nat :: nat \<Rightarrow> 'a)" by (simp add: comp_def)
```
```   331 qed
```
```   332
```
```   333 instance real_field < field_char_0 ..
```
```   334
```
```   335
```
```   336 subsection {* The Set of Real Numbers *}
```
```   337
```
```   338 definition Reals :: "'a::real_algebra_1 set" where
```
```   339   "Reals = range of_real"
```
```   340
```
```   341 notation (xsymbols)
```
```   342   Reals  ("\<real>")
```
```   343
```
```   344 lemma Reals_of_real [simp]: "of_real r \<in> Reals"
```
```   345 by (simp add: Reals_def)
```
```   346
```
```   347 lemma Reals_of_int [simp]: "of_int z \<in> Reals"
```
```   348 by (subst of_real_of_int_eq [symmetric], rule Reals_of_real)
```
```   349
```
```   350 lemma Reals_of_nat [simp]: "of_nat n \<in> Reals"
```
```   351 by (subst of_real_of_nat_eq [symmetric], rule Reals_of_real)
```
```   352
```
```   353 lemma Reals_numeral [simp]: "numeral w \<in> Reals"
```
```   354 by (subst of_real_numeral [symmetric], rule Reals_of_real)
```
```   355
```
```   356 lemma Reals_0 [simp]: "0 \<in> Reals"
```
```   357 apply (unfold Reals_def)
```
```   358 apply (rule range_eqI)
```
```   359 apply (rule of_real_0 [symmetric])
```
```   360 done
```
```   361
```
```   362 lemma Reals_1 [simp]: "1 \<in> Reals"
```
```   363 apply (unfold Reals_def)
```
```   364 apply (rule range_eqI)
```
```   365 apply (rule of_real_1 [symmetric])
```
```   366 done
```
```   367
```
```   368 lemma Reals_add [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a + b \<in> Reals"
```
```   369 apply (auto simp add: Reals_def)
```
```   370 apply (rule range_eqI)
```
```   371 apply (rule of_real_add [symmetric])
```
```   372 done
```
```   373
```
```   374 lemma Reals_minus [simp]: "a \<in> Reals \<Longrightarrow> - a \<in> Reals"
```
```   375 apply (auto simp add: Reals_def)
```
```   376 apply (rule range_eqI)
```
```   377 apply (rule of_real_minus [symmetric])
```
```   378 done
```
```   379
```
```   380 lemma Reals_diff [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a - b \<in> Reals"
```
```   381 apply (auto simp add: Reals_def)
```
```   382 apply (rule range_eqI)
```
```   383 apply (rule of_real_diff [symmetric])
```
```   384 done
```
```   385
```
```   386 lemma Reals_mult [simp]: "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a * b \<in> Reals"
```
```   387 apply (auto simp add: Reals_def)
```
```   388 apply (rule range_eqI)
```
```   389 apply (rule of_real_mult [symmetric])
```
```   390 done
```
```   391
```
```   392 lemma nonzero_Reals_inverse:
```
```   393   fixes a :: "'a::real_div_algebra"
```
```   394   shows "\<lbrakk>a \<in> Reals; a \<noteq> 0\<rbrakk> \<Longrightarrow> inverse a \<in> Reals"
```
```   395 apply (auto simp add: Reals_def)
```
```   396 apply (rule range_eqI)
```
```   397 apply (erule nonzero_of_real_inverse [symmetric])
```
```   398 done
```
```   399
```
```   400 lemma Reals_inverse:
```
```   401   fixes a :: "'a::{real_div_algebra, division_ring}"
```
```   402   shows "a \<in> Reals \<Longrightarrow> inverse a \<in> Reals"
```
```   403 apply (auto simp add: Reals_def)
```
```   404 apply (rule range_eqI)
```
```   405 apply (rule of_real_inverse [symmetric])
```
```   406 done
```
```   407
```
```   408 lemma Reals_inverse_iff [simp]:
```
```   409   fixes x:: "'a :: {real_div_algebra, division_ring}"
```
```   410   shows "inverse x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
```
```   411 by (metis Reals_inverse inverse_inverse_eq)
```
```   412
```
```   413 lemma nonzero_Reals_divide:
```
```   414   fixes a b :: "'a::real_field"
```
```   415   shows "\<lbrakk>a \<in> Reals; b \<in> Reals; b \<noteq> 0\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   416 apply (auto simp add: Reals_def)
```
```   417 apply (rule range_eqI)
```
```   418 apply (erule nonzero_of_real_divide [symmetric])
```
```   419 done
```
```   420
```
```   421 lemma Reals_divide [simp]:
```
```   422   fixes a b :: "'a::{real_field, field}"
```
```   423   shows "\<lbrakk>a \<in> Reals; b \<in> Reals\<rbrakk> \<Longrightarrow> a / b \<in> Reals"
```
```   424 apply (auto simp add: Reals_def)
```
```   425 apply (rule range_eqI)
```
```   426 apply (rule of_real_divide [symmetric])
```
```   427 done
```
```   428
```
```   429 lemma Reals_power [simp]:
```
```   430   fixes a :: "'a::{real_algebra_1}"
```
```   431   shows "a \<in> Reals \<Longrightarrow> a ^ n \<in> Reals"
```
```   432 apply (auto simp add: Reals_def)
```
```   433 apply (rule range_eqI)
```
```   434 apply (rule of_real_power [symmetric])
```
```   435 done
```
```   436
```
```   437 lemma Reals_cases [cases set: Reals]:
```
```   438   assumes "q \<in> \<real>"
```
```   439   obtains (of_real) r where "q = of_real r"
```
```   440   unfolding Reals_def
```
```   441 proof -
```
```   442   from `q \<in> \<real>` have "q \<in> range of_real" unfolding Reals_def .
```
```   443   then obtain r where "q = of_real r" ..
```
```   444   then show thesis ..
```
```   445 qed
```
```   446
```
```   447 lemma setsum_in_Reals [intro,simp]:
```
```   448   assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setsum f s \<in> \<real>"
```
```   449 proof (cases "finite s")
```
```   450   case True then show ?thesis using assms
```
```   451     by (induct s rule: finite_induct) auto
```
```   452 next
```
```   453   case False then show ?thesis using assms
```
```   454     by (metis Reals_0 setsum.infinite)
```
```   455 qed
```
```   456
```
```   457 lemma setprod_in_Reals [intro,simp]:
```
```   458   assumes "\<And>i. i \<in> s \<Longrightarrow> f i \<in> \<real>" shows "setprod f s \<in> \<real>"
```
```   459 proof (cases "finite s")
```
```   460   case True then show ?thesis using assms
```
```   461     by (induct s rule: finite_induct) auto
```
```   462 next
```
```   463   case False then show ?thesis using assms
```
```   464     by (metis Reals_1 setprod.infinite)
```
```   465 qed
```
```   466
```
```   467 lemma Reals_induct [case_names of_real, induct set: Reals]:
```
```   468   "q \<in> \<real> \<Longrightarrow> (\<And>r. P (of_real r)) \<Longrightarrow> P q"
```
```   469   by (rule Reals_cases) auto
```
```   470
```
```   471 subsection {* Ordered real vector spaces *}
```
```   472
```
```   473 class ordered_real_vector = real_vector + ordered_ab_group_add +
```
```   474   assumes scaleR_left_mono: "x \<le> y \<Longrightarrow> 0 \<le> a \<Longrightarrow> a *\<^sub>R x \<le> a *\<^sub>R y"
```
```   475   assumes scaleR_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> x \<Longrightarrow> a *\<^sub>R x \<le> b *\<^sub>R x"
```
```   476 begin
```
```   477
```
```   478 lemma scaleR_mono:
```
```   479   "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"
```
```   480 apply (erule scaleR_right_mono [THEN order_trans], assumption)
```
```   481 apply (erule scaleR_left_mono, assumption)
```
```   482 done
```
```   483
```
```   484 lemma scaleR_mono':
```
```   485   "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"
```
```   486   by (rule scaleR_mono) (auto intro: order.trans)
```
```   487
```
```   488 lemma pos_le_divideRI:
```
```   489   assumes "0 < c"
```
```   490   assumes "c *\<^sub>R a \<le> b"
```
```   491   shows "a \<le> b /\<^sub>R c"
```
```   492 proof -
```
```   493   from scaleR_left_mono[OF assms(2)] assms(1)
```
```   494   have "c *\<^sub>R a /\<^sub>R c \<le> b /\<^sub>R c"
```
```   495     by simp
```
```   496   with assms show ?thesis
```
```   497     by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
```
```   498 qed
```
```   499
```
```   500 lemma pos_le_divideR_eq:
```
```   501   assumes "0 < c"
```
```   502   shows "a \<le> b /\<^sub>R c \<longleftrightarrow> c *\<^sub>R a \<le> b"
```
```   503 proof rule
```
```   504   assume "a \<le> b /\<^sub>R c"
```
```   505   from scaleR_left_mono[OF this] assms
```
```   506   have "c *\<^sub>R a \<le> c *\<^sub>R (b /\<^sub>R c)"
```
```   507     by simp
```
```   508   with assms show "c *\<^sub>R a \<le> b"
```
```   509     by (simp add: scaleR_one scaleR_scaleR inverse_eq_divide)
```
```   510 qed (rule pos_le_divideRI[OF assms])
```
```   511
```
```   512 lemma scaleR_image_atLeastAtMost:
```
```   513   "c > 0 \<Longrightarrow> scaleR c ` {x..y} = {c *\<^sub>R x..c *\<^sub>R y}"
```
```   514   apply (auto intro!: scaleR_left_mono)
```
```   515   apply (rule_tac x = "inverse c *\<^sub>R xa" in image_eqI)
```
```   516   apply (simp_all add: pos_le_divideR_eq[symmetric] scaleR_scaleR scaleR_one)
```
```   517   done
```
```   518
```
```   519 end
```
```   520
```
```   521 lemma scaleR_nonneg_nonneg: "0 \<le> a \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> 0 \<le> a *\<^sub>R x"
```
```   522   using scaleR_left_mono [of 0 x a]
```
```   523   by simp
```
```   524
```
```   525 lemma scaleR_nonneg_nonpos: "0 \<le> a \<Longrightarrow> (x::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> a *\<^sub>R x \<le> 0"
```
```   526   using scaleR_left_mono [of x 0 a] by simp
```
```   527
```
```   528 lemma scaleR_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> (x::'a::ordered_real_vector) \<Longrightarrow> a *\<^sub>R x \<le> 0"
```
```   529   using scaleR_right_mono [of a 0 x] by simp
```
```   530
```
```   531 lemma split_scaleR_neg_le: "(0 \<le> a & x \<le> 0) | (a \<le> 0 & 0 \<le> x) \<Longrightarrow>
```
```   532   a *\<^sub>R (x::'a::ordered_real_vector) \<le> 0"
```
```   533   by (auto simp add: scaleR_nonneg_nonpos scaleR_nonpos_nonneg)
```
```   534
```
```   535 lemma le_add_iff1:
```
```   536   fixes c d e::"'a::ordered_real_vector"
```
```   537   shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> (a - b) *\<^sub>R e + c \<le> d"
```
```   538   by (simp add: algebra_simps)
```
```   539
```
```   540 lemma le_add_iff2:
```
```   541   fixes c d e::"'a::ordered_real_vector"
```
```   542   shows "a *\<^sub>R e + c \<le> b *\<^sub>R e + d \<longleftrightarrow> c \<le> (b - a) *\<^sub>R e + d"
```
```   543   by (simp add: algebra_simps)
```
```   544
```
```   545 lemma scaleR_left_mono_neg:
```
```   546   fixes a b::"'a::ordered_real_vector"
```
```   547   shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b"
```
```   548   apply (drule scaleR_left_mono [of _ _ "- c"])
```
```   549   apply simp_all
```
```   550   done
```
```   551
```
```   552 lemma scaleR_right_mono_neg:
```
```   553   fixes c::"'a::ordered_real_vector"
```
```   554   shows "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a *\<^sub>R c \<le> b *\<^sub>R c"
```
```   555   apply (drule scaleR_right_mono [of _ _ "- c"])
```
```   556   apply simp_all
```
```   557   done
```
```   558
```
```   559 lemma scaleR_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> (b::'a::ordered_real_vector) \<le> 0 \<Longrightarrow> 0 \<le> a *\<^sub>R b"
```
```   560 using scaleR_right_mono_neg [of a 0 b] by simp
```
```   561
```
```   562 lemma split_scaleR_pos_le:
```
```   563   fixes b::"'a::ordered_real_vector"
```
```   564   shows "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a *\<^sub>R b"
```
```   565   by (auto simp add: scaleR_nonneg_nonneg scaleR_nonpos_nonpos)
```
```   566
```
```   567 lemma zero_le_scaleR_iff:
```
```   568   fixes b::"'a::ordered_real_vector"
```
```   569   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")
```
```   570 proof cases
```
```   571   assume "a \<noteq> 0"
```
```   572   show ?thesis
```
```   573   proof
```
```   574     assume lhs: ?lhs
```
```   575     {
```
```   576       assume "0 < a"
```
```   577       with lhs have "inverse a *\<^sub>R 0 \<le> inverse a *\<^sub>R (a *\<^sub>R b)"
```
```   578         by (intro scaleR_mono) auto
```
```   579       hence ?rhs using `0 < a`
```
```   580         by simp
```
```   581     } moreover {
```
```   582       assume "0 > a"
```
```   583       with lhs have "- inverse a *\<^sub>R 0 \<le> - inverse a *\<^sub>R (a *\<^sub>R b)"
```
```   584         by (intro scaleR_mono) auto
```
```   585       hence ?rhs using `0 > a`
```
```   586         by simp
```
```   587     } ultimately show ?rhs using `a \<noteq> 0` by arith
```
```   588   qed (auto simp: not_le `a \<noteq> 0` intro!: split_scaleR_pos_le)
```
```   589 qed simp
```
```   590
```
```   591 lemma scaleR_le_0_iff:
```
```   592   fixes b::"'a::ordered_real_vector"
```
```   593   shows "a *\<^sub>R b \<le> 0 \<longleftrightarrow> 0 < a \<and> b \<le> 0 \<or> a < 0 \<and> 0 \<le> b \<or> a = 0"
```
```   594   by (insert zero_le_scaleR_iff [of "-a" b]) force
```
```   595
```
```   596 lemma scaleR_le_cancel_left:
```
```   597   fixes b::"'a::ordered_real_vector"
```
```   598   shows "c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
```
```   599   by (auto simp add: neq_iff scaleR_left_mono scaleR_left_mono_neg
```
```   600     dest: scaleR_left_mono[where a="inverse c"] scaleR_left_mono_neg[where c="inverse c"])
```
```   601
```
```   602 lemma scaleR_le_cancel_left_pos:
```
```   603   fixes b::"'a::ordered_real_vector"
```
```   604   shows "0 < c \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> a \<le> b"
```
```   605   by (auto simp: scaleR_le_cancel_left)
```
```   606
```
```   607 lemma scaleR_le_cancel_left_neg:
```
```   608   fixes b::"'a::ordered_real_vector"
```
```   609   shows "c < 0 \<Longrightarrow> c *\<^sub>R a \<le> c *\<^sub>R b \<longleftrightarrow> b \<le> a"
```
```   610   by (auto simp: scaleR_le_cancel_left)
```
```   611
```
```   612 lemma scaleR_left_le_one_le:
```
```   613   fixes x::"'a::ordered_real_vector" and a::real
```
```   614   shows "0 \<le> x \<Longrightarrow> a \<le> 1 \<Longrightarrow> a *\<^sub>R x \<le> x"
```
```   615   using scaleR_right_mono[of a 1 x] by simp
```
```   616
```
```   617
```
```   618 subsection {* Real normed vector spaces *}
```
```   619
```
```   620 class dist =
```
```   621   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```   622
```
```   623 class norm =
```
```   624   fixes norm :: "'a \<Rightarrow> real"
```
```   625
```
```   626 class sgn_div_norm = scaleR + norm + sgn +
```
```   627   assumes sgn_div_norm: "sgn x = x /\<^sub>R norm x"
```
```   628
```
```   629 class dist_norm = dist + norm + minus +
```
```   630   assumes dist_norm: "dist x y = norm (x - y)"
```
```   631
```
```   632 class open_dist = "open" + dist +
```
```   633   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   634
```
```   635 class real_normed_vector = real_vector + sgn_div_norm + dist_norm + open_dist +
```
```   636   assumes norm_eq_zero [simp]: "norm x = 0 \<longleftrightarrow> x = 0"
```
```   637   and norm_triangle_ineq: "norm (x + y) \<le> norm x + norm y"
```
```   638   and norm_scaleR [simp]: "norm (scaleR a x) = \<bar>a\<bar> * norm x"
```
```   639 begin
```
```   640
```
```   641 lemma norm_ge_zero [simp]: "0 \<le> norm x"
```
```   642 proof -
```
```   643   have "0 = norm (x + -1 *\<^sub>R x)"
```
```   644     using scaleR_add_left[of 1 "-1" x] norm_scaleR[of 0 x] by (simp add: scaleR_one)
```
```   645   also have "\<dots> \<le> norm x + norm (-1 *\<^sub>R x)" by (rule norm_triangle_ineq)
```
```   646   finally show ?thesis by simp
```
```   647 qed
```
```   648
```
```   649 end
```
```   650
```
```   651 class real_normed_algebra = real_algebra + real_normed_vector +
```
```   652   assumes norm_mult_ineq: "norm (x * y) \<le> norm x * norm y"
```
```   653
```
```   654 class real_normed_algebra_1 = real_algebra_1 + real_normed_algebra +
```
```   655   assumes norm_one [simp]: "norm 1 = 1"
```
```   656
```
```   657 class real_normed_div_algebra = real_div_algebra + real_normed_vector +
```
```   658   assumes norm_mult: "norm (x * y) = norm x * norm y"
```
```   659
```
```   660 class real_normed_field = real_field + real_normed_div_algebra
```
```   661
```
```   662 instance real_normed_div_algebra < real_normed_algebra_1
```
```   663 proof
```
```   664   fix x y :: 'a
```
```   665   show "norm (x * y) \<le> norm x * norm y"
```
```   666     by (simp add: norm_mult)
```
```   667 next
```
```   668   have "norm (1 * 1::'a) = norm (1::'a) * norm (1::'a)"
```
```   669     by (rule norm_mult)
```
```   670   thus "norm (1::'a) = 1" by simp
```
```   671 qed
```
```   672
```
```   673 lemma norm_zero [simp]: "norm (0::'a::real_normed_vector) = 0"
```
```   674 by simp
```
```   675
```
```   676 lemma zero_less_norm_iff [simp]:
```
```   677   fixes x :: "'a::real_normed_vector"
```
```   678   shows "(0 < norm x) = (x \<noteq> 0)"
```
```   679 by (simp add: order_less_le)
```
```   680
```
```   681 lemma norm_not_less_zero [simp]:
```
```   682   fixes x :: "'a::real_normed_vector"
```
```   683   shows "\<not> norm x < 0"
```
```   684 by (simp add: linorder_not_less)
```
```   685
```
```   686 lemma norm_le_zero_iff [simp]:
```
```   687   fixes x :: "'a::real_normed_vector"
```
```   688   shows "(norm x \<le> 0) = (x = 0)"
```
```   689 by (simp add: order_le_less)
```
```   690
```
```   691 lemma norm_minus_cancel [simp]:
```
```   692   fixes x :: "'a::real_normed_vector"
```
```   693   shows "norm (- x) = norm x"
```
```   694 proof -
```
```   695   have "norm (- x) = norm (scaleR (- 1) x)"
```
```   696     by (simp only: scaleR_minus_left scaleR_one)
```
```   697   also have "\<dots> = \<bar>- 1\<bar> * norm x"
```
```   698     by (rule norm_scaleR)
```
```   699   finally show ?thesis by simp
```
```   700 qed
```
```   701
```
```   702 lemma norm_minus_commute:
```
```   703   fixes a b :: "'a::real_normed_vector"
```
```   704   shows "norm (a - b) = norm (b - a)"
```
```   705 proof -
```
```   706   have "norm (- (b - a)) = norm (b - a)"
```
```   707     by (rule norm_minus_cancel)
```
```   708   thus ?thesis by simp
```
```   709 qed
```
```   710
```
```   711 lemma norm_triangle_ineq2:
```
```   712   fixes a b :: "'a::real_normed_vector"
```
```   713   shows "norm a - norm b \<le> norm (a - b)"
```
```   714 proof -
```
```   715   have "norm (a - b + b) \<le> norm (a - b) + norm b"
```
```   716     by (rule norm_triangle_ineq)
```
```   717   thus ?thesis by simp
```
```   718 qed
```
```   719
```
```   720 lemma norm_triangle_ineq3:
```
```   721   fixes a b :: "'a::real_normed_vector"
```
```   722   shows "\<bar>norm a - norm b\<bar> \<le> norm (a - b)"
```
```   723 apply (subst abs_le_iff)
```
```   724 apply auto
```
```   725 apply (rule norm_triangle_ineq2)
```
```   726 apply (subst norm_minus_commute)
```
```   727 apply (rule norm_triangle_ineq2)
```
```   728 done
```
```   729
```
```   730 lemma norm_triangle_ineq4:
```
```   731   fixes a b :: "'a::real_normed_vector"
```
```   732   shows "norm (a - b) \<le> norm a + norm b"
```
```   733 proof -
```
```   734   have "norm (a + - b) \<le> norm a + norm (- b)"
```
```   735     by (rule norm_triangle_ineq)
```
```   736   then show ?thesis by simp
```
```   737 qed
```
```   738
```
```   739 lemma norm_diff_ineq:
```
```   740   fixes a b :: "'a::real_normed_vector"
```
```   741   shows "norm a - norm b \<le> norm (a + b)"
```
```   742 proof -
```
```   743   have "norm a - norm (- b) \<le> norm (a - - b)"
```
```   744     by (rule norm_triangle_ineq2)
```
```   745   thus ?thesis by simp
```
```   746 qed
```
```   747
```
```   748 lemma norm_diff_triangle_ineq:
```
```   749   fixes a b c d :: "'a::real_normed_vector"
```
```   750   shows "norm ((a + b) - (c + d)) \<le> norm (a - c) + norm (b - d)"
```
```   751 proof -
```
```   752   have "norm ((a + b) - (c + d)) = norm ((a - c) + (b - d))"
```
```   753     by (simp add: algebra_simps)
```
```   754   also have "\<dots> \<le> norm (a - c) + norm (b - d)"
```
```   755     by (rule norm_triangle_ineq)
```
```   756   finally show ?thesis .
```
```   757 qed
```
```   758
```
```   759 lemma norm_triangle_mono:
```
```   760   fixes a b :: "'a::real_normed_vector"
```
```   761   shows "\<lbrakk>norm a \<le> r; norm b \<le> s\<rbrakk> \<Longrightarrow> norm (a + b) \<le> r + s"
```
```   762 by (metis add_mono_thms_linordered_semiring(1) norm_triangle_ineq order.trans)
```
```   763
```
```   764 lemma norm_setsum:
```
```   765   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   766   shows "norm (setsum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
```
```   767   by (induct A rule: infinite_finite_induct) (auto intro: norm_triangle_mono)
```
```   768
```
```   769 lemma setsum_norm_le:
```
```   770   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
```
```   771   assumes fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
```
```   772   shows "norm (setsum f S) \<le> setsum g S"
```
```   773   by (rule order_trans [OF norm_setsum setsum_mono]) (simp add: fg)
```
```   774
```
```   775 lemma abs_norm_cancel [simp]:
```
```   776   fixes a :: "'a::real_normed_vector"
```
```   777   shows "\<bar>norm a\<bar> = norm a"
```
```   778 by (rule abs_of_nonneg [OF norm_ge_zero])
```
```   779
```
```   780 lemma norm_add_less:
```
```   781   fixes x y :: "'a::real_normed_vector"
```
```   782   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x + y) < r + s"
```
```   783 by (rule order_le_less_trans [OF norm_triangle_ineq add_strict_mono])
```
```   784
```
```   785 lemma norm_mult_less:
```
```   786   fixes x y :: "'a::real_normed_algebra"
```
```   787   shows "\<lbrakk>norm x < r; norm y < s\<rbrakk> \<Longrightarrow> norm (x * y) < r * s"
```
```   788 apply (rule order_le_less_trans [OF norm_mult_ineq])
```
```   789 apply (simp add: mult_strict_mono')
```
```   790 done
```
```   791
```
```   792 lemma norm_of_real [simp]:
```
```   793   "norm (of_real r :: 'a::real_normed_algebra_1) = \<bar>r\<bar>"
```
```   794 unfolding of_real_def by simp
```
```   795
```
```   796 lemma norm_numeral [simp]:
```
```   797   "norm (numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   798 by (subst of_real_numeral [symmetric], subst norm_of_real, simp)
```
```   799
```
```   800 lemma norm_neg_numeral [simp]:
```
```   801   "norm (- numeral w::'a::real_normed_algebra_1) = numeral w"
```
```   802 by (subst of_real_neg_numeral [symmetric], subst norm_of_real, simp)
```
```   803
```
```   804 lemma norm_of_int [simp]:
```
```   805   "norm (of_int z::'a::real_normed_algebra_1) = \<bar>of_int z\<bar>"
```
```   806 by (subst of_real_of_int_eq [symmetric], rule norm_of_real)
```
```   807
```
```   808 lemma norm_of_nat [simp]:
```
```   809   "norm (of_nat n::'a::real_normed_algebra_1) = of_nat n"
```
```   810 apply (subst of_real_of_nat_eq [symmetric])
```
```   811 apply (subst norm_of_real, simp)
```
```   812 done
```
```   813
```
```   814 lemma nonzero_norm_inverse:
```
```   815   fixes a :: "'a::real_normed_div_algebra"
```
```   816   shows "a \<noteq> 0 \<Longrightarrow> norm (inverse a) = inverse (norm a)"
```
```   817 apply (rule inverse_unique [symmetric])
```
```   818 apply (simp add: norm_mult [symmetric])
```
```   819 done
```
```   820
```
```   821 lemma norm_inverse:
```
```   822   fixes a :: "'a::{real_normed_div_algebra, division_ring}"
```
```   823   shows "norm (inverse a) = inverse (norm a)"
```
```   824 apply (case_tac "a = 0", simp)
```
```   825 apply (erule nonzero_norm_inverse)
```
```   826 done
```
```   827
```
```   828 lemma nonzero_norm_divide:
```
```   829   fixes a b :: "'a::real_normed_field"
```
```   830   shows "b \<noteq> 0 \<Longrightarrow> norm (a / b) = norm a / norm b"
```
```   831 by (simp add: divide_inverse norm_mult nonzero_norm_inverse)
```
```   832
```
```   833 lemma norm_divide:
```
```   834   fixes a b :: "'a::{real_normed_field, field}"
```
```   835   shows "norm (a / b) = norm a / norm b"
```
```   836 by (simp add: divide_inverse norm_mult norm_inverse)
```
```   837
```
```   838 lemma norm_power_ineq:
```
```   839   fixes x :: "'a::{real_normed_algebra_1}"
```
```   840   shows "norm (x ^ n) \<le> norm x ^ n"
```
```   841 proof (induct n)
```
```   842   case 0 show "norm (x ^ 0) \<le> norm x ^ 0" by simp
```
```   843 next
```
```   844   case (Suc n)
```
```   845   have "norm (x * x ^ n) \<le> norm x * norm (x ^ n)"
```
```   846     by (rule norm_mult_ineq)
```
```   847   also from Suc have "\<dots> \<le> norm x * norm x ^ n"
```
```   848     using norm_ge_zero by (rule mult_left_mono)
```
```   849   finally show "norm (x ^ Suc n) \<le> norm x ^ Suc n"
```
```   850     by simp
```
```   851 qed
```
```   852
```
```   853 lemma norm_power:
```
```   854   fixes x :: "'a::{real_normed_div_algebra}"
```
```   855   shows "norm (x ^ n) = norm x ^ n"
```
```   856 by (induct n) (simp_all add: norm_mult)
```
```   857
```
```   858 text{*Despite a superficial resemblance, @{text norm_eq_1} is not relevant.*}
```
```   859 lemma square_norm_one:
```
```   860   fixes x :: "'a::real_normed_div_algebra"
```
```   861   assumes "x^2 = 1" shows "norm x = 1"
```
```   862   by (metis assms norm_minus_cancel norm_one power2_eq_1_iff)
```
```   863
```
```   864 lemma norm_less_p1:
```
```   865   fixes x :: "'a::real_normed_algebra_1"
```
```   866   shows "norm x < norm (of_real (norm x) + 1 :: 'a)"
```
```   867 proof -
```
```   868   have "norm x < norm (of_real (norm x + 1) :: 'a)"
```
```   869     by (simp add: of_real_def)
```
```   870   then show ?thesis
```
```   871     by simp
```
```   872 qed
```
```   873
```
```   874 lemma setprod_norm:
```
```   875   fixes f :: "'a \<Rightarrow> 'b::{comm_semiring_1,real_normed_div_algebra}"
```
```   876   shows "setprod (\<lambda>x. norm(f x)) A = norm (setprod f A)"
```
```   877   by (induct A rule: infinite_finite_induct) (auto simp: norm_mult)
```
```   878
```
```   879 lemma norm_setprod_le:
```
```   880   "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a :: 'a :: {real_normed_algebra_1, comm_monoid_mult}))"
```
```   881 proof (induction A rule: infinite_finite_induct)
```
```   882   case (insert a A)
```
```   883   then have "norm (setprod f (insert a A)) \<le> norm (f a) * norm (setprod f A)"
```
```   884     by (simp add: norm_mult_ineq)
```
```   885   also have "norm (setprod f A) \<le> (\<Prod>a\<in>A. norm (f a))"
```
```   886     by (rule insert)
```
```   887   finally show ?case
```
```   888     by (simp add: insert mult_left_mono)
```
```   889 qed simp_all
```
```   890
```
```   891 lemma norm_setprod_diff:
```
```   892   fixes z w :: "'i \<Rightarrow> 'a::{real_normed_algebra_1, comm_monoid_mult}"
```
```   893   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>
```
```   894     norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) \<le> (\<Sum>i\<in>I. norm (z i - w i))"
```
```   895 proof (induction I rule: infinite_finite_induct)
```
```   896   case (insert i I)
```
```   897   note insert.hyps[simp]
```
```   898
```
```   899   have "norm ((\<Prod>i\<in>insert i I. z i) - (\<Prod>i\<in>insert i I. w i)) =
```
```   900     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)"
```
```   901     (is "_ = norm (?t1 + ?t2)")
```
```   902     by (auto simp add: field_simps)
```
```   903   also have "... \<le> norm ?t1 + norm ?t2"
```
```   904     by (rule norm_triangle_ineq)
```
```   905   also have "norm ?t1 \<le> norm (\<Prod>i\<in>I. z i) * norm (z i - w i)"
```
```   906     by (rule norm_mult_ineq)
```
```   907   also have "\<dots> \<le> (\<Prod>i\<in>I. norm (z i)) * norm(z i - w i)"
```
```   908     by (rule mult_right_mono) (auto intro: norm_setprod_le)
```
```   909   also have "(\<Prod>i\<in>I. norm (z i)) \<le> (\<Prod>i\<in>I. 1)"
```
```   910     by (intro setprod_mono) (auto intro!: insert)
```
```   911   also have "norm ?t2 \<le> norm ((\<Prod>i\<in>I. z i) - (\<Prod>i\<in>I. w i)) * norm (w i)"
```
```   912     by (rule norm_mult_ineq)
```
```   913   also have "norm (w i) \<le> 1"
```
```   914     by (auto intro: insert)
```
```   915   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))"
```
```   916     using insert by auto
```
```   917   finally show ?case
```
```   918     by (auto simp add: ac_simps mult_right_mono mult_left_mono)
```
```   919 qed simp_all
```
```   920
```
```   921 lemma norm_power_diff:
```
```   922   fixes z w :: "'a::{real_normed_algebra_1, comm_monoid_mult}"
```
```   923   assumes "norm z \<le> 1" "norm w \<le> 1"
```
```   924   shows "norm (z^m - w^m) \<le> m * norm (z - w)"
```
```   925 proof -
```
```   926   have "norm (z^m - w^m) = norm ((\<Prod> i < m. z) - (\<Prod> i < m. w))"
```
```   927     by (simp add: setprod_constant)
```
```   928   also have "\<dots> \<le> (\<Sum>i<m. norm (z - w))"
```
```   929     by (intro norm_setprod_diff) (auto simp add: assms)
```
```   930   also have "\<dots> = m * norm (z - w)"
```
```   931     by (simp add: real_of_nat_def)
```
```   932   finally show ?thesis .
```
```   933 qed
```
```   934
```
```   935 subsection {* Metric spaces *}
```
```   936
```
```   937 class metric_space = open_dist +
```
```   938   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```   939   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```   940 begin
```
```   941
```
```   942 lemma dist_self [simp]: "dist x x = 0"
```
```   943 by simp
```
```   944
```
```   945 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```   946 using dist_triangle2 [of x x y] by simp
```
```   947
```
```   948 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```   949 by (simp add: less_le)
```
```   950
```
```   951 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```   952 by (simp add: not_less)
```
```   953
```
```   954 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```   955 by (simp add: le_less)
```
```   956
```
```   957 lemma dist_commute: "dist x y = dist y x"
```
```   958 proof (rule order_antisym)
```
```   959   show "dist x y \<le> dist y x"
```
```   960     using dist_triangle2 [of x y x] by simp
```
```   961   show "dist y x \<le> dist x y"
```
```   962     using dist_triangle2 [of y x y] by simp
```
```   963 qed
```
```   964
```
```   965 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```   966 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```   967
```
```   968 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```   969 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```   970
```
```   971 lemma dist_triangle_alt:
```
```   972   shows "dist y z <= dist x y + dist x z"
```
```   973 by (rule dist_triangle3)
```
```   974
```
```   975 lemma dist_pos_lt:
```
```   976   shows "x \<noteq> y ==> 0 < dist x y"
```
```   977 by (simp add: zero_less_dist_iff)
```
```   978
```
```   979 lemma dist_nz:
```
```   980   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
```
```   981 by (simp add: zero_less_dist_iff)
```
```   982
```
```   983 lemma dist_triangle_le:
```
```   984   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
```
```   985 by (rule order_trans [OF dist_triangle2])
```
```   986
```
```   987 lemma dist_triangle_lt:
```
```   988   shows "dist x z + dist y z < e ==> dist x y < e"
```
```   989 by (rule le_less_trans [OF dist_triangle2])
```
```   990
```
```   991 lemma dist_triangle_half_l:
```
```   992   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   993 by (rule dist_triangle_lt [where z=y], simp)
```
```   994
```
```   995 lemma dist_triangle_half_r:
```
```   996   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```   997 by (rule dist_triangle_half_l, simp_all add: dist_commute)
```
```   998
```
```   999 subclass topological_space
```
```  1000 proof
```
```  1001   have "\<exists>e::real. 0 < e"
```
```  1002     by (fast intro: zero_less_one)
```
```  1003   then show "open UNIV"
```
```  1004     unfolding open_dist by simp
```
```  1005 next
```
```  1006   fix S T assume "open S" "open T"
```
```  1007   then show "open (S \<inter> T)"
```
```  1008     unfolding open_dist
```
```  1009     apply clarify
```
```  1010     apply (drule (1) bspec)+
```
```  1011     apply (clarify, rename_tac r s)
```
```  1012     apply (rule_tac x="min r s" in exI, simp)
```
```  1013     done
```
```  1014 next
```
```  1015   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```  1016     unfolding open_dist by fast
```
```  1017 qed
```
```  1018
```
```  1019 lemma open_ball: "open {y. dist x y < d}"
```
```  1020 proof (unfold open_dist, intro ballI)
```
```  1021   fix y assume *: "y \<in> {y. dist x y < d}"
```
```  1022   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
```
```  1023     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
```
```  1024 qed
```
```  1025
```
```  1026 subclass first_countable_topology
```
```  1027 proof
```
```  1028   fix x
```
```  1029   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))"
```
```  1030   proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
```
```  1031     fix S assume "open S" "x \<in> S"
```
```  1032     then obtain e where e: "0 < e" and "{y. dist x y < e} \<subseteq> S"
```
```  1033       by (auto simp: open_dist subset_eq dist_commute)
```
```  1034     moreover
```
```  1035     from e obtain i where "inverse (Suc i) < e"
```
```  1036       by (auto dest!: reals_Archimedean)
```
```  1037     then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
```
```  1038       by auto
```
```  1039     ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
```
```  1040       by blast
```
```  1041   qed (auto intro: open_ball)
```
```  1042 qed
```
```  1043
```
```  1044 end
```
```  1045
```
```  1046 instance metric_space \<subseteq> t2_space
```
```  1047 proof
```
```  1048   fix x y :: "'a::metric_space"
```
```  1049   assume xy: "x \<noteq> y"
```
```  1050   let ?U = "{y'. dist x y' < dist x y / 2}"
```
```  1051   let ?V = "{x'. dist y x' < dist x y / 2}"
```
```  1052   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
```
```  1053                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```  1054   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```  1055     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
```
```  1056     using open_ball[of _ "dist x y / 2"] by auto
```
```  1057   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```  1058     by blast
```
```  1059 qed
```
```  1060
```
```  1061 text {* Every normed vector space is a metric space. *}
```
```  1062
```
```  1063 instance real_normed_vector < metric_space
```
```  1064 proof
```
```  1065   fix x y :: 'a show "dist x y = 0 \<longleftrightarrow> x = y"
```
```  1066     unfolding dist_norm by simp
```
```  1067 next
```
```  1068   fix x y z :: 'a show "dist x y \<le> dist x z + dist y z"
```
```  1069     unfolding dist_norm
```
```  1070     using norm_triangle_ineq4 [of "x - z" "y - z"] by simp
```
```  1071 qed
```
```  1072
```
```  1073 subsection {* Class instances for real numbers *}
```
```  1074
```
```  1075 instantiation real :: real_normed_field
```
```  1076 begin
```
```  1077
```
```  1078 definition dist_real_def:
```
```  1079   "dist x y = \<bar>x - y\<bar>"
```
```  1080
```
```  1081 definition open_real_def [code del]:
```
```  1082   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```  1083
```
```  1084 definition real_norm_def [simp]:
```
```  1085   "norm r = \<bar>r\<bar>"
```
```  1086
```
```  1087 instance
```
```  1088 apply (intro_classes, unfold real_norm_def real_scaleR_def)
```
```  1089 apply (rule dist_real_def)
```
```  1090 apply (rule open_real_def)
```
```  1091 apply (simp add: sgn_real_def)
```
```  1092 apply (rule abs_eq_0)
```
```  1093 apply (rule abs_triangle_ineq)
```
```  1094 apply (rule abs_mult)
```
```  1095 apply (rule abs_mult)
```
```  1096 done
```
```  1097
```
```  1098 end
```
```  1099
```
```  1100 declare [[code abort: "open :: real set \<Rightarrow> bool"]]
```
```  1101
```
```  1102 instance real :: linorder_topology
```
```  1103 proof
```
```  1104   show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
```
```  1105   proof (rule ext, safe)
```
```  1106     fix S :: "real set" assume "open S"
```
```  1107     then obtain f where "\<forall>x\<in>S. 0 < f x \<and> (\<forall>y. dist y x < f x \<longrightarrow> y \<in> S)"
```
```  1108       unfolding open_real_def bchoice_iff ..
```
```  1109     then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
```
```  1110       by (fastforce simp: dist_real_def)
```
```  1111     show "generate_topology (range lessThan \<union> range greaterThan) S"
```
```  1112       apply (subst *)
```
```  1113       apply (intro generate_topology_Union generate_topology.Int)
```
```  1114       apply (auto intro: generate_topology.Basis)
```
```  1115       done
```
```  1116   next
```
```  1117     fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
```
```  1118     moreover have "\<And>a::real. open {..<a}"
```
```  1119       unfolding open_real_def dist_real_def
```
```  1120     proof clarify
```
```  1121       fix x a :: real assume "x < a"
```
```  1122       hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```  1123       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```  1124     qed
```
```  1125     moreover have "\<And>a::real. open {a <..}"
```
```  1126       unfolding open_real_def dist_real_def
```
```  1127     proof clarify
```
```  1128       fix x a :: real assume "a < x"
```
```  1129       hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```  1130       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```  1131     qed
```
```  1132     ultimately show "open S"
```
```  1133       by induct auto
```
```  1134   qed
```
```  1135 qed
```
```  1136
```
```  1137 instance real :: linear_continuum_topology ..
```
```  1138
```
```  1139 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
```
```  1140 lemmas open_real_lessThan = open_lessThan[where 'a=real]
```
```  1141 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
```
```  1142 lemmas closed_real_atMost = closed_atMost[where 'a=real]
```
```  1143 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
```
```  1144 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
```
```  1145
```
```  1146 subsection {* Extra type constraints *}
```
```  1147
```
```  1148 text {* Only allow @{term "open"} in class @{text topological_space}. *}
```
```  1149
```
```  1150 setup {* Sign.add_const_constraint
```
```  1151   (@{const_name "open"}, SOME @{typ "'a::topological_space set \<Rightarrow> bool"}) *}
```
```  1152
```
```  1153 text {* Only allow @{term dist} in class @{text metric_space}. *}
```
```  1154
```
```  1155 setup {* Sign.add_const_constraint
```
```  1156   (@{const_name dist}, SOME @{typ "'a::metric_space \<Rightarrow> 'a \<Rightarrow> real"}) *}
```
```  1157
```
```  1158 text {* Only allow @{term norm} in class @{text real_normed_vector}. *}
```
```  1159
```
```  1160 setup {* Sign.add_const_constraint
```
```  1161   (@{const_name norm}, SOME @{typ "'a::real_normed_vector \<Rightarrow> real"}) *}
```
```  1162
```
```  1163 subsection {* Sign function *}
```
```  1164
```
```  1165 lemma norm_sgn:
```
```  1166   "norm (sgn(x::'a::real_normed_vector)) = (if x = 0 then 0 else 1)"
```
```  1167 by (simp add: sgn_div_norm)
```
```  1168
```
```  1169 lemma sgn_zero [simp]: "sgn(0::'a::real_normed_vector) = 0"
```
```  1170 by (simp add: sgn_div_norm)
```
```  1171
```
```  1172 lemma sgn_zero_iff: "(sgn(x::'a::real_normed_vector) = 0) = (x = 0)"
```
```  1173 by (simp add: sgn_div_norm)
```
```  1174
```
```  1175 lemma sgn_minus: "sgn (- x) = - sgn(x::'a::real_normed_vector)"
```
```  1176 by (simp add: sgn_div_norm)
```
```  1177
```
```  1178 lemma sgn_scaleR:
```
```  1179   "sgn (scaleR r x) = scaleR (sgn r) (sgn(x::'a::real_normed_vector))"
```
```  1180 by (simp add: sgn_div_norm ac_simps)
```
```  1181
```
```  1182 lemma sgn_one [simp]: "sgn (1::'a::real_normed_algebra_1) = 1"
```
```  1183 by (simp add: sgn_div_norm)
```
```  1184
```
```  1185 lemma sgn_of_real:
```
```  1186   "sgn (of_real r::'a::real_normed_algebra_1) = of_real (sgn r)"
```
```  1187 unfolding of_real_def by (simp only: sgn_scaleR sgn_one)
```
```  1188
```
```  1189 lemma sgn_mult:
```
```  1190   fixes x y :: "'a::real_normed_div_algebra"
```
```  1191   shows "sgn (x * y) = sgn x * sgn y"
```
```  1192 by (simp add: sgn_div_norm norm_mult mult.commute)
```
```  1193
```
```  1194 lemma real_sgn_eq: "sgn (x::real) = x / \<bar>x\<bar>"
```
```  1195 by (simp add: sgn_div_norm divide_inverse)
```
```  1196
```
```  1197 lemma real_sgn_pos: "0 < (x::real) \<Longrightarrow> sgn x = 1"
```
```  1198 unfolding real_sgn_eq by simp
```
```  1199
```
```  1200 lemma real_sgn_neg: "(x::real) < 0 \<Longrightarrow> sgn x = -1"
```
```  1201 unfolding real_sgn_eq by simp
```
```  1202
```
```  1203 lemma zero_le_sgn_iff [simp]: "0 \<le> sgn x \<longleftrightarrow> 0 \<le> (x::real)"
```
```  1204   by (cases "0::real" x rule: linorder_cases) simp_all
```
```  1205
```
```  1206 lemma zero_less_sgn_iff [simp]: "0 < sgn x \<longleftrightarrow> 0 < (x::real)"
```
```  1207   by (cases "0::real" x rule: linorder_cases) simp_all
```
```  1208
```
```  1209 lemma sgn_le_0_iff [simp]: "sgn x \<le> 0 \<longleftrightarrow> (x::real) \<le> 0"
```
```  1210   by (cases "0::real" x rule: linorder_cases) simp_all
```
```  1211
```
```  1212 lemma sgn_less_0_iff [simp]: "sgn x < 0 \<longleftrightarrow> (x::real) < 0"
```
```  1213   by (cases "0::real" x rule: linorder_cases) simp_all
```
```  1214
```
```  1215 lemma norm_conv_dist: "norm x = dist x 0"
```
```  1216   unfolding dist_norm by simp
```
```  1217
```
```  1218 subsection {* Bounded Linear and Bilinear Operators *}
```
```  1219
```
```  1220 locale linear = additive f for f :: "'a::real_vector \<Rightarrow> 'b::real_vector" +
```
```  1221   assumes scaleR: "f (scaleR r x) = scaleR r (f x)"
```
```  1222
```
```  1223 lemma linearI:
```
```  1224   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1225   assumes "\<And>c x. f (c *\<^sub>R x) = c *\<^sub>R f x"
```
```  1226   shows "linear f"
```
```  1227   by default (rule assms)+
```
```  1228
```
```  1229 locale bounded_linear = linear f for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector" +
```
```  1230   assumes bounded: "\<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1231 begin
```
```  1232
```
```  1233 lemma pos_bounded:
```
```  1234   "\<exists>K>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1235 proof -
```
```  1236   obtain K where K: "\<And>x. norm (f x) \<le> norm x * K"
```
```  1237     using bounded by fast
```
```  1238   show ?thesis
```
```  1239   proof (intro exI impI conjI allI)
```
```  1240     show "0 < max 1 K"
```
```  1241       by (rule order_less_le_trans [OF zero_less_one max.cobounded1])
```
```  1242   next
```
```  1243     fix x
```
```  1244     have "norm (f x) \<le> norm x * K" using K .
```
```  1245     also have "\<dots> \<le> norm x * max 1 K"
```
```  1246       by (rule mult_left_mono [OF max.cobounded2 norm_ge_zero])
```
```  1247     finally show "norm (f x) \<le> norm x * max 1 K" .
```
```  1248   qed
```
```  1249 qed
```
```  1250
```
```  1251 lemma nonneg_bounded:
```
```  1252   "\<exists>K\<ge>0. \<forall>x. norm (f x) \<le> norm x * K"
```
```  1253 proof -
```
```  1254   from pos_bounded
```
```  1255   show ?thesis by (auto intro: order_less_imp_le)
```
```  1256 qed
```
```  1257
```
```  1258 lemma linear: "linear f" ..
```
```  1259
```
```  1260 end
```
```  1261
```
```  1262 lemma bounded_linear_intro:
```
```  1263   assumes "\<And>x y. f (x + y) = f x + f y"
```
```  1264   assumes "\<And>r x. f (scaleR r x) = scaleR r (f x)"
```
```  1265   assumes "\<And>x. norm (f x) \<le> norm x * K"
```
```  1266   shows "bounded_linear f"
```
```  1267   by default (fast intro: assms)+
```
```  1268
```
```  1269 locale bounded_bilinear =
```
```  1270   fixes prod :: "['a::real_normed_vector, 'b::real_normed_vector]
```
```  1271                  \<Rightarrow> 'c::real_normed_vector"
```
```  1272     (infixl "**" 70)
```
```  1273   assumes add_left: "prod (a + a') b = prod a b + prod a' b"
```
```  1274   assumes add_right: "prod a (b + b') = prod a b + prod a b'"
```
```  1275   assumes scaleR_left: "prod (scaleR r a) b = scaleR r (prod a b)"
```
```  1276   assumes scaleR_right: "prod a (scaleR r b) = scaleR r (prod a b)"
```
```  1277   assumes bounded: "\<exists>K. \<forall>a b. norm (prod a b) \<le> norm a * norm b * K"
```
```  1278 begin
```
```  1279
```
```  1280 lemma pos_bounded:
```
```  1281   "\<exists>K>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1282 apply (cut_tac bounded, erule exE)
```
```  1283 apply (rule_tac x="max 1 K" in exI, safe)
```
```  1284 apply (rule order_less_le_trans [OF zero_less_one max.cobounded1])
```
```  1285 apply (drule spec, drule spec, erule order_trans)
```
```  1286 apply (rule mult_left_mono [OF max.cobounded2])
```
```  1287 apply (intro mult_nonneg_nonneg norm_ge_zero)
```
```  1288 done
```
```  1289
```
```  1290 lemma nonneg_bounded:
```
```  1291   "\<exists>K\<ge>0. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
```
```  1292 proof -
```
```  1293   from pos_bounded
```
```  1294   show ?thesis by (auto intro: order_less_imp_le)
```
```  1295 qed
```
```  1296
```
```  1297 lemma additive_right: "additive (\<lambda>b. prod a b)"
```
```  1298 by (rule additive.intro, rule add_right)
```
```  1299
```
```  1300 lemma additive_left: "additive (\<lambda>a. prod a b)"
```
```  1301 by (rule additive.intro, rule add_left)
```
```  1302
```
```  1303 lemma zero_left: "prod 0 b = 0"
```
```  1304 by (rule additive.zero [OF additive_left])
```
```  1305
```
```  1306 lemma zero_right: "prod a 0 = 0"
```
```  1307 by (rule additive.zero [OF additive_right])
```
```  1308
```
```  1309 lemma minus_left: "prod (- a) b = - prod a b"
```
```  1310 by (rule additive.minus [OF additive_left])
```
```  1311
```
```  1312 lemma minus_right: "prod a (- b) = - prod a b"
```
```  1313 by (rule additive.minus [OF additive_right])
```
```  1314
```
```  1315 lemma diff_left:
```
```  1316   "prod (a - a') b = prod a b - prod a' b"
```
```  1317 by (rule additive.diff [OF additive_left])
```
```  1318
```
```  1319 lemma diff_right:
```
```  1320   "prod a (b - b') = prod a b - prod a b'"
```
```  1321 by (rule additive.diff [OF additive_right])
```
```  1322
```
```  1323 lemma bounded_linear_left:
```
```  1324   "bounded_linear (\<lambda>a. a ** b)"
```
```  1325 apply (cut_tac bounded, safe)
```
```  1326 apply (rule_tac K="norm b * K" in bounded_linear_intro)
```
```  1327 apply (rule add_left)
```
```  1328 apply (rule scaleR_left)
```
```  1329 apply (simp add: ac_simps)
```
```  1330 done
```
```  1331
```
```  1332 lemma bounded_linear_right:
```
```  1333   "bounded_linear (\<lambda>b. a ** b)"
```
```  1334 apply (cut_tac bounded, safe)
```
```  1335 apply (rule_tac K="norm a * K" in bounded_linear_intro)
```
```  1336 apply (rule add_right)
```
```  1337 apply (rule scaleR_right)
```
```  1338 apply (simp add: ac_simps)
```
```  1339 done
```
```  1340
```
```  1341 lemma prod_diff_prod:
```
```  1342   "(x ** y - a ** b) = (x - a) ** (y - b) + (x - a) ** b + a ** (y - b)"
```
```  1343 by (simp add: diff_left diff_right)
```
```  1344
```
```  1345 end
```
```  1346
```
```  1347 lemma bounded_linear_ident[simp]: "bounded_linear (\<lambda>x. x)"
```
```  1348   by default (auto intro!: exI[of _ 1])
```
```  1349
```
```  1350 lemma bounded_linear_zero[simp]: "bounded_linear (\<lambda>x. 0)"
```
```  1351   by default (auto intro!: exI[of _ 1])
```
```  1352
```
```  1353 lemma bounded_linear_add:
```
```  1354   assumes "bounded_linear f"
```
```  1355   assumes "bounded_linear g"
```
```  1356   shows "bounded_linear (\<lambda>x. f x + g x)"
```
```  1357 proof -
```
```  1358   interpret f: bounded_linear f by fact
```
```  1359   interpret g: bounded_linear g by fact
```
```  1360   show ?thesis
```
```  1361   proof
```
```  1362     from f.bounded obtain Kf where Kf: "\<And>x. norm (f x) \<le> norm x * Kf" by blast
```
```  1363     from g.bounded obtain Kg where Kg: "\<And>x. norm (g x) \<le> norm x * Kg" by blast
```
```  1364     show "\<exists>K. \<forall>x. norm (f x + g x) \<le> norm x * K"
```
```  1365       using add_mono[OF Kf Kg]
```
```  1366       by (intro exI[of _ "Kf + Kg"]) (auto simp: field_simps intro: norm_triangle_ineq order_trans)
```
```  1367   qed (simp_all add: f.add g.add f.scaleR g.scaleR scaleR_right_distrib)
```
```  1368 qed
```
```  1369
```
```  1370 lemma bounded_linear_minus:
```
```  1371   assumes "bounded_linear f"
```
```  1372   shows "bounded_linear (\<lambda>x. - f x)"
```
```  1373 proof -
```
```  1374   interpret f: bounded_linear f by fact
```
```  1375   show ?thesis apply (unfold_locales)
```
```  1376     apply (simp add: f.add)
```
```  1377     apply (simp add: f.scaleR)
```
```  1378     apply (simp add: f.bounded)
```
```  1379     done
```
```  1380 qed
```
```  1381
```
```  1382 lemma bounded_linear_compose:
```
```  1383   assumes "bounded_linear f"
```
```  1384   assumes "bounded_linear g"
```
```  1385   shows "bounded_linear (\<lambda>x. f (g x))"
```
```  1386 proof -
```
```  1387   interpret f: bounded_linear f by fact
```
```  1388   interpret g: bounded_linear g by fact
```
```  1389   show ?thesis proof (unfold_locales)
```
```  1390     fix x y show "f (g (x + y)) = f (g x) + f (g y)"
```
```  1391       by (simp only: f.add g.add)
```
```  1392   next
```
```  1393     fix r x show "f (g (scaleR r x)) = scaleR r (f (g x))"
```
```  1394       by (simp only: f.scaleR g.scaleR)
```
```  1395   next
```
```  1396     from f.pos_bounded
```
```  1397     obtain Kf where f: "\<And>x. norm (f x) \<le> norm x * Kf" and Kf: "0 < Kf" by fast
```
```  1398     from g.pos_bounded
```
```  1399     obtain Kg where g: "\<And>x. norm (g x) \<le> norm x * Kg" by fast
```
```  1400     show "\<exists>K. \<forall>x. norm (f (g x)) \<le> norm x * K"
```
```  1401     proof (intro exI allI)
```
```  1402       fix x
```
```  1403       have "norm (f (g x)) \<le> norm (g x) * Kf"
```
```  1404         using f .
```
```  1405       also have "\<dots> \<le> (norm x * Kg) * Kf"
```
```  1406         using g Kf [THEN order_less_imp_le] by (rule mult_right_mono)
```
```  1407       also have "(norm x * Kg) * Kf = norm x * (Kg * Kf)"
```
```  1408         by (rule mult.assoc)
```
```  1409       finally show "norm (f (g x)) \<le> norm x * (Kg * Kf)" .
```
```  1410     qed
```
```  1411   qed
```
```  1412 qed
```
```  1413
```
```  1414 lemma bounded_bilinear_mult:
```
```  1415   "bounded_bilinear (op * :: 'a \<Rightarrow> 'a \<Rightarrow> 'a::real_normed_algebra)"
```
```  1416 apply (rule bounded_bilinear.intro)
```
```  1417 apply (rule distrib_right)
```
```  1418 apply (rule distrib_left)
```
```  1419 apply (rule mult_scaleR_left)
```
```  1420 apply (rule mult_scaleR_right)
```
```  1421 apply (rule_tac x="1" in exI)
```
```  1422 apply (simp add: norm_mult_ineq)
```
```  1423 done
```
```  1424
```
```  1425 lemma bounded_linear_mult_left:
```
```  1426   "bounded_linear (\<lambda>x::'a::real_normed_algebra. x * y)"
```
```  1427   using bounded_bilinear_mult
```
```  1428   by (rule bounded_bilinear.bounded_linear_left)
```
```  1429
```
```  1430 lemma bounded_linear_mult_right:
```
```  1431   "bounded_linear (\<lambda>y::'a::real_normed_algebra. x * y)"
```
```  1432   using bounded_bilinear_mult
```
```  1433   by (rule bounded_bilinear.bounded_linear_right)
```
```  1434
```
```  1435 lemmas bounded_linear_mult_const =
```
```  1436   bounded_linear_mult_left [THEN bounded_linear_compose]
```
```  1437
```
```  1438 lemmas bounded_linear_const_mult =
```
```  1439   bounded_linear_mult_right [THEN bounded_linear_compose]
```
```  1440
```
```  1441 lemma bounded_linear_divide:
```
```  1442   "bounded_linear (\<lambda>x::'a::real_normed_field. x / y)"
```
```  1443   unfolding divide_inverse by (rule bounded_linear_mult_left)
```
```  1444
```
```  1445 lemma bounded_bilinear_scaleR: "bounded_bilinear scaleR"
```
```  1446 apply (rule bounded_bilinear.intro)
```
```  1447 apply (rule scaleR_left_distrib)
```
```  1448 apply (rule scaleR_right_distrib)
```
```  1449 apply simp
```
```  1450 apply (rule scaleR_left_commute)
```
```  1451 apply (rule_tac x="1" in exI, simp)
```
```  1452 done
```
```  1453
```
```  1454 lemma bounded_linear_scaleR_left: "bounded_linear (\<lambda>r. scaleR r x)"
```
```  1455   using bounded_bilinear_scaleR
```
```  1456   by (rule bounded_bilinear.bounded_linear_left)
```
```  1457
```
```  1458 lemma bounded_linear_scaleR_right: "bounded_linear (\<lambda>x. scaleR r x)"
```
```  1459   using bounded_bilinear_scaleR
```
```  1460   by (rule bounded_bilinear.bounded_linear_right)
```
```  1461
```
```  1462 lemma bounded_linear_of_real: "bounded_linear (\<lambda>r. of_real r)"
```
```  1463   unfolding of_real_def by (rule bounded_linear_scaleR_left)
```
```  1464
```
```  1465 lemma real_bounded_linear:
```
```  1466   fixes f :: "real \<Rightarrow> real"
```
```  1467   shows "bounded_linear f \<longleftrightarrow> (\<exists>c::real. f = (\<lambda>x. x * c))"
```
```  1468 proof -
```
```  1469   { fix x assume "bounded_linear f"
```
```  1470     then interpret bounded_linear f .
```
```  1471     from scaleR[of x 1] have "f x = x * f 1"
```
```  1472       by simp }
```
```  1473   then show ?thesis
```
```  1474     by (auto intro: exI[of _ "f 1"] bounded_linear_mult_left)
```
```  1475 qed
```
```  1476
```
```  1477 instance real_normed_algebra_1 \<subseteq> perfect_space
```
```  1478 proof
```
```  1479   fix x::'a
```
```  1480   show "\<not> open {x}"
```
```  1481     unfolding open_dist dist_norm
```
```  1482     by (clarsimp, rule_tac x="x + of_real (e/2)" in exI, simp)
```
```  1483 qed
```
```  1484
```
```  1485 subsection {* Filters and Limits on Metric Space *}
```
```  1486
```
```  1487 lemma (in metric_space) nhds_metric: "nhds x = (INF e:{0 <..}. principal {y. dist y x < e})"
```
```  1488   unfolding nhds_def
```
```  1489 proof (safe intro!: INF_eq)
```
```  1490   fix S assume "open S" "x \<in> S"
```
```  1491   then obtain e where "{y. dist y x < e} \<subseteq> S" "0 < e"
```
```  1492     by (auto simp: open_dist subset_eq)
```
```  1493   then show "\<exists>e\<in>{0<..}. principal {y. dist y x < e} \<le> principal S"
```
```  1494     by auto
```
```  1495 qed (auto intro!: exI[of _ "{y. dist x y < e}" for e] open_ball simp: dist_commute)
```
```  1496
```
```  1497 lemma (in metric_space) tendsto_iff:
```
```  1498   "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
```
```  1499   unfolding nhds_metric filterlim_INF filterlim_principal by auto
```
```  1500
```
```  1501 lemma (in metric_space) tendstoI: "(\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F) \<Longrightarrow> (f ---> l) F"
```
```  1502   by (auto simp: tendsto_iff)
```
```  1503
```
```  1504 lemma (in metric_space) tendstoD: "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```  1505   by (auto simp: tendsto_iff)
```
```  1506
```
```  1507 lemma (in metric_space) eventually_nhds_metric:
```
```  1508   "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```  1509   unfolding nhds_metric
```
```  1510   by (subst eventually_INF_base)
```
```  1511      (auto simp: eventually_principal Bex_def subset_eq intro: exI[of _ "min a b" for a b])
```
```  1512
```
```  1513 lemma eventually_at:
```
```  1514   fixes a :: "'a :: metric_space"
```
```  1515   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)"
```
```  1516   unfolding eventually_at_filter eventually_nhds_metric by (auto simp: dist_nz)
```
```  1517
```
```  1518 lemma eventually_at_le:
```
```  1519   fixes a :: "'a::metric_space"
```
```  1520   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)"
```
```  1521   unfolding eventually_at_filter eventually_nhds_metric
```
```  1522   apply auto
```
```  1523   apply (rule_tac x="d / 2" in exI)
```
```  1524   apply auto
```
```  1525   done
```
```  1526
```
```  1527 lemma metric_tendsto_imp_tendsto:
```
```  1528   fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
```
```  1529   assumes f: "(f ---> a) F"
```
```  1530   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
```
```  1531   shows "(g ---> b) F"
```
```  1532 proof (rule tendstoI)
```
```  1533   fix e :: real assume "0 < e"
```
```  1534   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
```
```  1535   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
```
```  1536     using le_less_trans by (rule eventually_elim2)
```
```  1537 qed
```
```  1538
```
```  1539 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
```
```  1540   unfolding filterlim_at_top
```
```  1541   apply (intro allI)
```
```  1542   apply (rule_tac c="nat(ceiling (Z + 1))" in eventually_sequentiallyI)
```
```  1543   by linarith
```
```  1544
```
```  1545 subsubsection {* Limits of Sequences *}
```
```  1546
```
```  1547 lemma lim_sequentially: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```  1548   unfolding tendsto_iff eventually_sequentially ..
```
```  1549
```
```  1550 lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
```
```  1551   unfolding lim_sequentially by (metis Suc_leD zero_less_Suc)
```
```  1552
```
```  1553 lemma metric_LIMSEQ_I:
```
```  1554   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
```
```  1555 by (simp add: lim_sequentially)
```
```  1556
```
```  1557 lemma metric_LIMSEQ_D:
```
```  1558   "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```  1559 by (simp add: lim_sequentially)
```
```  1560
```
```  1561
```
```  1562 subsubsection {* Limits of Functions *}
```
```  1563
```
```  1564 lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
```
```  1565      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
```
```  1566         --> dist (f x) L < r)"
```
```  1567   unfolding tendsto_iff eventually_at by simp
```
```  1568
```
```  1569 lemma metric_LIM_I:
```
```  1570   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
```
```  1571     \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
```
```  1572 by (simp add: LIM_def)
```
```  1573
```
```  1574 lemma metric_LIM_D:
```
```  1575   "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
```
```  1576     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
```
```  1577 by (simp add: LIM_def)
```
```  1578
```
```  1579 lemma metric_LIM_imp_LIM:
```
```  1580   assumes f: "f -- a --> (l::'a::metric_space)"
```
```  1581   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
```
```  1582   shows "g -- a --> (m::'b::metric_space)"
```
```  1583   by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
```
```  1584
```
```  1585 lemma metric_LIM_equal2:
```
```  1586   assumes 1: "0 < R"
```
```  1587   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1588   shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
```
```  1589 apply (rule topological_tendstoI)
```
```  1590 apply (drule (2) topological_tendstoD)
```
```  1591 apply (simp add: eventually_at, safe)
```
```  1592 apply (rule_tac x="min d R" in exI, safe)
```
```  1593 apply (simp add: 1)
```
```  1594 apply (simp add: 2)
```
```  1595 done
```
```  1596
```
```  1597 lemma metric_LIM_compose2:
```
```  1598   assumes f: "f -- (a::'a::metric_space) --> b"
```
```  1599   assumes g: "g -- b --> c"
```
```  1600   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```  1601   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  1602   using inj
```
```  1603   by (intro tendsto_compose_eventually[OF g f]) (auto simp: eventually_at)
```
```  1604
```
```  1605 lemma metric_isCont_LIM_compose2:
```
```  1606   fixes f :: "'a :: metric_space \<Rightarrow> _"
```
```  1607   assumes f [unfolded isCont_def]: "isCont f a"
```
```  1608   assumes g: "g -- f a --> l"
```
```  1609   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```  1610   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```  1611 by (rule metric_LIM_compose2 [OF f g inj])
```
```  1612
```
```  1613 subsection {* Complete metric spaces *}
```
```  1614
```
```  1615 subsection {* Cauchy sequences *}
```
```  1616
```
```  1617 definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```  1618   "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```  1619
```
```  1620 subsection {* Cauchy Sequences *}
```
```  1621
```
```  1622 lemma metric_CauchyI:
```
```  1623   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```  1624   by (simp add: Cauchy_def)
```
```  1625
```
```  1626 lemma metric_CauchyD:
```
```  1627   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```  1628   by (simp add: Cauchy_def)
```
```  1629
```
```  1630 lemma metric_Cauchy_iff2:
```
```  1631   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
```
```  1632 apply (simp add: Cauchy_def, auto)
```
```  1633 apply (drule reals_Archimedean, safe)
```
```  1634 apply (drule_tac x = n in spec, auto)
```
```  1635 apply (rule_tac x = M in exI, auto)
```
```  1636 apply (drule_tac x = m in spec, simp)
```
```  1637 apply (drule_tac x = na in spec, auto)
```
```  1638 done
```
```  1639
```
```  1640 lemma Cauchy_iff2:
```
```  1641   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
```
```  1642   unfolding metric_Cauchy_iff2 dist_real_def ..
```
```  1643
```
```  1644 lemma Cauchy_subseq_Cauchy:
```
```  1645   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```  1646 apply (auto simp add: Cauchy_def)
```
```  1647 apply (drule_tac x=e in spec, clarify)
```
```  1648 apply (rule_tac x=M in exI, clarify)
```
```  1649 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```  1650 done
```
```  1651
```
```  1652 theorem LIMSEQ_imp_Cauchy:
```
```  1653   assumes X: "X ----> a" shows "Cauchy X"
```
```  1654 proof (rule metric_CauchyI)
```
```  1655   fix e::real assume "0 < e"
```
```  1656   hence "0 < e/2" by simp
```
```  1657   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```  1658   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```  1659   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```  1660   proof (intro exI allI impI)
```
```  1661     fix m assume "N \<le> m"
```
```  1662     hence m: "dist (X m) a < e/2" using N by fast
```
```  1663     fix n assume "N \<le> n"
```
```  1664     hence n: "dist (X n) a < e/2" using N by fast
```
```  1665     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```  1666       by (rule dist_triangle2)
```
```  1667     also from m n have "\<dots> < e" by simp
```
```  1668     finally show "dist (X m) (X n) < e" .
```
```  1669   qed
```
```  1670 qed
```
```  1671
```
```  1672 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```  1673 unfolding convergent_def
```
```  1674 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```  1675
```
```  1676 subsubsection {* Cauchy Sequences are Convergent *}
```
```  1677
```
```  1678 class complete_space = metric_space +
```
```  1679   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```  1680
```
```  1681 lemma Cauchy_convergent_iff:
```
```  1682   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```  1683   shows "Cauchy X = convergent X"
```
```  1684 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```  1685
```
```  1686 subsection {* The set of real numbers is a complete metric space *}
```
```  1687
```
```  1688 text {*
```
```  1689 Proof that Cauchy sequences converge based on the one from
```
```  1690 @{url "http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html"}
```
```  1691 *}
```
```  1692
```
```  1693 text {*
```
```  1694   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```  1695   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```  1696 *}
```
```  1697
```
```  1698 lemma increasing_LIMSEQ:
```
```  1699   fixes f :: "nat \<Rightarrow> real"
```
```  1700   assumes inc: "\<And>n. f n \<le> f (Suc n)"
```
```  1701       and bdd: "\<And>n. f n \<le> l"
```
```  1702       and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
```
```  1703   shows "f ----> l"
```
```  1704 proof (rule increasing_tendsto)
```
```  1705   fix x assume "x < l"
```
```  1706   with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
```
```  1707     by auto
```
```  1708   from en[OF `0 < e`] obtain n where "l - e \<le> f n"
```
```  1709     by (auto simp: field_simps)
```
```  1710   with `e < l - x` `0 < e` have "x < f n" by simp
```
```  1711   with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
```
```  1712     by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
```
```  1713 qed (insert bdd, auto)
```
```  1714
```
```  1715 lemma real_Cauchy_convergent:
```
```  1716   fixes X :: "nat \<Rightarrow> real"
```
```  1717   assumes X: "Cauchy X"
```
```  1718   shows "convergent X"
```
```  1719 proof -
```
```  1720   def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```  1721   then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
```
```  1722
```
```  1723   { fix N x assume N: "\<forall>n\<ge>N. X n < x"
```
```  1724   fix y::real assume "y \<in> S"
```
```  1725   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```  1726     by (simp add: S_def)
```
```  1727   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```  1728   hence "y < X (max M N)" by simp
```
```  1729   also have "\<dots> < x" using N by simp
```
```  1730   finally have "y \<le> x"
```
```  1731     by (rule order_less_imp_le) }
```
```  1732   note bound_isUb = this
```
```  1733
```
```  1734   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
```
```  1735     using X[THEN metric_CauchyD, OF zero_less_one] by auto
```
```  1736   hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
```
```  1737   have [simp]: "S \<noteq> {}"
```
```  1738   proof (intro exI ex_in_conv[THEN iffD1])
```
```  1739     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```  1740       by (simp add: abs_diff_less_iff dist_real_def)
```
```  1741     thus "X N - 1 \<in> S" by (rule mem_S)
```
```  1742   qed
```
```  1743   have [simp]: "bdd_above S"
```
```  1744   proof
```
```  1745     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```  1746       by (simp add: abs_diff_less_iff dist_real_def)
```
```  1747     thus "\<And>s. s \<in> S \<Longrightarrow>  s \<le> X N + 1"
```
```  1748       by (rule bound_isUb)
```
```  1749   qed
```
```  1750   have "X ----> Sup S"
```
```  1751   proof (rule metric_LIMSEQ_I)
```
```  1752   fix r::real assume "0 < r"
```
```  1753   hence r: "0 < r/2" by simp
```
```  1754   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
```
```  1755     using metric_CauchyD [OF X r] by auto
```
```  1756   hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
```
```  1757   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```  1758     by (simp only: dist_real_def abs_diff_less_iff)
```
```  1759
```
```  1760   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```  1761   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```  1762   hence 1: "X N - r/2 \<le> Sup S" by (simp add: cSup_upper)
```
```  1763
```
```  1764   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```  1765   from bound_isUb[OF this]
```
```  1766   have 2: "Sup S \<le> X N + r/2"
```
```  1767     by (intro cSup_least) simp_all
```
```  1768
```
```  1769   show "\<exists>N. \<forall>n\<ge>N. dist (X n) (Sup S) < r"
```
```  1770   proof (intro exI allI impI)
```
```  1771     fix n assume n: "N \<le> n"
```
```  1772     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```  1773     thus "dist (X n) (Sup S) < r" using 1 2
```
```  1774       by (simp add: abs_diff_less_iff dist_real_def)
```
```  1775   qed
```
```  1776   qed
```
```  1777   then show ?thesis unfolding convergent_def by auto
```
```  1778 qed
```
```  1779
```
```  1780 instance real :: complete_space
```
```  1781   by intro_classes (rule real_Cauchy_convergent)
```
```  1782
```
```  1783 class banach = real_normed_vector + complete_space
```
```  1784
```
```  1785 instance real :: banach by default
```
```  1786
```
```  1787 lemma tendsto_at_topI_sequentially:
```
```  1788   fixes f :: "real \<Rightarrow> 'b::first_countable_topology"
```
```  1789   assumes *: "\<And>X. filterlim X at_top sequentially \<Longrightarrow> (\<lambda>n. f (X n)) ----> y"
```
```  1790   shows "(f ---> y) at_top"
```
```  1791 proof -
```
```  1792   from nhds_countable[of y] guess A . note A = this
```
```  1793
```
```  1794   have "\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m"
```
```  1795   proof (rule ccontr)
```
```  1796     assume "\<not> (\<forall>m. \<exists>k. \<forall>x\<ge>k. f x \<in> A m)"
```
```  1797     then obtain m where "\<And>k. \<exists>x\<ge>k. f x \<notin> A m"
```
```  1798       by auto
```
```  1799     then have "\<exists>X. \<forall>n. (f (X n) \<notin> A m) \<and> max n (X n) + 1 \<le> X (Suc n)"
```
```  1800       by (intro dependent_nat_choice) (auto simp del: max.bounded_iff)
```
```  1801     then obtain X where X: "\<And>n. f (X n) \<notin> A m" "\<And>n. max n (X n) + 1 \<le> X (Suc n)"
```
```  1802       by auto
```
```  1803     { fix n have "1 \<le> n \<longrightarrow> real n \<le> X n"
```
```  1804         using X[of "n - 1"] by auto }
```
```  1805     then have "filterlim X at_top sequentially"
```
```  1806       by (force intro!: filterlim_at_top_mono[OF filterlim_real_sequentially]
```
```  1807                 simp: eventually_sequentially)
```
```  1808     from topological_tendstoD[OF *[OF this] A(2, 3), of m] X(1) show False
```
```  1809       by auto
```
```  1810   qed
```
```  1811   then obtain k where "\<And>m x. k m \<le> x \<Longrightarrow> f x \<in> A m"
```
```  1812     by metis
```
```  1813   then show ?thesis
```
```  1814     unfolding at_top_def A
```
```  1815     by (intro filterlim_base[where i=k]) auto
```
```  1816 qed
```
```  1817
```
```  1818 lemma tendsto_at_topI_sequentially_real:
```
```  1819   fixes f :: "real \<Rightarrow> real"
```
```  1820   assumes mono: "mono f"
```
```  1821   assumes limseq: "(\<lambda>n. f (real n)) ----> y"
```
```  1822   shows "(f ---> y) at_top"
```
```  1823 proof (rule tendstoI)
```
```  1824   fix e :: real assume "0 < e"
```
```  1825   with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
```
```  1826     by (auto simp: lim_sequentially dist_real_def)
```
```  1827   { fix x :: real
```
```  1828     obtain n where "x \<le> real_of_nat n"
```
```  1829       using ex_le_of_nat[of x] ..
```
```  1830     note monoD[OF mono this]
```
```  1831     also have "f (real_of_nat n) \<le> y"
```
```  1832       by (rule LIMSEQ_le_const[OF limseq])
```
```  1833          (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
```
```  1834     finally have "f x \<le> y" . }
```
```  1835   note le = this
```
```  1836   have "eventually (\<lambda>x. real N \<le> x) at_top"
```
```  1837     by (rule eventually_ge_at_top)
```
```  1838   then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
```
```  1839   proof eventually_elim
```
```  1840     fix x assume N': "real N \<le> x"
```
```  1841     with N[of N] le have "y - f (real N) < e" by auto
```
```  1842     moreover note monoD[OF mono N']
```
```  1843     ultimately show "dist (f x) y < e"
```
```  1844       using le[of x] by (auto simp: dist_real_def field_simps)
```
```  1845   qed
```
```  1846 qed
```
```  1847
```
```  1848 end
```
```  1849
```