src/HOL/Metric_Spaces.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51478 270b21f3ae0a parent 51474 1e9e68247ad1 child 51521 36fa825e0ea7 permissions -rw-r--r--
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
```     1 (*  Title:      HOL/Metric_Spaces.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Johannes Hölzl
```
```     4 *)
```
```     5
```
```     6 header {* Metric Spaces *}
```
```     7
```
```     8 theory Metric_Spaces
```
```     9 imports RComplete Topological_Spaces
```
```    10 begin
```
```    11
```
```    12
```
```    13 subsection {* Metric spaces *}
```
```    14
```
```    15 class dist =
```
```    16   fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"
```
```    17
```
```    18 class open_dist = "open" + dist +
```
```    19   assumes open_dist: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```    20
```
```    21 class metric_space = open_dist +
```
```    22   assumes dist_eq_0_iff [simp]: "dist x y = 0 \<longleftrightarrow> x = y"
```
```    23   assumes dist_triangle2: "dist x y \<le> dist x z + dist y z"
```
```    24 begin
```
```    25
```
```    26 lemma dist_self [simp]: "dist x x = 0"
```
```    27 by simp
```
```    28
```
```    29 lemma zero_le_dist [simp]: "0 \<le> dist x y"
```
```    30 using dist_triangle2 [of x x y] by simp
```
```    31
```
```    32 lemma zero_less_dist_iff: "0 < dist x y \<longleftrightarrow> x \<noteq> y"
```
```    33 by (simp add: less_le)
```
```    34
```
```    35 lemma dist_not_less_zero [simp]: "\<not> dist x y < 0"
```
```    36 by (simp add: not_less)
```
```    37
```
```    38 lemma dist_le_zero_iff [simp]: "dist x y \<le> 0 \<longleftrightarrow> x = y"
```
```    39 by (simp add: le_less)
```
```    40
```
```    41 lemma dist_commute: "dist x y = dist y x"
```
```    42 proof (rule order_antisym)
```
```    43   show "dist x y \<le> dist y x"
```
```    44     using dist_triangle2 [of x y x] by simp
```
```    45   show "dist y x \<le> dist x y"
```
```    46     using dist_triangle2 [of y x y] by simp
```
```    47 qed
```
```    48
```
```    49 lemma dist_triangle: "dist x z \<le> dist x y + dist y z"
```
```    50 using dist_triangle2 [of x z y] by (simp add: dist_commute)
```
```    51
```
```    52 lemma dist_triangle3: "dist x y \<le> dist a x + dist a y"
```
```    53 using dist_triangle2 [of x y a] by (simp add: dist_commute)
```
```    54
```
```    55 lemma dist_triangle_alt:
```
```    56   shows "dist y z <= dist x y + dist x z"
```
```    57 by (rule dist_triangle3)
```
```    58
```
```    59 lemma dist_pos_lt:
```
```    60   shows "x \<noteq> y ==> 0 < dist x y"
```
```    61 by (simp add: zero_less_dist_iff)
```
```    62
```
```    63 lemma dist_nz:
```
```    64   shows "x \<noteq> y \<longleftrightarrow> 0 < dist x y"
```
```    65 by (simp add: zero_less_dist_iff)
```
```    66
```
```    67 lemma dist_triangle_le:
```
```    68   shows "dist x z + dist y z <= e \<Longrightarrow> dist x y <= e"
```
```    69 by (rule order_trans [OF dist_triangle2])
```
```    70
```
```    71 lemma dist_triangle_lt:
```
```    72   shows "dist x z + dist y z < e ==> dist x y < e"
```
```    73 by (rule le_less_trans [OF dist_triangle2])
```
```    74
```
```    75 lemma dist_triangle_half_l:
```
```    76   shows "dist x1 y < e / 2 \<Longrightarrow> dist x2 y < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```    77 by (rule dist_triangle_lt [where z=y], simp)
```
```    78
```
```    79 lemma dist_triangle_half_r:
```
```    80   shows "dist y x1 < e / 2 \<Longrightarrow> dist y x2 < e / 2 \<Longrightarrow> dist x1 x2 < e"
```
```    81 by (rule dist_triangle_half_l, simp_all add: dist_commute)
```
```    82
```
```    83 subclass topological_space
```
```    84 proof
```
```    85   have "\<exists>e::real. 0 < e"
```
```    86     by (fast intro: zero_less_one)
```
```    87   then show "open UNIV"
```
```    88     unfolding open_dist by simp
```
```    89 next
```
```    90   fix S T assume "open S" "open T"
```
```    91   then show "open (S \<inter> T)"
```
```    92     unfolding open_dist
```
```    93     apply clarify
```
```    94     apply (drule (1) bspec)+
```
```    95     apply (clarify, rename_tac r s)
```
```    96     apply (rule_tac x="min r s" in exI, simp)
```
```    97     done
```
```    98 next
```
```    99   fix K assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
```
```   100     unfolding open_dist by fast
```
```   101 qed
```
```   102
```
```   103 lemma open_ball: "open {y. dist x y < d}"
```
```   104 proof (unfold open_dist, intro ballI)
```
```   105   fix y assume *: "y \<in> {y. dist x y < d}"
```
```   106   then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
```
```   107     by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
```
```   108 qed
```
```   109
```
```   110 subclass first_countable_topology
```
```   111 proof
```
```   112   fix x
```
```   113   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))"
```
```   114   proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
```
```   115     fix S assume "open S" "x \<in> S"
```
```   116     then obtain e where "0 < e" "{y. dist x y < e} \<subseteq> S"
```
```   117       by (auto simp: open_dist subset_eq dist_commute)
```
```   118     moreover
```
```   119     then obtain i where "inverse (Suc i) < e"
```
```   120       by (auto dest!: reals_Archimedean)
```
```   121     then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
```
```   122       by auto
```
```   123     ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
```
```   124       by blast
```
```   125   qed (auto intro: open_ball)
```
```   126 qed
```
```   127
```
```   128 end
```
```   129
```
```   130 instance metric_space \<subseteq> t2_space
```
```   131 proof
```
```   132   fix x y :: "'a::metric_space"
```
```   133   assume xy: "x \<noteq> y"
```
```   134   let ?U = "{y'. dist x y' < dist x y / 2}"
```
```   135   let ?V = "{x'. dist y x' < dist x y / 2}"
```
```   136   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
```
```   137                \<Longrightarrow> \<not>(d x y * 2 < d x z \<and> d z y * 2 < d x z)" by arith
```
```   138   have "open ?U \<and> open ?V \<and> x \<in> ?U \<and> y \<in> ?V \<and> ?U \<inter> ?V = {}"
```
```   139     using dist_pos_lt[OF xy] th0[of dist, OF dist_triangle dist_commute]
```
```   140     using open_ball[of _ "dist x y / 2"] by auto
```
```   141   then show "\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
```
```   142     by blast
```
```   143 qed
```
```   144
```
```   145 lemma eventually_nhds_metric:
```
```   146   fixes a :: "'a :: metric_space"
```
```   147   shows "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
```
```   148 unfolding eventually_nhds open_dist
```
```   149 apply safe
```
```   150 apply fast
```
```   151 apply (rule_tac x="{x. dist x a < d}" in exI, simp)
```
```   152 apply clarsimp
```
```   153 apply (rule_tac x="d - dist x a" in exI, clarsimp)
```
```   154 apply (simp only: less_diff_eq)
```
```   155 apply (erule le_less_trans [OF dist_triangle])
```
```   156 done
```
```   157
```
```   158 lemma eventually_at:
```
```   159   fixes a :: "'a::metric_space"
```
```   160   shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
```
```   161 unfolding at_def eventually_within eventually_nhds_metric by auto
```
```   162
```
```   163 lemma eventually_within_less: (* COPY FROM Topo/eventually_within *)
```
```   164   fixes a :: "'a :: metric_space"
```
```   165   shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
```
```   166   unfolding eventually_within eventually_at dist_nz by auto
```
```   167
```
```   168 lemma eventually_within_le: (* COPY FROM Topo/eventually_within_le *)
```
```   169   fixes a :: "'a :: metric_space"
```
```   170   shows "eventually P (at a within S) \<longleftrightarrow> (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a <= d \<longrightarrow> P x)"
```
```   171   unfolding eventually_within_less by auto (metis dense order_le_less_trans)
```
```   172
```
```   173 lemma tendstoI:
```
```   174   fixes l :: "'a :: metric_space"
```
```   175   assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```   176   shows "(f ---> l) F"
```
```   177   apply (rule topological_tendstoI)
```
```   178   apply (simp add: open_dist)
```
```   179   apply (drule (1) bspec, clarify)
```
```   180   apply (drule assms)
```
```   181   apply (erule eventually_elim1, simp)
```
```   182   done
```
```   183
```
```   184 lemma tendstoD:
```
```   185   fixes l :: "'a :: metric_space"
```
```   186   shows "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
```
```   187   apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
```
```   188   apply (clarsimp simp add: open_dist)
```
```   189   apply (rule_tac x="e - dist x l" in exI, clarsimp)
```
```   190   apply (simp only: less_diff_eq)
```
```   191   apply (erule le_less_trans [OF dist_triangle])
```
```   192   apply simp
```
```   193   apply simp
```
```   194   done
```
```   195
```
```   196 lemma tendsto_iff:
```
```   197   fixes l :: "'a :: metric_space"
```
```   198   shows "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
```
```   199   using tendstoI tendstoD by fast
```
```   200
```
```   201 lemma metric_tendsto_imp_tendsto:
```
```   202   fixes a :: "'a :: metric_space" and b :: "'b :: metric_space"
```
```   203   assumes f: "(f ---> a) F"
```
```   204   assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
```
```   205   shows "(g ---> b) F"
```
```   206 proof (rule tendstoI)
```
```   207   fix e :: real assume "0 < e"
```
```   208   with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
```
```   209   with le show "eventually (\<lambda>x. dist (g x) b < e) F"
```
```   210     using le_less_trans by (rule eventually_elim2)
```
```   211 qed
```
```   212
```
```   213 lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
```
```   214   unfolding filterlim_at_top
```
```   215   apply (intro allI)
```
```   216   apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
```
```   217   apply (auto simp: natceiling_le_eq)
```
```   218   done
```
```   219
```
```   220 subsubsection {* Limits of Sequences *}
```
```   221
```
```   222 lemma LIMSEQ_def: "X ----> (L::'a::metric_space) \<longleftrightarrow> (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
```
```   223   unfolding tendsto_iff eventually_sequentially ..
```
```   224
```
```   225 lemma LIMSEQ_iff_nz: "X ----> (L::'a::metric_space) = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
```
```   226   unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
```
```   227
```
```   228 lemma metric_LIMSEQ_I:
```
```   229   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> (L::'a::metric_space)"
```
```   230 by (simp add: LIMSEQ_def)
```
```   231
```
```   232 lemma metric_LIMSEQ_D:
```
```   233   "\<lbrakk>X ----> (L::'a::metric_space); 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
```
```   234 by (simp add: LIMSEQ_def)
```
```   235
```
```   236
```
```   237 subsubsection {* Limits of Functions *}
```
```   238
```
```   239 lemma LIM_def: "f -- (a::'a::metric_space) --> (L::'b::metric_space) =
```
```   240      (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
```
```   241         --> dist (f x) L < r)"
```
```   242 unfolding tendsto_iff eventually_at ..
```
```   243
```
```   244 lemma metric_LIM_I:
```
```   245   "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
```
```   246     \<Longrightarrow> f -- (a::'a::metric_space) --> (L::'b::metric_space)"
```
```   247 by (simp add: LIM_def)
```
```   248
```
```   249 lemma metric_LIM_D:
```
```   250   "\<lbrakk>f -- (a::'a::metric_space) --> (L::'b::metric_space); 0 < r\<rbrakk>
```
```   251     \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
```
```   252 by (simp add: LIM_def)
```
```   253
```
```   254 lemma metric_LIM_imp_LIM:
```
```   255   assumes f: "f -- a --> (l::'a::metric_space)"
```
```   256   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
```
```   257   shows "g -- a --> (m::'b::metric_space)"
```
```   258   by (rule metric_tendsto_imp_tendsto [OF f]) (auto simp add: eventually_at_topological le)
```
```   259
```
```   260 lemma metric_LIM_equal2:
```
```   261   assumes 1: "0 < R"
```
```   262   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```   263   shows "g -- a --> l \<Longrightarrow> f -- (a::'a::metric_space) --> l"
```
```   264 apply (rule topological_tendstoI)
```
```   265 apply (drule (2) topological_tendstoD)
```
```   266 apply (simp add: eventually_at, safe)
```
```   267 apply (rule_tac x="min d R" in exI, safe)
```
```   268 apply (simp add: 1)
```
```   269 apply (simp add: 2)
```
```   270 done
```
```   271
```
```   272 lemma metric_LIM_compose2:
```
```   273   assumes f: "f -- (a::'a::metric_space) --> b"
```
```   274   assumes g: "g -- b --> c"
```
```   275   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
```
```   276   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```   277   using g f inj [folded eventually_at]
```
```   278   by (rule tendsto_compose_eventually)
```
```   279
```
```   280 lemma metric_isCont_LIM_compose2:
```
```   281   fixes f :: "'a :: metric_space \<Rightarrow> _"
```
```   282   assumes f [unfolded isCont_def]: "isCont f a"
```
```   283   assumes g: "g -- f a --> l"
```
```   284   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
```
```   285   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```   286 by (rule metric_LIM_compose2 [OF f g inj])
```
```   287
```
```   288 subsubsection {* Boundedness *}
```
```   289
```
```   290 definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```   291   Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
```
```   292
```
```   293 abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```   294   "Bseq X \<equiv> Bfun X sequentially"
```
```   295
```
```   296 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
```
```   297
```
```   298 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```   299   unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
```
```   300
```
```   301 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```   302   unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
```
```   303
```
```   304 subsection {* Complete metric spaces *}
```
```   305
```
```   306 subsection {* Cauchy sequences *}
```
```   307
```
```   308 definition (in metric_space) Cauchy :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
```
```   309   "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
```
```   310
```
```   311 subsection {* Cauchy Sequences *}
```
```   312
```
```   313 lemma metric_CauchyI:
```
```   314   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
```
```   315   by (simp add: Cauchy_def)
```
```   316
```
```   317 lemma metric_CauchyD:
```
```   318   "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
```
```   319   by (simp add: Cauchy_def)
```
```   320
```
```   321 lemma metric_Cauchy_iff2:
```
```   322   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < inverse(real (Suc j))))"
```
```   323 apply (simp add: Cauchy_def, auto)
```
```   324 apply (drule reals_Archimedean, safe)
```
```   325 apply (drule_tac x = n in spec, auto)
```
```   326 apply (rule_tac x = M in exI, auto)
```
```   327 apply (drule_tac x = m in spec, simp)
```
```   328 apply (drule_tac x = na in spec, auto)
```
```   329 done
```
```   330
```
```   331 lemma Cauchy_subseq_Cauchy:
```
```   332   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
```
```   333 apply (auto simp add: Cauchy_def)
```
```   334 apply (drule_tac x=e in spec, clarify)
```
```   335 apply (rule_tac x=M in exI, clarify)
```
```   336 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
```
```   337 done
```
```   338
```
```   339 lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
```
```   340   unfolding Cauchy_def Bfun_metric_def eventually_sequentially
```
```   341   apply (erule_tac x=1 in allE)
```
```   342   apply simp
```
```   343   apply safe
```
```   344   apply (rule_tac x="X M" in exI)
```
```   345   apply (rule_tac x=1 in exI)
```
```   346   apply (erule_tac x=M in allE)
```
```   347   apply simp
```
```   348   apply (rule_tac x=M in exI)
```
```   349   apply (auto simp: dist_commute)
```
```   350   done
```
```   351
```
```   352 subsubsection {* Cauchy Sequences are Convergent *}
```
```   353
```
```   354 class complete_space = metric_space +
```
```   355   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
```
```   356
```
```   357 theorem LIMSEQ_imp_Cauchy:
```
```   358   assumes X: "X ----> a" shows "Cauchy X"
```
```   359 proof (rule metric_CauchyI)
```
```   360   fix e::real assume "0 < e"
```
```   361   hence "0 < e/2" by simp
```
```   362   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
```
```   363   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
```
```   364   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
```
```   365   proof (intro exI allI impI)
```
```   366     fix m assume "N \<le> m"
```
```   367     hence m: "dist (X m) a < e/2" using N by fast
```
```   368     fix n assume "N \<le> n"
```
```   369     hence n: "dist (X n) a < e/2" using N by fast
```
```   370     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
```
```   371       by (rule dist_triangle2)
```
```   372     also from m n have "\<dots> < e" by simp
```
```   373     finally show "dist (X m) (X n) < e" .
```
```   374   qed
```
```   375 qed
```
```   376
```
```   377 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
```
```   378 unfolding convergent_def
```
```   379 by (erule exE, erule LIMSEQ_imp_Cauchy)
```
```   380
```
```   381 lemma Cauchy_convergent_iff:
```
```   382   fixes X :: "nat \<Rightarrow> 'a::complete_space"
```
```   383   shows "Cauchy X = convergent X"
```
```   384 by (fast intro: Cauchy_convergent convergent_Cauchy)
```
```   385
```
```   386 subsection {* Uniform Continuity *}
```
```   387
```
```   388 definition
```
```   389   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
```
```   390   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
```
```   391
```
```   392 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```   393 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```   394
```
```   395 lemma isUCont_Cauchy:
```
```   396   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```   397 unfolding isUCont_def
```
```   398 apply (rule metric_CauchyI)
```
```   399 apply (drule_tac x=e in spec, safe)
```
```   400 apply (drule_tac e=s in metric_CauchyD, safe)
```
```   401 apply (rule_tac x=M in exI, simp)
```
```   402 done
```
```   403
```
```   404 subsection {* The set of real numbers is a complete metric space *}
```
```   405
```
```   406 instantiation real :: metric_space
```
```   407 begin
```
```   408
```
```   409 definition dist_real_def:
```
```   410   "dist x y = \<bar>x - y\<bar>"
```
```   411
```
```   412 definition open_real_def:
```
```   413   "open (S :: real set) \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
```
```   414
```
```   415 instance
```
```   416   by default (auto simp: open_real_def dist_real_def)
```
```   417 end
```
```   418
```
```   419 instance real :: linorder_topology
```
```   420 proof
```
```   421   show "(open :: real set \<Rightarrow> bool) = generate_topology (range lessThan \<union> range greaterThan)"
```
```   422   proof (rule ext, safe)
```
```   423     fix S :: "real set" assume "open S"
```
```   424     then guess f unfolding open_real_def bchoice_iff ..
```
```   425     then have *: "S = (\<Union>x\<in>S. {x - f x <..} \<inter> {..< x + f x})"
```
```   426       by (fastforce simp: dist_real_def)
```
```   427     show "generate_topology (range lessThan \<union> range greaterThan) S"
```
```   428       apply (subst *)
```
```   429       apply (intro generate_topology_Union generate_topology.Int)
```
```   430       apply (auto intro: generate_topology.Basis)
```
```   431       done
```
```   432   next
```
```   433     fix S :: "real set" assume "generate_topology (range lessThan \<union> range greaterThan) S"
```
```   434     moreover have "\<And>a::real. open {..<a}"
```
```   435       unfolding open_real_def dist_real_def
```
```   436     proof clarify
```
```   437       fix x a :: real assume "x < a"
```
```   438       hence "0 < a - x \<and> (\<forall>y. \<bar>y - x\<bar> < a - x \<longrightarrow> y \<in> {..<a})" by auto
```
```   439       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {..<a}" ..
```
```   440     qed
```
```   441     moreover have "\<And>a::real. open {a <..}"
```
```   442       unfolding open_real_def dist_real_def
```
```   443     proof clarify
```
```   444       fix x a :: real assume "a < x"
```
```   445       hence "0 < x - a \<and> (\<forall>y. \<bar>y - x\<bar> < x - a \<longrightarrow> y \<in> {a<..})" by auto
```
```   446       thus "\<exists>e>0. \<forall>y. \<bar>y - x\<bar> < e \<longrightarrow> y \<in> {a<..}" ..
```
```   447     qed
```
```   448     ultimately show "open S"
```
```   449       by induct auto
```
```   450   qed
```
```   451 qed
```
```   452
```
```   453 lemmas open_real_greaterThan = open_greaterThan[where 'a=real]
```
```   454 lemmas open_real_lessThan = open_lessThan[where 'a=real]
```
```   455 lemmas open_real_greaterThanLessThan = open_greaterThanLessThan[where 'a=real]
```
```   456 lemmas closed_real_atMost = closed_atMost[where 'a=real]
```
```   457 lemmas closed_real_atLeast = closed_atLeast[where 'a=real]
```
```   458 lemmas closed_real_atLeastAtMost = closed_atLeastAtMost[where 'a=real]
```
```   459
```
```   460 text {*
```
```   461 Proof that Cauchy sequences converge based on the one from
```
```   462 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
```
```   463 *}
```
```   464
```
```   465 text {*
```
```   466   If sequence @{term "X"} is Cauchy, then its limit is the lub of
```
```   467   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
```
```   468 *}
```
```   469
```
```   470 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
```
```   471 by (simp add: isUbI setleI)
```
```   472
```
```   473 lemma increasing_LIMSEQ:
```
```   474   fixes f :: "nat \<Rightarrow> real"
```
```   475   assumes inc: "\<And>n. f n \<le> f (Suc n)"
```
```   476       and bdd: "\<And>n. f n \<le> l"
```
```   477       and en: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
```
```   478   shows "f ----> l"
```
```   479 proof (rule increasing_tendsto)
```
```   480   fix x assume "x < l"
```
```   481   with dense[of 0 "l - x"] obtain e where "0 < e" "e < l - x"
```
```   482     by auto
```
```   483   from en[OF `0 < e`] obtain n where "l - e \<le> f n"
```
```   484     by (auto simp: field_simps)
```
```   485   with `e < l - x` `0 < e` have "x < f n" by simp
```
```   486   with incseq_SucI[of f, OF inc] show "eventually (\<lambda>n. x < f n) sequentially"
```
```   487     by (auto simp: eventually_sequentially incseq_def intro: less_le_trans)
```
```   488 qed (insert bdd, auto)
```
```   489
```
```   490 lemma real_Cauchy_convergent:
```
```   491   fixes X :: "nat \<Rightarrow> real"
```
```   492   assumes X: "Cauchy X"
```
```   493   shows "convergent X"
```
```   494 proof -
```
```   495   def S \<equiv> "{x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
```
```   496   then have mem_S: "\<And>N x. \<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S" by auto
```
```   497
```
```   498   { fix N x assume N: "\<forall>n\<ge>N. X n < x"
```
```   499   have "isUb UNIV S x"
```
```   500   proof (rule isUb_UNIV_I)
```
```   501   fix y::real assume "y \<in> S"
```
```   502   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
```
```   503     by (simp add: S_def)
```
```   504   then obtain M where "\<forall>n\<ge>M. y < X n" ..
```
```   505   hence "y < X (max M N)" by simp
```
```   506   also have "\<dots> < x" using N by simp
```
```   507   finally show "y \<le> x"
```
```   508     by (rule order_less_imp_le)
```
```   509   qed }
```
```   510   note bound_isUb = this
```
```   511
```
```   512   have "\<exists>u. isLub UNIV S u"
```
```   513   proof (rule reals_complete)
```
```   514   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < 1"
```
```   515     using X[THEN metric_CauchyD, OF zero_less_one] by auto
```
```   516   hence N: "\<forall>n\<ge>N. dist (X n) (X N) < 1" by simp
```
```   517   show "\<exists>x. x \<in> S"
```
```   518   proof
```
```   519     from N have "\<forall>n\<ge>N. X N - 1 < X n"
```
```   520       by (simp add: abs_diff_less_iff dist_real_def)
```
```   521     thus "X N - 1 \<in> S" by (rule mem_S)
```
```   522   qed
```
```   523   show "\<exists>u. isUb UNIV S u"
```
```   524   proof
```
```   525     from N have "\<forall>n\<ge>N. X n < X N + 1"
```
```   526       by (simp add: abs_diff_less_iff dist_real_def)
```
```   527     thus "isUb UNIV S (X N + 1)"
```
```   528       by (rule bound_isUb)
```
```   529   qed
```
```   530   qed
```
```   531   then obtain x where x: "isLub UNIV S x" ..
```
```   532   have "X ----> x"
```
```   533   proof (rule metric_LIMSEQ_I)
```
```   534   fix r::real assume "0 < r"
```
```   535   hence r: "0 < r/2" by simp
```
```   536   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. dist (X n) (X m) < r/2"
```
```   537     using metric_CauchyD [OF X r] by auto
```
```   538   hence "\<forall>n\<ge>N. dist (X n) (X N) < r/2" by simp
```
```   539   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
```
```   540     by (simp only: dist_real_def abs_diff_less_iff)
```
```   541
```
```   542   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
```
```   543   hence "X N - r/2 \<in> S" by (rule mem_S)
```
```   544   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
```
```   545
```
```   546   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
```
```   547   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
```
```   548   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
```
```   549
```
```   550   show "\<exists>N. \<forall>n\<ge>N. dist (X n) x < r"
```
```   551   proof (intro exI allI impI)
```
```   552     fix n assume n: "N \<le> n"
```
```   553     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
```
```   554     thus "dist (X n) x < r" using 1 2
```
```   555       by (simp add: abs_diff_less_iff dist_real_def)
```
```   556   qed
```
```   557   qed
```
```   558   then show ?thesis unfolding convergent_def by auto
```
```   559 qed
```
```   560
```
```   561 instance real :: complete_space
```
```   562   by intro_classes (rule real_Cauchy_convergent)
```
```   563
```
```   564 lemma tendsto_dist [tendsto_intros]:
```
```   565   fixes l m :: "'a :: metric_space"
```
```   566   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
```
```   567   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
```
```   568 proof (rule tendstoI)
```
```   569   fix e :: real assume "0 < e"
```
```   570   hence e2: "0 < e/2" by simp
```
```   571   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   572   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
```
```   573   proof (eventually_elim)
```
```   574     case (elim x)
```
```   575     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   576       unfolding dist_real_def
```
```   577       using dist_triangle2 [of "f x" "g x" "l"]
```
```   578       using dist_triangle2 [of "g x" "l" "m"]
```
```   579       using dist_triangle3 [of "l" "m" "f x"]
```
```   580       using dist_triangle [of "f x" "m" "g x"]
```
```   581       by arith
```
```   582   qed
```
```   583 qed
```
```   584
```
```   585 lemma continuous_dist[continuous_intros]:
```
```   586   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   587   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
```
```   588   unfolding continuous_def by (rule tendsto_dist)
```
```   589
```
```   590 lemma continuous_on_dist[continuous_on_intros]:
```
```   591   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   592   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
```
```   593   unfolding continuous_on_def by (auto intro: tendsto_dist)
```
```   594
```
```   595 lemma tendsto_at_topI_sequentially:
```
```   596   fixes f :: "real \<Rightarrow> real"
```
```   597   assumes mono: "mono f"
```
```   598   assumes limseq: "(\<lambda>n. f (real n)) ----> y"
```
```   599   shows "(f ---> y) at_top"
```
```   600 proof (rule tendstoI)
```
```   601   fix e :: real assume "0 < e"
```
```   602   with limseq obtain N :: nat where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>f (real n) - y\<bar> < e"
```
```   603     by (auto simp: LIMSEQ_def dist_real_def)
```
```   604   { fix x :: real
```
```   605     from ex_le_of_nat[of x] guess n ..
```
```   606     note monoD[OF mono this]
```
```   607     also have "f (real_of_nat n) \<le> y"
```
```   608       by (rule LIMSEQ_le_const[OF limseq])
```
```   609          (auto intro: exI[of _ n] monoD[OF mono] simp: real_eq_of_nat[symmetric])
```
```   610     finally have "f x \<le> y" . }
```
```   611   note le = this
```
```   612   have "eventually (\<lambda>x. real N \<le> x) at_top"
```
```   613     by (rule eventually_ge_at_top)
```
```   614   then show "eventually (\<lambda>x. dist (f x) y < e) at_top"
```
```   615   proof eventually_elim
```
```   616     fix x assume N': "real N \<le> x"
```
```   617     with N[of N] le have "y - f (real N) < e" by auto
```
```   618     moreover note monoD[OF mono N']
```
```   619     ultimately show "dist (f x) y < e"
```
```   620       using le[of x] by (auto simp: dist_real_def field_simps)
```
```   621   qed
```
```   622 qed
```
```   623
```
```   624 lemma Cauchy_iff2:
```
```   625   "Cauchy X = (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
```
```   626   unfolding metric_Cauchy_iff2 dist_real_def ..
```
```   627
```
```   628 end
```
```   629
```