src/HOL/Limits.thy
 changeset 44195 f5363511b212 parent 44194 0639898074ae child 44205 18da2a87421c
```     1.1 --- a/src/HOL/Limits.thy	Sun Aug 14 07:54:24 2011 -0700
1.2 +++ b/src/HOL/Limits.thy	Sun Aug 14 08:45:38 2011 -0700
1.3 @@ -25,8 +25,8 @@
1.4    show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
1.5  qed
1.6
1.7 -lemma is_filter_Rep_filter: "is_filter (Rep_filter A)"
1.8 -  using Rep_filter [of A] by simp
1.9 +lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
1.10 +  using Rep_filter [of F] by simp
1.11
1.12  lemma Abs_filter_inverse':
1.13    assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
1.14 @@ -36,84 +36,84 @@
1.15  subsection {* Eventually *}
1.16
1.17  definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
1.18 -  where "eventually P A \<longleftrightarrow> Rep_filter A P"
1.19 +  where "eventually P F \<longleftrightarrow> Rep_filter F P"
1.20
1.21  lemma eventually_Abs_filter:
1.22    assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
1.23    unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
1.24
1.25  lemma filter_eq_iff:
1.26 -  shows "A = B \<longleftrightarrow> (\<forall>P. eventually P A = eventually P B)"
1.27 +  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
1.28    unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
1.29
1.30 -lemma eventually_True [simp]: "eventually (\<lambda>x. True) A"
1.31 +lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
1.32    unfolding eventually_def
1.33    by (rule is_filter.True [OF is_filter_Rep_filter])
1.34
1.35 -lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P A"
1.36 +lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
1.37  proof -
1.38    assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
1.39 -  thus "eventually P A" by simp
1.40 +  thus "eventually P F" by simp
1.41  qed
1.42
1.43  lemma eventually_mono:
1.44 -  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P A \<Longrightarrow> eventually Q A"
1.45 +  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
1.46    unfolding eventually_def
1.47    by (rule is_filter.mono [OF is_filter_Rep_filter])
1.48
1.49  lemma eventually_conj:
1.50 -  assumes P: "eventually (\<lambda>x. P x) A"
1.51 -  assumes Q: "eventually (\<lambda>x. Q x) A"
1.52 -  shows "eventually (\<lambda>x. P x \<and> Q x) A"
1.53 +  assumes P: "eventually (\<lambda>x. P x) F"
1.54 +  assumes Q: "eventually (\<lambda>x. Q x) F"
1.55 +  shows "eventually (\<lambda>x. P x \<and> Q x) F"
1.56    using assms unfolding eventually_def
1.57    by (rule is_filter.conj [OF is_filter_Rep_filter])
1.58
1.59  lemma eventually_mp:
1.60 -  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
1.61 -  assumes "eventually (\<lambda>x. P x) A"
1.62 -  shows "eventually (\<lambda>x. Q x) A"
1.63 +  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
1.64 +  assumes "eventually (\<lambda>x. P x) F"
1.65 +  shows "eventually (\<lambda>x. Q x) F"
1.66  proof (rule eventually_mono)
1.67    show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
1.68 -  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) A"
1.69 +  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
1.70      using assms by (rule eventually_conj)
1.71  qed
1.72
1.73  lemma eventually_rev_mp:
1.74 -  assumes "eventually (\<lambda>x. P x) A"
1.75 -  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
1.76 -  shows "eventually (\<lambda>x. Q x) A"
1.77 +  assumes "eventually (\<lambda>x. P x) F"
1.78 +  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
1.79 +  shows "eventually (\<lambda>x. Q x) F"
1.80  using assms(2) assms(1) by (rule eventually_mp)
1.81
1.82  lemma eventually_conj_iff:
1.83 -  "eventually (\<lambda>x. P x \<and> Q x) A \<longleftrightarrow> eventually P A \<and> eventually Q A"
1.84 +  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
1.85    by (auto intro: eventually_conj elim: eventually_rev_mp)
1.86
1.87  lemma eventually_elim1:
1.88 -  assumes "eventually (\<lambda>i. P i) A"
1.89 +  assumes "eventually (\<lambda>i. P i) F"
1.90    assumes "\<And>i. P i \<Longrightarrow> Q i"
1.91 -  shows "eventually (\<lambda>i. Q i) A"
1.92 +  shows "eventually (\<lambda>i. Q i) F"
1.93    using assms by (auto elim!: eventually_rev_mp)
1.94
1.95  lemma eventually_elim2:
1.96 -  assumes "eventually (\<lambda>i. P i) A"
1.97 -  assumes "eventually (\<lambda>i. Q i) A"
1.98 +  assumes "eventually (\<lambda>i. P i) F"
1.99 +  assumes "eventually (\<lambda>i. Q i) F"
1.100    assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
1.101 -  shows "eventually (\<lambda>i. R i) A"
1.102 +  shows "eventually (\<lambda>i. R i) F"
1.103    using assms by (auto elim!: eventually_rev_mp)
1.104
1.105  subsection {* Finer-than relation *}
1.106
1.107 -text {* @{term "A \<le> B"} means that filter @{term A} is finer than
1.108 -filter @{term B}. *}
1.109 +text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
1.110 +filter @{term F'}. *}
1.111
1.112  instantiation filter :: (type) complete_lattice
1.113  begin
1.114
1.115  definition le_filter_def:
1.116 -  "A \<le> B \<longleftrightarrow> (\<forall>P. eventually P B \<longrightarrow> eventually P A)"
1.117 +  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
1.118
1.119  definition
1.120 -  "(A :: 'a filter) < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
1.121 +  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
1.122
1.123  definition
1.124    "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
1.125 @@ -122,17 +122,17 @@
1.126    "bot = Abs_filter (\<lambda>P. True)"
1.127
1.128  definition
1.129 -  "sup A B = Abs_filter (\<lambda>P. eventually P A \<and> eventually P B)"
1.130 +  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
1.131
1.132  definition
1.133 -  "inf A B = Abs_filter
1.134 -      (\<lambda>P. \<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
1.135 +  "inf F F' = Abs_filter
1.136 +      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
1.137
1.138  definition
1.139 -  "Sup S = Abs_filter (\<lambda>P. \<forall>A\<in>S. eventually P A)"
1.140 +  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
1.141
1.142  definition
1.143 -  "Inf S = Sup {A::'a filter. \<forall>B\<in>S. A \<le> B}"
1.144 +  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
1.145
1.146  lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
1.147    unfolding top_filter_def
1.148 @@ -143,14 +143,14 @@
1.149    by (subst eventually_Abs_filter, rule is_filter.intro, auto)
1.150
1.151  lemma eventually_sup:
1.152 -  "eventually P (sup A B) \<longleftrightarrow> eventually P A \<and> eventually P B"
1.153 +  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
1.154    unfolding sup_filter_def
1.155    by (rule eventually_Abs_filter, rule is_filter.intro)
1.156       (auto elim!: eventually_rev_mp)
1.157
1.158  lemma eventually_inf:
1.159 -  "eventually P (inf A B) \<longleftrightarrow>
1.160 -   (\<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
1.161 +  "eventually P (inf F F') \<longleftrightarrow>
1.162 +   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
1.163    unfolding inf_filter_def
1.164    apply (rule eventually_Abs_filter, rule is_filter.intro)
1.165    apply (fast intro: eventually_True)
1.166 @@ -163,92 +163,80 @@
1.167    done
1.168
1.169  lemma eventually_Sup:
1.170 -  "eventually P (Sup S) \<longleftrightarrow> (\<forall>A\<in>S. eventually P A)"
1.171 +  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
1.172    unfolding Sup_filter_def
1.173    apply (rule eventually_Abs_filter, rule is_filter.intro)
1.174    apply (auto intro: eventually_conj elim!: eventually_rev_mp)
1.175    done
1.176
1.177  instance proof
1.178 -  fix A B :: "'a filter" show "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
1.179 -    by (rule less_filter_def)
1.180 -next
1.181 -  fix A :: "'a filter" show "A \<le> A"
1.182 -    unfolding le_filter_def by simp
1.183 -next
1.184 -  fix A B C :: "'a filter" assume "A \<le> B" and "B \<le> C" thus "A \<le> C"
1.185 -    unfolding le_filter_def by simp
1.186 -next
1.187 -  fix A B :: "'a filter" assume "A \<le> B" and "B \<le> A" thus "A = B"
1.188 -    unfolding le_filter_def filter_eq_iff by fast
1.189 -next
1.190 -  fix A :: "'a filter" show "A \<le> top"
1.191 -    unfolding le_filter_def eventually_top by (simp add: always_eventually)
1.192 -next
1.193 -  fix A :: "'a filter" show "bot \<le> A"
1.194 -    unfolding le_filter_def by simp
1.195 -next
1.196 -  fix A B :: "'a filter" show "A \<le> sup A B" and "B \<le> sup A B"
1.197 -    unfolding le_filter_def eventually_sup by simp_all
1.198 -next
1.199 -  fix A B C :: "'a filter" assume "A \<le> C" and "B \<le> C" thus "sup A B \<le> C"
1.200 -    unfolding le_filter_def eventually_sup by simp
1.201 -next
1.202 -  fix A B :: "'a filter" show "inf A B \<le> A" and "inf A B \<le> B"
1.203 -    unfolding le_filter_def eventually_inf by (auto intro: eventually_True)
1.204 -next
1.205 -  fix A B C :: "'a filter" assume "A \<le> B" and "A \<le> C" thus "A \<le> inf B C"
1.206 +  fix F F' F'' :: "'a filter" and S :: "'a filter set"
1.207 +  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
1.208 +    by (rule less_filter_def) }
1.209 +  { show "F \<le> F"
1.210 +    unfolding le_filter_def by simp }
1.211 +  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
1.212 +    unfolding le_filter_def by simp }
1.213 +  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
1.214 +    unfolding le_filter_def filter_eq_iff by fast }
1.215 +  { show "F \<le> top"
1.216 +    unfolding le_filter_def eventually_top by (simp add: always_eventually) }
1.217 +  { show "bot \<le> F"
1.218 +    unfolding le_filter_def by simp }
1.219 +  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
1.220 +    unfolding le_filter_def eventually_sup by simp_all }
1.221 +  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
1.222 +    unfolding le_filter_def eventually_sup by simp }
1.223 +  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
1.224 +    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
1.225 +  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
1.226      unfolding le_filter_def eventually_inf
1.227 -    by (auto elim!: eventually_mono intro: eventually_conj)
1.228 -next
1.229 -  fix A :: "'a filter" and S assume "A \<in> S" thus "A \<le> Sup S"
1.230 -    unfolding le_filter_def eventually_Sup by simp
1.231 -next
1.232 -  fix S and B :: "'a filter" assume "\<And>A. A \<in> S \<Longrightarrow> A \<le> B" thus "Sup S \<le> B"
1.233 -    unfolding le_filter_def eventually_Sup by simp
1.234 -next
1.235 -  fix C :: "'a filter" and S assume "C \<in> S" thus "Inf S \<le> C"
1.236 -    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
1.237 -next
1.238 -  fix S and A :: "'a filter" assume "\<And>B. B \<in> S \<Longrightarrow> A \<le> B" thus "A \<le> Inf S"
1.239 -    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
1.240 +    by (auto elim!: eventually_mono intro: eventually_conj) }
1.241 +  { assume "F \<in> S" thus "F \<le> Sup S"
1.242 +    unfolding le_filter_def eventually_Sup by simp }
1.243 +  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
1.244 +    unfolding le_filter_def eventually_Sup by simp }
1.245 +  { assume "F'' \<in> S" thus "Inf S \<le> F''"
1.246 +    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
1.247 +  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
1.248 +    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
1.249  qed
1.250
1.251  end
1.252
1.253  lemma filter_leD:
1.254 -  "A \<le> B \<Longrightarrow> eventually P B \<Longrightarrow> eventually P A"
1.255 +  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
1.256    unfolding le_filter_def by simp
1.257
1.258  lemma filter_leI:
1.259 -  "(\<And>P. eventually P B \<Longrightarrow> eventually P A) \<Longrightarrow> A \<le> B"
1.260 +  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
1.261    unfolding le_filter_def by simp
1.262
1.263  lemma eventually_False:
1.264 -  "eventually (\<lambda>x. False) A \<longleftrightarrow> A = bot"
1.265 +  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
1.266    unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
1.267
1.268  subsection {* Map function for filters *}
1.269
1.270  definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
1.271 -  where "filtermap f A = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) A)"
1.272 +  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
1.273
1.274  lemma eventually_filtermap:
1.275 -  "eventually P (filtermap f A) = eventually (\<lambda>x. P (f x)) A"
1.276 +  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
1.277    unfolding filtermap_def
1.278    apply (rule eventually_Abs_filter)
1.279    apply (rule is_filter.intro)
1.280    apply (auto elim!: eventually_rev_mp)
1.281    done
1.282
1.283 -lemma filtermap_ident: "filtermap (\<lambda>x. x) A = A"
1.284 +lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
1.285    by (simp add: filter_eq_iff eventually_filtermap)
1.286
1.287  lemma filtermap_filtermap:
1.288 -  "filtermap f (filtermap g A) = filtermap (\<lambda>x. f (g x)) A"
1.289 +  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
1.290    by (simp add: filter_eq_iff eventually_filtermap)
1.291
1.292 -lemma filtermap_mono: "A \<le> B \<Longrightarrow> filtermap f A \<le> filtermap f B"
1.293 +lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
1.294    unfolding le_filter_def eventually_filtermap by simp
1.295
1.296  lemma filtermap_bot [simp]: "filtermap f bot = bot"
1.297 @@ -279,13 +267,13 @@
1.299
1.300  lemma le_sequentially:
1.301 -  "A \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) A)"
1.302 +  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
1.303    unfolding le_filter_def eventually_sequentially
1.304    by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
1.305
1.306
1.307  definition trivial_limit :: "'a filter \<Rightarrow> bool"
1.308 -  where "trivial_limit A \<longleftrightarrow> eventually (\<lambda>x. False) A"
1.309 +  where "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
1.310
1.311  lemma trivial_limit_sequentially [intro]: "\<not> trivial_limit sequentially"
1.312    by (auto simp add: trivial_limit_def eventually_sequentially)
1.313 @@ -293,7 +281,7 @@
1.314  subsection {* Standard filters *}
1.315
1.316  definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
1.317 -  where "A within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A)"
1.318 +  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
1.319
1.320  definition nhds :: "'a::topological_space \<Rightarrow> 'a filter"
1.321    where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
1.322 @@ -302,12 +290,12 @@
1.323    where "at a = nhds a within - {a}"
1.324
1.325  lemma eventually_within:
1.326 -  "eventually P (A within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A"
1.327 +  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
1.328    unfolding within_def
1.329    by (rule eventually_Abs_filter, rule is_filter.intro)
1.330       (auto elim!: eventually_rev_mp)
1.331
1.332 -lemma within_UNIV: "A within UNIV = A"
1.333 +lemma within_UNIV: "F within UNIV = F"
1.334    unfolding filter_eq_iff eventually_within by simp
1.335
1.336  lemma eventually_nhds:
1.337 @@ -353,51 +341,51 @@
1.338  subsection {* Boundedness *}
1.339
1.340  definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
1.341 -  where "Bfun f A = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) A)"
1.342 +  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
1.343
1.344  lemma BfunI:
1.345 -  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) A" shows "Bfun f A"
1.346 +  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
1.347  unfolding Bfun_def
1.348  proof (intro exI conjI allI)
1.349    show "0 < max K 1" by simp
1.350  next
1.351 -  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) A"
1.352 +  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
1.353      using K by (rule eventually_elim1, simp)
1.354  qed
1.355
1.356  lemma BfunE:
1.357 -  assumes "Bfun f A"
1.358 -  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) A"
1.359 +  assumes "Bfun f F"
1.360 +  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
1.361  using assms unfolding Bfun_def by fast
1.362
1.363
1.364  subsection {* Convergence to Zero *}
1.365
1.366  definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
1.367 -  where "Zfun f A = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) A)"
1.368 +  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
1.369
1.370  lemma ZfunI:
1.371 -  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A) \<Longrightarrow> Zfun f A"
1.372 +  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
1.373    unfolding Zfun_def by simp
1.374
1.375  lemma ZfunD:
1.376 -  "\<lbrakk>Zfun f A; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A"
1.377 +  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
1.378    unfolding Zfun_def by simp
1.379
1.380  lemma Zfun_ssubst:
1.381 -  "eventually (\<lambda>x. f x = g x) A \<Longrightarrow> Zfun g A \<Longrightarrow> Zfun f A"
1.382 +  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
1.383    unfolding Zfun_def by (auto elim!: eventually_rev_mp)
1.384
1.385 -lemma Zfun_zero: "Zfun (\<lambda>x. 0) A"
1.386 +lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
1.387    unfolding Zfun_def by simp
1.388
1.389 -lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) A = Zfun (\<lambda>x. f x) A"
1.390 +lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
1.391    unfolding Zfun_def by simp
1.392
1.393  lemma Zfun_imp_Zfun:
1.394 -  assumes f: "Zfun f A"
1.395 -  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) A"
1.396 -  shows "Zfun (\<lambda>x. g x) A"
1.397 +  assumes f: "Zfun f F"
1.398 +  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
1.399 +  shows "Zfun (\<lambda>x. g x) F"
1.400  proof (cases)
1.401    assume K: "0 < K"
1.402    show ?thesis
1.403 @@ -405,9 +393,9 @@
1.404      fix r::real assume "0 < r"
1.405      hence "0 < r / K"
1.406        using K by (rule divide_pos_pos)
1.407 -    then have "eventually (\<lambda>x. norm (f x) < r / K) A"
1.408 +    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
1.409        using ZfunD [OF f] by fast
1.410 -    with g show "eventually (\<lambda>x. norm (g x) < r) A"
1.411 +    with g show "eventually (\<lambda>x. norm (g x) < r) F"
1.412      proof (rule eventually_elim2)
1.413        fix x
1.414        assume *: "norm (g x) \<le> norm (f x) * K"
1.415 @@ -425,7 +413,7 @@
1.416    proof (rule ZfunI)
1.417      fix r :: real
1.418      assume "0 < r"
1.419 -    from g show "eventually (\<lambda>x. norm (g x) < r) A"
1.420 +    from g show "eventually (\<lambda>x. norm (g x) < r) F"
1.421      proof (rule eventually_elim1)
1.422        fix x
1.423        assume "norm (g x) \<le> norm (f x) * K"
1.424 @@ -437,22 +425,22 @@
1.425    qed
1.426  qed
1.427
1.428 -lemma Zfun_le: "\<lbrakk>Zfun g A; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f A"
1.429 +lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
1.430    by (erule_tac K="1" in Zfun_imp_Zfun, simp)
1.431
1.433 -  assumes f: "Zfun f A" and g: "Zfun g A"
1.434 -  shows "Zfun (\<lambda>x. f x + g x) A"
1.435 +  assumes f: "Zfun f F" and g: "Zfun g F"
1.436 +  shows "Zfun (\<lambda>x. f x + g x) F"
1.437  proof (rule ZfunI)
1.438    fix r::real assume "0 < r"
1.439    hence r: "0 < r / 2" by simp
1.440 -  have "eventually (\<lambda>x. norm (f x) < r/2) A"
1.441 +  have "eventually (\<lambda>x. norm (f x) < r/2) F"
1.442      using f r by (rule ZfunD)
1.443    moreover
1.444 -  have "eventually (\<lambda>x. norm (g x) < r/2) A"
1.445 +  have "eventually (\<lambda>x. norm (g x) < r/2) F"
1.446      using g r by (rule ZfunD)
1.447    ultimately
1.448 -  show "eventually (\<lambda>x. norm (f x + g x) < r) A"
1.449 +  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
1.450    proof (rule eventually_elim2)
1.451      fix x
1.452      assume *: "norm (f x) < r/2" "norm (g x) < r/2"
1.453 @@ -465,28 +453,28 @@
1.454    qed
1.455  qed
1.456
1.457 -lemma Zfun_minus: "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. - f x) A"
1.458 +lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
1.459    unfolding Zfun_def by simp
1.460
1.461 -lemma Zfun_diff: "\<lbrakk>Zfun f A; Zfun g A\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) A"
1.462 +lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
1.463    by (simp only: diff_minus Zfun_add Zfun_minus)
1.464
1.465  lemma (in bounded_linear) Zfun:
1.466 -  assumes g: "Zfun g A"
1.467 -  shows "Zfun (\<lambda>x. f (g x)) A"
1.468 +  assumes g: "Zfun g F"
1.469 +  shows "Zfun (\<lambda>x. f (g x)) F"
1.470  proof -
1.471    obtain K where "\<And>x. norm (f x) \<le> norm x * K"
1.472      using bounded by fast
1.473 -  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) A"
1.474 +  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
1.475      by simp
1.476    with g show ?thesis
1.477      by (rule Zfun_imp_Zfun)
1.478  qed
1.479
1.480  lemma (in bounded_bilinear) Zfun:
1.481 -  assumes f: "Zfun f A"
1.482 -  assumes g: "Zfun g A"
1.483 -  shows "Zfun (\<lambda>x. f x ** g x) A"
1.484 +  assumes f: "Zfun f F"
1.485 +  assumes g: "Zfun g F"
1.486 +  shows "Zfun (\<lambda>x. f x ** g x) F"
1.487  proof (rule ZfunI)
1.488    fix r::real assume r: "0 < r"
1.489    obtain K where K: "0 < K"
1.490 @@ -494,13 +482,13 @@
1.491      using pos_bounded by fast
1.492    from K have K': "0 < inverse K"
1.493      by (rule positive_imp_inverse_positive)
1.494 -  have "eventually (\<lambda>x. norm (f x) < r) A"
1.495 +  have "eventually (\<lambda>x. norm (f x) < r) F"
1.496      using f r by (rule ZfunD)
1.497    moreover
1.498 -  have "eventually (\<lambda>x. norm (g x) < inverse K) A"
1.499 +  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
1.500      using g K' by (rule ZfunD)
1.501    ultimately
1.502 -  show "eventually (\<lambda>x. norm (f x ** g x) < r) A"
1.503 +  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
1.504    proof (rule eventually_elim2)
1.505      fix x
1.506      assume *: "norm (f x) < r" "norm (g x) < inverse K"
1.507 @@ -515,11 +503,11 @@
1.508  qed
1.509
1.510  lemma (in bounded_bilinear) Zfun_left:
1.511 -  "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. f x ** a) A"
1.512 +  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
1.513    by (rule bounded_linear_left [THEN bounded_linear.Zfun])
1.514
1.515  lemma (in bounded_bilinear) Zfun_right:
1.516 -  "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. a ** f x) A"
1.517 +  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
1.518    by (rule bounded_linear_right [THEN bounded_linear.Zfun])
1.519
1.520  lemmas Zfun_mult = mult.Zfun
1.521 @@ -531,7 +519,7 @@
1.522
1.523  definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool"
1.524      (infixr "--->" 55) where
1.525 -  "(f ---> l) A \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) A)"
1.526 +  "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
1.527
1.528  ML {*
1.529  structure Tendsto_Intros = Named_Thms
1.530 @@ -543,21 +531,21 @@
1.531
1.532  setup Tendsto_Intros.setup
1.533
1.534 -lemma tendsto_mono: "A \<le> A' \<Longrightarrow> (f ---> l) A' \<Longrightarrow> (f ---> l) A"
1.535 +lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
1.536    unfolding tendsto_def le_filter_def by fast
1.537
1.538  lemma topological_tendstoI:
1.539 -  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A)
1.540 -    \<Longrightarrow> (f ---> l) A"
1.541 +  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
1.542 +    \<Longrightarrow> (f ---> l) F"
1.543    unfolding tendsto_def by auto
1.544
1.545  lemma topological_tendstoD:
1.546 -  "(f ---> l) A \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A"
1.547 +  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
1.548    unfolding tendsto_def by auto
1.549
1.550  lemma tendstoI:
1.551 -  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
1.552 -  shows "(f ---> l) A"
1.553 +  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
1.554 +  shows "(f ---> l) F"
1.555    apply (rule topological_tendstoI)
1.557    apply (drule (1) bspec, clarify)
1.558 @@ -566,7 +554,7 @@
1.559    done
1.560
1.561  lemma tendstoD:
1.562 -  "(f ---> l) A \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
1.563 +  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
1.564    apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
1.565    apply (clarsimp simp add: open_dist)
1.566    apply (rule_tac x="e - dist x l" in exI, clarsimp)
1.567 @@ -577,10 +565,10 @@
1.568    done
1.569
1.570  lemma tendsto_iff:
1.571 -  "(f ---> l) A \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) A)"
1.572 +  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
1.573    using tendstoI tendstoD by fast
1.574
1.575 -lemma tendsto_Zfun_iff: "(f ---> a) A = Zfun (\<lambda>x. f x - a) A"
1.576 +lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
1.577    by (simp only: tendsto_iff Zfun_def dist_norm)
1.578
1.579  lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
1.580 @@ -590,12 +578,12 @@
1.581    "((\<lambda>x. x) ---> a) (at a within S)"
1.582    unfolding tendsto_def eventually_within eventually_at_topological by auto
1.583
1.584 -lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) A"
1.585 +lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
1.587
1.588  lemma tendsto_const_iff:
1.589    fixes k l :: "'a::metric_space"
1.590 -  assumes "A \<noteq> bot" shows "((\<lambda>n. k) ---> l) A \<longleftrightarrow> k = l"
1.591 +  assumes "F \<noteq> bot" shows "((\<lambda>n. k) ---> l) F \<longleftrightarrow> k = l"
1.592    apply (safe intro!: tendsto_const)
1.593    apply (rule ccontr)
1.594    apply (drule_tac e="dist k l" in tendstoD)
1.595 @@ -604,13 +592,13 @@
1.596    done
1.597
1.598  lemma tendsto_dist [tendsto_intros]:
1.599 -  assumes f: "(f ---> l) A" and g: "(g ---> m) A"
1.600 -  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) A"
1.601 +  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
1.602 +  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
1.603  proof (rule tendstoI)
1.604    fix e :: real assume "0 < e"
1.605    hence e2: "0 < e/2" by simp
1.606    from tendstoD [OF f e2] tendstoD [OF g e2]
1.607 -  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) A"
1.608 +  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
1.609    proof (rule eventually_elim2)
1.610      fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
1.611      then show "dist (dist (f x) (g x)) (dist l m) < e"
1.612 @@ -629,68 +617,68 @@
1.613    unfolding dist_norm by simp
1.614
1.615  lemma tendsto_norm [tendsto_intros]:
1.616 -  "(f ---> a) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) A"
1.617 +  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
1.618    unfolding norm_conv_dist by (intro tendsto_intros)
1.619
1.620  lemma tendsto_norm_zero:
1.621 -  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) A"
1.622 +  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
1.623    by (drule tendsto_norm, simp)
1.624
1.625  lemma tendsto_norm_zero_cancel:
1.626 -  "((\<lambda>x. norm (f x)) ---> 0) A \<Longrightarrow> (f ---> 0) A"
1.627 +  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
1.628    unfolding tendsto_iff dist_norm by simp
1.629
1.630  lemma tendsto_norm_zero_iff:
1.631 -  "((\<lambda>x. norm (f x)) ---> 0) A \<longleftrightarrow> (f ---> 0) A"
1.632 +  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
1.633    unfolding tendsto_iff dist_norm by simp
1.634
1.635  lemma tendsto_rabs [tendsto_intros]:
1.636 -  "(f ---> (l::real)) A \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) A"
1.637 +  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
1.638    by (fold real_norm_def, rule tendsto_norm)
1.639
1.640  lemma tendsto_rabs_zero:
1.641 -  "(f ---> (0::real)) A \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) A"
1.642 +  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
1.643    by (fold real_norm_def, rule tendsto_norm_zero)
1.644
1.645  lemma tendsto_rabs_zero_cancel:
1.646 -  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) A \<Longrightarrow> (f ---> 0) A"
1.647 +  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
1.648    by (fold real_norm_def, rule tendsto_norm_zero_cancel)
1.649
1.650  lemma tendsto_rabs_zero_iff:
1.651 -  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) A \<longleftrightarrow> (f ---> 0) A"
1.652 +  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
1.653    by (fold real_norm_def, rule tendsto_norm_zero_iff)
1.654
1.655  subsubsection {* Addition and subtraction *}
1.656
1.658    fixes a b :: "'a::real_normed_vector"
1.659 -  shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) A"
1.660 +  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
1.662
1.664    fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
1.665 -  shows "\<lbrakk>(f ---> 0) A; (g ---> 0) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) A"
1.666 +  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
1.667    by (drule (1) tendsto_add, simp)
1.668
1.669  lemma tendsto_minus [tendsto_intros]:
1.670    fixes a :: "'a::real_normed_vector"
1.671 -  shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) A"
1.672 +  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
1.673    by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
1.674
1.675  lemma tendsto_minus_cancel:
1.676    fixes a :: "'a::real_normed_vector"
1.677 -  shows "((\<lambda>x. - f x) ---> - a) A \<Longrightarrow> (f ---> a) A"
1.678 +  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
1.679    by (drule tendsto_minus, simp)
1.680
1.681  lemma tendsto_diff [tendsto_intros]:
1.682    fixes a b :: "'a::real_normed_vector"
1.683 -  shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) A"
1.684 +  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
1.686
1.687  lemma tendsto_setsum [tendsto_intros]:
1.688    fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
1.689 -  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) A"
1.690 -  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) A"
1.691 +  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
1.692 +  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
1.693  proof (cases "finite S")
1.694    assume "finite S" thus ?thesis using assms
1.696 @@ -702,43 +690,43 @@
1.697  subsubsection {* Linear operators and multiplication *}
1.698
1.699  lemma (in bounded_linear) tendsto [tendsto_intros]:
1.700 -  "(g ---> a) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) A"
1.701 +  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
1.702    by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
1.703
1.704  lemma (in bounded_linear) tendsto_zero:
1.705 -  "(g ---> 0) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) A"
1.706 +  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
1.707    by (drule tendsto, simp only: zero)
1.708
1.709  lemma (in bounded_bilinear) tendsto [tendsto_intros]:
1.710 -  "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) A"
1.711 +  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
1.712    by (simp only: tendsto_Zfun_iff prod_diff_prod
1.714
1.715  lemma (in bounded_bilinear) tendsto_zero:
1.716 -  assumes f: "(f ---> 0) A"
1.717 -  assumes g: "(g ---> 0) A"
1.718 -  shows "((\<lambda>x. f x ** g x) ---> 0) A"
1.719 +  assumes f: "(f ---> 0) F"
1.720 +  assumes g: "(g ---> 0) F"
1.721 +  shows "((\<lambda>x. f x ** g x) ---> 0) F"
1.722    using tendsto [OF f g] by (simp add: zero_left)
1.723
1.724  lemma (in bounded_bilinear) tendsto_left_zero:
1.725 -  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) A"
1.726 +  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
1.727    by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
1.728
1.729  lemma (in bounded_bilinear) tendsto_right_zero:
1.730 -  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) A"
1.731 +  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
1.732    by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
1.733
1.734  lemmas tendsto_mult = mult.tendsto
1.735
1.736  lemma tendsto_power [tendsto_intros]:
1.737    fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
1.738 -  shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) A"
1.739 +  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
1.740    by (induct n) (simp_all add: tendsto_const tendsto_mult)
1.741
1.742  lemma tendsto_setprod [tendsto_intros]:
1.743    fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
1.744 -  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) A"
1.745 -  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) A"
1.746 +  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
1.747 +  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
1.748  proof (cases "finite S")
1.749    assume "finite S" thus ?thesis using assms
1.751 @@ -750,17 +738,17 @@
1.752  subsubsection {* Inverse and division *}
1.753
1.754  lemma (in bounded_bilinear) Zfun_prod_Bfun:
1.755 -  assumes f: "Zfun f A"
1.756 -  assumes g: "Bfun g A"
1.757 -  shows "Zfun (\<lambda>x. f x ** g x) A"
1.758 +  assumes f: "Zfun f F"
1.759 +  assumes g: "Bfun g F"
1.760 +  shows "Zfun (\<lambda>x. f x ** g x) F"
1.761  proof -
1.762    obtain K where K: "0 \<le> K"
1.763      and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
1.764      using nonneg_bounded by fast
1.765    obtain B where B: "0 < B"
1.766 -    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) A"
1.767 +    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
1.768      using g by (rule BfunE)
1.769 -  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) A"
1.770 +  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
1.771    using norm_g proof (rule eventually_elim1)
1.772      fix x
1.773      assume *: "norm (g x) \<le> B"
1.774 @@ -788,9 +776,9 @@
1.775    using bounded by fast
1.776
1.777  lemma (in bounded_bilinear) Bfun_prod_Zfun:
1.778 -  assumes f: "Bfun f A"
1.779 -  assumes g: "Zfun g A"
1.780 -  shows "Zfun (\<lambda>x. f x ** g x) A"
1.781 +  assumes f: "Bfun f F"
1.782 +  assumes g: "Zfun g F"
1.783 +  shows "Zfun (\<lambda>x. f x ** g x) F"
1.784    using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
1.785
1.786  lemma Bfun_inverse_lemma:
1.787 @@ -802,16 +790,16 @@
1.788
1.789  lemma Bfun_inverse:
1.790    fixes a :: "'a::real_normed_div_algebra"
1.791 -  assumes f: "(f ---> a) A"
1.792 +  assumes f: "(f ---> a) F"
1.793    assumes a: "a \<noteq> 0"
1.794 -  shows "Bfun (\<lambda>x. inverse (f x)) A"
1.795 +  shows "Bfun (\<lambda>x. inverse (f x)) F"
1.796  proof -
1.797    from a have "0 < norm a" by simp
1.798    hence "\<exists>r>0. r < norm a" by (rule dense)
1.799    then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
1.800 -  have "eventually (\<lambda>x. dist (f x) a < r) A"
1.801 +  have "eventually (\<lambda>x. dist (f x) a < r) F"
1.802      using tendstoD [OF f r1] by fast
1.803 -  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) A"
1.804 +  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
1.805    proof (rule eventually_elim1)
1.806      fix x
1.807      assume "dist (f x) a < r"
1.808 @@ -838,8 +826,8 @@
1.809
1.810  lemma tendsto_inverse_lemma:
1.811    fixes a :: "'a::real_normed_div_algebra"
1.812 -  shows "\<lbrakk>(f ---> a) A; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) A\<rbrakk>
1.813 -         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) A"
1.814 +  shows "\<lbrakk>(f ---> a) F; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) F\<rbrakk>
1.815 +         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) F"
1.816    apply (subst tendsto_Zfun_iff)
1.817    apply (rule Zfun_ssubst)
1.818    apply (erule eventually_elim1)
1.819 @@ -853,14 +841,14 @@
1.820
1.821  lemma tendsto_inverse [tendsto_intros]:
1.822    fixes a :: "'a::real_normed_div_algebra"
1.823 -  assumes f: "(f ---> a) A"
1.824 +  assumes f: "(f ---> a) F"
1.825    assumes a: "a \<noteq> 0"
1.826 -  shows "((\<lambda>x. inverse (f x)) ---> inverse a) A"
1.827 +  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
1.828  proof -
1.829    from a have "0 < norm a" by simp
1.830 -  with f have "eventually (\<lambda>x. dist (f x) a < norm a) A"
1.831 +  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
1.832      by (rule tendstoD)
1.833 -  then have "eventually (\<lambda>x. f x \<noteq> 0) A"
1.834 +  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
1.835      unfolding dist_norm by (auto elim!: eventually_elim1)
1.836    with f a show ?thesis
1.837      by (rule tendsto_inverse_lemma)
1.838 @@ -868,39 +856,39 @@
1.839
1.840  lemma tendsto_divide [tendsto_intros]:
1.841    fixes a b :: "'a::real_normed_field"
1.842 -  shows "\<lbrakk>(f ---> a) A; (g ---> b) A; b \<noteq> 0\<rbrakk>
1.843 -    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) A"
1.844 +  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
1.845 +    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
1.846    by (simp add: mult.tendsto tendsto_inverse divide_inverse)
1.847
1.848  lemma tendsto_sgn [tendsto_intros]:
1.849    fixes l :: "'a::real_normed_vector"
1.850 -  shows "\<lbrakk>(f ---> l) A; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) A"
1.851 +  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
1.852    unfolding sgn_div_norm by (simp add: tendsto_intros)
1.853
1.854  subsubsection {* Uniqueness *}
1.855
1.856  lemma tendsto_unique:
1.857    fixes f :: "'a \<Rightarrow> 'b::t2_space"
1.858 -  assumes "\<not> trivial_limit A"  "(f ---> l) A"  "(f ---> l') A"
1.859 +  assumes "\<not> trivial_limit F"  "(f ---> l) F"  "(f ---> l') F"
1.860    shows "l = l'"
1.861  proof (rule ccontr)
1.862    assume "l \<noteq> l'"
1.863    obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
1.864      using hausdorff [OF `l \<noteq> l'`] by fast
1.865 -  have "eventually (\<lambda>x. f x \<in> U) A"
1.866 -    using `(f ---> l) A` `open U` `l \<in> U` by (rule topological_tendstoD)
1.867 +  have "eventually (\<lambda>x. f x \<in> U) F"
1.868 +    using `(f ---> l) F` `open U` `l \<in> U` by (rule topological_tendstoD)
1.869    moreover
1.870 -  have "eventually (\<lambda>x. f x \<in> V) A"
1.871 -    using `(f ---> l') A` `open V` `l' \<in> V` by (rule topological_tendstoD)
1.872 +  have "eventually (\<lambda>x. f x \<in> V) F"
1.873 +    using `(f ---> l') F` `open V` `l' \<in> V` by (rule topological_tendstoD)
1.874    ultimately
1.875 -  have "eventually (\<lambda>x. False) A"
1.876 +  have "eventually (\<lambda>x. False) F"
1.877    proof (rule eventually_elim2)
1.878      fix x
1.879      assume "f x \<in> U" "f x \<in> V"
1.880      hence "f x \<in> U \<inter> V" by simp
1.881      with `U \<inter> V = {}` show "False" by simp
1.882    qed
1.883 -  with `\<not> trivial_limit A` show "False"
1.884 +  with `\<not> trivial_limit F` show "False"