src/HOL/Limits.thy
changeset 63546 5f097087fa1e
parent 63301 d3c87eb0bad2
child 63548 6c2c16fef8f1
     1.1 --- a/src/HOL/Limits.thy	Fri Jul 22 21:43:56 2016 +0200
     1.2 +++ b/src/HOL/Limits.thy	Fri Jul 22 23:55:47 2016 +0200
     1.3 @@ -8,13 +8,13 @@
     1.4  section \<open>Limits on Real Vector Spaces\<close>
     1.5  
     1.6  theory Limits
     1.7 -imports Real_Vector_Spaces
     1.8 +  imports Real_Vector_Spaces
     1.9  begin
    1.10  
    1.11  subsection \<open>Filter going to infinity norm\<close>
    1.12  
    1.13 -definition at_infinity :: "'a::real_normed_vector filter" where
    1.14 -  "at_infinity = (INF r. principal {x. r \<le> norm x})"
    1.15 +definition at_infinity :: "'a::real_normed_vector filter"
    1.16 +  where "at_infinity = (INF r. principal {x. r \<le> norm x})"
    1.17  
    1.18  lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
    1.19    unfolding at_infinity_def
    1.20 @@ -22,21 +22,24 @@
    1.21       (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
    1.22  
    1.23  corollary eventually_at_infinity_pos:
    1.24 -   "eventually p at_infinity \<longleftrightarrow> (\<exists>b. 0 < b \<and> (\<forall>x. norm x \<ge> b \<longrightarrow> p x))"
    1.25 -apply (simp add: eventually_at_infinity, auto)
    1.26 -apply (case_tac "b \<le> 0")
    1.27 -using norm_ge_zero order_trans zero_less_one apply blast
    1.28 -apply (force simp:)
    1.29 -done
    1.30 -
    1.31 -lemma at_infinity_eq_at_top_bot:
    1.32 -  "(at_infinity :: real filter) = sup at_top at_bot"
    1.33 +  "eventually p at_infinity \<longleftrightarrow> (\<exists>b. 0 < b \<and> (\<forall>x. norm x \<ge> b \<longrightarrow> p x))"
    1.34 +  apply (simp add: eventually_at_infinity)
    1.35 +  apply auto
    1.36 +  apply (case_tac "b \<le> 0")
    1.37 +  using norm_ge_zero order_trans zero_less_one apply blast
    1.38 +  apply force
    1.39 +  done
    1.40 +
    1.41 +lemma at_infinity_eq_at_top_bot: "(at_infinity :: real filter) = sup at_top at_bot"
    1.42    apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
    1.43 -                   eventually_at_top_linorder eventually_at_bot_linorder)
    1.44 +      eventually_at_top_linorder eventually_at_bot_linorder)
    1.45    apply safe
    1.46 -  apply (rule_tac x="b" in exI, simp)
    1.47 -  apply (rule_tac x="- b" in exI, simp)
    1.48 -  apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
    1.49 +    apply (rule_tac x="b" in exI)
    1.50 +    apply simp
    1.51 +   apply (rule_tac x="- b" in exI)
    1.52 +   apply simp
    1.53 +  apply (rule_tac x="max (- Na) N" in exI)
    1.54 +  apply (auto simp: abs_real_def)
    1.55    done
    1.56  
    1.57  lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
    1.58 @@ -45,23 +48,21 @@
    1.59  lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
    1.60    unfolding at_infinity_eq_at_top_bot by simp
    1.61  
    1.62 -lemma filterlim_at_top_imp_at_infinity:
    1.63 -  fixes f :: "_ \<Rightarrow> real"
    1.64 -  shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
    1.65 +lemma filterlim_at_top_imp_at_infinity: "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
    1.66 +  for f :: "_ \<Rightarrow> real"
    1.67    by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
    1.68  
    1.69 -lemma lim_infinity_imp_sequentially:
    1.70 -  "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
    1.71 -by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
    1.72 +lemma lim_infinity_imp_sequentially: "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
    1.73 +  by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
    1.74  
    1.75  
    1.76  subsubsection \<open>Boundedness\<close>
    1.77  
    1.78 -definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
    1.79 -  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
    1.80 -
    1.81 -abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
    1.82 -  "Bseq X \<equiv> Bfun X sequentially"
    1.83 +definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool"
    1.84 +  where Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
    1.85 +
    1.86 +abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool"
    1.87 +  where "Bseq X \<equiv> Bfun X sequentially"
    1.88  
    1.89  lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
    1.90  
    1.91 @@ -71,11 +72,11 @@
    1.92  lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
    1.93    unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
    1.94  
    1.95 -lemma Bfun_def:
    1.96 -  "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
    1.97 +lemma Bfun_def: "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
    1.98    unfolding Bfun_metric_def norm_conv_dist
    1.99  proof safe
   1.100 -  fix y K assume K: "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
   1.101 +  fix y K
   1.102 +  assume K: "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
   1.103    moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
   1.104      by (intro always_eventually) (metis dist_commute dist_triangle)
   1.105    with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
   1.106 @@ -85,19 +86,19 @@
   1.107  qed (force simp del: norm_conv_dist [symmetric])
   1.108  
   1.109  lemma BfunI:
   1.110 -  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
   1.111 -unfolding Bfun_def
   1.112 +  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F"
   1.113 +  shows "Bfun f F"
   1.114 +  unfolding Bfun_def
   1.115  proof (intro exI conjI allI)
   1.116    show "0 < max K 1" by simp
   1.117 -next
   1.118    show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
   1.119 -    using K by (rule eventually_mono, simp)
   1.120 +    using K by (rule eventually_mono) simp
   1.121  qed
   1.122  
   1.123  lemma BfunE:
   1.124    assumes "Bfun f F"
   1.125    obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
   1.126 -using assms unfolding Bfun_def by blast
   1.127 +  using assms unfolding Bfun_def by blast
   1.128  
   1.129  lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
   1.130    unfolding Cauchy_def Bfun_metric_def eventually_sequentially
   1.131 @@ -124,57 +125,66 @@
   1.132  lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
   1.133    unfolding Bfun_def eventually_sequentially
   1.134  proof safe
   1.135 -  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
   1.136 +  fix N K
   1.137 +  assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
   1.138    then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
   1.139      by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
   1.140         (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
   1.141  qed auto
   1.142  
   1.143 -lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
   1.144 -unfolding Bseq_def by auto
   1.145 -
   1.146 -lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
   1.147 -by (simp add: Bseq_def)
   1.148 -
   1.149 -lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
   1.150 -by (auto simp add: Bseq_def)
   1.151 -
   1.152 -lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
   1.153 +lemma BseqE: "Bseq X \<Longrightarrow> (\<And>K. 0 < K \<Longrightarrow> \<forall>n. norm (X n) \<le> K \<Longrightarrow> Q) \<Longrightarrow> Q"
   1.154 +  unfolding Bseq_def by auto
   1.155 +
   1.156 +lemma BseqD: "Bseq X \<Longrightarrow> \<exists>K. 0 < K \<and> (\<forall>n. norm (X n) \<le> K)"
   1.157 +  by (simp add: Bseq_def)
   1.158 +
   1.159 +lemma BseqI: "0 < K \<Longrightarrow> \<forall>n. norm (X n) \<le> K \<Longrightarrow> Bseq X"
   1.160 +  by (auto simp add: Bseq_def)
   1.161 +
   1.162 +lemma Bseq_bdd_above: "Bseq X \<Longrightarrow> bdd_above (range X)"
   1.163 +  for X :: "nat \<Rightarrow> real"
   1.164  proof (elim BseqE, intro bdd_aboveI2)
   1.165 -  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
   1.166 +  fix K n
   1.167 +  assume "0 < K" "\<forall>n. norm (X n) \<le> K"
   1.168 +  then show "X n \<le> K"
   1.169      by (auto elim!: allE[of _ n])
   1.170  qed
   1.171  
   1.172 -lemma Bseq_bdd_above':
   1.173 -  "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
   1.174 +lemma Bseq_bdd_above': "Bseq X \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
   1.175 +  for X :: "nat \<Rightarrow> 'a :: real_normed_vector"
   1.176  proof (elim BseqE, intro bdd_aboveI2)
   1.177 -  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"
   1.178 +  fix K n
   1.179 +  assume "0 < K" "\<forall>n. norm (X n) \<le> K"
   1.180 +  then show "norm (X n) \<le> K"
   1.181      by (auto elim!: allE[of _ n])
   1.182  qed
   1.183  
   1.184 -lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
   1.185 +lemma Bseq_bdd_below: "Bseq X \<Longrightarrow> bdd_below (range X)"
   1.186 +  for X :: "nat \<Rightarrow> real"
   1.187  proof (elim BseqE, intro bdd_belowI2)
   1.188 -  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
   1.189 +  fix K n
   1.190 +  assume "0 < K" "\<forall>n. norm (X n) \<le> K"
   1.191 +  then show "- K \<le> X n"
   1.192      by (auto elim!: allE[of _ n])
   1.193  qed
   1.194  
   1.195  lemma Bseq_eventually_mono:
   1.196    assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
   1.197 -  shows   "Bseq f"
   1.198 +  shows "Bseq f"
   1.199  proof -
   1.200    from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"
   1.201      by (auto simp: eventually_at_top_linorder)
   1.202 -  moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)
   1.203 +  moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K"
   1.204 +    by (blast elim!: BseqE)
   1.205    ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n
   1.206      apply (cases "n < N")
   1.207 -    apply (rule max.coboundedI2, rule Max.coboundedI, auto) []
   1.208 -    apply (rule max.coboundedI1, force intro: order.trans[OF N K])
   1.209 +    subgoal by (rule max.coboundedI2, rule Max.coboundedI) auto
   1.210 +    subgoal by (rule max.coboundedI1) (force intro: order.trans[OF N K])
   1.211      done
   1.212 -  thus ?thesis by (blast intro: BseqI')
   1.213 +  then show ?thesis by (blast intro: BseqI')
   1.214  qed
   1.215  
   1.216 -lemma lemma_NBseq_def:
   1.217 -  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
   1.218 +lemma lemma_NBseq_def: "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
   1.219  proof safe
   1.220    fix K :: real
   1.221    from reals_Archimedean2 obtain n :: nat where "K < real n" ..
   1.222 @@ -188,47 +198,50 @@
   1.223      using of_nat_0_less_iff by blast
   1.224  qed
   1.225  
   1.226 -text\<open>alternative definition for Bseq\<close>
   1.227 -lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
   1.228 -apply (simp add: Bseq_def)
   1.229 -apply (simp (no_asm) add: lemma_NBseq_def)
   1.230 -done
   1.231 -
   1.232 -lemma lemma_NBseq_def2:
   1.233 -     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
   1.234 -apply (subst lemma_NBseq_def, auto)
   1.235 -apply (rule_tac x = "Suc N" in exI)
   1.236 -apply (rule_tac [2] x = N in exI)
   1.237 -apply (auto simp add: of_nat_Suc)
   1.238 - prefer 2 apply (blast intro: order_less_imp_le)
   1.239 -apply (drule_tac x = n in spec, simp)
   1.240 -done
   1.241 -
   1.242 -(* yet another definition for Bseq *)
   1.243 -lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
   1.244 -by (simp add: Bseq_def lemma_NBseq_def2)
   1.245 -
   1.246 -subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
   1.247 -
   1.248 -text\<open>alternative formulation for boundedness\<close>
   1.249 -lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
   1.250 -apply (unfold Bseq_def, safe)
   1.251 -apply (rule_tac [2] x = "k + norm x" in exI)
   1.252 -apply (rule_tac x = K in exI, simp)
   1.253 -apply (rule exI [where x = 0], auto)
   1.254 -apply (erule order_less_le_trans, simp)
   1.255 -apply (drule_tac x=n in spec)
   1.256 -apply (drule order_trans [OF norm_triangle_ineq2])
   1.257 -apply simp
   1.258 -done
   1.259 -
   1.260 -text\<open>alternative formulation for boundedness\<close>
   1.261 -lemma Bseq_iff3:
   1.262 -  "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
   1.263 +text \<open>Alternative definition for \<open>Bseq\<close>.\<close>
   1.264 +lemma Bseq_iff: "Bseq X \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
   1.265 +  by (simp add: Bseq_def) (simp add: lemma_NBseq_def)
   1.266 +
   1.267 +lemma lemma_NBseq_def2: "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
   1.268 +  apply (subst lemma_NBseq_def)
   1.269 +  apply auto
   1.270 +   apply (rule_tac x = "Suc N" in exI)
   1.271 +   apply (rule_tac [2] x = N in exI)
   1.272 +   apply auto
   1.273 +   prefer 2 apply (blast intro: order_less_imp_le)
   1.274 +  apply (drule_tac x = n in spec)
   1.275 +  apply simp
   1.276 +  done
   1.277 +
   1.278 +text \<open>Yet another definition for Bseq.\<close>
   1.279 +lemma Bseq_iff1a: "Bseq X \<longleftrightarrow> (\<exists>N. \<forall>n. norm (X n) < real (Suc N))"
   1.280 +  by (simp add: Bseq_def lemma_NBseq_def2)
   1.281 +
   1.282 +subsubsection \<open>A Few More Equivalence Theorems for Boundedness\<close>
   1.283 +
   1.284 +text \<open>Alternative formulation for boundedness.\<close>
   1.285 +lemma Bseq_iff2: "Bseq X \<longleftrightarrow> (\<exists>k > 0. \<exists>x. \<forall>n. norm (X n + - x) \<le> k)"
   1.286 +  apply (unfold Bseq_def)
   1.287 +  apply safe
   1.288 +   apply (rule_tac [2] x = "k + norm x" in exI)
   1.289 +   apply (rule_tac x = K in exI)
   1.290 +   apply simp
   1.291 +   apply (rule exI [where x = 0])
   1.292 +   apply auto
   1.293 +   apply (erule order_less_le_trans)
   1.294 +   apply simp
   1.295 +  apply (drule_tac x=n in spec)
   1.296 +  apply (drule order_trans [OF norm_triangle_ineq2])
   1.297 +  apply simp
   1.298 +  done
   1.299 +
   1.300 +text \<open>Alternative formulation for boundedness.\<close>
   1.301 +lemma Bseq_iff3: "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)"
   1.302 +  (is "?P \<longleftrightarrow> ?Q")
   1.303  proof
   1.304    assume ?P
   1.305 -  then obtain K
   1.306 -    where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
   1.307 +  then obtain K where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K"
   1.308 +    by (auto simp add: Bseq_def)
   1.309    from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
   1.310    from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
   1.311      by (auto intro: order_trans norm_triangle_ineq4)
   1.312 @@ -236,129 +249,150 @@
   1.313      by simp
   1.314    with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
   1.315  next
   1.316 -  assume ?Q then show ?P by (auto simp add: Bseq_iff2)
   1.317 +  assume ?Q
   1.318 +  then show ?P by (auto simp add: Bseq_iff2)
   1.319  qed
   1.320  
   1.321 -lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
   1.322 -apply (simp add: Bseq_def)
   1.323 -apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
   1.324 -apply (drule_tac x = n in spec, arith)
   1.325 -done
   1.326 -
   1.327 -
   1.328 -subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
   1.329 -
   1.330 -lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
   1.331 +lemma BseqI2: "\<forall>n. k \<le> f n \<and> f n \<le> K \<Longrightarrow> Bseq f"
   1.332 +  for k K :: real
   1.333 +  apply (simp add: Bseq_def)
   1.334 +  apply (rule_tac x = "(\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI)
   1.335 +  apply auto
   1.336 +  apply (drule_tac x = n in spec)
   1.337 +  apply arith
   1.338 +  done
   1.339 +
   1.340 +
   1.341 +subsubsection \<open>Upper Bounds and Lubs of Bounded Sequences\<close>
   1.342 +
   1.343 +lemma Bseq_minus_iff: "Bseq (\<lambda>n. - (X n) :: 'a::real_normed_vector) \<longleftrightarrow> Bseq X"
   1.344    by (simp add: Bseq_def)
   1.345  
   1.346  lemma Bseq_add:
   1.347 -  assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
   1.348 -  shows   "Bseq (\<lambda>x. f x + c)"
   1.349 +  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.350 +  assumes "Bseq f"
   1.351 +  shows "Bseq (\<lambda>x. f x + c)"
   1.352  proof -
   1.353 -  from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast
   1.354 +  from assms obtain K where K: "\<And>x. norm (f x) \<le> K"
   1.355 +    unfolding Bseq_def by blast
   1.356    {
   1.357      fix x :: nat
   1.358      have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
   1.359      also have "norm (f x) \<le> K" by (rule K)
   1.360      finally have "norm (f x + c) \<le> K + norm c" by simp
   1.361    }
   1.362 -  thus ?thesis by (rule BseqI')
   1.363 +  then show ?thesis by (rule BseqI')
   1.364  qed
   1.365  
   1.366 -lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
   1.367 +lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq f"
   1.368 +  for f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.369    using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
   1.370  
   1.371  lemma Bseq_mult:
   1.372 -  assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"
   1.373 -  assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"
   1.374 -  shows   "Bseq (\<lambda>x. f x * g x)"
   1.375 +  fixes f g :: "nat \<Rightarrow> 'a::real_normed_field"
   1.376 +  assumes "Bseq f" and "Bseq g"
   1.377 +  shows "Bseq (\<lambda>x. f x * g x)"
   1.378  proof -
   1.379 -  from assms obtain K1 K2 where K: "\<And>x. norm (f x) \<le> K1" "K1 > 0" "\<And>x. norm (g x) \<le> K2" "K2 > 0"
   1.380 +  from assms obtain K1 K2 where K: "norm (f x) \<le> K1" "K1 > 0" "norm (g x) \<le> K2" "K2 > 0"
   1.381 +    for x
   1.382      unfolding Bseq_def by blast
   1.383 -  hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)
   1.384 -  thus ?thesis by (rule BseqI')
   1.385 +  then have "norm (f x * g x) \<le> K1 * K2" for x
   1.386 +    by (auto simp: norm_mult intro!: mult_mono)
   1.387 +  then show ?thesis by (rule BseqI')
   1.388  qed
   1.389  
   1.390  lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
   1.391    unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
   1.392  
   1.393 -lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
   1.394 +lemma Bseq_cmult_iff:
   1.395 +  fixes c :: "'a::real_normed_field"
   1.396 +  assumes "c \<noteq> 0"
   1.397 +  shows "Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
   1.398  proof
   1.399 -  assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
   1.400 -  find_theorems "Bfun (\<lambda>_. ?c) _"
   1.401 -  from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
   1.402 -  with \<open>c \<noteq> 0\<close> show "Bseq f" by (simp add: divide_simps)
   1.403 +  assume "Bseq (\<lambda>x. c * f x)"
   1.404 +  with Bfun_const have "Bseq (\<lambda>x. inverse c * (c * f x))"
   1.405 +    by (rule Bseq_mult)
   1.406 +  with \<open>c \<noteq> 0\<close> show "Bseq f"
   1.407 +    by (simp add: divide_simps)
   1.408  qed (intro Bseq_mult Bfun_const)
   1.409  
   1.410 -lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
   1.411 +lemma Bseq_subseq: "Bseq f \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
   1.412 +  for f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.413    unfolding Bseq_def by auto
   1.414  
   1.415 -lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
   1.416 +lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq f"
   1.417 +  for f :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.418    using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
   1.419  
   1.420  lemma increasing_Bseq_subseq_iff:
   1.421 -  assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
   1.422 -  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
   1.423 +  assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a::real_normed_vector) \<le> norm (f y)" "subseq g"
   1.424 +  shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
   1.425  proof
   1.426    assume "Bseq (\<lambda>x. f (g x))"
   1.427 -  then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
   1.428 +  then obtain K where K: "\<And>x. norm (f (g x)) \<le> K"
   1.429 +    unfolding Bseq_def by auto
   1.430    {
   1.431      fix x :: nat
   1.432      from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
   1.433        by (auto simp: filterlim_at_top eventually_at_top_linorder)
   1.434 -    hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
   1.435 +    then have "norm (f x) \<le> norm (f (g y))"
   1.436 +      using assms(1) by blast
   1.437      also have "norm (f (g y)) \<le> K" by (rule K)
   1.438      finally have "norm (f x) \<le> K" .
   1.439    }
   1.440 -  thus "Bseq f" by (rule BseqI')
   1.441 -qed (insert Bseq_subseq[of f g], simp_all)
   1.442 +  then show "Bseq f" by (rule BseqI')
   1.443 +qed (use Bseq_subseq[of f g] in simp_all)
   1.444  
   1.445  lemma nonneg_incseq_Bseq_subseq_iff:
   1.446 -  assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
   1.447 -  shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
   1.448 +  fixes f :: "nat \<Rightarrow> real"
   1.449 +    and g :: "nat \<Rightarrow> nat"
   1.450 +  assumes "\<And>x. f x \<ge> 0" "incseq f" "subseq g"
   1.451 +  shows "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
   1.452    using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
   1.453  
   1.454 -lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
   1.455 +lemma Bseq_eq_bounded: "range f \<subseteq> {a..b} \<Longrightarrow> Bseq f"
   1.456 +  for a b :: real
   1.457    apply (simp add: subset_eq)
   1.458    apply (rule BseqI'[where K="max (norm a) (norm b)"])
   1.459    apply (erule_tac x=n in allE)
   1.460    apply auto
   1.461    done
   1.462  
   1.463 -lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
   1.464 +lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> B \<Longrightarrow> Bseq X"
   1.465 +  for B :: real
   1.466    by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
   1.467  
   1.468 -lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
   1.469 +lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. B \<le> X i \<Longrightarrow> Bseq X"
   1.470 +  for B :: real
   1.471    by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
   1.472  
   1.473 +
   1.474  subsection \<open>Bounded Monotonic Sequences\<close>
   1.475  
   1.476 -subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
   1.477 +subsubsection \<open>A Bounded and Monotonic Sequence Converges\<close>
   1.478  
   1.479  (* TODO: delete *)
   1.480  (* FIXME: one use in NSA/HSEQ.thy *)
   1.481 -lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
   1.482 +lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n \<longrightarrow> X n = X m \<Longrightarrow> \<exists>L. X \<longlonglongrightarrow> L"
   1.483    apply (rule_tac x="X m" in exI)
   1.484    apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
   1.485    unfolding eventually_sequentially
   1.486    apply blast
   1.487    done
   1.488  
   1.489 +
   1.490  subsection \<open>Convergence to Zero\<close>
   1.491  
   1.492  definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
   1.493    where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
   1.494  
   1.495 -lemma ZfunI:
   1.496 -  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
   1.497 -  unfolding Zfun_def by simp
   1.498 -
   1.499 -lemma ZfunD:
   1.500 -  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
   1.501 -  unfolding Zfun_def by simp
   1.502 -
   1.503 -lemma Zfun_ssubst:
   1.504 -  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
   1.505 +lemma ZfunI: "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
   1.506 +  by (simp add: Zfun_def)
   1.507 +
   1.508 +lemma ZfunD: "Zfun f F \<Longrightarrow> 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
   1.509 +  by (simp add: Zfun_def)
   1.510 +
   1.511 +lemma Zfun_ssubst: "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
   1.512    unfolding Zfun_def by (auto elim!: eventually_rev_mp)
   1.513  
   1.514  lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
   1.515 @@ -369,28 +403,29 @@
   1.516  
   1.517  lemma Zfun_imp_Zfun:
   1.518    assumes f: "Zfun f F"
   1.519 -  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
   1.520 +    and g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
   1.521    shows "Zfun (\<lambda>x. g x) F"
   1.522 -proof (cases)
   1.523 -  assume K: "0 < K"
   1.524 +proof (cases "0 < K")
   1.525 +  case K: True
   1.526    show ?thesis
   1.527    proof (rule ZfunI)
   1.528 -    fix r::real assume "0 < r"
   1.529 -    hence "0 < r / K" using K by simp
   1.530 +    fix r :: real
   1.531 +    assume "0 < r"
   1.532 +    then have "0 < r / K" using K by simp
   1.533      then have "eventually (\<lambda>x. norm (f x) < r / K) F"
   1.534        using ZfunD [OF f] by blast
   1.535      with g show "eventually (\<lambda>x. norm (g x) < r) F"
   1.536      proof eventually_elim
   1.537        case (elim x)
   1.538 -      hence "norm (f x) * K < r"
   1.539 +      then have "norm (f x) * K < r"
   1.540          by (simp add: pos_less_divide_eq K)
   1.541 -      thus ?case
   1.542 +      then show ?case
   1.543          by (simp add: order_le_less_trans [OF elim(1)])
   1.544      qed
   1.545    qed
   1.546  next
   1.547 -  assume "\<not> 0 < K"
   1.548 -  hence K: "K \<le> 0" by (simp only: not_less)
   1.549 +  case False
   1.550 +  then have K: "K \<le> 0" by (simp only: not_less)
   1.551    show ?thesis
   1.552    proof (rule ZfunI)
   1.553      fix r :: real
   1.554 @@ -406,15 +441,17 @@
   1.555    qed
   1.556  qed
   1.557  
   1.558 -lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
   1.559 -  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
   1.560 +lemma Zfun_le: "Zfun g F \<Longrightarrow> \<forall>x. norm (f x) \<le> norm (g x) \<Longrightarrow> Zfun f F"
   1.561 +  by (erule Zfun_imp_Zfun [where K = 1]) simp
   1.562  
   1.563  lemma Zfun_add:
   1.564 -  assumes f: "Zfun f F" and g: "Zfun g F"
   1.565 +  assumes f: "Zfun f F"
   1.566 +    and g: "Zfun g F"
   1.567    shows "Zfun (\<lambda>x. f x + g x) F"
   1.568  proof (rule ZfunI)
   1.569 -  fix r::real assume "0 < r"
   1.570 -  hence r: "0 < r / 2" by simp
   1.571 +  fix r :: real
   1.572 +  assume "0 < r"
   1.573 +  then have r: "0 < r / 2" by simp
   1.574    have "eventually (\<lambda>x. norm (f x) < r/2) F"
   1.575      using f r by (rule ZfunD)
   1.576    moreover
   1.577 @@ -436,14 +473,14 @@
   1.578  lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
   1.579    unfolding Zfun_def by simp
   1.580  
   1.581 -lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
   1.582 +lemma Zfun_diff: "Zfun f F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
   1.583    using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
   1.584  
   1.585  lemma (in bounded_linear) Zfun:
   1.586    assumes g: "Zfun g F"
   1.587    shows "Zfun (\<lambda>x. f (g x)) F"
   1.588  proof -
   1.589 -  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
   1.590 +  obtain K where "norm (f x) \<le> norm x * K" for x
   1.591      using bounded by blast
   1.592    then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
   1.593      by simp
   1.594 @@ -453,12 +490,13 @@
   1.595  
   1.596  lemma (in bounded_bilinear) Zfun:
   1.597    assumes f: "Zfun f F"
   1.598 -  assumes g: "Zfun g F"
   1.599 +    and g: "Zfun g F"
   1.600    shows "Zfun (\<lambda>x. f x ** g x) F"
   1.601  proof (rule ZfunI)
   1.602 -  fix r::real assume r: "0 < r"
   1.603 +  fix r :: real
   1.604 +  assume r: "0 < r"
   1.605    obtain K where K: "0 < K"
   1.606 -    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
   1.607 +    and norm_le: "norm (x ** y) \<le> norm x * norm y * K" for x y
   1.608      using pos_bounded by blast
   1.609    from K have K': "0 < inverse K"
   1.610      by (rule positive_imp_inverse_positive)
   1.611 @@ -481,12 +519,10 @@
   1.612    qed
   1.613  qed
   1.614  
   1.615 -lemma (in bounded_bilinear) Zfun_left:
   1.616 -  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
   1.617 +lemma (in bounded_bilinear) Zfun_left: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
   1.618    by (rule bounded_linear_left [THEN bounded_linear.Zfun])
   1.619  
   1.620 -lemma (in bounded_bilinear) Zfun_right:
   1.621 -  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
   1.622 +lemma (in bounded_bilinear) Zfun_right: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
   1.623    by (rule bounded_linear_right [THEN bounded_linear.Zfun])
   1.624  
   1.625  lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
   1.626 @@ -496,19 +532,22 @@
   1.627  lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
   1.628    by (simp only: tendsto_iff Zfun_def dist_norm)
   1.629  
   1.630 -lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
   1.631 -                     \<Longrightarrow> (g \<longlongrightarrow> 0) F"
   1.632 +lemma tendsto_0_le:
   1.633 +  "(f \<longlongrightarrow> 0) F \<Longrightarrow> eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F \<Longrightarrow> (g \<longlongrightarrow> 0) F"
   1.634    by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
   1.635  
   1.636 +
   1.637  subsubsection \<open>Distance and norms\<close>
   1.638  
   1.639  lemma tendsto_dist [tendsto_intros]:
   1.640 -  fixes l m :: "'a :: metric_space"
   1.641 -  assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
   1.642 +  fixes l m :: "'a::metric_space"
   1.643 +  assumes f: "(f \<longlongrightarrow> l) F"
   1.644 +    and g: "(g \<longlongrightarrow> m) F"
   1.645    shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
   1.646  proof (rule tendstoI)
   1.647 -  fix e :: real assume "0 < e"
   1.648 -  hence e2: "0 < e/2" by simp
   1.649 +  fix e :: real
   1.650 +  assume "0 < e"
   1.651 +  then have e2: "0 < e/2" by simp
   1.652    from tendstoD [OF f e2] tendstoD [OF g e2]
   1.653    show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
   1.654    proof (eventually_elim)
   1.655 @@ -516,9 +555,9 @@
   1.656      then show "dist (dist (f x) (g x)) (dist l m) < e"
   1.657        unfolding dist_real_def
   1.658        using dist_triangle2 [of "f x" "g x" "l"]
   1.659 -      using dist_triangle2 [of "g x" "l" "m"]
   1.660 -      using dist_triangle3 [of "l" "m" "f x"]
   1.661 -      using dist_triangle [of "f x" "m" "g x"]
   1.662 +        and dist_triangle2 [of "g x" "l" "m"]
   1.663 +        and dist_triangle3 [of "l" "m" "f x"]
   1.664 +        and dist_triangle [of "f x" "m" "g x"]
   1.665        by arith
   1.666    qed
   1.667  qed
   1.668 @@ -533,33 +572,28 @@
   1.669    shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
   1.670    unfolding continuous_on_def by (auto intro: tendsto_dist)
   1.671  
   1.672 -lemma tendsto_norm [tendsto_intros]:
   1.673 -  "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
   1.674 +lemma tendsto_norm [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
   1.675    unfolding norm_conv_dist by (intro tendsto_intros)
   1.676  
   1.677 -lemma continuous_norm [continuous_intros]:
   1.678 -  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
   1.679 +lemma continuous_norm [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
   1.680    unfolding continuous_def by (rule tendsto_norm)
   1.681  
   1.682  lemma continuous_on_norm [continuous_intros]:
   1.683    "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
   1.684    unfolding continuous_on_def by (auto intro: tendsto_norm)
   1.685  
   1.686 -lemma tendsto_norm_zero:
   1.687 -  "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
   1.688 -  by (drule tendsto_norm, simp)
   1.689 -
   1.690 -lemma tendsto_norm_zero_cancel:
   1.691 -  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
   1.692 +lemma tendsto_norm_zero: "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
   1.693 +  by (drule tendsto_norm) simp
   1.694 +
   1.695 +lemma tendsto_norm_zero_cancel: "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
   1.696    unfolding tendsto_iff dist_norm by simp
   1.697  
   1.698 -lemma tendsto_norm_zero_iff:
   1.699 -  "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
   1.700 +lemma tendsto_norm_zero_iff: "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
   1.701    unfolding tendsto_iff dist_norm by simp
   1.702  
   1.703 -lemma tendsto_rabs [tendsto_intros]:
   1.704 -  "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
   1.705 -  by (fold real_norm_def, rule tendsto_norm)
   1.706 +lemma tendsto_rabs [tendsto_intros]: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
   1.707 +  for l :: real
   1.708 +  by (fold real_norm_def) (rule tendsto_norm)
   1.709  
   1.710  lemma continuous_rabs [continuous_intros]:
   1.711    "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
   1.712 @@ -569,17 +603,15 @@
   1.713    "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
   1.714    unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
   1.715  
   1.716 -lemma tendsto_rabs_zero:
   1.717 -  "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
   1.718 -  by (fold real_norm_def, rule tendsto_norm_zero)
   1.719 -
   1.720 -lemma tendsto_rabs_zero_cancel:
   1.721 -  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
   1.722 -  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
   1.723 -
   1.724 -lemma tendsto_rabs_zero_iff:
   1.725 -  "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
   1.726 -  by (fold real_norm_def, rule tendsto_norm_zero_iff)
   1.727 +lemma tendsto_rabs_zero: "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
   1.728 +  by (fold real_norm_def) (rule tendsto_norm_zero)
   1.729 +
   1.730 +lemma tendsto_rabs_zero_cancel: "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
   1.731 +  by (fold real_norm_def) (rule tendsto_norm_zero_cancel)
   1.732 +
   1.733 +lemma tendsto_rabs_zero_iff: "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
   1.734 +  by (fold real_norm_def) (rule tendsto_norm_zero_iff)
   1.735 +
   1.736  
   1.737  subsection \<open>Topological Monoid\<close>
   1.738  
   1.739 @@ -606,17 +638,22 @@
   1.740  
   1.741  lemma tendsto_add_zero:
   1.742    fixes f g :: "_ \<Rightarrow> 'b::topological_monoid_add"
   1.743 -  shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
   1.744 -  by (drule (1) tendsto_add, simp)
   1.745 +  shows "(f \<longlongrightarrow> 0) F \<Longrightarrow> (g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
   1.746 +  by (drule (1) tendsto_add) simp
   1.747  
   1.748  lemma tendsto_setsum [tendsto_intros]:
   1.749    fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::topological_comm_monoid_add"
   1.750    assumes "\<And>i. i \<in> I \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
   1.751    shows "((\<lambda>x. \<Sum>i\<in>I. f i x) \<longlongrightarrow> (\<Sum>i\<in>I. a i)) F"
   1.752  proof (cases "finite I")
   1.753 -  assume "finite I" thus ?thesis using assms
   1.754 -    by (induct, simp, simp add: tendsto_add)
   1.755 -qed simp
   1.756 +  case True
   1.757 +  then show ?thesis
   1.758 +    using assms by induct (simp_all add: tendsto_add)
   1.759 +next
   1.760 +  case False
   1.761 +  then show ?thesis
   1.762 +    by simp
   1.763 +qed
   1.764  
   1.765  lemma continuous_setsum [continuous_intros]:
   1.766    fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
   1.767 @@ -629,10 +666,13 @@
   1.768    unfolding continuous_on_def by (auto intro: tendsto_setsum)
   1.769  
   1.770  instance nat :: topological_comm_monoid_add
   1.771 -  proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
   1.772 +  by standard
   1.773 +    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
   1.774  
   1.775  instance int :: topological_comm_monoid_add
   1.776 -  proof qed (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
   1.777 +  by standard
   1.778 +    (simp add: nhds_discrete principal_prod_principal filterlim_principal eventually_principal)
   1.779 +
   1.780  
   1.781  subsubsection \<open>Topological group\<close>
   1.782  
   1.783 @@ -640,7 +680,7 @@
   1.784    assumes tendsto_uminus_nhds: "(uminus \<longlongrightarrow> - a) (nhds a)"
   1.785  begin
   1.786  
   1.787 -lemma tendsto_minus [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> -a) F"
   1.788 +lemma tendsto_minus [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
   1.789    by (rule filterlim_compose[OF tendsto_uminus_nhds])
   1.790  
   1.791  end
   1.792 @@ -649,29 +689,26 @@
   1.793  
   1.794  instance topological_ab_group_add < topological_comm_monoid_add ..
   1.795  
   1.796 -lemma continuous_minus [continuous_intros]:
   1.797 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
   1.798 -  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
   1.799 +lemma continuous_minus [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
   1.800 +  for f :: "'a::t2_space \<Rightarrow> 'b::topological_group_add"
   1.801    unfolding continuous_def by (rule tendsto_minus)
   1.802  
   1.803 -lemma continuous_on_minus [continuous_intros]:
   1.804 -  fixes f :: "_ \<Rightarrow> 'b::topological_group_add"
   1.805 -  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
   1.806 +lemma continuous_on_minus [continuous_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
   1.807 +  for f :: "_ \<Rightarrow> 'b::topological_group_add"
   1.808    unfolding continuous_on_def by (auto intro: tendsto_minus)
   1.809  
   1.810 -lemma tendsto_minus_cancel:
   1.811 -  fixes a :: "'a::topological_group_add"
   1.812 -  shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
   1.813 -  by (drule tendsto_minus, simp)
   1.814 +lemma tendsto_minus_cancel: "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
   1.815 +  for a :: "'a::topological_group_add"
   1.816 +  by (drule tendsto_minus) simp
   1.817  
   1.818  lemma tendsto_minus_cancel_left:
   1.819 -    "(f \<longlongrightarrow> - (y::_::topological_group_add)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
   1.820 +  "(f \<longlongrightarrow> - (y::_::topological_group_add)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
   1.821    using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
   1.822    by auto
   1.823  
   1.824  lemma tendsto_diff [tendsto_intros]:
   1.825    fixes a b :: "'a::topological_group_add"
   1.826 -  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
   1.827 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
   1.828    using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
   1.829  
   1.830  lemma continuous_diff [continuous_intros]:
   1.831 @@ -689,7 +726,8 @@
   1.832  
   1.833  instance real_normed_vector < topological_ab_group_add
   1.834  proof
   1.835 -  fix a b :: 'a show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
   1.836 +  fix a b :: 'a
   1.837 +  show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>F nhds b)"
   1.838      unfolding tendsto_Zfun_iff add_diff_add
   1.839      using tendsto_fst[OF filterlim_ident, of "(a,b)"] tendsto_snd[OF filterlim_ident, of "(a,b)"]
   1.840      by (intro Zfun_add)
   1.841 @@ -702,32 +740,28 @@
   1.842  
   1.843  lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
   1.844  
   1.845 +
   1.846  subsubsection \<open>Linear operators and multiplication\<close>
   1.847  
   1.848 -lemma linear_times:
   1.849 -  fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
   1.850 +lemma linear_times: "linear (\<lambda>x. c * x)"
   1.851 +  for c :: "'a::real_algebra"
   1.852    by (auto simp: linearI distrib_left)
   1.853  
   1.854 -lemma (in bounded_linear) tendsto:
   1.855 -  "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
   1.856 +lemma (in bounded_linear) tendsto: "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
   1.857    by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
   1.858  
   1.859 -lemma (in bounded_linear) continuous:
   1.860 -  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
   1.861 +lemma (in bounded_linear) continuous: "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
   1.862    using tendsto[of g _ F] by (auto simp: continuous_def)
   1.863  
   1.864 -lemma (in bounded_linear) continuous_on:
   1.865 -  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
   1.866 +lemma (in bounded_linear) continuous_on: "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
   1.867    using tendsto[of g] by (auto simp: continuous_on_def)
   1.868  
   1.869 -lemma (in bounded_linear) tendsto_zero:
   1.870 -  "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
   1.871 -  by (drule tendsto, simp only: zero)
   1.872 +lemma (in bounded_linear) tendsto_zero: "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
   1.873 +  by (drule tendsto) (simp only: zero)
   1.874  
   1.875  lemma (in bounded_bilinear) tendsto:
   1.876 -  "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
   1.877 -  by (simp only: tendsto_Zfun_iff prod_diff_prod
   1.878 -                 Zfun_add Zfun Zfun_left Zfun_right)
   1.879 +  "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
   1.880 +  by (simp only: tendsto_Zfun_iff prod_diff_prod Zfun_add Zfun Zfun_left Zfun_right)
   1.881  
   1.882  lemma (in bounded_bilinear) continuous:
   1.883    "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
   1.884 @@ -739,7 +773,7 @@
   1.885  
   1.886  lemma (in bounded_bilinear) tendsto_zero:
   1.887    assumes f: "(f \<longlongrightarrow> 0) F"
   1.888 -  assumes g: "(g \<longlongrightarrow> 0) F"
   1.889 +    and g: "(g \<longlongrightarrow> 0) F"
   1.890    shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
   1.891    using tendsto [OF f g] by (simp add: zero_left)
   1.892  
   1.893 @@ -760,15 +794,13 @@
   1.894  lemmas tendsto_mult [tendsto_intros] =
   1.895    bounded_bilinear.tendsto [OF bounded_bilinear_mult]
   1.896  
   1.897 -lemma tendsto_mult_left:
   1.898 -  fixes c::"'a::real_normed_algebra"
   1.899 -  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
   1.900 -by (rule tendsto_mult [OF tendsto_const])
   1.901 -
   1.902 -lemma tendsto_mult_right:
   1.903 -  fixes c::"'a::real_normed_algebra"
   1.904 -  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
   1.905 -by (rule tendsto_mult [OF _ tendsto_const])
   1.906 +lemma tendsto_mult_left: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
   1.907 +  for c :: "'a::real_normed_algebra"
   1.908 +  by (rule tendsto_mult [OF tendsto_const])
   1.909 +
   1.910 +lemma tendsto_mult_right: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
   1.911 +  for c :: "'a::real_normed_algebra"
   1.912 +  by (rule tendsto_mult [OF _ tendsto_const])
   1.913  
   1.914  lemmas continuous_of_real [continuous_intros] =
   1.915    bounded_linear.continuous [OF bounded_linear_of_real]
   1.916 @@ -797,14 +829,12 @@
   1.917  lemmas tendsto_mult_right_zero =
   1.918    bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
   1.919  
   1.920 -lemma tendsto_power [tendsto_intros]:
   1.921 -  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
   1.922 -  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
   1.923 +lemma tendsto_power [tendsto_intros]: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
   1.924 +  for f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
   1.925    by (induct n) (simp_all add: tendsto_mult)
   1.926  
   1.927 -lemma continuous_power [continuous_intros]:
   1.928 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
   1.929 -  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
   1.930 +lemma continuous_power [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
   1.931 +  for f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
   1.932    unfolding continuous_def by (rule tendsto_power)
   1.933  
   1.934  lemma continuous_on_power [continuous_intros]:
   1.935 @@ -817,9 +847,13 @@
   1.936    assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
   1.937    shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
   1.938  proof (cases "finite S")
   1.939 -  assume "finite S" thus ?thesis using assms
   1.940 -    by (induct, simp, simp add: tendsto_mult)
   1.941 -qed simp
   1.942 +  case True
   1.943 +  then show ?thesis using assms
   1.944 +    by induct (simp_all add: tendsto_mult)
   1.945 +next
   1.946 +  case False
   1.947 +  then show ?thesis by simp
   1.948 +qed
   1.949  
   1.950  lemma continuous_setprod [continuous_intros]:
   1.951    fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
   1.952 @@ -832,20 +866,20 @@
   1.953    unfolding continuous_on_def by (auto intro: tendsto_setprod)
   1.954  
   1.955  lemma tendsto_of_real_iff:
   1.956 -  "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
   1.957 +  "((\<lambda>x. of_real (f x) :: 'a::real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
   1.958    unfolding tendsto_iff by simp
   1.959  
   1.960  lemma tendsto_add_const_iff:
   1.961 -  "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
   1.962 +  "((\<lambda>x. c + f x :: 'a::real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
   1.963    using tendsto_add[OF tendsto_const[of c], of f d]
   1.964 -        tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
   1.965 +    and tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
   1.966  
   1.967  
   1.968  subsubsection \<open>Inverse and division\<close>
   1.969  
   1.970  lemma (in bounded_bilinear) Zfun_prod_Bfun:
   1.971    assumes f: "Zfun f F"
   1.972 -  assumes g: "Bfun g F"
   1.973 +    and g: "Bfun g F"
   1.974    shows "Zfun (\<lambda>x. f x ** g x) F"
   1.975  proof -
   1.976    obtain K where K: "0 \<le> K"
   1.977 @@ -860,8 +894,7 @@
   1.978      have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
   1.979        by (rule norm_le)
   1.980      also have "\<dots> \<le> norm (f x) * B * K"
   1.981 -      by (intro mult_mono' order_refl norm_g norm_ge_zero
   1.982 -                mult_nonneg_nonneg K elim)
   1.983 +      by (intro mult_mono' order_refl norm_g norm_ge_zero mult_nonneg_nonneg K elim)
   1.984      also have "\<dots> = norm (f x) * (B * K)"
   1.985        by (rule mult.assoc)
   1.986      finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
   1.987 @@ -872,14 +905,15 @@
   1.988  
   1.989  lemma (in bounded_bilinear) Bfun_prod_Zfun:
   1.990    assumes f: "Bfun f F"
   1.991 -  assumes g: "Zfun g F"
   1.992 +    and g: "Zfun g F"
   1.993    shows "Zfun (\<lambda>x. f x ** g x) F"
   1.994    using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
   1.995  
   1.996  lemma Bfun_inverse_lemma:
   1.997    fixes x :: "'a::real_normed_div_algebra"
   1.998 -  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
   1.999 -  apply (subst nonzero_norm_inverse, clarsimp)
  1.1000 +  shows "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
  1.1001 +  apply (subst nonzero_norm_inverse)
  1.1002 +  apply clarsimp
  1.1003    apply (erule (1) le_imp_inverse_le)
  1.1004    done
  1.1005  
  1.1006 @@ -890,38 +924,40 @@
  1.1007    shows "Bfun (\<lambda>x. inverse (f x)) F"
  1.1008  proof -
  1.1009    from a have "0 < norm a" by simp
  1.1010 -  hence "\<exists>r>0. r < norm a" by (rule dense)
  1.1011 -  then obtain r where r1: "0 < r" and r2: "r < norm a" by blast
  1.1012 +  then have "\<exists>r>0. r < norm a" by (rule dense)
  1.1013 +  then obtain r where r1: "0 < r" and r2: "r < norm a"
  1.1014 +    by blast
  1.1015    have "eventually (\<lambda>x. dist (f x) a < r) F"
  1.1016      using tendstoD [OF f r1] by blast
  1.1017 -  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
  1.1018 +  then have "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
  1.1019    proof eventually_elim
  1.1020      case (elim x)
  1.1021 -    hence 1: "norm (f x - a) < r"
  1.1022 +    then have 1: "norm (f x - a) < r"
  1.1023        by (simp add: dist_norm)
  1.1024 -    hence 2: "f x \<noteq> 0" using r2 by auto
  1.1025 -    hence "norm (inverse (f x)) = inverse (norm (f x))"
  1.1026 +    then have 2: "f x \<noteq> 0" using r2 by auto
  1.1027 +    then have "norm (inverse (f x)) = inverse (norm (f x))"
  1.1028        by (rule nonzero_norm_inverse)
  1.1029      also have "\<dots> \<le> inverse (norm a - r)"
  1.1030      proof (rule le_imp_inverse_le)
  1.1031 -      show "0 < norm a - r" using r2 by simp
  1.1032 -    next
  1.1033 +      show "0 < norm a - r"
  1.1034 +        using r2 by simp
  1.1035        have "norm a - norm (f x) \<le> norm (a - f x)"
  1.1036          by (rule norm_triangle_ineq2)
  1.1037        also have "\<dots> = norm (f x - a)"
  1.1038          by (rule norm_minus_commute)
  1.1039        also have "\<dots> < r" using 1 .
  1.1040 -      finally show "norm a - r \<le> norm (f x)" by simp
  1.1041 +      finally show "norm a - r \<le> norm (f x)"
  1.1042 +        by simp
  1.1043      qed
  1.1044      finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
  1.1045    qed
  1.1046 -  thus ?thesis by (rule BfunI)
  1.1047 +  then show ?thesis by (rule BfunI)
  1.1048  qed
  1.1049  
  1.1050  lemma tendsto_inverse [tendsto_intros]:
  1.1051    fixes a :: "'a::real_normed_div_algebra"
  1.1052    assumes f: "(f \<longlongrightarrow> a) F"
  1.1053 -  assumes a: "a \<noteq> 0"
  1.1054 +    and a: "a \<noteq> 0"
  1.1055    shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
  1.1056  proof -
  1.1057    from a have "0 < norm a" by simp
  1.1058 @@ -942,43 +978,49 @@
  1.1059  
  1.1060  lemma continuous_inverse:
  1.1061    fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  1.1062 -  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
  1.1063 +  assumes "continuous F f"
  1.1064 +    and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
  1.1065    shows "continuous F (\<lambda>x. inverse (f x))"
  1.1066    using assms unfolding continuous_def by (rule tendsto_inverse)
  1.1067  
  1.1068  lemma continuous_at_within_inverse[continuous_intros]:
  1.1069    fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  1.1070 -  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
  1.1071 +  assumes "continuous (at a within s) f"
  1.1072 +    and "f a \<noteq> 0"
  1.1073    shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
  1.1074    using assms unfolding continuous_within by (rule tendsto_inverse)
  1.1075  
  1.1076  lemma isCont_inverse[continuous_intros, simp]:
  1.1077    fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
  1.1078 -  assumes "isCont f a" and "f a \<noteq> 0"
  1.1079 +  assumes "isCont f a"
  1.1080 +    and "f a \<noteq> 0"
  1.1081    shows "isCont (\<lambda>x. inverse (f x)) a"
  1.1082    using assms unfolding continuous_at by (rule tendsto_inverse)
  1.1083  
  1.1084  lemma continuous_on_inverse[continuous_intros]:
  1.1085    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
  1.1086 -  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
  1.1087 +  assumes "continuous_on s f"
  1.1088 +    and "\<forall>x\<in>s. f x \<noteq> 0"
  1.1089    shows "continuous_on s (\<lambda>x. inverse (f x))"
  1.1090    using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
  1.1091  
  1.1092  lemma tendsto_divide [tendsto_intros]:
  1.1093    fixes a b :: "'a::real_normed_field"
  1.1094 -  shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
  1.1095 -    \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
  1.1096 +  shows "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
  1.1097    by (simp add: tendsto_mult tendsto_inverse divide_inverse)
  1.1098  
  1.1099  lemma continuous_divide:
  1.1100    fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
  1.1101 -  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
  1.1102 +  assumes "continuous F f"
  1.1103 +    and "continuous F g"
  1.1104 +    and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
  1.1105    shows "continuous F (\<lambda>x. (f x) / (g x))"
  1.1106    using assms unfolding continuous_def by (rule tendsto_divide)
  1.1107  
  1.1108  lemma continuous_at_within_divide[continuous_intros]:
  1.1109    fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
  1.1110 -  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
  1.1111 +  assumes "continuous (at a within s) f" "continuous (at a within s) g"
  1.1112 +    and "g a \<noteq> 0"
  1.1113    shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
  1.1114    using assms unfolding continuous_within by (rule tendsto_divide)
  1.1115  
  1.1116 @@ -990,36 +1032,40 @@
  1.1117  
  1.1118  lemma continuous_on_divide[continuous_intros]:
  1.1119    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
  1.1120 -  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
  1.1121 +  assumes "continuous_on s f" "continuous_on s g"
  1.1122 +    and "\<forall>x\<in>s. g x \<noteq> 0"
  1.1123    shows "continuous_on s (\<lambda>x. (f x) / (g x))"
  1.1124    using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
  1.1125  
  1.1126 -lemma tendsto_sgn [tendsto_intros]:
  1.1127 -  fixes l :: "'a::real_normed_vector"
  1.1128 -  shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
  1.1129 +lemma tendsto_sgn [tendsto_intros]: "(f \<longlongrightarrow> l) F \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
  1.1130 +  for l :: "'a::real_normed_vector"
  1.1131    unfolding sgn_div_norm by (simp add: tendsto_intros)
  1.1132  
  1.1133  lemma continuous_sgn:
  1.1134    fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.1135 -  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
  1.1136 +  assumes "continuous F f"
  1.1137 +    and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
  1.1138    shows "continuous F (\<lambda>x. sgn (f x))"
  1.1139    using assms unfolding continuous_def by (rule tendsto_sgn)
  1.1140  
  1.1141  lemma continuous_at_within_sgn[continuous_intros]:
  1.1142    fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.1143 -  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
  1.1144 +  assumes "continuous (at a within s) f"
  1.1145 +    and "f a \<noteq> 0"
  1.1146    shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
  1.1147    using assms unfolding continuous_within by (rule tendsto_sgn)
  1.1148  
  1.1149  lemma isCont_sgn[continuous_intros]:
  1.1150    fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.1151 -  assumes "isCont f a" and "f a \<noteq> 0"
  1.1152 +  assumes "isCont f a"
  1.1153 +    and "f a \<noteq> 0"
  1.1154    shows "isCont (\<lambda>x. sgn (f x)) a"
  1.1155    using assms unfolding continuous_at by (rule tendsto_sgn)
  1.1156  
  1.1157  lemma continuous_on_sgn[continuous_intros]:
  1.1158    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  1.1159 -  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
  1.1160 +  assumes "continuous_on s f"
  1.1161 +    and "\<forall>x\<in>s. f x \<noteq> 0"
  1.1162    shows "continuous_on s (\<lambda>x. sgn (f x))"
  1.1163    using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
  1.1164  
  1.1165 @@ -1029,35 +1075,40 @@
  1.1166    shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
  1.1167    unfolding filterlim_iff eventually_at_infinity
  1.1168  proof safe
  1.1169 -  fix P :: "'a \<Rightarrow> bool" and b
  1.1170 +  fix P :: "'a \<Rightarrow> bool"
  1.1171 +  fix b
  1.1172    assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
  1.1173 -    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
  1.1174 +  assume P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
  1.1175    have "max b (c + 1) > c" by auto
  1.1176    with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
  1.1177      by auto
  1.1178    then show "eventually (\<lambda>x. P (f x)) F"
  1.1179    proof eventually_elim
  1.1180 -    fix x assume "max b (c + 1) \<le> norm (f x)"
  1.1181 +    case (elim x)
  1.1182      with P show "P (f x)" by auto
  1.1183    qed
  1.1184  qed force
  1.1185  
  1.1186  lemma not_tendsto_and_filterlim_at_infinity:
  1.1187 +  fixes c :: "'a::real_normed_vector"
  1.1188    assumes "F \<noteq> bot"
  1.1189 -  assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
  1.1190 -  assumes "filterlim f at_infinity F"
  1.1191 -  shows   False
  1.1192 +    and "(f \<longlongrightarrow> c) F"
  1.1193 +    and "filterlim f at_infinity F"
  1.1194 +  shows False
  1.1195  proof -
  1.1196    from tendstoD[OF assms(2), of "1/2"]
  1.1197 -    have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
  1.1198 -  moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
  1.1199 -    have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
  1.1200 +  have "eventually (\<lambda>x. dist (f x) c < 1/2) F"
  1.1201 +    by simp
  1.1202 +  moreover
  1.1203 +  from filterlim_at_infinity[of "norm c" f F] assms(3)
  1.1204 +  have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
  1.1205    ultimately have "eventually (\<lambda>x. False) F"
  1.1206    proof eventually_elim
  1.1207 -    fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
  1.1208 -    note B
  1.1209 +    fix x
  1.1210 +    assume A: "dist (f x) c < 1/2"
  1.1211 +    assume "norm (f x) \<ge> norm c + 1"
  1.1212      also have "norm (f x) = dist (f x) 0" by simp
  1.1213 -    also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
  1.1214 +    also have "\<dots> \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
  1.1215      finally show False using A by simp
  1.1216    qed
  1.1217    with assms show False by simp
  1.1218 @@ -1065,83 +1116,97 @@
  1.1219  
  1.1220  lemma filterlim_at_infinity_imp_not_convergent:
  1.1221    assumes "filterlim f at_infinity sequentially"
  1.1222 -  shows   "\<not>convergent f"
  1.1223 +  shows "\<not> convergent f"
  1.1224    by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
  1.1225       (simp_all add: convergent_LIMSEQ_iff)
  1.1226  
  1.1227  lemma filterlim_at_infinity_imp_eventually_ne:
  1.1228    assumes "filterlim f at_infinity F"
  1.1229 -  shows   "eventually (\<lambda>z. f z \<noteq> c) F"
  1.1230 +  shows "eventually (\<lambda>z. f z \<noteq> c) F"
  1.1231  proof -
  1.1232 -  have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
  1.1233 +  have "norm c + 1 > 0"
  1.1234 +    by (intro add_nonneg_pos) simp_all
  1.1235    with filterlim_at_infinity[OF order.refl, of f F] assms
  1.1236 -    have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
  1.1237 -  thus ?thesis by eventually_elim auto
  1.1238 +  have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F"
  1.1239 +    by blast
  1.1240 +  then show ?thesis
  1.1241 +    by eventually_elim auto
  1.1242  qed
  1.1243  
  1.1244  lemma tendsto_of_nat [tendsto_intros]:
  1.1245 -  "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
  1.1246 +  "filterlim (of_nat :: nat \<Rightarrow> 'a::real_normed_algebra_1) at_infinity sequentially"
  1.1247  proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
  1.1248    fix r :: real
  1.1249    assume r: "r > 0"
  1.1250    define n where "n = nat \<lceil>r\<rceil>"
  1.1251 -  from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
  1.1252 +  from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r"
  1.1253 +    unfolding n_def by linarith
  1.1254    from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
  1.1255 -    by eventually_elim (insert n, simp_all)
  1.1256 +    by eventually_elim (use n in simp_all)
  1.1257  qed
  1.1258  
  1.1259  
  1.1260  subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
  1.1261  
  1.1262  text \<open>
  1.1263 -
  1.1264 -This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
  1.1265 -@{term "at_right x"} and also @{term "at_right 0"}.
  1.1266 -
  1.1267 +  This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
  1.1268 +  @{term "at_right x"} and also @{term "at_right 0"}.
  1.1269  \<close>
  1.1270  
  1.1271  lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
  1.1272  
  1.1273 -lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
  1.1274 +lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d)"
  1.1275 +  for a d :: "'a::real_normed_vector"
  1.1276    by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
  1.1277 -     (auto intro!: tendsto_eq_intros filterlim_ident)
  1.1278 -
  1.1279 -lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
  1.1280 +    (auto intro!: tendsto_eq_intros filterlim_ident)
  1.1281 +
  1.1282 +lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a)"
  1.1283 +  for a :: "'a::real_normed_vector"
  1.1284    by (rule filtermap_fun_inverse[where g=uminus])
  1.1285 -     (auto intro!: tendsto_eq_intros filterlim_ident)
  1.1286 -
  1.1287 -lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
  1.1288 +    (auto intro!: tendsto_eq_intros filterlim_ident)
  1.1289 +
  1.1290 +lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d)"
  1.1291 +  for a d :: "'a::real_normed_vector"
  1.1292    by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
  1.1293  
  1.1294 -lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
  1.1295 +lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d)"
  1.1296 +  for a d :: "real"
  1.1297    by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
  1.1298  
  1.1299 -lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
  1.1300 +lemma at_right_to_0: "at_right a = filtermap (\<lambda>x. x + a) (at_right 0)"
  1.1301 +  for a :: real
  1.1302    using filtermap_at_right_shift[of "-a" 0] by simp
  1.1303  
  1.1304  lemma filterlim_at_right_to_0:
  1.1305 -  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
  1.1306 +  "filterlim f F (at_right a) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
  1.1307 +  for a :: real
  1.1308    unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
  1.1309  
  1.1310  lemma eventually_at_right_to_0:
  1.1311 -  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
  1.1312 +  "eventually P (at_right a) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
  1.1313 +  for a :: real
  1.1314    unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
  1.1315  
  1.1316 -lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
  1.1317 +lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a)"
  1.1318 +  for a :: "'a::real_normed_vector"
  1.1319    by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
  1.1320  
  1.1321 -lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
  1.1322 +lemma at_left_minus: "at_left a = filtermap (\<lambda>x. - x) (at_right (- a))"
  1.1323 +  for a :: real
  1.1324    by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
  1.1325  
  1.1326 -lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
  1.1327 +lemma at_right_minus: "at_right a = filtermap (\<lambda>x. - x) (at_left (- a))"
  1.1328 +  for a :: real
  1.1329    by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
  1.1330  
  1.1331  lemma filterlim_at_left_to_right:
  1.1332 -  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
  1.1333 +  "filterlim f F (at_left a) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
  1.1334 +  for a :: real
  1.1335    unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
  1.1336  
  1.1337  lemma eventually_at_left_to_right:
  1.1338 -  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
  1.1339 +  "eventually P (at_left a) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
  1.1340 +  for a :: real
  1.1341    unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
  1.1342  
  1.1343  lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
  1.1344 @@ -1167,7 +1232,7 @@
  1.1345  
  1.1346  lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
  1.1347    using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
  1.1348 -  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
  1.1349 +    and filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
  1.1350    by auto
  1.1351  
  1.1352  lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
  1.1353 @@ -1176,7 +1241,8 @@
  1.1354  lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
  1.1355    unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
  1.1356  proof safe
  1.1357 -  fix Z :: real assume [arith]: "0 < Z"
  1.1358 +  fix Z :: real
  1.1359 +  assume [arith]: "0 < Z"
  1.1360    then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
  1.1361      by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
  1.1362    then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
  1.1363 @@ -1188,41 +1254,56 @@
  1.1364    shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
  1.1365    unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
  1.1366  proof safe
  1.1367 -  fix r :: real assume "0 < r"
  1.1368 +  fix r :: real
  1.1369 +  assume "0 < r"
  1.1370    show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
  1.1371    proof (intro exI[of _ "inverse (r / 2)"] allI impI)
  1.1372      fix x :: 'a
  1.1373      from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
  1.1374      also assume *: "inverse (r / 2) \<le> norm x"
  1.1375      finally show "norm (inverse x) < r"
  1.1376 -      using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
  1.1377 +      using * \<open>0 < r\<close>
  1.1378 +      by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
  1.1379    qed
  1.1380  qed
  1.1381  
  1.1382  lemma tendsto_add_filterlim_at_infinity:
  1.1383 -  assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
  1.1384 -  assumes "filterlim g at_infinity F"
  1.1385 -  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
  1.1386 +  fixes c :: "'b::real_normed_vector"
  1.1387 +    and F :: "'a filter"
  1.1388 +  assumes "(f \<longlongrightarrow> c) F"
  1.1389 +    and "filterlim g at_infinity F"
  1.1390 +  shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
  1.1391  proof (subst filterlim_at_infinity[OF order_refl], safe)
  1.1392 -  fix r :: real assume r: "r > 0"
  1.1393 -  from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
  1.1394 -  hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
  1.1395 -  moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
  1.1396 +  fix r :: real
  1.1397 +  assume r: "r > 0"
  1.1398 +  from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F"
  1.1399 +    by (rule tendsto_norm)
  1.1400 +  then have "eventually (\<lambda>x. norm (f x) < norm c + 1) F"
  1.1401 +    by (rule order_tendstoD) simp_all
  1.1402 +  moreover from r have "r + norm c + 1 > 0"
  1.1403 +    by (intro add_pos_nonneg) simp_all
  1.1404    with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
  1.1405 -    unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
  1.1406 +    unfolding filterlim_at_infinity[OF order_refl]
  1.1407 +    by (elim allE[of _ "r + norm c + 1"]) simp_all
  1.1408    ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
  1.1409    proof eventually_elim
  1.1410 -    fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
  1.1411 -    from A B have "r \<le> norm (g x) - norm (f x)" by simp
  1.1412 -    also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
  1.1413 -    finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
  1.1414 +    fix x :: 'a
  1.1415 +    assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
  1.1416 +    from A B have "r \<le> norm (g x) - norm (f x)"
  1.1417 +      by simp
  1.1418 +    also have "norm (g x) - norm (f x) \<le> norm (g x + f x)"
  1.1419 +      by (rule norm_diff_ineq)
  1.1420 +    finally show "r \<le> norm (f x + g x)"
  1.1421 +      by (simp add: add_ac)
  1.1422    qed
  1.1423  qed
  1.1424  
  1.1425  lemma tendsto_add_filterlim_at_infinity':
  1.1426 +  fixes c :: "'b::real_normed_vector"
  1.1427 +    and F :: "'a filter"
  1.1428    assumes "filterlim f at_infinity F"
  1.1429 -  assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
  1.1430 -  shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
  1.1431 +    and "(g \<longlongrightarrow> c) F"
  1.1432 +  shows "filterlim (\<lambda>x. f x + g x) at_infinity F"
  1.1433    by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
  1.1434  
  1.1435  lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
  1.1436 @@ -1272,7 +1353,8 @@
  1.1437    shows "filterlim inverse at_infinity (at (0::'a))"
  1.1438    unfolding filterlim_at_infinity[OF order_refl]
  1.1439  proof safe
  1.1440 -  fix r :: real assume "0 < r"
  1.1441 +  fix r :: real
  1.1442 +  assume "0 < r"
  1.1443    then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
  1.1444      unfolding eventually_at norm_inverse
  1.1445      by (intro exI[of _ "inverse r"])
  1.1446 @@ -1290,7 +1372,7 @@
  1.1447    also have "\<dots> \<le> at 0"
  1.1448      using tendsto_inverse_0[where 'a='b]
  1.1449      by (auto intro!: exI[of _ 1]
  1.1450 -             simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
  1.1451 +        simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
  1.1452    finally show "filtermap inverse (filtermap g F) \<le> at 0" .
  1.1453  next
  1.1454    assume "filtermap inverse (filtermap g F) \<le> at 0"
  1.1455 @@ -1301,36 +1383,40 @@
  1.1456  qed
  1.1457  
  1.1458  lemma tendsto_mult_filterlim_at_infinity:
  1.1459 -  assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
  1.1460 +  fixes c :: "'a::real_normed_field"
  1.1461 +  assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> 0"
  1.1462    assumes "filterlim g at_infinity F"
  1.1463 -  shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
  1.1464 +  shows "filterlim (\<lambda>x. f x * g x) at_infinity F"
  1.1465  proof -
  1.1466    have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
  1.1467      by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
  1.1468 -  hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
  1.1469 -    unfolding filterlim_at using assms
  1.1470 +  then have "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
  1.1471 +    unfolding filterlim_at
  1.1472 +    using assms
  1.1473      by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
  1.1474 -  thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
  1.1475 +  then show ?thesis
  1.1476 +    by (subst filterlim_inverse_at_iff[symmetric]) simp_all
  1.1477  qed
  1.1478  
  1.1479  lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
  1.1480   by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
  1.1481  
  1.1482 -lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
  1.1483 -  by (rule filterlim_subseq) (auto simp: subseq_def)
  1.1484 -
  1.1485 -lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
  1.1486 +lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x) at_top sequentially"
  1.1487 +  for c :: nat
  1.1488    by (rule filterlim_subseq) (auto simp: subseq_def)
  1.1489  
  1.1490 -lemma at_to_infinity:
  1.1491 -  fixes x :: "'a :: {real_normed_field,field}"
  1.1492 -  shows "(at (0::'a)) = filtermap inverse at_infinity"
  1.1493 +lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c) at_top sequentially"
  1.1494 +  for c :: nat
  1.1495 +  by (rule filterlim_subseq) (auto simp: subseq_def)
  1.1496 +
  1.1497 +lemma at_to_infinity: "(at (0::'a::{real_normed_field,field})) = filtermap inverse at_infinity"
  1.1498  proof (rule antisym)
  1.1499    have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
  1.1500      by (fact tendsto_inverse_0)
  1.1501    then show "filtermap inverse at_infinity \<le> at (0::'a)"
  1.1502      apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
  1.1503 -    apply (rule_tac x="1" in exI, auto)
  1.1504 +    apply (rule_tac x="1" in exI)
  1.1505 +    apply auto
  1.1506      done
  1.1507  next
  1.1508    have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
  1.1509 @@ -1341,38 +1427,39 @@
  1.1510  qed
  1.1511  
  1.1512  lemma lim_at_infinity_0:
  1.1513 -  fixes l :: "'a :: {real_normed_field,field}"
  1.1514 -  shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
  1.1515 -by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
  1.1516 +  fixes l :: "'a::{real_normed_field,field}"
  1.1517 +  shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f \<circ> inverse) \<longlongrightarrow> l) (at (0::'a))"
  1.1518 +  by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
  1.1519  
  1.1520  lemma lim_zero_infinity:
  1.1521 -  fixes l :: "'a :: {real_normed_field,field}"
  1.1522 +  fixes l :: "'a::{real_normed_field,field}"
  1.1523    shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
  1.1524 -by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
  1.1525 +  by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
  1.1526  
  1.1527  
  1.1528  text \<open>
  1.1529 -
  1.1530 -We only show rules for multiplication and addition when the functions are either against a real
  1.1531 -value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
  1.1532 -
  1.1533 +  We only show rules for multiplication and addition when the functions are either against a real
  1.1534 +  value or against infinity. Further rules are easy to derive by using @{thm
  1.1535 +  filterlim_uminus_at_top}.
  1.1536  \<close>
  1.1537  
  1.1538  lemma filterlim_tendsto_pos_mult_at_top:
  1.1539 -  assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
  1.1540 -  assumes g: "LIM x F. g x :> at_top"
  1.1541 +  assumes f: "(f \<longlongrightarrow> c) F"
  1.1542 +    and c: "0 < c"
  1.1543 +    and g: "LIM x F. g x :> at_top"
  1.1544    shows "LIM x F. (f x * g x :: real) :> at_top"
  1.1545    unfolding filterlim_at_top_gt[where c=0]
  1.1546  proof safe
  1.1547 -  fix Z :: real assume "0 < Z"
  1.1548 +  fix Z :: real
  1.1549 +  assume "0 < Z"
  1.1550    from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
  1.1551      by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
  1.1552 -             simp: dist_real_def abs_real_def split: if_split_asm)
  1.1553 +        simp: dist_real_def abs_real_def split: if_split_asm)
  1.1554    moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
  1.1555      unfolding filterlim_at_top by auto
  1.1556    ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
  1.1557    proof eventually_elim
  1.1558 -    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
  1.1559 +    case (elim x)
  1.1560      with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
  1.1561        by (intro mult_mono) (auto simp: zero_le_divide_iff)
  1.1562      with \<open>0 < c\<close> show "Z \<le> f x * g x"
  1.1563 @@ -1382,18 +1469,19 @@
  1.1564  
  1.1565  lemma filterlim_at_top_mult_at_top:
  1.1566    assumes f: "LIM x F. f x :> at_top"
  1.1567 -  assumes g: "LIM x F. g x :> at_top"
  1.1568 +    and g: "LIM x F. g x :> at_top"
  1.1569    shows "LIM x F. (f x * g x :: real) :> at_top"
  1.1570    unfolding filterlim_at_top_gt[where c=0]
  1.1571  proof safe
  1.1572 -  fix Z :: real assume "0 < Z"
  1.1573 +  fix Z :: real
  1.1574 +  assume "0 < Z"
  1.1575    from f have "eventually (\<lambda>x. 1 \<le> f x) F"
  1.1576      unfolding filterlim_at_top by auto
  1.1577    moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
  1.1578      unfolding filterlim_at_top by auto
  1.1579    ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
  1.1580    proof eventually_elim
  1.1581 -    fix x assume "1 \<le> f x" "Z \<le> g x"
  1.1582 +    case (elim x)
  1.1583      with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
  1.1584        by (intro mult_mono) (auto simp: zero_le_divide_iff)
  1.1585      then show "Z \<le> f x * g x"
  1.1586 @@ -1402,25 +1490,32 @@
  1.1587  qed
  1.1588  
  1.1589  lemma filterlim_tendsto_pos_mult_at_bot:
  1.1590 -  assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
  1.1591 +  fixes c :: real
  1.1592 +  assumes "(f \<longlongrightarrow> c) F" "0 < c" "filterlim g at_bot F"
  1.1593    shows "LIM x F. f x * g x :> at_bot"
  1.1594    using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
  1.1595    unfolding filterlim_uminus_at_bot by simp
  1.1596  
  1.1597  lemma filterlim_tendsto_neg_mult_at_bot:
  1.1598 -  assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
  1.1599 +  fixes c :: real
  1.1600 +  assumes c: "(f \<longlongrightarrow> c) F" "c < 0" and g: "filterlim g at_top F"
  1.1601    shows "LIM x F. f x * g x :> at_bot"
  1.1602    using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
  1.1603    unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
  1.1604  
  1.1605  lemma filterlim_pow_at_top:
  1.1606    fixes f :: "real \<Rightarrow> real"
  1.1607 -  assumes "0 < n" and f: "LIM x F. f x :> at_top"
  1.1608 +  assumes "0 < n"
  1.1609 +    and f: "LIM x F. f x :> at_top"
  1.1610    shows "LIM x F. (f x)^n :: real :> at_top"
  1.1611 -using \<open>0 < n\<close> proof (induct n)
  1.1612 +  using \<open>0 < n\<close>
  1.1613 +proof (induct n)
  1.1614 +  case 0
  1.1615 +  then show ?case by simp
  1.1616 +next
  1.1617    case (Suc n) with f show ?case
  1.1618      by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
  1.1619 -qed simp
  1.1620 +qed
  1.1621  
  1.1622  lemma filterlim_pow_at_bot_even:
  1.1623    fixes f :: "real \<Rightarrow> real"
  1.1624 @@ -1434,11 +1529,12 @@
  1.1625  
  1.1626  lemma filterlim_tendsto_add_at_top:
  1.1627    assumes f: "(f \<longlongrightarrow> c) F"
  1.1628 -  assumes g: "LIM x F. g x :> at_top"
  1.1629 +    and g: "LIM x F. g x :> at_top"
  1.1630    shows "LIM x F. (f x + g x :: real) :> at_top"
  1.1631    unfolding filterlim_at_top_gt[where c=0]
  1.1632  proof safe
  1.1633 -  fix Z :: real assume "0 < Z"
  1.1634 +  fix Z :: real
  1.1635 +  assume "0 < Z"
  1.1636    from f have "eventually (\<lambda>x. c - 1 < f x) F"
  1.1637      by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
  1.1638    moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
  1.1639 @@ -1450,18 +1546,19 @@
  1.1640  lemma LIM_at_top_divide:
  1.1641    fixes f g :: "'a \<Rightarrow> real"
  1.1642    assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
  1.1643 -  assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
  1.1644 +    and g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
  1.1645    shows "LIM x F. f x / g x :> at_top"
  1.1646    unfolding divide_inverse
  1.1647    by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
  1.1648  
  1.1649  lemma filterlim_at_top_add_at_top:
  1.1650    assumes f: "LIM x F. f x :> at_top"
  1.1651 -  assumes g: "LIM x F. g x :> at_top"
  1.1652 +    and g: "LIM x F. g x :> at_top"
  1.1653    shows "LIM x F. (f x + g x :: real) :> at_top"
  1.1654    unfolding filterlim_at_top_gt[where c=0]
  1.1655  proof safe
  1.1656 -  fix Z :: real assume "0 < Z"
  1.1657 +  fix Z :: real
  1.1658 +  assume "0 < Z"
  1.1659    from f have "eventually (\<lambda>x. 0 \<le> f x) F"
  1.1660      unfolding filterlim_at_top by auto
  1.1661    moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
  1.1662 @@ -1473,34 +1570,43 @@
  1.1663  lemma tendsto_divide_0:
  1.1664    fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
  1.1665    assumes f: "(f \<longlongrightarrow> c) F"
  1.1666 -  assumes g: "LIM x F. g x :> at_infinity"
  1.1667 +    and g: "LIM x F. g x :> at_infinity"
  1.1668    shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
  1.1669 -  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
  1.1670 +  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]]
  1.1671 +  by (simp add: divide_inverse)
  1.1672  
  1.1673  lemma linear_plus_1_le_power:
  1.1674    fixes x :: real
  1.1675    assumes x: "0 \<le> x"
  1.1676    shows "real n * x + 1 \<le> (x + 1) ^ n"
  1.1677  proof (induct n)
  1.1678 +  case 0
  1.1679 +  then show ?case by simp
  1.1680 +next
  1.1681    case (Suc n)
  1.1682 -  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
  1.1683 -    by (simp add: field_simps of_nat_Suc x)
  1.1684 +  from x have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
  1.1685 +    by (simp add: field_simps)
  1.1686    also have "\<dots> \<le> (x + 1)^Suc n"
  1.1687      using Suc x by (simp add: mult_left_mono)
  1.1688    finally show ?case .
  1.1689 -qed simp
  1.1690 +qed
  1.1691  
  1.1692  lemma filterlim_realpow_sequentially_gt1:
  1.1693    fixes x :: "'a :: real_normed_div_algebra"
  1.1694    assumes x[arith]: "1 < norm x"
  1.1695    shows "LIM n sequentially. x ^ n :> at_infinity"
  1.1696  proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
  1.1697 -  fix y :: real assume "0 < y"
  1.1698 +  fix y :: real
  1.1699 +  assume "0 < y"
  1.1700    have "0 < norm x - 1" by simp
  1.1701 -  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
  1.1702 -  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
  1.1703 -  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
  1.1704 -  also have "\<dots> = norm x ^ N" by simp
  1.1705 +  then obtain N :: nat where "y < real N * (norm x - 1)"
  1.1706 +    by (blast dest: reals_Archimedean3)
  1.1707 +  also have "\<dots> \<le> real N * (norm x - 1) + 1"
  1.1708 +    by simp
  1.1709 +  also have "\<dots> \<le> (norm x - 1 + 1) ^ N"
  1.1710 +    by (rule linear_plus_1_le_power) simp
  1.1711 +  also have "\<dots> = norm x ^ N"
  1.1712 +    by simp
  1.1713    finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
  1.1714      by (metis order_less_le_trans power_increasing order_less_imp_le x)
  1.1715    then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
  1.1716 @@ -1512,48 +1618,48 @@
  1.1717  subsection \<open>Floor and Ceiling\<close>
  1.1718  
  1.1719  lemma eventually_floor_less:
  1.1720 -  fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
  1.1721 +  fixes f :: "'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
  1.1722    assumes f: "(f \<longlongrightarrow> l) F"
  1.1723 -  assumes l: "l \<notin> \<int>"
  1.1724 +    and l: "l \<notin> \<int>"
  1.1725    shows "\<forall>\<^sub>F x in F. of_int (floor l) < f x"
  1.1726    by (intro order_tendstoD[OF f]) (metis Ints_of_int antisym_conv2 floor_correct l)
  1.1727  
  1.1728  lemma eventually_less_ceiling:
  1.1729 -  fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
  1.1730 +  fixes f :: "'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
  1.1731    assumes f: "(f \<longlongrightarrow> l) F"
  1.1732 -  assumes l: "l \<notin> \<int>"
  1.1733 +    and l: "l \<notin> \<int>"
  1.1734    shows "\<forall>\<^sub>F x in F. f x < of_int (ceiling l)"
  1.1735    by (intro order_tendstoD[OF f]) (metis Ints_of_int l le_of_int_ceiling less_le)
  1.1736  
  1.1737  lemma eventually_floor_eq:
  1.1738 -  fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
  1.1739 +  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
  1.1740    assumes f: "(f \<longlongrightarrow> l) F"
  1.1741 -  assumes l: "l \<notin> \<int>"
  1.1742 +    and l: "l \<notin> \<int>"
  1.1743    shows "\<forall>\<^sub>F x in F. floor (f x) = floor l"
  1.1744    using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
  1.1745    by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)
  1.1746  
  1.1747  lemma eventually_ceiling_eq:
  1.1748 -  fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
  1.1749 +  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
  1.1750    assumes f: "(f \<longlongrightarrow> l) F"
  1.1751 -  assumes l: "l \<notin> \<int>"
  1.1752 +    and l: "l \<notin> \<int>"
  1.1753    shows "\<forall>\<^sub>F x in F. ceiling (f x) = ceiling l"
  1.1754    using eventually_floor_less[OF assms] eventually_less_ceiling[OF assms]
  1.1755    by eventually_elim (meson floor_less_iff less_ceiling_iff not_less_iff_gr_or_eq)
  1.1756  
  1.1757  lemma tendsto_of_int_floor:
  1.1758 -  fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
  1.1759 +  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
  1.1760    assumes "(f \<longlongrightarrow> l) F"
  1.1761 -  assumes "l \<notin> \<int>"
  1.1762 -  shows "((\<lambda>x. of_int (floor (f x))::'c::{ring_1, topological_space}) \<longlongrightarrow> of_int (floor l)) F"
  1.1763 +    and "l \<notin> \<int>"
  1.1764 +  shows "((\<lambda>x. of_int (floor (f x)) :: 'c::{ring_1,topological_space}) \<longlongrightarrow> of_int (floor l)) F"
  1.1765    using eventually_floor_eq[OF assms]
  1.1766    by (simp add: eventually_mono topological_tendstoI)
  1.1767  
  1.1768  lemma tendsto_of_int_ceiling:
  1.1769 -  fixes f::"'a \<Rightarrow> 'b::{order_topology, floor_ceiling}"
  1.1770 +  fixes f::"'a \<Rightarrow> 'b::{order_topology,floor_ceiling}"
  1.1771    assumes "(f \<longlongrightarrow> l) F"
  1.1772 -  assumes "l \<notin> \<int>"
  1.1773 -  shows "((\<lambda>x. of_int (ceiling (f x))::'c::{ring_1, topological_space}) \<longlongrightarrow> of_int (ceiling l)) F"
  1.1774 +    and "l \<notin> \<int>"
  1.1775 +  shows "((\<lambda>x. of_int (ceiling (f x)):: 'c::{ring_1,topological_space}) \<longlongrightarrow> of_int (ceiling l)) F"
  1.1776    using eventually_ceiling_eq[OF assms]
  1.1777    by (simp add: eventually_mono topological_tendstoI)
  1.1778  
  1.1779 @@ -1580,69 +1686,64 @@
  1.1780    shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
  1.1781  unfolding lim_sequentially dist_norm ..
  1.1782  
  1.1783 -lemma LIMSEQ_I:
  1.1784 -  fixes L :: "'a::real_normed_vector"
  1.1785 -  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
  1.1786 -by (simp add: LIMSEQ_iff)
  1.1787 -
  1.1788 -lemma LIMSEQ_D:
  1.1789 -  fixes L :: "'a::real_normed_vector"
  1.1790 -  shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
  1.1791 -by (simp add: LIMSEQ_iff)
  1.1792 -
  1.1793 -lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
  1.1794 +lemma LIMSEQ_I: "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
  1.1795 +  for L :: "'a::real_normed_vector"
  1.1796 +  by (simp add: LIMSEQ_iff)
  1.1797 +
  1.1798 +lemma LIMSEQ_D: "X \<longlonglongrightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
  1.1799 +  for L :: "'a::real_normed_vector"
  1.1800 +  by (simp add: LIMSEQ_iff)
  1.1801 +
  1.1802 +lemma LIMSEQ_linear: "X \<longlonglongrightarrow> x \<Longrightarrow> l > 0 \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
  1.1803    unfolding tendsto_def eventually_sequentially
  1.1804    by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
  1.1805  
  1.1806 -lemma Bseq_inverse_lemma:
  1.1807 -  fixes x :: "'a::real_normed_div_algebra"
  1.1808 -  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
  1.1809 -apply (subst nonzero_norm_inverse, clarsimp)
  1.1810 -apply (erule (1) le_imp_inverse_le)
  1.1811 -done
  1.1812 -
  1.1813 -lemma Bseq_inverse:
  1.1814 -  fixes a :: "'a::real_normed_div_algebra"
  1.1815 -  shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
  1.1816 +lemma Bseq_inverse_lemma: "r \<le> norm x \<Longrightarrow> 0 < r \<Longrightarrow> norm (inverse x) \<le> inverse r"
  1.1817 +  for x :: "'a::real_normed_div_algebra"
  1.1818 +  apply (subst nonzero_norm_inverse, clarsimp)
  1.1819 +  apply (erule (1) le_imp_inverse_le)
  1.1820 +  done
  1.1821 +
  1.1822 +lemma Bseq_inverse: "X \<longlonglongrightarrow> a \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
  1.1823 +  for a :: "'a::real_normed_div_algebra"
  1.1824    by (rule Bfun_inverse)
  1.1825  
  1.1826 -text\<open>Transformation of limit.\<close>
  1.1827 -
  1.1828 -lemma Lim_transform:
  1.1829 -  fixes a b :: "'a::real_normed_vector"
  1.1830 -  shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
  1.1831 +
  1.1832 +text \<open>Transformation of limit.\<close>
  1.1833 +
  1.1834 +lemma Lim_transform: "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
  1.1835 +  for a b :: "'a::real_normed_vector"
  1.1836    using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
  1.1837  
  1.1838 -lemma Lim_transform2:
  1.1839 -  fixes a b :: "'a::real_normed_vector"
  1.1840 -  shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
  1.1841 +lemma Lim_transform2: "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (g \<longlongrightarrow> a) F"
  1.1842 +  for a b :: "'a::real_normed_vector"
  1.1843    by (erule Lim_transform) (simp add: tendsto_minus_cancel)
  1.1844  
  1.1845 -proposition Lim_transform_eq:
  1.1846 -  fixes a :: "'a::real_normed_vector"
  1.1847 -  shows "((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F \<longleftrightarrow> (g \<longlongrightarrow> a) F"
  1.1848 -using Lim_transform Lim_transform2 by blast
  1.1849 +proposition Lim_transform_eq: "((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> a) F \<longleftrightarrow> (g \<longlongrightarrow> a) F"
  1.1850 +  for a :: "'a::real_normed_vector"
  1.1851 +  using Lim_transform Lim_transform2 by blast
  1.1852  
  1.1853  lemma Lim_transform_eventually:
  1.1854    "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
  1.1855    apply (rule topological_tendstoI)
  1.1856    apply (drule (2) topological_tendstoD)
  1.1857 -  apply (erule (1) eventually_elim2, simp)
  1.1858 +  apply (erule (1) eventually_elim2)
  1.1859 +  apply simp
  1.1860    done
  1.1861  
  1.1862  lemma Lim_transform_within:
  1.1863    assumes "(f \<longlongrightarrow> l) (at x within S)"
  1.1864      and "0 < d"
  1.1865 -    and "\<And>x'. \<lbrakk>x'\<in>S; 0 < dist x' x; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
  1.1866 +    and "\<And>x'. x'\<in>S \<Longrightarrow> 0 < dist x' x \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x'"
  1.1867    shows "(g \<longlongrightarrow> l) (at x within S)"
  1.1868  proof (rule Lim_transform_eventually)
  1.1869    show "eventually (\<lambda>x. f x = g x) (at x within S)"
  1.1870      using assms by (auto simp: eventually_at)
  1.1871 -  show "(f \<longlongrightarrow> l) (at x within S)" by fact
  1.1872 +  show "(f \<longlongrightarrow> l) (at x within S)"
  1.1873 +    by fact
  1.1874  qed
  1.1875  
  1.1876 -text\<open>Common case assuming being away from some crucial point like 0.\<close>
  1.1877 -
  1.1878 +text \<open>Common case assuming being away from some crucial point like 0.\<close>
  1.1879  lemma Lim_transform_away_within:
  1.1880    fixes a b :: "'a::t1_space"
  1.1881    assumes "a \<noteq> b"
  1.1882 @@ -1650,26 +1751,26 @@
  1.1883      and "(f \<longlongrightarrow> l) (at a within S)"
  1.1884    shows "(g \<longlongrightarrow> l) (at a within S)"
  1.1885  proof (rule Lim_transform_eventually)
  1.1886 -  show "(f \<longlongrightarrow> l) (at a within S)" by fact
  1.1887 +  show "(f \<longlongrightarrow> l) (at a within S)"
  1.1888 +    by fact
  1.1889    show "eventually (\<lambda>x. f x = g x) (at a within S)"
  1.1890      unfolding eventually_at_topological
  1.1891 -    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
  1.1892 +    by (rule exI [where x="- {b}"]) (simp add: open_Compl assms)
  1.1893  qed
  1.1894  
  1.1895  lemma Lim_transform_away_at:
  1.1896    fixes a b :: "'a::t1_space"
  1.1897 -  assumes ab: "a\<noteq>b"
  1.1898 +  assumes ab: "a \<noteq> b"
  1.1899      and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
  1.1900      and fl: "(f \<longlongrightarrow> l) (at a)"
  1.1901    shows "(g \<longlongrightarrow> l) (at a)"
  1.1902    using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
  1.1903  
  1.1904 -text\<open>Alternatively, within an open set.\<close>
  1.1905 -
  1.1906 +text \<open>Alternatively, within an open set.\<close>
  1.1907  lemma Lim_transform_within_open:
  1.1908    assumes "(f \<longlongrightarrow> l) (at a within T)"
  1.1909      and "open s" and "a \<in> s"
  1.1910 -    and "\<And>x. \<lbrakk>x\<in>s; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
  1.1911 +    and "\<And>x. x\<in>s \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x = g x"
  1.1912    shows "(g \<longlongrightarrow> l) (at a within T)"
  1.1913  proof (rule Lim_transform_eventually)
  1.1914    show "eventually (\<lambda>x. f x = g x) (at a within T)"
  1.1915 @@ -1678,7 +1779,8 @@
  1.1916    show "(f \<longlongrightarrow> l) (at a within T)" by fact
  1.1917  qed
  1.1918  
  1.1919 -text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
  1.1920 +
  1.1921 +text \<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
  1.1922  
  1.1923  (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
  1.1924  
  1.1925 @@ -1697,35 +1799,32 @@
  1.1926    shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
  1.1927    unfolding tendsto_def eventually_at_topological
  1.1928    using assms by simp
  1.1929 -text\<open>An unbounded sequence's inverse tends to 0\<close>
  1.1930 -
  1.1931 -lemma LIMSEQ_inverse_zero:
  1.1932 -  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
  1.1933 +
  1.1934 +text \<open>An unbounded sequence's inverse tends to 0.\<close>
  1.1935 +lemma LIMSEQ_inverse_zero: "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
  1.1936    apply (rule filterlim_compose[OF tendsto_inverse_0])
  1.1937    apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
  1.1938    apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
  1.1939    done
  1.1940  
  1.1941 -text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
  1.1942 -
  1.1943 -lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
  1.1944 +text \<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity.\<close>
  1.1945 +lemma LIMSEQ_inverse_real_of_nat: "(\<lambda>n. inverse (real (Suc n))) \<longlonglongrightarrow> 0"
  1.1946    by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
  1.1947 -            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
  1.1948 -
  1.1949 -text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
  1.1950 -infinity is now easily proved\<close>
  1.1951 -
  1.1952 -lemma LIMSEQ_inverse_real_of_nat_add:
  1.1953 -     "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
  1.1954 +      filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
  1.1955 +
  1.1956 +text \<open>
  1.1957 +  The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
  1.1958 +  infinity is now easily proved.
  1.1959 +\<close>
  1.1960 +
  1.1961 +lemma LIMSEQ_inverse_real_of_nat_add: "(\<lambda>n. r + inverse (real (Suc n))) \<longlonglongrightarrow> r"
  1.1962    using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
  1.1963  
  1.1964 -lemma LIMSEQ_inverse_real_of_nat_add_minus:
  1.1965 -     "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
  1.1966 +lemma LIMSEQ_inverse_real_of_nat_add_minus: "(\<lambda>n. r + -inverse (real (Suc n))) \<longlonglongrightarrow> r"
  1.1967    using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
  1.1968    by auto
  1.1969  
  1.1970 -lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
  1.1971 -     "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
  1.1972 +lemma LIMSEQ_inverse_real_of_nat_add_minus_mult: "(\<lambda>n. r * (1 + - inverse (real (Suc n)))) \<longlonglongrightarrow> r"
  1.1973    using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
  1.1974    by auto
  1.1975  
  1.1976 @@ -1735,46 +1834,57 @@
  1.1977  lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
  1.1978  proof (rule Lim_transform_eventually)
  1.1979    show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
  1.1980 -    using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
  1.1981 +    using eventually_gt_at_top[of "0::nat"]
  1.1982 +    by eventually_elim (simp add: field_simps)
  1.1983    have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
  1.1984      by (intro tendsto_add tendsto_const lim_inverse_n)
  1.1985 -  thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
  1.1986 +  then show "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1"
  1.1987 +    by simp
  1.1988  qed
  1.1989  
  1.1990  lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
  1.1991  proof (rule Lim_transform_eventually)
  1.1992    show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
  1.1993 -                        of_nat n / of_nat (Suc n)) sequentially"
  1.1994 +      of_nat n / of_nat (Suc n)) sequentially"
  1.1995      using eventually_gt_at_top[of "0::nat"]
  1.1996      by eventually_elim (simp add: field_simps del: of_nat_Suc)
  1.1997    have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
  1.1998      by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
  1.1999 -  thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
  1.2000 +  then show "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1"
  1.2001 +    by simp
  1.2002  qed
  1.2003  
  1.2004 +
  1.2005  subsection \<open>Convergence on sequences\<close>
  1.2006  
  1.2007  lemma convergent_cong:
  1.2008    assumes "eventually (\<lambda>x. f x = g x) sequentially"
  1.2009 -  shows   "convergent f \<longleftrightarrow> convergent g"
  1.2010 -  unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
  1.2011 +  shows "convergent f \<longleftrightarrow> convergent g"
  1.2012 +  unfolding convergent_def
  1.2013 +  by (subst filterlim_cong[OF refl refl assms]) (rule refl)
  1.2014  
  1.2015  lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
  1.2016    by (auto simp: convergent_def LIMSEQ_Suc_iff)
  1.2017  
  1.2018  lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
  1.2019 -proof (induction m arbitrary: f)
  1.2020 +proof (induct m arbitrary: f)
  1.2021 +  case 0
  1.2022 +  then show ?case by simp
  1.2023 +next
  1.2024    case (Suc m)
  1.2025 -  have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
  1.2026 -  also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
  1.2027 -  also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
  1.2028 +  have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))"
  1.2029 +    by simp
  1.2030 +  also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))"
  1.2031 +    by (rule convergent_Suc_iff)
  1.2032 +  also have "\<dots> \<longleftrightarrow> convergent f"
  1.2033 +    by (rule Suc)
  1.2034    finally show ?case .
  1.2035 -qed simp_all
  1.2036 +qed
  1.2037  
  1.2038  lemma convergent_add:
  1.2039    fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2040    assumes "convergent (\<lambda>n. X n)"
  1.2041 -  assumes "convergent (\<lambda>n. Y n)"
  1.2042 +    and "convergent (\<lambda>n. Y n)"
  1.2043    shows "convergent (\<lambda>n. X n + Y n)"
  1.2044    using assms unfolding convergent_def by (blast intro: tendsto_add)
  1.2045  
  1.2046 @@ -1783,9 +1893,14 @@
  1.2047    assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
  1.2048    shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
  1.2049  proof (cases "finite A")
  1.2050 -  case True from this and assms show ?thesis
  1.2051 -    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
  1.2052 -qed (simp add: convergent_const)
  1.2053 +  case True
  1.2054 +  then show ?thesis
  1.2055 +    using assms by (induct A set: finite) (simp_all add: convergent_const convergent_add)
  1.2056 +next
  1.2057 +  case False
  1.2058 +  then show ?thesis
  1.2059 +    by (simp add: convergent_const)
  1.2060 +qed
  1.2061  
  1.2062  lemma (in bounded_linear) convergent:
  1.2063    assumes "convergent (\<lambda>n. X n)"
  1.2064 @@ -1793,17 +1908,18 @@
  1.2065    using assms unfolding convergent_def by (blast intro: tendsto)
  1.2066  
  1.2067  lemma (in bounded_bilinear) convergent:
  1.2068 -  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
  1.2069 +  assumes "convergent (\<lambda>n. X n)"
  1.2070 +    and "convergent (\<lambda>n. Y n)"
  1.2071    shows "convergent (\<lambda>n. X n ** Y n)"
  1.2072    using assms unfolding convergent_def by (blast intro: tendsto)
  1.2073  
  1.2074 -lemma convergent_minus_iff:
  1.2075 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2076 -  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
  1.2077 -apply (simp add: convergent_def)
  1.2078 -apply (auto dest: tendsto_minus)
  1.2079 -apply (drule tendsto_minus, auto)
  1.2080 -done
  1.2081 +lemma convergent_minus_iff: "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
  1.2082 +  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2083 +  apply (simp add: convergent_def)
  1.2084 +  apply (auto dest: tendsto_minus)
  1.2085 +  apply (drule tendsto_minus)
  1.2086 +  apply auto
  1.2087 +  done
  1.2088  
  1.2089  lemma convergent_diff:
  1.2090    fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2091 @@ -1814,57 +1930,64 @@
  1.2092  
  1.2093  lemma convergent_norm:
  1.2094    assumes "convergent f"
  1.2095 -  shows   "convergent (\<lambda>n. norm (f n))"
  1.2096 +  shows "convergent (\<lambda>n. norm (f n))"
  1.2097  proof -
  1.2098 -  from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
  1.2099 -  hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
  1.2100 -  thus ?thesis by (auto simp: convergent_def)
  1.2101 +  from assms have "f \<longlonglongrightarrow> lim f"
  1.2102 +    by (simp add: convergent_LIMSEQ_iff)
  1.2103 +  then have "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)"
  1.2104 +    by (rule tendsto_norm)
  1.2105 +  then show ?thesis
  1.2106 +    by (auto simp: convergent_def)
  1.2107  qed
  1.2108  
  1.2109  lemma convergent_of_real:
  1.2110 -  "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
  1.2111 +  "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a::real_normed_algebra_1)"
  1.2112    unfolding convergent_def by (blast intro!: tendsto_of_real)
  1.2113  
  1.2114  lemma convergent_add_const_iff:
  1.2115 -  "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
  1.2116 +  "convergent (\<lambda>n. c + f n :: 'a::real_normed_vector) \<longleftrightarrow> convergent f"
  1.2117  proof
  1.2118    assume "convergent (\<lambda>n. c + f n)"
  1.2119 -  from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
  1.2120 +  from convergent_diff[OF this convergent_const[of c]] show "convergent f"
  1.2121 +    by simp
  1.2122  next
  1.2123    assume "convergent f"
  1.2124 -  from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
  1.2125 +  from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)"
  1.2126 +    by simp
  1.2127  qed
  1.2128  
  1.2129  lemma convergent_add_const_right_iff:
  1.2130 -  "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
  1.2131 +  "convergent (\<lambda>n. f n + c :: 'a::real_normed_vector) \<longleftrightarrow> convergent f"
  1.2132    using convergent_add_const_iff[of c f] by (simp add: add_ac)
  1.2133  
  1.2134  lemma convergent_diff_const_right_iff:
  1.2135 -  "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
  1.2136 +  "convergent (\<lambda>n. f n - c :: 'a::real_normed_vector) \<longleftrightarrow> convergent f"
  1.2137    using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
  1.2138  
  1.2139  lemma convergent_mult:
  1.2140    fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
  1.2141    assumes "convergent (\<lambda>n. X n)"
  1.2142 -  assumes "convergent (\<lambda>n. Y n)"
  1.2143 +    and "convergent (\<lambda>n. Y n)"
  1.2144    shows "convergent (\<lambda>n. X n * Y n)"
  1.2145    using assms unfolding convergent_def by (blast intro: tendsto_mult)
  1.2146  
  1.2147  lemma convergent_mult_const_iff:
  1.2148    assumes "c \<noteq> 0"
  1.2149 -  shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
  1.2150 +  shows "convergent (\<lambda>n. c * f n :: 'a::real_normed_field) \<longleftrightarrow> convergent f"
  1.2151  proof
  1.2152    assume "convergent (\<lambda>n. c * f n)"
  1.2153    from assms convergent_mult[OF this convergent_const[of "inverse c"]]
  1.2154      show "convergent f" by (simp add: field_simps)
  1.2155  next
  1.2156    assume "convergent f"
  1.2157 -  from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
  1.2158 +  from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)"
  1.2159 +    by simp
  1.2160  qed
  1.2161  
  1.2162  lemma convergent_mult_const_right_iff:
  1.2163 +  fixes c :: "'a::real_normed_field"
  1.2164    assumes "c \<noteq> 0"
  1.2165 -  shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
  1.2166 +  shows "convergent (\<lambda>n. f n * c) \<longleftrightarrow> convergent f"
  1.2167    using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
  1.2168  
  1.2169  lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
  1.2170 @@ -1874,60 +1997,66 @@
  1.2171  text \<open>A monotone sequence converges to its least upper bound.\<close>
  1.2172  
  1.2173  lemma LIMSEQ_incseq_SUP:
  1.2174 -  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
  1.2175 +  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder,linorder_topology}"
  1.2176    assumes u: "bdd_above (range X)"
  1.2177 -  assumes X: "incseq X"
  1.2178 +    and X: "incseq X"
  1.2179    shows "X \<longlonglongrightarrow> (SUP i. X i)"
  1.2180    by (rule order_tendstoI)
  1.2181 -     (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
  1.2182 +    (auto simp: eventually_sequentially u less_cSUP_iff
  1.2183 +      intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
  1.2184  
  1.2185  lemma LIMSEQ_decseq_INF:
  1.2186    fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
  1.2187    assumes u: "bdd_below (range X)"
  1.2188 -  assumes X: "decseq X"
  1.2189 +    and X: "decseq X"
  1.2190    shows "X \<longlonglongrightarrow> (INF i. X i)"
  1.2191    by (rule order_tendstoI)
  1.2192 -     (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
  1.2193 -
  1.2194 -text\<open>Main monotonicity theorem\<close>
  1.2195 -
  1.2196 -lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
  1.2197 -  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
  1.2198 -
  1.2199 -lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
  1.2200 +     (auto simp: eventually_sequentially u cINF_less_iff
  1.2201 +       intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
  1.2202 +
  1.2203 +text \<open>Main monotonicity theorem.\<close>
  1.2204 +
  1.2205 +lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent X"
  1.2206 +  for X :: "nat \<Rightarrow> real"
  1.2207 +  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP
  1.2208 +      dest: Bseq_bdd_above Bseq_bdd_below)
  1.2209 +
  1.2210 +lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent X"
  1.2211 +  for X :: "nat \<Rightarrow> real"
  1.2212    by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
  1.2213  
  1.2214 -lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
  1.2215 +lemma monoseq_imp_convergent_iff_Bseq: "monoseq f \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
  1.2216 +  for f :: "nat \<Rightarrow> real"
  1.2217    using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
  1.2218  
  1.2219  lemma Bseq_monoseq_convergent'_inc:
  1.2220 -  "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
  1.2221 +  fixes f :: "nat \<Rightarrow> real"
  1.2222 +  shows "Bseq (\<lambda>n. f (n + M)) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<le> f n) \<Longrightarrow> convergent f"
  1.2223    by (subst convergent_ignore_initial_segment [symmetric, of _ M])
  1.2224       (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
  1.2225  
  1.2226  lemma Bseq_monoseq_convergent'_dec:
  1.2227 -  "Bseq (\<lambda>n. f (n + M) :: real) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
  1.2228 +  fixes f :: "nat \<Rightarrow> real"
  1.2229 +  shows "Bseq (\<lambda>n. f (n + M)) \<Longrightarrow> (\<And>m n. M \<le> m \<Longrightarrow> m \<le> n \<Longrightarrow> f m \<ge> f n) \<Longrightarrow> convergent f"
  1.2230    by (subst convergent_ignore_initial_segment [symmetric, of _ M])
  1.2231 -     (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
  1.2232 -
  1.2233 -lemma Cauchy_iff:
  1.2234 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2235 -  shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
  1.2236 +    (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
  1.2237 +
  1.2238 +lemma Cauchy_iff: "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
  1.2239 +  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2240    unfolding Cauchy_def dist_norm ..
  1.2241  
  1.2242 -lemma CauchyI:
  1.2243 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2244 -  shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
  1.2245 -by (simp add: Cauchy_iff)
  1.2246 -
  1.2247 -lemma CauchyD:
  1.2248 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2249 -  shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
  1.2250 -by (simp add: Cauchy_iff)
  1.2251 +lemma CauchyI: "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
  1.2252 +  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2253 +  by (simp add: Cauchy_iff)
  1.2254 +
  1.2255 +lemma CauchyD: "Cauchy X \<Longrightarrow> 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
  1.2256 +  for X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1.2257 +  by (simp add: Cauchy_iff)
  1.2258  
  1.2259  lemma incseq_convergent:
  1.2260    fixes X :: "nat \<Rightarrow> real"
  1.2261 -  assumes "incseq X" and "\<forall>i. X i \<le> B"
  1.2262 +  assumes "incseq X"
  1.2263 +    and "\<forall>i. X i \<le> B"
  1.2264    obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
  1.2265  proof atomize_elim
  1.2266    from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
  1.2267 @@ -1939,7 +2068,8 @@
  1.2268  
  1.2269  lemma decseq_convergent:
  1.2270    fixes X :: "nat \<Rightarrow> real"
  1.2271 -  assumes "decseq X" and "\<forall>i. B \<le> X i"
  1.2272 +  assumes "decseq X"
  1.2273 +    and "\<forall>i. B \<le> X i"
  1.2274    obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
  1.2275  proof atomize_elim
  1.2276    from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
  1.2277 @@ -1949,69 +2079,85 @@
  1.2278      by (auto intro!: exI[of _ L] decseq_le)
  1.2279  qed
  1.2280  
  1.2281 +
  1.2282  subsubsection \<open>Cauchy Sequences are Bounded\<close>
  1.2283  
  1.2284 -text\<open>A Cauchy sequence is bounded -- this is the standard
  1.2285 -  proof mechanization rather than the nonstandard proof\<close>
  1.2286 -
  1.2287 -lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
  1.2288 -          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
  1.2289 -apply (clarify, drule spec, drule (1) mp)
  1.2290 -apply (simp only: norm_minus_commute)
  1.2291 -apply (drule order_le_less_trans [OF norm_triangle_ineq2])
  1.2292 -apply simp
  1.2293 -done
  1.2294 +text \<open>
  1.2295 +  A Cauchy sequence is bounded -- this is the standard
  1.2296 +  proof mechanization rather than the nonstandard proof.
  1.2297 +\<close>
  1.2298 +
  1.2299 +lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real) \<Longrightarrow>
  1.2300 +  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
  1.2301 +  apply clarify
  1.2302 +  apply (drule spec)
  1.2303 +  apply (drule (1) mp)
  1.2304 +  apply (simp only: norm_minus_commute)
  1.2305 +  apply (drule order_le_less_trans [OF norm_triangle_ineq2])
  1.2306 +  apply simp
  1.2307 +  done
  1.2308 +
  1.2309  
  1.2310  subsection \<open>Power Sequences\<close>
  1.2311  
  1.2312 -text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
  1.2313 -"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
  1.2314 -  also fact that bounded and monotonic sequence converges.\<close>
  1.2315 -
  1.2316 -lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
  1.2317 -apply (simp add: Bseq_def)
  1.2318 -apply (rule_tac x = 1 in exI)
  1.2319 -apply (simp add: power_abs)
  1.2320 -apply (auto dest: power_mono)
  1.2321 -done
  1.2322 -
  1.2323 -lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
  1.2324 -apply (clarify intro!: mono_SucI2)
  1.2325 -apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
  1.2326 -done
  1.2327 -
  1.2328 -lemma convergent_realpow:
  1.2329 -  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
  1.2330 -by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
  1.2331 -
  1.2332 -lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
  1.2333 +text \<open>
  1.2334 +  The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
  1.2335 +  "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
  1.2336 +  also fact that bounded and monotonic sequence converges.
  1.2337 +\<close>
  1.2338 +
  1.2339 +lemma Bseq_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> Bseq (\<lambda>n. x ^ n)"
  1.2340 +  for x :: real
  1.2341 +  apply (simp add: Bseq_def)
  1.2342 +  apply (rule_tac x = 1 in exI)
  1.2343 +  apply (simp add: power_abs)
  1.2344 +  apply (auto dest: power_mono)
  1.2345 +  done
  1.2346 +
  1.2347 +lemma monoseq_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> monoseq (\<lambda>n. x ^ n)"
  1.2348 +  for x :: real
  1.2349 +  apply (clarify intro!: mono_SucI2)
  1.2350 +  apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing)
  1.2351 +     apply auto
  1.2352 +  done
  1.2353 +
  1.2354 +lemma convergent_realpow: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> convergent (\<lambda>n. x ^ n)"
  1.2355 +  for x :: real
  1.2356 +  by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
  1.2357 +
  1.2358 +lemma LIMSEQ_inverse_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
  1.2359 +  for x :: real
  1.2360    by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
  1.2361  
  1.2362  lemma LIMSEQ_realpow_zero:
  1.2363 -  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
  1.2364 -proof cases
  1.2365 -  assume "0 \<le> x" and "x \<noteq> 0"
  1.2366 -  hence x0: "0 < x" by simp
  1.2367 -  assume x1: "x < 1"
  1.2368 -  from x0 x1 have "1 < inverse x"
  1.2369 -    by (rule one_less_inverse)
  1.2370 -  hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
  1.2371 +  fixes x :: real
  1.2372 +  assumes "0 \<le> x" "x < 1"
  1.2373 +  shows "(\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
  1.2374 +proof (cases "x = 0")
  1.2375 +  case False
  1.2376 +  with \<open>0 \<le> x\<close> have x0: "0 < x" by simp
  1.2377 +  then have "1 < inverse x"
  1.2378 +    using \<open>x < 1\<close> by (rule one_less_inverse)
  1.2379 +  then have "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
  1.2380      by (rule LIMSEQ_inverse_realpow_zero)
  1.2381 -  thus ?thesis by (simp add: power_inverse)
  1.2382 -qed (rule LIMSEQ_imp_Suc, simp)
  1.2383 -
  1.2384 -lemma LIMSEQ_power_zero:
  1.2385 -  fixes x :: "'a::{real_normed_algebra_1}"
  1.2386 -  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
  1.2387 -apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
  1.2388 -apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
  1.2389 -apply (simp add: power_abs norm_power_ineq)
  1.2390 -done
  1.2391 +  then show ?thesis by (simp add: power_inverse)
  1.2392 +next
  1.2393 +  case True
  1.2394 +  show ?thesis
  1.2395 +    by (rule LIMSEQ_imp_Suc) (simp add: True)
  1.2396 +qed
  1.2397 +
  1.2398 +lemma LIMSEQ_power_zero: "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
  1.2399 +  for x :: "'a::real_normed_algebra_1"
  1.2400 +  apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
  1.2401 +  apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
  1.2402 +  apply (simp add: power_abs norm_power_ineq)
  1.2403 +  done
  1.2404  
  1.2405  lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
  1.2406    by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
  1.2407  
  1.2408 -text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
  1.2409 +text \<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}.\<close>
  1.2410  
  1.2411  lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
  1.2412    by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
  1.2413 @@ -2022,92 +2168,81 @@
  1.2414  
  1.2415  subsection \<open>Limits of Functions\<close>
  1.2416  
  1.2417 -lemma LIM_eq:
  1.2418 -  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  1.2419 -  shows "f \<midarrow>a\<rightarrow> L =
  1.2420 -     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
  1.2421 -by (simp add: LIM_def dist_norm)
  1.2422 +lemma LIM_eq: "f \<midarrow>a\<rightarrow> L = (\<forall>r>0. \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r)"
  1.2423 +  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  1.2424 +  by (simp add: LIM_def dist_norm)
  1.2425  
  1.2426  lemma LIM_I:
  1.2427 -  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  1.2428 -  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
  1.2429 -      ==> f \<midarrow>a\<rightarrow> L"
  1.2430 -by (simp add: LIM_eq)
  1.2431 -
  1.2432 -lemma LIM_D:
  1.2433 -  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  1.2434 -  shows "[| f \<midarrow>a\<rightarrow> L; 0<r |]
  1.2435 -      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
  1.2436 -by (simp add: LIM_eq)
  1.2437 -
  1.2438 -lemma LIM_offset:
  1.2439 -  fixes a :: "'a::real_normed_vector"
  1.2440 -  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
  1.2441 -  unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
  1.2442 -
  1.2443 -lemma LIM_offset_zero:
  1.2444 -  fixes a :: "'a::real_normed_vector"
  1.2445 -  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
  1.2446 -by (drule_tac k="a" in LIM_offset, simp add: add.commute)
  1.2447 -
  1.2448 -lemma LIM_offset_zero_cancel:
  1.2449 -  fixes a :: "'a::real_normed_vector"
  1.2450 -  shows "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
  1.2451 -by (drule_tac k="- a" in LIM_offset, simp)
  1.2452 -
  1.2453 -lemma LIM_offset_zero_iff:
  1.2454 -  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
  1.2455 -  shows  "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
  1.2456 +  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r) \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
  1.2457 +  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  1.2458 +  by (simp add: LIM_eq)
  1.2459 +
  1.2460 +lemma LIM_D: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> 0 < r \<Longrightarrow> \<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
  1.2461 +  for a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
  1.2462 +  by (simp add: LIM_eq)
  1.2463 +
  1.2464 +lemma LIM_offset: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>x. f (x + k)) \<midarrow>(a - k)\<rightarrow> L"
  1.2465 +  for a :: "'a::real_normed_vector"
  1.2466 +  by (simp add: filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap)
  1.2467 +
  1.2468 +lemma LIM_offset_zero: "f \<midarrow>a\<rightarrow> L \<Longrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
  1.2469 +  for a :: "'a::real_normed_vector"
  1.2470 +  by (drule LIM_offset [where k = a]) (simp add: add.commute)
  1.2471 +
  1.2472 +lemma LIM_offset_zero_cancel: "(\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> L"
  1.2473 +  for a :: "'a::real_normed_vector"
  1.2474 +  by (drule LIM_offset [where k = "- a"]) simp
  1.2475 +
  1.2476 +lemma LIM_offset_zero_iff: "f \<midarrow>a\<rightarrow> L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> L"
  1.2477 +  for f :: "'a :: real_normed_vector \<Rightarrow> _"
  1.2478    using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
  1.2479  
  1.2480 -lemma LIM_zero:
  1.2481 -  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  1.2482 -  shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
  1.2483 -unfolding tendsto_iff dist_norm by simp
  1.2484 +lemma LIM_zero: "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
  1.2485 +  for f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  1.2486 +  unfolding tendsto_iff dist_norm by simp
  1.2487  
  1.2488  lemma LIM_zero_cancel:
  1.2489    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  1.2490    shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
  1.2491  unfolding tendsto_iff dist_norm by simp
  1.2492  
  1.2493 -lemma LIM_zero_iff:
  1.2494 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  1.2495 -  shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
  1.2496 -unfolding tendsto_iff dist_norm by simp
  1.2497 +lemma LIM_zero_iff: "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
  1.2498 +  for f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
  1.2499 +  unfolding tendsto_iff dist_norm by simp
  1.2500  
  1.2501  lemma LIM_imp_LIM:
  1.2502    fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
  1.2503    fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
  1.2504    assumes f: "f \<midarrow>a\<rightarrow> l"
  1.2505 -  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
  1.2506 +    and le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
  1.2507    shows "g \<midarrow>a\<rightarrow> m"
  1.2508 -  by (rule metric_LIM_imp_LIM [OF f],
  1.2509 -    simp add: dist_norm le)
  1.2510 +  by (rule metric_LIM_imp_LIM [OF f]) (simp add: dist_norm le)
  1.2511  
  1.2512  lemma LIM_equal2:
  1.2513    fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
  1.2514 -  assumes 1: "0 < R"
  1.2515 -  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
  1.2516 +  assumes "0 < R"
  1.2517 +    and "\<And>x. x \<noteq> a \<Longrightarrow> norm (x - a) < R \<Longrightarrow> f x = g x"
  1.2518    shows "g \<midarrow>a\<rightarrow> l \<Longrightarrow> f \<midarrow>a\<rightarrow> l"
  1.2519 -by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
  1.2520 +  by (rule metric_LIM_equal2 [OF assms]) (simp_all add: dist_norm)
  1.2521  
  1.2522  lemma LIM_compose2:
  1.2523    fixes a :: "'a::real_normed_vector"
  1.2524    assumes f: "f \<midarrow>a\<rightarrow> b"
  1.2525 -  assumes g: "g \<midarrow>b\<rightarrow> c"
  1.2526 -  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
  1.2527 +    and g: "g \<midarrow>b\<rightarrow> c"
  1.2528 +    and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
  1.2529    shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
  1.2530 -by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
  1.2531 +  by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
  1.2532  
  1.2533  lemma real_LIM_sandwich_zero:
  1.2534    fixes f g :: "'a::topological_space \<Rightarrow> real"
  1.2535    assumes f: "f \<midarrow>a\<rightarrow> 0"
  1.2536 -  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
  1.2537 -  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
  1.2538 +    and 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
  1.2539 +    and 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
  1.2540    shows "g \<midarrow>a\<rightarrow> 0"
  1.2541  proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
  1.2542 -  fix x assume x: "x \<noteq> a"
  1.2543 -  have "norm (g x - 0) = g x" by (simp add: 1 x)
  1.2544 +  fix x
  1.2545 +  assume x: "x \<noteq> a"
  1.2546 +  with 1 have "norm (g x - 0) = g x" by simp
  1.2547    also have "g x \<le> f x" by (rule 2 [OF x])
  1.2548    also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
  1.2549    also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
  1.2550 @@ -2117,61 +2252,50 @@
  1.2551  
  1.2552  subsection \<open>Continuity\<close>
  1.2553  
  1.2554 -lemma LIM_isCont_iff:
  1.2555 -  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
  1.2556 -  shows "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
  1.2557 -by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
  1.2558 -
  1.2559 -lemma isCont_iff:
  1.2560 -  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
  1.2561 -  shows "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
  1.2562 -by (simp add: isCont_def LIM_isCont_iff)
  1.2563 +lemma LIM_isCont_iff: "(f \<midarrow>a\<rightarrow> f a) = ((\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow> f a)"
  1.2564 +  for f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
  1.2565 +  by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
  1.2566 +
  1.2567 +lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) \<midarrow>0\<rightarrow> f x"
  1.2568 +  for f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
  1.2569 +  by (simp add: isCont_def LIM_isCont_iff)
  1.2570  
  1.2571  lemma isCont_LIM_compose2:
  1.2572    fixes a :: "'a::real_normed_vector"
  1.2573    assumes f [unfolded isCont_def]: "isCont f a"
  1.2574 -  assumes g: "g \<midarrow>f a\<rightarrow> l"
  1.2575 -  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
  1.2576 +    and g: "g \<midarrow>f a\<rightarrow> l"
  1.2577 +    and inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
  1.2578    shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> l"
  1.2579 -by (rule LIM_compose2 [OF f g inj])
  1.2580 -
  1.2581 -
  1.2582 -lemma isCont_norm [simp]:
  1.2583 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.2584 -  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
  1.2585 +  by (rule LIM_compose2 [OF f g inj])
  1.2586 +
  1.2587 +lemma isCont_norm [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
  1.2588 +  for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.2589    by (fact continuous_norm)
  1.2590  
  1.2591 -lemma isCont_rabs [simp]:
  1.2592 -  fixes f :: "'a::t2_space \<Rightarrow> real"
  1.2593 -  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
  1.2594 +lemma isCont_rabs [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
  1.2595 +  for f :: "'a::t2_space \<Rightarrow> real"
  1.2596    by (fact continuous_rabs)
  1.2597  
  1.2598 -lemma isCont_add [simp]:
  1.2599 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
  1.2600 -  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
  1.2601 +lemma isCont_add [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
  1.2602 +  for f :: "'a::t2_space \<Rightarrow> 'b::topological_monoid_add"
  1.2603    by (fact continuous_add)
  1.2604  
  1.2605 -lemma isCont_minus [simp]:
  1.2606 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.2607 -  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
  1.2608 +lemma isCont_minus [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
  1.2609 +  for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.2610    by (fact continuous_minus)
  1.2611  
  1.2612 -lemma isCont_diff [simp]:
  1.2613 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.2614 -  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
  1.2615 +lemma isCont_diff [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
  1.2616 +  for f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
  1.2617    by (fact continuous_diff)
  1.2618  
  1.2619 -lemma isCont_mult [simp]:
  1.2620 -  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
  1.2621 -  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
  1.2622 +lemma isCont_mult [simp]: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
  1.2623 +  for f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
  1.2624    by (fact continuous_mult)
  1.2625  
  1.2626 -lemma (in bounded_linear) isCont:
  1.2627 -  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
  1.2628 +lemma (in bounded_linear) isCont: "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
  1.2629    by (fact continuous)
  1.2630  
  1.2631 -lemma (in bounded_bilinear) isCont:
  1.2632 -  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
  1.2633 +lemma (in bounded_bilinear) isCont: "isCont f a \<Longrightarrow> isCont g a \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
  1.2634    by (fact continuous)
  1.2635  
  1.2636  lemmas isCont_scaleR [simp] =
  1.2637 @@ -2180,16 +2304,15 @@
  1.2638  lemmas isCont_of_real [simp] =
  1.2639    bounded_linear.isCont [OF bounded_linear_of_real]
  1.2640  
  1.2641 -lemma isCont_power [simp]:
  1.2642 -  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
  1.2643 -  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
  1.2644 +lemma isCont_power [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
  1.2645 +  for f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
  1.2646    by (fact continuous_power)
  1.2647  
  1.2648 -lemma isCont_setsum [simp]:
  1.2649 -  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
  1.2650 -  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
  1.2651 +lemma isCont_setsum [simp]: "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
  1.2652 +  for f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::topological_comm_monoid_add"
  1.2653    by (auto intro: continuous_setsum)
  1.2654  
  1.2655 +
  1.2656  subsection \<open>Uniform Continuity\<close>
  1.2657  
  1.2658  lemma uniformly_continuous_on_def:
  1.2659 @@ -2200,37 +2323,39 @@
  1.2660      uniformity_dist filterlim_INF filterlim_principal eventually_inf_principal
  1.2661    by (force simp: Ball_def uniformity_dist[symmetric] eventually_uniformity_metric)
  1.2662  
  1.2663 -abbreviation isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
  1.2664 -  "isUCont f \<equiv> uniformly_continuous_on UNIV f"
  1.2665 -
  1.2666 -lemma isUCont_def: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
  1.2667 +abbreviation isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool"
  1.2668 +  where "isUCont f \<equiv> uniformly_continuous_on UNIV f"
  1.2669 +
  1.2670 +lemma isUCont_def: "isUCont f \<longleftrightarrow> (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
  1.2671    by (auto simp: uniformly_continuous_on_def dist_commute)
  1.2672  
  1.2673 -lemma isUCont_isCont: "isUCont f ==> isCont f x"
  1.2674 +lemma isUCont_isCont: "isUCont f \<Longrightarrow> isCont f x"
  1.2675    by (drule uniformly_continuous_imp_continuous) (simp add: continuous_on_eq_continuous_at)
  1.2676  
  1.2677  lemma uniformly_continuous_on_Cauchy:
  1.2678 -  fixes f::"'a::metric_space \<Rightarrow> 'b::metric_space"
  1.2679 +  fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
  1.2680    assumes "uniformly_continuous_on S f" "Cauchy X" "\<And>n. X n \<in> S"
  1.2681    shows "Cauchy (\<lambda>n. f (X n))"
  1.2682    using assms
  1.2683 -  unfolding uniformly_continuous_on_def
  1.2684 -  apply -
  1.2685 +  apply (simp only: uniformly_continuous_on_def)
  1.2686    apply (rule metric_CauchyI)
  1.2687 -  apply (drule_tac x=e in spec, safe)
  1.2688 -  apply (drule_tac e=d in metric_CauchyD, safe)
  1.2689 -  apply (rule_tac x=M in exI, simp)
  1.2690 +  apply (drule_tac x=e in spec)
  1.2691 +  apply safe
  1.2692 +  apply (drule_tac e=d in metric_CauchyD)
  1.2693 +   apply safe
  1.2694 +  apply (rule_tac x=M in exI)
  1.2695 +  apply simp
  1.2696    done
  1.2697  
  1.2698 -lemma isUCont_Cauchy:
  1.2699 -  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
  1.2700 +lemma isUCont_Cauchy: "isUCont f \<Longrightarrow> Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
  1.2701    by (rule uniformly_continuous_on_Cauchy[where S=UNIV and f=f]) simp_all
  1.2702  
  1.2703  lemma (in bounded_linear) isUCont: "isUCont f"
  1.2704 -unfolding isUCont_def dist_norm
  1.2705 +  unfolding isUCont_def dist_norm
  1.2706  proof (intro allI impI)
  1.2707 -  fix r::real assume r: "0 < r"
  1.2708 -  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
  1.2709 +  fix r :: real
  1.2710 +  assume r: "0 < r"
  1.2711 +  obtain K where K: "0 < K" and norm_le: "norm (f x) \<le> norm x * K" for x
  1.2712      using pos_bounded by blast
  1.2713    show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
  1.2714    proof (rule exI, safe)
  1.2715 @@ -2246,7 +2371,7 @@
  1.2716  qed
  1.2717  
  1.2718  lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
  1.2719 -by (rule isUCont [THEN isUCont_Cauchy])
  1.2720 +  by (rule isUCont [THEN isUCont_Cauchy])
  1.2721  
  1.2722  lemma LIM_less_bound:
  1.2723    fixes f :: "real \<Rightarrow> real"
  1.2724 @@ -2268,16 +2393,21 @@
  1.2725  proof -
  1.2726    have "incseq f" unfolding incseq_Suc_iff by fact
  1.2727    have "decseq g" unfolding decseq_Suc_iff by fact
  1.2728 -
  1.2729 -  { fix n
  1.2730 -    from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
  1.2731 -    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
  1.2732 +  have "f n \<le> g 0" for n
  1.2733 +  proof -
  1.2734 +    from \<open>decseq g\<close> have "g n \<le> g 0"
  1.2735 +      by (rule decseqD) simp
  1.2736 +    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] show ?thesis
  1.2737 +      by auto
  1.2738 +  qed
  1.2739    then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
  1.2740      using incseq_convergent[OF \<open>incseq f\<close>] by auto
  1.2741 -  moreover
  1.2742 -  { fix n
  1.2743 +  moreover have "f 0 \<le> g n" for n
  1.2744 +  proof -
  1.2745      from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
  1.2746 -    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
  1.2747 +    with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] show ?thesis
  1.2748 +      by simp
  1.2749 +  qed
  1.2750    then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
  1.2751      using decseq_convergent[OF \<open>decseq g\<close>] by auto
  1.2752    moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
  1.2753 @@ -2287,8 +2417,8 @@
  1.2754  lemma Bolzano[consumes 1, case_names trans local]:
  1.2755    fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
  1.2756    assumes [arith]: "a \<le> b"
  1.2757 -  assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
  1.2758 -  assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
  1.2759 +    and trans: "\<And>a b c. P a b \<Longrightarrow> P b c \<Longrightarrow> a \<le> b \<Longrightarrow> b \<le> c \<Longrightarrow> P a c"
  1.2760 +    and local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
  1.2761    shows "P a b"
  1.2762  proof -
  1.2763    define bisect where "bisect =
  1.2764 @@ -2298,57 +2428,73 @@
  1.2765      and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
  1.2766      by (simp_all add: l_def u_def bisect_def split: prod.split)
  1.2767  
  1.2768 -  { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
  1.2769 +  have [simp]: "l n \<le> u n" for n by (induct n) auto
  1.2770  
  1.2771    have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
  1.2772    proof (safe intro!: nested_sequence_unique)
  1.2773 -    fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
  1.2774 +    show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" for n
  1.2775 +      by (induct n) auto
  1.2776    next
  1.2777 -    { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
  1.2778 -    then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
  1.2779 +    have "l n - u n = (a - b) / 2^n" for n
  1.2780 +      by (induct n) (auto simp: field_simps)
  1.2781 +    then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0"
  1.2782 +      by (simp add: LIMSEQ_divide_realpow_zero)
  1.2783    qed fact
  1.2784 -  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x" by auto
  1.2785 -  obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
  1.2786 +  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l \<longlonglongrightarrow> x" "u \<longlonglongrightarrow> x"
  1.2787 +    by auto
  1.2788 +  obtain d where "0 < d" and d: "a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b" for a b
  1.2789      using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
  1.2790  
  1.2791    show "P a b"
  1.2792    proof (rule ccontr)
  1.2793      assume "\<not> P a b"
  1.2794 -    { fix n have "\<not> P (l n) (u n)"
  1.2795 -      proof (induct n)
  1.2796 -        case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
  1.2797 -      qed (simp add: \<open>\<not> P a b\<close>) }
  1.2798 +    have "\<not> P (l n) (u n)" for n
  1.2799 +    proof (induct n)
  1.2800 +      case 0
  1.2801 +      then show ?case
  1.2802 +        by (simp add: \<open>\<not> P a b\<close>)
  1.2803 +    next
  1.2804 +      case (Suc n)
  1.2805 +      with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case
  1.2806 +        by auto
  1.2807 +    qed
  1.2808      moreover
  1.2809 -    { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
  1.2810 +    {
  1.2811 +      have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
  1.2812          using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
  1.2813        moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
  1.2814          using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
  1.2815        ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
  1.2816        proof eventually_elim
  1.2817 -        fix n assume "x - d / 2 < l n" "u n < x + d / 2"
  1.2818 +        case (elim n)
  1.2819          from add_strict_mono[OF this] have "u n - l n < d" by simp
  1.2820          with x show "P (l n) (u n)" by (rule d)
  1.2821 -      qed }
  1.2822 +      qed
  1.2823 +    }
  1.2824      ultimately show False by simp
  1.2825    qed
  1.2826  qed
  1.2827  
  1.2828  lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
  1.2829  proof (cases "a \<le> b", rule compactI)
  1.2830 -  fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
  1.2831 +  fix C
  1.2832 +  assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
  1.2833    define T where "T = {a .. b}"
  1.2834    from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
  1.2835    proof (induct rule: Bolzano)
  1.2836      case (trans a b c)
  1.2837 -    then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
  1.2838 -    from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
  1.2839 -      by (auto simp: *)
  1.2840 +    then have *: "{a..c} = {a..b} \<union> {b..c}"
  1.2841 +      by auto
  1.2842 +    with trans obtain C1 C2
  1.2843 +      where "C1\<subseteq>C" "finite C1" "{a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C" "finite C2" "{b..c} \<subseteq> \<Union>C2"
  1.2844 +      by auto
  1.2845      with trans show ?case
  1.2846        unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
  1.2847    next
  1.2848      case (local x)
  1.2849 -    then have "x \<in> \<Union>C" using C by auto
  1.2850 -    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
  1.2851 +    with C have "x \<in> \<Union>C" by auto
  1.2852 +    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C"
  1.2853 +      by auto
  1.2854      then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
  1.2855        by (auto simp: open_dist dist_real_def subset_eq Ball_def abs_less_iff)
  1.2856      with \<open>c \<in> C\<close> show ?case
  1.2857 @@ -2378,17 +2524,18 @@
  1.2858  qed
  1.2859  
  1.2860  lemma open_Collect_positive:
  1.2861 - fixes f :: "'a::t2_space \<Rightarrow> real"
  1.2862 - assumes f: "continuous_on s f"
  1.2863 - shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
  1.2864 - using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
  1.2865 - by (auto simp: Int_def field_simps)
  1.2866 +  fixes f :: "'a::t2_space \<Rightarrow> real"
  1.2867 +  assumes f: "continuous_on s f"
  1.2868 +  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
  1.2869 +  using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
  1.2870 +  by (auto simp: Int_def field_simps)
  1.2871  
  1.2872  lemma open_Collect_less_Int:
  1.2873 - fixes f g :: "'a::t2_space \<Rightarrow> real"
  1.2874 - assumes f: "continuous_on s f" and g: "continuous_on s g"
  1.2875 - shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
  1.2876 - using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
  1.2877 +  fixes f g :: "'a::t2_space \<Rightarrow> real"
  1.2878 +  assumes f: "continuous_on s f"
  1.2879 +    and g: "continuous_on s g"
  1.2880 +  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
  1.2881 +  using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
  1.2882  
  1.2883  
  1.2884  subsection \<open>Boundedness of continuous functions\<close>
  1.2885 @@ -2399,14 +2546,14 @@
  1.2886    fixes f :: "real \<Rightarrow> 'a::linorder_topology"
  1.2887    shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
  1.2888      \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
  1.2889 -  using continuous_attains_sup[of "{a .. b}" f]
  1.2890 +  using continuous_attains_sup[of "{a..b}" f]
  1.2891    by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
  1.2892  
  1.2893  lemma isCont_eq_Lb:
  1.2894    fixes f :: "real \<Rightarrow> 'a::linorder_topology"
  1.2895    shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
  1.2896      \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
  1.2897 -  using continuous_attains_inf[of "{a .. b}" f]
  1.2898 +  using continuous_attains_inf[of "{a..b}" f]
  1.2899    by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
  1.2900  
  1.2901  lemma isCont_bounded:
  1.2902 @@ -2421,21 +2568,23 @@
  1.2903    using isCont_eq_Ub[of a b f] by auto
  1.2904  
  1.2905  (*HOL style here: object-level formulations*)
  1.2906 -lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
  1.2907 -      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
  1.2908 -      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
  1.2909 +lemma IVT_objl:
  1.2910 +  "(f a \<le> y \<and> y \<le> f b \<and> a \<le> b \<and> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x)) \<longrightarrow>
  1.2911 +    (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y)"
  1.2912 +  for a y :: real
  1.2913    by (blast intro: IVT)
  1.2914  
  1.2915 -lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
  1.2916 -      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
  1.2917 -      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
  1.2918 +lemma IVT2_objl:
  1.2919 +  "(f b \<le> y \<and> y \<le> f a \<and> a \<le> b \<and> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x)) \<longrightarrow>
  1.2920 +    (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y)"
  1.2921 +  for b y :: real
  1.2922    by (blast intro: IVT2)
  1.2923  
  1.2924  lemma isCont_Lb_Ub:
  1.2925    fixes f :: "real \<Rightarrow> real"
  1.2926    assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
  1.2927    shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
  1.2928 -               (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
  1.2929 +    (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
  1.2930  proof -
  1.2931    obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
  1.2932      using isCont_eq_Ub[OF assms] by auto
  1.2933 @@ -2446,22 +2595,26 @@
  1.2934      apply (rule_tac x="f L" in exI)
  1.2935      apply (rule_tac x="f M" in exI)
  1.2936      apply (cases "L \<le> M")
  1.2937 -    apply (simp, metis order_trans)
  1.2938 -    apply (simp, metis order_trans)
  1.2939 +     apply simp
  1.2940 +     apply (metis order_trans)
  1.2941 +    apply simp
  1.2942 +    apply (metis order_trans)
  1.2943      done
  1.2944  qed
  1.2945  
  1.2946  
  1.2947 -text\<open>Continuity of inverse function\<close>
  1.2948 +text \<open>Continuity of inverse function.\<close>
  1.2949  
  1.2950  lemma isCont_inverse_function:
  1.2951    fixes f g :: "real \<Rightarrow> real"
  1.2952    assumes d: "0 < d"
  1.2953 -      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
  1.2954 -      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
  1.2955 +    and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
  1.2956 +    and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
  1.2957    shows "isCont g (f x)"
  1.2958  proof -
  1.2959 -  let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
  1.2960 +  let ?A = "f (x - d)"
  1.2961 +  let ?B = "f (x + d)"
  1.2962 +  let ?D = "{x - d..x + d}"
  1.2963  
  1.2964    have f: "continuous_on ?D f"
  1.2965      using cont by (intro continuous_at_imp_continuous_on ballI) auto
  1.2966 @@ -2483,45 +2636,42 @@
  1.2967  qed
  1.2968  
  1.2969  lemma isCont_inverse_function2:
  1.2970 -  fixes f g :: "real \<Rightarrow> real" shows
  1.2971 -  "\<lbrakk>a < x; x < b;
  1.2972 -    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
  1.2973 -    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
  1.2974 -   \<Longrightarrow> isCont g (f x)"
  1.2975 -apply (rule isCont_inverse_function
  1.2976 -       [where f=f and d="min (x - a) (b - x)"])
  1.2977 -apply (simp_all add: abs_le_iff)
  1.2978 -done
  1.2979 +  fixes f g :: "real \<Rightarrow> real"
  1.2980 +  shows
  1.2981 +    "a < x \<Longrightarrow> x < b \<Longrightarrow>
  1.2982 +      \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z \<Longrightarrow>
  1.2983 +      \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z \<Longrightarrow> isCont g (f x)"
  1.2984 +  apply (rule isCont_inverse_function [where f=f and d="min (x - a) (b - x)"])
  1.2985 +  apply (simp_all add: abs_le_iff)
  1.2986 +  done
  1.2987  
  1.2988  (* need to rename second isCont_inverse *)
  1.2989 -
  1.2990  lemma isCont_inv_fun:
  1.2991    fixes f g :: "real \<Rightarrow> real"
  1.2992 -  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
  1.2993 -         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
  1.2994 -      ==> isCont g (f x)"
  1.2995 -by (rule isCont_inverse_function)
  1.2996 -
  1.2997 -text\<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110\<close>
  1.2998 -lemma LIM_fun_gt_zero:
  1.2999 -  fixes f :: "real \<Rightarrow> real"
  1.3000 -  shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
  1.3001 -apply (drule (1) LIM_D, clarify)
  1.3002 -apply (rule_tac x = s in exI)
  1.3003 -apply (simp add: abs_less_iff)
  1.3004 -done
  1.3005 -
  1.3006 -lemma LIM_fun_less_zero:
  1.3007 -  fixes f :: "real \<Rightarrow> real"
  1.3008 -  shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
  1.3009 -apply (drule LIM_D [where r="-l"], simp, clarify)
  1.3010 -apply (rule_tac x = s in exI)
  1.3011 -apply (simp add: abs_less_iff)
  1.3012 -done
  1.3013 -
  1.3014 -lemma LIM_fun_not_zero:
  1.3015 -  fixes f :: "real \<Rightarrow> real"
  1.3016 -  shows "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
  1.3017 +  shows "0 < d \<Longrightarrow> (\<forall>z. \<bar>z - x\<bar> \<le> d \<longrightarrow> g (f z) = z) \<Longrightarrow>
  1.3018 +    \<forall>z. \<bar>z - x\<bar> \<le> d \<longrightarrow> isCont f z \<Longrightarrow> isCont g (f x)"
  1.3019 +  by (rule isCont_inverse_function)
  1.3020 +
  1.3021 +text \<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110.\<close>
  1.3022 +lemma LIM_fun_gt_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
  1.3023 +  for f :: "real \<Rightarrow> real"
  1.3024 +  apply (drule (1) LIM_D)
  1.3025 +  apply clarify
  1.3026 +  apply (rule_tac x = s in exI)
  1.3027 +  apply (simp add: abs_less_iff)
  1.3028 +  done
  1.3029 +
  1.3030 +lemma LIM_fun_less_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
  1.3031 +  for f :: "real \<Rightarrow> real"
  1.3032 +  apply (drule LIM_D [where r="-l"])
  1.3033 +   apply simp
  1.3034 +  apply clarify
  1.3035 +  apply (rule_tac x = s in exI)
  1.3036 +  apply (simp add: abs_less_iff)
  1.3037 +  done
  1.3038 +
  1.3039 +lemma LIM_fun_not_zero: "f \<midarrow>c\<rightarrow> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
  1.3040 +  for f :: "real \<Rightarrow> real"
  1.3041    using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
  1.3042  
  1.3043  end