src/HOL/Limits.thy
 author wenzelm Wed Dec 30 11:21:54 2015 +0100 (2015-12-30) changeset 61973 0c7e865fa7cb parent 61969 e01015e49041 child 61976 3a27957ac658 permissions -rw-r--r--
more symbols;
```     1 (*  Title:      HOL/Limits.thy
```
```     2     Author:     Brian Huffman
```
```     3     Author:     Jacques D. Fleuriot, University of Cambridge
```
```     4     Author:     Lawrence C Paulson
```
```     5     Author:     Jeremy Avigad
```
```     6 *)
```
```     7
```
```     8 section \<open>Limits on Real Vector Spaces\<close>
```
```     9
```
```    10 theory Limits
```
```    11 imports Real_Vector_Spaces
```
```    12 begin
```
```    13
```
```    14 subsection \<open>Filter going to infinity norm\<close>
```
```    15
```
```    16 definition at_infinity :: "'a::real_normed_vector filter" where
```
```    17   "at_infinity = (INF r. principal {x. r \<le> norm x})"
```
```    18
```
```    19 lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
```
```    20   unfolding at_infinity_def
```
```    21   by (subst eventually_INF_base)
```
```    22      (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
```
```    23
```
```    24 lemma at_infinity_eq_at_top_bot:
```
```    25   "(at_infinity :: real filter) = sup at_top at_bot"
```
```    26   apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
```
```    27                    eventually_at_top_linorder eventually_at_bot_linorder)
```
```    28   apply safe
```
```    29   apply (rule_tac x="b" in exI, simp)
```
```    30   apply (rule_tac x="- b" in exI, simp)
```
```    31   apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
```
```    32   done
```
```    33
```
```    34 lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
```
```    35   unfolding at_infinity_eq_at_top_bot by simp
```
```    36
```
```    37 lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
```
```    38   unfolding at_infinity_eq_at_top_bot by simp
```
```    39
```
```    40 lemma filterlim_at_top_imp_at_infinity:
```
```    41   fixes f :: "_ \<Rightarrow> real"
```
```    42   shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
```
```    43   by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
```
```    44
```
```    45 lemma lim_infinity_imp_sequentially:
```
```    46   "(f \<longlongrightarrow> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) \<longlongrightarrow> l) sequentially"
```
```    47 by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
```
```    48
```
```    49
```
```    50 subsubsection \<open>Boundedness\<close>
```
```    51
```
```    52 definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```    53   Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
```
```    54
```
```    55 abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    56   "Bseq X \<equiv> Bfun X sequentially"
```
```    57
```
```    58 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
```
```    59
```
```    60 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```    61   unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
```
```    62
```
```    63 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```    64   unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
```
```    65
```
```    66 lemma Bfun_def:
```
```    67   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
```
```    68   unfolding Bfun_metric_def norm_conv_dist
```
```    69 proof safe
```
```    70   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
```
```    71   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
```
```    72     by (intro always_eventually) (metis dist_commute dist_triangle)
```
```    73   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
```
```    74     by eventually_elim auto
```
```    75   with \<open>0 < K\<close> show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
```
```    76     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
```
```    77 qed auto
```
```    78
```
```    79 lemma BfunI:
```
```    80   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
```
```    81 unfolding Bfun_def
```
```    82 proof (intro exI conjI allI)
```
```    83   show "0 < max K 1" by simp
```
```    84 next
```
```    85   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
```
```    86     using K by (rule eventually_mono, simp)
```
```    87 qed
```
```    88
```
```    89 lemma BfunE:
```
```    90   assumes "Bfun f F"
```
```    91   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
```
```    92 using assms unfolding Bfun_def by blast
```
```    93
```
```    94 lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
```
```    95   unfolding Cauchy_def Bfun_metric_def eventually_sequentially
```
```    96   apply (erule_tac x=1 in allE)
```
```    97   apply simp
```
```    98   apply safe
```
```    99   apply (rule_tac x="X M" in exI)
```
```   100   apply (rule_tac x=1 in exI)
```
```   101   apply (erule_tac x=M in allE)
```
```   102   apply simp
```
```   103   apply (rule_tac x=M in exI)
```
```   104   apply (auto simp: dist_commute)
```
```   105   done
```
```   106
```
```   107
```
```   108 subsubsection \<open>Bounded Sequences\<close>
```
```   109
```
```   110 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
```
```   111   by (intro BfunI) (auto simp: eventually_sequentially)
```
```   112
```
```   113 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
```
```   114   by (intro BfunI) (auto simp: eventually_sequentially)
```
```   115
```
```   116 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
```
```   117   unfolding Bfun_def eventually_sequentially
```
```   118 proof safe
```
```   119   fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
```
```   120   then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
```
```   121     by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
```
```   122        (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
```
```   123 qed auto
```
```   124
```
```   125 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```   126 unfolding Bseq_def by auto
```
```   127
```
```   128 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   129 by (simp add: Bseq_def)
```
```   130
```
```   131 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   132 by (auto simp add: Bseq_def)
```
```   133
```
```   134 lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
```
```   135 proof (elim BseqE, intro bdd_aboveI2)
```
```   136   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
```
```   137     by (auto elim!: allE[of _ n])
```
```   138 qed
```
```   139
```
```   140 lemma Bseq_bdd_above':
```
```   141   "Bseq (X::nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> bdd_above (range (\<lambda>n. norm (X n)))"
```
```   142 proof (elim BseqE, intro bdd_aboveI2)
```
```   143   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "norm (X n) \<le> K"
```
```   144     by (auto elim!: allE[of _ n])
```
```   145 qed
```
```   146
```
```   147 lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
```
```   148 proof (elim BseqE, intro bdd_belowI2)
```
```   149   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
```
```   150     by (auto elim!: allE[of _ n])
```
```   151 qed
```
```   152
```
```   153 lemma Bseq_eventually_mono:
```
```   154   assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) sequentially" "Bseq g"
```
```   155   shows   "Bseq f"
```
```   156 proof -
```
```   157   from assms(1) obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> norm (g n)"
```
```   158     by (auto simp: eventually_at_top_linorder)
```
```   159   moreover from assms(2) obtain K where K: "\<And>n. norm (g n) \<le> K" by (blast elim!: BseqE)
```
```   160   ultimately have "norm (f n) \<le> max K (Max {norm (f n) |n. n < N})" for n
```
```   161     apply (cases "n < N")
```
```   162     apply (rule max.coboundedI2, rule Max.coboundedI, auto) []
```
```   163     apply (rule max.coboundedI1, force intro: order.trans[OF N K])
```
```   164     done
```
```   165   thus ?thesis by (blast intro: BseqI')
```
```   166 qed
```
```   167
```
```   168 lemma lemma_NBseq_def:
```
```   169   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   170 proof safe
```
```   171   fix K :: real
```
```   172   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
```
```   173   then have "K \<le> real (Suc n)" by auto
```
```   174   moreover assume "\<forall>m. norm (X m) \<le> K"
```
```   175   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
```
```   176     by (blast intro: order_trans)
```
```   177   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
```
```   178 next
```
```   179   show "\<And>N. \<forall>n. norm (X n) \<le> real (Suc N) \<Longrightarrow> \<exists>K>0. \<forall>n. norm (X n) \<le> K"
```
```   180     using of_nat_0_less_iff by blast
```
```   181 qed
```
```   182
```
```   183 text\<open>alternative definition for Bseq\<close>
```
```   184 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   185 apply (simp add: Bseq_def)
```
```   186 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   187 done
```
```   188
```
```   189 lemma lemma_NBseq_def2:
```
```   190      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   191 apply (subst lemma_NBseq_def, auto)
```
```   192 apply (rule_tac x = "Suc N" in exI)
```
```   193 apply (rule_tac  x = N in exI)
```
```   194 apply (auto simp add: of_nat_Suc)
```
```   195  prefer 2 apply (blast intro: order_less_imp_le)
```
```   196 apply (drule_tac x = n in spec, simp)
```
```   197 done
```
```   198
```
```   199 (* yet another definition for Bseq *)
```
```   200 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   201 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   202
```
```   203 subsubsection\<open>A Few More Equivalence Theorems for Boundedness\<close>
```
```   204
```
```   205 text\<open>alternative formulation for boundedness\<close>
```
```   206 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```   207 apply (unfold Bseq_def, safe)
```
```   208 apply (rule_tac  x = "k + norm x" in exI)
```
```   209 apply (rule_tac x = K in exI, simp)
```
```   210 apply (rule exI [where x = 0], auto)
```
```   211 apply (erule order_less_le_trans, simp)
```
```   212 apply (drule_tac x=n in spec)
```
```   213 apply (drule order_trans [OF norm_triangle_ineq2])
```
```   214 apply simp
```
```   215 done
```
```   216
```
```   217 text\<open>alternative formulation for boundedness\<close>
```
```   218 lemma Bseq_iff3:
```
```   219   "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
```
```   220 proof
```
```   221   assume ?P
```
```   222   then obtain K
```
```   223     where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
```
```   224   from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
```
```   225   from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
```
```   226     by (auto intro: order_trans norm_triangle_ineq4)
```
```   227   then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
```
```   228     by simp
```
```   229   with \<open>0 < K + norm (X 0)\<close> show ?Q by blast
```
```   230 next
```
```   231   assume ?Q then show ?P by (auto simp add: Bseq_iff2)
```
```   232 qed
```
```   233
```
```   234 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```   235 apply (simp add: Bseq_def)
```
```   236 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```   237 apply (drule_tac x = n in spec, arith)
```
```   238 done
```
```   239
```
```   240
```
```   241 subsubsection\<open>Upper Bounds and Lubs of Bounded Sequences\<close>
```
```   242
```
```   243 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
```
```   244   by (simp add: Bseq_def)
```
```   245
```
```   246 lemma Bseq_add:
```
```   247   assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
```
```   248   shows   "Bseq (\<lambda>x. f x + c)"
```
```   249 proof -
```
```   250   from assms obtain K where K: "\<And>x. norm (f x) \<le> K" unfolding Bseq_def by blast
```
```   251   {
```
```   252     fix x :: nat
```
```   253     have "norm (f x + c) \<le> norm (f x) + norm c" by (rule norm_triangle_ineq)
```
```   254     also have "norm (f x) \<le> K" by (rule K)
```
```   255     finally have "norm (f x + c) \<le> K + norm c" by simp
```
```   256   }
```
```   257   thus ?thesis by (rule BseqI')
```
```   258 qed
```
```   259
```
```   260 lemma Bseq_add_iff: "Bseq (\<lambda>x. f x + c) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
```
```   261   using Bseq_add[of f c] Bseq_add[of "\<lambda>x. f x + c" "-c"] by auto
```
```   262
```
```   263 lemma Bseq_mult:
```
```   264   assumes "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_field)"
```
```   265   assumes "Bseq (g :: nat \<Rightarrow> 'a :: real_normed_field)"
```
```   266   shows   "Bseq (\<lambda>x. f x * g x)"
```
```   267 proof -
```
```   268   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"
```
```   269     unfolding Bseq_def by blast
```
```   270   hence "\<And>x. norm (f x * g x) \<le> K1 * K2" by (auto simp: norm_mult intro!: mult_mono)
```
```   271   thus ?thesis by (rule BseqI')
```
```   272 qed
```
```   273
```
```   274 lemma Bfun_const [simp]: "Bfun (\<lambda>_. c) F"
```
```   275   unfolding Bfun_metric_def by (auto intro!: exI[of _ c] exI[of _ "1::real"])
```
```   276
```
```   277 lemma Bseq_cmult_iff: "(c :: 'a :: real_normed_field) \<noteq> 0 \<Longrightarrow> Bseq (\<lambda>x. c * f x) \<longleftrightarrow> Bseq f"
```
```   278 proof
```
```   279   assume "c \<noteq> 0" "Bseq (\<lambda>x. c * f x)"
```
```   280   find_theorems "Bfun (\<lambda>_. ?c) _"
```
```   281   from Bfun_const this(2) have "Bseq (\<lambda>x. inverse c * (c * f x))" by (rule Bseq_mult)
```
```   282   with \<open>c \<noteq> 0\<close> show "Bseq f" by (simp add: divide_simps)
```
```   283 qed (intro Bseq_mult Bfun_const)
```
```   284
```
```   285 lemma Bseq_subseq: "Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector) \<Longrightarrow> Bseq (\<lambda>x. f (g x))"
```
```   286   unfolding Bseq_def by auto
```
```   287
```
```   288 lemma Bseq_Suc_iff: "Bseq (\<lambda>n. f (Suc n)) \<longleftrightarrow> Bseq (f :: nat \<Rightarrow> 'a :: real_normed_vector)"
```
```   289   using Bseq_offset[of f 1] by (auto intro: Bseq_subseq)
```
```   290
```
```   291 lemma increasing_Bseq_subseq_iff:
```
```   292   assumes "\<And>x y. x \<le> y \<Longrightarrow> norm (f x :: 'a :: real_normed_vector) \<le> norm (f y)" "subseq g"
```
```   293   shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
```
```   294 proof
```
```   295   assume "Bseq (\<lambda>x. f (g x))"
```
```   296   then obtain K where K: "\<And>x. norm (f (g x)) \<le> K" unfolding Bseq_def by auto
```
```   297   {
```
```   298     fix x :: nat
```
```   299     from filterlim_subseq[OF assms(2)] obtain y where "g y \<ge> x"
```
```   300       by (auto simp: filterlim_at_top eventually_at_top_linorder)
```
```   301     hence "norm (f x) \<le> norm (f (g y))" using assms(1) by blast
```
```   302     also have "norm (f (g y)) \<le> K" by (rule K)
```
```   303     finally have "norm (f x) \<le> K" .
```
```   304   }
```
```   305   thus "Bseq f" by (rule BseqI')
```
```   306 qed (insert Bseq_subseq[of f g], simp_all)
```
```   307
```
```   308 lemma nonneg_incseq_Bseq_subseq_iff:
```
```   309   assumes "\<And>x. f x \<ge> 0" "incseq (f :: nat \<Rightarrow> real)" "subseq g"
```
```   310   shows   "Bseq (\<lambda>x. f (g x)) \<longleftrightarrow> Bseq f"
```
```   311   using assms by (intro increasing_Bseq_subseq_iff) (auto simp: incseq_def)
```
```   312
```
```   313 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
```
```   314   apply (simp add: subset_eq)
```
```   315   apply (rule BseqI'[where K="max (norm a) (norm b)"])
```
```   316   apply (erule_tac x=n in allE)
```
```   317   apply auto
```
```   318   done
```
```   319
```
```   320 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
```
```   321   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
```
```   322
```
```   323 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
```
```   324   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
```
```   325
```
```   326 subsection \<open>Bounded Monotonic Sequences\<close>
```
```   327
```
```   328 subsubsection\<open>A Bounded and Monotonic Sequence Converges\<close>
```
```   329
```
```   330 (* TODO: delete *)
```
```   331 (* FIXME: one use in NSA/HSEQ.thy *)
```
```   332 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X \<longlonglongrightarrow> L)"
```
```   333   apply (rule_tac x="X m" in exI)
```
```   334   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
```
```   335   unfolding eventually_sequentially
```
```   336   apply blast
```
```   337   done
```
```   338
```
```   339 subsection \<open>Convergence to Zero\<close>
```
```   340
```
```   341 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   342   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```   343
```
```   344 lemma ZfunI:
```
```   345   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```   346   unfolding Zfun_def by simp
```
```   347
```
```   348 lemma ZfunD:
```
```   349   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```   350   unfolding Zfun_def by simp
```
```   351
```
```   352 lemma Zfun_ssubst:
```
```   353   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```   354   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   355
```
```   356 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```   357   unfolding Zfun_def by simp
```
```   358
```
```   359 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   360   unfolding Zfun_def by simp
```
```   361
```
```   362 lemma Zfun_imp_Zfun:
```
```   363   assumes f: "Zfun f F"
```
```   364   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   365   shows "Zfun (\<lambda>x. g x) F"
```
```   366 proof (cases)
```
```   367   assume K: "0 < K"
```
```   368   show ?thesis
```
```   369   proof (rule ZfunI)
```
```   370     fix r::real assume "0 < r"
```
```   371     hence "0 < r / K" using K by simp
```
```   372     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   373       using ZfunD [OF f] by blast
```
```   374     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   375     proof eventually_elim
```
```   376       case (elim x)
```
```   377       hence "norm (f x) * K < r"
```
```   378         by (simp add: pos_less_divide_eq K)
```
```   379       thus ?case
```
```   380         by (simp add: order_le_less_trans [OF elim(1)])
```
```   381     qed
```
```   382   qed
```
```   383 next
```
```   384   assume "\<not> 0 < K"
```
```   385   hence K: "K \<le> 0" by (simp only: not_less)
```
```   386   show ?thesis
```
```   387   proof (rule ZfunI)
```
```   388     fix r :: real
```
```   389     assume "0 < r"
```
```   390     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   391     proof eventually_elim
```
```   392       case (elim x)
```
```   393       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   394         using K norm_ge_zero by (rule mult_left_mono)
```
```   395       finally show ?case
```
```   396         using \<open>0 < r\<close> by simp
```
```   397     qed
```
```   398   qed
```
```   399 qed
```
```   400
```
```   401 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   402   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   403
```
```   404 lemma Zfun_add:
```
```   405   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   406   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   407 proof (rule ZfunI)
```
```   408   fix r::real assume "0 < r"
```
```   409   hence r: "0 < r / 2" by simp
```
```   410   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   411     using f r by (rule ZfunD)
```
```   412   moreover
```
```   413   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   414     using g r by (rule ZfunD)
```
```   415   ultimately
```
```   416   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   417   proof eventually_elim
```
```   418     case (elim x)
```
```   419     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   420       by (rule norm_triangle_ineq)
```
```   421     also have "\<dots> < r/2 + r/2"
```
```   422       using elim by (rule add_strict_mono)
```
```   423     finally show ?case
```
```   424       by simp
```
```   425   qed
```
```   426 qed
```
```   427
```
```   428 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   429   unfolding Zfun_def by simp
```
```   430
```
```   431 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   432   using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
```
```   433
```
```   434 lemma (in bounded_linear) Zfun:
```
```   435   assumes g: "Zfun g F"
```
```   436   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   437 proof -
```
```   438   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   439     using bounded by blast
```
```   440   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   441     by simp
```
```   442   with g show ?thesis
```
```   443     by (rule Zfun_imp_Zfun)
```
```   444 qed
```
```   445
```
```   446 lemma (in bounded_bilinear) Zfun:
```
```   447   assumes f: "Zfun f F"
```
```   448   assumes g: "Zfun g F"
```
```   449   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   450 proof (rule ZfunI)
```
```   451   fix r::real assume r: "0 < r"
```
```   452   obtain K where K: "0 < K"
```
```   453     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   454     using pos_bounded by blast
```
```   455   from K have K': "0 < inverse K"
```
```   456     by (rule positive_imp_inverse_positive)
```
```   457   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   458     using f r by (rule ZfunD)
```
```   459   moreover
```
```   460   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   461     using g K' by (rule ZfunD)
```
```   462   ultimately
```
```   463   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   464   proof eventually_elim
```
```   465     case (elim x)
```
```   466     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   467       by (rule norm_le)
```
```   468     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   469       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   470     also from K have "r * inverse K * K = r"
```
```   471       by simp
```
```   472     finally show ?case .
```
```   473   qed
```
```   474 qed
```
```   475
```
```   476 lemma (in bounded_bilinear) Zfun_left:
```
```   477   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   478   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   479
```
```   480 lemma (in bounded_bilinear) Zfun_right:
```
```   481   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   482   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   483
```
```   484 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   485 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   486 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   487
```
```   488 lemma tendsto_Zfun_iff: "(f \<longlongrightarrow> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   489   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   490
```
```   491 lemma tendsto_0_le: "\<lbrakk>(f \<longlongrightarrow> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
```
```   492                      \<Longrightarrow> (g \<longlongrightarrow> 0) F"
```
```   493   by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
```
```   494
```
```   495 subsubsection \<open>Distance and norms\<close>
```
```   496
```
```   497 lemma tendsto_dist [tendsto_intros]:
```
```   498   fixes l m :: "'a :: metric_space"
```
```   499   assumes f: "(f \<longlongrightarrow> l) F" and g: "(g \<longlongrightarrow> m) F"
```
```   500   shows "((\<lambda>x. dist (f x) (g x)) \<longlongrightarrow> dist l m) F"
```
```   501 proof (rule tendstoI)
```
```   502   fix e :: real assume "0 < e"
```
```   503   hence e2: "0 < e/2" by simp
```
```   504   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   505   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
```
```   506   proof (eventually_elim)
```
```   507     case (elim x)
```
```   508     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   509       unfolding dist_real_def
```
```   510       using dist_triangle2 [of "f x" "g x" "l"]
```
```   511       using dist_triangle2 [of "g x" "l" "m"]
```
```   512       using dist_triangle3 [of "l" "m" "f x"]
```
```   513       using dist_triangle [of "f x" "m" "g x"]
```
```   514       by arith
```
```   515   qed
```
```   516 qed
```
```   517
```
```   518 lemma continuous_dist[continuous_intros]:
```
```   519   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   520   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
```
```   521   unfolding continuous_def by (rule tendsto_dist)
```
```   522
```
```   523 lemma continuous_on_dist[continuous_intros]:
```
```   524   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   525   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
```
```   526   unfolding continuous_on_def by (auto intro: tendsto_dist)
```
```   527
```
```   528 lemma tendsto_norm [tendsto_intros]:
```
```   529   "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> norm a) F"
```
```   530   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   531
```
```   532 lemma continuous_norm [continuous_intros]:
```
```   533   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
```
```   534   unfolding continuous_def by (rule tendsto_norm)
```
```   535
```
```   536 lemma continuous_on_norm [continuous_intros]:
```
```   537   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
```
```   538   unfolding continuous_on_def by (auto intro: tendsto_norm)
```
```   539
```
```   540 lemma tendsto_norm_zero:
```
```   541   "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F"
```
```   542   by (drule tendsto_norm, simp)
```
```   543
```
```   544 lemma tendsto_norm_zero_cancel:
```
```   545   "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
```
```   546   unfolding tendsto_iff dist_norm by simp
```
```   547
```
```   548 lemma tendsto_norm_zero_iff:
```
```   549   "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
```
```   550   unfolding tendsto_iff dist_norm by simp
```
```   551
```
```   552 lemma tendsto_rabs [tendsto_intros]:
```
```   553   "(f \<longlongrightarrow> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> \<bar>l\<bar>) F"
```
```   554   by (fold real_norm_def, rule tendsto_norm)
```
```   555
```
```   556 lemma continuous_rabs [continuous_intros]:
```
```   557   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   558   unfolding real_norm_def[symmetric] by (rule continuous_norm)
```
```   559
```
```   560 lemma continuous_on_rabs [continuous_intros]:
```
```   561   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   562   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
```
```   563
```
```   564 lemma tendsto_rabs_zero:
```
```   565   "(f \<longlongrightarrow> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> 0) F"
```
```   566   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   567
```
```   568 lemma tendsto_rabs_zero_cancel:
```
```   569   "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<Longrightarrow> (f \<longlongrightarrow> 0) F"
```
```   570   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   571
```
```   572 lemma tendsto_rabs_zero_iff:
```
```   573   "((\<lambda>x. \<bar>f x\<bar>) \<longlongrightarrow> (0::real)) F \<longleftrightarrow> (f \<longlongrightarrow> 0) F"
```
```   574   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   575
```
```   576 subsubsection \<open>Addition and subtraction\<close>
```
```   577
```
```   578 lemma tendsto_add [tendsto_intros]:
```
```   579   fixes a b :: "'a::real_normed_vector"
```
```   580   shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> a + b) F"
```
```   581   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   582
```
```   583 lemma continuous_add [continuous_intros]:
```
```   584   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   585   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
```
```   586   unfolding continuous_def by (rule tendsto_add)
```
```   587
```
```   588 lemma continuous_on_add [continuous_intros]:
```
```   589   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   590   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
```
```   591   unfolding continuous_on_def by (auto intro: tendsto_add)
```
```   592
```
```   593 lemma tendsto_add_zero:
```
```   594   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   595   shows "\<lbrakk>(f \<longlongrightarrow> 0) F; (g \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) \<longlongrightarrow> 0) F"
```
```   596   by (drule (1) tendsto_add, simp)
```
```   597
```
```   598 lemma tendsto_minus [tendsto_intros]:
```
```   599   fixes a :: "'a::real_normed_vector"
```
```   600   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> - a) F"
```
```   601   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   602
```
```   603 lemma continuous_minus [continuous_intros]:
```
```   604   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   605   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
```
```   606   unfolding continuous_def by (rule tendsto_minus)
```
```   607
```
```   608 lemma continuous_on_minus [continuous_intros]:
```
```   609   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   610   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
```
```   611   unfolding continuous_on_def by (auto intro: tendsto_minus)
```
```   612
```
```   613 lemma tendsto_minus_cancel:
```
```   614   fixes a :: "'a::real_normed_vector"
```
```   615   shows "((\<lambda>x. - f x) \<longlongrightarrow> - a) F \<Longrightarrow> (f \<longlongrightarrow> a) F"
```
```   616   by (drule tendsto_minus, simp)
```
```   617
```
```   618 lemma tendsto_minus_cancel_left:
```
```   619     "(f \<longlongrightarrow> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) \<longlongrightarrow> y) F"
```
```   620   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
```
```   621   by auto
```
```   622
```
```   623 lemma tendsto_diff [tendsto_intros]:
```
```   624   fixes a b :: "'a::real_normed_vector"
```
```   625   shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) \<longlongrightarrow> a - b) F"
```
```   626   using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
```
```   627
```
```   628 lemma continuous_diff [continuous_intros]:
```
```   629   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   630   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
```
```   631   unfolding continuous_def by (rule tendsto_diff)
```
```   632
```
```   633 lemma continuous_on_diff [continuous_intros]:
```
```   634   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   635   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
```
```   636   unfolding continuous_on_def by (auto intro: tendsto_diff)
```
```   637
```
```   638 lemma continuous_on_op_minus: "continuous_on (s::'a::real_normed_vector set) (op - x)"
```
```   639   by (rule continuous_intros | simp)+
```
```   640
```
```   641 lemma tendsto_setsum [tendsto_intros]:
```
```   642   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   643   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> a i) F"
```
```   644   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) \<longlongrightarrow> (\<Sum>i\<in>S. a i)) F"
```
```   645 proof (cases "finite S")
```
```   646   assume "finite S" thus ?thesis using assms
```
```   647     by (induct, simp, simp add: tendsto_add)
```
```   648 qed simp
```
```   649
```
```   650 lemma continuous_setsum [continuous_intros]:
```
```   651   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```   652   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
```
```   653   unfolding continuous_def by (rule tendsto_setsum)
```
```   654
```
```   655 lemma continuous_on_setsum [continuous_intros]:
```
```   656   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
```
```   657   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
```
```   658   unfolding continuous_on_def by (auto intro: tendsto_setsum)
```
```   659
```
```   660 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
```
```   661
```
```   662 subsubsection \<open>Linear operators and multiplication\<close>
```
```   663
```
```   664 lemma linear_times:
```
```   665   fixes c::"'a::real_algebra" shows "linear (\<lambda>x. c * x)"
```
```   666   by (auto simp: linearI distrib_left)
```
```   667
```
```   668 lemma (in bounded_linear) tendsto:
```
```   669   "(g \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> f a) F"
```
```   670   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   671
```
```   672 lemma (in bounded_linear) continuous:
```
```   673   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
```
```   674   using tendsto[of g _ F] by (auto simp: continuous_def)
```
```   675
```
```   676 lemma (in bounded_linear) continuous_on:
```
```   677   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
```
```   678   using tendsto[of g] by (auto simp: continuous_on_def)
```
```   679
```
```   680 lemma (in bounded_linear) tendsto_zero:
```
```   681   "(g \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) \<longlongrightarrow> 0) F"
```
```   682   by (drule tendsto, simp only: zero)
```
```   683
```
```   684 lemma (in bounded_bilinear) tendsto:
```
```   685   "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) \<longlongrightarrow> a ** b) F"
```
```   686   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   687                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   688
```
```   689 lemma (in bounded_bilinear) continuous:
```
```   690   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
```
```   691   using tendsto[of f _ F g] by (auto simp: continuous_def)
```
```   692
```
```   693 lemma (in bounded_bilinear) continuous_on:
```
```   694   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
```
```   695   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
```
```   696
```
```   697 lemma (in bounded_bilinear) tendsto_zero:
```
```   698   assumes f: "(f \<longlongrightarrow> 0) F"
```
```   699   assumes g: "(g \<longlongrightarrow> 0) F"
```
```   700   shows "((\<lambda>x. f x ** g x) \<longlongrightarrow> 0) F"
```
```   701   using tendsto [OF f g] by (simp add: zero_left)
```
```   702
```
```   703 lemma (in bounded_bilinear) tendsto_left_zero:
```
```   704   "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) \<longlongrightarrow> 0) F"
```
```   705   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```   706
```
```   707 lemma (in bounded_bilinear) tendsto_right_zero:
```
```   708   "(f \<longlongrightarrow> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) \<longlongrightarrow> 0) F"
```
```   709   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```   710
```
```   711 lemmas tendsto_of_real [tendsto_intros] =
```
```   712   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```   713
```
```   714 lemmas tendsto_scaleR [tendsto_intros] =
```
```   715   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```   716
```
```   717 lemmas tendsto_mult [tendsto_intros] =
```
```   718   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```   719
```
```   720 lemma tendsto_mult_left:
```
```   721   fixes c::"'a::real_normed_algebra"
```
```   722   shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) \<longlongrightarrow> c * l) F"
```
```   723 by (rule tendsto_mult [OF tendsto_const])
```
```   724
```
```   725 lemma tendsto_mult_right:
```
```   726   fixes c::"'a::real_normed_algebra"
```
```   727   shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) \<longlongrightarrow> l * c) F"
```
```   728 by (rule tendsto_mult [OF _ tendsto_const])
```
```   729
```
```   730 lemmas continuous_of_real [continuous_intros] =
```
```   731   bounded_linear.continuous [OF bounded_linear_of_real]
```
```   732
```
```   733 lemmas continuous_scaleR [continuous_intros] =
```
```   734   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
```
```   735
```
```   736 lemmas continuous_mult [continuous_intros] =
```
```   737   bounded_bilinear.continuous [OF bounded_bilinear_mult]
```
```   738
```
```   739 lemmas continuous_on_of_real [continuous_intros] =
```
```   740   bounded_linear.continuous_on [OF bounded_linear_of_real]
```
```   741
```
```   742 lemmas continuous_on_scaleR [continuous_intros] =
```
```   743   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
```
```   744
```
```   745 lemmas continuous_on_mult [continuous_intros] =
```
```   746   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
```
```   747
```
```   748 lemmas tendsto_mult_zero =
```
```   749   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```   750
```
```   751 lemmas tendsto_mult_left_zero =
```
```   752   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```   753
```
```   754 lemmas tendsto_mult_right_zero =
```
```   755   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```   756
```
```   757 lemma tendsto_power [tendsto_intros]:
```
```   758   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   759   shows "(f \<longlongrightarrow> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) \<longlongrightarrow> a ^ n) F"
```
```   760   by (induct n) (simp_all add: tendsto_mult)
```
```   761
```
```   762 lemma continuous_power [continuous_intros]:
```
```   763   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   764   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
```
```   765   unfolding continuous_def by (rule tendsto_power)
```
```   766
```
```   767 lemma continuous_on_power [continuous_intros]:
```
```   768   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   769   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
```
```   770   unfolding continuous_on_def by (auto intro: tendsto_power)
```
```   771
```
```   772 lemma tendsto_setprod [tendsto_intros]:
```
```   773   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   774   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i \<longlongrightarrow> L i) F"
```
```   775   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) \<longlongrightarrow> (\<Prod>i\<in>S. L i)) F"
```
```   776 proof (cases "finite S")
```
```   777   assume "finite S" thus ?thesis using assms
```
```   778     by (induct, simp, simp add: tendsto_mult)
```
```   779 qed simp
```
```   780
```
```   781 lemma continuous_setprod [continuous_intros]:
```
```   782   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   783   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
```
```   784   unfolding continuous_def by (rule tendsto_setprod)
```
```   785
```
```   786 lemma continuous_on_setprod [continuous_intros]:
```
```   787   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   788   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
```
```   789   unfolding continuous_on_def by (auto intro: tendsto_setprod)
```
```   790
```
```   791 lemma tendsto_of_real_iff:
```
```   792   "((\<lambda>x. of_real (f x) :: 'a :: real_normed_div_algebra) \<longlongrightarrow> of_real c) F \<longleftrightarrow> (f \<longlongrightarrow> c) F"
```
```   793   unfolding tendsto_iff by simp
```
```   794
```
```   795 lemma tendsto_add_const_iff:
```
```   796   "((\<lambda>x. c + f x :: 'a :: real_normed_vector) \<longlongrightarrow> c + d) F \<longleftrightarrow> (f \<longlongrightarrow> d) F"
```
```   797   using tendsto_add[OF tendsto_const[of c], of f d]
```
```   798         tendsto_add[OF tendsto_const[of "-c"], of "\<lambda>x. c + f x" "c + d"] by auto
```
```   799
```
```   800
```
```   801 subsubsection \<open>Inverse and division\<close>
```
```   802
```
```   803 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   804   assumes f: "Zfun f F"
```
```   805   assumes g: "Bfun g F"
```
```   806   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   807 proof -
```
```   808   obtain K where K: "0 \<le> K"
```
```   809     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   810     using nonneg_bounded by blast
```
```   811   obtain B where B: "0 < B"
```
```   812     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```   813     using g by (rule BfunE)
```
```   814   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```   815   using norm_g proof eventually_elim
```
```   816     case (elim x)
```
```   817     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   818       by (rule norm_le)
```
```   819     also have "\<dots> \<le> norm (f x) * B * K"
```
```   820       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   821                 mult_nonneg_nonneg K elim)
```
```   822     also have "\<dots> = norm (f x) * (B * K)"
```
```   823       by (rule mult.assoc)
```
```   824     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   825   qed
```
```   826   with f show ?thesis
```
```   827     by (rule Zfun_imp_Zfun)
```
```   828 qed
```
```   829
```
```   830 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   831   assumes f: "Bfun f F"
```
```   832   assumes g: "Zfun g F"
```
```   833   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   834   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   835
```
```   836 lemma Bfun_inverse_lemma:
```
```   837   fixes x :: "'a::real_normed_div_algebra"
```
```   838   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   839   apply (subst nonzero_norm_inverse, clarsimp)
```
```   840   apply (erule (1) le_imp_inverse_le)
```
```   841   done
```
```   842
```
```   843 lemma Bfun_inverse:
```
```   844   fixes a :: "'a::real_normed_div_algebra"
```
```   845   assumes f: "(f \<longlongrightarrow> a) F"
```
```   846   assumes a: "a \<noteq> 0"
```
```   847   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```   848 proof -
```
```   849   from a have "0 < norm a" by simp
```
```   850   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   851   then obtain r where r1: "0 < r" and r2: "r < norm a" by blast
```
```   852   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```   853     using tendstoD [OF f r1] by blast
```
```   854   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```   855   proof eventually_elim
```
```   856     case (elim x)
```
```   857     hence 1: "norm (f x - a) < r"
```
```   858       by (simp add: dist_norm)
```
```   859     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   860     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   861       by (rule nonzero_norm_inverse)
```
```   862     also have "\<dots> \<le> inverse (norm a - r)"
```
```   863     proof (rule le_imp_inverse_le)
```
```   864       show "0 < norm a - r" using r2 by simp
```
```   865     next
```
```   866       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   867         by (rule norm_triangle_ineq2)
```
```   868       also have "\<dots> = norm (f x - a)"
```
```   869         by (rule norm_minus_commute)
```
```   870       also have "\<dots> < r" using 1 .
```
```   871       finally show "norm a - r \<le> norm (f x)" by simp
```
```   872     qed
```
```   873     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   874   qed
```
```   875   thus ?thesis by (rule BfunI)
```
```   876 qed
```
```   877
```
```   878 lemma tendsto_inverse [tendsto_intros]:
```
```   879   fixes a :: "'a::real_normed_div_algebra"
```
```   880   assumes f: "(f \<longlongrightarrow> a) F"
```
```   881   assumes a: "a \<noteq> 0"
```
```   882   shows "((\<lambda>x. inverse (f x)) \<longlongrightarrow> inverse a) F"
```
```   883 proof -
```
```   884   from a have "0 < norm a" by simp
```
```   885   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```   886     by (rule tendstoD)
```
```   887   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```   888     unfolding dist_norm by (auto elim!: eventually_mono)
```
```   889   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```   890     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   891     by (auto elim!: eventually_mono simp: inverse_diff_inverse)
```
```   892   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   893     by (intro Zfun_minus Zfun_mult_left
```
```   894       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```   895       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```   896   ultimately show ?thesis
```
```   897     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```   898 qed
```
```   899
```
```   900 lemma continuous_inverse:
```
```   901   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   902   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   903   shows "continuous F (\<lambda>x. inverse (f x))"
```
```   904   using assms unfolding continuous_def by (rule tendsto_inverse)
```
```   905
```
```   906 lemma continuous_at_within_inverse[continuous_intros]:
```
```   907   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   908   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   909   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
```
```   910   using assms unfolding continuous_within by (rule tendsto_inverse)
```
```   911
```
```   912 lemma isCont_inverse[continuous_intros, simp]:
```
```   913   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   914   assumes "isCont f a" and "f a \<noteq> 0"
```
```   915   shows "isCont (\<lambda>x. inverse (f x)) a"
```
```   916   using assms unfolding continuous_at by (rule tendsto_inverse)
```
```   917
```
```   918 lemma continuous_on_inverse[continuous_intros]:
```
```   919   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   920   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   921   shows "continuous_on s (\<lambda>x. inverse (f x))"
```
```   922   using assms unfolding continuous_on_def by (blast intro: tendsto_inverse)
```
```   923
```
```   924 lemma tendsto_divide [tendsto_intros]:
```
```   925   fixes a b :: "'a::real_normed_field"
```
```   926   shows "\<lbrakk>(f \<longlongrightarrow> a) F; (g \<longlongrightarrow> b) F; b \<noteq> 0\<rbrakk>
```
```   927     \<Longrightarrow> ((\<lambda>x. f x / g x) \<longlongrightarrow> a / b) F"
```
```   928   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```   929
```
```   930 lemma continuous_divide:
```
```   931   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   932   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   933   shows "continuous F (\<lambda>x. (f x) / (g x))"
```
```   934   using assms unfolding continuous_def by (rule tendsto_divide)
```
```   935
```
```   936 lemma continuous_at_within_divide[continuous_intros]:
```
```   937   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   938   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
```
```   939   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
```
```   940   using assms unfolding continuous_within by (rule tendsto_divide)
```
```   941
```
```   942 lemma isCont_divide[continuous_intros, simp]:
```
```   943   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   944   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
```
```   945   shows "isCont (\<lambda>x. (f x) / g x) a"
```
```   946   using assms unfolding continuous_at by (rule tendsto_divide)
```
```   947
```
```   948 lemma continuous_on_divide[continuous_intros]:
```
```   949   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   950   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
```
```   951   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
```
```   952   using assms unfolding continuous_on_def by (blast intro: tendsto_divide)
```
```   953
```
```   954 lemma tendsto_sgn [tendsto_intros]:
```
```   955   fixes l :: "'a::real_normed_vector"
```
```   956   shows "\<lbrakk>(f \<longlongrightarrow> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
```
```   957   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```   958
```
```   959 lemma continuous_sgn:
```
```   960   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   961   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   962   shows "continuous F (\<lambda>x. sgn (f x))"
```
```   963   using assms unfolding continuous_def by (rule tendsto_sgn)
```
```   964
```
```   965 lemma continuous_at_within_sgn[continuous_intros]:
```
```   966   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   967   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   968   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
```
```   969   using assms unfolding continuous_within by (rule tendsto_sgn)
```
```   970
```
```   971 lemma isCont_sgn[continuous_intros]:
```
```   972   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   973   assumes "isCont f a" and "f a \<noteq> 0"
```
```   974   shows "isCont (\<lambda>x. sgn (f x)) a"
```
```   975   using assms unfolding continuous_at by (rule tendsto_sgn)
```
```   976
```
```   977 lemma continuous_on_sgn[continuous_intros]:
```
```   978   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   979   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   980   shows "continuous_on s (\<lambda>x. sgn (f x))"
```
```   981   using assms unfolding continuous_on_def by (blast intro: tendsto_sgn)
```
```   982
```
```   983 lemma filterlim_at_infinity:
```
```   984   fixes f :: "_ \<Rightarrow> 'a::real_normed_vector"
```
```   985   assumes "0 \<le> c"
```
```   986   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```   987   unfolding filterlim_iff eventually_at_infinity
```
```   988 proof safe
```
```   989   fix P :: "'a \<Rightarrow> bool" and b
```
```   990   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```   991     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```   992   have "max b (c + 1) > c" by auto
```
```   993   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```   994     by auto
```
```   995   then show "eventually (\<lambda>x. P (f x)) F"
```
```   996   proof eventually_elim
```
```   997     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```   998     with P show "P (f x)" by auto
```
```   999   qed
```
```  1000 qed force
```
```  1001
```
```  1002 lemma not_tendsto_and_filterlim_at_infinity:
```
```  1003   assumes "F \<noteq> bot"
```
```  1004   assumes "(f \<longlongrightarrow> (c :: 'a :: real_normed_vector)) F"
```
```  1005   assumes "filterlim f at_infinity F"
```
```  1006   shows   False
```
```  1007 proof -
```
```  1008   from tendstoD[OF assms(2), of "1/2"]
```
```  1009     have "eventually (\<lambda>x. dist (f x) c < 1/2) F" by simp
```
```  1010   moreover from filterlim_at_infinity[of "norm c" f F] assms(3)
```
```  1011     have "eventually (\<lambda>x. norm (f x) \<ge> norm c + 1) F" by simp
```
```  1012   ultimately have "eventually (\<lambda>x. False) F"
```
```  1013   proof eventually_elim
```
```  1014     fix x assume A: "dist (f x) c < 1/2" and B: "norm (f x) \<ge> norm c + 1"
```
```  1015     note B
```
```  1016     also have "norm (f x) = dist (f x) 0" by (simp add: norm_conv_dist)
```
```  1017     also have "... \<le> dist (f x) c + dist c 0" by (rule dist_triangle)
```
```  1018     also note A
```
```  1019     finally show False by (simp add: norm_conv_dist)
```
```  1020   qed
```
```  1021   with assms show False by simp
```
```  1022 qed
```
```  1023
```
```  1024 lemma filterlim_at_infinity_imp_not_convergent:
```
```  1025   assumes "filterlim f at_infinity sequentially"
```
```  1026   shows   "\<not>convergent f"
```
```  1027   by (rule notI, rule not_tendsto_and_filterlim_at_infinity[OF _ _ assms])
```
```  1028      (simp_all add: convergent_LIMSEQ_iff)
```
```  1029
```
```  1030 lemma filterlim_at_infinity_imp_eventually_ne:
```
```  1031   assumes "filterlim f at_infinity F"
```
```  1032   shows   "eventually (\<lambda>z. f z \<noteq> c) F"
```
```  1033 proof -
```
```  1034   have "norm c + 1 > 0" by (intro add_nonneg_pos) simp_all
```
```  1035   with filterlim_at_infinity[OF order.refl, of f F] assms
```
```  1036     have "eventually (\<lambda>z. norm (f z) \<ge> norm c + 1) F" by blast
```
```  1037   thus ?thesis by eventually_elim auto
```
```  1038 qed
```
```  1039
```
```  1040 lemma tendsto_of_nat [tendsto_intros]:
```
```  1041   "filterlim (of_nat :: nat \<Rightarrow> 'a :: real_normed_algebra_1) at_infinity sequentially"
```
```  1042 proof (subst filterlim_at_infinity[OF order.refl], intro allI impI)
```
```  1043   fix r :: real assume r: "r > 0"
```
```  1044   def n \<equiv> "nat \<lceil>r\<rceil>"
```
```  1045   from r have n: "\<forall>m\<ge>n. of_nat m \<ge> r" unfolding n_def by linarith
```
```  1046   from eventually_ge_at_top[of n] show "eventually (\<lambda>m. norm (of_nat m :: 'a) \<ge> r) sequentially"
```
```  1047     by eventually_elim (insert n, simp_all)
```
```  1048 qed
```
```  1049
```
```  1050
```
```  1051 subsection \<open>Relate @{const at}, @{const at_left} and @{const at_right}\<close>
```
```  1052
```
```  1053 text \<open>
```
```  1054
```
```  1055 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
```
```  1056 @{term "at_right x"} and also @{term "at_right 0"}.
```
```  1057
```
```  1058 \<close>
```
```  1059
```
```  1060 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
```
```  1061
```
```  1062 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
```
```  1063   by (rule filtermap_fun_inverse[where g="\<lambda>x. x + d"])
```
```  1064      (auto intro!: tendsto_eq_intros filterlim_ident)
```
```  1065
```
```  1066 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
```
```  1067   by (rule filtermap_fun_inverse[where g=uminus])
```
```  1068      (auto intro!: tendsto_eq_intros filterlim_ident)
```
```  1069
```
```  1070 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
```
```  1071   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
```
```  1072
```
```  1073 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
```
```  1074   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
```
```  1075
```
```  1076 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
```
```  1077   using filtermap_at_right_shift[of "-a" 0] by simp
```
```  1078
```
```  1079 lemma filterlim_at_right_to_0:
```
```  1080   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
```
```  1081   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
```
```  1082
```
```  1083 lemma eventually_at_right_to_0:
```
```  1084   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
```
```  1085   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
```
```  1086
```
```  1087 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
```
```  1088   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```  1089
```
```  1090 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
```
```  1091   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```  1092
```
```  1093 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
```
```  1094   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```  1095
```
```  1096 lemma filterlim_at_left_to_right:
```
```  1097   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
```
```  1098   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
```
```  1099
```
```  1100 lemma eventually_at_left_to_right:
```
```  1101   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
```
```  1102   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
```
```  1103
```
```  1104 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```  1105   unfolding filterlim_at_top eventually_at_bot_dense
```
```  1106   by (metis leI minus_less_iff order_less_asym)
```
```  1107
```
```  1108 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```  1109   unfolding filterlim_at_bot eventually_at_top_dense
```
```  1110   by (metis leI less_minus_iff order_less_asym)
```
```  1111
```
```  1112 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
```
```  1113   by (rule filtermap_fun_inverse[symmetric, of uminus])
```
```  1114      (auto intro: filterlim_uminus_at_bot_at_top filterlim_uminus_at_top_at_bot)
```
```  1115
```
```  1116 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
```
```  1117   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
```
```  1118
```
```  1119 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
```
```  1120   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
```
```  1121
```
```  1122 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
```
```  1123   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
```
```  1124
```
```  1125 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
```
```  1126   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
```
```  1127   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
```
```  1128   by auto
```
```  1129
```
```  1130 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
```
```  1131   unfolding filterlim_uminus_at_top by simp
```
```  1132
```
```  1133 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
```
```  1134   unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
```
```  1135 proof safe
```
```  1136   fix Z :: real assume [arith]: "0 < Z"
```
```  1137   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```  1138     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```  1139   then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
```
```  1140     by (auto elim!: eventually_mono simp: inverse_eq_divide field_simps)
```
```  1141 qed
```
```  1142
```
```  1143 lemma tendsto_inverse_0:
```
```  1144   fixes x :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
```
```  1145   shows "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
```
```  1146   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```  1147 proof safe
```
```  1148   fix r :: real assume "0 < r"
```
```  1149   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```  1150   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```  1151     fix x :: 'a
```
```  1152     from \<open>0 < r\<close> have "0 < inverse (r / 2)" by simp
```
```  1153     also assume *: "inverse (r / 2) \<le> norm x"
```
```  1154     finally show "norm (inverse x) < r"
```
```  1155       using * \<open>0 < r\<close> by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```  1156   qed
```
```  1157 qed
```
```  1158
```
```  1159 lemma tendsto_add_filterlim_at_infinity:
```
```  1160   assumes "(f \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
```
```  1161   assumes "filterlim g at_infinity F"
```
```  1162   shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
```
```  1163 proof (subst filterlim_at_infinity[OF order_refl], safe)
```
```  1164   fix r :: real assume r: "r > 0"
```
```  1165   from assms(1) have "((\<lambda>x. norm (f x)) \<longlongrightarrow> norm c) F" by (rule tendsto_norm)
```
```  1166   hence "eventually (\<lambda>x. norm (f x) < norm c + 1) F" by (rule order_tendstoD) simp_all
```
```  1167   moreover from r have "r + norm c + 1 > 0" by (intro add_pos_nonneg) simp_all
```
```  1168   with assms(2) have "eventually (\<lambda>x. norm (g x) \<ge> r + norm c + 1) F"
```
```  1169     unfolding filterlim_at_infinity[OF order_refl] by (elim allE[of _ "r + norm c + 1"]) simp_all
```
```  1170   ultimately show "eventually (\<lambda>x. norm (f x + g x) \<ge> r) F"
```
```  1171   proof eventually_elim
```
```  1172     fix x :: 'a assume A: "norm (f x) < norm c + 1" and B: "r + norm c + 1 \<le> norm (g x)"
```
```  1173     from A B have "r \<le> norm (g x) - norm (f x)" by simp
```
```  1174     also have "norm (g x) - norm (f x) \<le> norm (g x + f x)" by (rule norm_diff_ineq)
```
```  1175     finally show "r \<le> norm (f x + g x)" by (simp add: add_ac)
```
```  1176   qed
```
```  1177 qed
```
```  1178
```
```  1179 lemma tendsto_add_filterlim_at_infinity':
```
```  1180   assumes "filterlim f at_infinity F"
```
```  1181   assumes "(g \<longlongrightarrow> (c :: 'b :: real_normed_vector)) (F :: 'a filter)"
```
```  1182   shows   "filterlim (\<lambda>x. f x + g x) at_infinity F"
```
```  1183   by (subst add.commute) (rule tendsto_add_filterlim_at_infinity assms)+
```
```  1184
```
```  1185 lemma filterlim_inverse_at_right_top: "LIM x at_top. inverse x :> at_right (0::real)"
```
```  1186   unfolding filterlim_at
```
```  1187   by (auto simp: eventually_at_top_dense)
```
```  1188      (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
```
```  1189
```
```  1190 lemma filterlim_inverse_at_top:
```
```  1191   "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```  1192   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
```
```  1193      (simp add: filterlim_def eventually_filtermap eventually_mono at_within_def le_principal)
```
```  1194
```
```  1195 lemma filterlim_inverse_at_bot_neg:
```
```  1196   "LIM x (at_left (0::real)). inverse x :> at_bot"
```
```  1197   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
```
```  1198
```
```  1199 lemma filterlim_inverse_at_bot:
```
```  1200   "(f \<longlongrightarrow> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```  1201   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
```
```  1202   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
```
```  1203
```
```  1204 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
```
```  1205   by (intro filtermap_fun_inverse[symmetric, where g=inverse])
```
```  1206      (auto intro: filterlim_inverse_at_top_right filterlim_inverse_at_right_top)
```
```  1207
```
```  1208 lemma eventually_at_right_to_top:
```
```  1209   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
```
```  1210   unfolding at_right_to_top eventually_filtermap ..
```
```  1211
```
```  1212 lemma filterlim_at_right_to_top:
```
```  1213   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
```
```  1214   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
```
```  1215
```
```  1216 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
```
```  1217   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
```
```  1218
```
```  1219 lemma eventually_at_top_to_right:
```
```  1220   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
```
```  1221   unfolding at_top_to_right eventually_filtermap ..
```
```  1222
```
```  1223 lemma filterlim_at_top_to_right:
```
```  1224   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
```
```  1225   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
```
```  1226
```
```  1227 lemma filterlim_inverse_at_infinity:
```
```  1228   fixes x :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
```
```  1229   shows "filterlim inverse at_infinity (at (0::'a))"
```
```  1230   unfolding filterlim_at_infinity[OF order_refl]
```
```  1231 proof safe
```
```  1232   fix r :: real assume "0 < r"
```
```  1233   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```  1234     unfolding eventually_at norm_inverse
```
```  1235     by (intro exI[of _ "inverse r"])
```
```  1236        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```  1237 qed
```
```  1238
```
```  1239 lemma filterlim_inverse_at_iff:
```
```  1240   fixes g :: "'a \<Rightarrow> 'b::{real_normed_div_algebra, division_ring}"
```
```  1241   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```  1242   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```  1243 proof
```
```  1244   assume "filtermap g F \<le> at_infinity"
```
```  1245   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```  1246     by (rule filtermap_mono)
```
```  1247   also have "\<dots> \<le> at 0"
```
```  1248     using tendsto_inverse_0[where 'a='b]
```
```  1249     by (auto intro!: exI[of _ 1]
```
```  1250              simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
```
```  1251   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```  1252 next
```
```  1253   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```  1254   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```  1255     by (rule filtermap_mono)
```
```  1256   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```  1257     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```  1258 qed
```
```  1259
```
```  1260 lemma tendsto_mult_filterlim_at_infinity:
```
```  1261   assumes "F \<noteq> bot" "(f \<longlongrightarrow> (c :: 'a :: real_normed_field)) F" "c \<noteq> 0"
```
```  1262   assumes "filterlim g at_infinity F"
```
```  1263   shows   "filterlim (\<lambda>x. f x * g x) at_infinity F"
```
```  1264 proof -
```
```  1265   have "((\<lambda>x. inverse (f x) * inverse (g x)) \<longlongrightarrow> inverse c * 0) F"
```
```  1266     by (intro tendsto_mult tendsto_inverse assms filterlim_compose[OF tendsto_inverse_0])
```
```  1267   hence "filterlim (\<lambda>x. inverse (f x) * inverse (g x)) (at (inverse c * 0)) F"
```
```  1268     unfolding filterlim_at using assms
```
```  1269     by (auto intro: filterlim_at_infinity_imp_eventually_ne tendsto_imp_eventually_ne eventually_conj)
```
```  1270   thus ?thesis by (subst filterlim_inverse_at_iff[symmetric]) simp_all
```
```  1271 qed
```
```  1272
```
```  1273 lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) \<longlongrightarrow> 0) F"
```
```  1274  by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
```
```  1275
```
```  1276 lemma mult_nat_left_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. c * x :: nat) at_top sequentially"
```
```  1277   by (rule filterlim_subseq) (auto simp: subseq_def)
```
```  1278
```
```  1279 lemma mult_nat_right_at_top: "c > 0 \<Longrightarrow> filterlim (\<lambda>x. x * c :: nat) at_top sequentially"
```
```  1280   by (rule filterlim_subseq) (auto simp: subseq_def)
```
```  1281
```
```  1282 lemma at_to_infinity:
```
```  1283   fixes x :: "'a :: {real_normed_field,field}"
```
```  1284   shows "(at (0::'a)) = filtermap inverse at_infinity"
```
```  1285 proof (rule antisym)
```
```  1286   have "(inverse \<longlongrightarrow> (0::'a)) at_infinity"
```
```  1287     by (fact tendsto_inverse_0)
```
```  1288   then show "filtermap inverse at_infinity \<le> at (0::'a)"
```
```  1289     apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
```
```  1290     apply (rule_tac x="1" in exI, auto)
```
```  1291     done
```
```  1292 next
```
```  1293   have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
```
```  1294     using filterlim_inverse_at_infinity unfolding filterlim_def
```
```  1295     by (rule filtermap_mono)
```
```  1296   then show "at (0::'a) \<le> filtermap inverse at_infinity"
```
```  1297     by (simp add: filtermap_ident filtermap_filtermap)
```
```  1298 qed
```
```  1299
```
```  1300 lemma lim_at_infinity_0:
```
```  1301   fixes l :: "'a :: {real_normed_field,field}"
```
```  1302   shows "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> ((f o inverse) \<longlongrightarrow> l) (at (0::'a))"
```
```  1303 by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
```
```  1304
```
```  1305 lemma lim_zero_infinity:
```
```  1306   fixes l :: "'a :: {real_normed_field,field}"
```
```  1307   shows "((\<lambda>x. f(1 / x)) \<longlongrightarrow> l) (at (0::'a)) \<Longrightarrow> (f \<longlongrightarrow> l) at_infinity"
```
```  1308 by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
```
```  1309
```
```  1310
```
```  1311 text \<open>
```
```  1312
```
```  1313 We only show rules for multiplication and addition when the functions are either against a real
```
```  1314 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```  1315
```
```  1316 \<close>
```
```  1317
```
```  1318 lemma filterlim_tendsto_pos_mult_at_top:
```
```  1319   assumes f: "(f \<longlongrightarrow> c) F" and c: "0 < c"
```
```  1320   assumes g: "LIM x F. g x :> at_top"
```
```  1321   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1322   unfolding filterlim_at_top_gt[where c=0]
```
```  1323 proof safe
```
```  1324   fix Z :: real assume "0 < Z"
```
```  1325   from f \<open>0 < c\<close> have "eventually (\<lambda>x. c / 2 < f x) F"
```
```  1326     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_mono
```
```  1327              simp: dist_real_def abs_real_def split: split_if_asm)
```
```  1328   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
```
```  1329     unfolding filterlim_at_top by auto
```
```  1330   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1331   proof eventually_elim
```
```  1332     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
```
```  1333     with \<open>0 < Z\<close> \<open>0 < c\<close> have "c / 2 * (Z / c * 2) \<le> f x * g x"
```
```  1334       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1335     with \<open>0 < c\<close> show "Z \<le> f x * g x"
```
```  1336        by simp
```
```  1337   qed
```
```  1338 qed
```
```  1339
```
```  1340 lemma filterlim_at_top_mult_at_top:
```
```  1341   assumes f: "LIM x F. f x :> at_top"
```
```  1342   assumes g: "LIM x F. g x :> at_top"
```
```  1343   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1344   unfolding filterlim_at_top_gt[where c=0]
```
```  1345 proof safe
```
```  1346   fix Z :: real assume "0 < Z"
```
```  1347   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
```
```  1348     unfolding filterlim_at_top by auto
```
```  1349   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1350     unfolding filterlim_at_top by auto
```
```  1351   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1352   proof eventually_elim
```
```  1353     fix x assume "1 \<le> f x" "Z \<le> g x"
```
```  1354     with \<open>0 < Z\<close> have "1 * Z \<le> f x * g x"
```
```  1355       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1356     then show "Z \<le> f x * g x"
```
```  1357        by simp
```
```  1358   qed
```
```  1359 qed
```
```  1360
```
```  1361 lemma filterlim_tendsto_pos_mult_at_bot:
```
```  1362   assumes "(f \<longlongrightarrow> c) F" "0 < (c::real)" "filterlim g at_bot F"
```
```  1363   shows "LIM x F. f x * g x :> at_bot"
```
```  1364   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
```
```  1365   unfolding filterlim_uminus_at_bot by simp
```
```  1366
```
```  1367 lemma filterlim_tendsto_neg_mult_at_bot:
```
```  1368   assumes c: "(f \<longlongrightarrow> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
```
```  1369   shows "LIM x F. f x * g x :> at_bot"
```
```  1370   using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
```
```  1371   unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
```
```  1372
```
```  1373 lemma filterlim_pow_at_top:
```
```  1374   fixes f :: "real \<Rightarrow> real"
```
```  1375   assumes "0 < n" and f: "LIM x F. f x :> at_top"
```
```  1376   shows "LIM x F. (f x)^n :: real :> at_top"
```
```  1377 using \<open>0 < n\<close> proof (induct n)
```
```  1378   case (Suc n) with f show ?case
```
```  1379     by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
```
```  1380 qed simp
```
```  1381
```
```  1382 lemma filterlim_pow_at_bot_even:
```
```  1383   fixes f :: "real \<Rightarrow> real"
```
```  1384   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
```
```  1385   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
```
```  1386
```
```  1387 lemma filterlim_pow_at_bot_odd:
```
```  1388   fixes f :: "real \<Rightarrow> real"
```
```  1389   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
```
```  1390   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
```
```  1391
```
```  1392 lemma filterlim_tendsto_add_at_top:
```
```  1393   assumes f: "(f \<longlongrightarrow> c) F"
```
```  1394   assumes g: "LIM x F. g x :> at_top"
```
```  1395   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1396   unfolding filterlim_at_top_gt[where c=0]
```
```  1397 proof safe
```
```  1398   fix Z :: real assume "0 < Z"
```
```  1399   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```  1400     by (auto dest!: tendstoD[where e=1] elim!: eventually_mono simp: dist_real_def)
```
```  1401   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
```
```  1402     unfolding filterlim_at_top by auto
```
```  1403   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1404     by eventually_elim simp
```
```  1405 qed
```
```  1406
```
```  1407 lemma LIM_at_top_divide:
```
```  1408   fixes f g :: "'a \<Rightarrow> real"
```
```  1409   assumes f: "(f \<longlongrightarrow> a) F" "0 < a"
```
```  1410   assumes g: "(g \<longlongrightarrow> 0) F" "eventually (\<lambda>x. 0 < g x) F"
```
```  1411   shows "LIM x F. f x / g x :> at_top"
```
```  1412   unfolding divide_inverse
```
```  1413   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
```
```  1414
```
```  1415 lemma filterlim_at_top_add_at_top:
```
```  1416   assumes f: "LIM x F. f x :> at_top"
```
```  1417   assumes g: "LIM x F. g x :> at_top"
```
```  1418   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1419   unfolding filterlim_at_top_gt[where c=0]
```
```  1420 proof safe
```
```  1421   fix Z :: real assume "0 < Z"
```
```  1422   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
```
```  1423     unfolding filterlim_at_top by auto
```
```  1424   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1425     unfolding filterlim_at_top by auto
```
```  1426   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1427     by eventually_elim simp
```
```  1428 qed
```
```  1429
```
```  1430 lemma tendsto_divide_0:
```
```  1431   fixes f :: "_ \<Rightarrow> 'a::{real_normed_div_algebra, division_ring}"
```
```  1432   assumes f: "(f \<longlongrightarrow> c) F"
```
```  1433   assumes g: "LIM x F. g x :> at_infinity"
```
```  1434   shows "((\<lambda>x. f x / g x) \<longlongrightarrow> 0) F"
```
```  1435   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
```
```  1436
```
```  1437 lemma linear_plus_1_le_power:
```
```  1438   fixes x :: real
```
```  1439   assumes x: "0 \<le> x"
```
```  1440   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1441 proof (induct n)
```
```  1442   case (Suc n)
```
```  1443   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
```
```  1444     by (simp add: field_simps of_nat_Suc x)
```
```  1445   also have "\<dots> \<le> (x + 1)^Suc n"
```
```  1446     using Suc x by (simp add: mult_left_mono)
```
```  1447   finally show ?case .
```
```  1448 qed simp
```
```  1449
```
```  1450 lemma filterlim_realpow_sequentially_gt1:
```
```  1451   fixes x :: "'a :: real_normed_div_algebra"
```
```  1452   assumes x[arith]: "1 < norm x"
```
```  1453   shows "LIM n sequentially. x ^ n :> at_infinity"
```
```  1454 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
```
```  1455   fix y :: real assume "0 < y"
```
```  1456   have "0 < norm x - 1" by simp
```
```  1457   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
```
```  1458   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
```
```  1459   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
```
```  1460   also have "\<dots> = norm x ^ N" by simp
```
```  1461   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
```
```  1462     by (metis order_less_le_trans power_increasing order_less_imp_le x)
```
```  1463   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
```
```  1464     unfolding eventually_sequentially
```
```  1465     by (auto simp: norm_power)
```
```  1466 qed simp
```
```  1467
```
```  1468
```
```  1469 subsection \<open>Limits of Sequences\<close>
```
```  1470
```
```  1471 lemma [trans]: "X=Y ==> Y \<longlonglongrightarrow> z ==> X \<longlonglongrightarrow> z"
```
```  1472   by simp
```
```  1473
```
```  1474 lemma LIMSEQ_iff:
```
```  1475   fixes L :: "'a::real_normed_vector"
```
```  1476   shows "(X \<longlonglongrightarrow> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```  1477 unfolding lim_sequentially dist_norm ..
```
```  1478
```
```  1479 lemma LIMSEQ_I:
```
```  1480   fixes L :: "'a::real_normed_vector"
```
```  1481   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X \<longlonglongrightarrow> L"
```
```  1482 by (simp add: LIMSEQ_iff)
```
```  1483
```
```  1484 lemma LIMSEQ_D:
```
```  1485   fixes L :: "'a::real_normed_vector"
```
```  1486   shows "\<lbrakk>X \<longlonglongrightarrow> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```  1487 by (simp add: LIMSEQ_iff)
```
```  1488
```
```  1489 lemma LIMSEQ_linear: "\<lbrakk> X \<longlonglongrightarrow> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) \<longlonglongrightarrow> x"
```
```  1490   unfolding tendsto_def eventually_sequentially
```
```  1491   by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
```
```  1492
```
```  1493 lemma Bseq_inverse_lemma:
```
```  1494   fixes x :: "'a::real_normed_div_algebra"
```
```  1495   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```  1496 apply (subst nonzero_norm_inverse, clarsimp)
```
```  1497 apply (erule (1) le_imp_inverse_le)
```
```  1498 done
```
```  1499
```
```  1500 lemma Bseq_inverse:
```
```  1501   fixes a :: "'a::real_normed_div_algebra"
```
```  1502   shows "\<lbrakk>X \<longlonglongrightarrow> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```  1503   by (rule Bfun_inverse)
```
```  1504
```
```  1505 text\<open>Transformation of limit.\<close>
```
```  1506
```
```  1507 lemma eventually_at2:
```
```  1508   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
```
```  1509   unfolding eventually_at dist_nz by auto
```
```  1510
```
```  1511 lemma Lim_transform:
```
```  1512   fixes a b :: "'a::real_normed_vector"
```
```  1513   shows "\<lbrakk>(g \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (f \<longlongrightarrow> a) F"
```
```  1514   using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
```
```  1515
```
```  1516 lemma Lim_transform2:
```
```  1517   fixes a b :: "'a::real_normed_vector"
```
```  1518   shows "\<lbrakk>(f \<longlongrightarrow> a) F; ((\<lambda>x. f x - g x) \<longlongrightarrow> 0) F\<rbrakk> \<Longrightarrow> (g \<longlongrightarrow> a) F"
```
```  1519   by (erule Lim_transform) (simp add: tendsto_minus_cancel)
```
```  1520
```
```  1521 lemma Lim_transform_eventually:
```
```  1522   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> (g \<longlongrightarrow> l) net"
```
```  1523   apply (rule topological_tendstoI)
```
```  1524   apply (drule (2) topological_tendstoD)
```
```  1525   apply (erule (1) eventually_elim2, simp)
```
```  1526   done
```
```  1527
```
```  1528 lemma Lim_transform_within:
```
```  1529   assumes "0 < d"
```
```  1530     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
```
```  1531     and "(f \<longlongrightarrow> l) (at x within S)"
```
```  1532   shows "(g \<longlongrightarrow> l) (at x within S)"
```
```  1533 proof (rule Lim_transform_eventually)
```
```  1534   show "eventually (\<lambda>x. f x = g x) (at x within S)"
```
```  1535     using assms(1,2) by (auto simp: dist_nz eventually_at)
```
```  1536   show "(f \<longlongrightarrow> l) (at x within S)" by fact
```
```  1537 qed
```
```  1538
```
```  1539 lemma Lim_transform_at:
```
```  1540   assumes "0 < d"
```
```  1541     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
```
```  1542     and "(f \<longlongrightarrow> l) (at x)"
```
```  1543   shows "(g \<longlongrightarrow> l) (at x)"
```
```  1544   using _ assms(3)
```
```  1545 proof (rule Lim_transform_eventually)
```
```  1546   show "eventually (\<lambda>x. f x = g x) (at x)"
```
```  1547     unfolding eventually_at2
```
```  1548     using assms(1,2) by auto
```
```  1549 qed
```
```  1550
```
```  1551 text\<open>Common case assuming being away from some crucial point like 0.\<close>
```
```  1552
```
```  1553 lemma Lim_transform_away_within:
```
```  1554   fixes a b :: "'a::t1_space"
```
```  1555   assumes "a \<noteq> b"
```
```  1556     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
```
```  1557     and "(f \<longlongrightarrow> l) (at a within S)"
```
```  1558   shows "(g \<longlongrightarrow> l) (at a within S)"
```
```  1559 proof (rule Lim_transform_eventually)
```
```  1560   show "(f \<longlongrightarrow> l) (at a within S)" by fact
```
```  1561   show "eventually (\<lambda>x. f x = g x) (at a within S)"
```
```  1562     unfolding eventually_at_topological
```
```  1563     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
```
```  1564 qed
```
```  1565
```
```  1566 lemma Lim_transform_away_at:
```
```  1567   fixes a b :: "'a::t1_space"
```
```  1568   assumes ab: "a\<noteq>b"
```
```  1569     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
```
```  1570     and fl: "(f \<longlongrightarrow> l) (at a)"
```
```  1571   shows "(g \<longlongrightarrow> l) (at a)"
```
```  1572   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
```
```  1573
```
```  1574 text\<open>Alternatively, within an open set.\<close>
```
```  1575
```
```  1576 lemma Lim_transform_within_open:
```
```  1577   assumes "open S" and "a \<in> S"
```
```  1578     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
```
```  1579     and "(f \<longlongrightarrow> l) (at a)"
```
```  1580   shows "(g \<longlongrightarrow> l) (at a)"
```
```  1581 proof (rule Lim_transform_eventually)
```
```  1582   show "eventually (\<lambda>x. f x = g x) (at a)"
```
```  1583     unfolding eventually_at_topological
```
```  1584     using assms(1,2,3) by auto
```
```  1585   show "(f \<longlongrightarrow> l) (at a)" by fact
```
```  1586 qed
```
```  1587
```
```  1588 text\<open>A congruence rule allowing us to transform limits assuming not at point.\<close>
```
```  1589
```
```  1590 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
```
```  1591
```
```  1592 lemma Lim_cong_within(*[cong add]*):
```
```  1593   assumes "a = b"
```
```  1594     and "x = y"
```
```  1595     and "S = T"
```
```  1596     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
```
```  1597   shows "(f \<longlongrightarrow> x) (at a within S) \<longleftrightarrow> (g \<longlongrightarrow> y) (at b within T)"
```
```  1598   unfolding tendsto_def eventually_at_topological
```
```  1599   using assms by simp
```
```  1600
```
```  1601 lemma Lim_cong_at(*[cong add]*):
```
```  1602   assumes "a = b" "x = y"
```
```  1603     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
```
```  1604   shows "((\<lambda>x. f x) \<longlongrightarrow> x) (at a) \<longleftrightarrow> ((g \<longlongrightarrow> y) (at a))"
```
```  1605   unfolding tendsto_def eventually_at_topological
```
```  1606   using assms by simp
```
```  1607 text\<open>An unbounded sequence's inverse tends to 0\<close>
```
```  1608
```
```  1609 lemma LIMSEQ_inverse_zero:
```
```  1610   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) \<longlonglongrightarrow> 0"
```
```  1611   apply (rule filterlim_compose[OF tendsto_inverse_0])
```
```  1612   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
```
```  1613   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
```
```  1614   done
```
```  1615
```
```  1616 text\<open>The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity\<close>
```
```  1617
```
```  1618 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) \<longlonglongrightarrow> 0"
```
```  1619   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
```
```  1620             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
```
```  1621
```
```  1622 text\<open>The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```  1623 infinity is now easily proved\<close>
```
```  1624
```
```  1625 lemma LIMSEQ_inverse_real_of_nat_add:
```
```  1626      "(%n. r + inverse(real(Suc n))) \<longlonglongrightarrow> r"
```
```  1627   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
```
```  1628
```
```  1629 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```  1630      "(%n. r + -inverse(real(Suc n))) \<longlonglongrightarrow> r"
```
```  1631   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
```
```  1632   by auto
```
```  1633
```
```  1634 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```  1635      "(%n. r*( 1 + -inverse(real(Suc n)))) \<longlonglongrightarrow> r"
```
```  1636   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
```
```  1637   by auto
```
```  1638
```
```  1639 lemma lim_1_over_n: "((\<lambda>n. 1 / of_nat n) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
```
```  1640 proof (subst lim_sequentially, intro allI impI exI)
```
```  1641   fix e :: real assume e: "e > 0"
```
```  1642   fix n :: nat assume n: "n \<ge> nat \<lceil>inverse e + 1\<rceil>"
```
```  1643   have "inverse e < of_nat (nat \<lceil>inverse e + 1\<rceil>)" by linarith
```
```  1644   also note n
```
```  1645   finally show "dist (1 / of_nat n :: 'a) 0 < e" using e
```
```  1646     by (simp add: divide_simps mult.commute norm_conv_dist[symmetric] norm_divide)
```
```  1647 qed
```
```  1648
```
```  1649 lemma lim_inverse_n: "((\<lambda>n. inverse(of_nat n)) \<longlongrightarrow> (0::'a::real_normed_field)) sequentially"
```
```  1650   using lim_1_over_n by (simp add: inverse_eq_divide)
```
```  1651
```
```  1652 lemma LIMSEQ_Suc_n_over_n: "(\<lambda>n. of_nat (Suc n) / of_nat n :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
```
```  1653 proof (rule Lim_transform_eventually)
```
```  1654   show "eventually (\<lambda>n. 1 + inverse (of_nat n :: 'a) = of_nat (Suc n) / of_nat n) sequentially"
```
```  1655     using eventually_gt_at_top[of "0::nat"] by eventually_elim (simp add: field_simps)
```
```  1656   have "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1 + 0"
```
```  1657     by (intro tendsto_add tendsto_const lim_inverse_n)
```
```  1658   thus "(\<lambda>n. 1 + inverse (of_nat n) :: 'a) \<longlonglongrightarrow> 1" by simp
```
```  1659 qed
```
```  1660
```
```  1661 lemma LIMSEQ_n_over_Suc_n: "(\<lambda>n. of_nat n / of_nat (Suc n) :: 'a :: real_normed_field) \<longlonglongrightarrow> 1"
```
```  1662 proof (rule Lim_transform_eventually)
```
```  1663   show "eventually (\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a) =
```
```  1664                         of_nat n / of_nat (Suc n)) sequentially"
```
```  1665     using eventually_gt_at_top[of "0::nat"]
```
```  1666     by eventually_elim (simp add: field_simps del: of_nat_Suc)
```
```  1667   have "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> inverse 1"
```
```  1668     by (intro tendsto_inverse LIMSEQ_Suc_n_over_n) simp_all
```
```  1669   thus "(\<lambda>n. inverse (of_nat (Suc n) / of_nat n :: 'a)) \<longlonglongrightarrow> 1" by simp
```
```  1670 qed
```
```  1671
```
```  1672 subsection \<open>Convergence on sequences\<close>
```
```  1673
```
```  1674 lemma convergent_cong:
```
```  1675   assumes "eventually (\<lambda>x. f x = g x) sequentially"
```
```  1676   shows   "convergent f \<longleftrightarrow> convergent g"
```
```  1677   unfolding convergent_def by (subst filterlim_cong[OF refl refl assms]) (rule refl)
```
```  1678
```
```  1679 lemma convergent_Suc_iff: "convergent (\<lambda>n. f (Suc n)) \<longleftrightarrow> convergent f"
```
```  1680   by (auto simp: convergent_def LIMSEQ_Suc_iff)
```
```  1681
```
```  1682 lemma convergent_ignore_initial_segment: "convergent (\<lambda>n. f (n + m)) = convergent f"
```
```  1683 proof (induction m arbitrary: f)
```
```  1684   case (Suc m)
```
```  1685   have "convergent (\<lambda>n. f (n + Suc m)) \<longleftrightarrow> convergent (\<lambda>n. f (Suc n + m))" by simp
```
```  1686   also have "\<dots> \<longleftrightarrow> convergent (\<lambda>n. f (n + m))" by (rule convergent_Suc_iff)
```
```  1687   also have "\<dots> \<longleftrightarrow> convergent f" by (rule Suc)
```
```  1688   finally show ?case .
```
```  1689 qed simp_all
```
```  1690
```
```  1691 lemma convergent_add:
```
```  1692   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1693   assumes "convergent (\<lambda>n. X n)"
```
```  1694   assumes "convergent (\<lambda>n. Y n)"
```
```  1695   shows "convergent (\<lambda>n. X n + Y n)"
```
```  1696   using assms unfolding convergent_def by (blast intro: tendsto_add)
```
```  1697
```
```  1698 lemma convergent_setsum:
```
```  1699   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
```
```  1700   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
```
```  1701   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
```
```  1702 proof (cases "finite A")
```
```  1703   case True from this and assms show ?thesis
```
```  1704     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
```
```  1705 qed (simp add: convergent_const)
```
```  1706
```
```  1707 lemma (in bounded_linear) convergent:
```
```  1708   assumes "convergent (\<lambda>n. X n)"
```
```  1709   shows "convergent (\<lambda>n. f (X n))"
```
```  1710   using assms unfolding convergent_def by (blast intro: tendsto)
```
```  1711
```
```  1712 lemma (in bounded_bilinear) convergent:
```
```  1713   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
```
```  1714   shows "convergent (\<lambda>n. X n ** Y n)"
```
```  1715   using assms unfolding convergent_def by (blast intro: tendsto)
```
```  1716
```
```  1717 lemma convergent_minus_iff:
```
```  1718   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1719   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```  1720 apply (simp add: convergent_def)
```
```  1721 apply (auto dest: tendsto_minus)
```
```  1722 apply (drule tendsto_minus, auto)
```
```  1723 done
```
```  1724
```
```  1725 lemma convergent_diff:
```
```  1726   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1727   assumes "convergent (\<lambda>n. X n)"
```
```  1728   assumes "convergent (\<lambda>n. Y n)"
```
```  1729   shows "convergent (\<lambda>n. X n - Y n)"
```
```  1730   using assms unfolding convergent_def by (blast intro: tendsto_diff)
```
```  1731
```
```  1732 lemma convergent_norm:
```
```  1733   assumes "convergent f"
```
```  1734   shows   "convergent (\<lambda>n. norm (f n))"
```
```  1735 proof -
```
```  1736   from assms have "f \<longlonglongrightarrow> lim f" by (simp add: convergent_LIMSEQ_iff)
```
```  1737   hence "(\<lambda>n. norm (f n)) \<longlonglongrightarrow> norm (lim f)" by (rule tendsto_norm)
```
```  1738   thus ?thesis by (auto simp: convergent_def)
```
```  1739 qed
```
```  1740
```
```  1741 lemma convergent_of_real:
```
```  1742   "convergent f \<Longrightarrow> convergent (\<lambda>n. of_real (f n) :: 'a :: real_normed_algebra_1)"
```
```  1743   unfolding convergent_def by (blast intro!: tendsto_of_real)
```
```  1744
```
```  1745 lemma convergent_add_const_iff:
```
```  1746   "convergent (\<lambda>n. c + f n :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
```
```  1747 proof
```
```  1748   assume "convergent (\<lambda>n. c + f n)"
```
```  1749   from convergent_diff[OF this convergent_const[of c]] show "convergent f" by simp
```
```  1750 next
```
```  1751   assume "convergent f"
```
```  1752   from convergent_add[OF convergent_const[of c] this] show "convergent (\<lambda>n. c + f n)" by simp
```
```  1753 qed
```
```  1754
```
```  1755 lemma convergent_add_const_right_iff:
```
```  1756   "convergent (\<lambda>n. f n + c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
```
```  1757   using convergent_add_const_iff[of c f] by (simp add: add_ac)
```
```  1758
```
```  1759 lemma convergent_diff_const_right_iff:
```
```  1760   "convergent (\<lambda>n. f n - c :: 'a :: real_normed_vector) \<longleftrightarrow> convergent f"
```
```  1761   using convergent_add_const_right_iff[of f "-c"] by (simp add: add_ac)
```
```  1762
```
```  1763 lemma convergent_mult:
```
```  1764   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_field"
```
```  1765   assumes "convergent (\<lambda>n. X n)"
```
```  1766   assumes "convergent (\<lambda>n. Y n)"
```
```  1767   shows "convergent (\<lambda>n. X n * Y n)"
```
```  1768   using assms unfolding convergent_def by (blast intro: tendsto_mult)
```
```  1769
```
```  1770 lemma convergent_mult_const_iff:
```
```  1771   assumes "c \<noteq> 0"
```
```  1772   shows   "convergent (\<lambda>n. c * f n :: 'a :: real_normed_field) \<longleftrightarrow> convergent f"
```
```  1773 proof
```
```  1774   assume "convergent (\<lambda>n. c * f n)"
```
```  1775   from assms convergent_mult[OF this convergent_const[of "inverse c"]]
```
```  1776     show "convergent f" by (simp add: field_simps)
```
```  1777 next
```
```  1778   assume "convergent f"
```
```  1779   from convergent_mult[OF convergent_const[of c] this] show "convergent (\<lambda>n. c * f n)" by simp
```
```  1780 qed
```
```  1781
```
```  1782 lemma convergent_mult_const_right_iff:
```
```  1783   assumes "c \<noteq> 0"
```
```  1784   shows   "convergent (\<lambda>n. (f n :: 'a :: real_normed_field) * c) \<longleftrightarrow> convergent f"
```
```  1785   using convergent_mult_const_iff[OF assms, of f] by (simp add: mult_ac)
```
```  1786
```
```  1787 lemma convergent_imp_Bseq: "convergent f \<Longrightarrow> Bseq f"
```
```  1788   by (simp add: Cauchy_Bseq convergent_Cauchy)
```
```  1789
```
```  1790
```
```  1791 text \<open>A monotone sequence converges to its least upper bound.\<close>
```
```  1792
```
```  1793 lemma LIMSEQ_incseq_SUP:
```
```  1794   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
```
```  1795   assumes u: "bdd_above (range X)"
```
```  1796   assumes X: "incseq X"
```
```  1797   shows "X \<longlonglongrightarrow> (SUP i. X i)"
```
```  1798   by (rule order_tendstoI)
```
```  1799      (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
```
```  1800
```
```  1801 lemma LIMSEQ_decseq_INF:
```
```  1802   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
```
```  1803   assumes u: "bdd_below (range X)"
```
```  1804   assumes X: "decseq X"
```
```  1805   shows "X \<longlonglongrightarrow> (INF i. X i)"
```
```  1806   by (rule order_tendstoI)
```
```  1807      (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
```
```  1808
```
```  1809 text\<open>Main monotonicity theorem\<close>
```
```  1810
```
```  1811 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
```
```  1812   by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
```
```  1813
```
```  1814 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)"
```
```  1815   by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
```
```  1816
```
```  1817 lemma monoseq_imp_convergent_iff_Bseq: "monoseq (f :: nat \<Rightarrow> real) \<Longrightarrow> convergent f \<longleftrightarrow> Bseq f"
```
```  1818   using Bseq_monoseq_convergent[of f] convergent_imp_Bseq[of f] by blast
```
```  1819
```
```  1820 lemma Bseq_monoseq_convergent'_inc:
```
```  1821   "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"
```
```  1822   by (subst convergent_ignore_initial_segment [symmetric, of _ M])
```
```  1823      (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
```
```  1824
```
```  1825 lemma Bseq_monoseq_convergent'_dec:
```
```  1826   "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"
```
```  1827   by (subst convergent_ignore_initial_segment [symmetric, of _ M])
```
```  1828      (auto intro!: Bseq_monoseq_convergent simp: monoseq_def)
```
```  1829
```
```  1830 lemma Cauchy_iff:
```
```  1831   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1832   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```  1833   unfolding Cauchy_def dist_norm ..
```
```  1834
```
```  1835 lemma CauchyI:
```
```  1836   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1837   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"
```
```  1838 by (simp add: Cauchy_iff)
```
```  1839
```
```  1840 lemma CauchyD:
```
```  1841   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1842   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1843 by (simp add: Cauchy_iff)
```
```  1844
```
```  1845 lemma incseq_convergent:
```
```  1846   fixes X :: "nat \<Rightarrow> real"
```
```  1847   assumes "incseq X" and "\<forall>i. X i \<le> B"
```
```  1848   obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. X i \<le> L"
```
```  1849 proof atomize_elim
```
```  1850   from incseq_bounded[OF assms] \<open>incseq X\<close> Bseq_monoseq_convergent[of X]
```
```  1851   obtain L where "X \<longlonglongrightarrow> L"
```
```  1852     by (auto simp: convergent_def monoseq_def incseq_def)
```
```  1853   with \<open>incseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. X i \<le> L)"
```
```  1854     by (auto intro!: exI[of _ L] incseq_le)
```
```  1855 qed
```
```  1856
```
```  1857 lemma decseq_convergent:
```
```  1858   fixes X :: "nat \<Rightarrow> real"
```
```  1859   assumes "decseq X" and "\<forall>i. B \<le> X i"
```
```  1860   obtains L where "X \<longlonglongrightarrow> L" "\<forall>i. L \<le> X i"
```
```  1861 proof atomize_elim
```
```  1862   from decseq_bounded[OF assms] \<open>decseq X\<close> Bseq_monoseq_convergent[of X]
```
```  1863   obtain L where "X \<longlonglongrightarrow> L"
```
```  1864     by (auto simp: convergent_def monoseq_def decseq_def)
```
```  1865   with \<open>decseq X\<close> show "\<exists>L. X \<longlonglongrightarrow> L \<and> (\<forall>i. L \<le> X i)"
```
```  1866     by (auto intro!: exI[of _ L] decseq_le)
```
```  1867 qed
```
```  1868
```
```  1869 subsubsection \<open>Cauchy Sequences are Bounded\<close>
```
```  1870
```
```  1871 text\<open>A Cauchy sequence is bounded -- this is the standard
```
```  1872   proof mechanization rather than the nonstandard proof\<close>
```
```  1873
```
```  1874 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1875           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1876 apply (clarify, drule spec, drule (1) mp)
```
```  1877 apply (simp only: norm_minus_commute)
```
```  1878 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1879 apply simp
```
```  1880 done
```
```  1881
```
```  1882 subsection \<open>Power Sequences\<close>
```
```  1883
```
```  1884 text\<open>The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1885 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1886   also fact that bounded and monotonic sequence converges.\<close>
```
```  1887
```
```  1888 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1889 apply (simp add: Bseq_def)
```
```  1890 apply (rule_tac x = 1 in exI)
```
```  1891 apply (simp add: power_abs)
```
```  1892 apply (auto dest: power_mono)
```
```  1893 done
```
```  1894
```
```  1895 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1896 apply (clarify intro!: mono_SucI2)
```
```  1897 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1898 done
```
```  1899
```
```  1900 lemma convergent_realpow:
```
```  1901   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1902 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1903
```
```  1904 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) \<longlonglongrightarrow> 0"
```
```  1905   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
```
```  1906
```
```  1907 lemma LIMSEQ_realpow_zero:
```
```  1908   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
```
```  1909 proof cases
```
```  1910   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1911   hence x0: "0 < x" by simp
```
```  1912   assume x1: "x < 1"
```
```  1913   from x0 x1 have "1 < inverse x"
```
```  1914     by (rule one_less_inverse)
```
```  1915   hence "(\<lambda>n. inverse (inverse x ^ n)) \<longlonglongrightarrow> 0"
```
```  1916     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1917   thus ?thesis by (simp add: power_inverse)
```
```  1918 qed (rule LIMSEQ_imp_Suc, simp)
```
```  1919
```
```  1920 lemma LIMSEQ_power_zero:
```
```  1921   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1922   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) \<longlonglongrightarrow> 0"
```
```  1923 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1924 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
```
```  1925 apply (simp add: power_abs norm_power_ineq)
```
```  1926 done
```
```  1927
```
```  1928 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) \<longlonglongrightarrow> 0"
```
```  1929   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
```
```  1930
```
```  1931 text\<open>Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}\<close>
```
```  1932
```
```  1933 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) \<longlonglongrightarrow> 0"
```
```  1934   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1935
```
```  1936 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) \<longlonglongrightarrow> 0"
```
```  1937   by (rule LIMSEQ_power_zero) simp
```
```  1938
```
```  1939
```
```  1940 subsection \<open>Limits of Functions\<close>
```
```  1941
```
```  1942 lemma LIM_eq:
```
```  1943   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1944   shows "f -- a --> L =
```
```  1945      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```  1946 by (simp add: LIM_def dist_norm)
```
```  1947
```
```  1948 lemma LIM_I:
```
```  1949   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1950   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```  1951       ==> f -- a --> L"
```
```  1952 by (simp add: LIM_eq)
```
```  1953
```
```  1954 lemma LIM_D:
```
```  1955   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1956   shows "[| f -- a --> L; 0<r |]
```
```  1957       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```  1958 by (simp add: LIM_eq)
```
```  1959
```
```  1960 lemma LIM_offset:
```
```  1961   fixes a :: "'a::real_normed_vector"
```
```  1962   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```  1963   unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
```
```  1964
```
```  1965 lemma LIM_offset_zero:
```
```  1966   fixes a :: "'a::real_normed_vector"
```
```  1967   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1968 by (drule_tac k="a" in LIM_offset, simp add: add.commute)
```
```  1969
```
```  1970 lemma LIM_offset_zero_cancel:
```
```  1971   fixes a :: "'a::real_normed_vector"
```
```  1972   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```  1973 by (drule_tac k="- a" in LIM_offset, simp)
```
```  1974
```
```  1975 lemma LIM_offset_zero_iff:
```
```  1976   fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
```
```  1977   shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1978   using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
```
```  1979
```
```  1980 lemma LIM_zero:
```
```  1981   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1982   shows "(f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. f x - l) \<longlongrightarrow> 0) F"
```
```  1983 unfolding tendsto_iff dist_norm by simp
```
```  1984
```
```  1985 lemma LIM_zero_cancel:
```
```  1986   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1987   shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F \<Longrightarrow> (f \<longlongrightarrow> l) F"
```
```  1988 unfolding tendsto_iff dist_norm by simp
```
```  1989
```
```  1990 lemma LIM_zero_iff:
```
```  1991   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  1992   shows "((\<lambda>x. f x - l) \<longlongrightarrow> 0) F = (f \<longlongrightarrow> l) F"
```
```  1993 unfolding tendsto_iff dist_norm by simp
```
```  1994
```
```  1995 lemma LIM_imp_LIM:
```
```  1996   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1997   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```  1998   assumes f: "f -- a --> l"
```
```  1999   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```  2000   shows "g -- a --> m"
```
```  2001   by (rule metric_LIM_imp_LIM [OF f],
```
```  2002     simp add: dist_norm le)
```
```  2003
```
```  2004 lemma LIM_equal2:
```
```  2005   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  2006   assumes 1: "0 < R"
```
```  2007   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  2008   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```  2009 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
```
```  2010
```
```  2011 lemma LIM_compose2:
```
```  2012   fixes a :: "'a::real_normed_vector"
```
```  2013   assumes f: "f -- a --> b"
```
```  2014   assumes g: "g -- b --> c"
```
```  2015   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```  2016   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  2017 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```  2018
```
```  2019 lemma real_LIM_sandwich_zero:
```
```  2020   fixes f g :: "'a::topological_space \<Rightarrow> real"
```
```  2021   assumes f: "f -- a --> 0"
```
```  2022   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```  2023   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```  2024   shows "g -- a --> 0"
```
```  2025 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
```
```  2026   fix x assume x: "x \<noteq> a"
```
```  2027   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```  2028   also have "g x \<le> f x" by (rule 2 [OF x])
```
```  2029   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```  2030   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```  2031   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```  2032 qed
```
```  2033
```
```  2034
```
```  2035 subsection \<open>Continuity\<close>
```
```  2036
```
```  2037 lemma LIM_isCont_iff:
```
```  2038   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  2039   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```  2040 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```  2041
```
```  2042 lemma isCont_iff:
```
```  2043   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  2044   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```  2045 by (simp add: isCont_def LIM_isCont_iff)
```
```  2046
```
```  2047 lemma isCont_LIM_compose2:
```
```  2048   fixes a :: "'a::real_normed_vector"
```
```  2049   assumes f [unfolded isCont_def]: "isCont f a"
```
```  2050   assumes g: "g -- f a --> l"
```
```  2051   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```  2052   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```  2053 by (rule LIM_compose2 [OF f g inj])
```
```  2054
```
```  2055
```
```  2056 lemma isCont_norm [simp]:
```
```  2057   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  2058   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```  2059   by (fact continuous_norm)
```
```  2060
```
```  2061 lemma isCont_rabs [simp]:
```
```  2062   fixes f :: "'a::t2_space \<Rightarrow> real"
```
```  2063   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
```
```  2064   by (fact continuous_rabs)
```
```  2065
```
```  2066 lemma isCont_add [simp]:
```
```  2067   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  2068   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```  2069   by (fact continuous_add)
```
```  2070
```
```  2071 lemma isCont_minus [simp]:
```
```  2072   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  2073   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```  2074   by (fact continuous_minus)
```
```  2075
```
```  2076 lemma isCont_diff [simp]:
```
```  2077   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  2078   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```  2079   by (fact continuous_diff)
```
```  2080
```
```  2081 lemma isCont_mult [simp]:
```
```  2082   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
```
```  2083   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```  2084   by (fact continuous_mult)
```
```  2085
```
```  2086 lemma (in bounded_linear) isCont:
```
```  2087   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
```
```  2088   by (fact continuous)
```
```  2089
```
```  2090 lemma (in bounded_bilinear) isCont:
```
```  2091   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```  2092   by (fact continuous)
```
```  2093
```
```  2094 lemmas isCont_scaleR [simp] =
```
```  2095   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
```
```  2096
```
```  2097 lemmas isCont_of_real [simp] =
```
```  2098   bounded_linear.isCont [OF bounded_linear_of_real]
```
```  2099
```
```  2100 lemma isCont_power [simp]:
```
```  2101   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  2102   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```  2103   by (fact continuous_power)
```
```  2104
```
```  2105 lemma isCont_setsum [simp]:
```
```  2106   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```  2107   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
```
```  2108   by (auto intro: continuous_setsum)
```
```  2109
```
```  2110 subsection \<open>Uniform Continuity\<close>
```
```  2111
```
```  2112 definition
```
```  2113   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
```
```  2114   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
```
```  2115
```
```  2116 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```  2117 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```  2118
```
```  2119 lemma isUCont_Cauchy:
```
```  2120   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  2121 unfolding isUCont_def
```
```  2122 apply (rule metric_CauchyI)
```
```  2123 apply (drule_tac x=e in spec, safe)
```
```  2124 apply (drule_tac e=s in metric_CauchyD, safe)
```
```  2125 apply (rule_tac x=M in exI, simp)
```
```  2126 done
```
```  2127
```
```  2128 lemma (in bounded_linear) isUCont: "isUCont f"
```
```  2129 unfolding isUCont_def dist_norm
```
```  2130 proof (intro allI impI)
```
```  2131   fix r::real assume r: "0 < r"
```
```  2132   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```  2133     using pos_bounded by blast
```
```  2134   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```  2135   proof (rule exI, safe)
```
```  2136     from r K show "0 < r / K" by simp
```
```  2137   next
```
```  2138     fix x y :: 'a
```
```  2139     assume xy: "norm (x - y) < r / K"
```
```  2140     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```  2141     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```  2142     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```  2143     finally show "norm (f x - f y) < r" .
```
```  2144   qed
```
```  2145 qed
```
```  2146
```
```  2147 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  2148 by (rule isUCont [THEN isUCont_Cauchy])
```
```  2149
```
```  2150 lemma LIM_less_bound:
```
```  2151   fixes f :: "real \<Rightarrow> real"
```
```  2152   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
```
```  2153   shows "0 \<le> f x"
```
```  2154 proof (rule tendsto_le_const)
```
```  2155   show "(f \<longlongrightarrow> f x) (at_left x)"
```
```  2156     using \<open>isCont f x\<close> by (simp add: filterlim_at_split isCont_def)
```
```  2157   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
```
```  2158     using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
```
```  2159 qed simp
```
```  2160
```
```  2161
```
```  2162 subsection \<open>Nested Intervals and Bisection -- Needed for Compactness\<close>
```
```  2163
```
```  2164 lemma nested_sequence_unique:
```
```  2165   assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) \<longlonglongrightarrow> 0"
```
```  2166   shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f \<longlonglongrightarrow> l) \<and> ((\<forall>n. l \<le> g n) \<and> g \<longlonglongrightarrow> l)"
```
```  2167 proof -
```
```  2168   have "incseq f" unfolding incseq_Suc_iff by fact
```
```  2169   have "decseq g" unfolding decseq_Suc_iff by fact
```
```  2170
```
```  2171   { fix n
```
```  2172     from \<open>decseq g\<close> have "g n \<le> g 0" by (rule decseqD) simp
```
```  2173     with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f n \<le> g 0" by auto }
```
```  2174   then obtain u where "f \<longlonglongrightarrow> u" "\<forall>i. f i \<le> u"
```
```  2175     using incseq_convergent[OF \<open>incseq f\<close>] by auto
```
```  2176   moreover
```
```  2177   { fix n
```
```  2178     from \<open>incseq f\<close> have "f 0 \<le> f n" by (rule incseqD) simp
```
```  2179     with \<open>\<forall>n. f n \<le> g n\<close>[THEN spec, of n] have "f 0 \<le> g n" by simp }
```
```  2180   then obtain l where "g \<longlonglongrightarrow> l" "\<forall>i. l \<le> g i"
```
```  2181     using decseq_convergent[OF \<open>decseq g\<close>] by auto
```
```  2182   moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF \<open>f \<longlonglongrightarrow> u\<close> \<open>g \<longlonglongrightarrow> l\<close>]]
```
```  2183   ultimately show ?thesis by auto
```
```  2184 qed
```
```  2185
```
```  2186 lemma Bolzano[consumes 1, case_names trans local]:
```
```  2187   fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
```
```  2188   assumes [arith]: "a \<le> b"
```
```  2189   assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
```
```  2190   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"
```
```  2191   shows "P a b"
```
```  2192 proof -
```
```  2193   def bisect \<equiv> "rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
```
```  2194   def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
```
```  2195   have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
```
```  2196     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)"
```
```  2197     by (simp_all add: l_def u_def bisect_def split: prod.split)
```
```  2198
```
```  2199   { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
```
```  2200
```
```  2201   have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l \<longlonglongrightarrow> x) \<and> ((\<forall>n. x \<le> u n) \<and> u \<longlonglongrightarrow> x)"
```
```  2202   proof (safe intro!: nested_sequence_unique)
```
```  2203     fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
```
```  2204   next
```
```  2205     { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
```
```  2206     then show "(\<lambda>n. l n - u n) \<longlonglongrightarrow> 0" by (simp add: LIMSEQ_divide_realpow_zero)
```
```  2207   qed fact
```
```  2208   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
```
```  2209   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"
```
```  2210     using \<open>l 0 \<le> x\<close> \<open>x \<le> u 0\<close> local[of x] by auto
```
```  2211
```
```  2212   show "P a b"
```
```  2213   proof (rule ccontr)
```
```  2214     assume "\<not> P a b"
```
```  2215     { fix n have "\<not> P (l n) (u n)"
```
```  2216       proof (induct n)
```
```  2217         case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
```
```  2218       qed (simp add: \<open>\<not> P a b\<close>) }
```
```  2219     moreover
```
```  2220     { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
```
```  2221         using \<open>0 < d\<close> \<open>l \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
```
```  2222       moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
```
```  2223         using \<open>0 < d\<close> \<open>u \<longlonglongrightarrow> x\<close> by (intro order_tendstoD[of _ x]) auto
```
```  2224       ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
```
```  2225       proof eventually_elim
```
```  2226         fix n assume "x - d / 2 < l n" "u n < x + d / 2"
```
```  2227         from add_strict_mono[OF this] have "u n - l n < d" by simp
```
```  2228         with x show "P (l n) (u n)" by (rule d)
```
```  2229       qed }
```
```  2230     ultimately show False by simp
```
```  2231   qed
```
```  2232 qed
```
```  2233
```
```  2234 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
```
```  2235 proof (cases "a \<le> b", rule compactI)
```
```  2236   fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
```
```  2237   def T == "{a .. b}"
```
```  2238   from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
```
```  2239   proof (induct rule: Bolzano)
```
```  2240     case (trans a b c)
```
```  2241     then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
```
```  2242     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"
```
```  2243       by (auto simp: *)
```
```  2244     with trans show ?case
```
```  2245       unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
```
```  2246   next
```
```  2247     case (local x)
```
```  2248     then have "x \<in> \<Union>C" using C by auto
```
```  2249     with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
```
```  2250     then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
```
```  2251       by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
```
```  2252     with \<open>c \<in> C\<close> show ?case
```
```  2253       by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
```
```  2254   qed
```
```  2255 qed simp
```
```  2256
```
```  2257
```
```  2258 lemma continuous_image_closed_interval:
```
```  2259   fixes a b and f :: "real \<Rightarrow> real"
```
```  2260   defines "S \<equiv> {a..b}"
```
```  2261   assumes "a \<le> b" and f: "continuous_on S f"
```
```  2262   shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
```
```  2263 proof -
```
```  2264   have S: "compact S" "S \<noteq> {}"
```
```  2265     using \<open>a \<le> b\<close> by (auto simp: S_def)
```
```  2266   obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
```
```  2267     using continuous_attains_sup[OF S f] by auto
```
```  2268   moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
```
```  2269     using continuous_attains_inf[OF S f] by auto
```
```  2270   moreover have "connected (f`S)"
```
```  2271     using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
```
```  2272   ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
```
```  2273     by (auto simp: connected_iff_interval)
```
```  2274   then show ?thesis
```
```  2275     by auto
```
```  2276 qed
```
```  2277
```
```  2278 lemma open_Collect_positive:
```
```  2279  fixes f :: "'a::t2_space \<Rightarrow> real"
```
```  2280  assumes f: "continuous_on s f"
```
```  2281  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. 0 < f x}"
```
```  2282  using continuous_on_open_invariant[THEN iffD1, OF f, rule_format, of "{0 <..}"]
```
```  2283  by (auto simp: Int_def field_simps)
```
```  2284
```
```  2285 lemma open_Collect_less_Int:
```
```  2286  fixes f g :: "'a::t2_space \<Rightarrow> real"
```
```  2287  assumes f: "continuous_on s f" and g: "continuous_on s g"
```
```  2288  shows "\<exists>A. open A \<and> A \<inter> s = {x\<in>s. f x < g x}"
```
```  2289  using open_Collect_positive[OF continuous_on_diff[OF g f]] by (simp add: field_simps)
```
```  2290
```
```  2291
```
```  2292 subsection \<open>Boundedness of continuous functions\<close>
```
```  2293
```
```  2294 text\<open>By bisection, function continuous on closed interval is bounded above\<close>
```
```  2295
```
```  2296 lemma isCont_eq_Ub:
```
```  2297   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  2298   shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  2299     \<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)"
```
```  2300   using continuous_attains_sup[of "{a .. b}" f]
```
```  2301   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  2302
```
```  2303 lemma isCont_eq_Lb:
```
```  2304   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  2305   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  2306     \<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)"
```
```  2307   using continuous_attains_inf[of "{a .. b}" f]
```
```  2308   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  2309
```
```  2310 lemma isCont_bounded:
```
```  2311   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  2312   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
```
```  2313   using isCont_eq_Ub[of a b f] by auto
```
```  2314
```
```  2315 lemma isCont_has_Ub:
```
```  2316   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  2317   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  2318     \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
```
```  2319   using isCont_eq_Ub[of a b f] by auto
```
```  2320
```
```  2321 (*HOL style here: object-level formulations*)
```
```  2322 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```  2323       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  2324       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  2325   by (blast intro: IVT)
```
```  2326
```
```  2327 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```  2328       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  2329       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  2330   by (blast intro: IVT2)
```
```  2331
```
```  2332 lemma isCont_Lb_Ub:
```
```  2333   fixes f :: "real \<Rightarrow> real"
```
```  2334   assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  2335   shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
```
```  2336                (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
```
```  2337 proof -
```
```  2338   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"
```
```  2339     using isCont_eq_Ub[OF assms] by auto
```
```  2340   obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
```
```  2341     using isCont_eq_Lb[OF assms] by auto
```
```  2342   show ?thesis
```
```  2343     using IVT[of f L _ M] IVT2[of f L _ M] M L assms
```
```  2344     apply (rule_tac x="f L" in exI)
```
```  2345     apply (rule_tac x="f M" in exI)
```
```  2346     apply (cases "L \<le> M")
```
```  2347     apply (simp, metis order_trans)
```
```  2348     apply (simp, metis order_trans)
```
```  2349     done
```
```  2350 qed
```
```  2351
```
```  2352
```
```  2353 text\<open>Continuity of inverse function\<close>
```
```  2354
```
```  2355 lemma isCont_inverse_function:
```
```  2356   fixes f g :: "real \<Rightarrow> real"
```
```  2357   assumes d: "0 < d"
```
```  2358       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
```
```  2359       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
```
```  2360   shows "isCont g (f x)"
```
```  2361 proof -
```
```  2362   let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
```
```  2363
```
```  2364   have f: "continuous_on ?D f"
```
```  2365     using cont by (intro continuous_at_imp_continuous_on ballI) auto
```
```  2366   then have g: "continuous_on (f`?D) g"
```
```  2367     using inj by (intro continuous_on_inv) auto
```
```  2368
```
```  2369   from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
```
```  2370     by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
```
```  2371   with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
```
```  2372     by (rule continuous_on_subset)
```
```  2373   moreover
```
```  2374   have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
```
```  2375     using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
```
```  2376   then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
```
```  2377     by auto
```
```  2378   ultimately
```
```  2379   show ?thesis
```
```  2380     by (simp add: continuous_on_eq_continuous_at)
```
```  2381 qed
```
```  2382
```
```  2383 lemma isCont_inverse_function2:
```
```  2384   fixes f g :: "real \<Rightarrow> real" shows
```
```  2385   "\<lbrakk>a < x; x < b;
```
```  2386     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
```
```  2387     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
```
```  2388    \<Longrightarrow> isCont g (f x)"
```
```  2389 apply (rule isCont_inverse_function
```
```  2390        [where f=f and d="min (x - a) (b - x)"])
```
```  2391 apply (simp_all add: abs_le_iff)
```
```  2392 done
```
```  2393
```
```  2394 (* need to rename second isCont_inverse *)
```
```  2395
```
```  2396 lemma isCont_inv_fun:
```
```  2397   fixes f g :: "real \<Rightarrow> real"
```
```  2398   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  2399          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  2400       ==> isCont g (f x)"
```
```  2401 by (rule isCont_inverse_function)
```
```  2402
```
```  2403 text\<open>Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110\<close>
```
```  2404 lemma LIM_fun_gt_zero:
```
```  2405   fixes f :: "real \<Rightarrow> real"
```
```  2406   shows "f -- c --> 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)"
```
```  2407 apply (drule (1) LIM_D, clarify)
```
```  2408 apply (rule_tac x = s in exI)
```
```  2409 apply (simp add: abs_less_iff)
```
```  2410 done
```
```  2411
```
```  2412 lemma LIM_fun_less_zero:
```
```  2413   fixes f :: "real \<Rightarrow> real"
```
```  2414   shows "f -- c --> 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)"
```
```  2415 apply (drule LIM_D [where r="-l"], simp, clarify)
```
```  2416 apply (rule_tac x = s in exI)
```
```  2417 apply (simp add: abs_less_iff)
```
```  2418 done
```
```  2419
```
```  2420 lemma LIM_fun_not_zero:
```
```  2421   fixes f :: "real \<Rightarrow> real"
```
```  2422   shows "f -- c --> 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)"
```
```  2423   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
```
```  2424
```
```  2425 end
```
```  2426
```