src/HOL/Limits.thy
 author haftmann Fri Jun 19 07:53:35 2015 +0200 (2015-06-19) changeset 60517 f16e4fb20652 parent 60182 e1ea5a6379c9 child 60721 c1b7793c23a3 permissions -rw-r--r--
separate class for notions specific for integral (semi)domains, in contrast to fields where these are trivial
```     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 {* Limits on Real Vector Spaces *}
```
```     9
```
```    10 theory Limits
```
```    11 imports Real_Vector_Spaces
```
```    12 begin
```
```    13
```
```    14 subsection {* Filter going to infinity norm *}
```
```    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 \<Colon> 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 ---> l) at_infinity \<Longrightarrow> ((\<lambda>n. f(n)) ---> l) sequentially"
```
```    47 by (simp add: filterlim_at_top_imp_at_infinity filterlim_compose filterlim_real_sequentially)
```
```    48
```
```    49
```
```    50 subsubsection {* Boundedness *}
```
```    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 `0 < K` 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_elim1, 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 fast
```
```    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 {* Bounded Sequences *}
```
```   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_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
```
```   141 proof (elim BseqE, intro bdd_belowI2)
```
```   142   fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
```
```   143     by (auto elim!: allE[of _ n])
```
```   144 qed
```
```   145
```
```   146 lemma lemma_NBseq_def:
```
```   147   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   148 proof safe
```
```   149   fix K :: real
```
```   150   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
```
```   151   then have "K \<le> real (Suc n)" by auto
```
```   152   moreover assume "\<forall>m. norm (X m) \<le> K"
```
```   153   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
```
```   154     by (blast intro: order_trans)
```
```   155   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
```
```   156 qed (force simp add: real_of_nat_Suc)
```
```   157
```
```   158 text{* alternative definition for Bseq *}
```
```   159 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   160 apply (simp add: Bseq_def)
```
```   161 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   162 done
```
```   163
```
```   164 lemma lemma_NBseq_def2:
```
```   165      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   166 apply (subst lemma_NBseq_def, auto)
```
```   167 apply (rule_tac x = "Suc N" in exI)
```
```   168 apply (rule_tac [2] x = N in exI)
```
```   169 apply (auto simp add: real_of_nat_Suc)
```
```   170  prefer 2 apply (blast intro: order_less_imp_le)
```
```   171 apply (drule_tac x = n in spec, simp)
```
```   172 done
```
```   173
```
```   174 (* yet another definition for Bseq *)
```
```   175 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   176 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   177
```
```   178 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```   179
```
```   180 text{*alternative formulation for boundedness*}
```
```   181 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```   182 apply (unfold Bseq_def, safe)
```
```   183 apply (rule_tac [2] x = "k + norm x" in exI)
```
```   184 apply (rule_tac x = K in exI, simp)
```
```   185 apply (rule exI [where x = 0], auto)
```
```   186 apply (erule order_less_le_trans, simp)
```
```   187 apply (drule_tac x=n in spec)
```
```   188 apply (drule order_trans [OF norm_triangle_ineq2])
```
```   189 apply simp
```
```   190 done
```
```   191
```
```   192 text{*alternative formulation for boundedness*}
```
```   193 lemma Bseq_iff3:
```
```   194   "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
```
```   195 proof
```
```   196   assume ?P
```
```   197   then obtain K
```
```   198     where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
```
```   199   from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
```
```   200   from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
```
```   201     by (auto intro: order_trans norm_triangle_ineq4)
```
```   202   then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
```
```   203     by simp
```
```   204   with `0 < K + norm (X 0)` show ?Q by blast
```
```   205 next
```
```   206   assume ?Q then show ?P by (auto simp add: Bseq_iff2)
```
```   207 qed
```
```   208
```
```   209 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```   210 apply (simp add: Bseq_def)
```
```   211 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```   212 apply (drule_tac x = n in spec, arith)
```
```   213 done
```
```   214
```
```   215
```
```   216 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   217
```
```   218 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
```
```   219   by (simp add: Bseq_def)
```
```   220
```
```   221 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
```
```   222   apply (simp add: subset_eq)
```
```   223   apply (rule BseqI'[where K="max (norm a) (norm b)"])
```
```   224   apply (erule_tac x=n in allE)
```
```   225   apply auto
```
```   226   done
```
```   227
```
```   228 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
```
```   229   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
```
```   230
```
```   231 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
```
```   232   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
```
```   233
```
```   234 subsection {* Bounded Monotonic Sequences *}
```
```   235
```
```   236 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   237
```
```   238 (* TODO: delete *)
```
```   239 (* FIXME: one use in NSA/HSEQ.thy *)
```
```   240 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   241   apply (rule_tac x="X m" in exI)
```
```   242   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
```
```   243   unfolding eventually_sequentially
```
```   244   apply blast
```
```   245   done
```
```   246
```
```   247 subsection {* Convergence to Zero *}
```
```   248
```
```   249 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   250   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```   251
```
```   252 lemma ZfunI:
```
```   253   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```   254   unfolding Zfun_def by simp
```
```   255
```
```   256 lemma ZfunD:
```
```   257   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```   258   unfolding Zfun_def by simp
```
```   259
```
```   260 lemma Zfun_ssubst:
```
```   261   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```   262   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   263
```
```   264 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```   265   unfolding Zfun_def by simp
```
```   266
```
```   267 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   268   unfolding Zfun_def by simp
```
```   269
```
```   270 lemma Zfun_imp_Zfun:
```
```   271   assumes f: "Zfun f F"
```
```   272   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   273   shows "Zfun (\<lambda>x. g x) F"
```
```   274 proof (cases)
```
```   275   assume K: "0 < K"
```
```   276   show ?thesis
```
```   277   proof (rule ZfunI)
```
```   278     fix r::real assume "0 < r"
```
```   279     hence "0 < r / K" using K by simp
```
```   280     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   281       using ZfunD [OF f] by fast
```
```   282     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   283     proof eventually_elim
```
```   284       case (elim x)
```
```   285       hence "norm (f x) * K < r"
```
```   286         by (simp add: pos_less_divide_eq K)
```
```   287       thus ?case
```
```   288         by (simp add: order_le_less_trans [OF elim(1)])
```
```   289     qed
```
```   290   qed
```
```   291 next
```
```   292   assume "\<not> 0 < K"
```
```   293   hence K: "K \<le> 0" by (simp only: not_less)
```
```   294   show ?thesis
```
```   295   proof (rule ZfunI)
```
```   296     fix r :: real
```
```   297     assume "0 < r"
```
```   298     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   299     proof eventually_elim
```
```   300       case (elim x)
```
```   301       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   302         using K norm_ge_zero by (rule mult_left_mono)
```
```   303       finally show ?case
```
```   304         using `0 < r` by simp
```
```   305     qed
```
```   306   qed
```
```   307 qed
```
```   308
```
```   309 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   310   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   311
```
```   312 lemma Zfun_add:
```
```   313   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   314   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   315 proof (rule ZfunI)
```
```   316   fix r::real assume "0 < r"
```
```   317   hence r: "0 < r / 2" by simp
```
```   318   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   319     using f r by (rule ZfunD)
```
```   320   moreover
```
```   321   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   322     using g r by (rule ZfunD)
```
```   323   ultimately
```
```   324   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   325   proof eventually_elim
```
```   326     case (elim x)
```
```   327     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   328       by (rule norm_triangle_ineq)
```
```   329     also have "\<dots> < r/2 + r/2"
```
```   330       using elim by (rule add_strict_mono)
```
```   331     finally show ?case
```
```   332       by simp
```
```   333   qed
```
```   334 qed
```
```   335
```
```   336 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   337   unfolding Zfun_def by simp
```
```   338
```
```   339 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   340   using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
```
```   341
```
```   342 lemma (in bounded_linear) Zfun:
```
```   343   assumes g: "Zfun g F"
```
```   344   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   345 proof -
```
```   346   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   347     using bounded by fast
```
```   348   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   349     by simp
```
```   350   with g show ?thesis
```
```   351     by (rule Zfun_imp_Zfun)
```
```   352 qed
```
```   353
```
```   354 lemma (in bounded_bilinear) Zfun:
```
```   355   assumes f: "Zfun f F"
```
```   356   assumes g: "Zfun g F"
```
```   357   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   358 proof (rule ZfunI)
```
```   359   fix r::real assume r: "0 < r"
```
```   360   obtain K where K: "0 < K"
```
```   361     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   362     using pos_bounded by fast
```
```   363   from K have K': "0 < inverse K"
```
```   364     by (rule positive_imp_inverse_positive)
```
```   365   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   366     using f r by (rule ZfunD)
```
```   367   moreover
```
```   368   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   369     using g K' by (rule ZfunD)
```
```   370   ultimately
```
```   371   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   372   proof eventually_elim
```
```   373     case (elim x)
```
```   374     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   375       by (rule norm_le)
```
```   376     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   377       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   378     also from K have "r * inverse K * K = r"
```
```   379       by simp
```
```   380     finally show ?case .
```
```   381   qed
```
```   382 qed
```
```   383
```
```   384 lemma (in bounded_bilinear) Zfun_left:
```
```   385   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   386   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   387
```
```   388 lemma (in bounded_bilinear) Zfun_right:
```
```   389   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   390   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   391
```
```   392 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   393 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   394 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   395
```
```   396 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   397   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   398
```
```   399 lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk>
```
```   400                      \<Longrightarrow> (g ---> 0) F"
```
```   401   by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
```
```   402
```
```   403 subsubsection {* Distance and norms *}
```
```   404
```
```   405 lemma tendsto_dist [tendsto_intros]:
```
```   406   fixes l m :: "'a :: metric_space"
```
```   407   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
```
```   408   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
```
```   409 proof (rule tendstoI)
```
```   410   fix e :: real assume "0 < e"
```
```   411   hence e2: "0 < e/2" by simp
```
```   412   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   413   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
```
```   414   proof (eventually_elim)
```
```   415     case (elim x)
```
```   416     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   417       unfolding dist_real_def
```
```   418       using dist_triangle2 [of "f x" "g x" "l"]
```
```   419       using dist_triangle2 [of "g x" "l" "m"]
```
```   420       using dist_triangle3 [of "l" "m" "f x"]
```
```   421       using dist_triangle [of "f x" "m" "g x"]
```
```   422       by arith
```
```   423   qed
```
```   424 qed
```
```   425
```
```   426 lemma continuous_dist[continuous_intros]:
```
```   427   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   428   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
```
```   429   unfolding continuous_def by (rule tendsto_dist)
```
```   430
```
```   431 lemma continuous_on_dist[continuous_intros]:
```
```   432   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   433   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
```
```   434   unfolding continuous_on_def by (auto intro: tendsto_dist)
```
```   435
```
```   436 lemma tendsto_norm [tendsto_intros]:
```
```   437   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
```
```   438   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   439
```
```   440 lemma continuous_norm [continuous_intros]:
```
```   441   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
```
```   442   unfolding continuous_def by (rule tendsto_norm)
```
```   443
```
```   444 lemma continuous_on_norm [continuous_intros]:
```
```   445   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
```
```   446   unfolding continuous_on_def by (auto intro: tendsto_norm)
```
```   447
```
```   448 lemma tendsto_norm_zero:
```
```   449   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
```
```   450   by (drule tendsto_norm, simp)
```
```   451
```
```   452 lemma tendsto_norm_zero_cancel:
```
```   453   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
```
```   454   unfolding tendsto_iff dist_norm by simp
```
```   455
```
```   456 lemma tendsto_norm_zero_iff:
```
```   457   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
```
```   458   unfolding tendsto_iff dist_norm by simp
```
```   459
```
```   460 lemma tendsto_rabs [tendsto_intros]:
```
```   461   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
```
```   462   by (fold real_norm_def, rule tendsto_norm)
```
```   463
```
```   464 lemma continuous_rabs [continuous_intros]:
```
```   465   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   466   unfolding real_norm_def[symmetric] by (rule continuous_norm)
```
```   467
```
```   468 lemma continuous_on_rabs [continuous_intros]:
```
```   469   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   470   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
```
```   471
```
```   472 lemma tendsto_rabs_zero:
```
```   473   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
```
```   474   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   475
```
```   476 lemma tendsto_rabs_zero_cancel:
```
```   477   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
```
```   478   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   479
```
```   480 lemma tendsto_rabs_zero_iff:
```
```   481   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
```
```   482   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   483
```
```   484 subsubsection {* Addition and subtraction *}
```
```   485
```
```   486 lemma tendsto_add [tendsto_intros]:
```
```   487   fixes a b :: "'a::real_normed_vector"
```
```   488   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
```
```   489   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   490
```
```   491 lemma continuous_add [continuous_intros]:
```
```   492   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   493   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
```
```   494   unfolding continuous_def by (rule tendsto_add)
```
```   495
```
```   496 lemma continuous_on_add [continuous_intros]:
```
```   497   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   498   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
```
```   499   unfolding continuous_on_def by (auto intro: tendsto_add)
```
```   500
```
```   501 lemma tendsto_add_zero:
```
```   502   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   503   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
```
```   504   by (drule (1) tendsto_add, simp)
```
```   505
```
```   506 lemma tendsto_minus [tendsto_intros]:
```
```   507   fixes a :: "'a::real_normed_vector"
```
```   508   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
```
```   509   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   510
```
```   511 lemma continuous_minus [continuous_intros]:
```
```   512   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   513   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
```
```   514   unfolding continuous_def by (rule tendsto_minus)
```
```   515
```
```   516 lemma continuous_on_minus [continuous_intros]:
```
```   517   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   518   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
```
```   519   unfolding continuous_on_def by (auto intro: tendsto_minus)
```
```   520
```
```   521 lemma tendsto_minus_cancel:
```
```   522   fixes a :: "'a::real_normed_vector"
```
```   523   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
```
```   524   by (drule tendsto_minus, simp)
```
```   525
```
```   526 lemma tendsto_minus_cancel_left:
```
```   527     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
```
```   528   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
```
```   529   by auto
```
```   530
```
```   531 lemma tendsto_diff [tendsto_intros]:
```
```   532   fixes a b :: "'a::real_normed_vector"
```
```   533   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
```
```   534   using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
```
```   535
```
```   536 lemma continuous_diff [continuous_intros]:
```
```   537   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   538   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
```
```   539   unfolding continuous_def by (rule tendsto_diff)
```
```   540
```
```   541 lemma continuous_on_diff [continuous_intros]:
```
```   542   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   543   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
```
```   544   unfolding continuous_on_def by (auto intro: tendsto_diff)
```
```   545
```
```   546 lemma tendsto_setsum [tendsto_intros]:
```
```   547   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   548   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
```
```   549   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
```
```   550 proof (cases "finite S")
```
```   551   assume "finite S" thus ?thesis using assms
```
```   552     by (induct, simp, simp add: tendsto_add)
```
```   553 qed simp
```
```   554
```
```   555 lemma continuous_setsum [continuous_intros]:
```
```   556   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```   557   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
```
```   558   unfolding continuous_def by (rule tendsto_setsum)
```
```   559
```
```   560 lemma continuous_on_setsum [continuous_intros]:
```
```   561   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
```
```   562   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)"
```
```   563   unfolding continuous_on_def by (auto intro: tendsto_setsum)
```
```   564
```
```   565 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
```
```   566
```
```   567 subsubsection {* Linear operators and multiplication *}
```
```   568
```
```   569 lemma (in bounded_linear) tendsto:
```
```   570   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
```
```   571   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   572
```
```   573 lemma (in bounded_linear) continuous:
```
```   574   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
```
```   575   using tendsto[of g _ F] by (auto simp: continuous_def)
```
```   576
```
```   577 lemma (in bounded_linear) continuous_on:
```
```   578   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
```
```   579   using tendsto[of g] by (auto simp: continuous_on_def)
```
```   580
```
```   581 lemma (in bounded_linear) tendsto_zero:
```
```   582   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
```
```   583   by (drule tendsto, simp only: zero)
```
```   584
```
```   585 lemma (in bounded_bilinear) tendsto:
```
```   586   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
```
```   587   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   588                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   589
```
```   590 lemma (in bounded_bilinear) continuous:
```
```   591   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
```
```   592   using tendsto[of f _ F g] by (auto simp: continuous_def)
```
```   593
```
```   594 lemma (in bounded_bilinear) continuous_on:
```
```   595   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
```
```   596   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
```
```   597
```
```   598 lemma (in bounded_bilinear) tendsto_zero:
```
```   599   assumes f: "(f ---> 0) F"
```
```   600   assumes g: "(g ---> 0) F"
```
```   601   shows "((\<lambda>x. f x ** g x) ---> 0) F"
```
```   602   using tendsto [OF f g] by (simp add: zero_left)
```
```   603
```
```   604 lemma (in bounded_bilinear) tendsto_left_zero:
```
```   605   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
```
```   606   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```   607
```
```   608 lemma (in bounded_bilinear) tendsto_right_zero:
```
```   609   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
```
```   610   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```   611
```
```   612 lemmas tendsto_of_real [tendsto_intros] =
```
```   613   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```   614
```
```   615 lemmas tendsto_scaleR [tendsto_intros] =
```
```   616   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```   617
```
```   618 lemmas tendsto_mult [tendsto_intros] =
```
```   619   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```   620
```
```   621 lemmas continuous_of_real [continuous_intros] =
```
```   622   bounded_linear.continuous [OF bounded_linear_of_real]
```
```   623
```
```   624 lemmas continuous_scaleR [continuous_intros] =
```
```   625   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
```
```   626
```
```   627 lemmas continuous_mult [continuous_intros] =
```
```   628   bounded_bilinear.continuous [OF bounded_bilinear_mult]
```
```   629
```
```   630 lemmas continuous_on_of_real [continuous_intros] =
```
```   631   bounded_linear.continuous_on [OF bounded_linear_of_real]
```
```   632
```
```   633 lemmas continuous_on_scaleR [continuous_intros] =
```
```   634   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
```
```   635
```
```   636 lemmas continuous_on_mult [continuous_intros] =
```
```   637   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
```
```   638
```
```   639 lemmas tendsto_mult_zero =
```
```   640   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```   641
```
```   642 lemmas tendsto_mult_left_zero =
```
```   643   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```   644
```
```   645 lemmas tendsto_mult_right_zero =
```
```   646   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```   647
```
```   648 lemma tendsto_power [tendsto_intros]:
```
```   649   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   650   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
```
```   651   by (induct n) (simp_all add: tendsto_mult)
```
```   652
```
```   653 lemma continuous_power [continuous_intros]:
```
```   654   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   655   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
```
```   656   unfolding continuous_def by (rule tendsto_power)
```
```   657
```
```   658 lemma continuous_on_power [continuous_intros]:
```
```   659   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   660   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
```
```   661   unfolding continuous_on_def by (auto intro: tendsto_power)
```
```   662
```
```   663 lemma tendsto_setprod [tendsto_intros]:
```
```   664   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   665   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
```
```   666   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
```
```   667 proof (cases "finite S")
```
```   668   assume "finite S" thus ?thesis using assms
```
```   669     by (induct, simp, simp add: tendsto_mult)
```
```   670 qed simp
```
```   671
```
```   672 lemma continuous_setprod [continuous_intros]:
```
```   673   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   674   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
```
```   675   unfolding continuous_def by (rule tendsto_setprod)
```
```   676
```
```   677 lemma continuous_on_setprod [continuous_intros]:
```
```   678   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   679   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)"
```
```   680   unfolding continuous_on_def by (auto intro: tendsto_setprod)
```
```   681
```
```   682 subsubsection {* Inverse and division *}
```
```   683
```
```   684 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   685   assumes f: "Zfun f F"
```
```   686   assumes g: "Bfun g F"
```
```   687   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   688 proof -
```
```   689   obtain K where K: "0 \<le> K"
```
```   690     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   691     using nonneg_bounded by fast
```
```   692   obtain B where B: "0 < B"
```
```   693     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```   694     using g by (rule BfunE)
```
```   695   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```   696   using norm_g proof eventually_elim
```
```   697     case (elim x)
```
```   698     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   699       by (rule norm_le)
```
```   700     also have "\<dots> \<le> norm (f x) * B * K"
```
```   701       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   702                 mult_nonneg_nonneg K elim)
```
```   703     also have "\<dots> = norm (f x) * (B * K)"
```
```   704       by (rule mult.assoc)
```
```   705     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   706   qed
```
```   707   with f show ?thesis
```
```   708     by (rule Zfun_imp_Zfun)
```
```   709 qed
```
```   710
```
```   711 lemma (in bounded_bilinear) flip:
```
```   712   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   713   apply default
```
```   714   apply (rule add_right)
```
```   715   apply (rule add_left)
```
```   716   apply (rule scaleR_right)
```
```   717   apply (rule scaleR_left)
```
```   718   apply (subst mult.commute)
```
```   719   using bounded by fast
```
```   720
```
```   721 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   722   assumes f: "Bfun f F"
```
```   723   assumes g: "Zfun g F"
```
```   724   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   725   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   726
```
```   727 lemma Bfun_inverse_lemma:
```
```   728   fixes x :: "'a::real_normed_div_algebra"
```
```   729   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   730   apply (subst nonzero_norm_inverse, clarsimp)
```
```   731   apply (erule (1) le_imp_inverse_le)
```
```   732   done
```
```   733
```
```   734 lemma Bfun_inverse:
```
```   735   fixes a :: "'a::real_normed_div_algebra"
```
```   736   assumes f: "(f ---> a) F"
```
```   737   assumes a: "a \<noteq> 0"
```
```   738   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```   739 proof -
```
```   740   from a have "0 < norm a" by simp
```
```   741   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   742   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   743   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```   744     using tendstoD [OF f r1] by fast
```
```   745   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```   746   proof eventually_elim
```
```   747     case (elim x)
```
```   748     hence 1: "norm (f x - a) < r"
```
```   749       by (simp add: dist_norm)
```
```   750     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   751     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   752       by (rule nonzero_norm_inverse)
```
```   753     also have "\<dots> \<le> inverse (norm a - r)"
```
```   754     proof (rule le_imp_inverse_le)
```
```   755       show "0 < norm a - r" using r2 by simp
```
```   756     next
```
```   757       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   758         by (rule norm_triangle_ineq2)
```
```   759       also have "\<dots> = norm (f x - a)"
```
```   760         by (rule norm_minus_commute)
```
```   761       also have "\<dots> < r" using 1 .
```
```   762       finally show "norm a - r \<le> norm (f x)" by simp
```
```   763     qed
```
```   764     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   765   qed
```
```   766   thus ?thesis by (rule BfunI)
```
```   767 qed
```
```   768
```
```   769 lemma tendsto_inverse [tendsto_intros]:
```
```   770   fixes a :: "'a::real_normed_div_algebra"
```
```   771   assumes f: "(f ---> a) F"
```
```   772   assumes a: "a \<noteq> 0"
```
```   773   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
```
```   774 proof -
```
```   775   from a have "0 < norm a" by simp
```
```   776   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```   777     by (rule tendstoD)
```
```   778   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```   779     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   780   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```   781     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   782     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
```
```   783   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   784     by (intro Zfun_minus Zfun_mult_left
```
```   785       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```   786       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```   787   ultimately show ?thesis
```
```   788     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```   789 qed
```
```   790
```
```   791 lemma continuous_inverse:
```
```   792   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   793   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   794   shows "continuous F (\<lambda>x. inverse (f x))"
```
```   795   using assms unfolding continuous_def by (rule tendsto_inverse)
```
```   796
```
```   797 lemma continuous_at_within_inverse[continuous_intros]:
```
```   798   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   799   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   800   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
```
```   801   using assms unfolding continuous_within by (rule tendsto_inverse)
```
```   802
```
```   803 lemma isCont_inverse[continuous_intros, simp]:
```
```   804   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   805   assumes "isCont f a" and "f a \<noteq> 0"
```
```   806   shows "isCont (\<lambda>x. inverse (f x)) a"
```
```   807   using assms unfolding continuous_at by (rule tendsto_inverse)
```
```   808
```
```   809 lemma continuous_on_inverse[continuous_intros]:
```
```   810   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   811   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   812   shows "continuous_on s (\<lambda>x. inverse (f x))"
```
```   813   using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
```
```   814
```
```   815 lemma tendsto_divide [tendsto_intros]:
```
```   816   fixes a b :: "'a::real_normed_field"
```
```   817   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
```
```   818     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
```
```   819   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```   820
```
```   821 lemma continuous_divide:
```
```   822   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   823   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   824   shows "continuous F (\<lambda>x. (f x) / (g x))"
```
```   825   using assms unfolding continuous_def by (rule tendsto_divide)
```
```   826
```
```   827 lemma continuous_at_within_divide[continuous_intros]:
```
```   828   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   829   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
```
```   830   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
```
```   831   using assms unfolding continuous_within by (rule tendsto_divide)
```
```   832
```
```   833 lemma isCont_divide[continuous_intros, simp]:
```
```   834   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   835   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
```
```   836   shows "isCont (\<lambda>x. (f x) / g x) a"
```
```   837   using assms unfolding continuous_at by (rule tendsto_divide)
```
```   838
```
```   839 lemma continuous_on_divide[continuous_intros]:
```
```   840   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   841   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
```
```   842   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
```
```   843   using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
```
```   844
```
```   845 lemma tendsto_sgn [tendsto_intros]:
```
```   846   fixes l :: "'a::real_normed_vector"
```
```   847   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
```
```   848   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```   849
```
```   850 lemma continuous_sgn:
```
```   851   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   852   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   853   shows "continuous F (\<lambda>x. sgn (f x))"
```
```   854   using assms unfolding continuous_def by (rule tendsto_sgn)
```
```   855
```
```   856 lemma continuous_at_within_sgn[continuous_intros]:
```
```   857   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   858   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   859   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
```
```   860   using assms unfolding continuous_within by (rule tendsto_sgn)
```
```   861
```
```   862 lemma isCont_sgn[continuous_intros]:
```
```   863   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   864   assumes "isCont f a" and "f a \<noteq> 0"
```
```   865   shows "isCont (\<lambda>x. sgn (f x)) a"
```
```   866   using assms unfolding continuous_at by (rule tendsto_sgn)
```
```   867
```
```   868 lemma continuous_on_sgn[continuous_intros]:
```
```   869   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   870   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   871   shows "continuous_on s (\<lambda>x. sgn (f x))"
```
```   872   using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
```
```   873
```
```   874 lemma filterlim_at_infinity:
```
```   875   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
```
```   876   assumes "0 \<le> c"
```
```   877   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```   878   unfolding filterlim_iff eventually_at_infinity
```
```   879 proof safe
```
```   880   fix P :: "'a \<Rightarrow> bool" and b
```
```   881   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```   882     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```   883   have "max b (c + 1) > c" by auto
```
```   884   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```   885     by auto
```
```   886   then show "eventually (\<lambda>x. P (f x)) F"
```
```   887   proof eventually_elim
```
```   888     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```   889     with P show "P (f x)" by auto
```
```   890   qed
```
```   891 qed force
```
```   892
```
```   893 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
```
```   894
```
```   895 text {*
```
```   896
```
```   897 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
```
```   898 @{term "at_right x"} and also @{term "at_right 0"}.
```
```   899
```
```   900 *}
```
```   901
```
```   902 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
```
```   903
```
```   904 lemma filtermap_homeomorph:
```
```   905   assumes f: "continuous (at a) f"
```
```   906   assumes g: "continuous (at (f a)) g"
```
```   907   assumes bij1: "\<forall>x. f (g x) = x" and bij2: "\<forall>x. g (f x) = x"
```
```   908   shows "filtermap f (nhds a) = nhds (f a)"
```
```   909   unfolding filter_eq_iff eventually_filtermap eventually_nhds
```
```   910 proof safe
```
```   911   fix P S assume S: "open S" "f a \<in> S" and P: "\<forall>x\<in>S. P x"
```
```   912   from continuous_within_topological[THEN iffD1, rule_format, OF f S] P
```
```   913   show "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P (f x))" by auto
```
```   914 next
```
```   915   fix P S assume S: "open S" "a \<in> S" and P: "\<forall>x\<in>S. P (f x)"
```
```   916   with continuous_within_topological[THEN iffD1, rule_format, OF g, of S] bij2
```
```   917   obtain A where "open A" "f a \<in> A" "(\<forall>y\<in>A. g y \<in> S)"
```
```   918     by (metis UNIV_I)
```
```   919   with P bij1 show "\<exists>S. open S \<and> f a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   920     by (force intro!: exI[of _ A])
```
```   921 qed
```
```   922
```
```   923 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
```
```   924   by (rule filtermap_homeomorph[where g="\<lambda>x. x + d"]) (auto intro: continuous_intros)
```
```   925
```
```   926 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
```
```   927   by (rule filtermap_homeomorph[where g=uminus]) (auto intro: continuous_minus)
```
```   928
```
```   929 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
```
```   930   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
```
```   931
```
```   932 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
```
```   933   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
```
```   934
```
```   935 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
```
```   936   using filtermap_at_right_shift[of "-a" 0] by simp
```
```   937
```
```   938 lemma filterlim_at_right_to_0:
```
```   939   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
```
```   940   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
```
```   941
```
```   942 lemma eventually_at_right_to_0:
```
```   943   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
```
```   944   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
```
```   945
```
```   946 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
```
```   947   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```   948
```
```   949 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
```
```   950   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```   951
```
```   952 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
```
```   953   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```   954
```
```   955 lemma filterlim_at_left_to_right:
```
```   956   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
```
```   957   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
```
```   958
```
```   959 lemma eventually_at_left_to_right:
```
```   960   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
```
```   961   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
```
```   962
```
```   963 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
```
```   964   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
```
```   965   by (metis le_minus_iff minus_minus)
```
```   966
```
```   967 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
```
```   968   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
```
```   969
```
```   970 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
```
```   971   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
```
```   972
```
```   973 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
```
```   974   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
```
```   975
```
```   976 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```   977   unfolding filterlim_at_top eventually_at_bot_dense
```
```   978   by (metis leI minus_less_iff order_less_asym)
```
```   979
```
```   980 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```   981   unfolding filterlim_at_bot eventually_at_top_dense
```
```   982   by (metis leI less_minus_iff order_less_asym)
```
```   983
```
```   984 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
```
```   985   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
```
```   986   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
```
```   987   by auto
```
```   988
```
```   989 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
```
```   990   unfolding filterlim_uminus_at_top by simp
```
```   991
```
```   992 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
```
```   993   unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
```
```   994 proof safe
```
```   995   fix Z :: real assume [arith]: "0 < Z"
```
```   996   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```   997     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```   998   then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
```
```   999     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```  1000 qed
```
```  1001
```
```  1002 lemma filterlim_inverse_at_top:
```
```  1003   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```  1004   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
```
```  1005      (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
```
```  1006
```
```  1007 lemma filterlim_inverse_at_bot_neg:
```
```  1008   "LIM x (at_left (0::real)). inverse x :> at_bot"
```
```  1009   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
```
```  1010
```
```  1011 lemma filterlim_inverse_at_bot:
```
```  1012   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```  1013   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
```
```  1014   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
```
```  1015
```
```  1016 lemma tendsto_inverse_0:
```
```  1017   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
```
```  1018   shows "(inverse ---> (0::'a)) at_infinity"
```
```  1019   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```  1020 proof safe
```
```  1021   fix r :: real assume "0 < r"
```
```  1022   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```  1023   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```  1024     fix x :: 'a
```
```  1025     from `0 < r` have "0 < inverse (r / 2)" by simp
```
```  1026     also assume *: "inverse (r / 2) \<le> norm x"
```
```  1027     finally show "norm (inverse x) < r"
```
```  1028       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```  1029   qed
```
```  1030 qed
```
```  1031
```
```  1032 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
```
```  1033 proof (rule antisym)
```
```  1034   have "(inverse ---> (0::real)) at_top"
```
```  1035     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
```
```  1036   then show "filtermap inverse at_top \<le> at_right (0::real)"
```
```  1037     by (simp add: le_principal eventually_filtermap eventually_gt_at_top filterlim_def at_within_def)
```
```  1038 next
```
```  1039   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
```
```  1040     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
```
```  1041   then show "at_right (0::real) \<le> filtermap inverse at_top"
```
```  1042     by (simp add: filtermap_ident filtermap_filtermap)
```
```  1043 qed
```
```  1044
```
```  1045 lemma eventually_at_right_to_top:
```
```  1046   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
```
```  1047   unfolding at_right_to_top eventually_filtermap ..
```
```  1048
```
```  1049 lemma filterlim_at_right_to_top:
```
```  1050   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
```
```  1051   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
```
```  1052
```
```  1053 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
```
```  1054   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
```
```  1055
```
```  1056 lemma eventually_at_top_to_right:
```
```  1057   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
```
```  1058   unfolding at_top_to_right eventually_filtermap ..
```
```  1059
```
```  1060 lemma filterlim_at_top_to_right:
```
```  1061   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
```
```  1062   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
```
```  1063
```
```  1064 lemma filterlim_inverse_at_infinity:
```
```  1065   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring}"
```
```  1066   shows "filterlim inverse at_infinity (at (0::'a))"
```
```  1067   unfolding filterlim_at_infinity[OF order_refl]
```
```  1068 proof safe
```
```  1069   fix r :: real assume "0 < r"
```
```  1070   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```  1071     unfolding eventually_at norm_inverse
```
```  1072     by (intro exI[of _ "inverse r"])
```
```  1073        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```  1074 qed
```
```  1075
```
```  1076 lemma filterlim_inverse_at_iff:
```
```  1077   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring}"
```
```  1078   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```  1079   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```  1080 proof
```
```  1081   assume "filtermap g F \<le> at_infinity"
```
```  1082   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```  1083     by (rule filtermap_mono)
```
```  1084   also have "\<dots> \<le> at 0"
```
```  1085     using tendsto_inverse_0[where 'a='b]
```
```  1086     by (auto intro!: exI[of _ 1]
```
```  1087              simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
```
```  1088   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```  1089 next
```
```  1090   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```  1091   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```  1092     by (rule filtermap_mono)
```
```  1093   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```  1094     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```  1095 qed
```
```  1096
```
```  1097 lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
```
```  1098  by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
```
```  1099
```
```  1100
```
```  1101 lemma at_to_infinity:
```
```  1102   fixes x :: "'a \<Colon> {real_normed_field,field}"
```
```  1103   shows "(at (0::'a)) = filtermap inverse at_infinity"
```
```  1104 proof (rule antisym)
```
```  1105   have "(inverse ---> (0::'a)) at_infinity"
```
```  1106     by (fact tendsto_inverse_0)
```
```  1107   then show "filtermap inverse at_infinity \<le> at (0::'a)"
```
```  1108     apply (simp add: le_principal eventually_filtermap eventually_at_infinity filterlim_def at_within_def)
```
```  1109     apply (rule_tac x="1" in exI, auto)
```
```  1110     done
```
```  1111 next
```
```  1112   have "filtermap inverse (filtermap inverse (at (0::'a))) \<le> filtermap inverse at_infinity"
```
```  1113     using filterlim_inverse_at_infinity unfolding filterlim_def
```
```  1114     by (rule filtermap_mono)
```
```  1115   then show "at (0::'a) \<le> filtermap inverse at_infinity"
```
```  1116     by (simp add: filtermap_ident filtermap_filtermap)
```
```  1117 qed
```
```  1118
```
```  1119 lemma lim_at_infinity_0:
```
```  1120   fixes l :: "'a :: {real_normed_field,field}"
```
```  1121   shows "(f ---> l) at_infinity \<longleftrightarrow> ((f o inverse) ---> l) (at (0::'a))"
```
```  1122 by (simp add: tendsto_compose_filtermap at_to_infinity filtermap_filtermap)
```
```  1123
```
```  1124 lemma lim_zero_infinity:
```
```  1125   fixes l :: "'a :: {real_normed_field,field}"
```
```  1126   shows "((\<lambda>x. f(1 / x)) ---> l) (at (0::'a)) \<Longrightarrow> (f ---> l) at_infinity"
```
```  1127 by (simp add: inverse_eq_divide lim_at_infinity_0 comp_def)
```
```  1128
```
```  1129
```
```  1130 text {*
```
```  1131
```
```  1132 We only show rules for multiplication and addition when the functions are either against a real
```
```  1133 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```  1134
```
```  1135 *}
```
```  1136
```
```  1137 lemma filterlim_tendsto_pos_mult_at_top:
```
```  1138   assumes f: "(f ---> c) F" and c: "0 < c"
```
```  1139   assumes g: "LIM x F. g x :> at_top"
```
```  1140   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1141   unfolding filterlim_at_top_gt[where c=0]
```
```  1142 proof safe
```
```  1143   fix Z :: real assume "0 < Z"
```
```  1144   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
```
```  1145     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
```
```  1146              simp: dist_real_def abs_real_def split: split_if_asm)
```
```  1147   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
```
```  1148     unfolding filterlim_at_top by auto
```
```  1149   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1150   proof eventually_elim
```
```  1151     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
```
```  1152     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
```
```  1153       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1154     with `0 < c` show "Z \<le> f x * g x"
```
```  1155        by simp
```
```  1156   qed
```
```  1157 qed
```
```  1158
```
```  1159 lemma filterlim_at_top_mult_at_top:
```
```  1160   assumes f: "LIM x F. f x :> at_top"
```
```  1161   assumes g: "LIM x F. g x :> at_top"
```
```  1162   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1163   unfolding filterlim_at_top_gt[where c=0]
```
```  1164 proof safe
```
```  1165   fix Z :: real assume "0 < Z"
```
```  1166   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
```
```  1167     unfolding filterlim_at_top by auto
```
```  1168   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1169     unfolding filterlim_at_top by auto
```
```  1170   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1171   proof eventually_elim
```
```  1172     fix x assume "1 \<le> f x" "Z \<le> g x"
```
```  1173     with `0 < Z` have "1 * Z \<le> f x * g x"
```
```  1174       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1175     then show "Z \<le> f x * g x"
```
```  1176        by simp
```
```  1177   qed
```
```  1178 qed
```
```  1179
```
```  1180 lemma filterlim_tendsto_pos_mult_at_bot:
```
```  1181   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
```
```  1182   shows "LIM x F. f x * g x :> at_bot"
```
```  1183   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
```
```  1184   unfolding filterlim_uminus_at_bot by simp
```
```  1185
```
```  1186 lemma filterlim_tendsto_neg_mult_at_bot:
```
```  1187   assumes c: "(f ---> c) F" "(c::real) < 0" and g: "filterlim g at_top F"
```
```  1188   shows "LIM x F. f x * g x :> at_bot"
```
```  1189   using c filterlim_tendsto_pos_mult_at_top[of "\<lambda>x. - f x" "- c" F, OF _ _ g]
```
```  1190   unfolding filterlim_uminus_at_bot tendsto_minus_cancel_left by simp
```
```  1191
```
```  1192 lemma filterlim_pow_at_top:
```
```  1193   fixes f :: "real \<Rightarrow> real"
```
```  1194   assumes "0 < n" and f: "LIM x F. f x :> at_top"
```
```  1195   shows "LIM x F. (f x)^n :: real :> at_top"
```
```  1196 using `0 < n` proof (induct n)
```
```  1197   case (Suc n) with f show ?case
```
```  1198     by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
```
```  1199 qed simp
```
```  1200
```
```  1201 lemma filterlim_pow_at_bot_even:
```
```  1202   fixes f :: "real \<Rightarrow> real"
```
```  1203   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
```
```  1204   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
```
```  1205
```
```  1206 lemma filterlim_pow_at_bot_odd:
```
```  1207   fixes f :: "real \<Rightarrow> real"
```
```  1208   shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
```
```  1209   using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
```
```  1210
```
```  1211 lemma filterlim_tendsto_add_at_top:
```
```  1212   assumes f: "(f ---> c) F"
```
```  1213   assumes g: "LIM x F. g x :> at_top"
```
```  1214   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1215   unfolding filterlim_at_top_gt[where c=0]
```
```  1216 proof safe
```
```  1217   fix Z :: real assume "0 < Z"
```
```  1218   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```  1219     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
```
```  1220   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
```
```  1221     unfolding filterlim_at_top by auto
```
```  1222   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1223     by eventually_elim simp
```
```  1224 qed
```
```  1225
```
```  1226 lemma LIM_at_top_divide:
```
```  1227   fixes f g :: "'a \<Rightarrow> real"
```
```  1228   assumes f: "(f ---> a) F" "0 < a"
```
```  1229   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
```
```  1230   shows "LIM x F. f x / g x :> at_top"
```
```  1231   unfolding divide_inverse
```
```  1232   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
```
```  1233
```
```  1234 lemma filterlim_at_top_add_at_top:
```
```  1235   assumes f: "LIM x F. f x :> at_top"
```
```  1236   assumes g: "LIM x F. g x :> at_top"
```
```  1237   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1238   unfolding filterlim_at_top_gt[where c=0]
```
```  1239 proof safe
```
```  1240   fix Z :: real assume "0 < Z"
```
```  1241   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
```
```  1242     unfolding filterlim_at_top by auto
```
```  1243   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1244     unfolding filterlim_at_top by auto
```
```  1245   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1246     by eventually_elim simp
```
```  1247 qed
```
```  1248
```
```  1249 lemma tendsto_divide_0:
```
```  1250   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring}"
```
```  1251   assumes f: "(f ---> c) F"
```
```  1252   assumes g: "LIM x F. g x :> at_infinity"
```
```  1253   shows "((\<lambda>x. f x / g x) ---> 0) F"
```
```  1254   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
```
```  1255
```
```  1256 lemma linear_plus_1_le_power:
```
```  1257   fixes x :: real
```
```  1258   assumes x: "0 \<le> x"
```
```  1259   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1260 proof (induct n)
```
```  1261   case (Suc n)
```
```  1262   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
```
```  1263     by (simp add: field_simps real_of_nat_Suc x)
```
```  1264   also have "\<dots> \<le> (x + 1)^Suc n"
```
```  1265     using Suc x by (simp add: mult_left_mono)
```
```  1266   finally show ?case .
```
```  1267 qed simp
```
```  1268
```
```  1269 lemma filterlim_realpow_sequentially_gt1:
```
```  1270   fixes x :: "'a :: real_normed_div_algebra"
```
```  1271   assumes x[arith]: "1 < norm x"
```
```  1272   shows "LIM n sequentially. x ^ n :> at_infinity"
```
```  1273 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
```
```  1274   fix y :: real assume "0 < y"
```
```  1275   have "0 < norm x - 1" by simp
```
```  1276   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
```
```  1277   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
```
```  1278   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
```
```  1279   also have "\<dots> = norm x ^ N" by simp
```
```  1280   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
```
```  1281     by (metis order_less_le_trans power_increasing order_less_imp_le x)
```
```  1282   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
```
```  1283     unfolding eventually_sequentially
```
```  1284     by (auto simp: norm_power)
```
```  1285 qed simp
```
```  1286
```
```  1287
```
```  1288 subsection {* Limits of Sequences *}
```
```  1289
```
```  1290 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
```
```  1291   by simp
```
```  1292
```
```  1293 lemma LIMSEQ_iff:
```
```  1294   fixes L :: "'a::real_normed_vector"
```
```  1295   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```  1296 unfolding lim_sequentially dist_norm ..
```
```  1297
```
```  1298 lemma LIMSEQ_I:
```
```  1299   fixes L :: "'a::real_normed_vector"
```
```  1300   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```  1301 by (simp add: LIMSEQ_iff)
```
```  1302
```
```  1303 lemma LIMSEQ_D:
```
```  1304   fixes L :: "'a::real_normed_vector"
```
```  1305   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```  1306 by (simp add: LIMSEQ_iff)
```
```  1307
```
```  1308 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```  1309   unfolding tendsto_def eventually_sequentially
```
```  1310   by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
```
```  1311
```
```  1312 lemma Bseq_inverse_lemma:
```
```  1313   fixes x :: "'a::real_normed_div_algebra"
```
```  1314   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```  1315 apply (subst nonzero_norm_inverse, clarsimp)
```
```  1316 apply (erule (1) le_imp_inverse_le)
```
```  1317 done
```
```  1318
```
```  1319 lemma Bseq_inverse:
```
```  1320   fixes a :: "'a::real_normed_div_algebra"
```
```  1321   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```  1322   by (rule Bfun_inverse)
```
```  1323
```
```  1324 text{* Transformation of limit. *}
```
```  1325
```
```  1326 lemma eventually_at2:
```
```  1327   "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
```
```  1328   unfolding eventually_at dist_nz by auto
```
```  1329
```
```  1330 lemma Lim_transform:
```
```  1331   fixes a b :: "'a::real_normed_vector"
```
```  1332   shows "\<lbrakk>(g ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (f ---> a) F"
```
```  1333   using tendsto_add [of g a F "\<lambda>x. f x - g x" 0] by simp
```
```  1334
```
```  1335 lemma Lim_transform2:
```
```  1336   fixes a b :: "'a::real_normed_vector"
```
```  1337   shows "\<lbrakk>(f ---> a) F; ((\<lambda>x. f x - g x) ---> 0) F\<rbrakk> \<Longrightarrow> (g ---> a) F"
```
```  1338   by (erule Lim_transform) (simp add: tendsto_minus_cancel)
```
```  1339
```
```  1340 lemma Lim_transform_eventually:
```
```  1341   "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
```
```  1342   apply (rule topological_tendstoI)
```
```  1343   apply (drule (2) topological_tendstoD)
```
```  1344   apply (erule (1) eventually_elim2, simp)
```
```  1345   done
```
```  1346
```
```  1347 lemma Lim_transform_within:
```
```  1348   assumes "0 < d"
```
```  1349     and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
```
```  1350     and "(f ---> l) (at x within S)"
```
```  1351   shows "(g ---> l) (at x within S)"
```
```  1352 proof (rule Lim_transform_eventually)
```
```  1353   show "eventually (\<lambda>x. f x = g x) (at x within S)"
```
```  1354     using assms(1,2) by (auto simp: dist_nz eventually_at)
```
```  1355   show "(f ---> l) (at x within S)" by fact
```
```  1356 qed
```
```  1357
```
```  1358 lemma Lim_transform_at:
```
```  1359   assumes "0 < d"
```
```  1360     and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
```
```  1361     and "(f ---> l) (at x)"
```
```  1362   shows "(g ---> l) (at x)"
```
```  1363   using _ assms(3)
```
```  1364 proof (rule Lim_transform_eventually)
```
```  1365   show "eventually (\<lambda>x. f x = g x) (at x)"
```
```  1366     unfolding eventually_at2
```
```  1367     using assms(1,2) by auto
```
```  1368 qed
```
```  1369
```
```  1370 text{* Common case assuming being away from some crucial point like 0. *}
```
```  1371
```
```  1372 lemma Lim_transform_away_within:
```
```  1373   fixes a b :: "'a::t1_space"
```
```  1374   assumes "a \<noteq> b"
```
```  1375     and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
```
```  1376     and "(f ---> l) (at a within S)"
```
```  1377   shows "(g ---> l) (at a within S)"
```
```  1378 proof (rule Lim_transform_eventually)
```
```  1379   show "(f ---> l) (at a within S)" by fact
```
```  1380   show "eventually (\<lambda>x. f x = g x) (at a within S)"
```
```  1381     unfolding eventually_at_topological
```
```  1382     by (rule exI [where x="- {b}"], simp add: open_Compl assms)
```
```  1383 qed
```
```  1384
```
```  1385 lemma Lim_transform_away_at:
```
```  1386   fixes a b :: "'a::t1_space"
```
```  1387   assumes ab: "a\<noteq>b"
```
```  1388     and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
```
```  1389     and fl: "(f ---> l) (at a)"
```
```  1390   shows "(g ---> l) (at a)"
```
```  1391   using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
```
```  1392
```
```  1393 text{* Alternatively, within an open set. *}
```
```  1394
```
```  1395 lemma Lim_transform_within_open:
```
```  1396   assumes "open S" and "a \<in> S"
```
```  1397     and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
```
```  1398     and "(f ---> l) (at a)"
```
```  1399   shows "(g ---> l) (at a)"
```
```  1400 proof (rule Lim_transform_eventually)
```
```  1401   show "eventually (\<lambda>x. f x = g x) (at a)"
```
```  1402     unfolding eventually_at_topological
```
```  1403     using assms(1,2,3) by auto
```
```  1404   show "(f ---> l) (at a)" by fact
```
```  1405 qed
```
```  1406
```
```  1407 text{* A congruence rule allowing us to transform limits assuming not at point. *}
```
```  1408
```
```  1409 (* FIXME: Only one congruence rule for tendsto can be used at a time! *)
```
```  1410
```
```  1411 lemma Lim_cong_within(*[cong add]*):
```
```  1412   assumes "a = b"
```
```  1413     and "x = y"
```
```  1414     and "S = T"
```
```  1415     and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
```
```  1416   shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
```
```  1417   unfolding tendsto_def eventually_at_topological
```
```  1418   using assms by simp
```
```  1419
```
```  1420 lemma Lim_cong_at(*[cong add]*):
```
```  1421   assumes "a = b" "x = y"
```
```  1422     and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
```
```  1423   shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
```
```  1424   unfolding tendsto_def eventually_at_topological
```
```  1425   using assms by simp
```
```  1426 text{*An unbounded sequence's inverse tends to 0*}
```
```  1427
```
```  1428 lemma LIMSEQ_inverse_zero:
```
```  1429   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```  1430   apply (rule filterlim_compose[OF tendsto_inverse_0])
```
```  1431   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
```
```  1432   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
```
```  1433   done
```
```  1434
```
```  1435 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```  1436
```
```  1437 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```  1438   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
```
```  1439             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
```
```  1440
```
```  1441 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```  1442 infinity is now easily proved*}
```
```  1443
```
```  1444 lemma LIMSEQ_inverse_real_of_nat_add:
```
```  1445      "(%n. r + inverse(real(Suc n))) ----> r"
```
```  1446   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
```
```  1447
```
```  1448 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```  1449      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```  1450   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
```
```  1451   by auto
```
```  1452
```
```  1453 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```  1454      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```  1455   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
```
```  1456   by auto
```
```  1457
```
```  1458 subsection {* Convergence on sequences *}
```
```  1459
```
```  1460 lemma convergent_add:
```
```  1461   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1462   assumes "convergent (\<lambda>n. X n)"
```
```  1463   assumes "convergent (\<lambda>n. Y n)"
```
```  1464   shows "convergent (\<lambda>n. X n + Y n)"
```
```  1465   using assms unfolding convergent_def by (fast intro: tendsto_add)
```
```  1466
```
```  1467 lemma convergent_setsum:
```
```  1468   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
```
```  1469   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
```
```  1470   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
```
```  1471 proof (cases "finite A")
```
```  1472   case True from this and assms show ?thesis
```
```  1473     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
```
```  1474 qed (simp add: convergent_const)
```
```  1475
```
```  1476 lemma (in bounded_linear) convergent:
```
```  1477   assumes "convergent (\<lambda>n. X n)"
```
```  1478   shows "convergent (\<lambda>n. f (X n))"
```
```  1479   using assms unfolding convergent_def by (fast intro: tendsto)
```
```  1480
```
```  1481 lemma (in bounded_bilinear) convergent:
```
```  1482   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
```
```  1483   shows "convergent (\<lambda>n. X n ** Y n)"
```
```  1484   using assms unfolding convergent_def by (fast intro: tendsto)
```
```  1485
```
```  1486 lemma convergent_minus_iff:
```
```  1487   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1488   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```  1489 apply (simp add: convergent_def)
```
```  1490 apply (auto dest: tendsto_minus)
```
```  1491 apply (drule tendsto_minus, auto)
```
```  1492 done
```
```  1493
```
```  1494
```
```  1495 text {* A monotone sequence converges to its least upper bound. *}
```
```  1496
```
```  1497 lemma LIMSEQ_incseq_SUP:
```
```  1498   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
```
```  1499   assumes u: "bdd_above (range X)"
```
```  1500   assumes X: "incseq X"
```
```  1501   shows "X ----> (SUP i. X i)"
```
```  1502   by (rule order_tendstoI)
```
```  1503      (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
```
```  1504
```
```  1505 lemma LIMSEQ_decseq_INF:
```
```  1506   fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
```
```  1507   assumes u: "bdd_below (range X)"
```
```  1508   assumes X: "decseq X"
```
```  1509   shows "X ----> (INF i. X i)"
```
```  1510   by (rule order_tendstoI)
```
```  1511      (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
```
```  1512
```
```  1513 text{*Main monotonicity theorem*}
```
```  1514
```
```  1515 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
```
```  1516   by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
```
```  1517
```
```  1518 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)"
```
```  1519   by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
```
```  1520
```
```  1521 lemma Cauchy_iff:
```
```  1522   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1523   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```  1524   unfolding Cauchy_def dist_norm ..
```
```  1525
```
```  1526 lemma CauchyI:
```
```  1527   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1528   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"
```
```  1529 by (simp add: Cauchy_iff)
```
```  1530
```
```  1531 lemma CauchyD:
```
```  1532   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1533   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1534 by (simp add: Cauchy_iff)
```
```  1535
```
```  1536 lemma incseq_convergent:
```
```  1537   fixes X :: "nat \<Rightarrow> real"
```
```  1538   assumes "incseq X" and "\<forall>i. X i \<le> B"
```
```  1539   obtains L where "X ----> L" "\<forall>i. X i \<le> L"
```
```  1540 proof atomize_elim
```
```  1541   from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
```
```  1542   obtain L where "X ----> L"
```
```  1543     by (auto simp: convergent_def monoseq_def incseq_def)
```
```  1544   with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
```
```  1545     by (auto intro!: exI[of _ L] incseq_le)
```
```  1546 qed
```
```  1547
```
```  1548 lemma decseq_convergent:
```
```  1549   fixes X :: "nat \<Rightarrow> real"
```
```  1550   assumes "decseq X" and "\<forall>i. B \<le> X i"
```
```  1551   obtains L where "X ----> L" "\<forall>i. L \<le> X i"
```
```  1552 proof atomize_elim
```
```  1553   from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
```
```  1554   obtain L where "X ----> L"
```
```  1555     by (auto simp: convergent_def monoseq_def decseq_def)
```
```  1556   with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
```
```  1557     by (auto intro!: exI[of _ L] decseq_le)
```
```  1558 qed
```
```  1559
```
```  1560 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1561
```
```  1562 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1563   proof mechanization rather than the nonstandard proof*}
```
```  1564
```
```  1565 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1566           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1567 apply (clarify, drule spec, drule (1) mp)
```
```  1568 apply (simp only: norm_minus_commute)
```
```  1569 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1570 apply simp
```
```  1571 done
```
```  1572
```
```  1573 subsection {* Power Sequences *}
```
```  1574
```
```  1575 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1576 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1577   also fact that bounded and monotonic sequence converges.*}
```
```  1578
```
```  1579 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1580 apply (simp add: Bseq_def)
```
```  1581 apply (rule_tac x = 1 in exI)
```
```  1582 apply (simp add: power_abs)
```
```  1583 apply (auto dest: power_mono)
```
```  1584 done
```
```  1585
```
```  1586 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1587 apply (clarify intro!: mono_SucI2)
```
```  1588 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1589 done
```
```  1590
```
```  1591 lemma convergent_realpow:
```
```  1592   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1593 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1594
```
```  1595 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1596   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
```
```  1597
```
```  1598 lemma LIMSEQ_realpow_zero:
```
```  1599   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1600 proof cases
```
```  1601   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1602   hence x0: "0 < x" by simp
```
```  1603   assume x1: "x < 1"
```
```  1604   from x0 x1 have "1 < inverse x"
```
```  1605     by (rule one_less_inverse)
```
```  1606   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1607     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1608   thus ?thesis by (simp add: power_inverse)
```
```  1609 qed (rule LIMSEQ_imp_Suc, simp)
```
```  1610
```
```  1611 lemma LIMSEQ_power_zero:
```
```  1612   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1613   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1614 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1615 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
```
```  1616 apply (simp add: power_abs norm_power_ineq)
```
```  1617 done
```
```  1618
```
```  1619 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
```
```  1620   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
```
```  1621
```
```  1622 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1623
```
```  1624 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
```
```  1625   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1626
```
```  1627 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
```
```  1628   by (rule LIMSEQ_power_zero) simp
```
```  1629
```
```  1630
```
```  1631 subsection {* Limits of Functions *}
```
```  1632
```
```  1633 lemma LIM_eq:
```
```  1634   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1635   shows "f -- a --> L =
```
```  1636      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```  1637 by (simp add: LIM_def dist_norm)
```
```  1638
```
```  1639 lemma LIM_I:
```
```  1640   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1641   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```  1642       ==> f -- a --> L"
```
```  1643 by (simp add: LIM_eq)
```
```  1644
```
```  1645 lemma LIM_D:
```
```  1646   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1647   shows "[| f -- a --> L; 0<r |]
```
```  1648       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```  1649 by (simp add: LIM_eq)
```
```  1650
```
```  1651 lemma LIM_offset:
```
```  1652   fixes a :: "'a::real_normed_vector"
```
```  1653   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```  1654   unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
```
```  1655
```
```  1656 lemma LIM_offset_zero:
```
```  1657   fixes a :: "'a::real_normed_vector"
```
```  1658   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1659 by (drule_tac k="a" in LIM_offset, simp add: add.commute)
```
```  1660
```
```  1661 lemma LIM_offset_zero_cancel:
```
```  1662   fixes a :: "'a::real_normed_vector"
```
```  1663   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```  1664 by (drule_tac k="- a" in LIM_offset, simp)
```
```  1665
```
```  1666 lemma LIM_offset_zero_iff:
```
```  1667   fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
```
```  1668   shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1669   using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
```
```  1670
```
```  1671 lemma LIM_zero:
```
```  1672   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1673   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
```
```  1674 unfolding tendsto_iff dist_norm by simp
```
```  1675
```
```  1676 lemma LIM_zero_cancel:
```
```  1677   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1678   shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
```
```  1679 unfolding tendsto_iff dist_norm by simp
```
```  1680
```
```  1681 lemma LIM_zero_iff:
```
```  1682   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  1683   shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
```
```  1684 unfolding tendsto_iff dist_norm by simp
```
```  1685
```
```  1686 lemma LIM_imp_LIM:
```
```  1687   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1688   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```  1689   assumes f: "f -- a --> l"
```
```  1690   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```  1691   shows "g -- a --> m"
```
```  1692   by (rule metric_LIM_imp_LIM [OF f],
```
```  1693     simp add: dist_norm le)
```
```  1694
```
```  1695 lemma LIM_equal2:
```
```  1696   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1697   assumes 1: "0 < R"
```
```  1698   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1699   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```  1700 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
```
```  1701
```
```  1702 lemma LIM_compose2:
```
```  1703   fixes a :: "'a::real_normed_vector"
```
```  1704   assumes f: "f -- a --> b"
```
```  1705   assumes g: "g -- b --> c"
```
```  1706   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```  1707   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  1708 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```  1709
```
```  1710 lemma real_LIM_sandwich_zero:
```
```  1711   fixes f g :: "'a::topological_space \<Rightarrow> real"
```
```  1712   assumes f: "f -- a --> 0"
```
```  1713   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```  1714   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```  1715   shows "g -- a --> 0"
```
```  1716 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
```
```  1717   fix x assume x: "x \<noteq> a"
```
```  1718   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```  1719   also have "g x \<le> f x" by (rule 2 [OF x])
```
```  1720   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```  1721   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```  1722   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```  1723 qed
```
```  1724
```
```  1725
```
```  1726 subsection {* Continuity *}
```
```  1727
```
```  1728 lemma LIM_isCont_iff:
```
```  1729   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1730   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```  1731 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```  1732
```
```  1733 lemma isCont_iff:
```
```  1734   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1735   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```  1736 by (simp add: isCont_def LIM_isCont_iff)
```
```  1737
```
```  1738 lemma isCont_LIM_compose2:
```
```  1739   fixes a :: "'a::real_normed_vector"
```
```  1740   assumes f [unfolded isCont_def]: "isCont f a"
```
```  1741   assumes g: "g -- f a --> l"
```
```  1742   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```  1743   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```  1744 by (rule LIM_compose2 [OF f g inj])
```
```  1745
```
```  1746
```
```  1747 lemma isCont_norm [simp]:
```
```  1748   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1749   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```  1750   by (fact continuous_norm)
```
```  1751
```
```  1752 lemma isCont_rabs [simp]:
```
```  1753   fixes f :: "'a::t2_space \<Rightarrow> real"
```
```  1754   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
```
```  1755   by (fact continuous_rabs)
```
```  1756
```
```  1757 lemma isCont_add [simp]:
```
```  1758   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1759   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```  1760   by (fact continuous_add)
```
```  1761
```
```  1762 lemma isCont_minus [simp]:
```
```  1763   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1764   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```  1765   by (fact continuous_minus)
```
```  1766
```
```  1767 lemma isCont_diff [simp]:
```
```  1768   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1769   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```  1770   by (fact continuous_diff)
```
```  1771
```
```  1772 lemma isCont_mult [simp]:
```
```  1773   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
```
```  1774   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```  1775   by (fact continuous_mult)
```
```  1776
```
```  1777 lemma (in bounded_linear) isCont:
```
```  1778   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
```
```  1779   by (fact continuous)
```
```  1780
```
```  1781 lemma (in bounded_bilinear) isCont:
```
```  1782   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```  1783   by (fact continuous)
```
```  1784
```
```  1785 lemmas isCont_scaleR [simp] =
```
```  1786   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
```
```  1787
```
```  1788 lemmas isCont_of_real [simp] =
```
```  1789   bounded_linear.isCont [OF bounded_linear_of_real]
```
```  1790
```
```  1791 lemma isCont_power [simp]:
```
```  1792   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1793   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```  1794   by (fact continuous_power)
```
```  1795
```
```  1796 lemma isCont_setsum [simp]:
```
```  1797   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```  1798   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
```
```  1799   by (auto intro: continuous_setsum)
```
```  1800
```
```  1801 subsection {* Uniform Continuity *}
```
```  1802
```
```  1803 definition
```
```  1804   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
```
```  1805   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
```
```  1806
```
```  1807 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```  1808 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```  1809
```
```  1810 lemma isUCont_Cauchy:
```
```  1811   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  1812 unfolding isUCont_def
```
```  1813 apply (rule metric_CauchyI)
```
```  1814 apply (drule_tac x=e in spec, safe)
```
```  1815 apply (drule_tac e=s in metric_CauchyD, safe)
```
```  1816 apply (rule_tac x=M in exI, simp)
```
```  1817 done
```
```  1818
```
```  1819 lemma (in bounded_linear) isUCont: "isUCont f"
```
```  1820 unfolding isUCont_def dist_norm
```
```  1821 proof (intro allI impI)
```
```  1822   fix r::real assume r: "0 < r"
```
```  1823   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```  1824     using pos_bounded by fast
```
```  1825   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```  1826   proof (rule exI, safe)
```
```  1827     from r K show "0 < r / K" by simp
```
```  1828   next
```
```  1829     fix x y :: 'a
```
```  1830     assume xy: "norm (x - y) < r / K"
```
```  1831     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```  1832     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```  1833     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```  1834     finally show "norm (f x - f y) < r" .
```
```  1835   qed
```
```  1836 qed
```
```  1837
```
```  1838 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  1839 by (rule isUCont [THEN isUCont_Cauchy])
```
```  1840
```
```  1841 lemma LIM_less_bound:
```
```  1842   fixes f :: "real \<Rightarrow> real"
```
```  1843   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
```
```  1844   shows "0 \<le> f x"
```
```  1845 proof (rule tendsto_le_const)
```
```  1846   show "(f ---> f x) (at_left x)"
```
```  1847     using `isCont f x` by (simp add: filterlim_at_split isCont_def)
```
```  1848   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
```
```  1849     using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
```
```  1850 qed simp
```
```  1851
```
```  1852
```
```  1853 subsection {* Nested Intervals and Bisection -- Needed for Compactness *}
```
```  1854
```
```  1855 lemma nested_sequence_unique:
```
```  1856   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) ----> 0"
```
```  1857   shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
```
```  1858 proof -
```
```  1859   have "incseq f" unfolding incseq_Suc_iff by fact
```
```  1860   have "decseq g" unfolding decseq_Suc_iff by fact
```
```  1861
```
```  1862   { fix n
```
```  1863     from `decseq g` have "g n \<le> g 0" by (rule decseqD) simp
```
```  1864     with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f n \<le> g 0" by auto }
```
```  1865   then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
```
```  1866     using incseq_convergent[OF `incseq f`] by auto
```
```  1867   moreover
```
```  1868   { fix n
```
```  1869     from `incseq f` have "f 0 \<le> f n" by (rule incseqD) simp
```
```  1870     with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f 0 \<le> g n" by simp }
```
```  1871   then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
```
```  1872     using decseq_convergent[OF `decseq g`] by auto
```
```  1873   moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF `f ----> u` `g ----> l`]]
```
```  1874   ultimately show ?thesis by auto
```
```  1875 qed
```
```  1876
```
```  1877 lemma Bolzano[consumes 1, case_names trans local]:
```
```  1878   fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
```
```  1879   assumes [arith]: "a \<le> b"
```
```  1880   assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
```
```  1881   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"
```
```  1882   shows "P a b"
```
```  1883 proof -
```
```  1884   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))"
```
```  1885   def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
```
```  1886   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)"
```
```  1887     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)"
```
```  1888     by (simp_all add: l_def u_def bisect_def split: prod.split)
```
```  1889
```
```  1890   { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
```
```  1891
```
```  1892   have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
```
```  1893   proof (safe intro!: nested_sequence_unique)
```
```  1894     fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
```
```  1895   next
```
```  1896     { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
```
```  1897     then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
```
```  1898   qed fact
```
```  1899   then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
```
```  1900   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"
```
```  1901     using `l 0 \<le> x` `x \<le> u 0` local[of x] by auto
```
```  1902
```
```  1903   show "P a b"
```
```  1904   proof (rule ccontr)
```
```  1905     assume "\<not> P a b"
```
```  1906     { fix n have "\<not> P (l n) (u n)"
```
```  1907       proof (induct n)
```
```  1908         case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
```
```  1909       qed (simp add: `\<not> P a b`) }
```
```  1910     moreover
```
```  1911     { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
```
```  1912         using `0 < d` `l ----> x` by (intro order_tendstoD[of _ x]) auto
```
```  1913       moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
```
```  1914         using `0 < d` `u ----> x` by (intro order_tendstoD[of _ x]) auto
```
```  1915       ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
```
```  1916       proof eventually_elim
```
```  1917         fix n assume "x - d / 2 < l n" "u n < x + d / 2"
```
```  1918         from add_strict_mono[OF this] have "u n - l n < d" by simp
```
```  1919         with x show "P (l n) (u n)" by (rule d)
```
```  1920       qed }
```
```  1921     ultimately show False by simp
```
```  1922   qed
```
```  1923 qed
```
```  1924
```
```  1925 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
```
```  1926 proof (cases "a \<le> b", rule compactI)
```
```  1927   fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
```
```  1928   def T == "{a .. b}"
```
```  1929   from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
```
```  1930   proof (induct rule: Bolzano)
```
```  1931     case (trans a b c)
```
```  1932     then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
```
```  1933     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"
```
```  1934       by (auto simp: *)
```
```  1935     with trans show ?case
```
```  1936       unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
```
```  1937   next
```
```  1938     case (local x)
```
```  1939     then have "x \<in> \<Union>C" using C by auto
```
```  1940     with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
```
```  1941     then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
```
```  1942       by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
```
```  1943     with `c \<in> C` show ?case
```
```  1944       by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
```
```  1945   qed
```
```  1946 qed simp
```
```  1947
```
```  1948
```
```  1949 lemma continuous_image_closed_interval:
```
```  1950   fixes a b and f :: "real \<Rightarrow> real"
```
```  1951   defines "S \<equiv> {a..b}"
```
```  1952   assumes "a \<le> b" and f: "continuous_on S f"
```
```  1953   shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
```
```  1954 proof -
```
```  1955   have S: "compact S" "S \<noteq> {}"
```
```  1956     using `a \<le> b` by (auto simp: S_def)
```
```  1957   obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
```
```  1958     using continuous_attains_sup[OF S f] by auto
```
```  1959   moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
```
```  1960     using continuous_attains_inf[OF S f] by auto
```
```  1961   moreover have "connected (f`S)"
```
```  1962     using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
```
```  1963   ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
```
```  1964     by (auto simp: connected_iff_interval)
```
```  1965   then show ?thesis
```
```  1966     by auto
```
```  1967 qed
```
```  1968
```
```  1969 subsection {* Boundedness of continuous functions *}
```
```  1970
```
```  1971 text{*By bisection, function continuous on closed interval is bounded above*}
```
```  1972
```
```  1973 lemma isCont_eq_Ub:
```
```  1974   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1975   shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1976     \<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)"
```
```  1977   using continuous_attains_sup[of "{a .. b}" f]
```
```  1978   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  1979
```
```  1980 lemma isCont_eq_Lb:
```
```  1981   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1982   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1983     \<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)"
```
```  1984   using continuous_attains_inf[of "{a .. b}" f]
```
```  1985   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  1986
```
```  1987 lemma isCont_bounded:
```
```  1988   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1989   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"
```
```  1990   using isCont_eq_Ub[of a b f] by auto
```
```  1991
```
```  1992 lemma isCont_has_Ub:
```
```  1993   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1994   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1995     \<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))"
```
```  1996   using isCont_eq_Ub[of a b f] by auto
```
```  1997
```
```  1998 (*HOL style here: object-level formulations*)
```
```  1999 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```  2000       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  2001       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  2002   by (blast intro: IVT)
```
```  2003
```
```  2004 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```  2005       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  2006       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  2007   by (blast intro: IVT2)
```
```  2008
```
```  2009 lemma isCont_Lb_Ub:
```
```  2010   fixes f :: "real \<Rightarrow> real"
```
```  2011   assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  2012   shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
```
```  2013                (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
```
```  2014 proof -
```
```  2015   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"
```
```  2016     using isCont_eq_Ub[OF assms] by auto
```
```  2017   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"
```
```  2018     using isCont_eq_Lb[OF assms] by auto
```
```  2019   show ?thesis
```
```  2020     using IVT[of f L _ M] IVT2[of f L _ M] M L assms
```
```  2021     apply (rule_tac x="f L" in exI)
```
```  2022     apply (rule_tac x="f M" in exI)
```
```  2023     apply (cases "L \<le> M")
```
```  2024     apply (simp, metis order_trans)
```
```  2025     apply (simp, metis order_trans)
```
```  2026     done
```
```  2027 qed
```
```  2028
```
```  2029
```
```  2030 text{*Continuity of inverse function*}
```
```  2031
```
```  2032 lemma isCont_inverse_function:
```
```  2033   fixes f g :: "real \<Rightarrow> real"
```
```  2034   assumes d: "0 < d"
```
```  2035       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
```
```  2036       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
```
```  2037   shows "isCont g (f x)"
```
```  2038 proof -
```
```  2039   let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
```
```  2040
```
```  2041   have f: "continuous_on ?D f"
```
```  2042     using cont by (intro continuous_at_imp_continuous_on ballI) auto
```
```  2043   then have g: "continuous_on (f`?D) g"
```
```  2044     using inj by (intro continuous_on_inv) auto
```
```  2045
```
```  2046   from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
```
```  2047     by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
```
```  2048   with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
```
```  2049     by (rule continuous_on_subset)
```
```  2050   moreover
```
```  2051   have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
```
```  2052     using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
```
```  2053   then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
```
```  2054     by auto
```
```  2055   ultimately
```
```  2056   show ?thesis
```
```  2057     by (simp add: continuous_on_eq_continuous_at)
```
```  2058 qed
```
```  2059
```
```  2060 lemma isCont_inverse_function2:
```
```  2061   fixes f g :: "real \<Rightarrow> real" shows
```
```  2062   "\<lbrakk>a < x; x < b;
```
```  2063     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
```
```  2064     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
```
```  2065    \<Longrightarrow> isCont g (f x)"
```
```  2066 apply (rule isCont_inverse_function
```
```  2067        [where f=f and d="min (x - a) (b - x)"])
```
```  2068 apply (simp_all add: abs_le_iff)
```
```  2069 done
```
```  2070
```
```  2071 (* need to rename second isCont_inverse *)
```
```  2072
```
```  2073 lemma isCont_inv_fun:
```
```  2074   fixes f g :: "real \<Rightarrow> real"
```
```  2075   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  2076          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  2077       ==> isCont g (f x)"
```
```  2078 by (rule isCont_inverse_function)
```
```  2079
```
```  2080 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
```
```  2081 lemma LIM_fun_gt_zero:
```
```  2082   fixes f :: "real \<Rightarrow> real"
```
```  2083   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)"
```
```  2084 apply (drule (1) LIM_D, clarify)
```
```  2085 apply (rule_tac x = s in exI)
```
```  2086 apply (simp add: abs_less_iff)
```
```  2087 done
```
```  2088
```
```  2089 lemma LIM_fun_less_zero:
```
```  2090   fixes f :: "real \<Rightarrow> real"
```
```  2091   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)"
```
```  2092 apply (drule LIM_D [where r="-l"], simp, clarify)
```
```  2093 apply (rule_tac x = s in exI)
```
```  2094 apply (simp add: abs_less_iff)
```
```  2095 done
```
```  2096
```
```  2097 lemma LIM_fun_not_zero:
```
```  2098   fixes f :: "real \<Rightarrow> real"
```
```  2099   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)"
```
```  2100   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
```
```  2101
```
```  2102 end
```
```  2103
```