src/HOL/Limits.thy
 author wenzelm Wed Apr 10 21:20:35 2013 +0200 (2013-04-10) changeset 51692 ecd34f863242 parent 51642 400ec5ae7f8f child 52265 bb907eba5902 permissions -rw-r--r--
tuned pretty layout: avoid nested Pretty.string_of, which merely happens to work with Isabelle/jEdit since formatting is delegated to Scala side;
declare command "print_case_translations" where it is actually defined;
```     1 (*  Title:      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
```
```     9 header {* Limits on Real Vector Spaces *}
```
```    10
```
```    11 theory Limits
```
```    12 imports Real_Vector_Spaces
```
```    13 begin
```
```    14
```
```    15 subsection {* Filter going to infinity norm *}
```
```    16
```
```    17 definition at_infinity :: "'a::real_normed_vector filter" where
```
```    18   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
```
```    19
```
```    20 lemma eventually_at_infinity:
```
```    21   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
```
```    22 unfolding at_infinity_def
```
```    23 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```    24   fix P Q :: "'a \<Rightarrow> bool"
```
```    25   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
```
```    26   then obtain r s where
```
```    27     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
```
```    28   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
```
```    29   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
```
```    30 qed auto
```
```    31
```
```    32 lemma at_infinity_eq_at_top_bot:
```
```    33   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
```
```    34   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
```
```    35 proof (intro arg_cong[where f=Abs_filter] ext iffI)
```
```    36   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```    37   then guess r ..
```
```    38   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
```
```    39   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
```
```    40 next
```
```    41   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
```
```    42   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
```
```    43   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```    44     by (intro exI[of _ "max p (-q)"])
```
```    45        (auto simp: abs_real_def)
```
```    46 qed
```
```    47
```
```    48 lemma at_top_le_at_infinity:
```
```    49   "at_top \<le> (at_infinity :: real filter)"
```
```    50   unfolding at_infinity_eq_at_top_bot by simp
```
```    51
```
```    52 lemma at_bot_le_at_infinity:
```
```    53   "at_bot \<le> (at_infinity :: real filter)"
```
```    54   unfolding at_infinity_eq_at_top_bot by simp
```
```    55
```
```    56 subsubsection {* Boundedness *}
```
```    57
```
```    58 definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
```
```    59   Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
```
```    60
```
```    61 abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
```
```    62   "Bseq X \<equiv> Bfun X sequentially"
```
```    63
```
```    64 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
```
```    65
```
```    66 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
```
```    67   unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
```
```    68
```
```    69 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
```
```    70   unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
```
```    71
```
```    72 lemma Bfun_def:
```
```    73   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
```
```    74   unfolding Bfun_metric_def norm_conv_dist
```
```    75 proof safe
```
```    76   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
```
```    77   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
```
```    78     by (intro always_eventually) (metis dist_commute dist_triangle)
```
```    79   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
```
```    80     by eventually_elim auto
```
```    81   with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
```
```    82     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
```
```    83 qed auto
```
```    84
```
```    85 lemma BfunI:
```
```    86   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
```
```    87 unfolding Bfun_def
```
```    88 proof (intro exI conjI allI)
```
```    89   show "0 < max K 1" by simp
```
```    90 next
```
```    91   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
```
```    92     using K by (rule eventually_elim1, simp)
```
```    93 qed
```
```    94
```
```    95 lemma BfunE:
```
```    96   assumes "Bfun f F"
```
```    97   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
```
```    98 using assms unfolding Bfun_def by fast
```
```    99
```
```   100 lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
```
```   101   unfolding Cauchy_def Bfun_metric_def eventually_sequentially
```
```   102   apply (erule_tac x=1 in allE)
```
```   103   apply simp
```
```   104   apply safe
```
```   105   apply (rule_tac x="X M" in exI)
```
```   106   apply (rule_tac x=1 in exI)
```
```   107   apply (erule_tac x=M in allE)
```
```   108   apply simp
```
```   109   apply (rule_tac x=M in exI)
```
```   110   apply (auto simp: dist_commute)
```
```   111   done
```
```   112
```
```   113
```
```   114 subsubsection {* Bounded Sequences *}
```
```   115
```
```   116 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
```
```   117   by (intro BfunI) (auto simp: eventually_sequentially)
```
```   118
```
```   119 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
```
```   120   by (intro BfunI) (auto simp: eventually_sequentially)
```
```   121
```
```   122 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
```
```   123   unfolding Bfun_def eventually_sequentially
```
```   124 proof safe
```
```   125   fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
```
```   126   then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
```
```   127     by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
```
```   128        (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
```
```   129 qed auto
```
```   130
```
```   131 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```   132 unfolding Bseq_def by auto
```
```   133
```
```   134 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```   135 by (simp add: Bseq_def)
```
```   136
```
```   137 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```   138 by (auto simp add: Bseq_def)
```
```   139
```
```   140 lemma lemma_NBseq_def:
```
```   141   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   142 proof safe
```
```   143   fix K :: real
```
```   144   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
```
```   145   then have "K \<le> real (Suc n)" by auto
```
```   146   moreover assume "\<forall>m. norm (X m) \<le> K"
```
```   147   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
```
```   148     by (blast intro: order_trans)
```
```   149   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
```
```   150 qed (force simp add: real_of_nat_Suc)
```
```   151
```
```   152 text{* alternative definition for Bseq *}
```
```   153 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```   154 apply (simp add: Bseq_def)
```
```   155 apply (simp (no_asm) add: lemma_NBseq_def)
```
```   156 done
```
```   157
```
```   158 lemma lemma_NBseq_def2:
```
```   159      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   160 apply (subst lemma_NBseq_def, auto)
```
```   161 apply (rule_tac x = "Suc N" in exI)
```
```   162 apply (rule_tac [2] x = N in exI)
```
```   163 apply (auto simp add: real_of_nat_Suc)
```
```   164  prefer 2 apply (blast intro: order_less_imp_le)
```
```   165 apply (drule_tac x = n in spec, simp)
```
```   166 done
```
```   167
```
```   168 (* yet another definition for Bseq *)
```
```   169 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```   170 by (simp add: Bseq_def lemma_NBseq_def2)
```
```   171
```
```   172 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```   173
```
```   174 text{*alternative formulation for boundedness*}
```
```   175 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```   176 apply (unfold Bseq_def, safe)
```
```   177 apply (rule_tac [2] x = "k + norm x" in exI)
```
```   178 apply (rule_tac x = K in exI, simp)
```
```   179 apply (rule exI [where x = 0], auto)
```
```   180 apply (erule order_less_le_trans, simp)
```
```   181 apply (drule_tac x=n in spec, fold diff_minus)
```
```   182 apply (drule order_trans [OF norm_triangle_ineq2])
```
```   183 apply simp
```
```   184 done
```
```   185
```
```   186 text{*alternative formulation for boundedness*}
```
```   187 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```   188 apply safe
```
```   189 apply (simp add: Bseq_def, safe)
```
```   190 apply (rule_tac x = "K + norm (X N)" in exI)
```
```   191 apply auto
```
```   192 apply (erule order_less_le_trans, simp)
```
```   193 apply (rule_tac x = N in exI, safe)
```
```   194 apply (drule_tac x = n in spec)
```
```   195 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```   196 apply (auto simp add: Bseq_iff2)
```
```   197 done
```
```   198
```
```   199 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```   200 apply (simp add: Bseq_def)
```
```   201 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```   202 apply (drule_tac x = n in spec, arith)
```
```   203 done
```
```   204
```
```   205 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```   206
```
```   207 lemma Bseq_isUb:
```
```   208   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   209 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```   210
```
```   211 text{* Use completeness of reals (supremum property)
```
```   212    to show that any bounded sequence has a least upper bound*}
```
```   213
```
```   214 lemma Bseq_isLub:
```
```   215   "!!(X::nat=>real). Bseq X ==>
```
```   216    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```   217 by (blast intro: reals_complete Bseq_isUb)
```
```   218
```
```   219 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
```
```   220   by (simp add: Bseq_def)
```
```   221
```
```   222 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
```
```   223   apply (simp add: subset_eq)
```
```   224   apply (rule BseqI'[where K="max (norm a) (norm b)"])
```
```   225   apply (erule_tac x=n in allE)
```
```   226   apply auto
```
```   227   done
```
```   228
```
```   229 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
```
```   230   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
```
```   231
```
```   232 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
```
```   233   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
```
```   234
```
```   235 subsection {* Bounded Monotonic Sequences *}
```
```   236
```
```   237 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```   238
```
```   239 (* TODO: delete *)
```
```   240 (* FIXME: one use in NSA/HSEQ.thy *)
```
```   241 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```   242   apply (rule_tac x="X m" in exI)
```
```   243   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
```
```   244   unfolding eventually_sequentially
```
```   245   apply blast
```
```   246   done
```
```   247
```
```   248 subsection {* Convergence to Zero *}
```
```   249
```
```   250 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```   251   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```   252
```
```   253 lemma ZfunI:
```
```   254   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```   255   unfolding Zfun_def by simp
```
```   256
```
```   257 lemma ZfunD:
```
```   258   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```   259   unfolding Zfun_def by simp
```
```   260
```
```   261 lemma Zfun_ssubst:
```
```   262   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```   263   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   264
```
```   265 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```   266   unfolding Zfun_def by simp
```
```   267
```
```   268 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   269   unfolding Zfun_def by simp
```
```   270
```
```   271 lemma Zfun_imp_Zfun:
```
```   272   assumes f: "Zfun f F"
```
```   273   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   274   shows "Zfun (\<lambda>x. g x) F"
```
```   275 proof (cases)
```
```   276   assume K: "0 < K"
```
```   277   show ?thesis
```
```   278   proof (rule ZfunI)
```
```   279     fix r::real assume "0 < r"
```
```   280     hence "0 < r / K"
```
```   281       using K by (rule divide_pos_pos)
```
```   282     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   283       using ZfunD [OF f] by fast
```
```   284     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   285     proof eventually_elim
```
```   286       case (elim x)
```
```   287       hence "norm (f x) * K < r"
```
```   288         by (simp add: pos_less_divide_eq K)
```
```   289       thus ?case
```
```   290         by (simp add: order_le_less_trans [OF elim(1)])
```
```   291     qed
```
```   292   qed
```
```   293 next
```
```   294   assume "\<not> 0 < K"
```
```   295   hence K: "K \<le> 0" by (simp only: not_less)
```
```   296   show ?thesis
```
```   297   proof (rule ZfunI)
```
```   298     fix r :: real
```
```   299     assume "0 < r"
```
```   300     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   301     proof eventually_elim
```
```   302       case (elim x)
```
```   303       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   304         using K norm_ge_zero by (rule mult_left_mono)
```
```   305       finally show ?case
```
```   306         using `0 < r` by simp
```
```   307     qed
```
```   308   qed
```
```   309 qed
```
```   310
```
```   311 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   312   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   313
```
```   314 lemma Zfun_add:
```
```   315   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   316   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   317 proof (rule ZfunI)
```
```   318   fix r::real assume "0 < r"
```
```   319   hence r: "0 < r / 2" by simp
```
```   320   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   321     using f r by (rule ZfunD)
```
```   322   moreover
```
```   323   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   324     using g r by (rule ZfunD)
```
```   325   ultimately
```
```   326   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   327   proof eventually_elim
```
```   328     case (elim x)
```
```   329     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   330       by (rule norm_triangle_ineq)
```
```   331     also have "\<dots> < r/2 + r/2"
```
```   332       using elim by (rule add_strict_mono)
```
```   333     finally show ?case
```
```   334       by simp
```
```   335   qed
```
```   336 qed
```
```   337
```
```   338 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   339   unfolding Zfun_def by simp
```
```   340
```
```   341 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   342   by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   343
```
```   344 lemma (in bounded_linear) Zfun:
```
```   345   assumes g: "Zfun g F"
```
```   346   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   347 proof -
```
```   348   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   349     using bounded by fast
```
```   350   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   351     by simp
```
```   352   with g show ?thesis
```
```   353     by (rule Zfun_imp_Zfun)
```
```   354 qed
```
```   355
```
```   356 lemma (in bounded_bilinear) Zfun:
```
```   357   assumes f: "Zfun f F"
```
```   358   assumes g: "Zfun g F"
```
```   359   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   360 proof (rule ZfunI)
```
```   361   fix r::real assume r: "0 < r"
```
```   362   obtain K where K: "0 < K"
```
```   363     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   364     using pos_bounded by fast
```
```   365   from K have K': "0 < inverse K"
```
```   366     by (rule positive_imp_inverse_positive)
```
```   367   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   368     using f r by (rule ZfunD)
```
```   369   moreover
```
```   370   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   371     using g K' by (rule ZfunD)
```
```   372   ultimately
```
```   373   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   374   proof eventually_elim
```
```   375     case (elim x)
```
```   376     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   377       by (rule norm_le)
```
```   378     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   379       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   380     also from K have "r * inverse K * K = r"
```
```   381       by simp
```
```   382     finally show ?case .
```
```   383   qed
```
```   384 qed
```
```   385
```
```   386 lemma (in bounded_bilinear) Zfun_left:
```
```   387   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   388   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   389
```
```   390 lemma (in bounded_bilinear) Zfun_right:
```
```   391   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   392   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   393
```
```   394 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   395 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   396 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   397
```
```   398 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   399   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   400
```
```   401 subsubsection {* Distance and norms *}
```
```   402
```
```   403 lemma tendsto_dist [tendsto_intros]:
```
```   404   fixes l m :: "'a :: metric_space"
```
```   405   assumes f: "(f ---> l) F" and g: "(g ---> m) F"
```
```   406   shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
```
```   407 proof (rule tendstoI)
```
```   408   fix e :: real assume "0 < e"
```
```   409   hence e2: "0 < e/2" by simp
```
```   410   from tendstoD [OF f e2] tendstoD [OF g e2]
```
```   411   show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
```
```   412   proof (eventually_elim)
```
```   413     case (elim x)
```
```   414     then show "dist (dist (f x) (g x)) (dist l m) < e"
```
```   415       unfolding dist_real_def
```
```   416       using dist_triangle2 [of "f x" "g x" "l"]
```
```   417       using dist_triangle2 [of "g x" "l" "m"]
```
```   418       using dist_triangle3 [of "l" "m" "f x"]
```
```   419       using dist_triangle [of "f x" "m" "g x"]
```
```   420       by arith
```
```   421   qed
```
```   422 qed
```
```   423
```
```   424 lemma continuous_dist[continuous_intros]:
```
```   425   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   426   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
```
```   427   unfolding continuous_def by (rule tendsto_dist)
```
```   428
```
```   429 lemma continuous_on_dist[continuous_on_intros]:
```
```   430   fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
```
```   431   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
```
```   432   unfolding continuous_on_def by (auto intro: tendsto_dist)
```
```   433
```
```   434 lemma tendsto_norm [tendsto_intros]:
```
```   435   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
```
```   436   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   437
```
```   438 lemma continuous_norm [continuous_intros]:
```
```   439   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
```
```   440   unfolding continuous_def by (rule tendsto_norm)
```
```   441
```
```   442 lemma continuous_on_norm [continuous_on_intros]:
```
```   443   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
```
```   444   unfolding continuous_on_def by (auto intro: tendsto_norm)
```
```   445
```
```   446 lemma tendsto_norm_zero:
```
```   447   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
```
```   448   by (drule tendsto_norm, simp)
```
```   449
```
```   450 lemma tendsto_norm_zero_cancel:
```
```   451   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
```
```   452   unfolding tendsto_iff dist_norm by simp
```
```   453
```
```   454 lemma tendsto_norm_zero_iff:
```
```   455   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
```
```   456   unfolding tendsto_iff dist_norm by simp
```
```   457
```
```   458 lemma tendsto_rabs [tendsto_intros]:
```
```   459   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
```
```   460   by (fold real_norm_def, rule tendsto_norm)
```
```   461
```
```   462 lemma continuous_rabs [continuous_intros]:
```
```   463   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   464   unfolding real_norm_def[symmetric] by (rule continuous_norm)
```
```   465
```
```   466 lemma continuous_on_rabs [continuous_on_intros]:
```
```   467   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   468   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
```
```   469
```
```   470 lemma tendsto_rabs_zero:
```
```   471   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
```
```   472   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   473
```
```   474 lemma tendsto_rabs_zero_cancel:
```
```   475   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
```
```   476   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   477
```
```   478 lemma tendsto_rabs_zero_iff:
```
```   479   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
```
```   480   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   481
```
```   482 subsubsection {* Addition and subtraction *}
```
```   483
```
```   484 lemma tendsto_add [tendsto_intros]:
```
```   485   fixes a b :: "'a::real_normed_vector"
```
```   486   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
```
```   487   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   488
```
```   489 lemma continuous_add [continuous_intros]:
```
```   490   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   491   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
```
```   492   unfolding continuous_def by (rule tendsto_add)
```
```   493
```
```   494 lemma continuous_on_add [continuous_on_intros]:
```
```   495   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   496   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
```
```   497   unfolding continuous_on_def by (auto intro: tendsto_add)
```
```   498
```
```   499 lemma tendsto_add_zero:
```
```   500   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   501   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
```
```   502   by (drule (1) tendsto_add, simp)
```
```   503
```
```   504 lemma tendsto_minus [tendsto_intros]:
```
```   505   fixes a :: "'a::real_normed_vector"
```
```   506   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
```
```   507   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   508
```
```   509 lemma continuous_minus [continuous_intros]:
```
```   510   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   511   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
```
```   512   unfolding continuous_def by (rule tendsto_minus)
```
```   513
```
```   514 lemma continuous_on_minus [continuous_on_intros]:
```
```   515   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   516   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
```
```   517   unfolding continuous_on_def by (auto intro: tendsto_minus)
```
```   518
```
```   519 lemma tendsto_minus_cancel:
```
```   520   fixes a :: "'a::real_normed_vector"
```
```   521   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
```
```   522   by (drule tendsto_minus, simp)
```
```   523
```
```   524 lemma tendsto_minus_cancel_left:
```
```   525     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
```
```   526   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
```
```   527   by auto
```
```   528
```
```   529 lemma tendsto_diff [tendsto_intros]:
```
```   530   fixes a b :: "'a::real_normed_vector"
```
```   531   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
```
```   532   by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   533
```
```   534 lemma continuous_diff [continuous_intros]:
```
```   535   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   536   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
```
```   537   unfolding continuous_def by (rule tendsto_diff)
```
```   538
```
```   539 lemma continuous_on_diff [continuous_on_intros]:
```
```   540   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   541   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
```
```   542   unfolding continuous_on_def by (auto intro: tendsto_diff)
```
```   543
```
```   544 lemma tendsto_setsum [tendsto_intros]:
```
```   545   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   546   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
```
```   547   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
```
```   548 proof (cases "finite S")
```
```   549   assume "finite S" thus ?thesis using assms
```
```   550     by (induct, simp add: tendsto_const, simp add: tendsto_add)
```
```   551 next
```
```   552   assume "\<not> finite S" thus ?thesis
```
```   553     by (simp add: tendsto_const)
```
```   554 qed
```
```   555
```
```   556 lemma continuous_setsum [continuous_intros]:
```
```   557   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```   558   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
```
```   559   unfolding continuous_def by (rule tendsto_setsum)
```
```   560
```
```   561 lemma continuous_on_setsum [continuous_intros]:
```
```   562   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
```
```   563   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)"
```
```   564   unfolding continuous_on_def by (auto intro: tendsto_setsum)
```
```   565
```
```   566 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
```
```   567
```
```   568 subsubsection {* Linear operators and multiplication *}
```
```   569
```
```   570 lemma (in bounded_linear) tendsto:
```
```   571   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
```
```   572   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   573
```
```   574 lemma (in bounded_linear) continuous:
```
```   575   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
```
```   576   using tendsto[of g _ F] by (auto simp: continuous_def)
```
```   577
```
```   578 lemma (in bounded_linear) continuous_on:
```
```   579   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
```
```   580   using tendsto[of g] by (auto simp: continuous_on_def)
```
```   581
```
```   582 lemma (in bounded_linear) tendsto_zero:
```
```   583   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
```
```   584   by (drule tendsto, simp only: zero)
```
```   585
```
```   586 lemma (in bounded_bilinear) tendsto:
```
```   587   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
```
```   588   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   589                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   590
```
```   591 lemma (in bounded_bilinear) continuous:
```
```   592   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
```
```   593   using tendsto[of f _ F g] by (auto simp: continuous_def)
```
```   594
```
```   595 lemma (in bounded_bilinear) continuous_on:
```
```   596   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
```
```   597   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
```
```   598
```
```   599 lemma (in bounded_bilinear) tendsto_zero:
```
```   600   assumes f: "(f ---> 0) F"
```
```   601   assumes g: "(g ---> 0) F"
```
```   602   shows "((\<lambda>x. f x ** g x) ---> 0) F"
```
```   603   using tendsto [OF f g] by (simp add: zero_left)
```
```   604
```
```   605 lemma (in bounded_bilinear) tendsto_left_zero:
```
```   606   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
```
```   607   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```   608
```
```   609 lemma (in bounded_bilinear) tendsto_right_zero:
```
```   610   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
```
```   611   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```   612
```
```   613 lemmas tendsto_of_real [tendsto_intros] =
```
```   614   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```   615
```
```   616 lemmas tendsto_scaleR [tendsto_intros] =
```
```   617   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```   618
```
```   619 lemmas tendsto_mult [tendsto_intros] =
```
```   620   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```   621
```
```   622 lemmas continuous_of_real [continuous_intros] =
```
```   623   bounded_linear.continuous [OF bounded_linear_of_real]
```
```   624
```
```   625 lemmas continuous_scaleR [continuous_intros] =
```
```   626   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
```
```   627
```
```   628 lemmas continuous_mult [continuous_intros] =
```
```   629   bounded_bilinear.continuous [OF bounded_bilinear_mult]
```
```   630
```
```   631 lemmas continuous_on_of_real [continuous_on_intros] =
```
```   632   bounded_linear.continuous_on [OF bounded_linear_of_real]
```
```   633
```
```   634 lemmas continuous_on_scaleR [continuous_on_intros] =
```
```   635   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
```
```   636
```
```   637 lemmas continuous_on_mult [continuous_on_intros] =
```
```   638   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
```
```   639
```
```   640 lemmas tendsto_mult_zero =
```
```   641   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```   642
```
```   643 lemmas tendsto_mult_left_zero =
```
```   644   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```   645
```
```   646 lemmas tendsto_mult_right_zero =
```
```   647   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```   648
```
```   649 lemma tendsto_power [tendsto_intros]:
```
```   650   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   651   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
```
```   652   by (induct n) (simp_all add: tendsto_const tendsto_mult)
```
```   653
```
```   654 lemma continuous_power [continuous_intros]:
```
```   655   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   656   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
```
```   657   unfolding continuous_def by (rule tendsto_power)
```
```   658
```
```   659 lemma continuous_on_power [continuous_on_intros]:
```
```   660   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   661   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
```
```   662   unfolding continuous_on_def by (auto intro: tendsto_power)
```
```   663
```
```   664 lemma tendsto_setprod [tendsto_intros]:
```
```   665   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   666   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
```
```   667   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
```
```   668 proof (cases "finite S")
```
```   669   assume "finite S" thus ?thesis using assms
```
```   670     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
```
```   671 next
```
```   672   assume "\<not> finite S" thus ?thesis
```
```   673     by (simp add: tendsto_const)
```
```   674 qed
```
```   675
```
```   676 lemma continuous_setprod [continuous_intros]:
```
```   677   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   678   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
```
```   679   unfolding continuous_def by (rule tendsto_setprod)
```
```   680
```
```   681 lemma continuous_on_setprod [continuous_intros]:
```
```   682   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   683   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)"
```
```   684   unfolding continuous_on_def by (auto intro: tendsto_setprod)
```
```   685
```
```   686 subsubsection {* Inverse and division *}
```
```   687
```
```   688 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   689   assumes f: "Zfun f F"
```
```   690   assumes g: "Bfun g F"
```
```   691   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   692 proof -
```
```   693   obtain K where K: "0 \<le> K"
```
```   694     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   695     using nonneg_bounded by fast
```
```   696   obtain B where B: "0 < B"
```
```   697     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```   698     using g by (rule BfunE)
```
```   699   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```   700   using norm_g proof eventually_elim
```
```   701     case (elim x)
```
```   702     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   703       by (rule norm_le)
```
```   704     also have "\<dots> \<le> norm (f x) * B * K"
```
```   705       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   706                 mult_nonneg_nonneg K elim)
```
```   707     also have "\<dots> = norm (f x) * (B * K)"
```
```   708       by (rule mult_assoc)
```
```   709     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   710   qed
```
```   711   with f show ?thesis
```
```   712     by (rule Zfun_imp_Zfun)
```
```   713 qed
```
```   714
```
```   715 lemma (in bounded_bilinear) flip:
```
```   716   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   717   apply default
```
```   718   apply (rule add_right)
```
```   719   apply (rule add_left)
```
```   720   apply (rule scaleR_right)
```
```   721   apply (rule scaleR_left)
```
```   722   apply (subst mult_commute)
```
```   723   using bounded by fast
```
```   724
```
```   725 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   726   assumes f: "Bfun f F"
```
```   727   assumes g: "Zfun g F"
```
```   728   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   729   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   730
```
```   731 lemma Bfun_inverse_lemma:
```
```   732   fixes x :: "'a::real_normed_div_algebra"
```
```   733   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   734   apply (subst nonzero_norm_inverse, clarsimp)
```
```   735   apply (erule (1) le_imp_inverse_le)
```
```   736   done
```
```   737
```
```   738 lemma Bfun_inverse:
```
```   739   fixes a :: "'a::real_normed_div_algebra"
```
```   740   assumes f: "(f ---> a) F"
```
```   741   assumes a: "a \<noteq> 0"
```
```   742   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```   743 proof -
```
```   744   from a have "0 < norm a" by simp
```
```   745   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   746   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   747   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```   748     using tendstoD [OF f r1] by fast
```
```   749   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```   750   proof eventually_elim
```
```   751     case (elim x)
```
```   752     hence 1: "norm (f x - a) < r"
```
```   753       by (simp add: dist_norm)
```
```   754     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   755     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   756       by (rule nonzero_norm_inverse)
```
```   757     also have "\<dots> \<le> inverse (norm a - r)"
```
```   758     proof (rule le_imp_inverse_le)
```
```   759       show "0 < norm a - r" using r2 by simp
```
```   760     next
```
```   761       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   762         by (rule norm_triangle_ineq2)
```
```   763       also have "\<dots> = norm (f x - a)"
```
```   764         by (rule norm_minus_commute)
```
```   765       also have "\<dots> < r" using 1 .
```
```   766       finally show "norm a - r \<le> norm (f x)" by simp
```
```   767     qed
```
```   768     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   769   qed
```
```   770   thus ?thesis by (rule BfunI)
```
```   771 qed
```
```   772
```
```   773 lemma tendsto_inverse [tendsto_intros]:
```
```   774   fixes a :: "'a::real_normed_div_algebra"
```
```   775   assumes f: "(f ---> a) F"
```
```   776   assumes a: "a \<noteq> 0"
```
```   777   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
```
```   778 proof -
```
```   779   from a have "0 < norm a" by simp
```
```   780   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```   781     by (rule tendstoD)
```
```   782   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```   783     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   784   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```   785     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   786     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
```
```   787   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   788     by (intro Zfun_minus Zfun_mult_left
```
```   789       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```   790       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```   791   ultimately show ?thesis
```
```   792     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```   793 qed
```
```   794
```
```   795 lemma continuous_inverse:
```
```   796   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   797   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   798   shows "continuous F (\<lambda>x. inverse (f x))"
```
```   799   using assms unfolding continuous_def by (rule tendsto_inverse)
```
```   800
```
```   801 lemma continuous_at_within_inverse[continuous_intros]:
```
```   802   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   803   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   804   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
```
```   805   using assms unfolding continuous_within by (rule tendsto_inverse)
```
```   806
```
```   807 lemma isCont_inverse[continuous_intros, simp]:
```
```   808   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   809   assumes "isCont f a" and "f a \<noteq> 0"
```
```   810   shows "isCont (\<lambda>x. inverse (f x)) a"
```
```   811   using assms unfolding continuous_at by (rule tendsto_inverse)
```
```   812
```
```   813 lemma continuous_on_inverse[continuous_on_intros]:
```
```   814   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   815   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   816   shows "continuous_on s (\<lambda>x. inverse (f x))"
```
```   817   using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
```
```   818
```
```   819 lemma tendsto_divide [tendsto_intros]:
```
```   820   fixes a b :: "'a::real_normed_field"
```
```   821   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
```
```   822     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
```
```   823   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```   824
```
```   825 lemma continuous_divide:
```
```   826   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   827   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   828   shows "continuous F (\<lambda>x. (f x) / (g x))"
```
```   829   using assms unfolding continuous_def by (rule tendsto_divide)
```
```   830
```
```   831 lemma continuous_at_within_divide[continuous_intros]:
```
```   832   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   833   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
```
```   834   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
```
```   835   using assms unfolding continuous_within by (rule tendsto_divide)
```
```   836
```
```   837 lemma isCont_divide[continuous_intros, simp]:
```
```   838   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   839   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
```
```   840   shows "isCont (\<lambda>x. (f x) / g x) a"
```
```   841   using assms unfolding continuous_at by (rule tendsto_divide)
```
```   842
```
```   843 lemma continuous_on_divide[continuous_on_intros]:
```
```   844   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   845   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
```
```   846   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
```
```   847   using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
```
```   848
```
```   849 lemma tendsto_sgn [tendsto_intros]:
```
```   850   fixes l :: "'a::real_normed_vector"
```
```   851   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
```
```   852   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```   853
```
```   854 lemma continuous_sgn:
```
```   855   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   856   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   857   shows "continuous F (\<lambda>x. sgn (f x))"
```
```   858   using assms unfolding continuous_def by (rule tendsto_sgn)
```
```   859
```
```   860 lemma continuous_at_within_sgn[continuous_intros]:
```
```   861   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   862   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   863   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
```
```   864   using assms unfolding continuous_within by (rule tendsto_sgn)
```
```   865
```
```   866 lemma isCont_sgn[continuous_intros]:
```
```   867   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   868   assumes "isCont f a" and "f a \<noteq> 0"
```
```   869   shows "isCont (\<lambda>x. sgn (f x)) a"
```
```   870   using assms unfolding continuous_at by (rule tendsto_sgn)
```
```   871
```
```   872 lemma continuous_on_sgn[continuous_on_intros]:
```
```   873   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   874   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   875   shows "continuous_on s (\<lambda>x. sgn (f x))"
```
```   876   using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
```
```   877
```
```   878 lemma filterlim_at_infinity:
```
```   879   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
```
```   880   assumes "0 \<le> c"
```
```   881   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```   882   unfolding filterlim_iff eventually_at_infinity
```
```   883 proof safe
```
```   884   fix P :: "'a \<Rightarrow> bool" and b
```
```   885   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```   886     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```   887   have "max b (c + 1) > c" by auto
```
```   888   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```   889     by auto
```
```   890   then show "eventually (\<lambda>x. P (f x)) F"
```
```   891   proof eventually_elim
```
```   892     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```   893     with P show "P (f x)" by auto
```
```   894   qed
```
```   895 qed force
```
```   896
```
```   897
```
```   898 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
```
```   899
```
```   900 text {*
```
```   901
```
```   902 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
```
```   903 @{term "at_right x"} and also @{term "at_right 0"}.
```
```   904
```
```   905 *}
```
```   906
```
```   907 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
```
```   908
```
```   909 lemma filtermap_homeomorph:
```
```   910   assumes f: "continuous (at a) f"
```
```   911   assumes g: "continuous (at (f a)) g"
```
```   912   assumes bij1: "\<forall>x. f (g x) = x" and bij2: "\<forall>x. g (f x) = x"
```
```   913   shows "filtermap f (nhds a) = nhds (f a)"
```
```   914   unfolding filter_eq_iff eventually_filtermap eventually_nhds
```
```   915 proof safe
```
```   916   fix P S assume S: "open S" "f a \<in> S" and P: "\<forall>x\<in>S. P x"
```
```   917   from continuous_within_topological[THEN iffD1, rule_format, OF f S] P
```
```   918   show "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P (f x))" by auto
```
```   919 next
```
```   920   fix P S assume S: "open S" "a \<in> S" and P: "\<forall>x\<in>S. P (f x)"
```
```   921   with continuous_within_topological[THEN iffD1, rule_format, OF g, of S] bij2
```
```   922   obtain A where "open A" "f a \<in> A" "(\<forall>y\<in>A. g y \<in> S)"
```
```   923     by (metis UNIV_I)
```
```   924   with P bij1 show "\<exists>S. open S \<and> f a \<in> S \<and> (\<forall>x\<in>S. P x)"
```
```   925     by (force intro!: exI[of _ A])
```
```   926 qed
```
```   927
```
```   928 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
```
```   929   by (rule filtermap_homeomorph[where g="\<lambda>x. x + d"]) (auto intro: continuous_intros)
```
```   930
```
```   931 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
```
```   932   by (rule filtermap_homeomorph[where g=uminus]) (auto intro: continuous_minus)
```
```   933
```
```   934 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
```
```   935   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
```
```   936
```
```   937 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
```
```   938   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
```
```   939
```
```   940 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
```
```   941   using filtermap_at_right_shift[of "-a" 0] by simp
```
```   942
```
```   943 lemma filterlim_at_right_to_0:
```
```   944   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
```
```   945   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
```
```   946
```
```   947 lemma eventually_at_right_to_0:
```
```   948   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
```
```   949   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
```
```   950
```
```   951 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
```
```   952   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```   953
```
```   954 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
```
```   955   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```   956
```
```   957 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
```
```   958   by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
```
```   959
```
```   960 lemma filterlim_at_left_to_right:
```
```   961   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
```
```   962   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
```
```   963
```
```   964 lemma eventually_at_left_to_right:
```
```   965   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
```
```   966   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
```
```   967
```
```   968 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
```
```   969   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
```
```   970   by (metis le_minus_iff minus_minus)
```
```   971
```
```   972 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
```
```   973   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
```
```   974
```
```   975 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
```
```   976   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
```
```   977
```
```   978 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
```
```   979   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
```
```   980
```
```   981 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```   982   unfolding filterlim_at_top eventually_at_bot_dense
```
```   983   by (metis leI minus_less_iff order_less_asym)
```
```   984
```
```   985 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```   986   unfolding filterlim_at_bot eventually_at_top_dense
```
```   987   by (metis leI less_minus_iff order_less_asym)
```
```   988
```
```   989 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
```
```   990   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
```
```   991   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
```
```   992   by auto
```
```   993
```
```   994 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
```
```   995   unfolding filterlim_uminus_at_top by simp
```
```   996
```
```   997 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
```
```   998   unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
```
```   999 proof safe
```
```  1000   fix Z :: real assume [arith]: "0 < Z"
```
```  1001   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```  1002     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```  1003   then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
```
```  1004     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```  1005 qed
```
```  1006
```
```  1007 lemma filterlim_inverse_at_top:
```
```  1008   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```  1009   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
```
```  1010      (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
```
```  1011
```
```  1012 lemma filterlim_inverse_at_bot_neg:
```
```  1013   "LIM x (at_left (0::real)). inverse x :> at_bot"
```
```  1014   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
```
```  1015
```
```  1016 lemma filterlim_inverse_at_bot:
```
```  1017   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```  1018   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
```
```  1019   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
```
```  1020
```
```  1021 lemma tendsto_inverse_0:
```
```  1022   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
```
```  1023   shows "(inverse ---> (0::'a)) at_infinity"
```
```  1024   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```  1025 proof safe
```
```  1026   fix r :: real assume "0 < r"
```
```  1027   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```  1028   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```  1029     fix x :: 'a
```
```  1030     from `0 < r` have "0 < inverse (r / 2)" by simp
```
```  1031     also assume *: "inverse (r / 2) \<le> norm x"
```
```  1032     finally show "norm (inverse x) < r"
```
```  1033       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```  1034   qed
```
```  1035 qed
```
```  1036
```
```  1037 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
```
```  1038 proof (rule antisym)
```
```  1039   have "(inverse ---> (0::real)) at_top"
```
```  1040     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
```
```  1041   then show "filtermap inverse at_top \<le> at_right (0::real)"
```
```  1042     by (simp add: le_principal eventually_filtermap eventually_gt_at_top filterlim_def at_within_def)
```
```  1043 next
```
```  1044   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
```
```  1045     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
```
```  1046   then show "at_right (0::real) \<le> filtermap inverse at_top"
```
```  1047     by (simp add: filtermap_ident filtermap_filtermap)
```
```  1048 qed
```
```  1049
```
```  1050 lemma eventually_at_right_to_top:
```
```  1051   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
```
```  1052   unfolding at_right_to_top eventually_filtermap ..
```
```  1053
```
```  1054 lemma filterlim_at_right_to_top:
```
```  1055   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
```
```  1056   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
```
```  1057
```
```  1058 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
```
```  1059   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
```
```  1060
```
```  1061 lemma eventually_at_top_to_right:
```
```  1062   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
```
```  1063   unfolding at_top_to_right eventually_filtermap ..
```
```  1064
```
```  1065 lemma filterlim_at_top_to_right:
```
```  1066   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
```
```  1067   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
```
```  1068
```
```  1069 lemma filterlim_inverse_at_infinity:
```
```  1070   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1071   shows "filterlim inverse at_infinity (at (0::'a))"
```
```  1072   unfolding filterlim_at_infinity[OF order_refl]
```
```  1073 proof safe
```
```  1074   fix r :: real assume "0 < r"
```
```  1075   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```  1076     unfolding eventually_at norm_inverse
```
```  1077     by (intro exI[of _ "inverse r"])
```
```  1078        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```  1079 qed
```
```  1080
```
```  1081 lemma filterlim_inverse_at_iff:
```
```  1082   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1083   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```  1084   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```  1085 proof
```
```  1086   assume "filtermap g F \<le> at_infinity"
```
```  1087   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```  1088     by (rule filtermap_mono)
```
```  1089   also have "\<dots> \<le> at 0"
```
```  1090     using tendsto_inverse_0[where 'a='b]
```
```  1091     by (auto intro!: exI[of _ 1]
```
```  1092              simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
```
```  1093   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```  1094 next
```
```  1095   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```  1096   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```  1097     by (rule filtermap_mono)
```
```  1098   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```  1099     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```  1100 qed
```
```  1101
```
```  1102 lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
```
```  1103  by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
```
```  1104
```
```  1105 text {*
```
```  1106
```
```  1107 We only show rules for multiplication and addition when the functions are either against a real
```
```  1108 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```  1109
```
```  1110 *}
```
```  1111
```
```  1112 lemma filterlim_tendsto_pos_mult_at_top:
```
```  1113   assumes f: "(f ---> c) F" and c: "0 < c"
```
```  1114   assumes g: "LIM x F. g x :> at_top"
```
```  1115   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1116   unfolding filterlim_at_top_gt[where c=0]
```
```  1117 proof safe
```
```  1118   fix Z :: real assume "0 < Z"
```
```  1119   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
```
```  1120     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
```
```  1121              simp: dist_real_def abs_real_def split: split_if_asm)
```
```  1122   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
```
```  1123     unfolding filterlim_at_top by auto
```
```  1124   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1125   proof eventually_elim
```
```  1126     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
```
```  1127     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
```
```  1128       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1129     with `0 < c` show "Z \<le> f x * g x"
```
```  1130        by simp
```
```  1131   qed
```
```  1132 qed
```
```  1133
```
```  1134 lemma filterlim_at_top_mult_at_top:
```
```  1135   assumes f: "LIM x F. f x :> at_top"
```
```  1136   assumes g: "LIM x F. g x :> at_top"
```
```  1137   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```  1138   unfolding filterlim_at_top_gt[where c=0]
```
```  1139 proof safe
```
```  1140   fix Z :: real assume "0 < Z"
```
```  1141   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
```
```  1142     unfolding filterlim_at_top by auto
```
```  1143   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1144     unfolding filterlim_at_top by auto
```
```  1145   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```  1146   proof eventually_elim
```
```  1147     fix x assume "1 \<le> f x" "Z \<le> g x"
```
```  1148     with `0 < Z` have "1 * Z \<le> f x * g x"
```
```  1149       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```  1150     then show "Z \<le> f x * g x"
```
```  1151        by simp
```
```  1152   qed
```
```  1153 qed
```
```  1154
```
```  1155 lemma filterlim_tendsto_pos_mult_at_bot:
```
```  1156   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
```
```  1157   shows "LIM x F. f x * g x :> at_bot"
```
```  1158   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
```
```  1159   unfolding filterlim_uminus_at_bot by simp
```
```  1160
```
```  1161 lemma filterlim_tendsto_add_at_top:
```
```  1162   assumes f: "(f ---> c) F"
```
```  1163   assumes g: "LIM x F. g x :> at_top"
```
```  1164   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1165   unfolding filterlim_at_top_gt[where c=0]
```
```  1166 proof safe
```
```  1167   fix Z :: real assume "0 < Z"
```
```  1168   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```  1169     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
```
```  1170   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
```
```  1171     unfolding filterlim_at_top by auto
```
```  1172   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1173     by eventually_elim simp
```
```  1174 qed
```
```  1175
```
```  1176 lemma LIM_at_top_divide:
```
```  1177   fixes f g :: "'a \<Rightarrow> real"
```
```  1178   assumes f: "(f ---> a) F" "0 < a"
```
```  1179   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
```
```  1180   shows "LIM x F. f x / g x :> at_top"
```
```  1181   unfolding divide_inverse
```
```  1182   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
```
```  1183
```
```  1184 lemma filterlim_at_top_add_at_top:
```
```  1185   assumes f: "LIM x F. f x :> at_top"
```
```  1186   assumes g: "LIM x F. g x :> at_top"
```
```  1187   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```  1188   unfolding filterlim_at_top_gt[where c=0]
```
```  1189 proof safe
```
```  1190   fix Z :: real assume "0 < Z"
```
```  1191   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
```
```  1192     unfolding filterlim_at_top by auto
```
```  1193   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```  1194     unfolding filterlim_at_top by auto
```
```  1195   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```  1196     by eventually_elim simp
```
```  1197 qed
```
```  1198
```
```  1199 lemma tendsto_divide_0:
```
```  1200   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1201   assumes f: "(f ---> c) F"
```
```  1202   assumes g: "LIM x F. g x :> at_infinity"
```
```  1203   shows "((\<lambda>x. f x / g x) ---> 0) F"
```
```  1204   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
```
```  1205
```
```  1206 lemma linear_plus_1_le_power:
```
```  1207   fixes x :: real
```
```  1208   assumes x: "0 \<le> x"
```
```  1209   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1210 proof (induct n)
```
```  1211   case (Suc n)
```
```  1212   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
```
```  1213     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
```
```  1214   also have "\<dots> \<le> (x + 1)^Suc n"
```
```  1215     using Suc x by (simp add: mult_left_mono)
```
```  1216   finally show ?case .
```
```  1217 qed simp
```
```  1218
```
```  1219 lemma filterlim_realpow_sequentially_gt1:
```
```  1220   fixes x :: "'a :: real_normed_div_algebra"
```
```  1221   assumes x[arith]: "1 < norm x"
```
```  1222   shows "LIM n sequentially. x ^ n :> at_infinity"
```
```  1223 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
```
```  1224   fix y :: real assume "0 < y"
```
```  1225   have "0 < norm x - 1" by simp
```
```  1226   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
```
```  1227   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
```
```  1228   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
```
```  1229   also have "\<dots> = norm x ^ N" by simp
```
```  1230   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
```
```  1231     by (metis order_less_le_trans power_increasing order_less_imp_le x)
```
```  1232   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
```
```  1233     unfolding eventually_sequentially
```
```  1234     by (auto simp: norm_power)
```
```  1235 qed simp
```
```  1236
```
```  1237
```
```  1238 subsection {* Limits of Sequences *}
```
```  1239
```
```  1240 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
```
```  1241   by simp
```
```  1242
```
```  1243 lemma LIMSEQ_iff:
```
```  1244   fixes L :: "'a::real_normed_vector"
```
```  1245   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```  1246 unfolding LIMSEQ_def dist_norm ..
```
```  1247
```
```  1248 lemma LIMSEQ_I:
```
```  1249   fixes L :: "'a::real_normed_vector"
```
```  1250   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```  1251 by (simp add: LIMSEQ_iff)
```
```  1252
```
```  1253 lemma LIMSEQ_D:
```
```  1254   fixes L :: "'a::real_normed_vector"
```
```  1255   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```  1256 by (simp add: LIMSEQ_iff)
```
```  1257
```
```  1258 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```  1259   unfolding tendsto_def eventually_sequentially
```
```  1260   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```  1261
```
```  1262 lemma Bseq_inverse_lemma:
```
```  1263   fixes x :: "'a::real_normed_div_algebra"
```
```  1264   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```  1265 apply (subst nonzero_norm_inverse, clarsimp)
```
```  1266 apply (erule (1) le_imp_inverse_le)
```
```  1267 done
```
```  1268
```
```  1269 lemma Bseq_inverse:
```
```  1270   fixes a :: "'a::real_normed_div_algebra"
```
```  1271   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```  1272   by (rule Bfun_inverse)
```
```  1273
```
```  1274 lemma LIMSEQ_diff_approach_zero:
```
```  1275   fixes L :: "'a::real_normed_vector"
```
```  1276   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
```
```  1277   by (drule (1) tendsto_add, simp)
```
```  1278
```
```  1279 lemma LIMSEQ_diff_approach_zero2:
```
```  1280   fixes L :: "'a::real_normed_vector"
```
```  1281   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
```
```  1282   by (drule (1) tendsto_diff, simp)
```
```  1283
```
```  1284 text{*An unbounded sequence's inverse tends to 0*}
```
```  1285
```
```  1286 lemma LIMSEQ_inverse_zero:
```
```  1287   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```  1288   apply (rule filterlim_compose[OF tendsto_inverse_0])
```
```  1289   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
```
```  1290   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
```
```  1291   done
```
```  1292
```
```  1293 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```  1294
```
```  1295 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```  1296   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
```
```  1297             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
```
```  1298
```
```  1299 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```  1300 infinity is now easily proved*}
```
```  1301
```
```  1302 lemma LIMSEQ_inverse_real_of_nat_add:
```
```  1303      "(%n. r + inverse(real(Suc n))) ----> r"
```
```  1304   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
```
```  1305
```
```  1306 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```  1307      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```  1308   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
```
```  1309   by auto
```
```  1310
```
```  1311 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```  1312      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```  1313   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
```
```  1314   by auto
```
```  1315
```
```  1316 subsection {* Convergence on sequences *}
```
```  1317
```
```  1318 lemma convergent_add:
```
```  1319   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1320   assumes "convergent (\<lambda>n. X n)"
```
```  1321   assumes "convergent (\<lambda>n. Y n)"
```
```  1322   shows "convergent (\<lambda>n. X n + Y n)"
```
```  1323   using assms unfolding convergent_def by (fast intro: tendsto_add)
```
```  1324
```
```  1325 lemma convergent_setsum:
```
```  1326   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
```
```  1327   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
```
```  1328   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
```
```  1329 proof (cases "finite A")
```
```  1330   case True from this and assms show ?thesis
```
```  1331     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
```
```  1332 qed (simp add: convergent_const)
```
```  1333
```
```  1334 lemma (in bounded_linear) convergent:
```
```  1335   assumes "convergent (\<lambda>n. X n)"
```
```  1336   shows "convergent (\<lambda>n. f (X n))"
```
```  1337   using assms unfolding convergent_def by (fast intro: tendsto)
```
```  1338
```
```  1339 lemma (in bounded_bilinear) convergent:
```
```  1340   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
```
```  1341   shows "convergent (\<lambda>n. X n ** Y n)"
```
```  1342   using assms unfolding convergent_def by (fast intro: tendsto)
```
```  1343
```
```  1344 lemma convergent_minus_iff:
```
```  1345   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1346   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```  1347 apply (simp add: convergent_def)
```
```  1348 apply (auto dest: tendsto_minus)
```
```  1349 apply (drule tendsto_minus, auto)
```
```  1350 done
```
```  1351
```
```  1352
```
```  1353 text {* A monotone sequence converges to its least upper bound. *}
```
```  1354
```
```  1355 lemma isLub_mono_imp_LIMSEQ:
```
```  1356   fixes X :: "nat \<Rightarrow> real"
```
```  1357   assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
```
```  1358   assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
```
```  1359   shows "X ----> u"
```
```  1360 proof (rule LIMSEQ_I)
```
```  1361   have 1: "\<forall>n. X n \<le> u"
```
```  1362     using isLubD2 [OF u] by auto
```
```  1363   have "\<forall>y. (\<forall>n. X n \<le> y) \<longrightarrow> u \<le> y"
```
```  1364     using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
```
```  1365   hence 2: "\<forall>y<u. \<exists>n. y < X n"
```
```  1366     by (metis not_le)
```
```  1367   fix r :: real assume "0 < r"
```
```  1368   hence "u - r < u" by simp
```
```  1369   hence "\<exists>m. u - r < X m" using 2 by simp
```
```  1370   then obtain m where "u - r < X m" ..
```
```  1371   with X have "\<forall>n\<ge>m. u - r < X n"
```
```  1372     by (fast intro: less_le_trans)
```
```  1373   hence "\<exists>m. \<forall>n\<ge>m. u - r < X n" ..
```
```  1374   thus "\<exists>m. \<forall>n\<ge>m. norm (X n - u) < r"
```
```  1375     using 1 by (simp add: diff_less_eq add_commute)
```
```  1376 qed
```
```  1377
```
```  1378 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```  1379
```
```  1380 lemma Bseq_mono_convergent:
```
```  1381    "Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
```
```  1382   by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
```
```  1383
```
```  1384 text{*Main monotonicity theorem*}
```
```  1385
```
```  1386 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
```
```  1387   by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
```
```  1388             Bseq_mono_convergent)
```
```  1389
```
```  1390 lemma Cauchy_iff:
```
```  1391   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1392   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```  1393   unfolding Cauchy_def dist_norm ..
```
```  1394
```
```  1395 lemma CauchyI:
```
```  1396   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1397   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"
```
```  1398 by (simp add: Cauchy_iff)
```
```  1399
```
```  1400 lemma CauchyD:
```
```  1401   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1402   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1403 by (simp add: Cauchy_iff)
```
```  1404
```
```  1405 lemma incseq_convergent:
```
```  1406   fixes X :: "nat \<Rightarrow> real"
```
```  1407   assumes "incseq X" and "\<forall>i. X i \<le> B"
```
```  1408   obtains L where "X ----> L" "\<forall>i. X i \<le> L"
```
```  1409 proof atomize_elim
```
```  1410   from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
```
```  1411   obtain L where "X ----> L"
```
```  1412     by (auto simp: convergent_def monoseq_def incseq_def)
```
```  1413   with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
```
```  1414     by (auto intro!: exI[of _ L] incseq_le)
```
```  1415 qed
```
```  1416
```
```  1417 lemma decseq_convergent:
```
```  1418   fixes X :: "nat \<Rightarrow> real"
```
```  1419   assumes "decseq X" and "\<forall>i. B \<le> X i"
```
```  1420   obtains L where "X ----> L" "\<forall>i. L \<le> X i"
```
```  1421 proof atomize_elim
```
```  1422   from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
```
```  1423   obtain L where "X ----> L"
```
```  1424     by (auto simp: convergent_def monoseq_def decseq_def)
```
```  1425   with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
```
```  1426     by (auto intro!: exI[of _ L] decseq_le)
```
```  1427 qed
```
```  1428
```
```  1429 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1430
```
```  1431 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1432   proof mechanization rather than the nonstandard proof*}
```
```  1433
```
```  1434 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1435           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1436 apply (clarify, drule spec, drule (1) mp)
```
```  1437 apply (simp only: norm_minus_commute)
```
```  1438 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1439 apply simp
```
```  1440 done
```
```  1441
```
```  1442 subsection {* Power Sequences *}
```
```  1443
```
```  1444 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1445 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1446   also fact that bounded and monotonic sequence converges.*}
```
```  1447
```
```  1448 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1449 apply (simp add: Bseq_def)
```
```  1450 apply (rule_tac x = 1 in exI)
```
```  1451 apply (simp add: power_abs)
```
```  1452 apply (auto dest: power_mono)
```
```  1453 done
```
```  1454
```
```  1455 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1456 apply (clarify intro!: mono_SucI2)
```
```  1457 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1458 done
```
```  1459
```
```  1460 lemma convergent_realpow:
```
```  1461   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1462 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1463
```
```  1464 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1465   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
```
```  1466
```
```  1467 lemma LIMSEQ_realpow_zero:
```
```  1468   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1469 proof cases
```
```  1470   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1471   hence x0: "0 < x" by simp
```
```  1472   assume x1: "x < 1"
```
```  1473   from x0 x1 have "1 < inverse x"
```
```  1474     by (rule one_less_inverse)
```
```  1475   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1476     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1477   thus ?thesis by (simp add: power_inverse)
```
```  1478 qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
```
```  1479
```
```  1480 lemma LIMSEQ_power_zero:
```
```  1481   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1482   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1483 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1484 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
```
```  1485 apply (simp add: power_abs norm_power_ineq)
```
```  1486 done
```
```  1487
```
```  1488 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
```
```  1489   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
```
```  1490
```
```  1491 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1492
```
```  1493 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
```
```  1494   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1495
```
```  1496 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
```
```  1497   by (rule LIMSEQ_power_zero) simp
```
```  1498
```
```  1499
```
```  1500 subsection {* Limits of Functions *}
```
```  1501
```
```  1502 lemma LIM_eq:
```
```  1503   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1504   shows "f -- a --> L =
```
```  1505      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```  1506 by (simp add: LIM_def dist_norm)
```
```  1507
```
```  1508 lemma LIM_I:
```
```  1509   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1510   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```  1511       ==> f -- a --> L"
```
```  1512 by (simp add: LIM_eq)
```
```  1513
```
```  1514 lemma LIM_D:
```
```  1515   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1516   shows "[| f -- a --> L; 0<r |]
```
```  1517       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```  1518 by (simp add: LIM_eq)
```
```  1519
```
```  1520 lemma LIM_offset:
```
```  1521   fixes a :: "'a::real_normed_vector"
```
```  1522   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```  1523   unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
```
```  1524
```
```  1525 lemma LIM_offset_zero:
```
```  1526   fixes a :: "'a::real_normed_vector"
```
```  1527   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1528 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
```
```  1529
```
```  1530 lemma LIM_offset_zero_cancel:
```
```  1531   fixes a :: "'a::real_normed_vector"
```
```  1532   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```  1533 by (drule_tac k="- a" in LIM_offset, simp)
```
```  1534
```
```  1535 lemma LIM_offset_zero_iff:
```
```  1536   fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
```
```  1537   shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1538   using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
```
```  1539
```
```  1540 lemma LIM_zero:
```
```  1541   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1542   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
```
```  1543 unfolding tendsto_iff dist_norm by simp
```
```  1544
```
```  1545 lemma LIM_zero_cancel:
```
```  1546   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1547   shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
```
```  1548 unfolding tendsto_iff dist_norm by simp
```
```  1549
```
```  1550 lemma LIM_zero_iff:
```
```  1551   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  1552   shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
```
```  1553 unfolding tendsto_iff dist_norm by simp
```
```  1554
```
```  1555 lemma LIM_imp_LIM:
```
```  1556   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1557   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```  1558   assumes f: "f -- a --> l"
```
```  1559   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```  1560   shows "g -- a --> m"
```
```  1561   by (rule metric_LIM_imp_LIM [OF f],
```
```  1562     simp add: dist_norm le)
```
```  1563
```
```  1564 lemma LIM_equal2:
```
```  1565   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1566   assumes 1: "0 < R"
```
```  1567   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1568   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```  1569 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
```
```  1570
```
```  1571 lemma LIM_compose2:
```
```  1572   fixes a :: "'a::real_normed_vector"
```
```  1573   assumes f: "f -- a --> b"
```
```  1574   assumes g: "g -- b --> c"
```
```  1575   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```  1576   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  1577 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```  1578
```
```  1579 lemma real_LIM_sandwich_zero:
```
```  1580   fixes f g :: "'a::topological_space \<Rightarrow> real"
```
```  1581   assumes f: "f -- a --> 0"
```
```  1582   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```  1583   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```  1584   shows "g -- a --> 0"
```
```  1585 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
```
```  1586   fix x assume x: "x \<noteq> a"
```
```  1587   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```  1588   also have "g x \<le> f x" by (rule 2 [OF x])
```
```  1589   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```  1590   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```  1591   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```  1592 qed
```
```  1593
```
```  1594
```
```  1595 subsection {* Continuity *}
```
```  1596
```
```  1597 lemma LIM_isCont_iff:
```
```  1598   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1599   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```  1600 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```  1601
```
```  1602 lemma isCont_iff:
```
```  1603   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1604   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```  1605 by (simp add: isCont_def LIM_isCont_iff)
```
```  1606
```
```  1607 lemma isCont_LIM_compose2:
```
```  1608   fixes a :: "'a::real_normed_vector"
```
```  1609   assumes f [unfolded isCont_def]: "isCont f a"
```
```  1610   assumes g: "g -- f a --> l"
```
```  1611   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```  1612   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```  1613 by (rule LIM_compose2 [OF f g inj])
```
```  1614
```
```  1615
```
```  1616 lemma isCont_norm [simp]:
```
```  1617   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1618   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```  1619   by (fact continuous_norm)
```
```  1620
```
```  1621 lemma isCont_rabs [simp]:
```
```  1622   fixes f :: "'a::t2_space \<Rightarrow> real"
```
```  1623   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
```
```  1624   by (fact continuous_rabs)
```
```  1625
```
```  1626 lemma isCont_add [simp]:
```
```  1627   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1628   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```  1629   by (fact continuous_add)
```
```  1630
```
```  1631 lemma isCont_minus [simp]:
```
```  1632   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1633   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```  1634   by (fact continuous_minus)
```
```  1635
```
```  1636 lemma isCont_diff [simp]:
```
```  1637   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1638   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```  1639   by (fact continuous_diff)
```
```  1640
```
```  1641 lemma isCont_mult [simp]:
```
```  1642   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
```
```  1643   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```  1644   by (fact continuous_mult)
```
```  1645
```
```  1646 lemma (in bounded_linear) isCont:
```
```  1647   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
```
```  1648   by (fact continuous)
```
```  1649
```
```  1650 lemma (in bounded_bilinear) isCont:
```
```  1651   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```  1652   by (fact continuous)
```
```  1653
```
```  1654 lemmas isCont_scaleR [simp] =
```
```  1655   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
```
```  1656
```
```  1657 lemmas isCont_of_real [simp] =
```
```  1658   bounded_linear.isCont [OF bounded_linear_of_real]
```
```  1659
```
```  1660 lemma isCont_power [simp]:
```
```  1661   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1662   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```  1663   by (fact continuous_power)
```
```  1664
```
```  1665 lemma isCont_setsum [simp]:
```
```  1666   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```  1667   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
```
```  1668   by (auto intro: continuous_setsum)
```
```  1669
```
```  1670 lemmas isCont_intros =
```
```  1671   isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
```
```  1672   isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
```
```  1673   isCont_of_real isCont_power isCont_sgn isCont_setsum
```
```  1674
```
```  1675 subsection {* Uniform Continuity *}
```
```  1676
```
```  1677 definition
```
```  1678   isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
```
```  1679   "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
```
```  1680
```
```  1681 lemma isUCont_isCont: "isUCont f ==> isCont f x"
```
```  1682 by (simp add: isUCont_def isCont_def LIM_def, force)
```
```  1683
```
```  1684 lemma isUCont_Cauchy:
```
```  1685   "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  1686 unfolding isUCont_def
```
```  1687 apply (rule metric_CauchyI)
```
```  1688 apply (drule_tac x=e in spec, safe)
```
```  1689 apply (drule_tac e=s in metric_CauchyD, safe)
```
```  1690 apply (rule_tac x=M in exI, simp)
```
```  1691 done
```
```  1692
```
```  1693 lemma (in bounded_linear) isUCont: "isUCont f"
```
```  1694 unfolding isUCont_def dist_norm
```
```  1695 proof (intro allI impI)
```
```  1696   fix r::real assume r: "0 < r"
```
```  1697   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```  1698     using pos_bounded by fast
```
```  1699   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```  1700   proof (rule exI, safe)
```
```  1701     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```  1702   next
```
```  1703     fix x y :: 'a
```
```  1704     assume xy: "norm (x - y) < r / K"
```
```  1705     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```  1706     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```  1707     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```  1708     finally show "norm (f x - f y) < r" .
```
```  1709   qed
```
```  1710 qed
```
```  1711
```
```  1712 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  1713 by (rule isUCont [THEN isUCont_Cauchy])
```
```  1714
```
```  1715 lemma LIM_less_bound:
```
```  1716   fixes f :: "real \<Rightarrow> real"
```
```  1717   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
```
```  1718   shows "0 \<le> f x"
```
```  1719 proof (rule tendsto_le_const)
```
```  1720   show "(f ---> f x) (at_left x)"
```
```  1721     using `isCont f x` by (simp add: filterlim_at_split isCont_def)
```
```  1722   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
```
```  1723     using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
```
```  1724 qed simp
```
```  1725
```
```  1726
```
```  1727 subsection {* Nested Intervals and Bisection -- Needed for Compactness *}
```
```  1728
```
```  1729 lemma nested_sequence_unique:
```
```  1730   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"
```
```  1731   shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
```
```  1732 proof -
```
```  1733   have "incseq f" unfolding incseq_Suc_iff by fact
```
```  1734   have "decseq g" unfolding decseq_Suc_iff by fact
```
```  1735
```
```  1736   { fix n
```
```  1737     from `decseq g` have "g n \<le> g 0" by (rule decseqD) simp
```
```  1738     with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f n \<le> g 0" by auto }
```
```  1739   then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
```
```  1740     using incseq_convergent[OF `incseq f`] by auto
```
```  1741   moreover
```
```  1742   { fix n
```
```  1743     from `incseq f` have "f 0 \<le> f n" by (rule incseqD) simp
```
```  1744     with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f 0 \<le> g n" by simp }
```
```  1745   then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
```
```  1746     using decseq_convergent[OF `decseq g`] by auto
```
```  1747   moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF `f ----> u` `g ----> l`]]
```
```  1748   ultimately show ?thesis by auto
```
```  1749 qed
```
```  1750
```
```  1751 lemma Bolzano[consumes 1, case_names trans local]:
```
```  1752   fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
```
```  1753   assumes [arith]: "a \<le> b"
```
```  1754   assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
```
```  1755   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"
```
```  1756   shows "P a b"
```
```  1757 proof -
```
```  1758   def bisect \<equiv> "nat_rec (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
```
```  1759   def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
```
```  1760   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)"
```
```  1761     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)"
```
```  1762     by (simp_all add: l_def u_def bisect_def split: prod.split)
```
```  1763
```
```  1764   { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
```
```  1765
```
```  1766   have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
```
```  1767   proof (safe intro!: nested_sequence_unique)
```
```  1768     fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
```
```  1769   next
```
```  1770     { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
```
```  1771     then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
```
```  1772   qed fact
```
```  1773   then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
```
```  1774   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"
```
```  1775     using `l 0 \<le> x` `x \<le> u 0` local[of x] by auto
```
```  1776
```
```  1777   show "P a b"
```
```  1778   proof (rule ccontr)
```
```  1779     assume "\<not> P a b"
```
```  1780     { fix n have "\<not> P (l n) (u n)"
```
```  1781       proof (induct n)
```
```  1782         case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
```
```  1783       qed (simp add: `\<not> P a b`) }
```
```  1784     moreover
```
```  1785     { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
```
```  1786         using `0 < d` `l ----> x` by (intro order_tendstoD[of _ x]) auto
```
```  1787       moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
```
```  1788         using `0 < d` `u ----> x` by (intro order_tendstoD[of _ x]) auto
```
```  1789       ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
```
```  1790       proof eventually_elim
```
```  1791         fix n assume "x - d / 2 < l n" "u n < x + d / 2"
```
```  1792         from add_strict_mono[OF this] have "u n - l n < d" by simp
```
```  1793         with x show "P (l n) (u n)" by (rule d)
```
```  1794       qed }
```
```  1795     ultimately show False by simp
```
```  1796   qed
```
```  1797 qed
```
```  1798
```
```  1799 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
```
```  1800 proof (cases "a \<le> b", rule compactI)
```
```  1801   fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
```
```  1802   def T == "{a .. b}"
```
```  1803   from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
```
```  1804   proof (induct rule: Bolzano)
```
```  1805     case (trans a b c)
```
```  1806     then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
```
```  1807     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"
```
```  1808       by (auto simp: *)
```
```  1809     with trans show ?case
```
```  1810       unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
```
```  1811   next
```
```  1812     case (local x)
```
```  1813     then have "x \<in> \<Union>C" using C by auto
```
```  1814     with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
```
```  1815     then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
```
```  1816       by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
```
```  1817     with `c \<in> C` show ?case
```
```  1818       by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
```
```  1819   qed
```
```  1820 qed simp
```
```  1821
```
```  1822
```
```  1823 subsection {* Boundedness of continuous functions *}
```
```  1824
```
```  1825 text{*By bisection, function continuous on closed interval is bounded above*}
```
```  1826
```
```  1827 lemma isCont_eq_Ub:
```
```  1828   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1829   shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1830     \<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)"
```
```  1831   using continuous_attains_sup[of "{a .. b}" f]
```
```  1832   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  1833
```
```  1834 lemma isCont_eq_Lb:
```
```  1835   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1836   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1837     \<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)"
```
```  1838   using continuous_attains_inf[of "{a .. b}" f]
```
```  1839   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  1840
```
```  1841 lemma isCont_bounded:
```
```  1842   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1843   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"
```
```  1844   using isCont_eq_Ub[of a b f] by auto
```
```  1845
```
```  1846 lemma isCont_has_Ub:
```
```  1847   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1848   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1849     \<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))"
```
```  1850   using isCont_eq_Ub[of a b f] by auto
```
```  1851
```
```  1852 (*HOL style here: object-level formulations*)
```
```  1853 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```  1854       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  1855       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  1856   by (blast intro: IVT)
```
```  1857
```
```  1858 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```  1859       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  1860       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  1861   by (blast intro: IVT2)
```
```  1862
```
```  1863 lemma isCont_Lb_Ub:
```
```  1864   fixes f :: "real \<Rightarrow> real"
```
```  1865   assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1866   shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
```
```  1867                (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
```
```  1868 proof -
```
```  1869   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"
```
```  1870     using isCont_eq_Ub[OF assms] by auto
```
```  1871   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"
```
```  1872     using isCont_eq_Lb[OF assms] by auto
```
```  1873   show ?thesis
```
```  1874     using IVT[of f L _ M] IVT2[of f L _ M] M L assms
```
```  1875     apply (rule_tac x="f L" in exI)
```
```  1876     apply (rule_tac x="f M" in exI)
```
```  1877     apply (cases "L \<le> M")
```
```  1878     apply (simp, metis order_trans)
```
```  1879     apply (simp, metis order_trans)
```
```  1880     done
```
```  1881 qed
```
```  1882
```
```  1883
```
```  1884 text{*Continuity of inverse function*}
```
```  1885
```
```  1886 lemma isCont_inverse_function:
```
```  1887   fixes f g :: "real \<Rightarrow> real"
```
```  1888   assumes d: "0 < d"
```
```  1889       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
```
```  1890       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
```
```  1891   shows "isCont g (f x)"
```
```  1892 proof -
```
```  1893   let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
```
```  1894
```
```  1895   have f: "continuous_on ?D f"
```
```  1896     using cont by (intro continuous_at_imp_continuous_on ballI) auto
```
```  1897   then have g: "continuous_on (f`?D) g"
```
```  1898     using inj by (intro continuous_on_inv) auto
```
```  1899
```
```  1900   from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
```
```  1901     by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
```
```  1902   with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
```
```  1903     by (rule continuous_on_subset)
```
```  1904   moreover
```
```  1905   have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
```
```  1906     using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
```
```  1907   then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
```
```  1908     by auto
```
```  1909   ultimately
```
```  1910   show ?thesis
```
```  1911     by (simp add: continuous_on_eq_continuous_at)
```
```  1912 qed
```
```  1913
```
```  1914 lemma isCont_inverse_function2:
```
```  1915   fixes f g :: "real \<Rightarrow> real" shows
```
```  1916   "\<lbrakk>a < x; x < b;
```
```  1917     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
```
```  1918     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
```
```  1919    \<Longrightarrow> isCont g (f x)"
```
```  1920 apply (rule isCont_inverse_function
```
```  1921        [where f=f and d="min (x - a) (b - x)"])
```
```  1922 apply (simp_all add: abs_le_iff)
```
```  1923 done
```
```  1924
```
```  1925 (* need to rename second isCont_inverse *)
```
```  1926
```
```  1927 lemma isCont_inv_fun:
```
```  1928   fixes f g :: "real \<Rightarrow> real"
```
```  1929   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1930          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1931       ==> isCont g (f x)"
```
```  1932 by (rule isCont_inverse_function)
```
```  1933
```
```  1934 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
```
```  1935 lemma LIM_fun_gt_zero:
```
```  1936   fixes f :: "real \<Rightarrow> real"
```
```  1937   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)"
```
```  1938 apply (drule (1) LIM_D, clarify)
```
```  1939 apply (rule_tac x = s in exI)
```
```  1940 apply (simp add: abs_less_iff)
```
```  1941 done
```
```  1942
```
```  1943 lemma LIM_fun_less_zero:
```
```  1944   fixes f :: "real \<Rightarrow> real"
```
```  1945   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)"
```
```  1946 apply (drule LIM_D [where r="-l"], simp, clarify)
```
```  1947 apply (rule_tac x = s in exI)
```
```  1948 apply (simp add: abs_less_iff)
```
```  1949 done
```
```  1950
```
```  1951 lemma LIM_fun_not_zero:
```
```  1952   fixes f :: "real \<Rightarrow> real"
```
```  1953   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)"
```
```  1954   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
```
```  1955
```
```  1956 end
```
```  1957
```