src/HOL/Limits.thy
 author hoelzl Tue Mar 26 12:21:00 2013 +0100 (2013-03-26) changeset 51529 2d2f59e6055a parent 51526 155263089e7b child 51531 f415febf4234 permissions -rw-r--r--
move theorems about compactness of real closed intervals, the intermediate value theorem, and lemmas about continuity of bijective functions from Deriv.thy to Limits.thy
```     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 (* Unfortunately eventually_within was overwritten by Multivariate_Analysis.
```
```    16    Hence it was references as Limits.eventually_within, but now it is Basic_Topology.eventually_within *)
```
```    17 lemmas eventually_within = eventually_within
```
```    18
```
```    19 subsection {* Filter going to infinity norm *}
```
```    20
```
```    21 definition at_infinity :: "'a::real_normed_vector filter" where
```
```    22   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
```
```    23
```
```    24 lemma eventually_at_infinity:
```
```    25   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
```
```    26 unfolding at_infinity_def
```
```    27 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```    28   fix P Q :: "'a \<Rightarrow> bool"
```
```    29   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
```
```    30   then obtain r s where
```
```    31     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
```
```    32   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
```
```    33   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
```
```    34 qed auto
```
```    35
```
```    36 lemma at_infinity_eq_at_top_bot:
```
```    37   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
```
```    38   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
```
```    39 proof (intro arg_cong[where f=Abs_filter] ext iffI)
```
```    40   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```    41   then guess r ..
```
```    42   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
```
```    43   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
```
```    44 next
```
```    45   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
```
```    46   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
```
```    47   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```    48     by (intro exI[of _ "max p (-q)"])
```
```    49        (auto simp: abs_real_def)
```
```    50 qed
```
```    51
```
```    52 lemma at_top_le_at_infinity:
```
```    53   "at_top \<le> (at_infinity :: real filter)"
```
```    54   unfolding at_infinity_eq_at_top_bot by simp
```
```    55
```
```    56 lemma at_bot_le_at_infinity:
```
```    57   "at_bot \<le> (at_infinity :: real filter)"
```
```    58   unfolding at_infinity_eq_at_top_bot by simp
```
```    59
```
```    60 subsection {* Boundedness *}
```
```    61
```
```    62 lemma Bfun_def:
```
```    63   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
```
```    64   unfolding Bfun_metric_def norm_conv_dist
```
```    65 proof safe
```
```    66   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
```
```    67   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
```
```    68     by (intro always_eventually) (metis dist_commute dist_triangle)
```
```    69   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
```
```    70     by eventually_elim auto
```
```    71   with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
```
```    72     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
```
```    73 qed auto
```
```    74
```
```    75 lemma BfunI:
```
```    76   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
```
```    77 unfolding Bfun_def
```
```    78 proof (intro exI conjI allI)
```
```    79   show "0 < max K 1" by simp
```
```    80 next
```
```    81   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
```
```    82     using K by (rule eventually_elim1, simp)
```
```    83 qed
```
```    84
```
```    85 lemma BfunE:
```
```    86   assumes "Bfun f F"
```
```    87   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
```
```    88 using assms unfolding Bfun_def by fast
```
```    89
```
```    90 subsection {* Convergence to Zero *}
```
```    91
```
```    92 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```    93   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```    94
```
```    95 lemma ZfunI:
```
```    96   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```    97   unfolding Zfun_def by simp
```
```    98
```
```    99 lemma ZfunD:
```
```   100   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```   101   unfolding Zfun_def by simp
```
```   102
```
```   103 lemma Zfun_ssubst:
```
```   104   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```   105   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```   106
```
```   107 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```   108   unfolding Zfun_def by simp
```
```   109
```
```   110 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   111   unfolding Zfun_def by simp
```
```   112
```
```   113 lemma Zfun_imp_Zfun:
```
```   114   assumes f: "Zfun f F"
```
```   115   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   116   shows "Zfun (\<lambda>x. g x) F"
```
```   117 proof (cases)
```
```   118   assume K: "0 < K"
```
```   119   show ?thesis
```
```   120   proof (rule ZfunI)
```
```   121     fix r::real assume "0 < r"
```
```   122     hence "0 < r / K"
```
```   123       using K by (rule divide_pos_pos)
```
```   124     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   125       using ZfunD [OF f] by fast
```
```   126     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   127     proof eventually_elim
```
```   128       case (elim x)
```
```   129       hence "norm (f x) * K < r"
```
```   130         by (simp add: pos_less_divide_eq K)
```
```   131       thus ?case
```
```   132         by (simp add: order_le_less_trans [OF elim(1)])
```
```   133     qed
```
```   134   qed
```
```   135 next
```
```   136   assume "\<not> 0 < K"
```
```   137   hence K: "K \<le> 0" by (simp only: not_less)
```
```   138   show ?thesis
```
```   139   proof (rule ZfunI)
```
```   140     fix r :: real
```
```   141     assume "0 < r"
```
```   142     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   143     proof eventually_elim
```
```   144       case (elim x)
```
```   145       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   146         using K norm_ge_zero by (rule mult_left_mono)
```
```   147       finally show ?case
```
```   148         using `0 < r` by simp
```
```   149     qed
```
```   150   qed
```
```   151 qed
```
```   152
```
```   153 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   154   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   155
```
```   156 lemma Zfun_add:
```
```   157   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   158   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   159 proof (rule ZfunI)
```
```   160   fix r::real assume "0 < r"
```
```   161   hence r: "0 < r / 2" by simp
```
```   162   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   163     using f r by (rule ZfunD)
```
```   164   moreover
```
```   165   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   166     using g r by (rule ZfunD)
```
```   167   ultimately
```
```   168   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   169   proof eventually_elim
```
```   170     case (elim x)
```
```   171     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   172       by (rule norm_triangle_ineq)
```
```   173     also have "\<dots> < r/2 + r/2"
```
```   174       using elim by (rule add_strict_mono)
```
```   175     finally show ?case
```
```   176       by simp
```
```   177   qed
```
```   178 qed
```
```   179
```
```   180 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   181   unfolding Zfun_def by simp
```
```   182
```
```   183 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   184   by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   185
```
```   186 lemma (in bounded_linear) Zfun:
```
```   187   assumes g: "Zfun g F"
```
```   188   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   189 proof -
```
```   190   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   191     using bounded by fast
```
```   192   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   193     by simp
```
```   194   with g show ?thesis
```
```   195     by (rule Zfun_imp_Zfun)
```
```   196 qed
```
```   197
```
```   198 lemma (in bounded_bilinear) Zfun:
```
```   199   assumes f: "Zfun f F"
```
```   200   assumes g: "Zfun g F"
```
```   201   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   202 proof (rule ZfunI)
```
```   203   fix r::real assume r: "0 < r"
```
```   204   obtain K where K: "0 < K"
```
```   205     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   206     using pos_bounded by fast
```
```   207   from K have K': "0 < inverse K"
```
```   208     by (rule positive_imp_inverse_positive)
```
```   209   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   210     using f r by (rule ZfunD)
```
```   211   moreover
```
```   212   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   213     using g K' by (rule ZfunD)
```
```   214   ultimately
```
```   215   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   216   proof eventually_elim
```
```   217     case (elim x)
```
```   218     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   219       by (rule norm_le)
```
```   220     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   221       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   222     also from K have "r * inverse K * K = r"
```
```   223       by simp
```
```   224     finally show ?case .
```
```   225   qed
```
```   226 qed
```
```   227
```
```   228 lemma (in bounded_bilinear) Zfun_left:
```
```   229   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   230   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   231
```
```   232 lemma (in bounded_bilinear) Zfun_right:
```
```   233   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   234   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   235
```
```   236 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   237 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   238 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   239
```
```   240 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   241   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   242
```
```   243 subsubsection {* Distance and norms *}
```
```   244
```
```   245 lemma tendsto_norm [tendsto_intros]:
```
```   246   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
```
```   247   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   248
```
```   249 lemma continuous_norm [continuous_intros]:
```
```   250   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
```
```   251   unfolding continuous_def by (rule tendsto_norm)
```
```   252
```
```   253 lemma continuous_on_norm [continuous_on_intros]:
```
```   254   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
```
```   255   unfolding continuous_on_def by (auto intro: tendsto_norm)
```
```   256
```
```   257 lemma tendsto_norm_zero:
```
```   258   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
```
```   259   by (drule tendsto_norm, simp)
```
```   260
```
```   261 lemma tendsto_norm_zero_cancel:
```
```   262   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
```
```   263   unfolding tendsto_iff dist_norm by simp
```
```   264
```
```   265 lemma tendsto_norm_zero_iff:
```
```   266   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
```
```   267   unfolding tendsto_iff dist_norm by simp
```
```   268
```
```   269 lemma tendsto_rabs [tendsto_intros]:
```
```   270   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
```
```   271   by (fold real_norm_def, rule tendsto_norm)
```
```   272
```
```   273 lemma continuous_rabs [continuous_intros]:
```
```   274   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   275   unfolding real_norm_def[symmetric] by (rule continuous_norm)
```
```   276
```
```   277 lemma continuous_on_rabs [continuous_on_intros]:
```
```   278   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   279   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
```
```   280
```
```   281 lemma tendsto_rabs_zero:
```
```   282   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
```
```   283   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   284
```
```   285 lemma tendsto_rabs_zero_cancel:
```
```   286   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
```
```   287   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   288
```
```   289 lemma tendsto_rabs_zero_iff:
```
```   290   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
```
```   291   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   292
```
```   293 subsubsection {* Addition and subtraction *}
```
```   294
```
```   295 lemma tendsto_add [tendsto_intros]:
```
```   296   fixes a b :: "'a::real_normed_vector"
```
```   297   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
```
```   298   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   299
```
```   300 lemma continuous_add [continuous_intros]:
```
```   301   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   302   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
```
```   303   unfolding continuous_def by (rule tendsto_add)
```
```   304
```
```   305 lemma continuous_on_add [continuous_on_intros]:
```
```   306   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   307   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
```
```   308   unfolding continuous_on_def by (auto intro: tendsto_add)
```
```   309
```
```   310 lemma tendsto_add_zero:
```
```   311   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   312   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
```
```   313   by (drule (1) tendsto_add, simp)
```
```   314
```
```   315 lemma tendsto_minus [tendsto_intros]:
```
```   316   fixes a :: "'a::real_normed_vector"
```
```   317   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
```
```   318   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   319
```
```   320 lemma continuous_minus [continuous_intros]:
```
```   321   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   322   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
```
```   323   unfolding continuous_def by (rule tendsto_minus)
```
```   324
```
```   325 lemma continuous_on_minus [continuous_on_intros]:
```
```   326   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   327   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
```
```   328   unfolding continuous_on_def by (auto intro: tendsto_minus)
```
```   329
```
```   330 lemma tendsto_minus_cancel:
```
```   331   fixes a :: "'a::real_normed_vector"
```
```   332   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
```
```   333   by (drule tendsto_minus, simp)
```
```   334
```
```   335 lemma tendsto_minus_cancel_left:
```
```   336     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
```
```   337   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
```
```   338   by auto
```
```   339
```
```   340 lemma tendsto_diff [tendsto_intros]:
```
```   341   fixes a b :: "'a::real_normed_vector"
```
```   342   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
```
```   343   by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   344
```
```   345 lemma continuous_diff [continuous_intros]:
```
```   346   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   347   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
```
```   348   unfolding continuous_def by (rule tendsto_diff)
```
```   349
```
```   350 lemma continuous_on_diff [continuous_on_intros]:
```
```   351   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   352   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
```
```   353   unfolding continuous_on_def by (auto intro: tendsto_diff)
```
```   354
```
```   355 lemma tendsto_setsum [tendsto_intros]:
```
```   356   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   357   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
```
```   358   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
```
```   359 proof (cases "finite S")
```
```   360   assume "finite S" thus ?thesis using assms
```
```   361     by (induct, simp add: tendsto_const, simp add: tendsto_add)
```
```   362 next
```
```   363   assume "\<not> finite S" thus ?thesis
```
```   364     by (simp add: tendsto_const)
```
```   365 qed
```
```   366
```
```   367 lemma continuous_setsum [continuous_intros]:
```
```   368   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```   369   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
```
```   370   unfolding continuous_def by (rule tendsto_setsum)
```
```   371
```
```   372 lemma continuous_on_setsum [continuous_intros]:
```
```   373   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
```
```   374   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)"
```
```   375   unfolding continuous_on_def by (auto intro: tendsto_setsum)
```
```   376
```
```   377 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
```
```   378
```
```   379 subsubsection {* Linear operators and multiplication *}
```
```   380
```
```   381 lemma (in bounded_linear) tendsto:
```
```   382   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
```
```   383   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   384
```
```   385 lemma (in bounded_linear) continuous:
```
```   386   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
```
```   387   using tendsto[of g _ F] by (auto simp: continuous_def)
```
```   388
```
```   389 lemma (in bounded_linear) continuous_on:
```
```   390   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
```
```   391   using tendsto[of g] by (auto simp: continuous_on_def)
```
```   392
```
```   393 lemma (in bounded_linear) tendsto_zero:
```
```   394   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
```
```   395   by (drule tendsto, simp only: zero)
```
```   396
```
```   397 lemma (in bounded_bilinear) tendsto:
```
```   398   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
```
```   399   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   400                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   401
```
```   402 lemma (in bounded_bilinear) continuous:
```
```   403   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
```
```   404   using tendsto[of f _ F g] by (auto simp: continuous_def)
```
```   405
```
```   406 lemma (in bounded_bilinear) continuous_on:
```
```   407   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
```
```   408   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
```
```   409
```
```   410 lemma (in bounded_bilinear) tendsto_zero:
```
```   411   assumes f: "(f ---> 0) F"
```
```   412   assumes g: "(g ---> 0) F"
```
```   413   shows "((\<lambda>x. f x ** g x) ---> 0) F"
```
```   414   using tendsto [OF f g] by (simp add: zero_left)
```
```   415
```
```   416 lemma (in bounded_bilinear) tendsto_left_zero:
```
```   417   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
```
```   418   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```   419
```
```   420 lemma (in bounded_bilinear) tendsto_right_zero:
```
```   421   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
```
```   422   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```   423
```
```   424 lemmas tendsto_of_real [tendsto_intros] =
```
```   425   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```   426
```
```   427 lemmas tendsto_scaleR [tendsto_intros] =
```
```   428   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```   429
```
```   430 lemmas tendsto_mult [tendsto_intros] =
```
```   431   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```   432
```
```   433 lemmas continuous_of_real [continuous_intros] =
```
```   434   bounded_linear.continuous [OF bounded_linear_of_real]
```
```   435
```
```   436 lemmas continuous_scaleR [continuous_intros] =
```
```   437   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
```
```   438
```
```   439 lemmas continuous_mult [continuous_intros] =
```
```   440   bounded_bilinear.continuous [OF bounded_bilinear_mult]
```
```   441
```
```   442 lemmas continuous_on_of_real [continuous_on_intros] =
```
```   443   bounded_linear.continuous_on [OF bounded_linear_of_real]
```
```   444
```
```   445 lemmas continuous_on_scaleR [continuous_on_intros] =
```
```   446   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
```
```   447
```
```   448 lemmas continuous_on_mult [continuous_on_intros] =
```
```   449   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
```
```   450
```
```   451 lemmas tendsto_mult_zero =
```
```   452   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```   453
```
```   454 lemmas tendsto_mult_left_zero =
```
```   455   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```   456
```
```   457 lemmas tendsto_mult_right_zero =
```
```   458   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```   459
```
```   460 lemma tendsto_power [tendsto_intros]:
```
```   461   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   462   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
```
```   463   by (induct n) (simp_all add: tendsto_const tendsto_mult)
```
```   464
```
```   465 lemma continuous_power [continuous_intros]:
```
```   466   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   467   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
```
```   468   unfolding continuous_def by (rule tendsto_power)
```
```   469
```
```   470 lemma continuous_on_power [continuous_on_intros]:
```
```   471   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   472   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
```
```   473   unfolding continuous_on_def by (auto intro: tendsto_power)
```
```   474
```
```   475 lemma tendsto_setprod [tendsto_intros]:
```
```   476   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   477   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
```
```   478   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
```
```   479 proof (cases "finite S")
```
```   480   assume "finite S" thus ?thesis using assms
```
```   481     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
```
```   482 next
```
```   483   assume "\<not> finite S" thus ?thesis
```
```   484     by (simp add: tendsto_const)
```
```   485 qed
```
```   486
```
```   487 lemma continuous_setprod [continuous_intros]:
```
```   488   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   489   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
```
```   490   unfolding continuous_def by (rule tendsto_setprod)
```
```   491
```
```   492 lemma continuous_on_setprod [continuous_intros]:
```
```   493   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   494   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)"
```
```   495   unfolding continuous_on_def by (auto intro: tendsto_setprod)
```
```   496
```
```   497 subsubsection {* Inverse and division *}
```
```   498
```
```   499 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   500   assumes f: "Zfun f F"
```
```   501   assumes g: "Bfun g F"
```
```   502   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   503 proof -
```
```   504   obtain K where K: "0 \<le> K"
```
```   505     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   506     using nonneg_bounded by fast
```
```   507   obtain B where B: "0 < B"
```
```   508     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```   509     using g by (rule BfunE)
```
```   510   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```   511   using norm_g proof eventually_elim
```
```   512     case (elim x)
```
```   513     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   514       by (rule norm_le)
```
```   515     also have "\<dots> \<le> norm (f x) * B * K"
```
```   516       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   517                 mult_nonneg_nonneg K elim)
```
```   518     also have "\<dots> = norm (f x) * (B * K)"
```
```   519       by (rule mult_assoc)
```
```   520     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   521   qed
```
```   522   with f show ?thesis
```
```   523     by (rule Zfun_imp_Zfun)
```
```   524 qed
```
```   525
```
```   526 lemma (in bounded_bilinear) flip:
```
```   527   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   528   apply default
```
```   529   apply (rule add_right)
```
```   530   apply (rule add_left)
```
```   531   apply (rule scaleR_right)
```
```   532   apply (rule scaleR_left)
```
```   533   apply (subst mult_commute)
```
```   534   using bounded by fast
```
```   535
```
```   536 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   537   assumes f: "Bfun f F"
```
```   538   assumes g: "Zfun g F"
```
```   539   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   540   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   541
```
```   542 lemma Bfun_inverse_lemma:
```
```   543   fixes x :: "'a::real_normed_div_algebra"
```
```   544   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   545   apply (subst nonzero_norm_inverse, clarsimp)
```
```   546   apply (erule (1) le_imp_inverse_le)
```
```   547   done
```
```   548
```
```   549 lemma Bfun_inverse:
```
```   550   fixes a :: "'a::real_normed_div_algebra"
```
```   551   assumes f: "(f ---> a) F"
```
```   552   assumes a: "a \<noteq> 0"
```
```   553   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```   554 proof -
```
```   555   from a have "0 < norm a" by simp
```
```   556   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   557   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   558   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```   559     using tendstoD [OF f r1] by fast
```
```   560   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```   561   proof eventually_elim
```
```   562     case (elim x)
```
```   563     hence 1: "norm (f x - a) < r"
```
```   564       by (simp add: dist_norm)
```
```   565     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   566     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   567       by (rule nonzero_norm_inverse)
```
```   568     also have "\<dots> \<le> inverse (norm a - r)"
```
```   569     proof (rule le_imp_inverse_le)
```
```   570       show "0 < norm a - r" using r2 by simp
```
```   571     next
```
```   572       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   573         by (rule norm_triangle_ineq2)
```
```   574       also have "\<dots> = norm (f x - a)"
```
```   575         by (rule norm_minus_commute)
```
```   576       also have "\<dots> < r" using 1 .
```
```   577       finally show "norm a - r \<le> norm (f x)" by simp
```
```   578     qed
```
```   579     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   580   qed
```
```   581   thus ?thesis by (rule BfunI)
```
```   582 qed
```
```   583
```
```   584 lemma tendsto_inverse [tendsto_intros]:
```
```   585   fixes a :: "'a::real_normed_div_algebra"
```
```   586   assumes f: "(f ---> a) F"
```
```   587   assumes a: "a \<noteq> 0"
```
```   588   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
```
```   589 proof -
```
```   590   from a have "0 < norm a" by simp
```
```   591   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```   592     by (rule tendstoD)
```
```   593   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```   594     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   595   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```   596     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   597     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
```
```   598   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   599     by (intro Zfun_minus Zfun_mult_left
```
```   600       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```   601       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```   602   ultimately show ?thesis
```
```   603     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```   604 qed
```
```   605
```
```   606 lemma continuous_inverse:
```
```   607   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   608   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   609   shows "continuous F (\<lambda>x. inverse (f x))"
```
```   610   using assms unfolding continuous_def by (rule tendsto_inverse)
```
```   611
```
```   612 lemma continuous_at_within_inverse[continuous_intros]:
```
```   613   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   614   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   615   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
```
```   616   using assms unfolding continuous_within by (rule tendsto_inverse)
```
```   617
```
```   618 lemma isCont_inverse[continuous_intros, simp]:
```
```   619   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   620   assumes "isCont f a" and "f a \<noteq> 0"
```
```   621   shows "isCont (\<lambda>x. inverse (f x)) a"
```
```   622   using assms unfolding continuous_at by (rule tendsto_inverse)
```
```   623
```
```   624 lemma continuous_on_inverse[continuous_on_intros]:
```
```   625   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   626   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   627   shows "continuous_on s (\<lambda>x. inverse (f x))"
```
```   628   using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
```
```   629
```
```   630 lemma tendsto_divide [tendsto_intros]:
```
```   631   fixes a b :: "'a::real_normed_field"
```
```   632   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
```
```   633     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
```
```   634   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```   635
```
```   636 lemma continuous_divide:
```
```   637   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   638   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   639   shows "continuous F (\<lambda>x. (f x) / (g x))"
```
```   640   using assms unfolding continuous_def by (rule tendsto_divide)
```
```   641
```
```   642 lemma continuous_at_within_divide[continuous_intros]:
```
```   643   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   644   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
```
```   645   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
```
```   646   using assms unfolding continuous_within by (rule tendsto_divide)
```
```   647
```
```   648 lemma isCont_divide[continuous_intros, simp]:
```
```   649   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   650   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
```
```   651   shows "isCont (\<lambda>x. (f x) / g x) a"
```
```   652   using assms unfolding continuous_at by (rule tendsto_divide)
```
```   653
```
```   654 lemma continuous_on_divide[continuous_on_intros]:
```
```   655   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   656   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
```
```   657   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
```
```   658   using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
```
```   659
```
```   660 lemma tendsto_sgn [tendsto_intros]:
```
```   661   fixes l :: "'a::real_normed_vector"
```
```   662   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
```
```   663   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```   664
```
```   665 lemma continuous_sgn:
```
```   666   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   667   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   668   shows "continuous F (\<lambda>x. sgn (f x))"
```
```   669   using assms unfolding continuous_def by (rule tendsto_sgn)
```
```   670
```
```   671 lemma continuous_at_within_sgn[continuous_intros]:
```
```   672   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   673   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   674   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
```
```   675   using assms unfolding continuous_within by (rule tendsto_sgn)
```
```   676
```
```   677 lemma isCont_sgn[continuous_intros]:
```
```   678   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   679   assumes "isCont f a" and "f a \<noteq> 0"
```
```   680   shows "isCont (\<lambda>x. sgn (f x)) a"
```
```   681   using assms unfolding continuous_at by (rule tendsto_sgn)
```
```   682
```
```   683 lemma continuous_on_sgn[continuous_on_intros]:
```
```   684   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   685   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   686   shows "continuous_on s (\<lambda>x. sgn (f x))"
```
```   687   using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
```
```   688
```
```   689 lemma filterlim_at_infinity:
```
```   690   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
```
```   691   assumes "0 \<le> c"
```
```   692   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```   693   unfolding filterlim_iff eventually_at_infinity
```
```   694 proof safe
```
```   695   fix P :: "'a \<Rightarrow> bool" and b
```
```   696   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```   697     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```   698   have "max b (c + 1) > c" by auto
```
```   699   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```   700     by auto
```
```   701   then show "eventually (\<lambda>x. P (f x)) F"
```
```   702   proof eventually_elim
```
```   703     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```   704     with P show "P (f x)" by auto
```
```   705   qed
```
```   706 qed force
```
```   707
```
```   708
```
```   709 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
```
```   710
```
```   711 text {*
```
```   712
```
```   713 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
```
```   714 @{term "at_right x"} and also @{term "at_right 0"}.
```
```   715
```
```   716 *}
```
```   717
```
```   718 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
```
```   719
```
```   720 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
```
```   721   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
```
```   722   by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
```
```   723
```
```   724 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
```
```   725   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
```
```   726   apply (intro allI ex_cong)
```
```   727   apply (auto simp: dist_real_def field_simps)
```
```   728   apply (erule_tac x="-x" in allE)
```
```   729   apply simp
```
```   730   done
```
```   731
```
```   732 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
```
```   733   unfolding at_def filtermap_nhds_shift[symmetric]
```
```   734   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```   735
```
```   736 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
```
```   737   unfolding filtermap_at_shift[symmetric]
```
```   738   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```   739
```
```   740 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
```
```   741   using filtermap_at_right_shift[of "-a" 0] by simp
```
```   742
```
```   743 lemma filterlim_at_right_to_0:
```
```   744   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
```
```   745   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
```
```   746
```
```   747 lemma eventually_at_right_to_0:
```
```   748   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
```
```   749   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
```
```   750
```
```   751 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
```
```   752   unfolding at_def filtermap_nhds_minus[symmetric]
```
```   753   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```   754
```
```   755 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
```
```   756   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
```
```   757
```
```   758 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
```
```   759   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
```
```   760
```
```   761 lemma filterlim_at_left_to_right:
```
```   762   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
```
```   763   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
```
```   764
```
```   765 lemma eventually_at_left_to_right:
```
```   766   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
```
```   767   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
```
```   768
```
```   769 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
```
```   770   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
```
```   771   by (metis le_minus_iff minus_minus)
```
```   772
```
```   773 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
```
```   774   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
```
```   775
```
```   776 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
```
```   777   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
```
```   778
```
```   779 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
```
```   780   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
```
```   781
```
```   782 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```   783   unfolding filterlim_at_top eventually_at_bot_dense
```
```   784   by (metis leI minus_less_iff order_less_asym)
```
```   785
```
```   786 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```   787   unfolding filterlim_at_bot eventually_at_top_dense
```
```   788   by (metis leI less_minus_iff order_less_asym)
```
```   789
```
```   790 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
```
```   791   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
```
```   792   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
```
```   793   by auto
```
```   794
```
```   795 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
```
```   796   unfolding filterlim_uminus_at_top by simp
```
```   797
```
```   798 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
```
```   799   unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
```
```   800 proof safe
```
```   801   fix Z :: real assume [arith]: "0 < Z"
```
```   802   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```   803     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```   804   then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
```
```   805     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```   806 qed
```
```   807
```
```   808 lemma filterlim_inverse_at_top:
```
```   809   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```   810   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
```
```   811      (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
```
```   812
```
```   813 lemma filterlim_inverse_at_bot_neg:
```
```   814   "LIM x (at_left (0::real)). inverse x :> at_bot"
```
```   815   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
```
```   816
```
```   817 lemma filterlim_inverse_at_bot:
```
```   818   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```   819   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
```
```   820   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
```
```   821
```
```   822 lemma tendsto_inverse_0:
```
```   823   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
```
```   824   shows "(inverse ---> (0::'a)) at_infinity"
```
```   825   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```   826 proof safe
```
```   827   fix r :: real assume "0 < r"
```
```   828   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```   829   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```   830     fix x :: 'a
```
```   831     from `0 < r` have "0 < inverse (r / 2)" by simp
```
```   832     also assume *: "inverse (r / 2) \<le> norm x"
```
```   833     finally show "norm (inverse x) < r"
```
```   834       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```   835   qed
```
```   836 qed
```
```   837
```
```   838 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
```
```   839 proof (rule antisym)
```
```   840   have "(inverse ---> (0::real)) at_top"
```
```   841     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
```
```   842   then show "filtermap inverse at_top \<le> at_right (0::real)"
```
```   843     unfolding at_within_eq
```
```   844     by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
```
```   845 next
```
```   846   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
```
```   847     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
```
```   848   then show "at_right (0::real) \<le> filtermap inverse at_top"
```
```   849     by (simp add: filtermap_ident filtermap_filtermap)
```
```   850 qed
```
```   851
```
```   852 lemma eventually_at_right_to_top:
```
```   853   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
```
```   854   unfolding at_right_to_top eventually_filtermap ..
```
```   855
```
```   856 lemma filterlim_at_right_to_top:
```
```   857   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
```
```   858   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
```
```   859
```
```   860 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
```
```   861   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
```
```   862
```
```   863 lemma eventually_at_top_to_right:
```
```   864   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
```
```   865   unfolding at_top_to_right eventually_filtermap ..
```
```   866
```
```   867 lemma filterlim_at_top_to_right:
```
```   868   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
```
```   869   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
```
```   870
```
```   871 lemma filterlim_inverse_at_infinity:
```
```   872   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   873   shows "filterlim inverse at_infinity (at (0::'a))"
```
```   874   unfolding filterlim_at_infinity[OF order_refl]
```
```   875 proof safe
```
```   876   fix r :: real assume "0 < r"
```
```   877   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```   878     unfolding eventually_at norm_inverse
```
```   879     by (intro exI[of _ "inverse r"])
```
```   880        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```   881 qed
```
```   882
```
```   883 lemma filterlim_inverse_at_iff:
```
```   884   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   885   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```   886   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```   887 proof
```
```   888   assume "filtermap g F \<le> at_infinity"
```
```   889   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```   890     by (rule filtermap_mono)
```
```   891   also have "\<dots> \<le> at 0"
```
```   892     using tendsto_inverse_0
```
```   893     by (auto intro!: le_withinI exI[of _ 1]
```
```   894              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
```
```   895   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```   896 next
```
```   897   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```   898   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```   899     by (rule filtermap_mono)
```
```   900   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```   901     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```   902 qed
```
```   903
```
```   904 lemma tendsto_inverse_0_at_top:
```
```   905   "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
```
```   906  by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
```
```   907
```
```   908 text {*
```
```   909
```
```   910 We only show rules for multiplication and addition when the functions are either against a real
```
```   911 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```   912
```
```   913 *}
```
```   914
```
```   915 lemma filterlim_tendsto_pos_mult_at_top:
```
```   916   assumes f: "(f ---> c) F" and c: "0 < c"
```
```   917   assumes g: "LIM x F. g x :> at_top"
```
```   918   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```   919   unfolding filterlim_at_top_gt[where c=0]
```
```   920 proof safe
```
```   921   fix Z :: real assume "0 < Z"
```
```   922   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
```
```   923     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
```
```   924              simp: dist_real_def abs_real_def split: split_if_asm)
```
```   925   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
```
```   926     unfolding filterlim_at_top by auto
```
```   927   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```   928   proof eventually_elim
```
```   929     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
```
```   930     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
```
```   931       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```   932     with `0 < c` show "Z \<le> f x * g x"
```
```   933        by simp
```
```   934   qed
```
```   935 qed
```
```   936
```
```   937 lemma filterlim_at_top_mult_at_top:
```
```   938   assumes f: "LIM x F. f x :> at_top"
```
```   939   assumes g: "LIM x F. g x :> at_top"
```
```   940   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```   941   unfolding filterlim_at_top_gt[where c=0]
```
```   942 proof safe
```
```   943   fix Z :: real assume "0 < Z"
```
```   944   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
```
```   945     unfolding filterlim_at_top by auto
```
```   946   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```   947     unfolding filterlim_at_top by auto
```
```   948   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```   949   proof eventually_elim
```
```   950     fix x assume "1 \<le> f x" "Z \<le> g x"
```
```   951     with `0 < Z` have "1 * Z \<le> f x * g x"
```
```   952       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```   953     then show "Z \<le> f x * g x"
```
```   954        by simp
```
```   955   qed
```
```   956 qed
```
```   957
```
```   958 lemma filterlim_tendsto_pos_mult_at_bot:
```
```   959   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
```
```   960   shows "LIM x F. f x * g x :> at_bot"
```
```   961   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
```
```   962   unfolding filterlim_uminus_at_bot by simp
```
```   963
```
```   964 lemma filterlim_tendsto_add_at_top:
```
```   965   assumes f: "(f ---> c) F"
```
```   966   assumes g: "LIM x F. g x :> at_top"
```
```   967   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```   968   unfolding filterlim_at_top_gt[where c=0]
```
```   969 proof safe
```
```   970   fix Z :: real assume "0 < Z"
```
```   971   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```   972     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
```
```   973   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
```
```   974     unfolding filterlim_at_top by auto
```
```   975   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```   976     by eventually_elim simp
```
```   977 qed
```
```   978
```
```   979 lemma LIM_at_top_divide:
```
```   980   fixes f g :: "'a \<Rightarrow> real"
```
```   981   assumes f: "(f ---> a) F" "0 < a"
```
```   982   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
```
```   983   shows "LIM x F. f x / g x :> at_top"
```
```   984   unfolding divide_inverse
```
```   985   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
```
```   986
```
```   987 lemma filterlim_at_top_add_at_top:
```
```   988   assumes f: "LIM x F. f x :> at_top"
```
```   989   assumes g: "LIM x F. g x :> at_top"
```
```   990   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```   991   unfolding filterlim_at_top_gt[where c=0]
```
```   992 proof safe
```
```   993   fix Z :: real assume "0 < Z"
```
```   994   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
```
```   995     unfolding filterlim_at_top by auto
```
```   996   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```   997     unfolding filterlim_at_top by auto
```
```   998   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```   999     by eventually_elim simp
```
```  1000 qed
```
```  1001
```
```  1002 lemma tendsto_divide_0:
```
```  1003   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```  1004   assumes f: "(f ---> c) F"
```
```  1005   assumes g: "LIM x F. g x :> at_infinity"
```
```  1006   shows "((\<lambda>x. f x / g x) ---> 0) F"
```
```  1007   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
```
```  1008
```
```  1009 lemma linear_plus_1_le_power:
```
```  1010   fixes x :: real
```
```  1011   assumes x: "0 \<le> x"
```
```  1012   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1013 proof (induct n)
```
```  1014   case (Suc n)
```
```  1015   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
```
```  1016     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
```
```  1017   also have "\<dots> \<le> (x + 1)^Suc n"
```
```  1018     using Suc x by (simp add: mult_left_mono)
```
```  1019   finally show ?case .
```
```  1020 qed simp
```
```  1021
```
```  1022 lemma filterlim_realpow_sequentially_gt1:
```
```  1023   fixes x :: "'a :: real_normed_div_algebra"
```
```  1024   assumes x[arith]: "1 < norm x"
```
```  1025   shows "LIM n sequentially. x ^ n :> at_infinity"
```
```  1026 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
```
```  1027   fix y :: real assume "0 < y"
```
```  1028   have "0 < norm x - 1" by simp
```
```  1029   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
```
```  1030   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
```
```  1031   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
```
```  1032   also have "\<dots> = norm x ^ N" by simp
```
```  1033   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
```
```  1034     by (metis order_less_le_trans power_increasing order_less_imp_le x)
```
```  1035   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
```
```  1036     unfolding eventually_sequentially
```
```  1037     by (auto simp: norm_power)
```
```  1038 qed simp
```
```  1039
```
```  1040
```
```  1041 subsection {* Limits of Sequences *}
```
```  1042
```
```  1043 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
```
```  1044   by simp
```
```  1045
```
```  1046 lemma LIMSEQ_iff:
```
```  1047   fixes L :: "'a::real_normed_vector"
```
```  1048   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
```
```  1049 unfolding LIMSEQ_def dist_norm ..
```
```  1050
```
```  1051 lemma LIMSEQ_I:
```
```  1052   fixes L :: "'a::real_normed_vector"
```
```  1053   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
```
```  1054 by (simp add: LIMSEQ_iff)
```
```  1055
```
```  1056 lemma LIMSEQ_D:
```
```  1057   fixes L :: "'a::real_normed_vector"
```
```  1058   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
```
```  1059 by (simp add: LIMSEQ_iff)
```
```  1060
```
```  1061 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
```
```  1062   unfolding tendsto_def eventually_sequentially
```
```  1063   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
```
```  1064
```
```  1065 lemma Bseq_inverse_lemma:
```
```  1066   fixes x :: "'a::real_normed_div_algebra"
```
```  1067   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```  1068 apply (subst nonzero_norm_inverse, clarsimp)
```
```  1069 apply (erule (1) le_imp_inverse_le)
```
```  1070 done
```
```  1071
```
```  1072 lemma Bseq_inverse:
```
```  1073   fixes a :: "'a::real_normed_div_algebra"
```
```  1074   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
```
```  1075   by (rule Bfun_inverse)
```
```  1076
```
```  1077 lemma LIMSEQ_diff_approach_zero:
```
```  1078   fixes L :: "'a::real_normed_vector"
```
```  1079   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
```
```  1080   by (drule (1) tendsto_add, simp)
```
```  1081
```
```  1082 lemma LIMSEQ_diff_approach_zero2:
```
```  1083   fixes L :: "'a::real_normed_vector"
```
```  1084   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
```
```  1085   by (drule (1) tendsto_diff, simp)
```
```  1086
```
```  1087 text{*An unbounded sequence's inverse tends to 0*}
```
```  1088
```
```  1089 lemma LIMSEQ_inverse_zero:
```
```  1090   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
```
```  1091   apply (rule filterlim_compose[OF tendsto_inverse_0])
```
```  1092   apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
```
```  1093   apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
```
```  1094   done
```
```  1095
```
```  1096 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
```
```  1097
```
```  1098 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
```
```  1099   by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
```
```  1100             filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
```
```  1101
```
```  1102 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
```
```  1103 infinity is now easily proved*}
```
```  1104
```
```  1105 lemma LIMSEQ_inverse_real_of_nat_add:
```
```  1106      "(%n. r + inverse(real(Suc n))) ----> r"
```
```  1107   using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
```
```  1108
```
```  1109 lemma LIMSEQ_inverse_real_of_nat_add_minus:
```
```  1110      "(%n. r + -inverse(real(Suc n))) ----> r"
```
```  1111   using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
```
```  1112   by auto
```
```  1113
```
```  1114 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
```
```  1115      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
```
```  1116   using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
```
```  1117   by auto
```
```  1118
```
```  1119 subsection {* Convergence on sequences *}
```
```  1120
```
```  1121 lemma convergent_add:
```
```  1122   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1123   assumes "convergent (\<lambda>n. X n)"
```
```  1124   assumes "convergent (\<lambda>n. Y n)"
```
```  1125   shows "convergent (\<lambda>n. X n + Y n)"
```
```  1126   using assms unfolding convergent_def by (fast intro: tendsto_add)
```
```  1127
```
```  1128 lemma convergent_setsum:
```
```  1129   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
```
```  1130   assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
```
```  1131   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
```
```  1132 proof (cases "finite A")
```
```  1133   case True from this and assms show ?thesis
```
```  1134     by (induct A set: finite) (simp_all add: convergent_const convergent_add)
```
```  1135 qed (simp add: convergent_const)
```
```  1136
```
```  1137 lemma (in bounded_linear) convergent:
```
```  1138   assumes "convergent (\<lambda>n. X n)"
```
```  1139   shows "convergent (\<lambda>n. f (X n))"
```
```  1140   using assms unfolding convergent_def by (fast intro: tendsto)
```
```  1141
```
```  1142 lemma (in bounded_bilinear) convergent:
```
```  1143   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
```
```  1144   shows "convergent (\<lambda>n. X n ** Y n)"
```
```  1145   using assms unfolding convergent_def by (fast intro: tendsto)
```
```  1146
```
```  1147 lemma convergent_minus_iff:
```
```  1148   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1149   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
```
```  1150 apply (simp add: convergent_def)
```
```  1151 apply (auto dest: tendsto_minus)
```
```  1152 apply (drule tendsto_minus, auto)
```
```  1153 done
```
```  1154
```
```  1155 subsection {* Bounded Monotonic Sequences *}
```
```  1156
```
```  1157 subsubsection {* Bounded Sequences *}
```
```  1158
```
```  1159 lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
```
```  1160   by (intro BfunI) (auto simp: eventually_sequentially)
```
```  1161
```
```  1162 lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
```
```  1163   by (intro BfunI) (auto simp: eventually_sequentially)
```
```  1164
```
```  1165 lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
```
```  1166   unfolding Bfun_def eventually_sequentially
```
```  1167 proof safe
```
```  1168   fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
```
```  1169   then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
```
```  1170     by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] min_max.less_supI2)
```
```  1171        (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
```
```  1172 qed auto
```
```  1173
```
```  1174 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
```
```  1175 unfolding Bseq_def by auto
```
```  1176
```
```  1177 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
```
```  1178 by (simp add: Bseq_def)
```
```  1179
```
```  1180 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
```
```  1181 by (auto simp add: Bseq_def)
```
```  1182
```
```  1183 lemma lemma_NBseq_def:
```
```  1184   "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```  1185 proof safe
```
```  1186   fix K :: real
```
```  1187   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
```
```  1188   then have "K \<le> real (Suc n)" by auto
```
```  1189   moreover assume "\<forall>m. norm (X m) \<le> K"
```
```  1190   ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
```
```  1191     by (blast intro: order_trans)
```
```  1192   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
```
```  1193 qed (force simp add: real_of_nat_Suc)
```
```  1194
```
```  1195 text{* alternative definition for Bseq *}
```
```  1196 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
```
```  1197 apply (simp add: Bseq_def)
```
```  1198 apply (simp (no_asm) add: lemma_NBseq_def)
```
```  1199 done
```
```  1200
```
```  1201 lemma lemma_NBseq_def2:
```
```  1202      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```  1203 apply (subst lemma_NBseq_def, auto)
```
```  1204 apply (rule_tac x = "Suc N" in exI)
```
```  1205 apply (rule_tac [2] x = N in exI)
```
```  1206 apply (auto simp add: real_of_nat_Suc)
```
```  1207  prefer 2 apply (blast intro: order_less_imp_le)
```
```  1208 apply (drule_tac x = n in spec, simp)
```
```  1209 done
```
```  1210
```
```  1211 (* yet another definition for Bseq *)
```
```  1212 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
```
```  1213 by (simp add: Bseq_def lemma_NBseq_def2)
```
```  1214
```
```  1215 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
```
```  1216
```
```  1217 text{*alternative formulation for boundedness*}
```
```  1218 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
```
```  1219 apply (unfold Bseq_def, safe)
```
```  1220 apply (rule_tac [2] x = "k + norm x" in exI)
```
```  1221 apply (rule_tac x = K in exI, simp)
```
```  1222 apply (rule exI [where x = 0], auto)
```
```  1223 apply (erule order_less_le_trans, simp)
```
```  1224 apply (drule_tac x=n in spec, fold diff_minus)
```
```  1225 apply (drule order_trans [OF norm_triangle_ineq2])
```
```  1226 apply simp
```
```  1227 done
```
```  1228
```
```  1229 text{*alternative formulation for boundedness*}
```
```  1230 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
```
```  1231 apply safe
```
```  1232 apply (simp add: Bseq_def, safe)
```
```  1233 apply (rule_tac x = "K + norm (X N)" in exI)
```
```  1234 apply auto
```
```  1235 apply (erule order_less_le_trans, simp)
```
```  1236 apply (rule_tac x = N in exI, safe)
```
```  1237 apply (drule_tac x = n in spec)
```
```  1238 apply (rule order_trans [OF norm_triangle_ineq], simp)
```
```  1239 apply (auto simp add: Bseq_iff2)
```
```  1240 done
```
```  1241
```
```  1242 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
```
```  1243 apply (simp add: Bseq_def)
```
```  1244 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
```
```  1245 apply (drule_tac x = n in spec, arith)
```
```  1246 done
```
```  1247
```
```  1248 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
```
```  1249
```
```  1250 lemma Bseq_isUb:
```
```  1251   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```  1252 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
```
```  1253
```
```  1254 text{* Use completeness of reals (supremum property)
```
```  1255    to show that any bounded sequence has a least upper bound*}
```
```  1256
```
```  1257 lemma Bseq_isLub:
```
```  1258   "!!(X::nat=>real). Bseq X ==>
```
```  1259    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
```
```  1260 by (blast intro: reals_complete Bseq_isUb)
```
```  1261
```
```  1262 subsubsection{*A Bounded and Monotonic Sequence Converges*}
```
```  1263
```
```  1264 (* TODO: delete *)
```
```  1265 (* FIXME: one use in NSA/HSEQ.thy *)
```
```  1266 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
```
```  1267   apply (rule_tac x="X m" in exI)
```
```  1268   apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
```
```  1269   unfolding eventually_sequentially
```
```  1270   apply blast
```
```  1271   done
```
```  1272
```
```  1273 text {* A monotone sequence converges to its least upper bound. *}
```
```  1274
```
```  1275 lemma isLub_mono_imp_LIMSEQ:
```
```  1276   fixes X :: "nat \<Rightarrow> real"
```
```  1277   assumes u: "isLub UNIV {x. \<exists>n. X n = x} u" (* FIXME: use 'range X' *)
```
```  1278   assumes X: "\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n"
```
```  1279   shows "X ----> u"
```
```  1280 proof (rule LIMSEQ_I)
```
```  1281   have 1: "\<forall>n. X n \<le> u"
```
```  1282     using isLubD2 [OF u] by auto
```
```  1283   have "\<forall>y. (\<forall>n. X n \<le> y) \<longrightarrow> u \<le> y"
```
```  1284     using isLub_le_isUb [OF u] by (auto simp add: isUb_def setle_def)
```
```  1285   hence 2: "\<forall>y<u. \<exists>n. y < X n"
```
```  1286     by (metis not_le)
```
```  1287   fix r :: real assume "0 < r"
```
```  1288   hence "u - r < u" by simp
```
```  1289   hence "\<exists>m. u - r < X m" using 2 by simp
```
```  1290   then obtain m where "u - r < X m" ..
```
```  1291   with X have "\<forall>n\<ge>m. u - r < X n"
```
```  1292     by (fast intro: less_le_trans)
```
```  1293   hence "\<exists>m. \<forall>n\<ge>m. u - r < X n" ..
```
```  1294   thus "\<exists>m. \<forall>n\<ge>m. norm (X n - u) < r"
```
```  1295     using 1 by (simp add: diff_less_eq add_commute)
```
```  1296 qed
```
```  1297
```
```  1298 text{*A standard proof of the theorem for monotone increasing sequence*}
```
```  1299
```
```  1300 lemma Bseq_mono_convergent:
```
```  1301    "Bseq X \<Longrightarrow> \<forall>m. \<forall>n \<ge> m. X m \<le> X n \<Longrightarrow> convergent (X::nat=>real)"
```
```  1302   by (metis Bseq_isLub isLub_mono_imp_LIMSEQ convergentI)
```
```  1303
```
```  1304 lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
```
```  1305   by (simp add: Bseq_def)
```
```  1306
```
```  1307 text{*Main monotonicity theorem*}
```
```  1308
```
```  1309 lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
```
```  1310   by (metis monoseq_iff incseq_def decseq_eq_incseq convergent_minus_iff Bseq_minus_iff
```
```  1311             Bseq_mono_convergent)
```
```  1312
```
```  1313 lemma Cauchy_iff:
```
```  1314   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1315   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
```
```  1316   unfolding Cauchy_def dist_norm ..
```
```  1317
```
```  1318 lemma CauchyI:
```
```  1319   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1320   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"
```
```  1321 by (simp add: Cauchy_iff)
```
```  1322
```
```  1323 lemma CauchyD:
```
```  1324   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
```
```  1325   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
```
```  1326 by (simp add: Cauchy_iff)
```
```  1327
```
```  1328 lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
```
```  1329   apply (simp add: subset_eq)
```
```  1330   apply (rule BseqI'[where K="max (norm a) (norm b)"])
```
```  1331   apply (erule_tac x=n in allE)
```
```  1332   apply auto
```
```  1333   done
```
```  1334
```
```  1335 lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
```
```  1336   by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
```
```  1337
```
```  1338 lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
```
```  1339   by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
```
```  1340
```
```  1341 lemma incseq_convergent:
```
```  1342   fixes X :: "nat \<Rightarrow> real"
```
```  1343   assumes "incseq X" and "\<forall>i. X i \<le> B"
```
```  1344   obtains L where "X ----> L" "\<forall>i. X i \<le> L"
```
```  1345 proof atomize_elim
```
```  1346   from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
```
```  1347   obtain L where "X ----> L"
```
```  1348     by (auto simp: convergent_def monoseq_def incseq_def)
```
```  1349   with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
```
```  1350     by (auto intro!: exI[of _ L] incseq_le)
```
```  1351 qed
```
```  1352
```
```  1353 lemma decseq_convergent:
```
```  1354   fixes X :: "nat \<Rightarrow> real"
```
```  1355   assumes "decseq X" and "\<forall>i. B \<le> X i"
```
```  1356   obtains L where "X ----> L" "\<forall>i. L \<le> X i"
```
```  1357 proof atomize_elim
```
```  1358   from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
```
```  1359   obtain L where "X ----> L"
```
```  1360     by (auto simp: convergent_def monoseq_def decseq_def)
```
```  1361   with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
```
```  1362     by (auto intro!: exI[of _ L] decseq_le)
```
```  1363 qed
```
```  1364
```
```  1365 subsubsection {* Cauchy Sequences are Bounded *}
```
```  1366
```
```  1367 text{*A Cauchy sequence is bounded -- this is the standard
```
```  1368   proof mechanization rather than the nonstandard proof*}
```
```  1369
```
```  1370 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
```
```  1371           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
```
```  1372 apply (clarify, drule spec, drule (1) mp)
```
```  1373 apply (simp only: norm_minus_commute)
```
```  1374 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
```
```  1375 apply simp
```
```  1376 done
```
```  1377
```
```  1378 class banach = real_normed_vector + complete_space
```
```  1379
```
```  1380 instance real :: banach by default
```
```  1381
```
```  1382 subsection {* Power Sequences *}
```
```  1383
```
```  1384 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
```
```  1385 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
```
```  1386   also fact that bounded and monotonic sequence converges.*}
```
```  1387
```
```  1388 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
```
```  1389 apply (simp add: Bseq_def)
```
```  1390 apply (rule_tac x = 1 in exI)
```
```  1391 apply (simp add: power_abs)
```
```  1392 apply (auto dest: power_mono)
```
```  1393 done
```
```  1394
```
```  1395 lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
```
```  1396 apply (clarify intro!: mono_SucI2)
```
```  1397 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
```
```  1398 done
```
```  1399
```
```  1400 lemma convergent_realpow:
```
```  1401   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
```
```  1402 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
```
```  1403
```
```  1404 lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
```
```  1405   by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
```
```  1406
```
```  1407 lemma LIMSEQ_realpow_zero:
```
```  1408   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1409 proof cases
```
```  1410   assume "0 \<le> x" and "x \<noteq> 0"
```
```  1411   hence x0: "0 < x" by simp
```
```  1412   assume x1: "x < 1"
```
```  1413   from x0 x1 have "1 < inverse x"
```
```  1414     by (rule one_less_inverse)
```
```  1415   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
```
```  1416     by (rule LIMSEQ_inverse_realpow_zero)
```
```  1417   thus ?thesis by (simp add: power_inverse)
```
```  1418 qed (rule LIMSEQ_imp_Suc, simp add: tendsto_const)
```
```  1419
```
```  1420 lemma LIMSEQ_power_zero:
```
```  1421   fixes x :: "'a::{real_normed_algebra_1}"
```
```  1422   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
```
```  1423 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
```
```  1424 apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
```
```  1425 apply (simp add: power_abs norm_power_ineq)
```
```  1426 done
```
```  1427
```
```  1428 lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
```
```  1429   by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
```
```  1430
```
```  1431 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
```
```  1432
```
```  1433 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
```
```  1434   by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
```
```  1435
```
```  1436 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
```
```  1437   by (rule LIMSEQ_power_zero) simp
```
```  1438
```
```  1439
```
```  1440 subsection {* Limits of Functions *}
```
```  1441
```
```  1442 lemma LIM_eq:
```
```  1443   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1444   shows "f -- a --> L =
```
```  1445      (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
```
```  1446 by (simp add: LIM_def dist_norm)
```
```  1447
```
```  1448 lemma LIM_I:
```
```  1449   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1450   shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
```
```  1451       ==> f -- a --> L"
```
```  1452 by (simp add: LIM_eq)
```
```  1453
```
```  1454 lemma LIM_D:
```
```  1455   fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
```
```  1456   shows "[| f -- a --> L; 0<r |]
```
```  1457       ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
```
```  1458 by (simp add: LIM_eq)
```
```  1459
```
```  1460 lemma LIM_offset:
```
```  1461   fixes a :: "'a::real_normed_vector"
```
```  1462   shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
```
```  1463 apply (rule topological_tendstoI)
```
```  1464 apply (drule (2) topological_tendstoD)
```
```  1465 apply (simp only: eventually_at dist_norm)
```
```  1466 apply (clarify, rule_tac x=d in exI, safe)
```
```  1467 apply (drule_tac x="x + k" in spec)
```
```  1468 apply (simp add: algebra_simps)
```
```  1469 done
```
```  1470
```
```  1471 lemma LIM_offset_zero:
```
```  1472   fixes a :: "'a::real_normed_vector"
```
```  1473   shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
```
```  1474 by (drule_tac k="a" in LIM_offset, simp add: add_commute)
```
```  1475
```
```  1476 lemma LIM_offset_zero_cancel:
```
```  1477   fixes a :: "'a::real_normed_vector"
```
```  1478   shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
```
```  1479 by (drule_tac k="- a" in LIM_offset, simp)
```
```  1480
```
```  1481 lemma LIM_zero:
```
```  1482   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1483   shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
```
```  1484 unfolding tendsto_iff dist_norm by simp
```
```  1485
```
```  1486 lemma LIM_zero_cancel:
```
```  1487   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1488   shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
```
```  1489 unfolding tendsto_iff dist_norm by simp
```
```  1490
```
```  1491 lemma LIM_zero_iff:
```
```  1492   fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
```
```  1493   shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
```
```  1494 unfolding tendsto_iff dist_norm by simp
```
```  1495
```
```  1496 lemma LIM_imp_LIM:
```
```  1497   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```  1498   fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
```
```  1499   assumes f: "f -- a --> l"
```
```  1500   assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
```
```  1501   shows "g -- a --> m"
```
```  1502   by (rule metric_LIM_imp_LIM [OF f],
```
```  1503     simp add: dist_norm le)
```
```  1504
```
```  1505 lemma LIM_equal2:
```
```  1506   fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1507   assumes 1: "0 < R"
```
```  1508   assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
```
```  1509   shows "g -- a --> l \<Longrightarrow> f -- a --> l"
```
```  1510 by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
```
```  1511
```
```  1512 lemma LIM_compose2:
```
```  1513   fixes a :: "'a::real_normed_vector"
```
```  1514   assumes f: "f -- a --> b"
```
```  1515   assumes g: "g -- b --> c"
```
```  1516   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
```
```  1517   shows "(\<lambda>x. g (f x)) -- a --> c"
```
```  1518 by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
```
```  1519
```
```  1520 lemma real_LIM_sandwich_zero:
```
```  1521   fixes f g :: "'a::topological_space \<Rightarrow> real"
```
```  1522   assumes f: "f -- a --> 0"
```
```  1523   assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
```
```  1524   assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
```
```  1525   shows "g -- a --> 0"
```
```  1526 proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
```
```  1527   fix x assume x: "x \<noteq> a"
```
```  1528   have "norm (g x - 0) = g x" by (simp add: 1 x)
```
```  1529   also have "g x \<le> f x" by (rule 2 [OF x])
```
```  1530   also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
```
```  1531   also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
```
```  1532   finally show "norm (g x - 0) \<le> norm (f x - 0)" .
```
```  1533 qed
```
```  1534
```
```  1535
```
```  1536 subsection {* Continuity *}
```
```  1537
```
```  1538 lemma LIM_isCont_iff:
```
```  1539   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1540   shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
```
```  1541 by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
```
```  1542
```
```  1543 lemma isCont_iff:
```
```  1544   fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
```
```  1545   shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
```
```  1546 by (simp add: isCont_def LIM_isCont_iff)
```
```  1547
```
```  1548 lemma isCont_LIM_compose2:
```
```  1549   fixes a :: "'a::real_normed_vector"
```
```  1550   assumes f [unfolded isCont_def]: "isCont f a"
```
```  1551   assumes g: "g -- f a --> l"
```
```  1552   assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
```
```  1553   shows "(\<lambda>x. g (f x)) -- a --> l"
```
```  1554 by (rule LIM_compose2 [OF f g inj])
```
```  1555
```
```  1556
```
```  1557 lemma isCont_norm [simp]:
```
```  1558   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1559   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
```
```  1560   by (fact continuous_norm)
```
```  1561
```
```  1562 lemma isCont_rabs [simp]:
```
```  1563   fixes f :: "'a::t2_space \<Rightarrow> real"
```
```  1564   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
```
```  1565   by (fact continuous_rabs)
```
```  1566
```
```  1567 lemma isCont_add [simp]:
```
```  1568   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1569   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
```
```  1570   by (fact continuous_add)
```
```  1571
```
```  1572 lemma isCont_minus [simp]:
```
```  1573   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1574   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
```
```  1575   by (fact continuous_minus)
```
```  1576
```
```  1577 lemma isCont_diff [simp]:
```
```  1578   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```  1579   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
```
```  1580   by (fact continuous_diff)
```
```  1581
```
```  1582 lemma isCont_mult [simp]:
```
```  1583   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
```
```  1584   shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
```
```  1585   by (fact continuous_mult)
```
```  1586
```
```  1587 lemma (in bounded_linear) isCont:
```
```  1588   "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
```
```  1589   by (fact continuous)
```
```  1590
```
```  1591 lemma (in bounded_bilinear) isCont:
```
```  1592   "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
```
```  1593   by (fact continuous)
```
```  1594
```
```  1595 lemmas isCont_scaleR [simp] =
```
```  1596   bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
```
```  1597
```
```  1598 lemmas isCont_of_real [simp] =
```
```  1599   bounded_linear.isCont [OF bounded_linear_of_real]
```
```  1600
```
```  1601 lemma isCont_power [simp]:
```
```  1602   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```  1603   shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
```
```  1604   by (fact continuous_power)
```
```  1605
```
```  1606 lemma isCont_setsum [simp]:
```
```  1607   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```  1608   shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
```
```  1609   by (auto intro: continuous_setsum)
```
```  1610
```
```  1611 lemmas isCont_intros =
```
```  1612   isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
```
```  1613   isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
```
```  1614   isCont_of_real isCont_power isCont_sgn isCont_setsum
```
```  1615
```
```  1616 subsection {* Uniform Continuity *}
```
```  1617
```
```  1618 lemma (in bounded_linear) isUCont: "isUCont f"
```
```  1619 unfolding isUCont_def dist_norm
```
```  1620 proof (intro allI impI)
```
```  1621   fix r::real assume r: "0 < r"
```
```  1622   obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
```
```  1623     using pos_bounded by fast
```
```  1624   show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
```
```  1625   proof (rule exI, safe)
```
```  1626     from r K show "0 < r / K" by (rule divide_pos_pos)
```
```  1627   next
```
```  1628     fix x y :: 'a
```
```  1629     assume xy: "norm (x - y) < r / K"
```
```  1630     have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
```
```  1631     also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
```
```  1632     also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
```
```  1633     finally show "norm (f x - f y) < r" .
```
```  1634   qed
```
```  1635 qed
```
```  1636
```
```  1637 lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
```
```  1638 by (rule isUCont [THEN isUCont_Cauchy])
```
```  1639
```
```  1640
```
```  1641 lemma LIM_less_bound:
```
```  1642   fixes f :: "real \<Rightarrow> real"
```
```  1643   assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
```
```  1644   shows "0 \<le> f x"
```
```  1645 proof (rule tendsto_le_const)
```
```  1646   show "(f ---> f x) (at_left x)"
```
```  1647     using `isCont f x` by (simp add: filterlim_at_split isCont_def)
```
```  1648   show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
```
```  1649     using ev by (auto simp: eventually_within_less dist_real_def intro!: exI[of _ "x - b"])
```
```  1650 qed simp
```
```  1651
```
```  1652
```
```  1653 subsection {* Nested Intervals and Bisection -- Needed for Compactness *}
```
```  1654
```
```  1655 lemma nested_sequence_unique:
```
```  1656   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"
```
```  1657   shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
```
```  1658 proof -
```
```  1659   have "incseq f" unfolding incseq_Suc_iff by fact
```
```  1660   have "decseq g" unfolding decseq_Suc_iff by fact
```
```  1661
```
```  1662   { fix n
```
```  1663     from `decseq g` have "g n \<le> g 0" by (rule decseqD) simp
```
```  1664     with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f n \<le> g 0" by auto }
```
```  1665   then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
```
```  1666     using incseq_convergent[OF `incseq f`] by auto
```
```  1667   moreover
```
```  1668   { fix n
```
```  1669     from `incseq f` have "f 0 \<le> f n" by (rule incseqD) simp
```
```  1670     with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f 0 \<le> g n" by simp }
```
```  1671   then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
```
```  1672     using decseq_convergent[OF `decseq g`] by auto
```
```  1673   moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF `f ----> u` `g ----> l`]]
```
```  1674   ultimately show ?thesis by auto
```
```  1675 qed
```
```  1676
```
```  1677 lemma Bolzano[consumes 1, case_names trans local]:
```
```  1678   fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
```
```  1679   assumes [arith]: "a \<le> b"
```
```  1680   assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
```
```  1681   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"
```
```  1682   shows "P a b"
```
```  1683 proof -
```
```  1684   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))"
```
```  1685   def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
```
```  1686   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)"
```
```  1687     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)"
```
```  1688     by (simp_all add: l_def u_def bisect_def split: prod.split)
```
```  1689
```
```  1690   { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
```
```  1691
```
```  1692   have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
```
```  1693   proof (safe intro!: nested_sequence_unique)
```
```  1694     fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
```
```  1695   next
```
```  1696     { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
```
```  1697     then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
```
```  1698   qed fact
```
```  1699   then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
```
```  1700   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"
```
```  1701     using `l 0 \<le> x` `x \<le> u 0` local[of x] by auto
```
```  1702
```
```  1703   show "P a b"
```
```  1704   proof (rule ccontr)
```
```  1705     assume "\<not> P a b"
```
```  1706     { fix n have "\<not> P (l n) (u n)"
```
```  1707       proof (induct n)
```
```  1708         case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
```
```  1709       qed (simp add: `\<not> P a b`) }
```
```  1710     moreover
```
```  1711     { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
```
```  1712         using `0 < d` `l ----> x` by (intro order_tendstoD[of _ x]) auto
```
```  1713       moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
```
```  1714         using `0 < d` `u ----> x` by (intro order_tendstoD[of _ x]) auto
```
```  1715       ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
```
```  1716       proof eventually_elim
```
```  1717         fix n assume "x - d / 2 < l n" "u n < x + d / 2"
```
```  1718         from add_strict_mono[OF this] have "u n - l n < d" by simp
```
```  1719         with x show "P (l n) (u n)" by (rule d)
```
```  1720       qed }
```
```  1721     ultimately show False by simp
```
```  1722   qed
```
```  1723 qed
```
```  1724
```
```  1725 lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
```
```  1726 proof (cases "a \<le> b", rule compactI)
```
```  1727   fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
```
```  1728   def T == "{a .. b}"
```
```  1729   from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
```
```  1730   proof (induct rule: Bolzano)
```
```  1731     case (trans a b c)
```
```  1732     then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
```
```  1733     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"
```
```  1734       by (auto simp: *)
```
```  1735     with trans show ?case
```
```  1736       unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
```
```  1737   next
```
```  1738     case (local x)
```
```  1739     then have "x \<in> \<Union>C" using C by auto
```
```  1740     with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
```
```  1741     then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
```
```  1742       by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
```
```  1743     with `c \<in> C` show ?case
```
```  1744       by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
```
```  1745   qed
```
```  1746 qed simp
```
```  1747
```
```  1748
```
```  1749 subsection {* Boundedness of continuous functions *}
```
```  1750
```
```  1751 text{*By bisection, function continuous on closed interval is bounded above*}
```
```  1752
```
```  1753 lemma isCont_eq_Ub:
```
```  1754   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1755   shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1756     \<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)"
```
```  1757   using continuous_attains_sup[of "{a .. b}" f]
```
```  1758   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  1759
```
```  1760 lemma isCont_eq_Lb:
```
```  1761   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1762   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1763     \<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)"
```
```  1764   using continuous_attains_inf[of "{a .. b}" f]
```
```  1765   by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
```
```  1766
```
```  1767 lemma isCont_bounded:
```
```  1768   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1769   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"
```
```  1770   using isCont_eq_Ub[of a b f] by auto
```
```  1771
```
```  1772 lemma isCont_has_Ub:
```
```  1773   fixes f :: "real \<Rightarrow> 'a::linorder_topology"
```
```  1774   shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
```
```  1775     \<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))"
```
```  1776   using isCont_eq_Ub[of a b f] by auto
```
```  1777
```
```  1778 (*HOL style here: object-level formulations*)
```
```  1779 lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
```
```  1780       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  1781       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  1782   by (blast intro: IVT)
```
```  1783
```
```  1784 lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
```
```  1785       (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
```
```  1786       --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
```
```  1787   by (blast intro: IVT2)
```
```  1788
```
```  1789 lemma isCont_Lb_Ub:
```
```  1790   fixes f :: "real \<Rightarrow> real"
```
```  1791   assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
```
```  1792   shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and>
```
```  1793                (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
```
```  1794 proof -
```
```  1795   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"
```
```  1796     using isCont_eq_Ub[OF assms] by auto
```
```  1797   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"
```
```  1798     using isCont_eq_Lb[OF assms] by auto
```
```  1799   show ?thesis
```
```  1800     using IVT[of f L _ M] IVT2[of f L _ M] M L assms
```
```  1801     apply (rule_tac x="f L" in exI)
```
```  1802     apply (rule_tac x="f M" in exI)
```
```  1803     apply (cases "L \<le> M")
```
```  1804     apply (simp, metis order_trans)
```
```  1805     apply (simp, metis order_trans)
```
```  1806     done
```
```  1807 qed
```
```  1808
```
```  1809
```
```  1810 text{*Continuity of inverse function*}
```
```  1811
```
```  1812 lemma isCont_inverse_function:
```
```  1813   fixes f g :: "real \<Rightarrow> real"
```
```  1814   assumes d: "0 < d"
```
```  1815       and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
```
```  1816       and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
```
```  1817   shows "isCont g (f x)"
```
```  1818 proof -
```
```  1819   let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
```
```  1820
```
```  1821   have f: "continuous_on ?D f"
```
```  1822     using cont by (intro continuous_at_imp_continuous_on ballI) auto
```
```  1823   then have g: "continuous_on (f`?D) g"
```
```  1824     using inj by (intro continuous_on_inv) auto
```
```  1825
```
```  1826   from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
```
```  1827     by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
```
```  1828   with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
```
```  1829     by (rule continuous_on_subset)
```
```  1830   moreover
```
```  1831   have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
```
```  1832     using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
```
```  1833   then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
```
```  1834     by auto
```
```  1835   ultimately
```
```  1836   show ?thesis
```
```  1837     by (simp add: continuous_on_eq_continuous_at)
```
```  1838 qed
```
```  1839
```
```  1840 lemma isCont_inverse_function2:
```
```  1841   fixes f g :: "real \<Rightarrow> real" shows
```
```  1842   "\<lbrakk>a < x; x < b;
```
```  1843     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
```
```  1844     \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
```
```  1845    \<Longrightarrow> isCont g (f x)"
```
```  1846 apply (rule isCont_inverse_function
```
```  1847        [where f=f and d="min (x - a) (b - x)"])
```
```  1848 apply (simp_all add: abs_le_iff)
```
```  1849 done
```
```  1850
```
```  1851 (* need to rename second isCont_inverse *)
```
```  1852
```
```  1853 lemma isCont_inv_fun:
```
```  1854   fixes f g :: "real \<Rightarrow> real"
```
```  1855   shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
```
```  1856          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
```
```  1857       ==> isCont g (f x)"
```
```  1858 by (rule isCont_inverse_function)
```
```  1859
```
```  1860 text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
```
```  1861 lemma LIM_fun_gt_zero:
```
```  1862   fixes f :: "real \<Rightarrow> real"
```
```  1863   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)"
```
```  1864 apply (drule (1) LIM_D, clarify)
```
```  1865 apply (rule_tac x = s in exI)
```
```  1866 apply (simp add: abs_less_iff)
```
```  1867 done
```
```  1868
```
```  1869 lemma LIM_fun_less_zero:
```
```  1870   fixes f :: "real \<Rightarrow> real"
```
```  1871   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)"
```
```  1872 apply (drule LIM_D [where r="-l"], simp, clarify)
```
```  1873 apply (rule_tac x = s in exI)
```
```  1874 apply (simp add: abs_less_iff)
```
```  1875 done
```
```  1876
```
```  1877 lemma LIM_fun_not_zero:
```
```  1878   fixes f :: "real \<Rightarrow> real"
```
```  1879   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)"
```
```  1880   using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
```
```  1881 end
```
```  1882
```