src/HOL/Limits.thy
 author hoelzl Tue Mar 26 12:20:57 2013 +0100 (2013-03-26) changeset 51524 7cb5ac44ca9e parent 51478 270b21f3ae0a child 51526 155263089e7b permissions -rw-r--r--
rename RealVector.thy to Real_Vector_Spaces.thy
```     1 (*  Title       : Limits.thy
```
```     2     Author      : Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Filters and Limits *}
```
```     6
```
```     7 theory Limits
```
```     8 imports Real_Vector_Spaces
```
```     9 begin
```
```    10
```
```    11 definition at_infinity :: "'a::real_normed_vector filter" where
```
```    12   "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
```
```    13
```
```    14 lemma eventually_at_infinity:
```
```    15   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
```
```    16 unfolding at_infinity_def
```
```    17 proof (rule eventually_Abs_filter, rule is_filter.intro)
```
```    18   fix P Q :: "'a \<Rightarrow> bool"
```
```    19   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
```
```    20   then obtain r s where
```
```    21     "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
```
```    22   then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
```
```    23   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
```
```    24 qed auto
```
```    25
```
```    26 lemma at_infinity_eq_at_top_bot:
```
```    27   "(at_infinity \<Colon> real filter) = sup at_top at_bot"
```
```    28   unfolding sup_filter_def at_infinity_def eventually_at_top_linorder eventually_at_bot_linorder
```
```    29 proof (intro arg_cong[where f=Abs_filter] ext iffI)
```
```    30   fix P :: "real \<Rightarrow> bool" assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```    31   then guess r ..
```
```    32   then have "(\<forall>x\<ge>r. P x) \<and> (\<forall>x\<le>-r. P x)" by auto
```
```    33   then show "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)" by auto
```
```    34 next
```
```    35   fix P :: "real \<Rightarrow> bool" assume "(\<exists>r. \<forall>x\<ge>r. P x) \<and> (\<exists>r. \<forall>x\<le>r. P x)"
```
```    36   then obtain p q where "\<forall>x\<ge>p. P x" "\<forall>x\<le>q. P x" by auto
```
```    37   then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x"
```
```    38     by (intro exI[of _ "max p (-q)"])
```
```    39        (auto simp: abs_real_def)
```
```    40 qed
```
```    41
```
```    42 lemma at_top_le_at_infinity:
```
```    43   "at_top \<le> (at_infinity :: real filter)"
```
```    44   unfolding at_infinity_eq_at_top_bot by simp
```
```    45
```
```    46 lemma at_bot_le_at_infinity:
```
```    47   "at_bot \<le> (at_infinity :: real filter)"
```
```    48   unfolding at_infinity_eq_at_top_bot by simp
```
```    49
```
```    50 subsection {* Boundedness *}
```
```    51
```
```    52 lemma Bfun_def:
```
```    53   "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
```
```    54   unfolding Bfun_metric_def norm_conv_dist
```
```    55 proof safe
```
```    56   fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
```
```    57   moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
```
```    58     by (intro always_eventually) (metis dist_commute dist_triangle)
```
```    59   with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
```
```    60     by eventually_elim auto
```
```    61   with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
```
```    62     by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
```
```    63 qed auto
```
```    64
```
```    65 lemma BfunI:
```
```    66   assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
```
```    67 unfolding Bfun_def
```
```    68 proof (intro exI conjI allI)
```
```    69   show "0 < max K 1" by simp
```
```    70 next
```
```    71   show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
```
```    72     using K by (rule eventually_elim1, simp)
```
```    73 qed
```
```    74
```
```    75 lemma BfunE:
```
```    76   assumes "Bfun f F"
```
```    77   obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
```
```    78 using assms unfolding Bfun_def by fast
```
```    79
```
```    80 subsection {* Convergence to Zero *}
```
```    81
```
```    82 definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
```
```    83   where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
```
```    84
```
```    85 lemma ZfunI:
```
```    86   "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
```
```    87   unfolding Zfun_def by simp
```
```    88
```
```    89 lemma ZfunD:
```
```    90   "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
```
```    91   unfolding Zfun_def by simp
```
```    92
```
```    93 lemma Zfun_ssubst:
```
```    94   "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
```
```    95   unfolding Zfun_def by (auto elim!: eventually_rev_mp)
```
```    96
```
```    97 lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
```
```    98   unfolding Zfun_def by simp
```
```    99
```
```   100 lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
```
```   101   unfolding Zfun_def by simp
```
```   102
```
```   103 lemma Zfun_imp_Zfun:
```
```   104   assumes f: "Zfun f F"
```
```   105   assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
```
```   106   shows "Zfun (\<lambda>x. g x) F"
```
```   107 proof (cases)
```
```   108   assume K: "0 < K"
```
```   109   show ?thesis
```
```   110   proof (rule ZfunI)
```
```   111     fix r::real assume "0 < r"
```
```   112     hence "0 < r / K"
```
```   113       using K by (rule divide_pos_pos)
```
```   114     then have "eventually (\<lambda>x. norm (f x) < r / K) F"
```
```   115       using ZfunD [OF f] by fast
```
```   116     with g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   117     proof eventually_elim
```
```   118       case (elim x)
```
```   119       hence "norm (f x) * K < r"
```
```   120         by (simp add: pos_less_divide_eq K)
```
```   121       thus ?case
```
```   122         by (simp add: order_le_less_trans [OF elim(1)])
```
```   123     qed
```
```   124   qed
```
```   125 next
```
```   126   assume "\<not> 0 < K"
```
```   127   hence K: "K \<le> 0" by (simp only: not_less)
```
```   128   show ?thesis
```
```   129   proof (rule ZfunI)
```
```   130     fix r :: real
```
```   131     assume "0 < r"
```
```   132     from g show "eventually (\<lambda>x. norm (g x) < r) F"
```
```   133     proof eventually_elim
```
```   134       case (elim x)
```
```   135       also have "norm (f x) * K \<le> norm (f x) * 0"
```
```   136         using K norm_ge_zero by (rule mult_left_mono)
```
```   137       finally show ?case
```
```   138         using `0 < r` by simp
```
```   139     qed
```
```   140   qed
```
```   141 qed
```
```   142
```
```   143 lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
```
```   144   by (erule_tac K="1" in Zfun_imp_Zfun, simp)
```
```   145
```
```   146 lemma Zfun_add:
```
```   147   assumes f: "Zfun f F" and g: "Zfun g F"
```
```   148   shows "Zfun (\<lambda>x. f x + g x) F"
```
```   149 proof (rule ZfunI)
```
```   150   fix r::real assume "0 < r"
```
```   151   hence r: "0 < r / 2" by simp
```
```   152   have "eventually (\<lambda>x. norm (f x) < r/2) F"
```
```   153     using f r by (rule ZfunD)
```
```   154   moreover
```
```   155   have "eventually (\<lambda>x. norm (g x) < r/2) F"
```
```   156     using g r by (rule ZfunD)
```
```   157   ultimately
```
```   158   show "eventually (\<lambda>x. norm (f x + g x) < r) F"
```
```   159   proof eventually_elim
```
```   160     case (elim x)
```
```   161     have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
```
```   162       by (rule norm_triangle_ineq)
```
```   163     also have "\<dots> < r/2 + r/2"
```
```   164       using elim by (rule add_strict_mono)
```
```   165     finally show ?case
```
```   166       by simp
```
```   167   qed
```
```   168 qed
```
```   169
```
```   170 lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
```
```   171   unfolding Zfun_def by simp
```
```   172
```
```   173 lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
```
```   174   by (simp only: diff_minus Zfun_add Zfun_minus)
```
```   175
```
```   176 lemma (in bounded_linear) Zfun:
```
```   177   assumes g: "Zfun g F"
```
```   178   shows "Zfun (\<lambda>x. f (g x)) F"
```
```   179 proof -
```
```   180   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
```
```   181     using bounded by fast
```
```   182   then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
```
```   183     by simp
```
```   184   with g show ?thesis
```
```   185     by (rule Zfun_imp_Zfun)
```
```   186 qed
```
```   187
```
```   188 lemma (in bounded_bilinear) Zfun:
```
```   189   assumes f: "Zfun f F"
```
```   190   assumes g: "Zfun g F"
```
```   191   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   192 proof (rule ZfunI)
```
```   193   fix r::real assume r: "0 < r"
```
```   194   obtain K where K: "0 < K"
```
```   195     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   196     using pos_bounded by fast
```
```   197   from K have K': "0 < inverse K"
```
```   198     by (rule positive_imp_inverse_positive)
```
```   199   have "eventually (\<lambda>x. norm (f x) < r) F"
```
```   200     using f r by (rule ZfunD)
```
```   201   moreover
```
```   202   have "eventually (\<lambda>x. norm (g x) < inverse K) F"
```
```   203     using g K' by (rule ZfunD)
```
```   204   ultimately
```
```   205   show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
```
```   206   proof eventually_elim
```
```   207     case (elim x)
```
```   208     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   209       by (rule norm_le)
```
```   210     also have "norm (f x) * norm (g x) * K < r * inverse K * K"
```
```   211       by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
```
```   212     also from K have "r * inverse K * K = r"
```
```   213       by simp
```
```   214     finally show ?case .
```
```   215   qed
```
```   216 qed
```
```   217
```
```   218 lemma (in bounded_bilinear) Zfun_left:
```
```   219   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
```
```   220   by (rule bounded_linear_left [THEN bounded_linear.Zfun])
```
```   221
```
```   222 lemma (in bounded_bilinear) Zfun_right:
```
```   223   "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
```
```   224   by (rule bounded_linear_right [THEN bounded_linear.Zfun])
```
```   225
```
```   226 lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
```
```   227 lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
```
```   228 lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
```
```   229
```
```   230 lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
```
```   231   by (simp only: tendsto_iff Zfun_def dist_norm)
```
```   232
```
```   233 subsubsection {* Distance and norms *}
```
```   234
```
```   235 lemma tendsto_norm [tendsto_intros]:
```
```   236   "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
```
```   237   unfolding norm_conv_dist by (intro tendsto_intros)
```
```   238
```
```   239 lemma continuous_norm [continuous_intros]:
```
```   240   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
```
```   241   unfolding continuous_def by (rule tendsto_norm)
```
```   242
```
```   243 lemma continuous_on_norm [continuous_on_intros]:
```
```   244   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
```
```   245   unfolding continuous_on_def by (auto intro: tendsto_norm)
```
```   246
```
```   247 lemma tendsto_norm_zero:
```
```   248   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
```
```   249   by (drule tendsto_norm, simp)
```
```   250
```
```   251 lemma tendsto_norm_zero_cancel:
```
```   252   "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
```
```   253   unfolding tendsto_iff dist_norm by simp
```
```   254
```
```   255 lemma tendsto_norm_zero_iff:
```
```   256   "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
```
```   257   unfolding tendsto_iff dist_norm by simp
```
```   258
```
```   259 lemma tendsto_rabs [tendsto_intros]:
```
```   260   "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
```
```   261   by (fold real_norm_def, rule tendsto_norm)
```
```   262
```
```   263 lemma continuous_rabs [continuous_intros]:
```
```   264   "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   265   unfolding real_norm_def[symmetric] by (rule continuous_norm)
```
```   266
```
```   267 lemma continuous_on_rabs [continuous_on_intros]:
```
```   268   "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
```
```   269   unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
```
```   270
```
```   271 lemma tendsto_rabs_zero:
```
```   272   "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
```
```   273   by (fold real_norm_def, rule tendsto_norm_zero)
```
```   274
```
```   275 lemma tendsto_rabs_zero_cancel:
```
```   276   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
```
```   277   by (fold real_norm_def, rule tendsto_norm_zero_cancel)
```
```   278
```
```   279 lemma tendsto_rabs_zero_iff:
```
```   280   "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
```
```   281   by (fold real_norm_def, rule tendsto_norm_zero_iff)
```
```   282
```
```   283 subsubsection {* Addition and subtraction *}
```
```   284
```
```   285 lemma tendsto_add [tendsto_intros]:
```
```   286   fixes a b :: "'a::real_normed_vector"
```
```   287   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
```
```   288   by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
```
```   289
```
```   290 lemma continuous_add [continuous_intros]:
```
```   291   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   292   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
```
```   293   unfolding continuous_def by (rule tendsto_add)
```
```   294
```
```   295 lemma continuous_on_add [continuous_on_intros]:
```
```   296   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   297   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
```
```   298   unfolding continuous_on_def by (auto intro: tendsto_add)
```
```   299
```
```   300 lemma tendsto_add_zero:
```
```   301   fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   302   shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
```
```   303   by (drule (1) tendsto_add, simp)
```
```   304
```
```   305 lemma tendsto_minus [tendsto_intros]:
```
```   306   fixes a :: "'a::real_normed_vector"
```
```   307   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
```
```   308   by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
```
```   309
```
```   310 lemma continuous_minus [continuous_intros]:
```
```   311   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   312   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
```
```   313   unfolding continuous_def by (rule tendsto_minus)
```
```   314
```
```   315 lemma continuous_on_minus [continuous_on_intros]:
```
```   316   fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
```
```   317   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
```
```   318   unfolding continuous_on_def by (auto intro: tendsto_minus)
```
```   319
```
```   320 lemma tendsto_minus_cancel:
```
```   321   fixes a :: "'a::real_normed_vector"
```
```   322   shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
```
```   323   by (drule tendsto_minus, simp)
```
```   324
```
```   325 lemma tendsto_minus_cancel_left:
```
```   326     "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
```
```   327   using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
```
```   328   by auto
```
```   329
```
```   330 lemma tendsto_diff [tendsto_intros]:
```
```   331   fixes a b :: "'a::real_normed_vector"
```
```   332   shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
```
```   333   by (simp add: diff_minus tendsto_add tendsto_minus)
```
```   334
```
```   335 lemma continuous_diff [continuous_intros]:
```
```   336   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   337   shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
```
```   338   unfolding continuous_def by (rule tendsto_diff)
```
```   339
```
```   340 lemma continuous_on_diff [continuous_on_intros]:
```
```   341   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   342   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
```
```   343   unfolding continuous_on_def by (auto intro: tendsto_diff)
```
```   344
```
```   345 lemma tendsto_setsum [tendsto_intros]:
```
```   346   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
```
```   347   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
```
```   348   shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
```
```   349 proof (cases "finite S")
```
```   350   assume "finite S" thus ?thesis using assms
```
```   351     by (induct, simp add: tendsto_const, simp add: tendsto_add)
```
```   352 next
```
```   353   assume "\<not> finite S" thus ?thesis
```
```   354     by (simp add: tendsto_const)
```
```   355 qed
```
```   356
```
```   357 lemma continuous_setsum [continuous_intros]:
```
```   358   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
```
```   359   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
```
```   360   unfolding continuous_def by (rule tendsto_setsum)
```
```   361
```
```   362 lemma continuous_on_setsum [continuous_intros]:
```
```   363   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
```
```   364   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)"
```
```   365   unfolding continuous_on_def by (auto intro: tendsto_setsum)
```
```   366
```
```   367 lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
```
```   368
```
```   369 subsubsection {* Linear operators and multiplication *}
```
```   370
```
```   371 lemma (in bounded_linear) tendsto:
```
```   372   "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
```
```   373   by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
```
```   374
```
```   375 lemma (in bounded_linear) continuous:
```
```   376   "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
```
```   377   using tendsto[of g _ F] by (auto simp: continuous_def)
```
```   378
```
```   379 lemma (in bounded_linear) continuous_on:
```
```   380   "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
```
```   381   using tendsto[of g] by (auto simp: continuous_on_def)
```
```   382
```
```   383 lemma (in bounded_linear) tendsto_zero:
```
```   384   "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
```
```   385   by (drule tendsto, simp only: zero)
```
```   386
```
```   387 lemma (in bounded_bilinear) tendsto:
```
```   388   "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
```
```   389   by (simp only: tendsto_Zfun_iff prod_diff_prod
```
```   390                  Zfun_add Zfun Zfun_left Zfun_right)
```
```   391
```
```   392 lemma (in bounded_bilinear) continuous:
```
```   393   "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
```
```   394   using tendsto[of f _ F g] by (auto simp: continuous_def)
```
```   395
```
```   396 lemma (in bounded_bilinear) continuous_on:
```
```   397   "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
```
```   398   using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
```
```   399
```
```   400 lemma (in bounded_bilinear) tendsto_zero:
```
```   401   assumes f: "(f ---> 0) F"
```
```   402   assumes g: "(g ---> 0) F"
```
```   403   shows "((\<lambda>x. f x ** g x) ---> 0) F"
```
```   404   using tendsto [OF f g] by (simp add: zero_left)
```
```   405
```
```   406 lemma (in bounded_bilinear) tendsto_left_zero:
```
```   407   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
```
```   408   by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
```
```   409
```
```   410 lemma (in bounded_bilinear) tendsto_right_zero:
```
```   411   "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
```
```   412   by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
```
```   413
```
```   414 lemmas tendsto_of_real [tendsto_intros] =
```
```   415   bounded_linear.tendsto [OF bounded_linear_of_real]
```
```   416
```
```   417 lemmas tendsto_scaleR [tendsto_intros] =
```
```   418   bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
```
```   419
```
```   420 lemmas tendsto_mult [tendsto_intros] =
```
```   421   bounded_bilinear.tendsto [OF bounded_bilinear_mult]
```
```   422
```
```   423 lemmas continuous_of_real [continuous_intros] =
```
```   424   bounded_linear.continuous [OF bounded_linear_of_real]
```
```   425
```
```   426 lemmas continuous_scaleR [continuous_intros] =
```
```   427   bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
```
```   428
```
```   429 lemmas continuous_mult [continuous_intros] =
```
```   430   bounded_bilinear.continuous [OF bounded_bilinear_mult]
```
```   431
```
```   432 lemmas continuous_on_of_real [continuous_on_intros] =
```
```   433   bounded_linear.continuous_on [OF bounded_linear_of_real]
```
```   434
```
```   435 lemmas continuous_on_scaleR [continuous_on_intros] =
```
```   436   bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
```
```   437
```
```   438 lemmas continuous_on_mult [continuous_on_intros] =
```
```   439   bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
```
```   440
```
```   441 lemmas tendsto_mult_zero =
```
```   442   bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
```
```   443
```
```   444 lemmas tendsto_mult_left_zero =
```
```   445   bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
```
```   446
```
```   447 lemmas tendsto_mult_right_zero =
```
```   448   bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
```
```   449
```
```   450 lemma tendsto_power [tendsto_intros]:
```
```   451   fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   452   shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
```
```   453   by (induct n) (simp_all add: tendsto_const tendsto_mult)
```
```   454
```
```   455 lemma continuous_power [continuous_intros]:
```
```   456   fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   457   shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
```
```   458   unfolding continuous_def by (rule tendsto_power)
```
```   459
```
```   460 lemma continuous_on_power [continuous_on_intros]:
```
```   461   fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
```
```   462   shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
```
```   463   unfolding continuous_on_def by (auto intro: tendsto_power)
```
```   464
```
```   465 lemma tendsto_setprod [tendsto_intros]:
```
```   466   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   467   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
```
```   468   shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
```
```   469 proof (cases "finite S")
```
```   470   assume "finite S" thus ?thesis using assms
```
```   471     by (induct, simp add: tendsto_const, simp add: tendsto_mult)
```
```   472 next
```
```   473   assume "\<not> finite S" thus ?thesis
```
```   474     by (simp add: tendsto_const)
```
```   475 qed
```
```   476
```
```   477 lemma continuous_setprod [continuous_intros]:
```
```   478   fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   479   shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
```
```   480   unfolding continuous_def by (rule tendsto_setprod)
```
```   481
```
```   482 lemma continuous_on_setprod [continuous_intros]:
```
```   483   fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
```
```   484   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)"
```
```   485   unfolding continuous_on_def by (auto intro: tendsto_setprod)
```
```   486
```
```   487 subsubsection {* Inverse and division *}
```
```   488
```
```   489 lemma (in bounded_bilinear) Zfun_prod_Bfun:
```
```   490   assumes f: "Zfun f F"
```
```   491   assumes g: "Bfun g F"
```
```   492   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   493 proof -
```
```   494   obtain K where K: "0 \<le> K"
```
```   495     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
```
```   496     using nonneg_bounded by fast
```
```   497   obtain B where B: "0 < B"
```
```   498     and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
```
```   499     using g by (rule BfunE)
```
```   500   have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
```
```   501   using norm_g proof eventually_elim
```
```   502     case (elim x)
```
```   503     have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
```
```   504       by (rule norm_le)
```
```   505     also have "\<dots> \<le> norm (f x) * B * K"
```
```   506       by (intro mult_mono' order_refl norm_g norm_ge_zero
```
```   507                 mult_nonneg_nonneg K elim)
```
```   508     also have "\<dots> = norm (f x) * (B * K)"
```
```   509       by (rule mult_assoc)
```
```   510     finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
```
```   511   qed
```
```   512   with f show ?thesis
```
```   513     by (rule Zfun_imp_Zfun)
```
```   514 qed
```
```   515
```
```   516 lemma (in bounded_bilinear) flip:
```
```   517   "bounded_bilinear (\<lambda>x y. y ** x)"
```
```   518   apply default
```
```   519   apply (rule add_right)
```
```   520   apply (rule add_left)
```
```   521   apply (rule scaleR_right)
```
```   522   apply (rule scaleR_left)
```
```   523   apply (subst mult_commute)
```
```   524   using bounded by fast
```
```   525
```
```   526 lemma (in bounded_bilinear) Bfun_prod_Zfun:
```
```   527   assumes f: "Bfun f F"
```
```   528   assumes g: "Zfun g F"
```
```   529   shows "Zfun (\<lambda>x. f x ** g x) F"
```
```   530   using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
```
```   531
```
```   532 lemma Bfun_inverse_lemma:
```
```   533   fixes x :: "'a::real_normed_div_algebra"
```
```   534   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
```
```   535   apply (subst nonzero_norm_inverse, clarsimp)
```
```   536   apply (erule (1) le_imp_inverse_le)
```
```   537   done
```
```   538
```
```   539 lemma Bfun_inverse:
```
```   540   fixes a :: "'a::real_normed_div_algebra"
```
```   541   assumes f: "(f ---> a) F"
```
```   542   assumes a: "a \<noteq> 0"
```
```   543   shows "Bfun (\<lambda>x. inverse (f x)) F"
```
```   544 proof -
```
```   545   from a have "0 < norm a" by simp
```
```   546   hence "\<exists>r>0. r < norm a" by (rule dense)
```
```   547   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
```
```   548   have "eventually (\<lambda>x. dist (f x) a < r) F"
```
```   549     using tendstoD [OF f r1] by fast
```
```   550   hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
```
```   551   proof eventually_elim
```
```   552     case (elim x)
```
```   553     hence 1: "norm (f x - a) < r"
```
```   554       by (simp add: dist_norm)
```
```   555     hence 2: "f x \<noteq> 0" using r2 by auto
```
```   556     hence "norm (inverse (f x)) = inverse (norm (f x))"
```
```   557       by (rule nonzero_norm_inverse)
```
```   558     also have "\<dots> \<le> inverse (norm a - r)"
```
```   559     proof (rule le_imp_inverse_le)
```
```   560       show "0 < norm a - r" using r2 by simp
```
```   561     next
```
```   562       have "norm a - norm (f x) \<le> norm (a - f x)"
```
```   563         by (rule norm_triangle_ineq2)
```
```   564       also have "\<dots> = norm (f x - a)"
```
```   565         by (rule norm_minus_commute)
```
```   566       also have "\<dots> < r" using 1 .
```
```   567       finally show "norm a - r \<le> norm (f x)" by simp
```
```   568     qed
```
```   569     finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
```
```   570   qed
```
```   571   thus ?thesis by (rule BfunI)
```
```   572 qed
```
```   573
```
```   574 lemma tendsto_inverse [tendsto_intros]:
```
```   575   fixes a :: "'a::real_normed_div_algebra"
```
```   576   assumes f: "(f ---> a) F"
```
```   577   assumes a: "a \<noteq> 0"
```
```   578   shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
```
```   579 proof -
```
```   580   from a have "0 < norm a" by simp
```
```   581   with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
```
```   582     by (rule tendstoD)
```
```   583   then have "eventually (\<lambda>x. f x \<noteq> 0) F"
```
```   584     unfolding dist_norm by (auto elim!: eventually_elim1)
```
```   585   with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
```
```   586     - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   587     by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
```
```   588   moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
```
```   589     by (intro Zfun_minus Zfun_mult_left
```
```   590       bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
```
```   591       Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
```
```   592   ultimately show ?thesis
```
```   593     unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
```
```   594 qed
```
```   595
```
```   596 lemma continuous_inverse:
```
```   597   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   598   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   599   shows "continuous F (\<lambda>x. inverse (f x))"
```
```   600   using assms unfolding continuous_def by (rule tendsto_inverse)
```
```   601
```
```   602 lemma continuous_at_within_inverse[continuous_intros]:
```
```   603   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   604   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   605   shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
```
```   606   using assms unfolding continuous_within by (rule tendsto_inverse)
```
```   607
```
```   608 lemma isCont_inverse[continuous_intros, simp]:
```
```   609   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   610   assumes "isCont f a" and "f a \<noteq> 0"
```
```   611   shows "isCont (\<lambda>x. inverse (f x)) a"
```
```   612   using assms unfolding continuous_at by (rule tendsto_inverse)
```
```   613
```
```   614 lemma continuous_on_inverse[continuous_on_intros]:
```
```   615   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
```
```   616   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   617   shows "continuous_on s (\<lambda>x. inverse (f x))"
```
```   618   using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
```
```   619
```
```   620 lemma tendsto_divide [tendsto_intros]:
```
```   621   fixes a b :: "'a::real_normed_field"
```
```   622   shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
```
```   623     \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
```
```   624   by (simp add: tendsto_mult tendsto_inverse divide_inverse)
```
```   625
```
```   626 lemma continuous_divide:
```
```   627   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   628   assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   629   shows "continuous F (\<lambda>x. (f x) / (g x))"
```
```   630   using assms unfolding continuous_def by (rule tendsto_divide)
```
```   631
```
```   632 lemma continuous_at_within_divide[continuous_intros]:
```
```   633   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   634   assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
```
```   635   shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
```
```   636   using assms unfolding continuous_within by (rule tendsto_divide)
```
```   637
```
```   638 lemma isCont_divide[continuous_intros, simp]:
```
```   639   fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
```
```   640   assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
```
```   641   shows "isCont (\<lambda>x. (f x) / g x) a"
```
```   642   using assms unfolding continuous_at by (rule tendsto_divide)
```
```   643
```
```   644 lemma continuous_on_divide[continuous_on_intros]:
```
```   645   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
```
```   646   assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
```
```   647   shows "continuous_on s (\<lambda>x. (f x) / (g x))"
```
```   648   using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
```
```   649
```
```   650 lemma tendsto_sgn [tendsto_intros]:
```
```   651   fixes l :: "'a::real_normed_vector"
```
```   652   shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
```
```   653   unfolding sgn_div_norm by (simp add: tendsto_intros)
```
```   654
```
```   655 lemma continuous_sgn:
```
```   656   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   657   assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
```
```   658   shows "continuous F (\<lambda>x. sgn (f x))"
```
```   659   using assms unfolding continuous_def by (rule tendsto_sgn)
```
```   660
```
```   661 lemma continuous_at_within_sgn[continuous_intros]:
```
```   662   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   663   assumes "continuous (at a within s) f" and "f a \<noteq> 0"
```
```   664   shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
```
```   665   using assms unfolding continuous_within by (rule tendsto_sgn)
```
```   666
```
```   667 lemma isCont_sgn[continuous_intros]:
```
```   668   fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
```
```   669   assumes "isCont f a" and "f a \<noteq> 0"
```
```   670   shows "isCont (\<lambda>x. sgn (f x)) a"
```
```   671   using assms unfolding continuous_at by (rule tendsto_sgn)
```
```   672
```
```   673 lemma continuous_on_sgn[continuous_on_intros]:
```
```   674   fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
```
```   675   assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
```
```   676   shows "continuous_on s (\<lambda>x. sgn (f x))"
```
```   677   using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
```
```   678
```
```   679 lemma filterlim_at_infinity:
```
```   680   fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
```
```   681   assumes "0 \<le> c"
```
```   682   shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
```
```   683   unfolding filterlim_iff eventually_at_infinity
```
```   684 proof safe
```
```   685   fix P :: "'a \<Rightarrow> bool" and b
```
```   686   assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
```
```   687     and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
```
```   688   have "max b (c + 1) > c" by auto
```
```   689   with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
```
```   690     by auto
```
```   691   then show "eventually (\<lambda>x. P (f x)) F"
```
```   692   proof eventually_elim
```
```   693     fix x assume "max b (c + 1) \<le> norm (f x)"
```
```   694     with P show "P (f x)" by auto
```
```   695   qed
```
```   696 qed force
```
```   697
```
```   698 subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
```
```   699
```
```   700 text {*
```
```   701
```
```   702 This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
```
```   703 @{term "at_right x"} and also @{term "at_right 0"}.
```
```   704
```
```   705 *}
```
```   706
```
```   707 lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
```
```   708
```
```   709 lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::real)"
```
```   710   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
```
```   711   by (intro allI ex_cong) (auto simp: dist_real_def field_simps)
```
```   712
```
```   713 lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::real)"
```
```   714   unfolding filter_eq_iff eventually_filtermap eventually_nhds_metric
```
```   715   apply (intro allI ex_cong)
```
```   716   apply (auto simp: dist_real_def field_simps)
```
```   717   apply (erule_tac x="-x" in allE)
```
```   718   apply simp
```
```   719   done
```
```   720
```
```   721 lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::real)"
```
```   722   unfolding at_def filtermap_nhds_shift[symmetric]
```
```   723   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```   724
```
```   725 lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
```
```   726   unfolding filtermap_at_shift[symmetric]
```
```   727   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```   728
```
```   729 lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
```
```   730   using filtermap_at_right_shift[of "-a" 0] by simp
```
```   731
```
```   732 lemma filterlim_at_right_to_0:
```
```   733   "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
```
```   734   unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
```
```   735
```
```   736 lemma eventually_at_right_to_0:
```
```   737   "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
```
```   738   unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
```
```   739
```
```   740 lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::real)"
```
```   741   unfolding at_def filtermap_nhds_minus[symmetric]
```
```   742   by (simp add: filter_eq_iff eventually_filtermap eventually_within)
```
```   743
```
```   744 lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
```
```   745   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
```
```   746
```
```   747 lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
```
```   748   by (simp add: filter_eq_iff eventually_filtermap eventually_within filtermap_at_minus[symmetric])
```
```   749
```
```   750 lemma filterlim_at_left_to_right:
```
```   751   "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
```
```   752   unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
```
```   753
```
```   754 lemma eventually_at_left_to_right:
```
```   755   "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
```
```   756   unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
```
```   757
```
```   758 lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
```
```   759   unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
```
```   760   by (metis le_minus_iff minus_minus)
```
```   761
```
```   762 lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
```
```   763   unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
```
```   764
```
```   765 lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
```
```   766   unfolding filterlim_def at_top_mirror filtermap_filtermap ..
```
```   767
```
```   768 lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
```
```   769   unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
```
```   770
```
```   771 lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
```
```   772   unfolding filterlim_at_top eventually_at_bot_dense
```
```   773   by (metis leI minus_less_iff order_less_asym)
```
```   774
```
```   775 lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
```
```   776   unfolding filterlim_at_bot eventually_at_top_dense
```
```   777   by (metis leI less_minus_iff order_less_asym)
```
```   778
```
```   779 lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
```
```   780   using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
```
```   781   using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
```
```   782   by auto
```
```   783
```
```   784 lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
```
```   785   unfolding filterlim_uminus_at_top by simp
```
```   786
```
```   787 lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
```
```   788   unfolding filterlim_at_top_gt[where c=0] eventually_within at_def
```
```   789 proof safe
```
```   790   fix Z :: real assume [arith]: "0 < Z"
```
```   791   then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
```
```   792     by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
```
```   793   then show "eventually (\<lambda>x. x \<in> - {0} \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
```
```   794     by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
```
```   795 qed
```
```   796
```
```   797 lemma filterlim_inverse_at_top:
```
```   798   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
```
```   799   by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
```
```   800      (simp add: filterlim_def eventually_filtermap le_within_iff at_def eventually_elim1)
```
```   801
```
```   802 lemma filterlim_inverse_at_bot_neg:
```
```   803   "LIM x (at_left (0::real)). inverse x :> at_bot"
```
```   804   by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
```
```   805
```
```   806 lemma filterlim_inverse_at_bot:
```
```   807   "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
```
```   808   unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
```
```   809   by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
```
```   810
```
```   811 lemma tendsto_inverse_0:
```
```   812   fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
```
```   813   shows "(inverse ---> (0::'a)) at_infinity"
```
```   814   unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
```
```   815 proof safe
```
```   816   fix r :: real assume "0 < r"
```
```   817   show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
```
```   818   proof (intro exI[of _ "inverse (r / 2)"] allI impI)
```
```   819     fix x :: 'a
```
```   820     from `0 < r` have "0 < inverse (r / 2)" by simp
```
```   821     also assume *: "inverse (r / 2) \<le> norm x"
```
```   822     finally show "norm (inverse x) < r"
```
```   823       using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
```
```   824   qed
```
```   825 qed
```
```   826
```
```   827 lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
```
```   828 proof (rule antisym)
```
```   829   have "(inverse ---> (0::real)) at_top"
```
```   830     by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
```
```   831   then show "filtermap inverse at_top \<le> at_right (0::real)"
```
```   832     unfolding at_within_eq
```
```   833     by (intro le_withinI) (simp_all add: eventually_filtermap eventually_gt_at_top filterlim_def)
```
```   834 next
```
```   835   have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
```
```   836     using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
```
```   837   then show "at_right (0::real) \<le> filtermap inverse at_top"
```
```   838     by (simp add: filtermap_ident filtermap_filtermap)
```
```   839 qed
```
```   840
```
```   841 lemma eventually_at_right_to_top:
```
```   842   "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
```
```   843   unfolding at_right_to_top eventually_filtermap ..
```
```   844
```
```   845 lemma filterlim_at_right_to_top:
```
```   846   "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
```
```   847   unfolding filterlim_def at_right_to_top filtermap_filtermap ..
```
```   848
```
```   849 lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
```
```   850   unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
```
```   851
```
```   852 lemma eventually_at_top_to_right:
```
```   853   "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
```
```   854   unfolding at_top_to_right eventually_filtermap ..
```
```   855
```
```   856 lemma filterlim_at_top_to_right:
```
```   857   "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
```
```   858   unfolding filterlim_def at_top_to_right filtermap_filtermap ..
```
```   859
```
```   860 lemma filterlim_inverse_at_infinity:
```
```   861   fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   862   shows "filterlim inverse at_infinity (at (0::'a))"
```
```   863   unfolding filterlim_at_infinity[OF order_refl]
```
```   864 proof safe
```
```   865   fix r :: real assume "0 < r"
```
```   866   then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
```
```   867     unfolding eventually_at norm_inverse
```
```   868     by (intro exI[of _ "inverse r"])
```
```   869        (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
```
```   870 qed
```
```   871
```
```   872 lemma filterlim_inverse_at_iff:
```
```   873   fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   874   shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
```
```   875   unfolding filterlim_def filtermap_filtermap[symmetric]
```
```   876 proof
```
```   877   assume "filtermap g F \<le> at_infinity"
```
```   878   then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
```
```   879     by (rule filtermap_mono)
```
```   880   also have "\<dots> \<le> at 0"
```
```   881     using tendsto_inverse_0
```
```   882     by (auto intro!: le_withinI exI[of _ 1]
```
```   883              simp: eventually_filtermap eventually_at_infinity filterlim_def at_def)
```
```   884   finally show "filtermap inverse (filtermap g F) \<le> at 0" .
```
```   885 next
```
```   886   assume "filtermap inverse (filtermap g F) \<le> at 0"
```
```   887   then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
```
```   888     by (rule filtermap_mono)
```
```   889   with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
```
```   890     by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
```
```   891 qed
```
```   892
```
```   893 lemma tendsto_inverse_0_at_top:
```
```   894   "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
```
```   895  by (metis at_top_le_at_infinity filterlim_at filterlim_inverse_at_iff filterlim_mono order_refl)
```
```   896
```
```   897 text {*
```
```   898
```
```   899 We only show rules for multiplication and addition when the functions are either against a real
```
```   900 value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
```
```   901
```
```   902 *}
```
```   903
```
```   904 lemma filterlim_tendsto_pos_mult_at_top:
```
```   905   assumes f: "(f ---> c) F" and c: "0 < c"
```
```   906   assumes g: "LIM x F. g x :> at_top"
```
```   907   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```   908   unfolding filterlim_at_top_gt[where c=0]
```
```   909 proof safe
```
```   910   fix Z :: real assume "0 < Z"
```
```   911   from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
```
```   912     by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
```
```   913              simp: dist_real_def abs_real_def split: split_if_asm)
```
```   914   moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
```
```   915     unfolding filterlim_at_top by auto
```
```   916   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```   917   proof eventually_elim
```
```   918     fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
```
```   919     with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
```
```   920       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```   921     with `0 < c` show "Z \<le> f x * g x"
```
```   922        by simp
```
```   923   qed
```
```   924 qed
```
```   925
```
```   926 lemma filterlim_at_top_mult_at_top:
```
```   927   assumes f: "LIM x F. f x :> at_top"
```
```   928   assumes g: "LIM x F. g x :> at_top"
```
```   929   shows "LIM x F. (f x * g x :: real) :> at_top"
```
```   930   unfolding filterlim_at_top_gt[where c=0]
```
```   931 proof safe
```
```   932   fix Z :: real assume "0 < Z"
```
```   933   from f have "eventually (\<lambda>x. 1 \<le> f x) F"
```
```   934     unfolding filterlim_at_top by auto
```
```   935   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```   936     unfolding filterlim_at_top by auto
```
```   937   ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
```
```   938   proof eventually_elim
```
```   939     fix x assume "1 \<le> f x" "Z \<le> g x"
```
```   940     with `0 < Z` have "1 * Z \<le> f x * g x"
```
```   941       by (intro mult_mono) (auto simp: zero_le_divide_iff)
```
```   942     then show "Z \<le> f x * g x"
```
```   943        by simp
```
```   944   qed
```
```   945 qed
```
```   946
```
```   947 lemma filterlim_tendsto_pos_mult_at_bot:
```
```   948   assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
```
```   949   shows "LIM x F. f x * g x :> at_bot"
```
```   950   using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
```
```   951   unfolding filterlim_uminus_at_bot by simp
```
```   952
```
```   953 lemma filterlim_tendsto_add_at_top:
```
```   954   assumes f: "(f ---> c) F"
```
```   955   assumes g: "LIM x F. g x :> at_top"
```
```   956   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```   957   unfolding filterlim_at_top_gt[where c=0]
```
```   958 proof safe
```
```   959   fix Z :: real assume "0 < Z"
```
```   960   from f have "eventually (\<lambda>x. c - 1 < f x) F"
```
```   961     by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
```
```   962   moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
```
```   963     unfolding filterlim_at_top by auto
```
```   964   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```   965     by eventually_elim simp
```
```   966 qed
```
```   967
```
```   968 lemma LIM_at_top_divide:
```
```   969   fixes f g :: "'a \<Rightarrow> real"
```
```   970   assumes f: "(f ---> a) F" "0 < a"
```
```   971   assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
```
```   972   shows "LIM x F. f x / g x :> at_top"
```
```   973   unfolding divide_inverse
```
```   974   by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
```
```   975
```
```   976 lemma filterlim_at_top_add_at_top:
```
```   977   assumes f: "LIM x F. f x :> at_top"
```
```   978   assumes g: "LIM x F. g x :> at_top"
```
```   979   shows "LIM x F. (f x + g x :: real) :> at_top"
```
```   980   unfolding filterlim_at_top_gt[where c=0]
```
```   981 proof safe
```
```   982   fix Z :: real assume "0 < Z"
```
```   983   from f have "eventually (\<lambda>x. 0 \<le> f x) F"
```
```   984     unfolding filterlim_at_top by auto
```
```   985   moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
```
```   986     unfolding filterlim_at_top by auto
```
```   987   ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
```
```   988     by eventually_elim simp
```
```   989 qed
```
```   990
```
```   991 lemma tendsto_divide_0:
```
```   992   fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
```
```   993   assumes f: "(f ---> c) F"
```
```   994   assumes g: "LIM x F. g x :> at_infinity"
```
```   995   shows "((\<lambda>x. f x / g x) ---> 0) F"
```
```   996   using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
```
```   997
```
```   998 lemma linear_plus_1_le_power:
```
```   999   fixes x :: real
```
```  1000   assumes x: "0 \<le> x"
```
```  1001   shows "real n * x + 1 \<le> (x + 1) ^ n"
```
```  1002 proof (induct n)
```
```  1003   case (Suc n)
```
```  1004   have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
```
```  1005     by (simp add: field_simps real_of_nat_Suc mult_nonneg_nonneg x)
```
```  1006   also have "\<dots> \<le> (x + 1)^Suc n"
```
```  1007     using Suc x by (simp add: mult_left_mono)
```
```  1008   finally show ?case .
```
```  1009 qed simp
```
```  1010
```
```  1011 lemma filterlim_realpow_sequentially_gt1:
```
```  1012   fixes x :: "'a :: real_normed_div_algebra"
```
```  1013   assumes x[arith]: "1 < norm x"
```
```  1014   shows "LIM n sequentially. x ^ n :> at_infinity"
```
```  1015 proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
```
```  1016   fix y :: real assume "0 < y"
```
```  1017   have "0 < norm x - 1" by simp
```
```  1018   then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
```
```  1019   also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
```
```  1020   also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
```
```  1021   also have "\<dots> = norm x ^ N" by simp
```
```  1022   finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
```
```  1023     by (metis order_less_le_trans power_increasing order_less_imp_le x)
```
```  1024   then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
```
```  1025     unfolding eventually_sequentially
```
```  1026     by (auto simp: norm_power)
```
```  1027 qed simp
```
```  1028
```
```  1029
```
```  1030 (* Unfortunately eventually_within was overwritten by Multivariate_Analysis.
```
```  1031    Hence it was references as Limits.within, but now it is Basic_Topology.eventually_within *)
```
```  1032 lemmas eventually_within = eventually_within
```
```  1033
```
```  1034 end
```
```  1035
```