src/HOL/Lim.thy
author hoelzl
Wed, 23 Sep 2009 13:17:17 +0200
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(*  Title       : Lim.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{* Limits and Continuity *}
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theory Lim
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imports SEQ
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begin
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text{*Standard Definitions*}
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definition
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  LIM :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a, 'b] \<Rightarrow> bool"
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        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
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  [code del]: "f -- a --> L =
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     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & dist x a < s
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        --> dist (f x) L < r)"
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definition
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  isCont :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a] \<Rightarrow> bool" where
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  "isCont f a = (f -- a --> (f a))"
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definition
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  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
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  [code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
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subsection {* Limits of Functions *}
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lemma LIM_conv_tendsto: "(f -- a --> L) \<longleftrightarrow> (f ---> L) (at a)"
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unfolding LIM_def tendsto_iff eventually_at ..
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lemma metric_LIM_I:
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  "(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
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    \<Longrightarrow> f -- a --> L"
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by (simp add: LIM_def)
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lemma metric_LIM_D:
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  "\<lbrakk>f -- a --> L; 0 < r\<rbrakk>
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    \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r"
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by (simp add: LIM_def)
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lemma LIM_eq:
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  shows "f -- a --> L =
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     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
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by (simp add: LIM_def dist_norm)
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lemma LIM_I:
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  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
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      ==> f -- a --> L"
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by (simp add: LIM_eq)
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lemma LIM_D:
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  shows "[| f -- a --> L; 0<r |]
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      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
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by (simp add: LIM_eq)
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lemma LIM_offset:
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  fixes a :: "'a::real_normed_vector" and L :: "'b::metric_space"
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  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
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unfolding LIM_def dist_norm
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apply clarify
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apply (drule_tac x="r" in spec, safe)
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apply (rule_tac x="s" in exI, safe)
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apply (drule_tac x="x + k" in spec)
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apply (simp add: algebra_simps)
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done
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lemma LIM_offset_zero:
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  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
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by (drule_tac k="a" in LIM_offset, simp add: add_commute)
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lemma LIM_offset_zero_cancel:
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  shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
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by (drule_tac k="- a" in LIM_offset, simp)
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lemma LIM_const [simp]: "(%x. k) -- x --> k"
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by (simp add: LIM_def)
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lemma LIM_cong_limit: "\<lbrakk> f -- x --> L ; K = L \<rbrakk> \<Longrightarrow> f -- x --> K" by simp
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lemma LIM_add:
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  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  assumes f: "f -- a --> L" and g: "g -- a --> M"
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  shows "(\<lambda>x. f x + g x) -- a --> (L + M)"
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using assms unfolding LIM_conv_tendsto by (rule tendsto_add)
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lemma LIM_add_zero:
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  shows "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
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by (drule (1) LIM_add, simp)
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lemma minus_diff_minus:
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  fixes a b :: "'a::ab_group_add"
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  shows "(- a) - (- b) = - (a - b)"
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by simp
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lemma LIM_minus:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. - f x) -- a --> - L"
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unfolding LIM_conv_tendsto by (rule tendsto_minus)
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(* TODO: delete *)
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lemma LIM_add_minus:
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  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
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by (intro LIM_add LIM_minus)
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lemma LIM_diff:
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  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "\<lbrakk>f -- x --> l; g -- x --> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --> l - m"
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unfolding LIM_conv_tendsto by (rule tendsto_diff)
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lemma LIM_zero:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
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by (simp add: LIM_def dist_norm)
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lemma LIM_zero_cancel:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
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by (simp add: LIM_def dist_norm)
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lemma LIM_zero_iff:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
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by (simp add: LIM_def dist_norm)
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lemma metric_LIM_imp_LIM:
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  assumes f: "f -- a --> l"
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  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
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  shows "g -- a --> m"
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apply (rule metric_LIM_I, drule metric_LIM_D [OF f], safe)
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apply (rule_tac x="s" in exI, safe)
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   142
apply (drule_tac x="x" in spec, safe)
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apply (erule (1) order_le_less_trans [OF le])
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   144
done
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   145
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lemma LIM_imp_LIM:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  fixes g :: "'a::metric_space \<Rightarrow> 'c::real_normed_vector"
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  assumes f: "f -- a --> l"
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  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
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   151
  shows "g -- a --> m"
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apply (rule metric_LIM_imp_LIM [OF f])
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apply (simp add: dist_norm le)
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done
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lemma LIM_norm:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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   158
  shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
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   159
unfolding LIM_conv_tendsto by (rule tendsto_norm)
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   160
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lemma LIM_norm_zero:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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   163
  shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
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   164
by (drule LIM_norm, simp)
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   165
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lemma LIM_norm_zero_cancel:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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   168
  shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
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   169
by (erule LIM_imp_LIM, simp)
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   170
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lemma LIM_norm_zero_iff:
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  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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   173
  shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
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   174
by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
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   175
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   176
lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
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   177
by (fold real_norm_def, rule LIM_norm)
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   178
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   179
lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
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   180
by (fold real_norm_def, rule LIM_norm_zero)
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   181
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lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
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   183
by (fold real_norm_def, rule LIM_norm_zero_cancel)
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   184
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   185
lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
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   186
by (fold real_norm_def, rule LIM_norm_zero_iff)
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   187
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lemma LIM_const_not_eq:
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   189
  fixes a :: "'a::real_normed_algebra_1"
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   190
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
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apply (simp add: LIM_def)
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apply (rule_tac x="dist k L" in exI, simp add: zero_less_dist_iff, safe)
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   193
apply (rule_tac x="a + of_real (s/2)" in exI, simp add: dist_norm)
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done
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   195
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lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
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   197
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lemma LIM_const_eq:
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  fixes a :: "'a::real_normed_algebra_1"
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   200
  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
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   201
apply (rule ccontr)
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5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
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apply (blast dest: LIM_const_not_eq)
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done
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   204
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lemma LIM_unique:
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  fixes a :: "'a::real_normed_algebra_1" -- {* TODO: find a more appropriate class *}
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   207
  shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
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apply (rule ccontr)
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   209
apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)
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apply (drule_tac r="dist L M / 2" in metric_LIM_D, simp add: zero_less_dist_iff)
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   211
apply (clarify, rename_tac r s)
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   212
apply (subgoal_tac "min r s \<noteq> 0")
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apply (subgoal_tac "dist L M < dist L M / 2 + dist L M / 2", simp)
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apply (subgoal_tac "dist L M \<le> dist (f (a + of_real (min r s / 2))) L +
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   215
                               dist (f (a + of_real (min r s / 2))) M")
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   216
apply (erule le_less_trans, rule add_strict_mono)
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   217
apply (drule spec, erule mp, simp add: dist_norm)
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   218
apply (drule spec, erule mp, simp add: dist_norm)
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   219
apply (subst dist_commute, rule dist_triangle)
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apply simp
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   221
done
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   222
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   223
lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
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by (auto simp add: LIM_def)
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   225
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text{*Limits are equal for functions equal except at limit point*}
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lemma LIM_equal:
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     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
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   229
by (simp add: LIM_def)
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   230
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lemma LIM_cong:
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   232
  "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
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   233
   \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
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   234
by (simp add: LIM_def)
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   235
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   236
lemma metric_LIM_equal2:
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  assumes 1: "0 < R"
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   238
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
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   239
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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   240
apply (unfold LIM_def, safe)
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   241
apply (drule_tac x="r" in spec, safe)
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   242
apply (rule_tac x="min s R" in exI, safe)
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   243
apply (simp add: 1)
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   244
apply (simp add: 2)
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done
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   246
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   247
lemma LIM_equal2:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
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   249
  assumes 1: "0 < R"
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   250
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
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   251
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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   252
apply (unfold LIM_def dist_norm, safe)
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   253
apply (drule_tac x="r" in spec, safe)
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   254
apply (rule_tac x="min s R" in exI, safe)
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   255
apply (simp add: 1)
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   256
apply (simp add: 2)
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   257
done
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   258
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   259
text{*Two uses in Transcendental.ML*}
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lemma LIM_trans:
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   261
  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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   262
  shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
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   263
apply (drule LIM_add, assumption)
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   264
apply (auto simp add: add_assoc)
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   265
done
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   266
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d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   267
lemma LIM_compose:
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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  assumes g: "g -- l --> g l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   269
  assumes f: "f -- a --> l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   270
  shows "(\<lambda>x. g (f x)) -- a --> g l"
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   271
proof (rule metric_LIM_I)
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   272
  fix r::real assume r: "0 < r"
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   273
  obtain s where s: "0 < s"
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    and less_r: "\<And>y. \<lbrakk>y \<noteq> l; dist y l < s\<rbrakk> \<Longrightarrow> dist (g y) (g l) < r"
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   275
    using metric_LIM_D [OF g r] by fast
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   276
  obtain t where t: "0 < t"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   277
    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) l < s"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   278
    using metric_LIM_D [OF f s] by fast
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   279
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   280
  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) (g l) < r"
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   281
  proof (rule exI, safe)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   282
    show "0 < t" using t .
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   283
  next
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   284
    fix x assume "x \<noteq> a" and "dist x a < t"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   285
    hence "dist (f x) l < s" by (rule less_s)
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   286
    thus "dist (g (f x)) (g l) < r"
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   287
      using r less_r by (case_tac "f x = l", simp_all)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   288
  qed
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   289
qed
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   290
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   291
lemma metric_LIM_compose2:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   292
  assumes f: "f -- a --> b"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   293
  assumes g: "g -- b --> c"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   294
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   295
  shows "(\<lambda>x. g (f x)) -- a --> c"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   296
proof (rule metric_LIM_I)
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   297
  fix r :: real
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   298
  assume r: "0 < r"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   299
  obtain s where s: "0 < s"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   300
    and less_r: "\<And>y. \<lbrakk>y \<noteq> b; dist y b < s\<rbrakk> \<Longrightarrow> dist (g y) c < r"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   301
    using metric_LIM_D [OF g r] by fast
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   302
  obtain t where t: "0 < t"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   303
    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < t\<rbrakk> \<Longrightarrow> dist (f x) b < s"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   304
    using metric_LIM_D [OF f s] by fast
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   305
  obtain d where d: "0 < d"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   306
    and neq_b: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < d\<rbrakk> \<Longrightarrow> f x \<noteq> b"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   307
    using inj by fast
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   308
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   309
  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> dist x a < t \<longrightarrow> dist (g (f x)) c < r"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   310
  proof (safe intro!: exI)
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   311
    show "0 < min d t" using d t by simp
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   312
  next
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   313
    fix x
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   314
    assume "x \<noteq> a" and "dist x a < min d t"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   315
    hence "f x \<noteq> b" and "dist (f x) b < s"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   316
      using neq_b less_s by simp_all
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   317
    thus "dist (g (f x)) c < r"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   318
      by (rule less_r)
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   319
  qed
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   320
qed
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   321
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   322
lemma LIM_compose2:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   323
  fixes a :: "'a::real_normed_vector"
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   324
  assumes f: "f -- a --> b"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   325
  assumes g: "g -- b --> c"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   326
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   327
  shows "(\<lambda>x. g (f x)) -- a --> c"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   328
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   329
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   330
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   331
unfolding o_def by (rule LIM_compose)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   332
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   333
lemma real_LIM_sandwich_zero:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   334
  fixes f g :: "'a::metric_space \<Rightarrow> real"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   335
  assumes f: "f -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   336
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   337
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   338
  shows "g -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   339
proof (rule LIM_imp_LIM [OF f])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   340
  fix x assume x: "x \<noteq> a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   341
  have "norm (g x - 0) = g x" by (simp add: 1 x)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   342
  also have "g x \<le> f x" by (rule 2 [OF x])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   343
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   344
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   345
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   346
qed
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   347
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21810
diff changeset
   348
text {* Bounded Linear Operators *}
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   349
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   350
lemma (in bounded_linear) LIM:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   351
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   352
unfolding LIM_conv_tendsto by (rule tendsto)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   353
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   354
lemma (in bounded_linear) cont: "f -- a --> f a"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   355
by (rule LIM [OF LIM_ident])
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   356
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   357
lemma (in bounded_linear) LIM_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   358
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   359
by (drule LIM, simp only: zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   360
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21810
diff changeset
   361
text {* Bounded Bilinear Operators *}
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   362
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   363
lemma (in bounded_bilinear) LIM:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   364
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   365
unfolding LIM_conv_tendsto by (rule tendsto)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   366
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   367
lemma (in bounded_bilinear) LIM_prod_zero:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   368
  fixes a :: "'d::metric_space"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   369
  assumes f: "f -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   370
  assumes g: "g -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   371
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents: 31338
diff changeset
   372
using LIM [OF f g] by (simp add: zero_left)
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   373
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   374
lemma (in bounded_bilinear) LIM_left_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   375
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   376
by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   377
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   378
lemma (in bounded_bilinear) LIM_right_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   379
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   380
by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   381
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   382
lemmas LIM_mult = mult.LIM
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   383
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   384
lemmas LIM_mult_zero = mult.LIM_prod_zero
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   385
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   386
lemmas LIM_mult_left_zero = mult.LIM_left_zero
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   387
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   388
lemmas LIM_mult_right_zero = mult.LIM_right_zero
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   389
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   390
lemmas LIM_scaleR = scaleR.LIM
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   391
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   392
lemmas LIM_of_real = of_real.LIM
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   393
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   394
lemma LIM_power:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   395
  fixes f :: "'a::metric_space \<Rightarrow> 'b::{power,real_normed_algebra}"
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   396
  assumes f: "f -- a --> l"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   397
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
30273
ecd6f0ca62ea declare power_Suc [simp]; remove redundant type-specific versions of power_Suc
huffman
parents: 29885
diff changeset
   398
by (induct n, simp, simp add: LIM_mult f)
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   399
22641
a5dc96fad632 moved nonstandard limit stuff from Lim.thy into new theory HLim.thy
huffman
parents: 22637
diff changeset
   400
subsubsection {* Derived theorems about @{term LIM} *}
a5dc96fad632 moved nonstandard limit stuff from Lim.thy into new theory HLim.thy
huffman
parents: 22637
diff changeset
   401
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   402
lemma LIM_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   403
  fixes L :: "'a::real_normed_div_algebra"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   404
  shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   405
unfolding LIM_conv_tendsto
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   406
by (rule tendsto_inverse)
22637
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   407
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   408
lemma LIM_inverse_fun:
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   409
  assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   410
  shows "inverse -- a --> inverse a"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   411
by (rule LIM_inverse [OF LIM_ident a])
22637
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   412
29885
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   413
lemma LIM_sgn:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   414
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   415
  shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
29885
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   416
unfolding sgn_div_norm
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   417
by (simp add: LIM_scaleR LIM_inverse LIM_norm)
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   418
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   419
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   420
subsection {* Continuity *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   421
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   422
lemma LIM_isCont_iff:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   423
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   424
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   425
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   426
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   427
lemma isCont_iff:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   428
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::metric_space"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   429
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   430
by (simp add: isCont_def LIM_isCont_iff)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   431
23069
cdfff0241c12 rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents: 23040
diff changeset
   432
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
cdfff0241c12 rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents: 23040
diff changeset
   433
  unfolding isCont_def by (rule LIM_ident)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   434
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   435
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   436
  unfolding isCont_def by (rule LIM_const)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   437
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   438
lemma isCont_norm:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   439
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   440
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   441
  unfolding isCont_def by (rule LIM_norm)
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   442
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   443
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   444
  unfolding isCont_def by (rule LIM_rabs)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   445
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   446
lemma isCont_add:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   447
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   448
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   449
  unfolding isCont_def by (rule LIM_add)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   450
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   451
lemma isCont_minus:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   452
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   453
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   454
  unfolding isCont_def by (rule LIM_minus)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   455
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   456
lemma isCont_diff:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   457
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   458
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   459
  unfolding isCont_def by (rule LIM_diff)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   460
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   461
lemma isCont_mult:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   462
  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_algebra"
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   463
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   464
  unfolding isCont_def by (rule LIM_mult)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   465
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   466
lemma isCont_inverse:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   467
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_div_algebra"
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   468
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   469
  unfolding isCont_def by (rule LIM_inverse)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   470
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   471
lemma isCont_LIM_compose:
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   472
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   473
  unfolding isCont_def by (rule LIM_compose)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   474
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   475
lemma metric_isCont_LIM_compose2:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   476
  assumes f [unfolded isCont_def]: "isCont f a"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   477
  assumes g: "g -- f a --> l"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   478
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   479
  shows "(\<lambda>x. g (f x)) -- a --> l"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   480
by (rule metric_LIM_compose2 [OF f g inj])
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   481
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   482
lemma isCont_LIM_compose2:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   483
  fixes a :: "'a::real_normed_vector"
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   484
  assumes f [unfolded isCont_def]: "isCont f a"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   485
  assumes g: "g -- f a --> l"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   486
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   487
  shows "(\<lambda>x. g (f x)) -- a --> l"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   488
by (rule LIM_compose2 [OF f g inj])
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   489
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   490
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   491
  unfolding isCont_def by (rule LIM_compose)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   492
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   493
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   494
  unfolding o_def by (rule isCont_o2)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   495
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   496
lemma (in bounded_linear) isCont: "isCont f a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   497
  unfolding isCont_def by (rule cont)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   498
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   499
lemma (in bounded_bilinear) isCont:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   500
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   501
  unfolding isCont_def by (rule LIM)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   502
23127
56ee8105c002 simplify names of locale interpretations
huffman
parents: 23118
diff changeset
   503
lemmas isCont_scaleR = scaleR.isCont
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   504
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   505
lemma isCont_of_real:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   506
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   507
  unfolding isCont_def by (rule LIM_of_real)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   508
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   509
lemma isCont_power:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   510
  fixes f :: "'a::metric_space \<Rightarrow> 'b::{power,real_normed_algebra}"
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   511
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   512
  unfolding isCont_def by (rule LIM_power)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   513
29885
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   514
lemma isCont_sgn:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   515
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   516
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
29885
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   517
  unfolding isCont_def by (rule LIM_sgn)
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   518
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   519
lemma isCont_abs [simp]: "isCont abs (a::real)"
23069
cdfff0241c12 rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents: 23040
diff changeset
   520
by (rule isCont_rabs [OF isCont_ident])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   521
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   522
lemma isCont_setsum:
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   523
  fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::real_normed_vector"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   524
  fixes A :: "'a set" assumes "finite A"
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   525
  shows "\<forall> i \<in> A. isCont (f i) x \<Longrightarrow> isCont (\<lambda> x. \<Sum> i \<in> A. f i x) x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   526
  using `finite A`
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   527
proof induct
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   528
  case (insert a F) show "isCont (\<lambda> x. \<Sum> i \<in> (insert a F). f i x) x" 
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   529
    unfolding setsum_insert[OF `finite F` `a \<notin> F`] by (rule isCont_add, auto simp add: insert)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   530
qed auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   531
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   532
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   533
  and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   534
  shows "0 \<le> f x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   535
proof (rule ccontr)
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   536
  assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   537
  hence "0 < - f x / 2" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   538
  from isCont[unfolded isCont_def, THEN LIM_D, OF this]
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   539
  obtain s where "s > 0" and s_D: "\<And>x'. \<lbrakk> x' \<noteq> x ; \<bar> x' - x \<bar> < s \<rbrakk> \<Longrightarrow> \<bar> f x' - f x \<bar> < - f x / 2" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   540
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   541
  let ?x = "x - min (s / 2) ((x - b) / 2)"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   542
  have "?x < x" and "\<bar> ?x - x \<bar> < s"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   543
    using `b < x` and `0 < s` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   544
  have "b < ?x"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   545
  proof (cases "s < x - b")
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   546
    case True thus ?thesis using `0 < s` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   547
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   548
    case False hence "s / 2 \<ge> (x - b) / 2" by auto
32642
026e7c6a6d08 be more cautious wrt. simp rules: inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer
haftmann
parents: 32436
diff changeset
   549
    hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   550
    thus ?thesis using `b < x` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   551
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   552
  hence "0 \<le> f ?x" using all_le `?x < x` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   553
  moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   554
    using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   555
  hence "f ?x - f x < - f x / 2" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   556
  hence "f ?x < f x / 2" by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   557
  hence "f ?x < 0" using `f x < 0` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   558
  thus False using `0 \<le> f ?x` by auto
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   559
qed
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   560
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   561
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   562
subsection {* Uniform Continuity *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   563
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   564
lemma isUCont_isCont: "isUCont f ==> isCont f x"
23012
496b42cf588d remove dependence on Hilbert_Choice.thy
huffman
parents: 22641
diff changeset
   565
by (simp add: isUCont_def isCont_def LIM_def, force)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   566
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   567
lemma isUCont_Cauchy:
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   568
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   569
unfolding isUCont_def
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   570
apply (rule metric_CauchyI)
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   571
apply (drule_tac x=e in spec, safe)
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   572
apply (drule_tac e=s in metric_CauchyD, safe)
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   573
apply (rule_tac x=M in exI, simp)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   574
done
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   575
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   576
lemma (in bounded_linear) isUCont: "isUCont f"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   577
unfolding isUCont_def dist_norm
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   578
proof (intro allI impI)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   579
  fix r::real assume r: "0 < r"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   580
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   581
    using pos_bounded by fast
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   582
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   583
  proof (rule exI, safe)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   584
    from r K show "0 < r / K" by (rule divide_pos_pos)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   585
  next
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   586
    fix x y :: 'a
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   587
    assume xy: "norm (x - y) < r / K"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   588
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   589
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   590
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   591
    finally show "norm (f x - f y) < r" .
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   592
  qed
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   593
qed
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   594
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   595
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   596
by (rule isUCont [THEN isUCont_Cauchy])
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   597
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   598
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   599
subsection {* Relation of LIM and LIMSEQ *}
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   600
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   601
lemma LIMSEQ_SEQ_conv1:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   602
  fixes a :: "'a::metric_space"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   603
  assumes X: "X -- a --> L"
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   604
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   605
proof (safe intro!: metric_LIMSEQ_I)
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   606
  fix S :: "nat \<Rightarrow> 'a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   607
  fix r :: real
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   608
  assume rgz: "0 < r"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   609
  assume as: "\<forall>n. S n \<noteq> a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   610
  assume S: "S ----> a"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   611
  from metric_LIM_D [OF X rgz] obtain s
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   612
    where sgz: "0 < s"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   613
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   614
    by fast
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   615
  from metric_LIMSEQ_D [OF S sgz]
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   616
  obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   617
  hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   618
  thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   619
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   620
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   621
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   622
lemma LIMSEQ_SEQ_conv2:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   623
  fixes a :: real
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   624
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   625
  shows "X -- a --> L"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   626
proof (rule ccontr)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   627
  assume "\<not> (X -- a --> L)"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   628
  hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   629
    unfolding LIM_def dist_norm .
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   630
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> dist (X x) L < r)" by simp
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   631
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r)" by (simp add: not_less)
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   632
  then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> dist (X x) L \<ge> r))" by auto
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   633
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   634
  let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   635
  have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   636
    using rdef by simp
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   637
  hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   638
    by (rule someI_ex)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   639
  hence F1: "\<And>n. ?F n \<noteq> a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   640
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   641
    and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   642
    by fast+
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   643
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   644
  have "?F ----> a"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   645
  proof (rule LIMSEQ_I, unfold real_norm_def)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   646
      fix e::real
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   647
      assume "0 < e"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   648
        (* choose no such that inverse (real (Suc n)) < e *)
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23127
diff changeset
   649
      then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   650
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   651
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   652
      proof (intro exI allI impI)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   653
        fix n
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   654
        assume mlen: "m \<le> n"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   655
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   656
          by (rule F2)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   657
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23127
diff changeset
   658
          using mlen by auto
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   659
        also from nodef have
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   660
          "inverse (real (Suc m)) < e" .
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   661
        finally show "\<bar>?F n - a\<bar> < e" .
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   662
      qed
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   663
  qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   664
  
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   665
  moreover have "\<forall>n. ?F n \<noteq> a"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   666
    by (rule allI) (rule F1)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   667
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   668
  moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   669
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   670
  
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   671
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   672
  proof -
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   673
    {
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   674
      fix no::nat
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   675
      obtain n where "n = no + 1" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   676
      then have nolen: "no \<le> n" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   677
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   678
      have "dist (X (?F n)) L \<ge> r"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   679
        by (rule F3)
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   680
      with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   681
    }
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   682
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   683
    with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   684
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   685
  qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   686
  ultimately show False by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   687
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   688
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   689
lemma LIMSEQ_SEQ_conv:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   690
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   691
   (X -- a --> L)"
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   692
proof
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   693
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23127
diff changeset
   694
  thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   695
next
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   696
  assume "(X -- a --> L)"
23441
ee218296d635 avoid using implicit prems in assumption
huffman
parents: 23127
diff changeset
   697
  thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   698
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   699
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
   700
end