src/HOL/Lim.thy
author huffman
Sun Aug 14 07:54:24 2011 -0700 (2011-08-14)
changeset 44194 0639898074ae
parent 41550 efa734d9b221
child 44205 18da2a87421c
permissions -rw-r--r--
generalize lemmas about LIM and LIMSEQ to tendsto
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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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abbreviation
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  LIM :: "['a::topological_space \<Rightarrow> 'b::topological_space, 'a, 'b] \<Rightarrow> bool"
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        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
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  "f -- a --> L \<equiv> (f ---> L) (at a)"
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definition
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  isCont :: "['a::topological_space \<Rightarrow> 'b::topological_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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  "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_def: "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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unfolding 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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  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
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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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  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
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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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  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
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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"
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  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
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apply (rule topological_tendstoI)
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apply (drule (2) topological_tendstoD)
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apply (simp only: eventually_at dist_norm)
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apply (clarify, rule_tac x=d 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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  fixes a :: "'a::real_normed_vector"
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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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  fixes a :: "'a::real_normed_vector"
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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 (rule tendsto_const)
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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::topological_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 by (rule tendsto_add)
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lemma LIM_add_zero:
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  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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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 (rule tendsto_add_zero)
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lemma LIM_minus:
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  fixes f :: "'a::topological_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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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::topological_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::topological_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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by (rule tendsto_diff)
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lemma LIM_zero:
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  fixes f :: "'a::topological_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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unfolding tendsto_iff dist_norm by simp
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lemma LIM_zero_cancel:
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  fixes f :: "'a::topological_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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unfolding tendsto_iff dist_norm by simp
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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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unfolding tendsto_iff dist_norm by simp
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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 tendstoI, drule tendstoD [OF f])
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apply (simp add: eventually_at_topological, safe)
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apply (rule_tac x="S" in exI, safe)
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apply (drule_tac x="x" in bspec, safe)
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apply (erule (1) order_le_less_trans [OF le])
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done
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lemma LIM_imp_LIM:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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  fixes g :: "'a::topological_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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  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::topological_space \<Rightarrow> 'b::real_normed_vector"
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  shows "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
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by (rule tendsto_norm)
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lemma LIM_norm_zero:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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  shows "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
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by (rule tendsto_norm_zero)
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lemma LIM_norm_zero_cancel:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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  shows "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
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by (rule tendsto_norm_zero_cancel)
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lemma LIM_norm_zero_iff:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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  shows "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
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by (rule tendsto_norm_zero_iff)
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lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
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  by (rule tendsto_rabs)
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lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
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  by (rule tendsto_rabs_zero)
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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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  by (rule tendsto_rabs_zero_cancel)
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lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
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  by (rule tendsto_rabs_zero_iff)
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lemma at_neq_bot:
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  fixes a :: "'a::real_normed_algebra_1"
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  shows "at a \<noteq> bot"  -- {* TODO: find a more appropriate class *}
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unfolding eventually_False [symmetric]
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unfolding eventually_at dist_norm
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by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp)
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lemma LIM_const_not_eq:
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  fixes a :: "'a::real_normed_algebra_1"
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  fixes k L :: "'b::metric_space"
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  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
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by (simp add: tendsto_const_iff at_neq_bot)
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lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
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lemma LIM_const_eq:
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  fixes a :: "'a::real_normed_algebra_1"
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  fixes k L :: "'b::metric_space"
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  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
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by (simp add: tendsto_const_iff at_neq_bot)
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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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  fixes L M :: "'b::metric_space"
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  shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
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by (drule (1) tendsto_dist, simp add: tendsto_const_iff at_neq_bot)
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lemma LIM_ident [simp]: "(\<lambda>x. x) -- a --> a"
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by (rule tendsto_ident_at)
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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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unfolding tendsto_def eventually_at_topological by simp
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lemma LIM_cong:
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  "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
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   \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
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by (simp add: LIM_equal)
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lemma metric_LIM_equal2:
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  assumes 1: "0 < R"
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  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x"
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  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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apply (rule topological_tendstoI)
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apply (drule (2) topological_tendstoD)
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apply (simp add: eventually_at, safe)
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apply (rule_tac x="min d R" in exI, safe)
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apply (simp add: 1)
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apply (simp add: 2)
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done
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lemma LIM_equal2:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
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  assumes 1: "0 < R"
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  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
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  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
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text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *)
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lemma LIM_trans:
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  fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
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  shows "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
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apply (drule LIM_add, assumption)
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apply (auto simp add: add_assoc)
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done
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lemma LIM_compose:
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  assumes g: "g -- l --> g l"
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  assumes f: "f -- a --> l"
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  shows "(\<lambda>x. g (f x)) -- a --> g l"
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proof (rule topological_tendstoI)
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  fix C assume C: "open C" "g l \<in> C"
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  obtain B where B: "open B" "l \<in> B"
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    and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> C"
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    using topological_tendstoD [OF g C]
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    unfolding eventually_at_topological by fast
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  obtain A where A: "open A" "a \<in> A"
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    and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B"
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    using topological_tendstoD [OF f B]
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    unfolding eventually_at_topological by fast
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  show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
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  unfolding eventually_at_topological
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  proof (intro exI conjI ballI impI)
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    show "open A" and "a \<in> A" using A .
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  next
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    fix x assume "x \<in> A" and "x \<noteq> a"
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    then show "g (f x) \<in> C"
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      by (cases "f x = l", simp add: C, simp add: gC fB)
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  qed
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qed
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lemma LIM_compose_eventually:
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  assumes f: "f -- a --> b"
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  assumes g: "g -- b --> c"
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  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
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  shows "(\<lambda>x. g (f x)) -- a --> c"
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proof (rule topological_tendstoI)
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  fix C assume C: "open C" "c \<in> C"
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  obtain B where B: "open B" "b \<in> B"
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    and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> b \<Longrightarrow> g y \<in> C"
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    using topological_tendstoD [OF g C]
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    unfolding eventually_at_topological by fast
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  obtain A where A: "open A" "a \<in> A"
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   288
    and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B"
huffman@36662
   289
    using topological_tendstoD [OF f B]
huffman@36662
   290
    unfolding eventually_at_topological by fast
huffman@36662
   291
  have "eventually (\<lambda>x. f x \<noteq> b \<longrightarrow> g (f x) \<in> C) (at a)"
huffman@36662
   292
  unfolding eventually_at_topological
huffman@36662
   293
  proof (intro exI conjI ballI impI)
huffman@36662
   294
    show "open A" and "a \<in> A" using A .
huffman@21239
   295
  next
huffman@36662
   296
    fix x assume "x \<in> A" and "x \<noteq> a" and "f x \<noteq> b"
huffman@36662
   297
    then show "g (f x) \<in> C" by (simp add: gC fB)
huffman@21239
   298
  qed
huffman@36662
   299
  with inj show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
huffman@36662
   300
    by (rule eventually_rev_mp)
huffman@21239
   301
qed
huffman@21239
   302
huffman@31338
   303
lemma metric_LIM_compose2:
huffman@31338
   304
  assumes f: "f -- a --> b"
huffman@31338
   305
  assumes g: "g -- b --> c"
huffman@31338
   306
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
huffman@31338
   307
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@36662
   308
using f g inj [folded eventually_at]
huffman@36662
   309
by (rule LIM_compose_eventually)
huffman@31338
   310
huffman@23040
   311
lemma LIM_compose2:
huffman@31338
   312
  fixes a :: "'a::real_normed_vector"
huffman@23040
   313
  assumes f: "f -- a --> b"
huffman@23040
   314
  assumes g: "g -- b --> c"
huffman@23040
   315
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
huffman@23040
   316
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@31338
   317
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
huffman@23040
   318
huffman@21239
   319
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
huffman@21239
   320
unfolding o_def by (rule LIM_compose)
huffman@21239
   321
huffman@21282
   322
lemma real_LIM_sandwich_zero:
huffman@36662
   323
  fixes f g :: "'a::topological_space \<Rightarrow> real"
huffman@21282
   324
  assumes f: "f -- a --> 0"
huffman@21282
   325
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
huffman@21282
   326
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
huffman@21282
   327
  shows "g -- a --> 0"
huffman@21282
   328
proof (rule LIM_imp_LIM [OF f])
huffman@21282
   329
  fix x assume x: "x \<noteq> a"
huffman@21282
   330
  have "norm (g x - 0) = g x" by (simp add: 1 x)
huffman@21282
   331
  also have "g x \<le> f x" by (rule 2 [OF x])
huffman@21282
   332
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
huffman@21282
   333
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
huffman@21282
   334
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
huffman@21282
   335
qed
huffman@21282
   336
huffman@22442
   337
text {* Bounded Linear Operators *}
huffman@21282
   338
huffman@21282
   339
lemma (in bounded_linear) LIM:
huffman@21282
   340
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
huffman@36661
   341
by (rule tendsto)
huffman@31349
   342
huffman@31349
   343
lemma (in bounded_linear) cont: "f -- a --> f a"
huffman@31349
   344
by (rule LIM [OF LIM_ident])
huffman@21282
   345
huffman@21282
   346
lemma (in bounded_linear) LIM_zero:
huffman@21282
   347
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
huffman@44194
   348
  by (rule tendsto_zero)
huffman@21282
   349
huffman@22442
   350
text {* Bounded Bilinear Operators *}
huffman@21282
   351
huffman@31349
   352
lemma (in bounded_bilinear) LIM:
huffman@31349
   353
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
huffman@36661
   354
by (rule tendsto)
huffman@31349
   355
huffman@21282
   356
lemma (in bounded_bilinear) LIM_prod_zero:
huffman@31338
   357
  fixes a :: "'d::metric_space"
huffman@21282
   358
  assumes f: "f -- a --> 0"
huffman@21282
   359
  assumes g: "g -- a --> 0"
huffman@21282
   360
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
huffman@44194
   361
  using f g by (rule tendsto_zero)
huffman@21282
   362
huffman@21282
   363
lemma (in bounded_bilinear) LIM_left_zero:
huffman@21282
   364
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
huffman@44194
   365
  by (rule tendsto_left_zero)
huffman@21282
   366
huffman@21282
   367
lemma (in bounded_bilinear) LIM_right_zero:
huffman@21282
   368
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
huffman@44194
   369
  by (rule tendsto_right_zero)
huffman@21282
   370
huffman@23127
   371
lemmas LIM_mult = mult.LIM
huffman@21282
   372
huffman@23127
   373
lemmas LIM_mult_zero = mult.LIM_prod_zero
huffman@21282
   374
huffman@23127
   375
lemmas LIM_mult_left_zero = mult.LIM_left_zero
huffman@21282
   376
huffman@23127
   377
lemmas LIM_mult_right_zero = mult.LIM_right_zero
huffman@21282
   378
huffman@23127
   379
lemmas LIM_scaleR = scaleR.LIM
huffman@21282
   380
huffman@23127
   381
lemmas LIM_of_real = of_real.LIM
huffman@22627
   382
huffman@22627
   383
lemma LIM_power:
huffman@36665
   384
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   385
  assumes f: "f -- a --> l"
huffman@22627
   386
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
huffman@44194
   387
  using assms by (rule tendsto_power)
huffman@22627
   388
huffman@22641
   389
subsubsection {* Derived theorems about @{term LIM} *}
huffman@22641
   390
huffman@31355
   391
lemma LIM_inverse:
huffman@31355
   392
  fixes L :: "'a::real_normed_div_algebra"
huffman@31355
   393
  shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
huffman@31355
   394
by (rule tendsto_inverse)
huffman@22637
   395
huffman@22637
   396
lemma LIM_inverse_fun:
huffman@22637
   397
  assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
huffman@22637
   398
  shows "inverse -- a --> inverse a"
huffman@31355
   399
by (rule LIM_inverse [OF LIM_ident a])
huffman@22637
   400
huffman@29885
   401
lemma LIM_sgn:
huffman@36665
   402
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   403
  shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
huffman@44194
   404
  by (rule tendsto_sgn)
huffman@29885
   405
paulson@14477
   406
huffman@20755
   407
subsection {* Continuity *}
paulson@14477
   408
huffman@31338
   409
lemma LIM_isCont_iff:
huffman@36665
   410
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   411
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
huffman@21239
   412
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
huffman@21239
   413
huffman@31338
   414
lemma isCont_iff:
huffman@36665
   415
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   416
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
huffman@21239
   417
by (simp add: isCont_def LIM_isCont_iff)
huffman@21239
   418
huffman@23069
   419
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
huffman@23069
   420
  unfolding isCont_def by (rule LIM_ident)
huffman@21239
   421
huffman@21786
   422
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
huffman@21282
   423
  unfolding isCont_def by (rule LIM_const)
huffman@21239
   424
huffman@31338
   425
lemma isCont_norm:
huffman@36665
   426
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   427
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
huffman@21786
   428
  unfolding isCont_def by (rule LIM_norm)
huffman@21786
   429
huffman@22627
   430
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
huffman@22627
   431
  unfolding isCont_def by (rule LIM_rabs)
huffman@22627
   432
huffman@31338
   433
lemma isCont_add:
huffman@36665
   434
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   435
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
huffman@21282
   436
  unfolding isCont_def by (rule LIM_add)
huffman@21239
   437
huffman@31338
   438
lemma isCont_minus:
huffman@36665
   439
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   440
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
huffman@21282
   441
  unfolding isCont_def by (rule LIM_minus)
huffman@21239
   442
huffman@31338
   443
lemma isCont_diff:
huffman@36665
   444
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   445
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
huffman@21282
   446
  unfolding isCont_def by (rule LIM_diff)
huffman@21239
   447
huffman@21239
   448
lemma isCont_mult:
huffman@36665
   449
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
huffman@21786
   450
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
huffman@21282
   451
  unfolding isCont_def by (rule LIM_mult)
huffman@21239
   452
huffman@21239
   453
lemma isCont_inverse:
huffman@36665
   454
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
huffman@21786
   455
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
huffman@21282
   456
  unfolding isCont_def by (rule LIM_inverse)
huffman@21239
   457
huffman@21239
   458
lemma isCont_LIM_compose:
huffman@21239
   459
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
huffman@21282
   460
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   461
huffman@31338
   462
lemma metric_isCont_LIM_compose2:
huffman@31338
   463
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@31338
   464
  assumes g: "g -- f a --> l"
huffman@31338
   465
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
huffman@31338
   466
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@31338
   467
by (rule metric_LIM_compose2 [OF f g inj])
huffman@31338
   468
huffman@23040
   469
lemma isCont_LIM_compose2:
huffman@31338
   470
  fixes a :: "'a::real_normed_vector"
huffman@23040
   471
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@23040
   472
  assumes g: "g -- f a --> l"
huffman@23040
   473
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
huffman@23040
   474
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@23040
   475
by (rule LIM_compose2 [OF f g inj])
huffman@23040
   476
huffman@21239
   477
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
huffman@21282
   478
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   479
huffman@21239
   480
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
huffman@21282
   481
  unfolding o_def by (rule isCont_o2)
huffman@21282
   482
huffman@21282
   483
lemma (in bounded_linear) isCont: "isCont f a"
huffman@21282
   484
  unfolding isCont_def by (rule cont)
huffman@21282
   485
huffman@21282
   486
lemma (in bounded_bilinear) isCont:
huffman@21282
   487
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
huffman@21282
   488
  unfolding isCont_def by (rule LIM)
huffman@21282
   489
huffman@23127
   490
lemmas isCont_scaleR = scaleR.isCont
huffman@21239
   491
huffman@22627
   492
lemma isCont_of_real:
huffman@31338
   493
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
huffman@22627
   494
  unfolding isCont_def by (rule LIM_of_real)
huffman@22627
   495
huffman@22627
   496
lemma isCont_power:
huffman@36665
   497
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   498
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
huffman@22627
   499
  unfolding isCont_def by (rule LIM_power)
huffman@22627
   500
huffman@29885
   501
lemma isCont_sgn:
huffman@36665
   502
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   503
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
huffman@29885
   504
  unfolding isCont_def by (rule LIM_sgn)
huffman@29885
   505
huffman@20561
   506
lemma isCont_abs [simp]: "isCont abs (a::real)"
huffman@23069
   507
by (rule isCont_rabs [OF isCont_ident])
paulson@15228
   508
huffman@31338
   509
lemma isCont_setsum:
huffman@36665
   510
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
huffman@31338
   511
  fixes A :: "'a set" assumes "finite A"
hoelzl@29803
   512
  shows "\<forall> i \<in> A. isCont (f i) x \<Longrightarrow> isCont (\<lambda> x. \<Sum> i \<in> A. f i x) x"
huffman@44194
   513
  unfolding isCont_def by (simp add: tendsto_setsum)
hoelzl@29803
   514
hoelzl@29803
   515
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
hoelzl@29803
   516
  and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
hoelzl@29803
   517
  shows "0 \<le> f x"
hoelzl@29803
   518
proof (rule ccontr)
hoelzl@29803
   519
  assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
hoelzl@29803
   520
  hence "0 < - f x / 2" by auto
hoelzl@29803
   521
  from isCont[unfolded isCont_def, THEN LIM_D, OF this]
hoelzl@29803
   522
  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
hoelzl@29803
   523
hoelzl@29803
   524
  let ?x = "x - min (s / 2) ((x - b) / 2)"
hoelzl@29803
   525
  have "?x < x" and "\<bar> ?x - x \<bar> < s"
hoelzl@29803
   526
    using `b < x` and `0 < s` by auto
hoelzl@29803
   527
  have "b < ?x"
hoelzl@29803
   528
  proof (cases "s < x - b")
hoelzl@29803
   529
    case True thus ?thesis using `0 < s` by auto
hoelzl@29803
   530
  next
hoelzl@29803
   531
    case False hence "s / 2 \<ge> (x - b) / 2" by auto
haftmann@32642
   532
    hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
hoelzl@29803
   533
    thus ?thesis using `b < x` by auto
hoelzl@29803
   534
  qed
hoelzl@29803
   535
  hence "0 \<le> f ?x" using all_le `?x < x` by auto
hoelzl@29803
   536
  moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
hoelzl@29803
   537
    using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
hoelzl@29803
   538
  hence "f ?x - f x < - f x / 2" by auto
hoelzl@29803
   539
  hence "f ?x < f x / 2" by auto
hoelzl@29803
   540
  hence "f ?x < 0" using `f x < 0` by auto
hoelzl@29803
   541
  thus False using `0 \<le> f ?x` by auto
hoelzl@29803
   542
qed
huffman@31338
   543
paulson@14477
   544
huffman@20755
   545
subsection {* Uniform Continuity *}
huffman@20755
   546
paulson@14477
   547
lemma isUCont_isCont: "isUCont f ==> isCont f x"
huffman@23012
   548
by (simp add: isUCont_def isCont_def LIM_def, force)
paulson@14477
   549
huffman@23118
   550
lemma isUCont_Cauchy:
huffman@23118
   551
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   552
unfolding isUCont_def
huffman@31338
   553
apply (rule metric_CauchyI)
huffman@23118
   554
apply (drule_tac x=e in spec, safe)
huffman@31338
   555
apply (drule_tac e=s in metric_CauchyD, safe)
huffman@23118
   556
apply (rule_tac x=M in exI, simp)
huffman@23118
   557
done
huffman@23118
   558
huffman@23118
   559
lemma (in bounded_linear) isUCont: "isUCont f"
huffman@31338
   560
unfolding isUCont_def dist_norm
huffman@23118
   561
proof (intro allI impI)
huffman@23118
   562
  fix r::real assume r: "0 < r"
huffman@23118
   563
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
huffman@23118
   564
    using pos_bounded by fast
huffman@23118
   565
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@23118
   566
  proof (rule exI, safe)
huffman@23118
   567
    from r K show "0 < r / K" by (rule divide_pos_pos)
huffman@23118
   568
  next
huffman@23118
   569
    fix x y :: 'a
huffman@23118
   570
    assume xy: "norm (x - y) < r / K"
huffman@23118
   571
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
huffman@23118
   572
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
huffman@23118
   573
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
huffman@23118
   574
    finally show "norm (f x - f y) < r" .
huffman@23118
   575
  qed
huffman@23118
   576
qed
huffman@23118
   577
huffman@23118
   578
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   579
by (rule isUCont [THEN isUCont_Cauchy])
huffman@23118
   580
paulson@14477
   581
huffman@21165
   582
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   583
kleing@19023
   584
lemma LIMSEQ_SEQ_conv1:
huffman@36662
   585
  fixes a :: "'a::metric_space" and L :: "'b::metric_space"
huffman@21165
   586
  assumes X: "X -- a --> L"
kleing@19023
   587
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@31338
   588
proof (safe intro!: metric_LIMSEQ_I)
huffman@21165
   589
  fix S :: "nat \<Rightarrow> 'a"
huffman@21165
   590
  fix r :: real
huffman@21165
   591
  assume rgz: "0 < r"
huffman@21165
   592
  assume as: "\<forall>n. S n \<noteq> a"
huffman@21165
   593
  assume S: "S ----> a"
huffman@31338
   594
  from metric_LIM_D [OF X rgz] obtain s
huffman@21165
   595
    where sgz: "0 < s"
huffman@31338
   596
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"
huffman@21165
   597
    by fast
huffman@31338
   598
  from metric_LIMSEQ_D [OF S sgz]
huffman@31338
   599
  obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast
huffman@31338
   600
  hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)
huffman@31338
   601
  thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..
kleing@19023
   602
qed
kleing@19023
   603
huffman@31338
   604
kleing@19023
   605
lemma LIMSEQ_SEQ_conv2:
huffman@36662
   606
  fixes a :: real and L :: "'a::metric_space"
kleing@19023
   607
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   608
  shows "X -- a --> L"
kleing@19023
   609
proof (rule ccontr)
kleing@19023
   610
  assume "\<not> (X -- a --> L)"
huffman@31338
   611
  hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
huffman@31338
   612
    unfolding LIM_def dist_norm .
huffman@31338
   613
  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
huffman@31338
   614
  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)
huffman@31338
   615
  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
kleing@19023
   616
huffman@31338
   617
  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"
huffman@31338
   618
  have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r"
huffman@21165
   619
    using rdef by simp
huffman@31338
   620
  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"
huffman@21165
   621
    by (rule someI_ex)
huffman@21165
   622
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@21165
   623
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@31338
   624
    and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
huffman@21165
   625
    by fast+
huffman@21165
   626
kleing@19023
   627
  have "?F ----> a"
huffman@21165
   628
  proof (rule LIMSEQ_I, unfold real_norm_def)
kleing@19023
   629
      fix e::real
kleing@19023
   630
      assume "0 < e"
kleing@19023
   631
        (* choose no such that inverse (real (Suc n)) < e *)
huffman@23441
   632
      then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
   633
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
huffman@21165
   634
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
huffman@21165
   635
      proof (intro exI allI impI)
kleing@19023
   636
        fix n
kleing@19023
   637
        assume mlen: "m \<le> n"
huffman@21165
   638
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   639
          by (rule F2)
huffman@21165
   640
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
huffman@23441
   641
          using mlen by auto
huffman@21165
   642
        also from nodef have
kleing@19023
   643
          "inverse (real (Suc m)) < e" .
huffman@21165
   644
        finally show "\<bar>?F n - a\<bar> < e" .
huffman@21165
   645
      qed
kleing@19023
   646
  qed
kleing@19023
   647
  
kleing@19023
   648
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   649
    by (rule allI) (rule F1)
huffman@21165
   650
wenzelm@41550
   651
  moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
kleing@19023
   652
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
   653
  
kleing@19023
   654
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
   655
  proof -
kleing@19023
   656
    {
kleing@19023
   657
      fix no::nat
kleing@19023
   658
      obtain n where "n = no + 1" by simp
kleing@19023
   659
      then have nolen: "no \<le> n" by simp
kleing@19023
   660
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
huffman@31338
   661
      have "dist (X (?F n)) L \<ge> r"
huffman@21165
   662
        by (rule F3)
huffman@31338
   663
      with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
kleing@19023
   664
    }
huffman@31338
   665
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
huffman@31338
   666
    with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto
huffman@31338
   667
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
kleing@19023
   668
  qed
kleing@19023
   669
  ultimately show False by simp
kleing@19023
   670
qed
kleing@19023
   671
kleing@19023
   672
lemma LIMSEQ_SEQ_conv:
huffman@20561
   673
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@36662
   674
   (X -- a --> (L::'a::metric_space))"
kleing@19023
   675
proof
kleing@19023
   676
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@23441
   677
  thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
   678
next
kleing@19023
   679
  assume "(X -- a --> L)"
huffman@23441
   680
  thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
kleing@19023
   681
qed
kleing@19023
   682
paulson@10751
   683
end