src/HOL/Lim.thy
author huffman
Mon Aug 15 16:48:05 2011 -0700 (2011-08-15)
changeset 44218 f0e442e24816
parent 44217 5cdad94bdc29
child 44233 aa74ce315bae
permissions -rw-r--r--
add lemma tendsto_compose
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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 trivial_limit_at:
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  fixes a :: "'a::real_normed_algebra_1"
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  shows "\<not> trivial_limit (at a)"  -- {* TODO: find a more appropriate class *}
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unfolding trivial_limit_def
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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::t2_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 trivial_limit_at)
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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::t2_space"
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  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
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  by (simp add: tendsto_const_iff trivial_limit_at)
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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::t2_space"
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  shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
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  using trivial_limit_at by (rule tendsto_unique)
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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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  using assms by (rule tendsto_compose)
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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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    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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  have "eventually (\<lambda>x. f x \<noteq> b \<longrightarrow> 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" and "f x \<noteq> b"
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    then show "g (f x) \<in> C" by (simp add: gC fB)
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  qed
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  with inj show "eventually (\<lambda>x. g (f x) \<in> C) (at a)"
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    by (rule eventually_rev_mp)
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qed
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lemma metric_LIM_compose2:
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  assumes f: "f -- a --> b"
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  assumes g: "g -- b --> c"
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  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b"
huffman@31338
   288
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@36662
   289
using f g inj [folded eventually_at]
huffman@36662
   290
by (rule LIM_compose_eventually)
huffman@31338
   291
huffman@23040
   292
lemma LIM_compose2:
huffman@31338
   293
  fixes a :: "'a::real_normed_vector"
huffman@23040
   294
  assumes f: "f -- a --> b"
huffman@23040
   295
  assumes g: "g -- b --> c"
huffman@23040
   296
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
huffman@23040
   297
  shows "(\<lambda>x. g (f x)) -- a --> c"
huffman@31338
   298
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
huffman@23040
   299
huffman@21239
   300
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
huffman@21239
   301
unfolding o_def by (rule LIM_compose)
huffman@21239
   302
huffman@21282
   303
lemma real_LIM_sandwich_zero:
huffman@36662
   304
  fixes f g :: "'a::topological_space \<Rightarrow> real"
huffman@21282
   305
  assumes f: "f -- a --> 0"
huffman@21282
   306
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
huffman@21282
   307
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
huffman@21282
   308
  shows "g -- a --> 0"
huffman@21282
   309
proof (rule LIM_imp_LIM [OF f])
huffman@21282
   310
  fix x assume x: "x \<noteq> a"
huffman@21282
   311
  have "norm (g x - 0) = g x" by (simp add: 1 x)
huffman@21282
   312
  also have "g x \<le> f x" by (rule 2 [OF x])
huffman@21282
   313
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
huffman@21282
   314
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
huffman@21282
   315
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
huffman@21282
   316
qed
huffman@21282
   317
huffman@22442
   318
text {* Bounded Linear Operators *}
huffman@21282
   319
huffman@21282
   320
lemma (in bounded_linear) LIM:
huffman@21282
   321
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
huffman@36661
   322
by (rule tendsto)
huffman@31349
   323
huffman@31349
   324
lemma (in bounded_linear) cont: "f -- a --> f a"
huffman@31349
   325
by (rule LIM [OF LIM_ident])
huffman@21282
   326
huffman@21282
   327
lemma (in bounded_linear) LIM_zero:
huffman@21282
   328
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
huffman@44194
   329
  by (rule tendsto_zero)
huffman@21282
   330
huffman@22442
   331
text {* Bounded Bilinear Operators *}
huffman@21282
   332
huffman@31349
   333
lemma (in bounded_bilinear) LIM:
huffman@31349
   334
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
huffman@36661
   335
by (rule tendsto)
huffman@31349
   336
huffman@21282
   337
lemma (in bounded_bilinear) LIM_prod_zero:
huffman@31338
   338
  fixes a :: "'d::metric_space"
huffman@21282
   339
  assumes f: "f -- a --> 0"
huffman@21282
   340
  assumes g: "g -- a --> 0"
huffman@21282
   341
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
huffman@44194
   342
  using f g by (rule tendsto_zero)
huffman@21282
   343
huffman@21282
   344
lemma (in bounded_bilinear) LIM_left_zero:
huffman@21282
   345
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
huffman@44194
   346
  by (rule tendsto_left_zero)
huffman@21282
   347
huffman@21282
   348
lemma (in bounded_bilinear) LIM_right_zero:
huffman@21282
   349
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
huffman@44194
   350
  by (rule tendsto_right_zero)
huffman@21282
   351
huffman@23127
   352
lemmas LIM_mult = mult.LIM
huffman@21282
   353
huffman@23127
   354
lemmas LIM_mult_zero = mult.LIM_prod_zero
huffman@21282
   355
huffman@23127
   356
lemmas LIM_mult_left_zero = mult.LIM_left_zero
huffman@21282
   357
huffman@23127
   358
lemmas LIM_mult_right_zero = mult.LIM_right_zero
huffman@21282
   359
huffman@23127
   360
lemmas LIM_scaleR = scaleR.LIM
huffman@21282
   361
huffman@23127
   362
lemmas LIM_of_real = of_real.LIM
huffman@22627
   363
huffman@22627
   364
lemma LIM_power:
huffman@36665
   365
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   366
  assumes f: "f -- a --> l"
huffman@22627
   367
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
huffman@44194
   368
  using assms by (rule tendsto_power)
huffman@22627
   369
huffman@31355
   370
lemma LIM_inverse:
huffman@31355
   371
  fixes L :: "'a::real_normed_div_algebra"
huffman@31355
   372
  shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
huffman@31355
   373
by (rule tendsto_inverse)
huffman@22637
   374
huffman@22637
   375
lemma LIM_inverse_fun:
huffman@22637
   376
  assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
huffman@22637
   377
  shows "inverse -- a --> inverse a"
huffman@31355
   378
by (rule LIM_inverse [OF LIM_ident a])
huffman@22637
   379
huffman@29885
   380
lemma LIM_sgn:
huffman@36665
   381
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   382
  shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
huffman@44194
   383
  by (rule tendsto_sgn)
huffman@29885
   384
paulson@14477
   385
huffman@20755
   386
subsection {* Continuity *}
paulson@14477
   387
huffman@31338
   388
lemma LIM_isCont_iff:
huffman@36665
   389
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   390
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
huffman@21239
   391
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
huffman@21239
   392
huffman@31338
   393
lemma isCont_iff:
huffman@36665
   394
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   395
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
huffman@21239
   396
by (simp add: isCont_def LIM_isCont_iff)
huffman@21239
   397
huffman@23069
   398
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
huffman@23069
   399
  unfolding isCont_def by (rule LIM_ident)
huffman@21239
   400
huffman@21786
   401
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
huffman@21282
   402
  unfolding isCont_def by (rule LIM_const)
huffman@21239
   403
huffman@31338
   404
lemma isCont_norm:
huffman@36665
   405
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   406
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
huffman@21786
   407
  unfolding isCont_def by (rule LIM_norm)
huffman@21786
   408
huffman@22627
   409
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
huffman@22627
   410
  unfolding isCont_def by (rule LIM_rabs)
huffman@22627
   411
huffman@31338
   412
lemma isCont_add:
huffman@36665
   413
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   414
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
huffman@21282
   415
  unfolding isCont_def by (rule LIM_add)
huffman@21239
   416
huffman@31338
   417
lemma isCont_minus:
huffman@36665
   418
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   419
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
huffman@21282
   420
  unfolding isCont_def by (rule LIM_minus)
huffman@21239
   421
huffman@31338
   422
lemma isCont_diff:
huffman@36665
   423
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   424
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
huffman@21282
   425
  unfolding isCont_def by (rule LIM_diff)
huffman@21239
   426
huffman@21239
   427
lemma isCont_mult:
huffman@36665
   428
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
huffman@21786
   429
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
huffman@21282
   430
  unfolding isCont_def by (rule LIM_mult)
huffman@21239
   431
huffman@21239
   432
lemma isCont_inverse:
huffman@36665
   433
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
huffman@21786
   434
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
huffman@21282
   435
  unfolding isCont_def by (rule LIM_inverse)
huffman@21239
   436
huffman@21239
   437
lemma isCont_LIM_compose:
huffman@21239
   438
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
huffman@21282
   439
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   440
huffman@31338
   441
lemma metric_isCont_LIM_compose2:
huffman@31338
   442
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@31338
   443
  assumes g: "g -- f a --> l"
huffman@31338
   444
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
huffman@31338
   445
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@31338
   446
by (rule metric_LIM_compose2 [OF f g inj])
huffman@31338
   447
huffman@23040
   448
lemma isCont_LIM_compose2:
huffman@31338
   449
  fixes a :: "'a::real_normed_vector"
huffman@23040
   450
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@23040
   451
  assumes g: "g -- f a --> l"
huffman@23040
   452
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
huffman@23040
   453
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@23040
   454
by (rule LIM_compose2 [OF f g inj])
huffman@23040
   455
huffman@21239
   456
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
huffman@21282
   457
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   458
huffman@21239
   459
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
huffman@21282
   460
  unfolding o_def by (rule isCont_o2)
huffman@21282
   461
huffman@21282
   462
lemma (in bounded_linear) isCont: "isCont f a"
huffman@21282
   463
  unfolding isCont_def by (rule cont)
huffman@21282
   464
huffman@21282
   465
lemma (in bounded_bilinear) isCont:
huffman@21282
   466
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
huffman@21282
   467
  unfolding isCont_def by (rule LIM)
huffman@21282
   468
huffman@23127
   469
lemmas isCont_scaleR = scaleR.isCont
huffman@21239
   470
huffman@22627
   471
lemma isCont_of_real:
huffman@31338
   472
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
huffman@22627
   473
  unfolding isCont_def by (rule LIM_of_real)
huffman@22627
   474
huffman@22627
   475
lemma isCont_power:
huffman@36665
   476
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   477
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
huffman@22627
   478
  unfolding isCont_def by (rule LIM_power)
huffman@22627
   479
huffman@29885
   480
lemma isCont_sgn:
huffman@36665
   481
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   482
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
huffman@29885
   483
  unfolding isCont_def by (rule LIM_sgn)
huffman@29885
   484
huffman@20561
   485
lemma isCont_abs [simp]: "isCont abs (a::real)"
huffman@23069
   486
by (rule isCont_rabs [OF isCont_ident])
paulson@15228
   487
huffman@31338
   488
lemma isCont_setsum:
huffman@36665
   489
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
huffman@31338
   490
  fixes A :: "'a set" assumes "finite A"
hoelzl@29803
   491
  shows "\<forall> i \<in> A. isCont (f i) x \<Longrightarrow> isCont (\<lambda> x. \<Sum> i \<in> A. f i x) x"
huffman@44194
   492
  unfolding isCont_def by (simp add: tendsto_setsum)
hoelzl@29803
   493
hoelzl@29803
   494
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
hoelzl@29803
   495
  and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
hoelzl@29803
   496
  shows "0 \<le> f x"
hoelzl@29803
   497
proof (rule ccontr)
hoelzl@29803
   498
  assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
hoelzl@29803
   499
  hence "0 < - f x / 2" by auto
hoelzl@29803
   500
  from isCont[unfolded isCont_def, THEN LIM_D, OF this]
hoelzl@29803
   501
  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
   502
hoelzl@29803
   503
  let ?x = "x - min (s / 2) ((x - b) / 2)"
hoelzl@29803
   504
  have "?x < x" and "\<bar> ?x - x \<bar> < s"
hoelzl@29803
   505
    using `b < x` and `0 < s` by auto
hoelzl@29803
   506
  have "b < ?x"
hoelzl@29803
   507
  proof (cases "s < x - b")
hoelzl@29803
   508
    case True thus ?thesis using `0 < s` by auto
hoelzl@29803
   509
  next
hoelzl@29803
   510
    case False hence "s / 2 \<ge> (x - b) / 2" by auto
haftmann@32642
   511
    hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
hoelzl@29803
   512
    thus ?thesis using `b < x` by auto
hoelzl@29803
   513
  qed
hoelzl@29803
   514
  hence "0 \<le> f ?x" using all_le `?x < x` by auto
hoelzl@29803
   515
  moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
hoelzl@29803
   516
    using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
hoelzl@29803
   517
  hence "f ?x - f x < - f x / 2" by auto
hoelzl@29803
   518
  hence "f ?x < f x / 2" by auto
hoelzl@29803
   519
  hence "f ?x < 0" using `f x < 0` by auto
hoelzl@29803
   520
  thus False using `0 \<le> f ?x` by auto
hoelzl@29803
   521
qed
huffman@31338
   522
paulson@14477
   523
huffman@20755
   524
subsection {* Uniform Continuity *}
huffman@20755
   525
paulson@14477
   526
lemma isUCont_isCont: "isUCont f ==> isCont f x"
huffman@23012
   527
by (simp add: isUCont_def isCont_def LIM_def, force)
paulson@14477
   528
huffman@23118
   529
lemma isUCont_Cauchy:
huffman@23118
   530
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   531
unfolding isUCont_def
huffman@31338
   532
apply (rule metric_CauchyI)
huffman@23118
   533
apply (drule_tac x=e in spec, safe)
huffman@31338
   534
apply (drule_tac e=s in metric_CauchyD, safe)
huffman@23118
   535
apply (rule_tac x=M in exI, simp)
huffman@23118
   536
done
huffman@23118
   537
huffman@23118
   538
lemma (in bounded_linear) isUCont: "isUCont f"
huffman@31338
   539
unfolding isUCont_def dist_norm
huffman@23118
   540
proof (intro allI impI)
huffman@23118
   541
  fix r::real assume r: "0 < r"
huffman@23118
   542
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
huffman@23118
   543
    using pos_bounded by fast
huffman@23118
   544
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@23118
   545
  proof (rule exI, safe)
huffman@23118
   546
    from r K show "0 < r / K" by (rule divide_pos_pos)
huffman@23118
   547
  next
huffman@23118
   548
    fix x y :: 'a
huffman@23118
   549
    assume xy: "norm (x - y) < r / K"
huffman@23118
   550
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
huffman@23118
   551
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
huffman@23118
   552
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
huffman@23118
   553
    finally show "norm (f x - f y) < r" .
huffman@23118
   554
  qed
huffman@23118
   555
qed
huffman@23118
   556
huffman@23118
   557
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   558
by (rule isUCont [THEN isUCont_Cauchy])
huffman@23118
   559
paulson@14477
   560
huffman@21165
   561
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   562
kleing@19023
   563
lemma LIMSEQ_SEQ_conv1:
huffman@36662
   564
  fixes a :: "'a::metric_space" and L :: "'b::metric_space"
huffman@21165
   565
  assumes X: "X -- a --> L"
kleing@19023
   566
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@31338
   567
proof (safe intro!: metric_LIMSEQ_I)
huffman@21165
   568
  fix S :: "nat \<Rightarrow> 'a"
huffman@21165
   569
  fix r :: real
huffman@21165
   570
  assume rgz: "0 < r"
huffman@21165
   571
  assume as: "\<forall>n. S n \<noteq> a"
huffman@21165
   572
  assume S: "S ----> a"
huffman@31338
   573
  from metric_LIM_D [OF X rgz] obtain s
huffman@21165
   574
    where sgz: "0 < s"
huffman@31338
   575
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r"
huffman@21165
   576
    by fast
huffman@31338
   577
  from metric_LIMSEQ_D [OF S sgz]
huffman@31338
   578
  obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast
huffman@31338
   579
  hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as)
huffman@31338
   580
  thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" ..
kleing@19023
   581
qed
kleing@19023
   582
huffman@31338
   583
kleing@19023
   584
lemma LIMSEQ_SEQ_conv2:
huffman@36662
   585
  fixes a :: real and L :: "'a::metric_space"
kleing@19023
   586
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   587
  shows "X -- a --> L"
kleing@19023
   588
proof (rule ccontr)
kleing@19023
   589
  assume "\<not> (X -- a --> L)"
huffman@31338
   590
  hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> dist (X x) L < r)"
huffman@31338
   591
    unfolding LIM_def dist_norm .
huffman@31338
   592
  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
   593
  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
   594
  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
   595
huffman@31338
   596
  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
   597
  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
   598
    using rdef by simp
huffman@31338
   599
  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
   600
    by (rule someI_ex)
huffman@21165
   601
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@21165
   602
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@31338
   603
    and F3: "\<And>n. dist (X (?F n)) L \<ge> r"
huffman@21165
   604
    by fast+
huffman@21165
   605
kleing@19023
   606
  have "?F ----> a"
huffman@21165
   607
  proof (rule LIMSEQ_I, unfold real_norm_def)
kleing@19023
   608
      fix e::real
kleing@19023
   609
      assume "0 < e"
kleing@19023
   610
        (* choose no such that inverse (real (Suc n)) < e *)
huffman@23441
   611
      then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
   612
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
huffman@21165
   613
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
huffman@21165
   614
      proof (intro exI allI impI)
kleing@19023
   615
        fix n
kleing@19023
   616
        assume mlen: "m \<le> n"
huffman@21165
   617
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   618
          by (rule F2)
huffman@21165
   619
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
huffman@23441
   620
          using mlen by auto
huffman@21165
   621
        also from nodef have
kleing@19023
   622
          "inverse (real (Suc m)) < e" .
huffman@21165
   623
        finally show "\<bar>?F n - a\<bar> < e" .
huffman@21165
   624
      qed
kleing@19023
   625
  qed
kleing@19023
   626
  
kleing@19023
   627
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   628
    by (rule allI) (rule F1)
huffman@21165
   629
wenzelm@41550
   630
  moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L`
kleing@19023
   631
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
   632
  
kleing@19023
   633
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
   634
  proof -
kleing@19023
   635
    {
kleing@19023
   636
      fix no::nat
kleing@19023
   637
      obtain n where "n = no + 1" by simp
kleing@19023
   638
      then have nolen: "no \<le> n" by simp
kleing@19023
   639
        (* 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
   640
      have "dist (X (?F n)) L \<ge> r"
huffman@21165
   641
        by (rule F3)
huffman@31338
   642
      with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast
kleing@19023
   643
    }
huffman@31338
   644
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp
huffman@31338
   645
    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
   646
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less)
kleing@19023
   647
  qed
kleing@19023
   648
  ultimately show False by simp
kleing@19023
   649
qed
kleing@19023
   650
kleing@19023
   651
lemma LIMSEQ_SEQ_conv:
huffman@20561
   652
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@36662
   653
   (X -- a --> (L::'a::metric_space))"
kleing@19023
   654
proof
kleing@19023
   655
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@23441
   656
  thus "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
   657
next
kleing@19023
   658
  assume "(X -- a --> L)"
huffman@23441
   659
  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
   660
qed
kleing@19023
   661
paulson@10751
   662
end