src/HOL/Lim.thy
author huffman
Wed Aug 17 13:10:11 2011 -0700 (2011-08-17)
changeset 44254 336dd390e4a4
parent 44253 c073a0bd8458
child 44282 f0de18b62d63
permissions -rw-r--r--
Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
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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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  by (rule metric_tendsto_imp_tendsto [OF f],
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    auto simp add: eventually_at_topological le)
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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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  by (rule metric_LIM_imp_LIM [OF f],
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    simp add: dist_norm le)
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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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  using g f inj by (rule tendsto_compose_eventually)
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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"
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  shows "(\<lambda>x. g (f x)) -- a --> c"
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using f g inj [folded eventually_at]
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by (rule LIM_compose_eventually)
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lemma LIM_compose2:
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  fixes a :: "'a::real_normed_vector"
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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> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
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  shows "(\<lambda>x. g (f x)) -- a --> c"
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by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
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lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
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unfolding o_def by (rule LIM_compose)
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lemma real_LIM_sandwich_zero:
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  fixes f g :: "'a::topological_space \<Rightarrow> real"
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  assumes f: "f -- a --> 0"
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  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
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  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
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  shows "g -- a --> 0"
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proof (rule LIM_imp_LIM [OF f])
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  fix x assume x: "x \<noteq> a"
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  have "norm (g x - 0) = g x" by (simp add: 1 x)
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   287
  also have "g x \<le> f x" by (rule 2 [OF x])
huffman@21282
   288
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
huffman@21282
   289
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
huffman@21282
   290
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
huffman@21282
   291
qed
huffman@21282
   292
huffman@22442
   293
text {* Bounded Linear Operators *}
huffman@21282
   294
huffman@21282
   295
lemma (in bounded_linear) LIM:
huffman@21282
   296
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
huffman@36661
   297
by (rule tendsto)
huffman@31349
   298
huffman@21282
   299
lemma (in bounded_linear) LIM_zero:
huffman@21282
   300
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
huffman@44194
   301
  by (rule tendsto_zero)
huffman@21282
   302
huffman@22442
   303
text {* Bounded Bilinear Operators *}
huffman@21282
   304
huffman@31349
   305
lemma (in bounded_bilinear) LIM:
huffman@31349
   306
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
huffman@36661
   307
by (rule tendsto)
huffman@31349
   308
huffman@21282
   309
lemma (in bounded_bilinear) LIM_prod_zero:
huffman@31338
   310
  fixes a :: "'d::metric_space"
huffman@21282
   311
  assumes f: "f -- a --> 0"
huffman@21282
   312
  assumes g: "g -- a --> 0"
huffman@21282
   313
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
huffman@44194
   314
  using f g by (rule tendsto_zero)
huffman@21282
   315
huffman@21282
   316
lemma (in bounded_bilinear) LIM_left_zero:
huffman@21282
   317
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
huffman@44194
   318
  by (rule tendsto_left_zero)
huffman@21282
   319
huffman@21282
   320
lemma (in bounded_bilinear) LIM_right_zero:
huffman@21282
   321
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
huffman@44194
   322
  by (rule tendsto_right_zero)
huffman@21282
   323
huffman@23127
   324
lemmas LIM_mult = mult.LIM
huffman@21282
   325
huffman@23127
   326
lemmas LIM_mult_zero = mult.LIM_prod_zero
huffman@21282
   327
huffman@23127
   328
lemmas LIM_mult_left_zero = mult.LIM_left_zero
huffman@21282
   329
huffman@23127
   330
lemmas LIM_mult_right_zero = mult.LIM_right_zero
huffman@21282
   331
huffman@23127
   332
lemmas LIM_scaleR = scaleR.LIM
huffman@21282
   333
huffman@23127
   334
lemmas LIM_of_real = of_real.LIM
huffman@22627
   335
huffman@22627
   336
lemma LIM_power:
huffman@36665
   337
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   338
  assumes f: "f -- a --> l"
huffman@22627
   339
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
huffman@44194
   340
  using assms by (rule tendsto_power)
huffman@22627
   341
huffman@31355
   342
lemma LIM_inverse:
huffman@31355
   343
  fixes L :: "'a::real_normed_div_algebra"
huffman@31355
   344
  shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
huffman@31355
   345
by (rule tendsto_inverse)
huffman@22637
   346
huffman@22637
   347
lemma LIM_inverse_fun:
huffman@22637
   348
  assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
huffman@22637
   349
  shows "inverse -- a --> inverse a"
huffman@31355
   350
by (rule LIM_inverse [OF LIM_ident a])
huffman@22637
   351
huffman@29885
   352
lemma LIM_sgn:
huffman@36665
   353
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   354
  shows "\<lbrakk>f -- a --> l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x)) -- a --> sgn l"
huffman@44194
   355
  by (rule tendsto_sgn)
huffman@29885
   356
paulson@14477
   357
huffman@20755
   358
subsection {* Continuity *}
paulson@14477
   359
huffman@31338
   360
lemma LIM_isCont_iff:
huffman@36665
   361
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   362
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
huffman@21239
   363
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
huffman@21239
   364
huffman@31338
   365
lemma isCont_iff:
huffman@36665
   366
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
huffman@31338
   367
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
huffman@21239
   368
by (simp add: isCont_def LIM_isCont_iff)
huffman@21239
   369
huffman@23069
   370
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
huffman@23069
   371
  unfolding isCont_def by (rule LIM_ident)
huffman@21239
   372
huffman@21786
   373
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
huffman@21282
   374
  unfolding isCont_def by (rule LIM_const)
huffman@21239
   375
huffman@44233
   376
lemma isCont_norm [simp]:
huffman@36665
   377
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   378
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
huffman@21786
   379
  unfolding isCont_def by (rule LIM_norm)
huffman@21786
   380
huffman@44233
   381
lemma isCont_rabs [simp]:
huffman@44233
   382
  fixes f :: "'a::topological_space \<Rightarrow> real"
huffman@44233
   383
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
huffman@22627
   384
  unfolding isCont_def by (rule LIM_rabs)
huffman@22627
   385
huffman@44233
   386
lemma isCont_add [simp]:
huffman@36665
   387
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   388
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
huffman@21282
   389
  unfolding isCont_def by (rule LIM_add)
huffman@21239
   390
huffman@44233
   391
lemma isCont_minus [simp]:
huffman@36665
   392
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   393
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
huffman@21282
   394
  unfolding isCont_def by (rule LIM_minus)
huffman@21239
   395
huffman@44233
   396
lemma isCont_diff [simp]:
huffman@36665
   397
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   398
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
huffman@21282
   399
  unfolding isCont_def by (rule LIM_diff)
huffman@21239
   400
huffman@44233
   401
lemma isCont_mult [simp]:
huffman@36665
   402
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
huffman@21786
   403
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
huffman@21282
   404
  unfolding isCont_def by (rule LIM_mult)
huffman@21239
   405
huffman@44233
   406
lemma isCont_inverse [simp]:
huffman@36665
   407
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
huffman@21786
   408
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
huffman@21282
   409
  unfolding isCont_def by (rule LIM_inverse)
huffman@21239
   410
huffman@44233
   411
lemma isCont_divide [simp]:
huffman@44233
   412
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
huffman@44233
   413
  shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
huffman@44233
   414
  unfolding isCont_def by (rule tendsto_divide)
huffman@44233
   415
huffman@21239
   416
lemma isCont_LIM_compose:
huffman@21239
   417
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
huffman@21282
   418
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   419
huffman@31338
   420
lemma metric_isCont_LIM_compose2:
huffman@31338
   421
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@31338
   422
  assumes g: "g -- f a --> l"
huffman@31338
   423
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
huffman@31338
   424
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@31338
   425
by (rule metric_LIM_compose2 [OF f g inj])
huffman@31338
   426
huffman@23040
   427
lemma isCont_LIM_compose2:
huffman@31338
   428
  fixes a :: "'a::real_normed_vector"
huffman@23040
   429
  assumes f [unfolded isCont_def]: "isCont f a"
huffman@23040
   430
  assumes g: "g -- f a --> l"
huffman@23040
   431
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
huffman@23040
   432
  shows "(\<lambda>x. g (f x)) -- a --> l"
huffman@23040
   433
by (rule LIM_compose2 [OF f g inj])
huffman@23040
   434
huffman@21239
   435
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
huffman@21282
   436
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   437
huffman@21239
   438
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
huffman@21282
   439
  unfolding o_def by (rule isCont_o2)
huffman@21282
   440
huffman@44233
   441
lemma (in bounded_linear) isCont:
huffman@44233
   442
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
huffman@44233
   443
  unfolding isCont_def by (rule LIM)
huffman@21282
   444
huffman@21282
   445
lemma (in bounded_bilinear) isCont:
huffman@21282
   446
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
huffman@21282
   447
  unfolding isCont_def by (rule LIM)
huffman@21282
   448
huffman@44233
   449
lemmas isCont_scaleR [simp] = scaleR.isCont
huffman@21239
   450
huffman@44233
   451
lemma isCont_of_real [simp]:
huffman@31338
   452
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a"
huffman@44233
   453
  by (rule of_real.isCont)
huffman@22627
   454
huffman@44233
   455
lemma isCont_power [simp]:
huffman@36665
   456
  fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@22627
   457
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
huffman@22627
   458
  unfolding isCont_def by (rule LIM_power)
huffman@22627
   459
huffman@44233
   460
lemma isCont_sgn [simp]:
huffman@36665
   461
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
huffman@31338
   462
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
huffman@29885
   463
  unfolding isCont_def by (rule LIM_sgn)
huffman@29885
   464
huffman@44233
   465
lemma isCont_setsum [simp]:
huffman@44233
   466
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
huffman@44233
   467
  fixes A :: "'a set"
huffman@44233
   468
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
huffman@44233
   469
  unfolding isCont_def by (simp add: tendsto_setsum)
paulson@15228
   470
huffman@44233
   471
lemmas isCont_intros =
huffman@44233
   472
  isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
huffman@44233
   473
  isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
huffman@44233
   474
  isCont_of_real isCont_power isCont_sgn isCont_setsum
hoelzl@29803
   475
hoelzl@29803
   476
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x"
hoelzl@29803
   477
  and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x"
hoelzl@29803
   478
  shows "0 \<le> f x"
hoelzl@29803
   479
proof (rule ccontr)
hoelzl@29803
   480
  assume "\<not> 0 \<le> f x" hence "f x < 0" by auto
hoelzl@29803
   481
  hence "0 < - f x / 2" by auto
hoelzl@29803
   482
  from isCont[unfolded isCont_def, THEN LIM_D, OF this]
hoelzl@29803
   483
  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
   484
hoelzl@29803
   485
  let ?x = "x - min (s / 2) ((x - b) / 2)"
hoelzl@29803
   486
  have "?x < x" and "\<bar> ?x - x \<bar> < s"
hoelzl@29803
   487
    using `b < x` and `0 < s` by auto
hoelzl@29803
   488
  have "b < ?x"
hoelzl@29803
   489
  proof (cases "s < x - b")
hoelzl@29803
   490
    case True thus ?thesis using `0 < s` by auto
hoelzl@29803
   491
  next
hoelzl@29803
   492
    case False hence "s / 2 \<ge> (x - b) / 2" by auto
haftmann@32642
   493
    hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2)
hoelzl@29803
   494
    thus ?thesis using `b < x` by auto
hoelzl@29803
   495
  qed
hoelzl@29803
   496
  hence "0 \<le> f ?x" using all_le `?x < x` by auto
hoelzl@29803
   497
  moreover have "\<bar>f ?x - f x\<bar> < - f x / 2"
hoelzl@29803
   498
    using s_D[OF _ `\<bar> ?x - x \<bar> < s`] `?x < x` by auto
hoelzl@29803
   499
  hence "f ?x - f x < - f x / 2" by auto
hoelzl@29803
   500
  hence "f ?x < f x / 2" by auto
hoelzl@29803
   501
  hence "f ?x < 0" using `f x < 0` by auto
hoelzl@29803
   502
  thus False using `0 \<le> f ?x` by auto
hoelzl@29803
   503
qed
huffman@31338
   504
paulson@14477
   505
huffman@20755
   506
subsection {* Uniform Continuity *}
huffman@20755
   507
paulson@14477
   508
lemma isUCont_isCont: "isUCont f ==> isCont f x"
huffman@23012
   509
by (simp add: isUCont_def isCont_def LIM_def, force)
paulson@14477
   510
huffman@23118
   511
lemma isUCont_Cauchy:
huffman@23118
   512
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   513
unfolding isUCont_def
huffman@31338
   514
apply (rule metric_CauchyI)
huffman@23118
   515
apply (drule_tac x=e in spec, safe)
huffman@31338
   516
apply (drule_tac e=s in metric_CauchyD, safe)
huffman@23118
   517
apply (rule_tac x=M in exI, simp)
huffman@23118
   518
done
huffman@23118
   519
huffman@23118
   520
lemma (in bounded_linear) isUCont: "isUCont f"
huffman@31338
   521
unfolding isUCont_def dist_norm
huffman@23118
   522
proof (intro allI impI)
huffman@23118
   523
  fix r::real assume r: "0 < r"
huffman@23118
   524
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
huffman@23118
   525
    using pos_bounded by fast
huffman@23118
   526
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@23118
   527
  proof (rule exI, safe)
huffman@23118
   528
    from r K show "0 < r / K" by (rule divide_pos_pos)
huffman@23118
   529
  next
huffman@23118
   530
    fix x y :: 'a
huffman@23118
   531
    assume xy: "norm (x - y) < r / K"
huffman@23118
   532
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
huffman@23118
   533
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
huffman@23118
   534
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
huffman@23118
   535
    finally show "norm (f x - f y) < r" .
huffman@23118
   536
  qed
huffman@23118
   537
qed
huffman@23118
   538
huffman@23118
   539
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
huffman@23118
   540
by (rule isUCont [THEN isUCont_Cauchy])
huffman@23118
   541
paulson@14477
   542
huffman@21165
   543
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   544
kleing@19023
   545
lemma LIMSEQ_SEQ_conv1:
huffman@44254
   546
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
huffman@44254
   547
  assumes f: "f -- a --> l"
huffman@44254
   548
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
huffman@44254
   549
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
huffman@31338
   550
huffman@44254
   551
lemma LIMSEQ_SEQ_conv2_lemma:
huffman@44254
   552
  fixes a :: "'a::metric_space"
huffman@44254
   553
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> eventually (\<lambda>x. P (S x)) sequentially"
huffman@44254
   554
  shows "eventually P (at a)"
kleing@19023
   555
proof (rule ccontr)
huffman@44254
   556
  let ?I = "\<lambda>n. inverse (real (Suc n))"
huffman@44254
   557
  let ?F = "\<lambda>n::nat. SOME x. x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x"
huffman@44254
   558
  assume "\<not> eventually P (at a)"
huffman@44254
   559
  hence P: "\<forall>d>0. \<exists>x. x \<noteq> a \<and> dist x a < d \<and> \<not> P x"
huffman@44254
   560
    unfolding eventually_at by simp
huffman@44254
   561
  hence "\<And>n. \<exists>x. x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x" by simp
huffman@44254
   562
  hence F: "\<And>n. ?F n \<noteq> a \<and> dist (?F n) a < ?I n \<and> \<not> P (?F n)"
huffman@21165
   563
    by (rule someI_ex)
huffman@21165
   564
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@44254
   565
    and F2: "\<And>n. dist (?F n) a < ?I n"
huffman@44254
   566
    and F3: "\<And>n. \<not> P (?F n)"
huffman@21165
   567
    by fast+
kleing@19023
   568
  have "?F ----> a"
huffman@44254
   569
    using LIMSEQ_inverse_real_of_nat
huffman@44254
   570
    by (rule metric_tendsto_imp_tendsto,
huffman@44254
   571
      simp add: dist_norm F2 [THEN less_imp_le])
kleing@19023
   572
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   573
    by (rule allI) (rule F1)
huffman@44254
   574
  ultimately have "eventually (\<lambda>n. P (?F n)) sequentially"
huffman@44254
   575
    using assms by simp
huffman@44254
   576
  thus "False" by (simp add: F3)
kleing@19023
   577
qed
kleing@19023
   578
huffman@44254
   579
lemma LIMSEQ_SEQ_conv2:
huffman@44254
   580
  fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
huffman@44254
   581
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
huffman@44254
   582
  shows "f -- a --> l"
huffman@44254
   583
  using assms unfolding tendsto_def [where l=l]
huffman@44254
   584
  by (simp add: LIMSEQ_SEQ_conv2_lemma)
huffman@44254
   585
kleing@19023
   586
lemma LIMSEQ_SEQ_conv:
huffman@44254
   587
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::metric_space) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@44254
   588
   (X -- a --> (L::'b::topological_space))"
huffman@44253
   589
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
kleing@19023
   590
paulson@10751
   591
end