author  wenzelm 
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parent 45031  9583f2b56f85 
child 50331  4b6dc5077e98 
permissions  rwrr 
10751  1 
(* 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)" 
10751  27 

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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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shows "f  a > L = 
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(\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (xa) < 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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==> f  a > L" 
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by (simp add: LIM_eq) 
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lemma LIM_D: 
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shows "[ f  a > L; 0<r ] 
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==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (xa) < 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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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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shows "f  a > L \<Longrightarrow> (\<lambda>h. f (a + h))  0 > L" 
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by (drule_tac k="a" in LIM_offset, simp add: add_commute) 
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lemma LIM_offset_zero_cancel: 
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shows "(\<lambda>h. f (a + h))  0 > L \<Longrightarrow> f  a > L" 
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by (drule_tac k=" a" in LIM_offset, simp) 
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32650  84 
lemma LIM_cong_limit: "\<lbrakk> f  x > L ; K = L \<rbrakk> \<Longrightarrow> f  x > K" by simp 
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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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unfolding tendsto_iff dist_norm by simp 
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lemma LIM_zero_cancel: 
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lemma LIM_zero_iff: 
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lemma metric_LIM_imp_LIM: 
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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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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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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_const_not_eq: 
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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) 
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21786  123 
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] 
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lemma LIM_const_eq: 
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by (simp add: tendsto_const_iff) 
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lemma LIM_unique: 
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shows "\<lbrakk>f  a > L; f  a > M\<rbrakk> \<Longrightarrow> L = M" 
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using at_neq_bot by (rule tendsto_unique) 
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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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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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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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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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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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171 
using g f inj by (rule tendsto_compose_eventually) 
21239  172 

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lemma metric_LIM_compose2: 
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174 
assumes f: "f  a > b" 
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175 
assumes g: "g  b > c" 
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176 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b" 
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177 
shows "(\<lambda>x. g (f x))  a > c" 
44314  178 
using g f inj [folded eventually_at] 
179 
by (rule tendsto_compose_eventually) 

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180 

23040  181 
lemma LIM_compose2: 
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182 
fixes a :: "'a::real_normed_vector" 
23040  183 
assumes f: "f  a > b" 
184 
assumes g: "g  b > c" 

185 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x  a) < d \<longrightarrow> f x \<noteq> b" 

186 
shows "(\<lambda>x. g (f x))  a > c" 

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187 
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]]) 
23040  188 

21239  189 
lemma LIM_o: "\<lbrakk>g  l > g l; f  a > l\<rbrakk> \<Longrightarrow> (g \<circ> f)  a > g l" 
44314  190 
unfolding o_def by (rule tendsto_compose) 
21239  191 

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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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194 
assumes f: "f  a > 0" 
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195 
assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x" 
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196 
assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x" 
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197 
shows "g  a > 0" 
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198 
proof (rule LIM_imp_LIM [OF f]) 
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199 
fix x assume x: "x \<noteq> a" 
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200 
have "norm (g x  0) = g x" by (simp add: 1 x) 
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201 
also have "g x \<le> f x" by (rule 2 [OF x]) 
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202 
also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self) 
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203 
also have "\<bar>f x\<bar> = norm (f x  0)" by simp 
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204 
finally show "norm (g x  0) \<le> norm (f x  0)" . 
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205 
qed 
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206 

14477  207 

20755  208 
subsection {* Continuity *} 
14477  209 

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lemma LIM_isCont_iff: 
36665  211 
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space" 
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212 
shows "(f  a > f a) = ((\<lambda>h. f (a + h))  0 > f a)" 
21239  213 
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel]) 
214 

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lemma isCont_iff: 
36665  216 
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space" 
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217 
shows "isCont f x = (\<lambda>h. f (x + h))  0 > f x" 
21239  218 
by (simp add: isCont_def LIM_isCont_iff) 
219 

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lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a" 
44314  221 
unfolding isCont_def by (rule tendsto_ident_at) 
21239  222 

21786  223 
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a" 
44314  224 
unfolding isCont_def by (rule tendsto_const) 
21239  225 

44233  226 
lemma isCont_norm [simp]: 
36665  227 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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228 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a" 
44314  229 
unfolding isCont_def by (rule tendsto_norm) 
21786  230 

44233  231 
lemma isCont_rabs [simp]: 
232 
fixes f :: "'a::topological_space \<Rightarrow> real" 

233 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a" 

44314  234 
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44233  236 
lemma isCont_add [simp]: 
36665  237 
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238 
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a" 
44314  239 
unfolding isCont_def by (rule tendsto_add) 
21239  240 

44233  241 
lemma isCont_minus [simp]: 
36665  242 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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243 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x.  f x) a" 
44314  244 
unfolding isCont_def by (rule tendsto_minus) 
21239  245 

44233  246 
lemma isCont_diff [simp]: 
36665  247 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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248 
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x  g x) a" 
44314  249 
unfolding isCont_def by (rule tendsto_diff) 
21239  250 

44233  251 
lemma isCont_mult [simp]: 
36665  252 
fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra" 
21786  253 
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a" 
44314  254 
unfolding isCont_def by (rule tendsto_mult) 
21239  255 

44233  256 
lemma isCont_inverse [simp]: 
36665  257 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra" 
21786  258 
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a" 
44314  259 
unfolding isCont_def by (rule tendsto_inverse) 
21239  260 

44233  261 
lemma isCont_divide [simp]: 
262 
fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field" 

263 
shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a" 

264 
unfolding isCont_def by (rule tendsto_divide) 

265 

44310  266 
lemma isCont_tendsto_compose: 
267 
"\<lbrakk>isCont g l; (f > l) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. g (f x)) > g l) F" 

268 
unfolding isCont_def by (rule tendsto_compose) 

269 

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270 
lemma metric_isCont_LIM_compose2: 
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271 
assumes f [unfolded isCont_def]: "isCont f a" 
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272 
assumes g: "g  f a > l" 
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273 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a" 
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274 
shows "(\<lambda>x. g (f x))  a > l" 
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275 
by (rule metric_LIM_compose2 [OF f g inj]) 
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276 

23040  277 
lemma isCont_LIM_compose2: 
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278 
fixes a :: "'a::real_normed_vector" 
23040  279 
assumes f [unfolded isCont_def]: "isCont f a" 
280 
assumes g: "g  f a > l" 

281 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x  a) < d \<longrightarrow> f x \<noteq> f a" 

282 
shows "(\<lambda>x. g (f x))  a > l" 

283 
by (rule LIM_compose2 [OF f g inj]) 

284 

21239  285 
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a" 
44314  286 
unfolding isCont_def by (rule tendsto_compose) 
21239  287 

288 
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a" 

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289 
unfolding o_def by (rule isCont_o2) 
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290 

44233  291 
lemma (in bounded_linear) isCont: 
292 
"isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a" 

44314  293 
unfolding isCont_def by (rule tendsto) 
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294 

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295 
lemma (in bounded_bilinear) isCont: 
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296 
"\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a" 
44314  297 
unfolding isCont_def by (rule tendsto) 
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298 

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299 
lemmas isCont_scaleR [simp] = 
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300 
bounded_bilinear.isCont [OF bounded_bilinear_scaleR] 
21239  301 

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lemmas isCont_of_real [simp] = 
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303 
bounded_linear.isCont [OF bounded_linear_of_real] 
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304 

44233  305 
lemma isCont_power [simp]: 
36665  306 
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307 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a" 
44314  308 
unfolding isCont_def by (rule tendsto_power) 
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309 

44233  310 
lemma isCont_sgn [simp]: 
36665  311 
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312 
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a" 
44314  313 
unfolding isCont_def by (rule tendsto_sgn) 
29885  314 

44233  315 
lemma isCont_setsum [simp]: 
316 
fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector" 

317 
fixes A :: "'a set" 

318 
shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a" 

319 
unfolding isCont_def by (simp add: tendsto_setsum) 

15228  320 

44233  321 
lemmas isCont_intros = 
322 
isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus 

323 
isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR 

324 
isCont_of_real isCont_power isCont_sgn isCont_setsum 

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325 

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326 
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x" 
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327 
and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x" 
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328 
shows "0 \<le> f x" 
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329 
proof (rule ccontr) 
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330 
assume "\<not> 0 \<le> f x" hence "f x < 0" by auto 
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331 
hence "0 <  f x / 2" by auto 
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332 
from isCont[unfolded isCont_def, THEN LIM_D, OF this] 
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333 
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 
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334 

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335 
let ?x = "x  min (s / 2) ((x  b) / 2)" 
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336 
have "?x < x" and "\<bar> ?x  x \<bar> < s" 
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337 
using `b < x` and `0 < s` by auto 
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338 
have "b < ?x" 
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339 
proof (cases "s < x  b") 
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340 
case True thus ?thesis using `0 < s` by auto 
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341 
next 
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342 
case False hence "s / 2 \<ge> (x  b) / 2" by auto 
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343 
hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2) 
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344 
thus ?thesis using `b < x` by auto 
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345 
qed 
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346 
hence "0 \<le> f ?x" using all_le `?x < x` by auto 
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347 
moreover have "\<bar>f ?x  f x\<bar> <  f x / 2" 
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348 
using s_D[OF _ `\<bar> ?x  x \<bar> < s`] `?x < x` by auto 
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349 
hence "f ?x  f x <  f x / 2" by auto 
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350 
hence "f ?x < f x / 2" by auto 
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351 
hence "f ?x < 0" using `f x < 0` by auto 
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352 
thus False using `0 \<le> f ?x` by auto 
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qed 
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20755  356 
subsection {* Uniform Continuity *} 
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lemma isUCont_isCont: "isUCont f ==> isCont f x" 
23012  359 
by (simp add: isUCont_def isCont_def LIM_def, force) 
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lemma isUCont_Cauchy: 
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"\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" 

363 
unfolding isUCont_def 

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apply (rule metric_CauchyI) 
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apply (drule_tac x=e in spec, safe) 
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apply (drule_tac e=s in metric_CauchyD, safe) 
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apply (rule_tac x=M in exI, simp) 
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done 

369 

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lemma (in bounded_linear) isUCont: "isUCont f" 

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unfolding isUCont_def dist_norm 
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proof (intro allI impI) 
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fix r::real assume r: "0 < r" 

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obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K" 

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using pos_bounded by fast 

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show "\<exists>s>0. \<forall>x y. norm (x  y) < s \<longrightarrow> norm (f x  f y) < r" 

377 
proof (rule exI, safe) 

378 
from r K show "0 < r / K" by (rule divide_pos_pos) 

379 
next 

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fix x y :: 'a 

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assume xy: "norm (x  y) < r / K" 

382 
have "norm (f x  f y) = norm (f (x  y))" by (simp only: diff) 

383 
also have "\<dots> \<le> norm (x  y) * K" by (rule norm_le) 

384 
also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq) 

385 
finally show "norm (f x  f y) < r" . 

386 
qed 

387 
qed 

388 

389 
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" 

390 
by (rule isUCont [THEN isUCont_Cauchy]) 

391 

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subsection {* Relation of LIM and LIMSEQ *} 
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lemma sequentially_imp_eventually_within: 
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fixes a :: "'a::metric_space" 

397 
assumes "\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f > a \<longrightarrow> 

398 
eventually (\<lambda>n. P (f n)) sequentially" 

399 
shows "eventually P (at a within s)" 

400 
proof (rule ccontr) 

401 
let ?I = "\<lambda>n. inverse (real (Suc n))" 

402 
def F \<equiv> "\<lambda>n::nat. SOME x. x \<in> s \<and> x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x" 

403 
assume "\<not> eventually P (at a within s)" 

404 
hence P: "\<forall>d>0. \<exists>x. x \<in> s \<and> x \<noteq> a \<and> dist x a < d \<and> \<not> P x" 

405 
unfolding Limits.eventually_within Limits.eventually_at by fast 

406 
hence "\<And>n. \<exists>x. x \<in> s \<and> x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x" by simp 

407 
hence F: "\<And>n. F n \<in> s \<and> F n \<noteq> a \<and> dist (F n) a < ?I n \<and> \<not> P (F n)" 

408 
unfolding F_def by (rule someI_ex) 

409 
hence F0: "\<forall>n. F n \<in> s" and F1: "\<forall>n. F n \<noteq> a" 

410 
and F2: "\<forall>n. dist (F n) a < ?I n" and F3: "\<forall>n. \<not> P (F n)" 

411 
by fast+ 

412 
from LIMSEQ_inverse_real_of_nat have "F > a" 

413 
by (rule metric_tendsto_imp_tendsto, 

414 
simp add: dist_norm F2 less_imp_le) 

415 
hence "eventually (\<lambda>n. P (F n)) sequentially" 

416 
using assms F0 F1 by simp 

417 
thus "False" by (simp add: F3) 

418 
qed 

419 

420 
lemma sequentially_imp_eventually_at: 

421 
fixes a :: "'a::metric_space" 

422 
assumes "\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f > a \<longrightarrow> 

423 
eventually (\<lambda>n. P (f n)) sequentially" 

424 
shows "eventually P (at a)" 

45031  425 
using assms sequentially_imp_eventually_within [where s=UNIV] by simp 
44532  426 

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lemma LIMSEQ_SEQ_conv1: 
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fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" 
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assumes f: "f  a > l" 
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shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. f (S n)) > l" 
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using tendsto_compose_eventually [OF f, where F=sequentially] by simp 
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lemma LIMSEQ_SEQ_conv2: 
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fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space" 
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assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. f (S n)) > l" 
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shows "f  a > l" 
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using assms unfolding tendsto_def [where l=l] 
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by (simp add: sequentially_imp_eventually_at) 
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439 

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lemma LIMSEQ_SEQ_conv: 
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"(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > (a::'a::metric_space) \<longrightarrow> (\<lambda>n. X (S n)) > L) = 
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(X  a > (L::'b::topological_space))" 
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using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 .. 
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10751  445 
end 