author  huffman 
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parent 44194  0639898074ae 
child 44217  5cdad94bdc29 
permissions  rwrr 
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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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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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shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (xa) < s > norm (f x  L) < r) 
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==> f  a > L" 
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by (simp add: LIM_eq) 
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lemma LIM_D: 
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shows "[ f  a > L; 0<r ] 
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==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (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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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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shows "f  a > L \<Longrightarrow> (\<lambda>h. f (a + h))  0 > L" 
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by (drule_tac k="a" in LIM_offset, simp add: add_commute) 
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lemma LIM_offset_zero_cancel: 
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shows "(\<lambda>h. f (a + h))  0 > L \<Longrightarrow> f  a > L" 
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by (drule_tac k=" a" in LIM_offset, simp) 
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lemma LIM_const [simp]: "(%x. k)  x > k" 
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by (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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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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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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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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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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shows "f  a > l \<Longrightarrow> (\<lambda>x. f x  l)  a > 0" 
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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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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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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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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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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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163 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
31338
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164 
shows "(\<lambda>x. norm (f x))  a > 0 \<Longrightarrow> f  a > 0" 
36662
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165 
by (rule tendsto_norm_zero_cancel) 
21257  166 

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167 
lemma LIM_norm_zero_iff: 
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168 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
31338
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169 
shows "(\<lambda>x. norm (f x))  a > 0 = f  a > 0" 
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170 
by (rule tendsto_norm_zero_iff) 
21399  171 

22627
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172 
lemma LIM_rabs: "f  a > (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>)  a > \<bar>l\<bar>" 
44194
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173 
by (rule tendsto_rabs) 
22627
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174 

2b093ba973bc
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175 
lemma LIM_rabs_zero: "f  a > (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>)  a > 0" 
44194
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176 
by (rule tendsto_rabs_zero) 
22627
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177 

2b093ba973bc
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178 
lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>)  a > (0::real) \<Longrightarrow> f  a > 0" 
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179 
by (rule tendsto_rabs_zero_cancel) 
22627
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180 

2b093ba973bc
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181 
lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>)  a > (0::real) = f  a > 0" 
44194
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182 
by (rule tendsto_rabs_zero_iff) 
22627
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changeset

183 

44205
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184 
lemma trivial_limit_at: 
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185 
fixes a :: "'a::real_normed_algebra_1" 
44205
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186 
shows "\<not> trivial_limit (at a)"  {* TODO: find a more appropriate class *} 
18da2a87421c
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187 
unfolding trivial_limit_def 
36662
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188 
unfolding eventually_at dist_norm 
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189 
by (clarsimp, rule_tac x="a + of_real (d/2)" in exI, simp) 
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190 

20561
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191 
lemma LIM_const_not_eq: 
23076
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192 
fixes a :: "'a::real_normed_algebra_1" 
44205
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193 
fixes k L :: "'b::t2_space" 
23076
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194 
shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k)  a > L" 
44205
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195 
by (simp add: tendsto_const_iff trivial_limit_at) 
14477  196 

21786  197 
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] 
198 

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199 
lemma LIM_const_eq: 
23076
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200 
fixes a :: "'a::real_normed_algebra_1" 
44205
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201 
fixes k L :: "'b::t2_space" 
23076
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202 
shows "(\<lambda>x. k)  a > L \<Longrightarrow> k = L" 
44205
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203 
by (simp add: tendsto_const_iff trivial_limit_at) 
14477  204 

20561
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205 
lemma LIM_unique: 
31338
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206 
fixes a :: "'a::real_normed_algebra_1"  {* TODO: find a more appropriate class *} 
44205
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207 
fixes L M :: "'b::t2_space" 
23076
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208 
shows "\<lbrakk>f  a > L; f  a > M\<rbrakk> \<Longrightarrow> L = M" 
44205
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changeset

209 
using trivial_limit_at by (rule tendsto_unique) 
14477  210 

23069
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211 
lemma LIM_ident [simp]: "(\<lambda>x. x)  a > a" 
36661
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212 
by (rule tendsto_ident_at) 
14477  213 

214 
text{*Limits are equal for functions equal except at limit point*} 

215 
lemma LIM_equal: 

216 
"[ \<forall>x. x \<noteq> a > (f x = g x) ] ==> (f  a > l) = (g  a > l)" 

36662
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217 
unfolding tendsto_def eventually_at_topological by simp 
14477  218 

20796  219 
lemma LIM_cong: 
220 
"\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk> 

21399  221 
\<Longrightarrow> ((\<lambda>x. f x)  a > l) = ((\<lambda>x. g x)  b > m)" 
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222 
by (simp add: LIM_equal) 
20796  223 

31338
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224 
lemma metric_LIM_equal2: 
21282
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225 
assumes 1: "0 < R" 
31338
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226 
assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < R\<rbrakk> \<Longrightarrow> f x = g x" 
21282
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227 
shows "g  a > l \<Longrightarrow> f  a > l" 
36662
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228 
apply (rule topological_tendstoI) 
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229 
apply (drule (2) topological_tendstoD) 
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230 
apply (simp add: eventually_at, safe) 
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231 
apply (rule_tac x="min d R" in exI, safe) 
21282
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232 
apply (simp add: 1) 
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233 
apply (simp add: 2) 
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234 
done 
dd647b4d7952
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parents:
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235 

31338
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236 
lemma LIM_equal2: 
36662
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237 
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space" 
31338
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238 
assumes 1: "0 < R" 
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parents:
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changeset

239 
assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x  a) < R\<rbrakk> \<Longrightarrow> f x = g x" 
d41a8ba25b67
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parents:
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changeset

240 
shows "g  a > l \<Longrightarrow> f  a > l" 
36662
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changeset

241 
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm) 
31338
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huffman
parents:
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changeset

242 

36662
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243 
text{*Two uses in Transcendental.ML*} (* BH: no longer true; delete? *) 
14477  244 
lemma LIM_trans: 
31338
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245 
fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector" 
d41a8ba25b67
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parents:
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changeset

246 
shows "[ (%x. f(x) + g(x))  a > 0; g  a > l ] ==> f  a > l" 
14477  247 
apply (drule LIM_add, assumption) 
248 
apply (auto simp add: add_assoc) 

249 
done 

250 

21239  251 
lemma LIM_compose: 
252 
assumes g: "g  l > g l" 

253 
assumes f: "f  a > l" 

254 
shows "(\<lambda>x. g (f x))  a > g l" 

36662
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255 
proof (rule topological_tendstoI) 
621122eeb138
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256 
fix C assume C: "open C" "g l \<in> C" 
621122eeb138
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parents:
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changeset

257 
obtain B where B: "open B" "l \<in> B" 
621122eeb138
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parents:
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changeset

258 
and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> C" 
621122eeb138
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parents:
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changeset

259 
using topological_tendstoD [OF g C] 
621122eeb138
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huffman
parents:
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diff
changeset

260 
unfolding eventually_at_topological by fast 
621122eeb138
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huffman
parents:
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changeset

261 
obtain A where A: "open A" "a \<in> A" 
621122eeb138
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huffman
parents:
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changeset

262 
and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B" 
621122eeb138
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huffman
parents:
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diff
changeset

263 
using topological_tendstoD [OF f B] 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

264 
unfolding eventually_at_topological by fast 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

265 
show "eventually (\<lambda>x. g (f x) \<in> C) (at a)" 
621122eeb138
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huffman
parents:
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changeset

266 
unfolding eventually_at_topological 
621122eeb138
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huffman
parents:
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changeset

267 
proof (intro exI conjI ballI impI) 
621122eeb138
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huffman
parents:
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changeset

268 
show "open A" and "a \<in> A" using A . 
621122eeb138
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parents:
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269 
next 
621122eeb138
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parents:
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changeset

270 
fix x assume "x \<in> A" and "x \<noteq> a" 
621122eeb138
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parents:
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changeset

271 
then show "g (f x) \<in> C" 
621122eeb138
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parents:
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272 
by (cases "f x = l", simp add: C, simp add: gC fB) 
621122eeb138
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273 
qed 
621122eeb138
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huffman
parents:
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274 
qed 
21239  275 

36662
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276 
lemma LIM_compose_eventually: 
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277 
assumes f: "f  a > b" 
621122eeb138
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huffman
parents:
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changeset

278 
assumes g: "g  b > c" 
621122eeb138
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parents:
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changeset

279 
assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)" 
621122eeb138
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parents:
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changeset

280 
shows "(\<lambda>x. g (f x))  a > c" 
621122eeb138
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huffman
parents:
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diff
changeset

281 
proof (rule topological_tendstoI) 
621122eeb138
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huffman
parents:
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diff
changeset

282 
fix C assume C: "open C" "c \<in> C" 
621122eeb138
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huffman
parents:
36661
diff
changeset

283 
obtain B where B: "open B" "b \<in> B" 
621122eeb138
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huffman
parents:
36661
diff
changeset

284 
and gC: "\<And>y. y \<in> B \<Longrightarrow> y \<noteq> b \<Longrightarrow> g y \<in> C" 
621122eeb138
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huffman
parents:
36661
diff
changeset

285 
using topological_tendstoD [OF g C] 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

286 
unfolding eventually_at_topological by fast 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

287 
obtain A where A: "open A" "a \<in> A" 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

288 
and fB: "\<And>x. x \<in> A \<Longrightarrow> x \<noteq> a \<Longrightarrow> f x \<in> B" 
621122eeb138
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huffman
parents:
36661
diff
changeset

289 
using topological_tendstoD [OF f B] 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

290 
unfolding eventually_at_topological by fast 
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

291 
have "eventually (\<lambda>x. f x \<noteq> b \<longrightarrow> g (f x) \<in> C) (at a)" 
621122eeb138
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huffman
parents:
36661
diff
changeset

292 
unfolding eventually_at_topological 
621122eeb138
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huffman
parents:
36661
diff
changeset

293 
proof (intro exI conjI ballI impI) 
621122eeb138
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parents:
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changeset

294 
show "open A" and "a \<in> A" using A . 
21239  295 
next 
36662
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parents:
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diff
changeset

296 
fix x assume "x \<in> A" and "x \<noteq> a" and "f x \<noteq> b" 
621122eeb138
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parents:
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diff
changeset

297 
then show "g (f x) \<in> C" by (simp add: gC fB) 
21239  298 
qed 
36662
621122eeb138
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parents:
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diff
changeset

299 
with inj show "eventually (\<lambda>x. g (f x) \<in> C) (at a)" 
621122eeb138
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huffman
parents:
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diff
changeset

300 
by (rule eventually_rev_mp) 
21239  301 
qed 
302 

31338
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parents:
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diff
changeset

303 
lemma metric_LIM_compose2: 
d41a8ba25b67
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huffman
parents:
31336
diff
changeset

304 
assumes f: "f  a > b" 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

305 
assumes g: "g  b > c" 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

306 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> b" 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

307 
shows "(\<lambda>x. g (f x))  a > c" 
36662
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308 
using f g inj [folded eventually_at] 
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309 
by (rule LIM_compose_eventually) 
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310 

23040  311 
lemma LIM_compose2: 
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312 
fixes a :: "'a::real_normed_vector" 
23040  313 
assumes f: "f  a > b" 
314 
assumes g: "g  b > c" 

315 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x  a) < d \<longrightarrow> f x \<noteq> b" 

316 
shows "(\<lambda>x. g (f x))  a > c" 

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317 
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]]) 
23040  318 

21239  319 
lemma LIM_o: "\<lbrakk>g  l > g l; f  a > l\<rbrakk> \<Longrightarrow> (g \<circ> f)  a > g l" 
320 
unfolding o_def by (rule LIM_compose) 

321 

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322 
lemma real_LIM_sandwich_zero: 
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323 
fixes f g :: "'a::topological_space \<Rightarrow> real" 
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324 
assumes f: "f  a > 0" 
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325 
assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x" 
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326 
assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x" 
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327 
shows "g  a > 0" 
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328 
proof (rule LIM_imp_LIM [OF f]) 
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329 
fix x assume x: "x \<noteq> a" 
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330 
have "norm (g x  0) = g x" by (simp add: 1 x) 
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331 
also have "g x \<le> f x" by (rule 2 [OF x]) 
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332 
also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self) 
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333 
also have "\<bar>f x\<bar> = norm (f x  0)" by simp 
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334 
finally show "norm (g x  0) \<le> norm (f x  0)" . 
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335 
qed 
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336 

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337 
text {* Bounded Linear Operators *} 
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338 

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339 
lemma (in bounded_linear) LIM: 
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340 
"g  a > l \<Longrightarrow> (\<lambda>x. f (g x))  a > f l" 
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341 
by (rule tendsto) 
31349
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342 

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343 
lemma (in bounded_linear) cont: "f  a > f a" 
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344 
by (rule LIM [OF LIM_ident]) 
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345 

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346 
lemma (in bounded_linear) LIM_zero: 
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347 
"g  a > 0 \<Longrightarrow> (\<lambda>x. f (g x))  a > 0" 
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348 
by (rule tendsto_zero) 
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349 

22442
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350 
text {* Bounded Bilinear Operators *} 
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351 

31349
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352 
lemma (in bounded_bilinear) LIM: 
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353 
"\<lbrakk>f  a > L; g  a > M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x)  a > L ** M" 
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354 
by (rule tendsto) 
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355 

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356 
lemma (in bounded_bilinear) LIM_prod_zero: 
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357 
fixes a :: "'d::metric_space" 
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358 
assumes f: "f  a > 0" 
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359 
assumes g: "g  a > 0" 
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360 
shows "(\<lambda>x. f x ** g x)  a > 0" 
44194
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361 
using f g by (rule tendsto_zero) 
21282
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362 

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363 
lemma (in bounded_bilinear) LIM_left_zero: 
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364 
"f  a > 0 \<Longrightarrow> (\<lambda>x. f x ** c)  a > 0" 
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365 
by (rule tendsto_left_zero) 
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366 

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367 
lemma (in bounded_bilinear) LIM_right_zero: 
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368 
"f  a > 0 \<Longrightarrow> (\<lambda>x. c ** f x)  a > 0" 
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369 
by (rule tendsto_right_zero) 
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370 

23127  371 
lemmas LIM_mult = mult.LIM 
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372 

23127  373 
lemmas LIM_mult_zero = mult.LIM_prod_zero 
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374 

23127  375 
lemmas LIM_mult_left_zero = mult.LIM_left_zero 
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376 

23127  377 
lemmas LIM_mult_right_zero = mult.LIM_right_zero 
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378 

23127  379 
lemmas LIM_scaleR = scaleR.LIM 
21282
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380 

23127  381 
lemmas LIM_of_real = of_real.LIM 
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382 

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383 
lemma LIM_power: 
36665  384 
fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}" 
22627
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385 
assumes f: "f  a > l" 
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386 
shows "(\<lambda>x. f x ^ n)  a > l ^ n" 
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387 
using assms by (rule tendsto_power) 
22627
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388 

22641
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389 
subsubsection {* Derived theorems about @{term LIM} *} 
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390 

31355  391 
lemma LIM_inverse: 
392 
fixes L :: "'a::real_normed_div_algebra" 

393 
shows "\<lbrakk>f  a > L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x))  a > inverse L" 

394 
by (rule tendsto_inverse) 

22637  395 

396 
lemma LIM_inverse_fun: 

397 
assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)" 

398 
shows "inverse  a > inverse a" 

31355  399 
by (rule LIM_inverse [OF LIM_ident a]) 
22637  400 

29885  401 
lemma LIM_sgn: 
36665  402 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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403 
shows "\<lbrakk>f  a > l; l \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. sgn (f x))  a > sgn l" 
44194
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404 
by (rule tendsto_sgn) 
29885  405 

14477  406 

20755  407 
subsection {* Continuity *} 
14477  408 

31338
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409 
lemma LIM_isCont_iff: 
36665  410 
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space" 
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411 
shows "(f  a > f a) = ((\<lambda>h. f (a + h))  0 > f a)" 
21239  412 
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel]) 
413 

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414 
lemma isCont_iff: 
36665  415 
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space" 
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416 
shows "isCont f x = (\<lambda>h. f (x + h))  0 > f x" 
21239  417 
by (simp add: isCont_def LIM_isCont_iff) 
418 

23069
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diff
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419 
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a" 
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diff
changeset

420 
unfolding isCont_def by (rule LIM_ident) 
21239  421 

21786  422 
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a" 
21282
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423 
unfolding isCont_def by (rule LIM_const) 
21239  424 

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425 
lemma isCont_norm: 
36665  426 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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427 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a" 
21786  428 
unfolding isCont_def by (rule LIM_norm) 
429 

22627
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430 
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a" 
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431 
unfolding isCont_def by (rule LIM_rabs) 
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changeset

432 

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433 
lemma isCont_add: 
36665  434 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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435 
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a" 
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changeset

436 
unfolding isCont_def by (rule LIM_add) 
21239  437 

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438 
lemma isCont_minus: 
36665  439 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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440 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x.  f x) a" 
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changeset

441 
unfolding isCont_def by (rule LIM_minus) 
21239  442 

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443 
lemma isCont_diff: 
36665  444 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
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445 
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x  g x) a" 
21282
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changeset

446 
unfolding isCont_def by (rule LIM_diff) 
21239  447 

448 
lemma isCont_mult: 

36665  449 
fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra" 
21786  450 
shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a" 
21282
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changeset

451 
unfolding isCont_def by (rule LIM_mult) 
21239  452 

453 
lemma isCont_inverse: 

36665  454 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra" 
21786  455 
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a" 
21282
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456 
unfolding isCont_def by (rule LIM_inverse) 
21239  457 

458 
lemma isCont_LIM_compose: 

459 
"\<lbrakk>isCont g l; f  a > l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x))  a > g l" 

21282
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changeset

460 
unfolding isCont_def by (rule LIM_compose) 
21239  461 

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462 
lemma metric_isCont_LIM_compose2: 
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463 
assumes f [unfolded isCont_def]: "isCont f a" 
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464 
assumes g: "g  f a > l" 
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465 
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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466 
shows "(\<lambda>x. g (f x))  a > l" 
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changeset

467 
by (rule metric_LIM_compose2 [OF f g inj]) 
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468 

23040  469 
lemma isCont_LIM_compose2: 
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470 
fixes a :: "'a::real_normed_vector" 
23040  471 
assumes f [unfolded isCont_def]: "isCont f a" 
472 
assumes g: "g  f a > l" 

473 
assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x  a) < d \<longrightarrow> f x \<noteq> f a" 

474 
shows "(\<lambda>x. g (f x))  a > l" 

475 
by (rule LIM_compose2 [OF f g inj]) 

476 

21239  477 
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a" 
21282
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478 
unfolding isCont_def by (rule LIM_compose) 
21239  479 

480 
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a" 

21282
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changeset

481 
unfolding o_def by (rule isCont_o2) 
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changeset

482 

dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

483 
lemma (in bounded_linear) isCont: "isCont f a" 
dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

484 
unfolding isCont_def by (rule cont) 
dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

485 

dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

486 
lemma (in bounded_bilinear) isCont: 
dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

487 
"\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a" 
dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

488 
unfolding isCont_def by (rule LIM) 
dd647b4d7952
added bounded_linear and bounded_bilinear locales
huffman
parents:
21257
diff
changeset

489 

23127  490 
lemmas isCont_scaleR = scaleR.isCont 
21239  491 

22627
2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

492 
lemma isCont_of_real: 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

493 
"isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)::'b::real_normed_algebra_1) a" 
22627
2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

494 
unfolding isCont_def by (rule LIM_of_real) 
2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

495 

2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

496 
lemma isCont_power: 
36665  497 
fixes f :: "'a::topological_space \<Rightarrow> 'b::{power,real_normed_algebra}" 
22627
2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

498 
shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a" 
2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

499 
unfolding isCont_def by (rule LIM_power) 
2b093ba973bc
new LIM/isCont lemmas for abs, of_real, and power
huffman
parents:
22613
diff
changeset

500 

29885  501 
lemma isCont_sgn: 
36665  502 
fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

503 
shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a" 
29885  504 
unfolding isCont_def by (rule LIM_sgn) 
505 

20561
6a6d8004322f
generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents:
20552
diff
changeset

506 
lemma isCont_abs [simp]: "isCont abs (a::real)" 
23069
cdfff0241c12
rename lemmas LIM_ident, isCont_ident, DERIV_ident
huffman
parents:
23040
diff
changeset

507 
by (rule isCont_rabs [OF isCont_ident]) 
15228  508 

31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

509 
lemma isCont_setsum: 
36665  510 
fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

511 
fixes A :: "'a set" assumes "finite A" 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

512 
shows "\<forall> i \<in> A. isCont (f i) x \<Longrightarrow> isCont (\<lambda> x. \<Sum> i \<in> A. f i x) x" 
44194
0639898074ae
generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents:
41550
diff
changeset

513 
unfolding isCont_def by (simp add: tendsto_setsum) 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

514 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

515 
lemma LIM_less_bound: fixes f :: "real \<Rightarrow> real" assumes "b < x" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

516 
and all_le: "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and isCont: "isCont f x" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

517 
shows "0 \<le> f x" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

518 
proof (rule ccontr) 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

519 
assume "\<not> 0 \<le> f x" hence "f x < 0" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

520 
hence "0 <  f x / 2" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

521 
from isCont[unfolded isCont_def, THEN LIM_D, OF this] 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

522 
obtain s where "s > 0" and s_D: "\<And>x'. \<lbrakk> x' \<noteq> x ; \<bar> x'  x \<bar> < s \<rbrakk> \<Longrightarrow> \<bar> f x'  f x \<bar> <  f x / 2" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

523 

c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

524 
let ?x = "x  min (s / 2) ((x  b) / 2)" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

525 
have "?x < x" and "\<bar> ?x  x \<bar> < s" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

526 
using `b < x` and `0 < s` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

527 
have "b < ?x" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

528 
proof (cases "s < x  b") 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

529 
case True thus ?thesis using `0 < s` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

530 
next 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

531 
case False hence "s / 2 \<ge> (x  b) / 2" by auto 
32642
026e7c6a6d08
be more cautious wrt. simp rules: inf_absorb1, inf_absorb2, sup_absorb1, sup_absorb2 are no simp rules by default any longer
haftmann
parents:
32436
diff
changeset

532 
hence "?x = (x + b) / 2" by (simp add: field_simps min_max.inf_absorb2) 
29803
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

533 
thus ?thesis using `b < x` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

534 
qed 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

535 
hence "0 \<le> f ?x" using all_le `?x < x` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

536 
moreover have "\<bar>f ?x  f x\<bar> <  f x / 2" 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

537 
using s_D[OF _ `\<bar> ?x  x \<bar> < s`] `?x < x` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

538 
hence "f ?x  f x <  f x / 2" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

539 
hence "f ?x < f x / 2" by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

540 
hence "f ?x < 0" using `f x < 0` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

541 
thus False using `0 \<le> f ?x` by auto 
c56a5571f60a
Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents:
29667
diff
changeset

542 
qed 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

543 

14477  544 

20755  545 
subsection {* Uniform Continuity *} 
546 

14477  547 
lemma isUCont_isCont: "isUCont f ==> isCont f x" 
23012  548 
by (simp add: isUCont_def isCont_def LIM_def, force) 
14477  549 

23118  550 
lemma isUCont_Cauchy: 
551 
"\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" 

552 
unfolding isUCont_def 

31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

553 
apply (rule metric_CauchyI) 
23118  554 
apply (drule_tac x=e in spec, safe) 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

555 
apply (drule_tac e=s in metric_CauchyD, safe) 
23118  556 
apply (rule_tac x=M in exI, simp) 
557 
done 

558 

559 
lemma (in bounded_linear) isUCont: "isUCont f" 

31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

560 
unfolding isUCont_def dist_norm 
23118  561 
proof (intro allI impI) 
562 
fix r::real assume r: "0 < r" 

563 
obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K" 

564 
using pos_bounded by fast 

565 
show "\<exists>s>0. \<forall>x y. norm (x  y) < s \<longrightarrow> norm (f x  f y) < r" 

566 
proof (rule exI, safe) 

567 
from r K show "0 < r / K" by (rule divide_pos_pos) 

568 
next 

569 
fix x y :: 'a 

570 
assume xy: "norm (x  y) < r / K" 

571 
have "norm (f x  f y) = norm (f (x  y))" by (simp only: diff) 

572 
also have "\<dots> \<le> norm (x  y) * K" by (rule norm_le) 

573 
also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq) 

574 
finally show "norm (f x  f y) < r" . 

575 
qed 

576 
qed 

577 

578 
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))" 

579 
by (rule isUCont [THEN isUCont_Cauchy]) 

580 

14477  581 

21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

582 
subsection {* Relation of LIM and LIMSEQ *} 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

583 

5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

584 
lemma LIMSEQ_SEQ_conv1: 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

585 
fixes a :: "'a::metric_space" and L :: "'b::metric_space" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

586 
assumes X: "X  a > L" 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

587 
shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. X (S n)) > L" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

588 
proof (safe intro!: metric_LIMSEQ_I) 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

589 
fix S :: "nat \<Rightarrow> 'a" 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

590 
fix r :: real 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

591 
assume rgz: "0 < r" 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

592 
assume as: "\<forall>n. S n \<noteq> a" 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

593 
assume S: "S > a" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

594 
from metric_LIM_D [OF X rgz] obtain s 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

595 
where sgz: "0 < s" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

596 
and aux: "\<And>x. \<lbrakk>x \<noteq> a; dist x a < s\<rbrakk> \<Longrightarrow> dist (X x) L < r" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

597 
by fast 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

598 
from metric_LIMSEQ_D [OF S sgz] 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

599 
obtain no where "\<forall>n\<ge>no. dist (S n) a < s" by blast 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

600 
hence "\<forall>n\<ge>no. dist (X (S n)) L < r" by (simp add: aux as) 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

601 
thus "\<exists>no. \<forall>n\<ge>no. dist (X (S n)) L < r" .. 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

602 
qed 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

603 

31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

604 

19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

605 
lemma LIMSEQ_SEQ_conv2: 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

606 
fixes a :: real and L :: "'a::metric_space" 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

607 
assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. X (S n)) > L" 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

608 
shows "X  a > L" 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

609 
proof (rule ccontr) 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

610 
assume "\<not> (X  a > L)" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

611 
hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x  a) < s > dist (X x) L < r)" 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

612 
unfolding LIM_def dist_norm . 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

613 
hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x  a\<bar> < s > dist (X x) L < r)" by simp 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

614 
hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x  a\<bar> < s \<and> dist (X x) L \<ge> r)" by (simp add: not_less) 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

615 
then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x  a\<bar> < s \<and> dist (X x) L \<ge> r))" by auto 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

616 

31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

617 
let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x  a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r" 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

618 
have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x  a\<bar> < inverse (real (Suc n)) \<and> dist (X x) L \<ge> r" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

619 
using rdef by simp 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

620 
hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n  a\<bar> < inverse (real (Suc n)) \<and> dist (X (?F n)) L \<ge> r" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

621 
by (rule someI_ex) 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

622 
hence F1: "\<And>n. ?F n \<noteq> a" 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

623 
and F2: "\<And>n. \<bar>?F n  a\<bar> < inverse (real (Suc n))" 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

624 
and F3: "\<And>n. dist (X (?F n)) L \<ge> r" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

625 
by fast+ 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

626 

19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

627 
have "?F > a" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

628 
proof (rule LIMSEQ_I, unfold real_norm_def) 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

629 
fix e::real 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

630 
assume "0 < e" 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
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diff
changeset

631 
(* choose no such that inverse (real (Suc n)) < e *) 
23441  632 
then have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean) 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

633 
then obtain m where nodef: "inverse (real (Suc m)) < e" by auto 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

634 
show "\<exists>no. \<forall>n. no \<le> n > \<bar>?F n  a\<bar> < e" 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

635 
proof (intro exI allI impI) 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

636 
fix n 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
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diff
changeset

637 
assume mlen: "m \<le> n" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

638 
have "\<bar>?F n  a\<bar> < inverse (real (Suc n))" 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

639 
by (rule F2) 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

640 
also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))" 
23441  641 
using mlen by auto 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

642 
also from nodef have 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

643 
"inverse (real (Suc m)) < e" . 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

644 
finally show "\<bar>?F n  a\<bar> < e" . 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

645 
qed 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

646 
qed 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

647 

5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

648 
moreover have "\<forall>n. ?F n \<noteq> a" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

649 
by (rule allI) (rule F1) 
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

650 

41550  651 
moreover note `\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. X (S n)) > L` 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

652 
ultimately have "(\<lambda>n. X (?F n)) > L" by simp 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

653 

5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

654 
moreover have "\<not> ((\<lambda>n. X (?F n)) > L)" 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

655 
proof  
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

656 
{ 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

657 
fix no::nat 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

658 
obtain n where "n = no + 1" by simp 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
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diff
changeset

659 
then have nolen: "no \<le> n" by simp 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

660 
(* We prove this by showing that for any m there is an n\<ge>m such that X (?F n)  L \<ge> r *) 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

661 
have "dist (X (?F n)) L \<ge> r" 
21165
8fb49f668511
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents:
21141
diff
changeset

662 
by (rule F3) 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

663 
with nolen have "\<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r" by fast 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

664 
} 
31338
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

665 
then have "(\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> r)" by simp 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

666 
with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> dist (X (?F n)) L \<ge> e)" by auto 
d41a8ba25b67
generalize constants from Lim.thy to class metric_space
huffman
parents:
31336
diff
changeset

667 
thus ?thesis by (unfold LIMSEQ_def, auto simp add: not_less) 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

668 
qed 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

669 
ultimately show False by simp 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

670 
qed 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

671 

5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

672 
lemma LIMSEQ_SEQ_conv: 
20561
6a6d8004322f
generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents:
20552
diff
changeset

673 
"(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > (a::real) \<longrightarrow> (\<lambda>n. X (S n)) > L) = 
36662
621122eeb138
generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents:
36661
diff
changeset

674 
(X  a > (L::'a::metric_space))" 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

675 
proof 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

676 
assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. X (S n)) > L" 
23441  677 
thus "X  a > L" by (rule LIMSEQ_SEQ_conv2) 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

678 
next 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

679 
assume "(X  a > L)" 
23441  680 
thus "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S > a \<longrightarrow> (\<lambda>n. X (S n)) > L" by (rule LIMSEQ_SEQ_conv1) 
19023
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

681 
qed 
5652a536b7e8
* include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents:
17318
diff
changeset

682 

10751  683 
end 