src/HOL/Lim.thy
author wenzelm
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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 (x-a) < s --> norm (f x - L) < r)"
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by (simp add: LIM_def dist_norm)
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lemma LIM_I:
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  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
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      ==> f -- a --> L"
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by (simp add: LIM_eq)
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lemma LIM_D:
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  shows "[| f -- a --> L; 0<r |]
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      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
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by (simp add: LIM_eq)
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lemma LIM_offset:
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  fixes a :: "'a::real_normed_vector"
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  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
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apply (rule topological_tendstoI)
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apply (drule (2) topological_tendstoD)
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apply (simp only: eventually_at dist_norm)
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apply (clarify, rule_tac x=d in exI, safe)
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apply (drule_tac x="x + k" in spec)
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apply (simp add: algebra_simps)
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done
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lemma LIM_offset_zero:
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  fixes a :: "'a::real_normed_vector"
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  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
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by (drule_tac k="a" in LIM_offset, simp add: add_commute)
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lemma LIM_offset_zero_cancel:
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  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_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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  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
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unfolding tendsto_iff dist_norm by simp
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lemma LIM_zero_cancel:
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  shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
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lemma LIM_zero_iff:
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  shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
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unfolding tendsto_iff dist_norm by simp
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lemma metric_LIM_imp_LIM:
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  assumes f: "f -- a --> l"
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  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> dist (g x) m \<le> dist (f x) l"
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  shows "g -- a --> m"
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  by (rule metric_tendsto_imp_tendsto [OF f],
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    auto simp add: eventually_at_topological le)
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lemma LIM_imp_LIM:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
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  assumes f: "f -- a --> l"
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  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
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  shows "g -- a --> m"
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  by (rule metric_LIM_imp_LIM [OF f],
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    simp add: dist_norm le)
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lemma LIM_const_not_eq:
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  fixes a :: "'a::perfect_space"
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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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lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
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lemma LIM_const_eq:
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  fixes a :: "'a::perfect_space"
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  fixes k L :: "'b::t2_space"
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  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
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  by (simp add: tendsto_const_iff)
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lemma LIM_unique:
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  fixes a :: "'a::perfect_space"
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  fixes L M :: "'b::t2_space"
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  shows "\<lbrakk>f -- a --> L; f -- a --> M\<rbrakk> \<Longrightarrow> L = M"
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  using at_neq_bot by (rule tendsto_unique)
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   136
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   137
text{*Limits are equal for functions equal except at limit point*}
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lemma LIM_equal:
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   139
     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
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   140
unfolding tendsto_def eventually_at_topological by simp
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   141
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   142
lemma LIM_cong:
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   143
  "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
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   144
   \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
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   145
by (simp add: LIM_equal)
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   146
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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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   150
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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   151
apply (rule topological_tendstoI)
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   152
apply (drule (2) topological_tendstoD)
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   153
apply (simp add: eventually_at, safe)
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   154
apply (rule_tac x="min d R" in exI, safe)
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   155
apply (simp add: 1)
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   156
apply (simp add: 2)
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done
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   158
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   159
lemma LIM_equal2:
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   160
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
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   161
  assumes 1: "0 < R"
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   162
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
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   163
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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   164
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
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   165
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   166
lemma LIM_compose_eventually:
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   167
  assumes f: "f -- a --> b"
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   168
  assumes g: "g -- b --> c"
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   169
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
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   170
  shows "(\<lambda>x. g (f x)) -- a --> c"
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   171
  using g f inj by (rule tendsto_compose_eventually)
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   172
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   173
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"
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   178
  using g f inj [folded eventually_at]
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   179
  by (rule tendsto_compose_eventually)
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   180
23040
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   181
lemma LIM_compose2:
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   182
  fixes a :: "'a::real_normed_vector"
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   183
  assumes f: "f -- a --> b"
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   184
  assumes g: "g -- b --> c"
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   185
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
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   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]])
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   188
21239
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   189
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
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   190
  unfolding o_def by (rule tendsto_compose)
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   191
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   192
lemma real_LIM_sandwich_zero:
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   193
  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
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   207
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   208
subsection {* Continuity *}
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   209
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   210
lemma LIM_isCont_iff:
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   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)"
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   213
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
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diff changeset
   214
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   215
lemma isCont_iff:
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   216
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
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diff changeset
   217
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
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d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   218
by (simp add: isCont_def LIM_isCont_iff)
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diff changeset
   219
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   220
lemma isCont_ident [simp]: "isCont (\<lambda>x. x) a"
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   221
  unfolding isCont_def by (rule tendsto_ident_at)
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   222
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   223
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
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   224
  unfolding isCont_def by (rule tendsto_const)
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d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   225
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   226
lemma isCont_norm [simp]:
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   227
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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diff changeset
   228
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
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   229
  unfolding isCont_def by (rule tendsto_norm)
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   230
44233
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   231
lemma isCont_rabs [simp]:
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   232
  fixes f :: "'a::topological_space \<Rightarrow> real"
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   233
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
44314
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diff changeset
   234
  unfolding isCont_def by (rule tendsto_rabs)
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2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
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   235
44233
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   236
lemma isCont_add [simp]:
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   237
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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diff changeset
   238
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
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   239
  unfolding isCont_def by (rule tendsto_add)
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d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   240
44233
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   241
lemma isCont_minus [simp]:
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   242
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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diff changeset
   243
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
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   244
  unfolding isCont_def by (rule tendsto_minus)
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d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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diff changeset
   245
44233
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diff changeset
   246
lemma isCont_diff [simp]:
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   247
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
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diff changeset
   248
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
44314
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diff changeset
   249
  unfolding isCont_def by (rule tendsto_diff)
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d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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   250
44233
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   251
lemma isCont_mult [simp]:
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diff changeset
   252
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra"
21786
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parents: 21733
diff changeset
   253
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
44314
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diff changeset
   254
  unfolding isCont_def by (rule tendsto_mult)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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parents: 21165
diff changeset
   255
44233
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diff changeset
   256
lemma isCont_inverse [simp]:
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diff changeset
   257
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
21786
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parents: 21733
diff changeset
   258
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
44314
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diff changeset
   259
  unfolding isCont_def by (rule tendsto_inverse)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
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parents: 21165
diff changeset
   260
44233
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   261
lemma isCont_divide [simp]:
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   262
  fixes f g :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
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parents: 44218
diff changeset
   263
  shows "\<lbrakk>isCont f a; isCont g a; g a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x / g x) a"
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   264
  unfolding isCont_def by (rule tendsto_divide)
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parents: 44218
diff changeset
   265
44310
ba3d031b5dbb add lemma isCont_tendsto_compose
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   266
lemma isCont_tendsto_compose:
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   267
  "\<lbrakk>isCont g l; (f ---> l) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
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diff changeset
   268
  unfolding isCont_def by (rule tendsto_compose)
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parents: 44282
diff changeset
   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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diff changeset
   273
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> f x \<noteq> f a"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   274
  shows "(\<lambda>x. g (f x)) -- a --> l"
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   275
by (rule metric_LIM_compose2 [OF f g inj])
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   276
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   277
lemma isCont_LIM_compose2:
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   278
  fixes a :: "'a::real_normed_vector"
23040
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   279
  assumes f [unfolded isCont_def]: "isCont f a"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   280
  assumes g: "g -- f a --> l"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   281
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   282
  shows "(\<lambda>x. g (f x)) -- a --> l"
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   283
by (rule LIM_compose2 [OF f g inj])
d329182b5966 add lemmas LIM_compose2, isCont_LIM_compose2
huffman
parents: 23012
diff changeset
   284
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   285
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
44314
dbad46932536 Lim.thy: legacy theorems
huffman
parents: 44312
diff changeset
   286
  unfolding isCont_def by (rule tendsto_compose)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   287
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   288
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   289
  unfolding o_def by (rule isCont_o2)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   290
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   291
lemma (in bounded_linear) isCont:
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   292
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
44314
dbad46932536 Lim.thy: legacy theorems
huffman
parents: 44312
diff changeset
   293
  unfolding isCont_def by (rule tendsto)
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   294
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   295
lemma (in bounded_bilinear) isCont:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   296
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
44314
dbad46932536 Lim.thy: legacy theorems
huffman
parents: 44312
diff changeset
   297
  unfolding isCont_def by (rule tendsto)
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   298
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44254
diff changeset
   299
lemmas isCont_scaleR [simp] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44254
diff changeset
   300
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   301
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44254
diff changeset
   302
lemmas isCont_of_real [simp] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44254
diff changeset
   303
  bounded_linear.isCont [OF bounded_linear_of_real]
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   304
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   305
lemma isCont_power [simp]:
36665
5d37a96de20c generalize more lemmas about limits
huffman
parents: 36662
diff changeset
   306
  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
   307
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
44314
dbad46932536 Lim.thy: legacy theorems
huffman
parents: 44312
diff changeset
   308
  unfolding isCont_def by (rule tendsto_power)
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   309
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   310
lemma isCont_sgn [simp]:
36665
5d37a96de20c generalize more lemmas about limits
huffman
parents: 36662
diff changeset
   311
  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
   312
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. sgn (f x)) a"
44314
dbad46932536 Lim.thy: legacy theorems
huffman
parents: 44312
diff changeset
   313
  unfolding isCont_def by (rule tendsto_sgn)
29885
379e43e513c2 add lemmas about sgn
huffman
parents: 29803
diff changeset
   314
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   315
lemma isCont_setsum [simp]:
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   316
  fixes f :: "'a \<Rightarrow> 'b::topological_space \<Rightarrow> 'c::real_normed_vector"
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   317
  fixes A :: "'a set"
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   318
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   319
  unfolding isCont_def by (simp add: tendsto_setsum)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   320
44233
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   321
lemmas isCont_intros =
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   322
  isCont_ident isCont_const isCont_norm isCont_rabs isCont_add isCont_minus
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   323
  isCont_diff isCont_mult isCont_inverse isCont_divide isCont_scaleR
aa74ce315bae add simp rules for isCont
huffman
parents: 44218
diff changeset
   324
  isCont_of_real isCont_power isCont_sgn isCont_setsum
29803
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   325
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   326
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
   327
  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
   328
  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
   329
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
   330
  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
   331
  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
   332
  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
   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
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   334
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   335
  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
   336
  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
   337
    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
   338
  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
   339
  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
   340
    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
   341
  next
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   342
    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
   343
    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
   344
    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
   345
  qed
c56a5571f60a Added derivation lemmas for power series and theorems for the pi, arcus tangens and logarithm series
hoelzl
parents: 29667
diff changeset
   346
  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
   347
  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
   348
    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
   349
  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
   350
  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
   351
  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
   352
  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
   353
qed
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   354
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   355
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   356
subsection {* Uniform Continuity *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   357
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   358
lemma isUCont_isCont: "isUCont f ==> isCont f x"
23012
496b42cf588d remove dependence on Hilbert_Choice.thy
huffman
parents: 22641
diff changeset
   359
by (simp add: isUCont_def isCont_def LIM_def, force)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   360
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   361
lemma isUCont_Cauchy:
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   362
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   363
unfolding isUCont_def
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   364
apply (rule metric_CauchyI)
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   365
apply (drule_tac x=e in spec, safe)
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   366
apply (drule_tac e=s in metric_CauchyD, safe)
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   367
apply (rule_tac x=M in exI, simp)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   368
done
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   369
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   370
lemma (in bounded_linear) isUCont: "isUCont f"
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   371
unfolding isUCont_def dist_norm
23118
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   372
proof (intro allI impI)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   373
  fix r::real assume r: "0 < r"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   374
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   375
    using pos_bounded by fast
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   376
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   377
  proof (rule exI, safe)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   378
    from r K show "0 < r / K" by (rule divide_pos_pos)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   379
  next
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   380
    fix x y :: 'a
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   381
    assume xy: "norm (x - y) < r / K"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   382
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   383
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   384
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   385
    finally show "norm (f x - f y) < r" .
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   386
  qed
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   387
qed
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   388
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   389
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   390
by (rule isUCont [THEN isUCont_Cauchy])
ce3cf072ae14 add isUCont lemmas
huffman
parents: 23076
diff changeset
   391
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   392
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   393
subsection {* Relation of LIM and LIMSEQ *}
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   394
44532
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   395
lemma sequentially_imp_eventually_within:
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   396
  fixes a :: "'a::metric_space"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   397
  assumes "\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow>
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   398
    eventually (\<lambda>n. P (f n)) sequentially"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   399
  shows "eventually P (at a within s)"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   400
proof (rule ccontr)
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   401
  let ?I = "\<lambda>n. inverse (real (Suc n))"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   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"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   403
  assume "\<not> eventually P (at a within s)"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   404
  hence P: "\<forall>d>0. \<exists>x. x \<in> s \<and> x \<noteq> a \<and> dist x a < d \<and> \<not> P x"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   405
    unfolding Limits.eventually_within Limits.eventually_at by fast
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   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
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   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)"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   408
    unfolding F_def by (rule someI_ex)
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   409
  hence F0: "\<forall>n. F n \<in> s" and F1: "\<forall>n. F n \<noteq> a"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   410
    and F2: "\<forall>n. dist (F n) a < ?I n" and F3: "\<forall>n. \<not> P (F n)"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   411
    by fast+
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   412
  from LIMSEQ_inverse_real_of_nat have "F ----> a"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   413
    by (rule metric_tendsto_imp_tendsto,
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   414
      simp add: dist_norm F2 less_imp_le)
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   415
  hence "eventually (\<lambda>n. P (F n)) sequentially"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   416
    using assms F0 F1 by simp
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   417
  thus "False" by (simp add: F3)
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   418
qed
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   419
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   420
lemma sequentially_imp_eventually_at:
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   421
  fixes a :: "'a::metric_space"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   422
  assumes "\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow>
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   423
    eventually (\<lambda>n. P (f n)) sequentially"
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   424
  shows "eventually P (at a)"
45031
9583f2b56f85 add lemmas within_empty and tendsto_bot;
huffman
parents: 44571
diff changeset
   425
  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
44532
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   426
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   427
lemma LIMSEQ_SEQ_conv1:
44254
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   428
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   429
  assumes f: "f -- a --> l"
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   430
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   431
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
31338
d41a8ba25b67 generalize constants from Lim.thy to class metric_space
huffman
parents: 31336
diff changeset
   432
44254
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   433
lemma LIMSEQ_SEQ_conv2:
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   434
  fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   435
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   436
  shows "f -- a --> l"
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   437
  using assms unfolding tendsto_def [where l=l]
44532
a2e9b39df938 add lemma sequentially_imp_eventually_within;
huffman
parents: 44314
diff changeset
   438
  by (simp add: sequentially_imp_eventually_at)
44254
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   439
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   440
lemma LIMSEQ_SEQ_conv:
44254
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   441
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::metric_space) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
336dd390e4a4 Lim.thy: generalize and simplify proofs of LIM/LIMSEQ theorems
huffman
parents: 44253
diff changeset
   442
   (X -- a --> (L::'b::topological_space))"
44253
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   443
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   444
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
   445
end