src/HOL/Hyperreal/Lim.thy
author huffman
Wed Mar 14 21:40:28 2007 +0100 (2007-03-14)
changeset 22442 15d9ed9b5051
parent 21810 b2d23672b003
child 22613 2f119f54d150
permissions -rw-r--r--
move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
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(*  Title       : Lim.thy
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    ID          : $Id$
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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 and Nonstandard Definitions*}
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definition
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  LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
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        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
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  "f -- a --> L =
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     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s
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        --> norm (f x - L) < r)"
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definition
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  NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
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            ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60) where
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  "f -- a --NS> L =
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    (\<forall>x. (x \<noteq> star_of a & x @= star_of a --> ( *f* f) x @= star_of L))"
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definition
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  isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
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  "isCont f a = (f -- a --> (f a))"
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definition
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  isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
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    --{*NS definition dispenses with limit notions*}
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  "isNSCont f a = (\<forall>y. y @= star_of a -->
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         ( *f* f) y @= star_of (f a))"
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definition
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  isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
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  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
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definition
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  isNSUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
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  "isNSUCont f = (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
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subsection {* Limits of Functions *}
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subsubsection {* Purely standard proofs *}
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lemma LIM_eq:
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     "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 diff_minus)
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lemma LIM_I:
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     "(!!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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     "[| 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: "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
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apply (rule LIM_I)
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apply (drule_tac r="r" in LIM_D, safe)
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apply (rule_tac x="s" in exI, safe)
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apply (drule_tac x="x + k" in spec)
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apply (simp add: compare_rls)
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done
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lemma LIM_offset_zero: "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: "(\<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 (simp add: LIM_def)
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lemma LIM_add:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
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  assumes f: "f -- a --> L" and g: "g -- a --> M"
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  shows "(%x. f x + g(x)) -- a --> (L + M)"
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proof (rule LIM_I)
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  fix r :: real
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  assume r: "0 < r"
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  from LIM_D [OF f half_gt_zero [OF r]]
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  obtain fs
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    where fs:    "0 < fs"
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      and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x - L) < r/2"
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  by blast
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  from LIM_D [OF g half_gt_zero [OF r]]
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  obtain gs
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    where gs:    "0 < gs"
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      and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x - M) < r/2"
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  by blast
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  show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x + g x - (L + M)) < r"
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  proof (intro exI conjI strip)
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    show "0 < min fs gs"  by (simp add: fs gs)
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    fix x :: 'a
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    assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
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    hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
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    with fs_lt gs_lt
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    have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+
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    hence "norm (f x - L) + norm (g x - M) < r" by arith
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    thus "norm (f x + g x - (L + M)) < r"
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      by (blast intro: norm_diff_triangle_ineq order_le_less_trans)
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  qed
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qed
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lemma LIM_add_zero:
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  "\<lbrakk>f -- a --> 0; g -- a --> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. f x + g x) -- a --> 0"
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by (drule (1) LIM_add, simp)
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lemma minus_diff_minus:
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  fixes a b :: "'a::ab_group_add"
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  shows "(- a) - (- b) = - (a - b)"
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by simp
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lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
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by (simp only: LIM_eq minus_diff_minus norm_minus_cancel)
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lemma LIM_add_minus:
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    "[| 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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    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
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by (simp only: diff_minus LIM_add LIM_minus)
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lemma LIM_zero: "f -- a --> l \<Longrightarrow> (\<lambda>x. f x - l) -- a --> 0"
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by (simp add: LIM_def)
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lemma LIM_zero_cancel: "(\<lambda>x. f x - l) -- a --> 0 \<Longrightarrow> f -- a --> l"
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by (simp add: LIM_def)
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lemma LIM_zero_iff: "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
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by (simp add: LIM_def)
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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 LIM_I, drule LIM_D [OF f], safe)
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apply (rule_tac x="s" in exI, safe)
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apply (drule_tac x="x" in spec, 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_norm: "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
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by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3)
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lemma LIM_norm_zero: "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
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by (drule LIM_norm, simp)
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lemma LIM_norm_zero_cancel: "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
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by (erule LIM_imp_LIM, simp)
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lemma LIM_norm_zero_iff: "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
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by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
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lemma LIM_const_not_eq:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
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apply (simp add: LIM_eq)
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apply (rule_tac x="norm (k - L)" in exI, simp, safe)
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apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real)
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done
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lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
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lemma LIM_const_eq:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "(%x. k) -- a --> L ==> k = L"
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apply (rule ccontr)
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apply (blast dest: LIM_const_not_eq)
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done
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lemma LIM_unique:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
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apply (drule LIM_diff, assumption)
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apply (auto dest!: LIM_const_eq)
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done
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lemma LIM_mult_zero:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  assumes f: "f -- a --> 0" and g: "g -- a --> 0"
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  shows "(%x. f(x) * g(x)) -- a --> 0"
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proof (rule LIM_I, simp)
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  fix r :: real
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  assume r: "0<r"
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  from LIM_D [OF f zero_less_one]
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  obtain fs
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    where fs:    "0 < fs"
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      and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x) < 1"
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  by auto
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  from LIM_D [OF g r]
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  obtain gs
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    where gs:    "0 < gs"
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      and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x) < r"
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  by auto
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  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x * g x) < r)"
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  proof (intro exI conjI strip)
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    show "0 < min fs gs"  by (simp add: fs gs)
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    fix x :: 'a
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    assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
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    hence  "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
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    with fs_lt gs_lt
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    have "norm (f x) < 1" and "norm (g x) < r" by blast+
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    hence "norm (f x) * norm (g x) < 1*r"
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      by (rule mult_strict_mono' [OF _ _ norm_ge_zero norm_ge_zero])
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    thus "norm (f x * g x) < r"
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      by (simp add: order_le_less_trans [OF norm_mult_ineq])
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  qed
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qed
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lemma LIM_self: "(%x. x) -- a --> a"
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by (auto simp add: LIM_def)
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text{*Limits are equal for functions equal except at limit point*}
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lemma LIM_equal:
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     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
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by (simp add: LIM_def)
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lemma LIM_cong:
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  "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
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   \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
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by (simp add: LIM_def)
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lemma LIM_equal2:
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  assumes 1: "0 < R"
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  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
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  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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apply (unfold LIM_def, safe)
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apply (drule_tac x="r" in spec, safe)
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apply (rule_tac x="min s R" in exI, safe)
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apply (simp add: 1)
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apply (simp add: 2)
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done
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text{*Two uses in Hyperreal/Transcendental.ML*}
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lemma LIM_trans:
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     "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
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apply (drule LIM_add, assumption)
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apply (auto simp add: add_assoc)
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done
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lemma LIM_compose:
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  assumes g: "g -- l --> g l"
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  assumes f: "f -- a --> l"
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  shows "(\<lambda>x. g (f x)) -- a --> g l"
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proof (rule LIM_I)
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  fix r::real assume r: "0 < r"
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  obtain s where s: "0 < s"
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    and less_r: "\<And>y. \<lbrakk>y \<noteq> l; norm (y - l) < s\<rbrakk> \<Longrightarrow> norm (g y - g l) < r"
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    using LIM_D [OF g r] by fast
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  obtain t where t: "0 < t"
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    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (f x - l) < s"
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    using LIM_D [OF f s] by fast
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  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (g (f x) - g l) < r"
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  proof (rule exI, safe)
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    show "0 < t" using t .
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  next
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    fix x assume "x \<noteq> a" and "norm (x - a) < t"
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    hence "norm (f x - l) < s" by (rule less_s)
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    thus "norm (g (f x) - g l) < r"
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      using r less_r by (case_tac "f x = l", simp_all)
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  qed
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qed
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lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
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unfolding o_def by (rule LIM_compose)
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lemma real_LIM_sandwich_zero:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> real"
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  assumes f: "f -- a --> 0"
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  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
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  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
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  shows "g -- a --> 0"
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proof (rule LIM_imp_LIM [OF f])
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  fix x assume x: "x \<noteq> a"
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  have "norm (g x - 0) = g x" by (simp add: 1 x)
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  also have "g x \<le> f x" by (rule 2 [OF x])
huffman@21282
   290
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
huffman@21282
   291
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
huffman@21282
   292
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
huffman@21282
   293
qed
huffman@21282
   294
huffman@22442
   295
text {* Bounded Linear Operators *}
huffman@21282
   296
huffman@21282
   297
lemma (in bounded_linear) cont: "f -- a --> f a"
huffman@21282
   298
proof (rule LIM_I)
huffman@21282
   299
  fix r::real assume r: "0 < r"
huffman@21282
   300
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
huffman@21282
   301
    using pos_bounded by fast
huffman@21282
   302
  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"
huffman@21282
   303
  proof (rule exI, safe)
huffman@21282
   304
    from r K show "0 < r / K" by (rule divide_pos_pos)
huffman@21282
   305
  next
huffman@21282
   306
    fix x assume x: "norm (x - a) < r / K"
huffman@21282
   307
    have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)
huffman@21282
   308
    also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)
huffman@21282
   309
    also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)
huffman@21282
   310
    finally show "norm (f x - f a) < r" .
huffman@21282
   311
  qed
huffman@21282
   312
qed
huffman@21282
   313
huffman@21282
   314
lemma (in bounded_linear) LIM:
huffman@21282
   315
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
huffman@21282
   316
by (rule LIM_compose [OF cont])
huffman@21282
   317
huffman@21282
   318
lemma (in bounded_linear) LIM_zero:
huffman@21282
   319
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
huffman@21282
   320
by (drule LIM, simp only: zero)
huffman@21282
   321
huffman@22442
   322
text {* Bounded Bilinear Operators *}
huffman@21282
   323
huffman@21282
   324
lemma (in bounded_bilinear) LIM_prod_zero:
huffman@21282
   325
  assumes f: "f -- a --> 0"
huffman@21282
   326
  assumes g: "g -- a --> 0"
huffman@21282
   327
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
huffman@21282
   328
proof (rule LIM_I)
huffman@21282
   329
  fix r::real assume r: "0 < r"
huffman@21282
   330
  obtain K where K: "0 < K"
huffman@21282
   331
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@21282
   332
    using pos_bounded by fast
huffman@21282
   333
  from K have K': "0 < inverse K"
huffman@21282
   334
    by (rule positive_imp_inverse_positive)
huffman@21282
   335
  obtain s where s: "0 < s"
huffman@21282
   336
    and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x) < r"
huffman@21282
   337
    using LIM_D [OF f r] by auto
huffman@21282
   338
  obtain t where t: "0 < t"
huffman@21282
   339
    and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"
huffman@21282
   340
    using LIM_D [OF g K'] by auto
huffman@21282
   341
  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x ** g x - 0) < r"
huffman@21282
   342
  proof (rule exI, safe)
huffman@21282
   343
    from s t show "0 < min s t" by simp
huffman@21282
   344
  next
huffman@21282
   345
    fix x assume x: "x \<noteq> a"
huffman@21282
   346
    assume "norm (x - a) < min s t"
huffman@21282
   347
    hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all
huffman@21282
   348
    from x xs have 1: "norm (f x) < r" by (rule norm_f)
huffman@21282
   349
    from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)
huffman@21282
   350
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)
huffman@21282
   351
    also from 1 2 K have "\<dots> < r * inverse K * K"
huffman@21282
   352
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)
huffman@21282
   353
    also from K have "r * inverse K * K = r" by simp
huffman@21282
   354
    finally show "norm (f x ** g x - 0) < r" by simp
huffman@21282
   355
  qed
huffman@21282
   356
qed
huffman@21282
   357
huffman@21282
   358
lemma (in bounded_bilinear) LIM_left_zero:
huffman@21282
   359
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
huffman@21282
   360
by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
huffman@21282
   361
huffman@21282
   362
lemma (in bounded_bilinear) LIM_right_zero:
huffman@21282
   363
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
huffman@21282
   364
by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
huffman@21282
   365
huffman@21282
   366
lemma (in bounded_bilinear) LIM:
huffman@21282
   367
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
huffman@21282
   368
apply (drule LIM_zero)
huffman@21282
   369
apply (drule LIM_zero)
huffman@21282
   370
apply (rule LIM_zero_cancel)
huffman@21282
   371
apply (subst prod_diff_prod)
huffman@21282
   372
apply (rule LIM_add_zero)
huffman@21282
   373
apply (rule LIM_add_zero)
huffman@21282
   374
apply (erule (1) LIM_prod_zero)
huffman@21282
   375
apply (erule LIM_left_zero)
huffman@21282
   376
apply (erule LIM_right_zero)
huffman@21282
   377
done
huffman@21282
   378
huffman@21282
   379
lemmas LIM_mult = bounded_bilinear_mult.LIM
huffman@21282
   380
huffman@21282
   381
lemmas LIM_mult_zero = bounded_bilinear_mult.LIM_prod_zero
huffman@21282
   382
huffman@21282
   383
lemmas LIM_mult_left_zero = bounded_bilinear_mult.LIM_left_zero
huffman@21282
   384
huffman@21282
   385
lemmas LIM_mult_right_zero = bounded_bilinear_mult.LIM_right_zero
huffman@21282
   386
huffman@21282
   387
lemmas LIM_scaleR = bounded_bilinear_scaleR.LIM
huffman@21282
   388
huffman@20755
   389
subsubsection {* Purely nonstandard proofs *}
paulson@14477
   390
huffman@20754
   391
lemma NSLIM_I:
huffman@20754
   392
  "(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
huffman@20754
   393
   \<Longrightarrow> f -- a --NS> L"
huffman@20754
   394
by (simp add: NSLIM_def)
paulson@14477
   395
huffman@20754
   396
lemma NSLIM_D:
huffman@20754
   397
  "\<lbrakk>f -- a --NS> L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
huffman@20754
   398
   \<Longrightarrow> starfun f x \<approx> star_of L"
huffman@20754
   399
by (simp add: NSLIM_def)
paulson@14477
   400
huffman@20755
   401
text{*Proving properties of limits using nonstandard definition.
huffman@20755
   402
      The properties hold for standard limits as well!*}
huffman@20755
   403
huffman@20755
   404
lemma NSLIM_mult:
huffman@20755
   405
  fixes l m :: "'a::real_normed_algebra"
huffman@20755
   406
  shows "[| f -- x --NS> l; g -- x --NS> m |]
huffman@20755
   407
      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
huffman@20755
   408
by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
huffman@20755
   409
huffman@20794
   410
lemma starfun_scaleR [simp]:
huffman@20794
   411
  "starfun (\<lambda>x. f x *# g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
huffman@20794
   412
by transfer (rule refl)
huffman@20794
   413
huffman@20794
   414
lemma NSLIM_scaleR:
huffman@20794
   415
  "[| f -- x --NS> l; g -- x --NS> m |]
huffman@20794
   416
      ==> (%x. f(x) *# g(x)) -- x --NS> (l *# m)"
huffman@20794
   417
by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
huffman@20794
   418
huffman@20755
   419
lemma NSLIM_add:
huffman@20755
   420
     "[| f -- x --NS> l; g -- x --NS> m |]
huffman@20755
   421
      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
huffman@20755
   422
by (auto simp add: NSLIM_def intro!: approx_add)
huffman@20755
   423
huffman@20755
   424
lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
huffman@20755
   425
by (simp add: NSLIM_def)
huffman@20755
   426
huffman@20755
   427
lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
huffman@20755
   428
by (simp add: NSLIM_def)
huffman@20755
   429
huffman@21786
   430
lemma NSLIM_diff:
huffman@21786
   431
  "\<lbrakk>f -- x --NS> l; g -- x --NS> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --NS> (l - m)"
huffman@21786
   432
by (simp only: diff_def NSLIM_add NSLIM_minus)
huffman@21786
   433
huffman@20755
   434
lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
huffman@20755
   435
by (simp only: NSLIM_add NSLIM_minus)
huffman@20755
   436
huffman@20755
   437
lemma NSLIM_inverse:
huffman@20755
   438
  fixes L :: "'a::real_normed_div_algebra"
huffman@20755
   439
  shows "[| f -- a --NS> L;  L \<noteq> 0 |]
huffman@20755
   440
      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
huffman@20755
   441
apply (simp add: NSLIM_def, clarify)
huffman@20755
   442
apply (drule spec)
huffman@20755
   443
apply (auto simp add: star_of_approx_inverse)
huffman@20755
   444
done
huffman@20755
   445
huffman@20755
   446
lemma NSLIM_zero:
huffman@21786
   447
  assumes f: "f -- a --NS> l" shows "(%x. f(x) - l) -- a --NS> 0"
huffman@20755
   448
proof -
huffman@21786
   449
  have "(\<lambda>x. f x - l) -- a --NS> l - l"
huffman@21786
   450
    by (rule NSLIM_diff [OF f NSLIM_const])
huffman@20755
   451
  thus ?thesis by simp
huffman@20755
   452
qed
huffman@20755
   453
huffman@20755
   454
lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
huffman@20755
   455
apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
huffman@20755
   456
apply (auto simp add: diff_minus add_assoc)
huffman@20755
   457
done
huffman@20755
   458
huffman@20755
   459
lemma NSLIM_const_not_eq:
huffman@20755
   460
  fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
huffman@20755
   461
  shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
huffman@20755
   462
apply (simp add: NSLIM_def)
huffman@20755
   463
apply (rule_tac x="star_of a + epsilon" in exI)
huffman@20755
   464
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
huffman@20755
   465
            simp add: hypreal_epsilon_not_zero)
huffman@20755
   466
done
huffman@20755
   467
huffman@20755
   468
lemma NSLIM_not_zero:
huffman@20755
   469
  fixes a :: real
huffman@20755
   470
  shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
huffman@20755
   471
by (rule NSLIM_const_not_eq)
huffman@20755
   472
huffman@20755
   473
lemma NSLIM_const_eq:
huffman@20755
   474
  fixes a :: real
huffman@20755
   475
  shows "(%x. k) -- a --NS> L ==> k = L"
huffman@20755
   476
apply (rule ccontr)
huffman@20755
   477
apply (blast dest: NSLIM_const_not_eq)
huffman@20755
   478
done
huffman@20755
   479
huffman@20755
   480
text{* can actually be proved more easily by unfolding the definition!*}
huffman@20755
   481
lemma NSLIM_unique:
huffman@20755
   482
  fixes a :: real
huffman@20755
   483
  shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
huffman@20755
   484
apply (drule NSLIM_minus)
huffman@20755
   485
apply (drule NSLIM_add, assumption)
huffman@20755
   486
apply (auto dest!: NSLIM_const_eq [symmetric])
huffman@20755
   487
apply (simp add: diff_def [symmetric])
huffman@20755
   488
done
huffman@20755
   489
huffman@20755
   490
lemma NSLIM_mult_zero:
huffman@20755
   491
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20755
   492
  shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
huffman@20755
   493
by (drule NSLIM_mult, auto)
huffman@20755
   494
huffman@20755
   495
lemma NSLIM_self: "(%x. x) -- a --NS> a"
huffman@20755
   496
by (simp add: NSLIM_def)
huffman@20755
   497
huffman@20755
   498
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
huffman@20755
   499
huffman@20754
   500
lemma LIM_NSLIM:
huffman@20754
   501
  assumes f: "f -- a --> L" shows "f -- a --NS> L"
huffman@20754
   502
proof (rule NSLIM_I)
huffman@20754
   503
  fix x
huffman@20754
   504
  assume neq: "x \<noteq> star_of a"
huffman@20754
   505
  assume approx: "x \<approx> star_of a"
huffman@20754
   506
  have "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   507
  proof (rule InfinitesimalI2)
huffman@20754
   508
    fix r::real assume r: "0 < r"
huffman@20754
   509
    from LIM_D [OF f r]
huffman@20754
   510
    obtain s where s: "0 < s" and
huffman@20754
   511
      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
huffman@20754
   512
      by fast
huffman@20754
   513
    from less_r have less_r':
huffman@20754
   514
       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
huffman@20754
   515
        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   516
      by transfer
huffman@20754
   517
    from approx have "x - star_of a \<in> Infinitesimal"
huffman@20754
   518
      by (unfold approx_def)
huffman@20754
   519
    hence "hnorm (x - star_of a) < star_of s"
huffman@20754
   520
      using s by (rule InfinitesimalD2)
huffman@20754
   521
    with neq show "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   522
      by (rule less_r')
huffman@20754
   523
  qed
huffman@20754
   524
  thus "starfun f x \<approx> star_of L"
huffman@20754
   525
    by (unfold approx_def)
huffman@20754
   526
qed
huffman@20552
   527
huffman@20754
   528
lemma NSLIM_LIM:
huffman@20754
   529
  assumes f: "f -- a --NS> L" shows "f -- a --> L"
huffman@20754
   530
proof (rule LIM_I)
huffman@20754
   531
  fix r::real assume r: "0 < r"
huffman@20754
   532
  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
huffman@20754
   533
        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   534
  proof (rule exI, safe)
huffman@20754
   535
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   536
  next
huffman@20754
   537
    fix x assume neq: "x \<noteq> star_of a"
huffman@20754
   538
    assume "hnorm (x - star_of a) < epsilon"
huffman@20754
   539
    with Infinitesimal_epsilon
huffman@20754
   540
    have "x - star_of a \<in> Infinitesimal"
huffman@20754
   541
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   542
    hence "x \<approx> star_of a"
huffman@20754
   543
      by (unfold approx_def)
huffman@20754
   544
    with f neq have "starfun f x \<approx> star_of L"
huffman@20754
   545
      by (rule NSLIM_D)
huffman@20754
   546
    hence "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   547
      by (unfold approx_def)
huffman@20754
   548
    thus "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   549
      using r by (rule InfinitesimalD2)
huffman@20754
   550
  qed
huffman@20754
   551
  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
huffman@20754
   552
    by transfer
huffman@20754
   553
qed
paulson@14477
   554
paulson@15228
   555
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
paulson@14477
   556
by (blast intro: LIM_NSLIM NSLIM_LIM)
paulson@14477
   557
huffman@20755
   558
subsubsection {* Derived theorems about @{term LIM} *}
paulson@14477
   559
paulson@15228
   560
lemma LIM_mult2:
huffman@20552
   561
  fixes l m :: "'a::real_normed_algebra"
huffman@20552
   562
  shows "[| f -- x --> l; g -- x --> m |]
huffman@20552
   563
      ==> (%x. f(x) * g(x)) -- x --> (l * m)"
paulson@14477
   564
by (simp add: LIM_NSLIM_iff NSLIM_mult)
paulson@14477
   565
huffman@20794
   566
lemma LIM_scaleR:
huffman@20794
   567
  "[| f -- x --> l; g -- x --> m |]
huffman@20794
   568
      ==> (%x. f(x) *# g(x)) -- x --> (l *# m)"
huffman@20794
   569
by (simp add: LIM_NSLIM_iff NSLIM_scaleR)
huffman@20794
   570
paulson@15228
   571
lemma LIM_add2:
paulson@15228
   572
     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
paulson@14477
   573
by (simp add: LIM_NSLIM_iff NSLIM_add)
paulson@14477
   574
paulson@14477
   575
lemma LIM_const2: "(%x. k) -- x --> k"
paulson@14477
   576
by (simp add: LIM_NSLIM_iff)
paulson@14477
   577
paulson@14477
   578
lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
paulson@14477
   579
by (simp add: LIM_NSLIM_iff NSLIM_minus)
paulson@14477
   580
paulson@14477
   581
lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
paulson@14477
   582
by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
paulson@14477
   583
huffman@20552
   584
lemma LIM_inverse:
huffman@20653
   585
  fixes L :: "'a::real_normed_div_algebra"
huffman@20552
   586
  shows "[| f -- a --> L; L \<noteq> 0 |]
huffman@20552
   587
      ==> (%x. inverse(f(x))) -- a --> (inverse L)"
paulson@14477
   588
by (simp add: LIM_NSLIM_iff NSLIM_inverse)
paulson@14477
   589
huffman@21786
   590
lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) - l) -- a --> 0"
paulson@14477
   591
by (simp add: LIM_NSLIM_iff NSLIM_zero)
paulson@14477
   592
huffman@20561
   593
lemma LIM_unique2:
huffman@20561
   594
  fixes a :: real
huffman@20561
   595
  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
paulson@14477
   596
by (simp add: LIM_NSLIM_iff NSLIM_unique)
paulson@14477
   597
paulson@14477
   598
(* we can use the corresponding thm LIM_mult2 *)
paulson@14477
   599
(* for standard definition of limit           *)
paulson@14477
   600
huffman@20552
   601
lemma LIM_mult_zero2:
huffman@20561
   602
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20552
   603
  shows "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
paulson@14477
   604
by (drule LIM_mult2, auto)
paulson@14477
   605
paulson@14477
   606
huffman@20755
   607
subsection {* Continuity *}
paulson@14477
   608
huffman@21239
   609
subsubsection {* Purely standard proofs *}
huffman@21239
   610
huffman@21239
   611
lemma LIM_isCont_iff: "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
huffman@21239
   612
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
huffman@21239
   613
huffman@21239
   614
lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
huffman@21239
   615
by (simp add: isCont_def LIM_isCont_iff)
huffman@21239
   616
huffman@21239
   617
lemma isCont_Id: "isCont (\<lambda>x. x) a"
huffman@21282
   618
  unfolding isCont_def by (rule LIM_self)
huffman@21239
   619
huffman@21786
   620
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
huffman@21282
   621
  unfolding isCont_def by (rule LIM_const)
huffman@21239
   622
huffman@21786
   623
lemma isCont_norm: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
huffman@21786
   624
  unfolding isCont_def by (rule LIM_norm)
huffman@21786
   625
huffman@21239
   626
lemma isCont_add: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
huffman@21282
   627
  unfolding isCont_def by (rule LIM_add)
huffman@21239
   628
huffman@21239
   629
lemma isCont_minus: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
huffman@21282
   630
  unfolding isCont_def by (rule LIM_minus)
huffman@21239
   631
huffman@21239
   632
lemma isCont_diff: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
huffman@21282
   633
  unfolding isCont_def by (rule LIM_diff)
huffman@21239
   634
huffman@21239
   635
lemma isCont_mult:
huffman@21239
   636
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@21786
   637
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
huffman@21282
   638
  unfolding isCont_def by (rule LIM_mult)
huffman@21239
   639
huffman@21239
   640
lemma isCont_inverse:
huffman@21239
   641
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@21786
   642
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
huffman@21282
   643
  unfolding isCont_def by (rule LIM_inverse)
huffman@21239
   644
huffman@21239
   645
lemma isCont_LIM_compose:
huffman@21239
   646
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
huffman@21282
   647
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   648
huffman@21239
   649
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
huffman@21282
   650
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   651
huffman@21239
   652
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
huffman@21282
   653
  unfolding o_def by (rule isCont_o2)
huffman@21282
   654
huffman@21282
   655
lemma (in bounded_linear) isCont: "isCont f a"
huffman@21282
   656
  unfolding isCont_def by (rule cont)
huffman@21282
   657
huffman@21282
   658
lemma (in bounded_bilinear) isCont:
huffman@21282
   659
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
huffman@21282
   660
  unfolding isCont_def by (rule LIM)
huffman@21282
   661
huffman@21282
   662
lemmas isCont_scaleR = bounded_bilinear_scaleR.isCont
huffman@21239
   663
huffman@21239
   664
subsubsection {* Nonstandard proofs *}
huffman@21239
   665
huffman@21786
   666
lemma isNSContD:
huffman@21786
   667
  "\<lbrakk>isNSCont f a; y \<approx> star_of a\<rbrakk> \<Longrightarrow> ( *f* f) y \<approx> star_of (f a)"
paulson@14477
   668
by (simp add: isNSCont_def)
paulson@14477
   669
paulson@14477
   670
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
paulson@14477
   671
by (simp add: isNSCont_def NSLIM_def)
paulson@14477
   672
paulson@14477
   673
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
paulson@14477
   674
apply (simp add: isNSCont_def NSLIM_def, auto)
huffman@20561
   675
apply (case_tac "y = star_of a", auto)
paulson@14477
   676
done
paulson@14477
   677
paulson@15228
   678
text{*NS continuity can be defined using NS Limit in
paulson@15228
   679
    similar fashion to standard def of continuity*}
paulson@14477
   680
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
paulson@14477
   681
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
paulson@14477
   682
paulson@15228
   683
text{*Hence, NS continuity can be given
paulson@15228
   684
  in terms of standard limit*}
paulson@14477
   685
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
paulson@14477
   686
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
paulson@14477
   687
paulson@15228
   688
text{*Moreover, it's trivial now that NS continuity
paulson@15228
   689
  is equivalent to standard continuity*}
paulson@14477
   690
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
paulson@14477
   691
apply (simp add: isCont_def)
paulson@14477
   692
apply (rule isNSCont_LIM_iff)
paulson@14477
   693
done
paulson@14477
   694
paulson@15228
   695
text{*Standard continuity ==> NS continuity*}
paulson@14477
   696
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
paulson@14477
   697
by (erule isNSCont_isCont_iff [THEN iffD2])
paulson@14477
   698
paulson@15228
   699
text{*NS continuity ==> Standard continuity*}
paulson@14477
   700
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
paulson@14477
   701
by (erule isNSCont_isCont_iff [THEN iffD1])
paulson@14477
   702
paulson@14477
   703
text{*Alternative definition of continuity*}
paulson@14477
   704
(* Prove equivalence between NS limits - *)
paulson@14477
   705
(* seems easier than using standard def  *)
paulson@14477
   706
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
paulson@14477
   707
apply (simp add: NSLIM_def, auto)
huffman@20561
   708
apply (drule_tac x = "star_of a + x" in spec)
huffman@20561
   709
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
huffman@20561
   710
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
huffman@20561
   711
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
huffman@20561
   712
 prefer 2 apply (simp add: add_commute diff_def [symmetric])
huffman@20561
   713
apply (rule_tac x = x in star_cases)
huffman@17318
   714
apply (rule_tac [2] x = x in star_cases)
huffman@17318
   715
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
paulson@14477
   716
done
paulson@14477
   717
paulson@14477
   718
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
paulson@14477
   719
by (rule NSLIM_h_iff)
paulson@14477
   720
paulson@14477
   721
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
paulson@14477
   722
by (simp add: isNSCont_def)
paulson@14477
   723
huffman@20552
   724
lemma isNSCont_inverse:
huffman@20653
   725
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@20552
   726
  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
paulson@14477
   727
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
paulson@14477
   728
paulson@15228
   729
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
paulson@14477
   730
by (simp add: isNSCont_def)
paulson@14477
   731
huffman@20561
   732
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
paulson@14477
   733
apply (simp add: isNSCont_def)
huffman@21810
   734
apply (auto intro: approx_hrabs simp add: starfun_rabs_hrabs)
paulson@14477
   735
done
paulson@14477
   736
huffman@20561
   737
lemma isCont_abs [simp]: "isCont abs (a::real)"
paulson@14477
   738
by (auto simp add: isNSCont_isCont_iff [symmetric])
paulson@15228
   739
paulson@14477
   740
paulson@14477
   741
(****************************************************************
paulson@14477
   742
(%* Leave as commented until I add topology theory or remove? *%)
paulson@14477
   743
(%*------------------------------------------------------------
paulson@14477
   744
  Elementary topology proof for a characterisation of
paulson@14477
   745
  continuity now: a function f is continuous if and only
paulson@14477
   746
  if the inverse image, {x. f(x) \<in> A}, of any open set A
paulson@14477
   747
  is always an open set
paulson@14477
   748
 ------------------------------------------------------------*%)
paulson@14477
   749
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
paulson@14477
   750
               ==> isNSopen {x. f x \<in> A}"
paulson@14477
   751
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
paulson@14477
   752
by (dtac (mem_monad_approx RS approx_sym);
paulson@14477
   753
by (dres_inst_tac [("x","a")] spec 1);
paulson@14477
   754
by (dtac isNSContD 1 THEN assume_tac 1)
paulson@14477
   755
by (dtac bspec 1 THEN assume_tac 1)
paulson@14477
   756
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
paulson@14477
   757
by (blast_tac (claset() addIs [starfun_mem_starset]);
paulson@14477
   758
qed "isNSCont_isNSopen";
paulson@14477
   759
paulson@14477
   760
Goalw [isNSCont_def]
paulson@14477
   761
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
paulson@14477
   762
\              ==> isNSCont f x";
paulson@14477
   763
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
paulson@14477
   764
     (approx_minus_iff RS iffD2)],simpset() addsimps
paulson@14477
   765
      [Infinitesimal_def,SReal_iff]));
paulson@14477
   766
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
paulson@14477
   767
by (etac (isNSopen_open_interval RSN (2,impE));
paulson@14477
   768
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
paulson@14477
   769
by (dres_inst_tac [("x","x")] spec 1);
paulson@14477
   770
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
paulson@14477
   771
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
paulson@14477
   772
qed "isNSopen_isNSCont";
paulson@14477
   773
paulson@14477
   774
Goal "(\<forall>x. isNSCont f x) = \
paulson@14477
   775
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
paulson@14477
   776
by (blast_tac (claset() addIs [isNSCont_isNSopen,
paulson@14477
   777
    isNSopen_isNSCont]);
paulson@14477
   778
qed "isNSCont_isNSopen_iff";
paulson@14477
   779
paulson@14477
   780
(%*------- Standard version of same theorem --------*%)
paulson@14477
   781
Goal "(\<forall>x. isCont f x) = \
paulson@14477
   782
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
paulson@14477
   783
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
paulson@14477
   784
              simpset() addsimps [isNSopen_isopen_iff RS sym,
paulson@14477
   785
              isNSCont_isCont_iff RS sym]));
paulson@14477
   786
qed "isCont_isopen_iff";
paulson@14477
   787
*******************************************************************)
paulson@14477
   788
huffman@20755
   789
subsection {* Uniform Continuity *}
huffman@20755
   790
paulson@14477
   791
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
paulson@14477
   792
by (simp add: isNSUCont_def)
paulson@14477
   793
paulson@14477
   794
lemma isUCont_isCont: "isUCont f ==> isCont f x"
paulson@14477
   795
by (simp add: isUCont_def isCont_def LIM_def, meson)
paulson@14477
   796
huffman@20754
   797
lemma isUCont_isNSUCont:
huffman@20754
   798
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   799
  assumes f: "isUCont f" shows "isNSUCont f"
huffman@20754
   800
proof (unfold isNSUCont_def, safe)
huffman@20754
   801
  fix x y :: "'a star"
huffman@20754
   802
  assume approx: "x \<approx> y"
huffman@20754
   803
  have "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   804
  proof (rule InfinitesimalI2)
huffman@20754
   805
    fix r::real assume r: "0 < r"
huffman@20754
   806
    with f obtain s where s: "0 < s" and
huffman@20754
   807
      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
huffman@20754
   808
      by (auto simp add: isUCont_def)
huffman@20754
   809
    from less_r have less_r':
huffman@20754
   810
       "\<And>x y. hnorm (x - y) < star_of s
huffman@20754
   811
        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   812
      by transfer
huffman@20754
   813
    from approx have "x - y \<in> Infinitesimal"
huffman@20754
   814
      by (unfold approx_def)
huffman@20754
   815
    hence "hnorm (x - y) < star_of s"
huffman@20754
   816
      using s by (rule InfinitesimalD2)
huffman@20754
   817
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   818
      by (rule less_r')
huffman@20754
   819
  qed
huffman@20754
   820
  thus "starfun f x \<approx> starfun f y"
huffman@20754
   821
    by (unfold approx_def)
huffman@20754
   822
qed
paulson@14477
   823
paulson@14477
   824
lemma isNSUCont_isUCont:
huffman@20754
   825
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   826
  assumes f: "isNSUCont f" shows "isUCont f"
huffman@20754
   827
proof (unfold isUCont_def, safe)
huffman@20754
   828
  fix r::real assume r: "0 < r"
huffman@20754
   829
  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
huffman@20754
   830
        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   831
  proof (rule exI, safe)
huffman@20754
   832
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   833
  next
huffman@20754
   834
    fix x y :: "'a star"
huffman@20754
   835
    assume "hnorm (x - y) < epsilon"
huffman@20754
   836
    with Infinitesimal_epsilon
huffman@20754
   837
    have "x - y \<in> Infinitesimal"
huffman@20754
   838
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   839
    hence "x \<approx> y"
huffman@20754
   840
      by (unfold approx_def)
huffman@20754
   841
    with f have "starfun f x \<approx> starfun f y"
huffman@20754
   842
      by (simp add: isNSUCont_def)
huffman@20754
   843
    hence "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   844
      by (unfold approx_def)
huffman@20754
   845
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   846
      using r by (rule InfinitesimalD2)
huffman@20754
   847
  qed
huffman@20754
   848
  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@20754
   849
    by transfer
huffman@20754
   850
qed
paulson@14477
   851
huffman@21165
   852
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   853
kleing@19023
   854
lemma LIMSEQ_SEQ_conv1:
huffman@21165
   855
  fixes a :: "'a::real_normed_vector"
huffman@21165
   856
  assumes X: "X -- a --> L"
kleing@19023
   857
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@21165
   858
proof (safe intro!: LIMSEQ_I)
huffman@21165
   859
  fix S :: "nat \<Rightarrow> 'a"
huffman@21165
   860
  fix r :: real
huffman@21165
   861
  assume rgz: "0 < r"
huffman@21165
   862
  assume as: "\<forall>n. S n \<noteq> a"
huffman@21165
   863
  assume S: "S ----> a"
huffman@21165
   864
  from LIM_D [OF X rgz] obtain s
huffman@21165
   865
    where sgz: "0 < s"
huffman@21165
   866
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
huffman@21165
   867
    by fast
huffman@21165
   868
  from LIMSEQ_D [OF S sgz]
nipkow@21733
   869
  obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by blast
huffman@21165
   870
  hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
huffman@21165
   871
  thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
kleing@19023
   872
qed
kleing@19023
   873
kleing@19023
   874
lemma LIMSEQ_SEQ_conv2:
huffman@20561
   875
  fixes a :: real
kleing@19023
   876
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   877
  shows "X -- a --> L"
kleing@19023
   878
proof (rule ccontr)
kleing@19023
   879
  assume "\<not> (X -- a --> L)"
huffman@20563
   880
  hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (X x - L) < r)" by (unfold LIM_def)
huffman@20563
   881
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r)" by simp
huffman@20563
   882
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r)" by (simp add: linorder_not_less)
huffman@20563
   883
  then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r))" by auto
kleing@19023
   884
huffman@20563
   885
  let ?F = "\<lambda>n::nat. SOME x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
huffman@21165
   886
  have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
huffman@21165
   887
    using rdef by simp
huffman@21165
   888
  hence F: "\<And>n. ?F n \<noteq> a \<and> \<bar>?F n - a\<bar> < inverse (real (Suc n)) \<and> norm (X (?F n) - L) \<ge> r"
huffman@21165
   889
    by (rule someI_ex)
huffman@21165
   890
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@21165
   891
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   892
    and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
huffman@21165
   893
    by fast+
huffman@21165
   894
kleing@19023
   895
  have "?F ----> a"
huffman@21165
   896
  proof (rule LIMSEQ_I, unfold real_norm_def)
kleing@19023
   897
      fix e::real
kleing@19023
   898
      assume "0 < e"
kleing@19023
   899
        (* choose no such that inverse (real (Suc n)) < e *)
kleing@19023
   900
      have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
   901
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
huffman@21165
   902
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
huffman@21165
   903
      proof (intro exI allI impI)
kleing@19023
   904
        fix n
kleing@19023
   905
        assume mlen: "m \<le> n"
huffman@21165
   906
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   907
          by (rule F2)
huffman@21165
   908
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
kleing@19023
   909
          by auto
huffman@21165
   910
        also from nodef have
kleing@19023
   911
          "inverse (real (Suc m)) < e" .
huffman@21165
   912
        finally show "\<bar>?F n - a\<bar> < e" .
huffman@21165
   913
      qed
kleing@19023
   914
  qed
kleing@19023
   915
  
kleing@19023
   916
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   917
    by (rule allI) (rule F1)
huffman@21165
   918
kleing@19023
   919
  moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
kleing@19023
   920
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
   921
  
kleing@19023
   922
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
   923
  proof -
kleing@19023
   924
    {
kleing@19023
   925
      fix no::nat
kleing@19023
   926
      obtain n where "n = no + 1" by simp
kleing@19023
   927
      then have nolen: "no \<le> n" by simp
kleing@19023
   928
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
huffman@21165
   929
      have "norm (X (?F n) - L) \<ge> r"
huffman@21165
   930
        by (rule F3)
huffman@21165
   931
      with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
kleing@19023
   932
    }
huffman@20563
   933
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
huffman@20563
   934
    with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> e)" by auto
kleing@19023
   935
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
kleing@19023
   936
  qed
kleing@19023
   937
  ultimately show False by simp
kleing@19023
   938
qed
kleing@19023
   939
kleing@19023
   940
lemma LIMSEQ_SEQ_conv:
huffman@20561
   941
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@20561
   942
   (X -- a --> L)"
kleing@19023
   943
proof
kleing@19023
   944
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   945
  show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
   946
next
kleing@19023
   947
  assume "(X -- a --> L)"
kleing@19023
   948
  show "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
kleing@19023
   949
qed
kleing@19023
   950
paulson@10751
   951
end