src/HOL/Hyperreal/Lim.thy
author huffman
Wed Apr 11 04:13:06 2007 +0200 (2007-04-11)
changeset 22631 7ae5a6ab7bd6
parent 22627 2b093ba973bc
child 22637 3f158760b68f
permissions -rw-r--r--
moved nonstandard stuff from SEQ.thy into new file HSEQ.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 HSEQ
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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_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
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by (fold real_norm_def, rule LIM_norm)
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lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
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by (fold real_norm_def, rule LIM_norm_zero)
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lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
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by (fold real_norm_def, rule LIM_norm_zero_cancel)
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lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
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by (fold real_norm_def, rule LIM_norm_zero_iff)
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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_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])
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  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
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  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
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  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
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qed
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text {* Bounded Linear Operators *}
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lemma (in bounded_linear) cont: "f -- a --> f a"
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proof (rule LIM_I)
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  fix r::real assume r: "0 < r"
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  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
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    using pos_bounded by fast
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  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"
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  proof (rule exI, safe)
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    from r K show "0 < r / K" by (rule divide_pos_pos)
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  next
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    fix x assume x: "norm (x - a) < r / K"
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    have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)
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    also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)
huffman@21282
   289
    also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)
huffman@21282
   290
    finally show "norm (f x - f a) < r" .
huffman@21282
   291
  qed
huffman@21282
   292
qed
huffman@21282
   293
huffman@21282
   294
lemma (in bounded_linear) LIM:
huffman@21282
   295
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
huffman@21282
   296
by (rule LIM_compose [OF cont])
huffman@21282
   297
huffman@21282
   298
lemma (in bounded_linear) LIM_zero:
huffman@21282
   299
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
huffman@21282
   300
by (drule LIM, simp only: zero)
huffman@21282
   301
huffman@22442
   302
text {* Bounded Bilinear Operators *}
huffman@21282
   303
huffman@21282
   304
lemma (in bounded_bilinear) LIM_prod_zero:
huffman@21282
   305
  assumes f: "f -- a --> 0"
huffman@21282
   306
  assumes g: "g -- a --> 0"
huffman@21282
   307
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
huffman@21282
   308
proof (rule LIM_I)
huffman@21282
   309
  fix r::real assume r: "0 < r"
huffman@21282
   310
  obtain K where K: "0 < K"
huffman@21282
   311
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@21282
   312
    using pos_bounded by fast
huffman@21282
   313
  from K have K': "0 < inverse K"
huffman@21282
   314
    by (rule positive_imp_inverse_positive)
huffman@21282
   315
  obtain s where s: "0 < s"
huffman@21282
   316
    and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x) < r"
huffman@21282
   317
    using LIM_D [OF f r] by auto
huffman@21282
   318
  obtain t where t: "0 < t"
huffman@21282
   319
    and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"
huffman@21282
   320
    using LIM_D [OF g K'] by auto
huffman@21282
   321
  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x ** g x - 0) < r"
huffman@21282
   322
  proof (rule exI, safe)
huffman@21282
   323
    from s t show "0 < min s t" by simp
huffman@21282
   324
  next
huffman@21282
   325
    fix x assume x: "x \<noteq> a"
huffman@21282
   326
    assume "norm (x - a) < min s t"
huffman@21282
   327
    hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all
huffman@21282
   328
    from x xs have 1: "norm (f x) < r" by (rule norm_f)
huffman@21282
   329
    from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)
huffman@21282
   330
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)
huffman@21282
   331
    also from 1 2 K have "\<dots> < r * inverse K * K"
huffman@21282
   332
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)
huffman@21282
   333
    also from K have "r * inverse K * K = r" by simp
huffman@21282
   334
    finally show "norm (f x ** g x - 0) < r" by simp
huffman@21282
   335
  qed
huffman@21282
   336
qed
huffman@21282
   337
huffman@21282
   338
lemma (in bounded_bilinear) LIM_left_zero:
huffman@21282
   339
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
huffman@21282
   340
by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
huffman@21282
   341
huffman@21282
   342
lemma (in bounded_bilinear) LIM_right_zero:
huffman@21282
   343
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
huffman@21282
   344
by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
huffman@21282
   345
huffman@21282
   346
lemma (in bounded_bilinear) LIM:
huffman@21282
   347
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
huffman@21282
   348
apply (drule LIM_zero)
huffman@21282
   349
apply (drule LIM_zero)
huffman@21282
   350
apply (rule LIM_zero_cancel)
huffman@21282
   351
apply (subst prod_diff_prod)
huffman@21282
   352
apply (rule LIM_add_zero)
huffman@21282
   353
apply (rule LIM_add_zero)
huffman@21282
   354
apply (erule (1) LIM_prod_zero)
huffman@21282
   355
apply (erule LIM_left_zero)
huffman@21282
   356
apply (erule LIM_right_zero)
huffman@21282
   357
done
huffman@21282
   358
huffman@21282
   359
lemmas LIM_mult = bounded_bilinear_mult.LIM
huffman@21282
   360
huffman@21282
   361
lemmas LIM_mult_zero = bounded_bilinear_mult.LIM_prod_zero
huffman@21282
   362
huffman@21282
   363
lemmas LIM_mult_left_zero = bounded_bilinear_mult.LIM_left_zero
huffman@21282
   364
huffman@21282
   365
lemmas LIM_mult_right_zero = bounded_bilinear_mult.LIM_right_zero
huffman@21282
   366
huffman@21282
   367
lemmas LIM_scaleR = bounded_bilinear_scaleR.LIM
huffman@21282
   368
huffman@22627
   369
lemmas LIM_of_real = bounded_linear_of_real.LIM
huffman@22627
   370
huffman@22627
   371
lemma LIM_power:
huffman@22627
   372
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
huffman@22627
   373
  assumes f: "f -- a --> l"
huffman@22627
   374
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
huffman@22627
   375
by (induct n, simp, simp add: power_Suc LIM_mult f)
huffman@22627
   376
huffman@20755
   377
subsubsection {* Purely nonstandard proofs *}
paulson@14477
   378
huffman@20754
   379
lemma NSLIM_I:
huffman@20754
   380
  "(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
huffman@20754
   381
   \<Longrightarrow> f -- a --NS> L"
huffman@20754
   382
by (simp add: NSLIM_def)
paulson@14477
   383
huffman@20754
   384
lemma NSLIM_D:
huffman@20754
   385
  "\<lbrakk>f -- a --NS> L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
huffman@20754
   386
   \<Longrightarrow> starfun f x \<approx> star_of L"
huffman@20754
   387
by (simp add: NSLIM_def)
paulson@14477
   388
huffman@20755
   389
text{*Proving properties of limits using nonstandard definition.
huffman@20755
   390
      The properties hold for standard limits as well!*}
huffman@20755
   391
huffman@20755
   392
lemma NSLIM_mult:
huffman@20755
   393
  fixes l m :: "'a::real_normed_algebra"
huffman@20755
   394
  shows "[| f -- x --NS> l; g -- x --NS> m |]
huffman@20755
   395
      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
huffman@20755
   396
by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
huffman@20755
   397
huffman@20794
   398
lemma starfun_scaleR [simp]:
huffman@20794
   399
  "starfun (\<lambda>x. f x *# g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
huffman@20794
   400
by transfer (rule refl)
huffman@20794
   401
huffman@20794
   402
lemma NSLIM_scaleR:
huffman@20794
   403
  "[| f -- x --NS> l; g -- x --NS> m |]
huffman@20794
   404
      ==> (%x. f(x) *# g(x)) -- x --NS> (l *# m)"
huffman@20794
   405
by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
huffman@20794
   406
huffman@20755
   407
lemma NSLIM_add:
huffman@20755
   408
     "[| f -- x --NS> l; g -- x --NS> m |]
huffman@20755
   409
      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
huffman@20755
   410
by (auto simp add: NSLIM_def intro!: approx_add)
huffman@20755
   411
huffman@20755
   412
lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
huffman@20755
   413
by (simp add: NSLIM_def)
huffman@20755
   414
huffman@20755
   415
lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
huffman@20755
   416
by (simp add: NSLIM_def)
huffman@20755
   417
huffman@21786
   418
lemma NSLIM_diff:
huffman@21786
   419
  "\<lbrakk>f -- x --NS> l; g -- x --NS> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --NS> (l - m)"
huffman@21786
   420
by (simp only: diff_def NSLIM_add NSLIM_minus)
huffman@21786
   421
huffman@20755
   422
lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
huffman@20755
   423
by (simp only: NSLIM_add NSLIM_minus)
huffman@20755
   424
huffman@20755
   425
lemma NSLIM_inverse:
huffman@20755
   426
  fixes L :: "'a::real_normed_div_algebra"
huffman@20755
   427
  shows "[| f -- a --NS> L;  L \<noteq> 0 |]
huffman@20755
   428
      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
huffman@20755
   429
apply (simp add: NSLIM_def, clarify)
huffman@20755
   430
apply (drule spec)
huffman@20755
   431
apply (auto simp add: star_of_approx_inverse)
huffman@20755
   432
done
huffman@20755
   433
huffman@20755
   434
lemma NSLIM_zero:
huffman@21786
   435
  assumes f: "f -- a --NS> l" shows "(%x. f(x) - l) -- a --NS> 0"
huffman@20755
   436
proof -
huffman@21786
   437
  have "(\<lambda>x. f x - l) -- a --NS> l - l"
huffman@21786
   438
    by (rule NSLIM_diff [OF f NSLIM_const])
huffman@20755
   439
  thus ?thesis by simp
huffman@20755
   440
qed
huffman@20755
   441
huffman@20755
   442
lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
huffman@20755
   443
apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
huffman@20755
   444
apply (auto simp add: diff_minus add_assoc)
huffman@20755
   445
done
huffman@20755
   446
huffman@20755
   447
lemma NSLIM_const_not_eq:
huffman@20755
   448
  fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
huffman@20755
   449
  shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
huffman@20755
   450
apply (simp add: NSLIM_def)
huffman@20755
   451
apply (rule_tac x="star_of a + epsilon" in exI)
huffman@20755
   452
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
huffman@20755
   453
            simp add: hypreal_epsilon_not_zero)
huffman@20755
   454
done
huffman@20755
   455
huffman@20755
   456
lemma NSLIM_not_zero:
huffman@20755
   457
  fixes a :: real
huffman@20755
   458
  shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
huffman@20755
   459
by (rule NSLIM_const_not_eq)
huffman@20755
   460
huffman@20755
   461
lemma NSLIM_const_eq:
huffman@20755
   462
  fixes a :: real
huffman@20755
   463
  shows "(%x. k) -- a --NS> L ==> k = L"
huffman@20755
   464
apply (rule ccontr)
huffman@20755
   465
apply (blast dest: NSLIM_const_not_eq)
huffman@20755
   466
done
huffman@20755
   467
huffman@20755
   468
text{* can actually be proved more easily by unfolding the definition!*}
huffman@20755
   469
lemma NSLIM_unique:
huffman@20755
   470
  fixes a :: real
huffman@20755
   471
  shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
huffman@20755
   472
apply (drule NSLIM_minus)
huffman@20755
   473
apply (drule NSLIM_add, assumption)
huffman@20755
   474
apply (auto dest!: NSLIM_const_eq [symmetric])
huffman@20755
   475
apply (simp add: diff_def [symmetric])
huffman@20755
   476
done
huffman@20755
   477
huffman@20755
   478
lemma NSLIM_mult_zero:
huffman@20755
   479
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20755
   480
  shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
huffman@20755
   481
by (drule NSLIM_mult, auto)
huffman@20755
   482
huffman@20755
   483
lemma NSLIM_self: "(%x. x) -- a --NS> a"
huffman@20755
   484
by (simp add: NSLIM_def)
huffman@20755
   485
huffman@20755
   486
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
huffman@20755
   487
huffman@20754
   488
lemma LIM_NSLIM:
huffman@20754
   489
  assumes f: "f -- a --> L" shows "f -- a --NS> L"
huffman@20754
   490
proof (rule NSLIM_I)
huffman@20754
   491
  fix x
huffman@20754
   492
  assume neq: "x \<noteq> star_of a"
huffman@20754
   493
  assume approx: "x \<approx> star_of a"
huffman@20754
   494
  have "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   495
  proof (rule InfinitesimalI2)
huffman@20754
   496
    fix r::real assume r: "0 < r"
huffman@20754
   497
    from LIM_D [OF f r]
huffman@20754
   498
    obtain s where s: "0 < s" and
huffman@20754
   499
      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
huffman@20754
   500
      by fast
huffman@20754
   501
    from less_r have less_r':
huffman@20754
   502
       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
huffman@20754
   503
        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   504
      by transfer
huffman@20754
   505
    from approx have "x - star_of a \<in> Infinitesimal"
huffman@20754
   506
      by (unfold approx_def)
huffman@20754
   507
    hence "hnorm (x - star_of a) < star_of s"
huffman@20754
   508
      using s by (rule InfinitesimalD2)
huffman@20754
   509
    with neq show "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   510
      by (rule less_r')
huffman@20754
   511
  qed
huffman@20754
   512
  thus "starfun f x \<approx> star_of L"
huffman@20754
   513
    by (unfold approx_def)
huffman@20754
   514
qed
huffman@20552
   515
huffman@20754
   516
lemma NSLIM_LIM:
huffman@20754
   517
  assumes f: "f -- a --NS> L" shows "f -- a --> L"
huffman@20754
   518
proof (rule LIM_I)
huffman@20754
   519
  fix r::real assume r: "0 < r"
huffman@20754
   520
  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
huffman@20754
   521
        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   522
  proof (rule exI, safe)
huffman@20754
   523
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   524
  next
huffman@20754
   525
    fix x assume neq: "x \<noteq> star_of a"
huffman@20754
   526
    assume "hnorm (x - star_of a) < epsilon"
huffman@20754
   527
    with Infinitesimal_epsilon
huffman@20754
   528
    have "x - star_of a \<in> Infinitesimal"
huffman@20754
   529
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   530
    hence "x \<approx> star_of a"
huffman@20754
   531
      by (unfold approx_def)
huffman@20754
   532
    with f neq have "starfun f x \<approx> star_of L"
huffman@20754
   533
      by (rule NSLIM_D)
huffman@20754
   534
    hence "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   535
      by (unfold approx_def)
huffman@20754
   536
    thus "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   537
      using r by (rule InfinitesimalD2)
huffman@20754
   538
  qed
huffman@20754
   539
  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
huffman@20754
   540
    by transfer
huffman@20754
   541
qed
paulson@14477
   542
paulson@15228
   543
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
paulson@14477
   544
by (blast intro: LIM_NSLIM NSLIM_LIM)
paulson@14477
   545
huffman@20755
   546
subsubsection {* Derived theorems about @{term LIM} *}
paulson@14477
   547
huffman@20552
   548
lemma LIM_inverse:
huffman@20653
   549
  fixes L :: "'a::real_normed_div_algebra"
huffman@20552
   550
  shows "[| f -- a --> L; L \<noteq> 0 |]
huffman@20552
   551
      ==> (%x. inverse(f(x))) -- a --> (inverse L)"
paulson@14477
   552
by (simp add: LIM_NSLIM_iff NSLIM_inverse)
paulson@14477
   553
paulson@14477
   554
huffman@20755
   555
subsection {* Continuity *}
paulson@14477
   556
huffman@21239
   557
subsubsection {* Purely standard proofs *}
huffman@21239
   558
huffman@21239
   559
lemma LIM_isCont_iff: "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
huffman@21239
   560
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
huffman@21239
   561
huffman@21239
   562
lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
huffman@21239
   563
by (simp add: isCont_def LIM_isCont_iff)
huffman@21239
   564
huffman@21239
   565
lemma isCont_Id: "isCont (\<lambda>x. x) a"
huffman@21282
   566
  unfolding isCont_def by (rule LIM_self)
huffman@21239
   567
huffman@21786
   568
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
huffman@21282
   569
  unfolding isCont_def by (rule LIM_const)
huffman@21239
   570
huffman@21786
   571
lemma isCont_norm: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
huffman@21786
   572
  unfolding isCont_def by (rule LIM_norm)
huffman@21786
   573
huffman@22627
   574
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
huffman@22627
   575
  unfolding isCont_def by (rule LIM_rabs)
huffman@22627
   576
huffman@21239
   577
lemma isCont_add: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
huffman@21282
   578
  unfolding isCont_def by (rule LIM_add)
huffman@21239
   579
huffman@21239
   580
lemma isCont_minus: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
huffman@21282
   581
  unfolding isCont_def by (rule LIM_minus)
huffman@21239
   582
huffman@21239
   583
lemma isCont_diff: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
huffman@21282
   584
  unfolding isCont_def by (rule LIM_diff)
huffman@21239
   585
huffman@21239
   586
lemma isCont_mult:
huffman@21239
   587
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@21786
   588
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
huffman@21282
   589
  unfolding isCont_def by (rule LIM_mult)
huffman@21239
   590
huffman@21239
   591
lemma isCont_inverse:
huffman@21239
   592
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@21786
   593
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
huffman@21282
   594
  unfolding isCont_def by (rule LIM_inverse)
huffman@21239
   595
huffman@21239
   596
lemma isCont_LIM_compose:
huffman@21239
   597
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
huffman@21282
   598
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   599
huffman@21239
   600
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
huffman@21282
   601
  unfolding isCont_def by (rule LIM_compose)
huffman@21239
   602
huffman@21239
   603
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
huffman@21282
   604
  unfolding o_def by (rule isCont_o2)
huffman@21282
   605
huffman@21282
   606
lemma (in bounded_linear) isCont: "isCont f a"
huffman@21282
   607
  unfolding isCont_def by (rule cont)
huffman@21282
   608
huffman@21282
   609
lemma (in bounded_bilinear) isCont:
huffman@21282
   610
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
huffman@21282
   611
  unfolding isCont_def by (rule LIM)
huffman@21282
   612
huffman@21282
   613
lemmas isCont_scaleR = bounded_bilinear_scaleR.isCont
huffman@21239
   614
huffman@22627
   615
lemma isCont_of_real:
huffman@22627
   616
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)) a"
huffman@22627
   617
  unfolding isCont_def by (rule LIM_of_real)
huffman@22627
   618
huffman@22627
   619
lemma isCont_power:
huffman@22627
   620
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
huffman@22627
   621
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
huffman@22627
   622
  unfolding isCont_def by (rule LIM_power)
huffman@22627
   623
huffman@21239
   624
subsubsection {* Nonstandard proofs *}
huffman@21239
   625
huffman@21786
   626
lemma isNSContD:
huffman@21786
   627
  "\<lbrakk>isNSCont f a; y \<approx> star_of a\<rbrakk> \<Longrightarrow> ( *f* f) y \<approx> star_of (f a)"
paulson@14477
   628
by (simp add: isNSCont_def)
paulson@14477
   629
paulson@14477
   630
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
paulson@14477
   631
by (simp add: isNSCont_def NSLIM_def)
paulson@14477
   632
paulson@14477
   633
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
paulson@14477
   634
apply (simp add: isNSCont_def NSLIM_def, auto)
huffman@20561
   635
apply (case_tac "y = star_of a", auto)
paulson@14477
   636
done
paulson@14477
   637
paulson@15228
   638
text{*NS continuity can be defined using NS Limit in
paulson@15228
   639
    similar fashion to standard def of continuity*}
paulson@14477
   640
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
paulson@14477
   641
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
paulson@14477
   642
paulson@15228
   643
text{*Hence, NS continuity can be given
paulson@15228
   644
  in terms of standard limit*}
paulson@14477
   645
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
paulson@14477
   646
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
paulson@14477
   647
paulson@15228
   648
text{*Moreover, it's trivial now that NS continuity
paulson@15228
   649
  is equivalent to standard continuity*}
paulson@14477
   650
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
paulson@14477
   651
apply (simp add: isCont_def)
paulson@14477
   652
apply (rule isNSCont_LIM_iff)
paulson@14477
   653
done
paulson@14477
   654
paulson@15228
   655
text{*Standard continuity ==> NS continuity*}
paulson@14477
   656
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
paulson@14477
   657
by (erule isNSCont_isCont_iff [THEN iffD2])
paulson@14477
   658
paulson@15228
   659
text{*NS continuity ==> Standard continuity*}
paulson@14477
   660
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
paulson@14477
   661
by (erule isNSCont_isCont_iff [THEN iffD1])
paulson@14477
   662
paulson@14477
   663
text{*Alternative definition of continuity*}
paulson@14477
   664
(* Prove equivalence between NS limits - *)
paulson@14477
   665
(* seems easier than using standard def  *)
paulson@14477
   666
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
paulson@14477
   667
apply (simp add: NSLIM_def, auto)
huffman@20561
   668
apply (drule_tac x = "star_of a + x" in spec)
huffman@20561
   669
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
huffman@20561
   670
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
huffman@20561
   671
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
huffman@20561
   672
 prefer 2 apply (simp add: add_commute diff_def [symmetric])
huffman@20561
   673
apply (rule_tac x = x in star_cases)
huffman@17318
   674
apply (rule_tac [2] x = x in star_cases)
huffman@17318
   675
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
paulson@14477
   676
done
paulson@14477
   677
paulson@14477
   678
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
paulson@14477
   679
by (rule NSLIM_h_iff)
paulson@14477
   680
paulson@14477
   681
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
paulson@14477
   682
by (simp add: isNSCont_def)
paulson@14477
   683
huffman@20552
   684
lemma isNSCont_inverse:
huffman@20653
   685
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@20552
   686
  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
paulson@14477
   687
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
paulson@14477
   688
paulson@15228
   689
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
paulson@14477
   690
by (simp add: isNSCont_def)
paulson@14477
   691
huffman@20561
   692
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
paulson@14477
   693
apply (simp add: isNSCont_def)
huffman@21810
   694
apply (auto intro: approx_hrabs simp add: starfun_rabs_hrabs)
paulson@14477
   695
done
paulson@14477
   696
huffman@20561
   697
lemma isCont_abs [simp]: "isCont abs (a::real)"
paulson@14477
   698
by (auto simp add: isNSCont_isCont_iff [symmetric])
paulson@15228
   699
paulson@14477
   700
paulson@14477
   701
(****************************************************************
paulson@14477
   702
(%* Leave as commented until I add topology theory or remove? *%)
paulson@14477
   703
(%*------------------------------------------------------------
paulson@14477
   704
  Elementary topology proof for a characterisation of
paulson@14477
   705
  continuity now: a function f is continuous if and only
paulson@14477
   706
  if the inverse image, {x. f(x) \<in> A}, of any open set A
paulson@14477
   707
  is always an open set
paulson@14477
   708
 ------------------------------------------------------------*%)
paulson@14477
   709
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
paulson@14477
   710
               ==> isNSopen {x. f x \<in> A}"
paulson@14477
   711
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
paulson@14477
   712
by (dtac (mem_monad_approx RS approx_sym);
paulson@14477
   713
by (dres_inst_tac [("x","a")] spec 1);
paulson@14477
   714
by (dtac isNSContD 1 THEN assume_tac 1)
paulson@14477
   715
by (dtac bspec 1 THEN assume_tac 1)
paulson@14477
   716
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
paulson@14477
   717
by (blast_tac (claset() addIs [starfun_mem_starset]);
paulson@14477
   718
qed "isNSCont_isNSopen";
paulson@14477
   719
paulson@14477
   720
Goalw [isNSCont_def]
paulson@14477
   721
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
paulson@14477
   722
\              ==> isNSCont f x";
paulson@14477
   723
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
paulson@14477
   724
     (approx_minus_iff RS iffD2)],simpset() addsimps
paulson@14477
   725
      [Infinitesimal_def,SReal_iff]));
paulson@14477
   726
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
paulson@14477
   727
by (etac (isNSopen_open_interval RSN (2,impE));
paulson@14477
   728
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
paulson@14477
   729
by (dres_inst_tac [("x","x")] spec 1);
paulson@14477
   730
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
paulson@14477
   731
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
paulson@14477
   732
qed "isNSopen_isNSCont";
paulson@14477
   733
paulson@14477
   734
Goal "(\<forall>x. isNSCont f x) = \
paulson@14477
   735
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
paulson@14477
   736
by (blast_tac (claset() addIs [isNSCont_isNSopen,
paulson@14477
   737
    isNSopen_isNSCont]);
paulson@14477
   738
qed "isNSCont_isNSopen_iff";
paulson@14477
   739
paulson@14477
   740
(%*------- Standard version of same theorem --------*%)
paulson@14477
   741
Goal "(\<forall>x. isCont f x) = \
paulson@14477
   742
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
paulson@14477
   743
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
paulson@14477
   744
              simpset() addsimps [isNSopen_isopen_iff RS sym,
paulson@14477
   745
              isNSCont_isCont_iff RS sym]));
paulson@14477
   746
qed "isCont_isopen_iff";
paulson@14477
   747
*******************************************************************)
paulson@14477
   748
huffman@20755
   749
subsection {* Uniform Continuity *}
huffman@20755
   750
paulson@14477
   751
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
paulson@14477
   752
by (simp add: isNSUCont_def)
paulson@14477
   753
paulson@14477
   754
lemma isUCont_isCont: "isUCont f ==> isCont f x"
paulson@14477
   755
by (simp add: isUCont_def isCont_def LIM_def, meson)
paulson@14477
   756
huffman@20754
   757
lemma isUCont_isNSUCont:
huffman@20754
   758
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   759
  assumes f: "isUCont f" shows "isNSUCont f"
huffman@20754
   760
proof (unfold isNSUCont_def, safe)
huffman@20754
   761
  fix x y :: "'a star"
huffman@20754
   762
  assume approx: "x \<approx> y"
huffman@20754
   763
  have "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   764
  proof (rule InfinitesimalI2)
huffman@20754
   765
    fix r::real assume r: "0 < r"
huffman@20754
   766
    with f obtain s where s: "0 < s" and
huffman@20754
   767
      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
huffman@20754
   768
      by (auto simp add: isUCont_def)
huffman@20754
   769
    from less_r have less_r':
huffman@20754
   770
       "\<And>x y. hnorm (x - y) < star_of s
huffman@20754
   771
        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   772
      by transfer
huffman@20754
   773
    from approx have "x - y \<in> Infinitesimal"
huffman@20754
   774
      by (unfold approx_def)
huffman@20754
   775
    hence "hnorm (x - y) < star_of s"
huffman@20754
   776
      using s by (rule InfinitesimalD2)
huffman@20754
   777
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   778
      by (rule less_r')
huffman@20754
   779
  qed
huffman@20754
   780
  thus "starfun f x \<approx> starfun f y"
huffman@20754
   781
    by (unfold approx_def)
huffman@20754
   782
qed
paulson@14477
   783
paulson@14477
   784
lemma isNSUCont_isUCont:
huffman@20754
   785
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   786
  assumes f: "isNSUCont f" shows "isUCont f"
huffman@20754
   787
proof (unfold isUCont_def, safe)
huffman@20754
   788
  fix r::real assume r: "0 < r"
huffman@20754
   789
  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
huffman@20754
   790
        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   791
  proof (rule exI, safe)
huffman@20754
   792
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   793
  next
huffman@20754
   794
    fix x y :: "'a star"
huffman@20754
   795
    assume "hnorm (x - y) < epsilon"
huffman@20754
   796
    with Infinitesimal_epsilon
huffman@20754
   797
    have "x - y \<in> Infinitesimal"
huffman@20754
   798
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   799
    hence "x \<approx> y"
huffman@20754
   800
      by (unfold approx_def)
huffman@20754
   801
    with f have "starfun f x \<approx> starfun f y"
huffman@20754
   802
      by (simp add: isNSUCont_def)
huffman@20754
   803
    hence "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   804
      by (unfold approx_def)
huffman@20754
   805
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   806
      using r by (rule InfinitesimalD2)
huffman@20754
   807
  qed
huffman@20754
   808
  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@20754
   809
    by transfer
huffman@20754
   810
qed
paulson@14477
   811
huffman@21165
   812
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   813
kleing@19023
   814
lemma LIMSEQ_SEQ_conv1:
huffman@21165
   815
  fixes a :: "'a::real_normed_vector"
huffman@21165
   816
  assumes X: "X -- a --> L"
kleing@19023
   817
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@21165
   818
proof (safe intro!: LIMSEQ_I)
huffman@21165
   819
  fix S :: "nat \<Rightarrow> 'a"
huffman@21165
   820
  fix r :: real
huffman@21165
   821
  assume rgz: "0 < r"
huffman@21165
   822
  assume as: "\<forall>n. S n \<noteq> a"
huffman@21165
   823
  assume S: "S ----> a"
huffman@21165
   824
  from LIM_D [OF X rgz] obtain s
huffman@21165
   825
    where sgz: "0 < s"
huffman@21165
   826
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
huffman@21165
   827
    by fast
huffman@21165
   828
  from LIMSEQ_D [OF S sgz]
nipkow@21733
   829
  obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by blast
huffman@21165
   830
  hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
huffman@21165
   831
  thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
kleing@19023
   832
qed
kleing@19023
   833
kleing@19023
   834
lemma LIMSEQ_SEQ_conv2:
huffman@20561
   835
  fixes a :: real
kleing@19023
   836
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   837
  shows "X -- a --> L"
kleing@19023
   838
proof (rule ccontr)
kleing@19023
   839
  assume "\<not> (X -- a --> L)"
huffman@20563
   840
  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
   841
  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
   842
  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
   843
  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
   844
huffman@20563
   845
  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
   846
  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
   847
    using rdef by simp
huffman@21165
   848
  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
   849
    by (rule someI_ex)
huffman@21165
   850
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@21165
   851
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   852
    and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
huffman@21165
   853
    by fast+
huffman@21165
   854
kleing@19023
   855
  have "?F ----> a"
huffman@21165
   856
  proof (rule LIMSEQ_I, unfold real_norm_def)
kleing@19023
   857
      fix e::real
kleing@19023
   858
      assume "0 < e"
kleing@19023
   859
        (* choose no such that inverse (real (Suc n)) < e *)
kleing@19023
   860
      have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
   861
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
huffman@21165
   862
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
huffman@21165
   863
      proof (intro exI allI impI)
kleing@19023
   864
        fix n
kleing@19023
   865
        assume mlen: "m \<le> n"
huffman@21165
   866
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   867
          by (rule F2)
huffman@21165
   868
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
kleing@19023
   869
          by auto
huffman@21165
   870
        also from nodef have
kleing@19023
   871
          "inverse (real (Suc m)) < e" .
huffman@21165
   872
        finally show "\<bar>?F n - a\<bar> < e" .
huffman@21165
   873
      qed
kleing@19023
   874
  qed
kleing@19023
   875
  
kleing@19023
   876
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   877
    by (rule allI) (rule F1)
huffman@21165
   878
kleing@19023
   879
  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
   880
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
   881
  
kleing@19023
   882
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
   883
  proof -
kleing@19023
   884
    {
kleing@19023
   885
      fix no::nat
kleing@19023
   886
      obtain n where "n = no + 1" by simp
kleing@19023
   887
      then have nolen: "no \<le> n" by simp
kleing@19023
   888
        (* 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
   889
      have "norm (X (?F n) - L) \<ge> r"
huffman@21165
   890
        by (rule F3)
huffman@21165
   891
      with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
kleing@19023
   892
    }
huffman@20563
   893
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
huffman@20563
   894
    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
   895
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
kleing@19023
   896
  qed
kleing@19023
   897
  ultimately show False by simp
kleing@19023
   898
qed
kleing@19023
   899
kleing@19023
   900
lemma LIMSEQ_SEQ_conv:
huffman@20561
   901
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@20561
   902
   (X -- a --> L)"
kleing@19023
   903
proof
kleing@19023
   904
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   905
  show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
   906
next
kleing@19023
   907
  assume "(X -- a --> L)"
kleing@19023
   908
  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
   909
qed
kleing@19023
   910
paulson@10751
   911
end