src/HOL/Hyperreal/Lim.thy
author huffman
Sat Nov 04 00:12:06 2006 +0100 (2006-11-04)
changeset 21165 8fb49f668511
parent 21141 f0b5e6254a1f
child 21239 d4fbe2c87ef1
permissions -rw-r--r--
moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
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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)
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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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  NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
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            ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
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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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  isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool"
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  "isCont f a = (f -- a --> (f a))"
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  isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool"
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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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  isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool"
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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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  isNSUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool"
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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_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 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_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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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> (f -- a --> l) = (g -- b --> m)"
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by (simp add: LIM_def)
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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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subsubsection {* Purely nonstandard proofs *}
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lemma NSLIM_I:
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  "(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
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   \<Longrightarrow> f -- a --NS> L"
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by (simp add: NSLIM_def)
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lemma NSLIM_D:
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  "\<lbrakk>f -- a --NS> L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
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   \<Longrightarrow> starfun f x \<approx> star_of L"
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by (simp add: NSLIM_def)
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text{*Proving properties of limits using nonstandard definition.
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      The properties hold for standard limits as well!*}
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lemma NSLIM_mult:
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  fixes l m :: "'a::real_normed_algebra"
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  shows "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
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by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
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lemma starfun_scaleR [simp]:
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  "starfun (\<lambda>x. f x *# g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
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by transfer (rule refl)
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lemma NSLIM_scaleR:
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  "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) *# g(x)) -- x --NS> (l *# m)"
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by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
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lemma NSLIM_add:
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     "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
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by (auto simp add: NSLIM_def intro!: approx_add)
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lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
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by (simp add: NSLIM_def)
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lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
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by (simp add: NSLIM_def)
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lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
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by (simp only: NSLIM_add NSLIM_minus)
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lemma NSLIM_inverse:
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  fixes L :: "'a::real_normed_div_algebra"
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  shows "[| f -- a --NS> L;  L \<noteq> 0 |]
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      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
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apply (simp add: NSLIM_def, clarify)
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apply (drule spec)
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apply (auto simp add: star_of_approx_inverse)
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done
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lemma NSLIM_zero:
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  assumes f: "f -- a --NS> l" shows "(%x. f(x) + -l) -- a --NS> 0"
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proof -
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  have "(\<lambda>x. f x + - l) -- a --NS> l + -l"
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    by (rule NSLIM_add_minus [OF f NSLIM_const])
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  thus ?thesis by simp
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qed
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lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
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apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
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apply (auto simp add: diff_minus add_assoc)
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done
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lemma NSLIM_const_not_eq:
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  fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
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  shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
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apply (simp add: NSLIM_def)
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apply (rule_tac x="star_of a + epsilon" in exI)
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apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
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            simp add: hypreal_epsilon_not_zero)
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done
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lemma NSLIM_not_zero:
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  fixes a :: real
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  shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
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by (rule NSLIM_const_not_eq)
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lemma NSLIM_const_eq:
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  fixes a :: real
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  shows "(%x. k) -- a --NS> L ==> k = L"
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apply (rule ccontr)
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apply (blast dest: NSLIM_const_not_eq)
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done
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text{* can actually be proved more easily by unfolding the definition!*}
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lemma NSLIM_unique:
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  fixes a :: real
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  shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
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apply (drule NSLIM_minus)
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apply (drule NSLIM_add, assumption)
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apply (auto dest!: NSLIM_const_eq [symmetric])
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apply (simp add: diff_def [symmetric])
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done
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lemma NSLIM_mult_zero:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
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by (drule NSLIM_mult, auto)
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lemma NSLIM_self: "(%x. x) -- a --NS> a"
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   297
by (simp add: NSLIM_def)
huffman@20755
   298
huffman@20755
   299
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
huffman@20755
   300
huffman@20754
   301
lemma LIM_NSLIM:
huffman@20754
   302
  assumes f: "f -- a --> L" shows "f -- a --NS> L"
huffman@20754
   303
proof (rule NSLIM_I)
huffman@20754
   304
  fix x
huffman@20754
   305
  assume neq: "x \<noteq> star_of a"
huffman@20754
   306
  assume approx: "x \<approx> star_of a"
huffman@20754
   307
  have "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   308
  proof (rule InfinitesimalI2)
huffman@20754
   309
    fix r::real assume r: "0 < r"
huffman@20754
   310
    from LIM_D [OF f r]
huffman@20754
   311
    obtain s where s: "0 < s" and
huffman@20754
   312
      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
huffman@20754
   313
      by fast
huffman@20754
   314
    from less_r have less_r':
huffman@20754
   315
       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
huffman@20754
   316
        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   317
      by transfer
huffman@20754
   318
    from approx have "x - star_of a \<in> Infinitesimal"
huffman@20754
   319
      by (unfold approx_def)
huffman@20754
   320
    hence "hnorm (x - star_of a) < star_of s"
huffman@20754
   321
      using s by (rule InfinitesimalD2)
huffman@20754
   322
    with neq show "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   323
      by (rule less_r')
huffman@20754
   324
  qed
huffman@20754
   325
  thus "starfun f x \<approx> star_of L"
huffman@20754
   326
    by (unfold approx_def)
huffman@20754
   327
qed
huffman@20552
   328
huffman@20754
   329
lemma NSLIM_LIM:
huffman@20754
   330
  assumes f: "f -- a --NS> L" shows "f -- a --> L"
huffman@20754
   331
proof (rule LIM_I)
huffman@20754
   332
  fix r::real assume r: "0 < r"
huffman@20754
   333
  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
huffman@20754
   334
        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   335
  proof (rule exI, safe)
huffman@20754
   336
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   337
  next
huffman@20754
   338
    fix x assume neq: "x \<noteq> star_of a"
huffman@20754
   339
    assume "hnorm (x - star_of a) < epsilon"
huffman@20754
   340
    with Infinitesimal_epsilon
huffman@20754
   341
    have "x - star_of a \<in> Infinitesimal"
huffman@20754
   342
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   343
    hence "x \<approx> star_of a"
huffman@20754
   344
      by (unfold approx_def)
huffman@20754
   345
    with f neq have "starfun f x \<approx> star_of L"
huffman@20754
   346
      by (rule NSLIM_D)
huffman@20754
   347
    hence "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   348
      by (unfold approx_def)
huffman@20754
   349
    thus "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   350
      using r by (rule InfinitesimalD2)
huffman@20754
   351
  qed
huffman@20754
   352
  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
huffman@20754
   353
    by transfer
huffman@20754
   354
qed
paulson@14477
   355
paulson@15228
   356
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
paulson@14477
   357
by (blast intro: LIM_NSLIM NSLIM_LIM)
paulson@14477
   358
huffman@20755
   359
subsubsection {* Derived theorems about @{term LIM} *}
paulson@14477
   360
paulson@15228
   361
lemma LIM_mult2:
huffman@20552
   362
  fixes l m :: "'a::real_normed_algebra"
huffman@20552
   363
  shows "[| f -- x --> l; g -- x --> m |]
huffman@20552
   364
      ==> (%x. f(x) * g(x)) -- x --> (l * m)"
paulson@14477
   365
by (simp add: LIM_NSLIM_iff NSLIM_mult)
paulson@14477
   366
huffman@20794
   367
lemma LIM_scaleR:
huffman@20794
   368
  "[| f -- x --> l; g -- x --> m |]
huffman@20794
   369
      ==> (%x. f(x) *# g(x)) -- x --> (l *# m)"
huffman@20794
   370
by (simp add: LIM_NSLIM_iff NSLIM_scaleR)
huffman@20794
   371
paulson@15228
   372
lemma LIM_add2:
paulson@15228
   373
     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
paulson@14477
   374
by (simp add: LIM_NSLIM_iff NSLIM_add)
paulson@14477
   375
paulson@14477
   376
lemma LIM_const2: "(%x. k) -- x --> k"
paulson@14477
   377
by (simp add: LIM_NSLIM_iff)
paulson@14477
   378
paulson@14477
   379
lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
paulson@14477
   380
by (simp add: LIM_NSLIM_iff NSLIM_minus)
paulson@14477
   381
paulson@14477
   382
lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
paulson@14477
   383
by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
paulson@14477
   384
huffman@20552
   385
lemma LIM_inverse:
huffman@20653
   386
  fixes L :: "'a::real_normed_div_algebra"
huffman@20552
   387
  shows "[| f -- a --> L; L \<noteq> 0 |]
huffman@20552
   388
      ==> (%x. inverse(f(x))) -- a --> (inverse L)"
paulson@14477
   389
by (simp add: LIM_NSLIM_iff NSLIM_inverse)
paulson@14477
   390
paulson@14477
   391
lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0"
paulson@14477
   392
by (simp add: LIM_NSLIM_iff NSLIM_zero)
paulson@14477
   393
paulson@14477
   394
lemma LIM_zero_cancel: "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l"
paulson@14477
   395
apply (drule_tac g = "%x. l" and M = l in LIM_add)
paulson@14477
   396
apply (auto simp add: diff_minus add_assoc)
paulson@14477
   397
done
paulson@14477
   398
huffman@20561
   399
lemma LIM_unique2:
huffman@20561
   400
  fixes a :: real
huffman@20561
   401
  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
paulson@14477
   402
by (simp add: LIM_NSLIM_iff NSLIM_unique)
paulson@14477
   403
paulson@14477
   404
(* we can use the corresponding thm LIM_mult2 *)
paulson@14477
   405
(* for standard definition of limit           *)
paulson@14477
   406
huffman@20552
   407
lemma LIM_mult_zero2:
huffman@20561
   408
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20552
   409
  shows "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
paulson@14477
   410
by (drule LIM_mult2, auto)
paulson@14477
   411
paulson@14477
   412
huffman@20755
   413
subsection {* Continuity *}
paulson@14477
   414
paulson@14477
   415
lemma isNSContD: "[| isNSCont f a; y \<approx> hypreal_of_real a |] ==> ( *f* f) y \<approx> hypreal_of_real (f a)"
paulson@14477
   416
by (simp add: isNSCont_def)
paulson@14477
   417
paulson@14477
   418
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
paulson@14477
   419
by (simp add: isNSCont_def NSLIM_def)
paulson@14477
   420
paulson@14477
   421
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
paulson@14477
   422
apply (simp add: isNSCont_def NSLIM_def, auto)
huffman@20561
   423
apply (case_tac "y = star_of a", auto)
paulson@14477
   424
done
paulson@14477
   425
paulson@15228
   426
text{*NS continuity can be defined using NS Limit in
paulson@15228
   427
    similar fashion to standard def of continuity*}
paulson@14477
   428
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
paulson@14477
   429
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
paulson@14477
   430
paulson@15228
   431
text{*Hence, NS continuity can be given
paulson@15228
   432
  in terms of standard limit*}
paulson@14477
   433
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
paulson@14477
   434
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
paulson@14477
   435
paulson@15228
   436
text{*Moreover, it's trivial now that NS continuity
paulson@15228
   437
  is equivalent to standard continuity*}
paulson@14477
   438
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
paulson@14477
   439
apply (simp add: isCont_def)
paulson@14477
   440
apply (rule isNSCont_LIM_iff)
paulson@14477
   441
done
paulson@14477
   442
paulson@15228
   443
text{*Standard continuity ==> NS continuity*}
paulson@14477
   444
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
paulson@14477
   445
by (erule isNSCont_isCont_iff [THEN iffD2])
paulson@14477
   446
paulson@15228
   447
text{*NS continuity ==> Standard continuity*}
paulson@14477
   448
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
paulson@14477
   449
by (erule isNSCont_isCont_iff [THEN iffD1])
paulson@14477
   450
paulson@14477
   451
text{*Alternative definition of continuity*}
paulson@14477
   452
(* Prove equivalence between NS limits - *)
paulson@14477
   453
(* seems easier than using standard def  *)
paulson@14477
   454
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
paulson@14477
   455
apply (simp add: NSLIM_def, auto)
huffman@20561
   456
apply (drule_tac x = "star_of a + x" in spec)
huffman@20561
   457
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
huffman@20561
   458
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
huffman@20561
   459
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
huffman@20561
   460
 prefer 2 apply (simp add: add_commute diff_def [symmetric])
huffman@20561
   461
apply (rule_tac x = x in star_cases)
huffman@17318
   462
apply (rule_tac [2] x = x in star_cases)
huffman@17318
   463
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
paulson@14477
   464
done
paulson@14477
   465
paulson@14477
   466
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
paulson@14477
   467
by (rule NSLIM_h_iff)
paulson@14477
   468
paulson@14477
   469
lemma LIM_isCont_iff: "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))"
paulson@14477
   470
by (simp add: LIM_NSLIM_iff NSLIM_isCont_iff)
paulson@14477
   471
paulson@14477
   472
lemma isCont_iff: "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))"
paulson@14477
   473
by (simp add: isCont_def LIM_isCont_iff)
paulson@14477
   474
paulson@15228
   475
text{*Immediate application of nonstandard criterion for continuity can offer
paulson@15228
   476
   very simple proofs of some standard property of continuous functions*}
paulson@14477
   477
text{*sum continuous*}
paulson@14477
   478
lemma isCont_add: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"
paulson@14477
   479
by (auto intro: approx_add simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
paulson@14477
   480
paulson@14477
   481
text{*mult continuous*}
huffman@20552
   482
lemma isCont_mult:
huffman@20561
   483
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20552
   484
  shows "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
paulson@15228
   485
by (auto intro!: starfun_mult_HFinite_approx
paulson@15228
   486
            simp del: starfun_mult [symmetric]
paulson@14477
   487
            simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
paulson@14477
   488
paulson@15228
   489
text{*composition of continuous functions
paulson@15228
   490
     Note very short straightforard proof!*}
paulson@14477
   491
lemma isCont_o: "[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a"
paulson@14477
   492
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_def starfun_o [symmetric])
paulson@14477
   493
paulson@14477
   494
lemma isCont_o2: "[| isCont f a; isCont g (f a) |] ==> isCont (%x. g (f x)) a"
paulson@14477
   495
by (auto dest: isCont_o simp add: o_def)
paulson@14477
   496
paulson@14477
   497
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
paulson@14477
   498
by (simp add: isNSCont_def)
paulson@14477
   499
paulson@14477
   500
lemma isCont_minus: "isCont f a ==> isCont (%x. - f x) a"
paulson@14477
   501
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
paulson@14477
   502
paulson@14477
   503
lemma isCont_inverse:
huffman@20653
   504
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@20552
   505
  shows "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
paulson@14477
   506
apply (simp add: isCont_def)
paulson@14477
   507
apply (blast intro: LIM_inverse)
paulson@14477
   508
done
paulson@14477
   509
huffman@20552
   510
lemma isNSCont_inverse:
huffman@20653
   511
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@20552
   512
  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
paulson@14477
   513
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
paulson@14477
   514
paulson@14477
   515
lemma isCont_diff:
paulson@14477
   516
      "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"
paulson@14477
   517
apply (simp add: diff_minus)
paulson@14477
   518
apply (auto intro: isCont_add isCont_minus)
paulson@14477
   519
done
paulson@14477
   520
huffman@21140
   521
lemma isCont_Id: "isCont (\<lambda>x. x) a"
huffman@21140
   522
by (simp only: isCont_def LIM_self)
huffman@21140
   523
paulson@15228
   524
lemma isCont_const [simp]: "isCont (%x. k) a"
paulson@14477
   525
by (simp add: isCont_def)
paulson@14477
   526
paulson@15228
   527
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
paulson@14477
   528
by (simp add: isNSCont_def)
paulson@14477
   529
huffman@20561
   530
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
paulson@14477
   531
apply (simp add: isNSCont_def)
paulson@14477
   532
apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
paulson@14477
   533
done
paulson@14477
   534
huffman@20561
   535
lemma isCont_abs [simp]: "isCont abs (a::real)"
paulson@14477
   536
by (auto simp add: isNSCont_isCont_iff [symmetric])
paulson@15228
   537
paulson@14477
   538
paulson@14477
   539
(****************************************************************
paulson@14477
   540
(%* Leave as commented until I add topology theory or remove? *%)
paulson@14477
   541
(%*------------------------------------------------------------
paulson@14477
   542
  Elementary topology proof for a characterisation of
paulson@14477
   543
  continuity now: a function f is continuous if and only
paulson@14477
   544
  if the inverse image, {x. f(x) \<in> A}, of any open set A
paulson@14477
   545
  is always an open set
paulson@14477
   546
 ------------------------------------------------------------*%)
paulson@14477
   547
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
paulson@14477
   548
               ==> isNSopen {x. f x \<in> A}"
paulson@14477
   549
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
paulson@14477
   550
by (dtac (mem_monad_approx RS approx_sym);
paulson@14477
   551
by (dres_inst_tac [("x","a")] spec 1);
paulson@14477
   552
by (dtac isNSContD 1 THEN assume_tac 1)
paulson@14477
   553
by (dtac bspec 1 THEN assume_tac 1)
paulson@14477
   554
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
paulson@14477
   555
by (blast_tac (claset() addIs [starfun_mem_starset]);
paulson@14477
   556
qed "isNSCont_isNSopen";
paulson@14477
   557
paulson@14477
   558
Goalw [isNSCont_def]
paulson@14477
   559
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
paulson@14477
   560
\              ==> isNSCont f x";
paulson@14477
   561
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
paulson@14477
   562
     (approx_minus_iff RS iffD2)],simpset() addsimps
paulson@14477
   563
      [Infinitesimal_def,SReal_iff]));
paulson@14477
   564
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
paulson@14477
   565
by (etac (isNSopen_open_interval RSN (2,impE));
paulson@14477
   566
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
paulson@14477
   567
by (dres_inst_tac [("x","x")] spec 1);
paulson@14477
   568
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
paulson@14477
   569
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
paulson@14477
   570
qed "isNSopen_isNSCont";
paulson@14477
   571
paulson@14477
   572
Goal "(\<forall>x. isNSCont f x) = \
paulson@14477
   573
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
paulson@14477
   574
by (blast_tac (claset() addIs [isNSCont_isNSopen,
paulson@14477
   575
    isNSopen_isNSCont]);
paulson@14477
   576
qed "isNSCont_isNSopen_iff";
paulson@14477
   577
paulson@14477
   578
(%*------- Standard version of same theorem --------*%)
paulson@14477
   579
Goal "(\<forall>x. isCont f x) = \
paulson@14477
   580
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
paulson@14477
   581
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
paulson@14477
   582
              simpset() addsimps [isNSopen_isopen_iff RS sym,
paulson@14477
   583
              isNSCont_isCont_iff RS sym]));
paulson@14477
   584
qed "isCont_isopen_iff";
paulson@14477
   585
*******************************************************************)
paulson@14477
   586
huffman@20755
   587
subsection {* Uniform Continuity *}
huffman@20755
   588
paulson@14477
   589
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
paulson@14477
   590
by (simp add: isNSUCont_def)
paulson@14477
   591
paulson@14477
   592
lemma isUCont_isCont: "isUCont f ==> isCont f x"
paulson@14477
   593
by (simp add: isUCont_def isCont_def LIM_def, meson)
paulson@14477
   594
huffman@20754
   595
lemma isUCont_isNSUCont:
huffman@20754
   596
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   597
  assumes f: "isUCont f" shows "isNSUCont f"
huffman@20754
   598
proof (unfold isNSUCont_def, safe)
huffman@20754
   599
  fix x y :: "'a star"
huffman@20754
   600
  assume approx: "x \<approx> y"
huffman@20754
   601
  have "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   602
  proof (rule InfinitesimalI2)
huffman@20754
   603
    fix r::real assume r: "0 < r"
huffman@20754
   604
    with f obtain s where s: "0 < s" and
huffman@20754
   605
      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
huffman@20754
   606
      by (auto simp add: isUCont_def)
huffman@20754
   607
    from less_r have less_r':
huffman@20754
   608
       "\<And>x y. hnorm (x - y) < star_of s
huffman@20754
   609
        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   610
      by transfer
huffman@20754
   611
    from approx have "x - y \<in> Infinitesimal"
huffman@20754
   612
      by (unfold approx_def)
huffman@20754
   613
    hence "hnorm (x - y) < star_of s"
huffman@20754
   614
      using s by (rule InfinitesimalD2)
huffman@20754
   615
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   616
      by (rule less_r')
huffman@20754
   617
  qed
huffman@20754
   618
  thus "starfun f x \<approx> starfun f y"
huffman@20754
   619
    by (unfold approx_def)
huffman@20754
   620
qed
paulson@14477
   621
paulson@14477
   622
lemma isNSUCont_isUCont:
huffman@20754
   623
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   624
  assumes f: "isNSUCont f" shows "isUCont f"
huffman@20754
   625
proof (unfold isUCont_def, safe)
huffman@20754
   626
  fix r::real assume r: "0 < r"
huffman@20754
   627
  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
huffman@20754
   628
        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   629
  proof (rule exI, safe)
huffman@20754
   630
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   631
  next
huffman@20754
   632
    fix x y :: "'a star"
huffman@20754
   633
    assume "hnorm (x - y) < epsilon"
huffman@20754
   634
    with Infinitesimal_epsilon
huffman@20754
   635
    have "x - y \<in> Infinitesimal"
huffman@20754
   636
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   637
    hence "x \<approx> y"
huffman@20754
   638
      by (unfold approx_def)
huffman@20754
   639
    with f have "starfun f x \<approx> starfun f y"
huffman@20754
   640
      by (simp add: isNSUCont_def)
huffman@20754
   641
    hence "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   642
      by (unfold approx_def)
huffman@20754
   643
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   644
      using r by (rule InfinitesimalD2)
huffman@20754
   645
  qed
huffman@20754
   646
  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@20754
   647
    by transfer
huffman@20754
   648
qed
paulson@14477
   649
huffman@20805
   650
lemma LIM_compose:
huffman@21165
   651
  assumes g: "isCont g l"
huffman@20805
   652
  assumes f: "f -- a --> l"
huffman@20805
   653
  shows "(\<lambda>x. g (f x)) -- a --> g l"
huffman@20805
   654
proof (rule LIM_I)
huffman@20805
   655
  fix r::real assume r: "0 < r"
huffman@20805
   656
  obtain s where s: "0 < s"
huffman@20805
   657
    and less_r: "\<And>y. \<lbrakk>y \<noteq> l; norm (y - l) < s\<rbrakk> \<Longrightarrow> norm (g y - g l) < r"
huffman@20805
   658
    using LIM_D [OF g [unfolded isCont_def] r] by fast
huffman@20805
   659
  obtain t where t: "0 < t"
huffman@20805
   660
    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (f x - l) < s"
huffman@20805
   661
    using LIM_D [OF f s] by fast
huffman@20805
   662
huffman@20805
   663
  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (g (f x) - g l) < r"
huffman@20805
   664
  proof (rule exI, safe)
huffman@20805
   665
    show "0 < t" using t .
huffman@20805
   666
  next
huffman@20805
   667
    fix x assume "x \<noteq> a" and "norm (x - a) < t"
huffman@20805
   668
    hence "norm (f x - l) < s" by (rule less_s)
huffman@20805
   669
    thus "norm (g (f x) - g l) < r"
huffman@20805
   670
      using r less_r by (case_tac "f x = l", simp_all)
huffman@20805
   671
  qed
huffman@20805
   672
qed
huffman@20805
   673
huffman@21165
   674
subsection {* Relation of LIM and LIMSEQ *}
kleing@19023
   675
kleing@19023
   676
lemma LIMSEQ_SEQ_conv1:
huffman@21165
   677
  fixes a :: "'a::real_normed_vector"
huffman@21165
   678
  assumes X: "X -- a --> L"
kleing@19023
   679
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
huffman@21165
   680
proof (safe intro!: LIMSEQ_I)
huffman@21165
   681
  fix S :: "nat \<Rightarrow> 'a"
huffman@21165
   682
  fix r :: real
huffman@21165
   683
  assume rgz: "0 < r"
huffman@21165
   684
  assume as: "\<forall>n. S n \<noteq> a"
huffman@21165
   685
  assume S: "S ----> a"
huffman@21165
   686
  from LIM_D [OF X rgz] obtain s
huffman@21165
   687
    where sgz: "0 < s"
huffman@21165
   688
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
huffman@21165
   689
    by fast
huffman@21165
   690
  from LIMSEQ_D [OF S sgz]
huffman@21165
   691
  obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by fast
huffman@21165
   692
  hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
huffman@21165
   693
  thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
kleing@19023
   694
qed
kleing@19023
   695
kleing@19023
   696
lemma LIMSEQ_SEQ_conv2:
huffman@20561
   697
  fixes a :: real
kleing@19023
   698
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   699
  shows "X -- a --> L"
kleing@19023
   700
proof (rule ccontr)
kleing@19023
   701
  assume "\<not> (X -- a --> L)"
huffman@20563
   702
  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
   703
  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
   704
  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
   705
  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
   706
huffman@20563
   707
  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
   708
  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
   709
    using rdef by simp
huffman@21165
   710
  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
   711
    by (rule someI_ex)
huffman@21165
   712
  hence F1: "\<And>n. ?F n \<noteq> a"
huffman@21165
   713
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   714
    and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
huffman@21165
   715
    by fast+
huffman@21165
   716
kleing@19023
   717
  have "?F ----> a"
huffman@21165
   718
  proof (rule LIMSEQ_I, unfold real_norm_def)
kleing@19023
   719
      fix e::real
kleing@19023
   720
      assume "0 < e"
kleing@19023
   721
        (* choose no such that inverse (real (Suc n)) < e *)
kleing@19023
   722
      have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
   723
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
huffman@21165
   724
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
huffman@21165
   725
      proof (intro exI allI impI)
kleing@19023
   726
        fix n
kleing@19023
   727
        assume mlen: "m \<le> n"
huffman@21165
   728
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
huffman@21165
   729
          by (rule F2)
huffman@21165
   730
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
kleing@19023
   731
          by auto
huffman@21165
   732
        also from nodef have
kleing@19023
   733
          "inverse (real (Suc m)) < e" .
huffman@21165
   734
        finally show "\<bar>?F n - a\<bar> < e" .
huffman@21165
   735
      qed
kleing@19023
   736
  qed
kleing@19023
   737
  
kleing@19023
   738
  moreover have "\<forall>n. ?F n \<noteq> a"
huffman@21165
   739
    by (rule allI) (rule F1)
huffman@21165
   740
kleing@19023
   741
  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
   742
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
   743
  
kleing@19023
   744
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
   745
  proof -
kleing@19023
   746
    {
kleing@19023
   747
      fix no::nat
kleing@19023
   748
      obtain n where "n = no + 1" by simp
kleing@19023
   749
      then have nolen: "no \<le> n" by simp
kleing@19023
   750
        (* 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
   751
      have "norm (X (?F n) - L) \<ge> r"
huffman@21165
   752
        by (rule F3)
huffman@21165
   753
      with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
kleing@19023
   754
    }
huffman@20563
   755
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
huffman@20563
   756
    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
   757
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
kleing@19023
   758
  qed
kleing@19023
   759
  ultimately show False by simp
kleing@19023
   760
qed
kleing@19023
   761
kleing@19023
   762
lemma LIMSEQ_SEQ_conv:
huffman@20561
   763
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@20561
   764
   (X -- a --> L)"
kleing@19023
   765
proof
kleing@19023
   766
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
   767
  show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
   768
next
kleing@19023
   769
  assume "(X -- a --> L)"
kleing@19023
   770
  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
   771
qed
kleing@19023
   772
paulson@10751
   773
end