src/HOL/Hyperreal/Lim.thy
author huffman
Thu, 12 Apr 2007 01:53:02 +0200
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new standard proof of lemma LIM_inverse
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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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diff changeset
   141
lemma LIM_zero_iff: "(\<lambda>x. f x - l) -- a --> 0 = f -- a --> l"
700ae58d2273 add lemmas LIM_zero_iff, LIM_norm_zero_iff
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   142
by (simp add: LIM_def)
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huffman
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diff changeset
   143
21257
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   144
lemma LIM_imp_LIM:
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   145
  assumes f: "f -- a --> l"
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diff changeset
   146
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
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diff changeset
   147
  shows "g -- a --> m"
b7f090c5057d added LIM_norm and related lemmas
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diff changeset
   148
apply (rule LIM_I, drule LIM_D [OF f], safe)
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diff changeset
   149
apply (rule_tac x="s" in exI, safe)
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   150
apply (drule_tac x="x" in spec, safe)
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   151
apply (erule (1) order_le_less_trans [OF le])
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   152
done
b7f090c5057d added LIM_norm and related lemmas
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diff changeset
   153
b7f090c5057d added LIM_norm and related lemmas
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diff changeset
   154
lemma LIM_norm: "f -- a --> l \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> norm l"
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diff changeset
   155
by (erule LIM_imp_LIM, simp add: norm_triangle_ineq3)
b7f090c5057d added LIM_norm and related lemmas
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diff changeset
   156
b7f090c5057d added LIM_norm and related lemmas
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diff changeset
   157
lemma LIM_norm_zero: "f -- a --> 0 \<Longrightarrow> (\<lambda>x. norm (f x)) -- a --> 0"
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diff changeset
   158
by (drule LIM_norm, simp)
b7f090c5057d added LIM_norm and related lemmas
huffman
parents: 21239
diff changeset
   159
b7f090c5057d added LIM_norm and related lemmas
huffman
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diff changeset
   160
lemma LIM_norm_zero_cancel: "(\<lambda>x. norm (f x)) -- a --> 0 \<Longrightarrow> f -- a --> 0"
b7f090c5057d added LIM_norm and related lemmas
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diff changeset
   161
by (erule LIM_imp_LIM, simp)
b7f090c5057d added LIM_norm and related lemmas
huffman
parents: 21239
diff changeset
   162
21399
700ae58d2273 add lemmas LIM_zero_iff, LIM_norm_zero_iff
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diff changeset
   163
lemma LIM_norm_zero_iff: "(\<lambda>x. norm (f x)) -- a --> 0 = f -- a --> 0"
700ae58d2273 add lemmas LIM_zero_iff, LIM_norm_zero_iff
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parents: 21282
diff changeset
   164
by (rule iffI [OF LIM_norm_zero_cancel LIM_norm_zero])
700ae58d2273 add lemmas LIM_zero_iff, LIM_norm_zero_iff
huffman
parents: 21282
diff changeset
   165
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   166
lemma LIM_rabs: "f -- a --> (l::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> \<bar>l\<bar>"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
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diff changeset
   167
by (fold real_norm_def, rule LIM_norm)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   168
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   169
lemma LIM_rabs_zero: "f -- a --> (0::real) \<Longrightarrow> (\<lambda>x. \<bar>f x\<bar>) -- a --> 0"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   170
by (fold real_norm_def, rule LIM_norm_zero)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   171
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   172
lemma LIM_rabs_zero_cancel: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) \<Longrightarrow> f -- a --> 0"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   173
by (fold real_norm_def, rule LIM_norm_zero_cancel)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   174
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   175
lemma LIM_rabs_zero_iff: "(\<lambda>x. \<bar>f x\<bar>) -- a --> (0::real) = f -- a --> 0"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
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parents: 22613
diff changeset
   176
by (fold real_norm_def, rule LIM_norm_zero_iff)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   177
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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   178
lemma LIM_const_not_eq:
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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   179
  fixes a :: "'a::real_normed_div_algebra"
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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diff changeset
   180
  shows "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
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diff changeset
   181
apply (simp add: LIM_eq)
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
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diff changeset
   182
apply (rule_tac x="norm (k - L)" in exI, simp, safe)
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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parents: 20552
diff changeset
   183
apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real)
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
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parents: 20432
diff changeset
   184
done
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   185
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   186
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   187
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   188
lemma LIM_const_eq:
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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parents: 20552
diff changeset
   189
  fixes a :: "'a::real_normed_div_algebra"
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   190
  shows "(%x. k) -- a --> L ==> k = L"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   191
apply (rule ccontr)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   192
apply (blast dest: LIM_const_not_eq)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   193
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   194
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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diff changeset
   195
lemma LIM_unique:
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
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parents: 20552
diff changeset
   196
  fixes a :: "'a::real_normed_div_algebra"
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   197
  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   198
apply (drule LIM_diff, assumption)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   199
apply (auto dest!: LIM_const_eq)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   200
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   201
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
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diff changeset
   202
lemma LIM_self: "(%x. x) -- a --> a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
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diff changeset
   203
by (auto simp add: LIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   204
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
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diff changeset
   205
text{*Limits are equal for functions equal except at limit point*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
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diff changeset
   206
lemma LIM_equal:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   207
     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
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diff changeset
   208
by (simp add: LIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   209
20796
257e01fab4b7 generalize proofs of DERIV_isCont and DERIV_mult
huffman
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diff changeset
   210
lemma LIM_cong:
257e01fab4b7 generalize proofs of DERIV_isCont and DERIV_mult
huffman
parents: 20795
diff changeset
   211
  "\<lbrakk>a = b; \<And>x. x \<noteq> b \<Longrightarrow> f x = g x; l = m\<rbrakk>
21399
700ae58d2273 add lemmas LIM_zero_iff, LIM_norm_zero_iff
huffman
parents: 21282
diff changeset
   212
   \<Longrightarrow> ((\<lambda>x. f x) -- a --> l) = ((\<lambda>x. g x) -- b --> m)"
20796
257e01fab4b7 generalize proofs of DERIV_isCont and DERIV_mult
huffman
parents: 20795
diff changeset
   213
by (simp add: LIM_def)
257e01fab4b7 generalize proofs of DERIV_isCont and DERIV_mult
huffman
parents: 20795
diff changeset
   214
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
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diff changeset
   215
lemma LIM_equal2:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
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parents: 21257
diff changeset
   216
  assumes 1: "0 < R"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
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parents: 21257
diff changeset
   217
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   218
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   219
apply (unfold LIM_def, safe)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   220
apply (drule_tac x="r" in spec, safe)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   221
apply (rule_tac x="min s R" in exI, safe)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   222
apply (simp add: 1)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   223
apply (simp add: 2)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   224
done
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   225
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   226
text{*Two uses in Hyperreal/Transcendental.ML*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
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diff changeset
   227
lemma LIM_trans:
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diff changeset
   228
     "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   229
apply (drule LIM_add, assumption)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   230
apply (auto simp add: add_assoc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   231
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   232
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   233
lemma LIM_compose:
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   234
  assumes g: "g -- l --> g l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   235
  assumes f: "f -- a --> l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   236
  shows "(\<lambda>x. g (f x)) -- a --> g l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   237
proof (rule LIM_I)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   238
  fix r::real assume r: "0 < r"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   239
  obtain s where s: "0 < s"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   240
    and less_r: "\<And>y. \<lbrakk>y \<noteq> l; norm (y - l) < s\<rbrakk> \<Longrightarrow> norm (g y - g l) < r"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   241
    using LIM_D [OF g r] by fast
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   242
  obtain t where t: "0 < t"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   243
    and less_s: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (f x - l) < s"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   244
    using LIM_D [OF f s] by fast
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   245
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   246
  show "\<exists>t>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < t \<longrightarrow> norm (g (f x) - g l) < r"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   247
  proof (rule exI, safe)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   248
    show "0 < t" using t .
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   249
  next
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   250
    fix x assume "x \<noteq> a" and "norm (x - a) < t"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   251
    hence "norm (f x - l) < s" by (rule less_s)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   252
    thus "norm (g (f x) - g l) < r"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   253
      using r less_r by (case_tac "f x = l", simp_all)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   254
  qed
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   255
qed
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   256
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   257
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   258
unfolding o_def by (rule LIM_compose)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   259
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   260
lemma real_LIM_sandwich_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   261
  fixes f g :: "'a::real_normed_vector \<Rightarrow> real"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   262
  assumes f: "f -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   263
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   264
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   265
  shows "g -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   266
proof (rule LIM_imp_LIM [OF f])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   267
  fix x assume x: "x \<noteq> a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   268
  have "norm (g x - 0) = g x" by (simp add: 1 x)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   269
  also have "g x \<le> f x" by (rule 2 [OF x])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   270
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   271
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   272
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   273
qed
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   274
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21810
diff changeset
   275
text {* Bounded Linear Operators *}
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   276
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   277
lemma (in bounded_linear) cont: "f -- a --> f a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   278
proof (rule LIM_I)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   279
  fix r::real assume r: "0 < r"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   280
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   281
    using pos_bounded by fast
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   282
  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - f a) < r"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   283
  proof (rule exI, safe)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   284
    from r K show "0 < r / K" by (rule divide_pos_pos)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   285
  next
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   286
    fix x assume x: "norm (x - a) < r / K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   287
    have "norm (f x - f a) = norm (f (x - a))" by (simp only: diff)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   288
    also have "\<dots> \<le> norm (x - a) * K" by (rule norm_le)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   289
    also from K x have "\<dots> < r" by (simp only: pos_less_divide_eq)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   290
    finally show "norm (f x - f a) < r" .
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   291
  qed
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   292
qed
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   293
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   294
lemma (in bounded_linear) LIM:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   295
  "g -- a --> l \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> f l"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   296
by (rule LIM_compose [OF cont])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   297
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   298
lemma (in bounded_linear) LIM_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   299
  "g -- a --> 0 \<Longrightarrow> (\<lambda>x. f (g x)) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   300
by (drule LIM, simp only: zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   301
22442
15d9ed9b5051 move bounded (bi)linear operator locales from Lim.thy to RealVector.thy
huffman
parents: 21810
diff changeset
   302
text {* Bounded Bilinear Operators *}
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   303
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   304
lemma (in bounded_bilinear) LIM_prod_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   305
  assumes f: "f -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   306
  assumes g: "g -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   307
  shows "(\<lambda>x. f x ** g x) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   308
proof (rule LIM_I)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   309
  fix r::real assume r: "0 < r"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   310
  obtain K where K: "0 < K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   311
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   312
    using pos_bounded by fast
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   313
  from K have K': "0 < inverse K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   314
    by (rule positive_imp_inverse_positive)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   315
  obtain s where s: "0 < s"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   316
    and norm_f: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x) < r"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   317
    using LIM_D [OF f r] by auto
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   318
  obtain t where t: "0 < t"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   319
    and norm_g: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < t\<rbrakk> \<Longrightarrow> norm (g x) < inverse K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   320
    using LIM_D [OF g K'] by auto
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   321
  show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x ** g x - 0) < r"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   322
  proof (rule exI, safe)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   323
    from s t show "0 < min s t" by simp
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   324
  next
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   325
    fix x assume x: "x \<noteq> a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   326
    assume "norm (x - a) < min s t"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   327
    hence xs: "norm (x - a) < s" and xt: "norm (x - a) < t" by simp_all
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   328
    from x xs have 1: "norm (f x) < r" by (rule norm_f)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   329
    from x xt have 2: "norm (g x) < inverse K" by (rule norm_g)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   330
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K" by (rule norm_le)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   331
    also from 1 2 K have "\<dots> < r * inverse K * K"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   332
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   333
    also from K have "r * inverse K * K = r" by simp
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   334
    finally show "norm (f x ** g x - 0) < r" by simp
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   335
  qed
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   336
qed
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   337
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   338
lemma (in bounded_bilinear) LIM_left_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   339
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. f x ** c) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   340
by (rule bounded_linear.LIM_zero [OF bounded_linear_left])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   341
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   342
lemma (in bounded_bilinear) LIM_right_zero:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   343
  "f -- a --> 0 \<Longrightarrow> (\<lambda>x. c ** f x) -- a --> 0"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   344
by (rule bounded_linear.LIM_zero [OF bounded_linear_right])
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   345
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   346
lemma (in bounded_bilinear) LIM:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   347
  "\<lbrakk>f -- a --> L; g -- a --> M\<rbrakk> \<Longrightarrow> (\<lambda>x. f x ** g x) -- a --> L ** M"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   348
apply (drule LIM_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   349
apply (drule LIM_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   350
apply (rule LIM_zero_cancel)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   351
apply (subst prod_diff_prod)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   352
apply (rule LIM_add_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   353
apply (rule LIM_add_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   354
apply (erule (1) LIM_prod_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   355
apply (erule LIM_left_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   356
apply (erule LIM_right_zero)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   357
done
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   358
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   359
lemmas LIM_mult = bounded_bilinear_mult.LIM
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   360
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   361
lemmas LIM_mult_zero = bounded_bilinear_mult.LIM_prod_zero
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   362
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   363
lemmas LIM_mult_left_zero = bounded_bilinear_mult.LIM_left_zero
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   364
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   365
lemmas LIM_mult_right_zero = bounded_bilinear_mult.LIM_right_zero
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   366
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   367
lemmas LIM_scaleR = bounded_bilinear_scaleR.LIM
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   368
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   369
lemmas LIM_of_real = bounded_linear_of_real.LIM
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   370
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   371
lemma LIM_power:
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   372
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   373
  assumes f: "f -- a --> l"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   374
  shows "(\<lambda>x. f x ^ n) -- a --> l ^ n"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   375
by (induct n, simp, simp add: power_Suc LIM_mult f)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   376
22637
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   377
lemma LIM_inverse_lemma:
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   378
  fixes x :: "'a::real_normed_div_algebra"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   379
  assumes r: "0 < r"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   380
  assumes x: "norm (x - 1) < min (1/2) (r/2)"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   381
  shows "norm (inverse x - 1) < r"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   382
proof -
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   383
  from r have r2: "0 < r/2" by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   384
  from x have 0: "x \<noteq> 0" by clarsimp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   385
  from x have x': "norm (1 - x) < min (1/2) (r/2)"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   386
    by (simp only: norm_minus_commute)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   387
  hence less1: "norm (1 - x) < r/2" by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   388
  have "norm (1::'a) - norm x \<le> norm (1 - x)" by (rule norm_triangle_ineq2)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   389
  also from x' have "norm (1 - x) < 1/2" by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   390
  finally have "1/2 < norm x" by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   391
  hence "inverse (norm x) < inverse (1/2)"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   392
    by (rule less_imp_inverse_less, simp)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   393
  hence less2: "norm (inverse x) < 2"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   394
    by (simp add: nonzero_norm_inverse 0)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   395
  from less1 less2 r2 norm_ge_zero
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   396
  have "norm (1 - x) * norm (inverse x) < (r/2) * 2"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   397
    by (rule mult_strict_mono)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   398
  thus "norm (inverse x - 1) < r"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   399
    by (simp only: norm_mult [symmetric] left_diff_distrib, simp add: 0)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   400
qed
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   401
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   402
lemma LIM_inverse_fun:
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   403
  assumes a: "a \<noteq> (0::'a::real_normed_div_algebra)"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   404
  shows "inverse -- a --> inverse a"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   405
proof (rule LIM_equal2)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   406
  from a show "0 < norm a" by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   407
next
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   408
  fix x assume "norm (x - a) < norm a"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   409
  hence "x \<noteq> 0" by auto
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   410
  with a show "inverse x = inverse (inverse a * x) * inverse a"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   411
    by (simp add: nonzero_inverse_mult_distrib
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   412
                  nonzero_imp_inverse_nonzero
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   413
                  nonzero_inverse_inverse_eq mult_assoc)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   414
next
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   415
  have 1: "inverse -- 1 --> inverse (1::'a)"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   416
    apply (rule LIM_I)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   417
    apply (rule_tac x="min (1/2) (r/2)" in exI)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   418
    apply (simp add: LIM_inverse_lemma)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   419
    done
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   420
  have "(\<lambda>x. inverse a * x) -- a --> inverse a * a"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   421
    by (intro LIM_mult LIM_self LIM_const)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   422
  hence "(\<lambda>x. inverse a * x) -- a --> 1"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   423
    by (simp add: a)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   424
  with 1 have "(\<lambda>x. inverse (inverse a * x)) -- a --> inverse 1"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   425
    by (rule LIM_compose)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   426
  hence "(\<lambda>x. inverse (inverse a * x)) -- a --> 1"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   427
    by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   428
  hence "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> 1 * inverse a"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   429
    by (intro LIM_mult LIM_const)
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   430
  thus "(\<lambda>x. inverse (inverse a * x) * inverse a) -- a --> inverse a"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   431
    by simp
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   432
qed
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   433
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   434
lemma LIM_inverse:
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   435
  fixes L :: "'a::real_normed_div_algebra"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   436
  shows "\<lbrakk>f -- a --> L; L \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>x. inverse (f x)) -- a --> inverse L"
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   437
by (rule LIM_inverse_fun [THEN LIM_compose])
3f158760b68f new standard proof of lemma LIM_inverse
huffman
parents: 22631
diff changeset
   438
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   439
subsubsection {* Purely nonstandard proofs *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   440
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   441
lemma NSLIM_I:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   442
  "(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   443
   \<Longrightarrow> f -- a --NS> L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   444
by (simp add: NSLIM_def)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   445
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   446
lemma NSLIM_D:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   447
  "\<lbrakk>f -- a --NS> L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   448
   \<Longrightarrow> starfun f x \<approx> star_of L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   449
by (simp add: NSLIM_def)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   450
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   451
text{*Proving properties of limits using nonstandard definition.
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   452
      The properties hold for standard limits as well!*}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   453
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   454
lemma NSLIM_mult:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   455
  fixes l m :: "'a::real_normed_algebra"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   456
  shows "[| f -- x --NS> l; g -- x --NS> m |]
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   457
      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   458
by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   459
20794
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   460
lemma starfun_scaleR [simp]:
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   461
  "starfun (\<lambda>x. f x *# g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   462
by transfer (rule refl)
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   463
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   464
lemma NSLIM_scaleR:
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   465
  "[| f -- x --NS> l; g -- x --NS> m |]
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   466
      ==> (%x. f(x) *# g(x)) -- x --NS> (l *# m)"
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   467
by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
02482f9501ac add scaleR lemmas
huffman
parents: 20793
diff changeset
   468
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   469
lemma NSLIM_add:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   470
     "[| f -- x --NS> l; g -- x --NS> m |]
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   471
      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   472
by (auto simp add: NSLIM_def intro!: approx_add)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   473
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   474
lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   475
by (simp add: NSLIM_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   476
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   477
lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   478
by (simp add: NSLIM_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   479
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   480
lemma NSLIM_diff:
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   481
  "\<lbrakk>f -- x --NS> l; g -- x --NS> m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) -- x --NS> (l - m)"
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   482
by (simp only: diff_def NSLIM_add NSLIM_minus)
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   483
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   484
lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   485
by (simp only: NSLIM_add NSLIM_minus)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   486
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   487
lemma NSLIM_inverse:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   488
  fixes L :: "'a::real_normed_div_algebra"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   489
  shows "[| f -- a --NS> L;  L \<noteq> 0 |]
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   490
      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   491
apply (simp add: NSLIM_def, clarify)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   492
apply (drule spec)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   493
apply (auto simp add: star_of_approx_inverse)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   494
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   495
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   496
lemma NSLIM_zero:
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   497
  assumes f: "f -- a --NS> l" shows "(%x. f(x) - l) -- a --NS> 0"
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   498
proof -
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   499
  have "(\<lambda>x. f x - l) -- a --NS> l - l"
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   500
    by (rule NSLIM_diff [OF f NSLIM_const])
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   501
  thus ?thesis by simp
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   502
qed
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   503
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   504
lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   505
apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   506
apply (auto simp add: diff_minus add_assoc)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   507
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   508
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   509
lemma NSLIM_const_not_eq:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   510
  fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   511
  shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   512
apply (simp add: NSLIM_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   513
apply (rule_tac x="star_of a + epsilon" in exI)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   514
apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   515
            simp add: hypreal_epsilon_not_zero)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   516
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   517
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   518
lemma NSLIM_not_zero:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   519
  fixes a :: real
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   520
  shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   521
by (rule NSLIM_const_not_eq)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   522
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   523
lemma NSLIM_const_eq:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   524
  fixes a :: real
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   525
  shows "(%x. k) -- a --NS> L ==> k = L"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   526
apply (rule ccontr)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   527
apply (blast dest: NSLIM_const_not_eq)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   528
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   529
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   530
text{* can actually be proved more easily by unfolding the definition!*}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   531
lemma NSLIM_unique:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   532
  fixes a :: real
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   533
  shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   534
apply (drule NSLIM_minus)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   535
apply (drule NSLIM_add, assumption)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   536
apply (auto dest!: NSLIM_const_eq [symmetric])
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   537
apply (simp add: diff_def [symmetric])
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   538
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   539
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   540
lemma NSLIM_mult_zero:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   541
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   542
  shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   543
by (drule NSLIM_mult, auto)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   544
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   545
lemma NSLIM_self: "(%x. x) -- a --NS> a"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   546
by (simp add: NSLIM_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   547
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   548
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   549
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   550
lemma LIM_NSLIM:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   551
  assumes f: "f -- a --> L" shows "f -- a --NS> L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   552
proof (rule NSLIM_I)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   553
  fix x
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   554
  assume neq: "x \<noteq> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   555
  assume approx: "x \<approx> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   556
  have "starfun f x - star_of L \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   557
  proof (rule InfinitesimalI2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   558
    fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   559
    from LIM_D [OF f r]
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   560
    obtain s where s: "0 < s" and
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   561
      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   562
      by fast
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   563
    from less_r have less_r':
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   564
       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   565
        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   566
      by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   567
    from approx have "x - star_of a \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   568
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   569
    hence "hnorm (x - star_of a) < star_of s"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   570
      using s by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   571
    with neq show "hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   572
      by (rule less_r')
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   573
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   574
  thus "starfun f x \<approx> star_of L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   575
    by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   576
qed
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   577
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   578
lemma NSLIM_LIM:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   579
  assumes f: "f -- a --NS> L" shows "f -- a --> L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   580
proof (rule LIM_I)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   581
  fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   582
  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   583
        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   584
  proof (rule exI, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   585
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   586
  next
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   587
    fix x assume neq: "x \<noteq> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   588
    assume "hnorm (x - star_of a) < epsilon"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   589
    with Infinitesimal_epsilon
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   590
    have "x - star_of a \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   591
      by (rule hnorm_less_Infinitesimal)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   592
    hence "x \<approx> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   593
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   594
    with f neq have "starfun f x \<approx> star_of L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   595
      by (rule NSLIM_D)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   596
    hence "starfun f x - star_of L \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   597
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   598
    thus "hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   599
      using r by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   600
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   601
  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   602
    by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   603
qed
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   604
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   605
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   606
by (blast intro: LIM_NSLIM NSLIM_LIM)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   607
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   608
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   609
subsection {* Continuity *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   610
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   611
subsubsection {* Purely standard proofs *}
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   612
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   613
lemma LIM_isCont_iff: "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   614
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   615
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   616
lemma isCont_iff: "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   617
by (simp add: isCont_def LIM_isCont_iff)
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   618
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   619
lemma isCont_Id: "isCont (\<lambda>x. x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   620
  unfolding isCont_def by (rule LIM_self)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   621
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   622
lemma isCont_const [simp]: "isCont (\<lambda>x. k) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   623
  unfolding isCont_def by (rule LIM_const)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   624
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   625
lemma isCont_norm: "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   626
  unfolding isCont_def by (rule LIM_norm)
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   627
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   628
lemma isCont_rabs: "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x :: real\<bar>) a"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   629
  unfolding isCont_def by (rule LIM_rabs)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   630
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   631
lemma isCont_add: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   632
  unfolding isCont_def by (rule LIM_add)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   633
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   634
lemma isCont_minus: "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   635
  unfolding isCont_def by (rule LIM_minus)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   636
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   637
lemma isCont_diff: "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   638
  unfolding isCont_def by (rule LIM_diff)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   639
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   640
lemma isCont_mult:
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   641
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   642
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   643
  unfolding isCont_def by (rule LIM_mult)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   644
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   645
lemma isCont_inverse:
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   646
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   647
  shows "\<lbrakk>isCont f a; f a \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. inverse (f x)) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   648
  unfolding isCont_def by (rule LIM_inverse)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   649
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   650
lemma isCont_LIM_compose:
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   651
  "\<lbrakk>isCont g l; f -- a --> l\<rbrakk> \<Longrightarrow> (\<lambda>x. g (f x)) -- a --> g l"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   652
  unfolding isCont_def by (rule LIM_compose)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   653
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   654
lemma isCont_o2: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   655
  unfolding isCont_def by (rule LIM_compose)
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   656
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   657
lemma isCont_o: "\<lbrakk>isCont f a; isCont g (f a)\<rbrakk> \<Longrightarrow> isCont (g o f) a"
21282
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   658
  unfolding o_def by (rule isCont_o2)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   659
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   660
lemma (in bounded_linear) isCont: "isCont f a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   661
  unfolding isCont_def by (rule cont)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   662
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   663
lemma (in bounded_bilinear) isCont:
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   664
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   665
  unfolding isCont_def by (rule LIM)
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   666
dd647b4d7952 added bounded_linear and bounded_bilinear locales
huffman
parents: 21257
diff changeset
   667
lemmas isCont_scaleR = bounded_bilinear_scaleR.isCont
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   668
22627
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   669
lemma isCont_of_real:
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   670
  "isCont f a \<Longrightarrow> isCont (\<lambda>x. of_real (f x)) a"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   671
  unfolding isCont_def by (rule LIM_of_real)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   672
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   673
lemma isCont_power:
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   674
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::{recpower,real_normed_algebra}"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   675
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   676
  unfolding isCont_def by (rule LIM_power)
2b093ba973bc new LIM/isCont lemmas for abs, of_real, and power
huffman
parents: 22613
diff changeset
   677
21239
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   678
subsubsection {* Nonstandard proofs *}
d4fbe2c87ef1 LIM_compose -> isCont_LIM_compose;
huffman
parents: 21165
diff changeset
   679
21786
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   680
lemma isNSContD:
66da6af2b0c9 cleaned up; generalized some proofs
huffman
parents: 21733
diff changeset
   681
  "\<lbrakk>isNSCont f a; y \<approx> star_of a\<rbrakk> \<Longrightarrow> ( *f* f) y \<approx> star_of (f a)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   682
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   683
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   684
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   685
by (simp add: isNSCont_def NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   686
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   687
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   688
apply (simp add: isNSCont_def NSLIM_def, auto)
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   689
apply (case_tac "y = star_of a", auto)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   690
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   691
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   692
text{*NS continuity can be defined using NS Limit in
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   693
    similar fashion to standard def of continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   694
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   695
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   696
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   697
text{*Hence, NS continuity can be given
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   698
  in terms of standard limit*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   699
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   700
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   701
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   702
text{*Moreover, it's trivial now that NS continuity
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   703
  is equivalent to standard continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   704
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   705
apply (simp add: isCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   706
apply (rule isNSCont_LIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   707
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   708
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   709
text{*Standard continuity ==> NS continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   710
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   711
by (erule isNSCont_isCont_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   712
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   713
text{*NS continuity ==> Standard continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   714
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   715
by (erule isNSCont_isCont_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   716
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   717
text{*Alternative definition of continuity*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   718
(* Prove equivalence between NS limits - *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   719
(* seems easier than using standard def  *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   720
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   721
apply (simp add: NSLIM_def, auto)
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   722
apply (drule_tac x = "star_of a + x" in spec)
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   723
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   724
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   725
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   726
 prefer 2 apply (simp add: add_commute diff_def [symmetric])
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   727
apply (rule_tac x = x in star_cases)
17318
bc1c75855f3d starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents: 17298
diff changeset
   728
apply (rule_tac [2] x = x in star_cases)
bc1c75855f3d starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents: 17298
diff changeset
   729
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   730
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   731
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   732
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   733
by (rule NSLIM_h_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   734
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   735
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   736
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   737
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   738
lemma isNSCont_inverse:
20653
24cda2c5fd40 removed division_by_zero class requirements from several lemmas
huffman
parents: 20635
diff changeset
   739
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   740
  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   741
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   742
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   743
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   744
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   745
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   746
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   747
apply (simp add: isNSCont_def)
21810
b2d23672b003 generalized some lemmas; removed redundant lemmas; cleaned up some proofs
huffman
parents: 21786
diff changeset
   748
apply (auto intro: approx_hrabs simp add: starfun_rabs_hrabs)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   749
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   750
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   751
lemma isCont_abs [simp]: "isCont abs (a::real)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   752
by (auto simp add: isNSCont_isCont_iff [symmetric])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   753
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   754
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   755
(****************************************************************
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   756
(%* Leave as commented until I add topology theory or remove? *%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   757
(%*------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   758
  Elementary topology proof for a characterisation of
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   759
  continuity now: a function f is continuous if and only
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   760
  if the inverse image, {x. f(x) \<in> A}, of any open set A
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   761
  is always an open set
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   762
 ------------------------------------------------------------*%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   763
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   764
               ==> isNSopen {x. f x \<in> A}"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   765
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   766
by (dtac (mem_monad_approx RS approx_sym);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   767
by (dres_inst_tac [("x","a")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   768
by (dtac isNSContD 1 THEN assume_tac 1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   769
by (dtac bspec 1 THEN assume_tac 1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   770
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   771
by (blast_tac (claset() addIs [starfun_mem_starset]);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   772
qed "isNSCont_isNSopen";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   773
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   774
Goalw [isNSCont_def]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   775
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   776
\              ==> isNSCont f x";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   777
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   778
     (approx_minus_iff RS iffD2)],simpset() addsimps
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   779
      [Infinitesimal_def,SReal_iff]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   780
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   781
by (etac (isNSopen_open_interval RSN (2,impE));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   782
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   783
by (dres_inst_tac [("x","x")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   784
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   785
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   786
qed "isNSopen_isNSCont";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   787
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   788
Goal "(\<forall>x. isNSCont f x) = \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   789
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   790
by (blast_tac (claset() addIs [isNSCont_isNSopen,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   791
    isNSopen_isNSCont]);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   792
qed "isNSCont_isNSopen_iff";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   793
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   794
(%*------- Standard version of same theorem --------*%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   795
Goal "(\<forall>x. isCont f x) = \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   796
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   797
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   798
              simpset() addsimps [isNSopen_isopen_iff RS sym,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   799
              isNSCont_isCont_iff RS sym]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   800
qed "isCont_isopen_iff";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   801
*******************************************************************)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   802
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   803
subsection {* Uniform Continuity *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   804
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   805
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   806
by (simp add: isNSUCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   807
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   808
lemma isUCont_isCont: "isUCont f ==> isCont f x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   809
by (simp add: isUCont_def isCont_def LIM_def, meson)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   810
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   811
lemma isUCont_isNSUCont:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   812
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   813
  assumes f: "isUCont f" shows "isNSUCont f"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   814
proof (unfold isNSUCont_def, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   815
  fix x y :: "'a star"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   816
  assume approx: "x \<approx> y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   817
  have "starfun f x - starfun f y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   818
  proof (rule InfinitesimalI2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   819
    fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   820
    with f obtain s where s: "0 < s" and
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   821
      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   822
      by (auto simp add: isUCont_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   823
    from less_r have less_r':
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   824
       "\<And>x y. hnorm (x - y) < star_of s
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   825
        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   826
      by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   827
    from approx have "x - y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   828
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   829
    hence "hnorm (x - y) < star_of s"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   830
      using s by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   831
    thus "hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   832
      by (rule less_r')
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   833
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   834
  thus "starfun f x \<approx> starfun f y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   835
    by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   836
qed
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   837
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   838
lemma isNSUCont_isUCont:
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   839
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   840
  assumes f: "isNSUCont f" shows "isUCont f"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   841
proof (unfold isUCont_def, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   842
  fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   843
  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   844
        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   845
  proof (rule exI, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   846
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   847
  next
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   848
    fix x y :: "'a star"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   849
    assume "hnorm (x - y) < epsilon"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   850
    with Infinitesimal_epsilon
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   851
    have "x - y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   852
      by (rule hnorm_less_Infinitesimal)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   853
    hence "x \<approx> y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   854
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   855
    with f have "starfun f x \<approx> starfun f y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   856
      by (simp add: isNSUCont_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   857
    hence "starfun f x - starfun f y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   858
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   859
    thus "hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   860
      using r by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   861
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   862
  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   863
    by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   864
qed
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   865
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   866
subsection {* Relation of LIM and LIMSEQ *}
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   867
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   868
lemma LIMSEQ_SEQ_conv1:
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   869
  fixes a :: "'a::real_normed_vector"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   870
  assumes X: "X -- a --> L"
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   871
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   872
proof (safe intro!: LIMSEQ_I)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   873
  fix S :: "nat \<Rightarrow> 'a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   874
  fix r :: real
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   875
  assume rgz: "0 < r"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   876
  assume as: "\<forall>n. S n \<noteq> a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   877
  assume S: "S ----> a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   878
  from LIM_D [OF X rgz] obtain s
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   879
    where sgz: "0 < s"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   880
    and aux: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (X x - L) < r"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   881
    by fast
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   882
  from LIMSEQ_D [OF S sgz]
21733
131dd2a27137 Modified lattice locale
nipkow
parents: 21404
diff changeset
   883
  obtain no where "\<forall>n\<ge>no. norm (S n - a) < s" by blast
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   884
  hence "\<forall>n\<ge>no. norm (X (S n) - L) < r" by (simp add: aux as)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   885
  thus "\<exists>no. \<forall>n\<ge>no. norm (X (S n) - L) < r" ..
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   886
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   887
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   888
lemma LIMSEQ_SEQ_conv2:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   889
  fixes a :: real
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   890
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   891
  shows "X -- a --> L"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   892
proof (rule ccontr)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   893
  assume "\<not> (X -- a --> L)"
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   894
  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)
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   895
  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
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   896
  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)
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   897
  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
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   898
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   899
  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"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   900
  have "\<And>n. \<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   901
    using rdef by simp
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   902
  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"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   903
    by (rule someI_ex)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   904
  hence F1: "\<And>n. ?F n \<noteq> a"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   905
    and F2: "\<And>n. \<bar>?F n - a\<bar> < inverse (real (Suc n))"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   906
    and F3: "\<And>n. norm (X (?F n) - L) \<ge> r"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   907
    by fast+
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   908
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   909
  have "?F ----> a"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   910
  proof (rule LIMSEQ_I, unfold real_norm_def)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   911
      fix e::real
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   912
      assume "0 < e"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   913
        (* choose no such that inverse (real (Suc n)) < e *)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   914
      have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   915
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   916
      show "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   917
      proof (intro exI allI impI)
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   918
        fix n
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   919
        assume mlen: "m \<le> n"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   920
        have "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   921
          by (rule F2)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   922
        also have "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   923
          by auto
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   924
        also from nodef have
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   925
          "inverse (real (Suc m)) < e" .
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   926
        finally show "\<bar>?F n - a\<bar> < e" .
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   927
      qed
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   928
  qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   929
  
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   930
  moreover have "\<forall>n. ?F n \<noteq> a"
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   931
    by (rule allI) (rule F1)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   932
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   933
  moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   934
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   935
  
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   936
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   937
  proof -
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   938
    {
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   939
      fix no::nat
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   940
      obtain n where "n = no + 1" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   941
      then have nolen: "no \<le> n" by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   942
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
21165
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   943
      have "norm (X (?F n) - L) \<ge> r"
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   944
        by (rule F3)
8fb49f668511 moved DERIV stuff from Lim.thy to new Deriv.thy; cleaned up LIMSEQ_SEQ proofs
huffman
parents: 21141
diff changeset
   945
      with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by fast
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   946
    }
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   947
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   948
    with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> e)" by auto
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   949
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   950
  qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   951
  ultimately show False by simp
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   952
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   953
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   954
lemma LIMSEQ_SEQ_conv:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   955
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   956
   (X -- a --> L)"
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   957
proof
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   958
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   959
  show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   960
next
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   961
  assume "(X -- a --> L)"
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   962
  show "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   963
qed
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   964
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
   965
end