src/HOL/Hyperreal/Lim.thy
author huffman
Thu Sep 28 04:03:43 2006 +0200 (2006-09-28)
changeset 20755 956a0377a408
parent 20754 9c053a494dc6
child 20756 fec7f5834ffe
permissions -rw-r--r--
reorganize sections
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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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    GMVT by Benjamin Porter, 2005
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*)
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header{*Limits, Continuity and Differentiation*}
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theory Lim
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imports SEQ
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begin
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text{*Standard and Nonstandard Definitions*}
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definition
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  LIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
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        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60)
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  "f -- a --> L =
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     (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s
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        --> norm (f x - L) < r)"
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  NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
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            ("((_)/ -- (_)/ --NS> (_))" [60, 0, 60] 60)
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  "f -- a --NS> L =
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    (\<forall>x. (x \<noteq> star_of a & x @= star_of a --> ( *f* f) x @= star_of L))"
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  isCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool"
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  "isCont f a = (f -- a --> (f a))"
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  isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool"
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    --{*NS definition dispenses with limit notions*}
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  "isNSCont f a = (\<forall>y. y @= star_of a -->
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         ( *f* f) y @= star_of (f a))"
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  deriv:: "[real=>real,real,real] => bool"
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    --{*Differentiation: D is derivative of function f at x*}
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          ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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  "DERIV f x :> D = ((%h. (f(x + h) - f x)/h) -- 0 --> D)"
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  nsderiv :: "[real=>real,real,real] => bool"
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          ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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  "NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}.
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      (( *f* f)(hypreal_of_real x + h)
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       - hypreal_of_real (f x))/h @= hypreal_of_real D)"
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  differentiable :: "[real=>real,real] => bool"   (infixl "differentiable" 60)
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  "f differentiable x = (\<exists>D. DERIV f x :> D)"
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  NSdifferentiable :: "[real=>real,real] => bool"
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                       (infixl "NSdifferentiable" 60)
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  "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
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  increment :: "[real=>real,real,hypreal] => hypreal"
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  "increment f x h = (@inc. f NSdifferentiable x &
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           inc = ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x))"
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  isUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool"
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  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r)"
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  isNSUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool"
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  "isNSUCont f = (\<forall>x y. x @= y --> ( *f* f) x @= ( *f* f) y)"
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consts
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  Bolzano_bisect :: "[real*real=>bool, real, real, nat] => (real*real)"
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primrec
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  "Bolzano_bisect P a b 0 = (a,b)"
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  "Bolzano_bisect P a b (Suc n) =
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      (let (x,y) = Bolzano_bisect P a b n
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       in if P(x, (x+y)/2) then ((x+y)/2, y)
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                            else (x, (x+y)/2))"
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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_const [simp]: "(%x. k) -- x --> k"
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by (simp add: LIM_def)
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lemma LIM_add:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
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  assumes f: "f -- a --> L" and g: "g -- a --> M"
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  shows "(%x. f x + g(x)) -- a --> (L + M)"
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proof (rule LIM_I)
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  fix r :: real
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  assume r: "0 < r"
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  from LIM_D [OF f half_gt_zero [OF r]]
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  obtain fs
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    where fs:    "0 < fs"
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      and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x - L) < r/2"
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  by blast
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  from LIM_D [OF g half_gt_zero [OF r]]
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  obtain gs
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    where gs:    "0 < gs"
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      and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x - M) < r/2"
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  by blast
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  show "\<exists>s>0.\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x + g x - (L + M)) < r"
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  proof (intro exI conjI strip)
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    show "0 < min fs gs"  by (simp add: fs gs)
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    fix x :: 'a
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    assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
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    hence "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
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    with fs_lt gs_lt
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    have "norm (f x - L) < r/2" and "norm (g x - M) < r/2" by blast+
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    hence "norm (f x - L) + norm (g x - M) < r" by arith
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    thus "norm (f x + g x - (L + M)) < r"
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      by (blast intro: norm_diff_triangle_ineq order_le_less_trans)
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  qed
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qed
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lemma minus_diff_minus:
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  fixes a b :: "'a::ab_group_add"
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  shows "(- a) - (- b) = - (a - b)"
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by simp
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lemma LIM_minus: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
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by (simp only: LIM_eq minus_diff_minus norm_minus_cancel)
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lemma LIM_add_minus:
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    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
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by (intro LIM_add LIM_minus)
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lemma LIM_diff:
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    "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) - g(x)) -- x --> l-m"
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by (simp only: diff_minus LIM_add LIM_minus)
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lemma LIM_const_not_eq:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "k \<noteq> L ==> ~ ((%x. k) -- a --> L)"
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apply (simp add: LIM_eq)
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apply (rule_tac x="norm (k - L)" in exI, simp, safe)
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apply (rule_tac x="a + of_real (s/2)" in exI, simp add: norm_of_real)
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done
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lemma LIM_const_eq:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "(%x. k) -- a --> L ==> k = L"
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apply (rule ccontr)
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apply (blast dest: LIM_const_not_eq)
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done
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lemma LIM_unique:
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  fixes a :: "'a::real_normed_div_algebra"
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  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
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apply (drule LIM_diff, assumption)
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apply (auto dest!: LIM_const_eq)
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done
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lemma LIM_mult_zero:
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  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
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  assumes f: "f -- a --> 0" and g: "g -- a --> 0"
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  shows "(%x. f(x) * g(x)) -- a --> 0"
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proof (rule LIM_I, simp)
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  fix r :: real
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  assume r: "0<r"
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  from LIM_D [OF f zero_less_one]
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  obtain fs
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    where fs:    "0 < fs"
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      and fs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < fs --> norm (f x) < 1"
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  by auto
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  from LIM_D [OF g r]
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  obtain gs
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    where gs:    "0 < gs"
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      and gs_lt: "\<forall>x. x \<noteq> a & norm (x-a) < gs --> norm (g x) < r"
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  by auto
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  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm (f x * g x) < r)"
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  proof (intro exI conjI strip)
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    show "0 < min fs gs"  by (simp add: fs gs)
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    fix x :: 'a
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    assume "x \<noteq> a \<and> norm (x-a) < min fs gs"
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    hence  "x \<noteq> a \<and> norm (x-a) < fs \<and> norm (x-a) < gs" by simp
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    with fs_lt gs_lt
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    have "norm (f x) < 1" and "norm (g x) < r" by blast+
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    hence "norm (f x) * norm (g x) < 1*r"
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      by (rule mult_strict_mono' [OF _ _ norm_ge_zero norm_ge_zero])
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    thus "norm (f x * g x) < r"
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      by (simp add: order_le_less_trans [OF norm_mult_ineq])
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  qed
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qed
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lemma LIM_self: "(%x. x) -- a --> a"
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by (auto simp add: LIM_def)
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text{*Limits are equal for functions equal except at limit point*}
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lemma LIM_equal:
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     "[| \<forall>x. x \<noteq> a --> (f x = g x) |] ==> (f -- a --> l) = (g -- a --> l)"
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by (simp add: LIM_def)
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text{*Two uses in Hyperreal/Transcendental.ML*}
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lemma LIM_trans:
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     "[| (%x. f(x) + -g(x)) -- a --> 0;  g -- a --> l |] ==> f -- a --> l"
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apply (drule LIM_add, assumption)
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apply (auto simp add: add_assoc)
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done
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subsubsection {* Purely nonstandard proofs *}
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lemma NSLIM_I:
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  "(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
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   \<Longrightarrow> f -- a --NS> L"
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by (simp add: NSLIM_def)
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lemma NSLIM_D:
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  "\<lbrakk>f -- a --NS> L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
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   \<Longrightarrow> starfun f x \<approx> star_of L"
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by (simp add: NSLIM_def)
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text{*Proving properties of limits using nonstandard definition.
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      The properties hold for standard limits as well!*}
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lemma NSLIM_mult:
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  fixes l m :: "'a::real_normed_algebra"
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  shows "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) * g(x)) -- x --NS> (l * m)"
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by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
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lemma NSLIM_add:
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     "[| f -- x --NS> l; g -- x --NS> m |]
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      ==> (%x. f(x) + g(x)) -- x --NS> (l + m)"
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by (auto simp add: NSLIM_def intro!: approx_add)
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lemma NSLIM_const [simp]: "(%x. k) -- x --NS> k"
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by (simp add: NSLIM_def)
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lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
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by (simp add: NSLIM_def)
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lemma NSLIM_add_minus: "[| f -- x --NS> l; g -- x --NS> m |] ==> (%x. f(x) + -g(x)) -- x --NS> (l + -m)"
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by (simp only: NSLIM_add NSLIM_minus)
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lemma NSLIM_inverse:
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  fixes L :: "'a::real_normed_div_algebra"
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  shows "[| f -- a --NS> L;  L \<noteq> 0 |]
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      ==> (%x. inverse(f(x))) -- a --NS> (inverse L)"
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apply (simp add: NSLIM_def, clarify)
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apply (drule spec)
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apply (auto simp add: star_of_approx_inverse)
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done
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lemma NSLIM_zero:
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  assumes f: "f -- a --NS> l" shows "(%x. f(x) + -l) -- a --NS> 0"
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proof -
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  have "(\<lambda>x. f x + - l) -- a --NS> l + -l"
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    by (rule NSLIM_add_minus [OF f NSLIM_const])
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  thus ?thesis by simp
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qed
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lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
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apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
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apply (auto simp add: diff_minus add_assoc)
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done
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lemma NSLIM_const_not_eq:
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  fixes a :: real (* TODO: generalize to real_normed_div_algebra *)
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  shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
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apply (simp add: NSLIM_def)
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apply (rule_tac x="star_of a + epsilon" in exI)
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apply (auto intro: Infinitesimal_add_approx_self [THEN approx_sym]
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            simp add: hypreal_epsilon_not_zero)
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done
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lemma NSLIM_not_zero:
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  fixes a :: real
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  shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
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by (rule NSLIM_const_not_eq)
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lemma NSLIM_const_eq:
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  fixes a :: real
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  shows "(%x. k) -- a --NS> L ==> k = L"
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apply (rule ccontr)
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apply (blast dest: NSLIM_const_not_eq)
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done
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text{* can actually be proved more easily by unfolding the definition!*}
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lemma NSLIM_unique:
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  fixes a :: real
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  shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
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apply (drule NSLIM_minus)
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   297
apply (drule NSLIM_add, assumption)
huffman@20755
   298
apply (auto dest!: NSLIM_const_eq [symmetric])
huffman@20755
   299
apply (simp add: diff_def [symmetric])
huffman@20755
   300
done
huffman@20755
   301
huffman@20755
   302
lemma NSLIM_mult_zero:
huffman@20755
   303
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20755
   304
  shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
huffman@20755
   305
by (drule NSLIM_mult, auto)
huffman@20755
   306
huffman@20755
   307
lemma NSLIM_self: "(%x. x) -- a --NS> a"
huffman@20755
   308
by (simp add: NSLIM_def)
huffman@20755
   309
huffman@20755
   310
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
huffman@20755
   311
huffman@20754
   312
lemma LIM_NSLIM:
huffman@20754
   313
  assumes f: "f -- a --> L" shows "f -- a --NS> L"
huffman@20754
   314
proof (rule NSLIM_I)
huffman@20754
   315
  fix x
huffman@20754
   316
  assume neq: "x \<noteq> star_of a"
huffman@20754
   317
  assume approx: "x \<approx> star_of a"
huffman@20754
   318
  have "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   319
  proof (rule InfinitesimalI2)
huffman@20754
   320
    fix r::real assume r: "0 < r"
huffman@20754
   321
    from LIM_D [OF f r]
huffman@20754
   322
    obtain s where s: "0 < s" and
huffman@20754
   323
      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
huffman@20754
   324
      by fast
huffman@20754
   325
    from less_r have less_r':
huffman@20754
   326
       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
huffman@20754
   327
        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   328
      by transfer
huffman@20754
   329
    from approx have "x - star_of a \<in> Infinitesimal"
huffman@20754
   330
      by (unfold approx_def)
huffman@20754
   331
    hence "hnorm (x - star_of a) < star_of s"
huffman@20754
   332
      using s by (rule InfinitesimalD2)
huffman@20754
   333
    with neq show "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   334
      by (rule less_r')
huffman@20754
   335
  qed
huffman@20754
   336
  thus "starfun f x \<approx> star_of L"
huffman@20754
   337
    by (unfold approx_def)
huffman@20754
   338
qed
huffman@20552
   339
huffman@20754
   340
lemma NSLIM_LIM:
huffman@20754
   341
  assumes f: "f -- a --NS> L" shows "f -- a --> L"
huffman@20754
   342
proof (rule LIM_I)
huffman@20754
   343
  fix r::real assume r: "0 < r"
huffman@20754
   344
  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
huffman@20754
   345
        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   346
  proof (rule exI, safe)
huffman@20754
   347
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   348
  next
huffman@20754
   349
    fix x assume neq: "x \<noteq> star_of a"
huffman@20754
   350
    assume "hnorm (x - star_of a) < epsilon"
huffman@20754
   351
    with Infinitesimal_epsilon
huffman@20754
   352
    have "x - star_of a \<in> Infinitesimal"
huffman@20754
   353
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   354
    hence "x \<approx> star_of a"
huffman@20754
   355
      by (unfold approx_def)
huffman@20754
   356
    with f neq have "starfun f x \<approx> star_of L"
huffman@20754
   357
      by (rule NSLIM_D)
huffman@20754
   358
    hence "starfun f x - star_of L \<in> Infinitesimal"
huffman@20754
   359
      by (unfold approx_def)
huffman@20754
   360
    thus "hnorm (starfun f x - star_of L) < star_of r"
huffman@20754
   361
      using r by (rule InfinitesimalD2)
huffman@20754
   362
  qed
huffman@20754
   363
  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
huffman@20754
   364
    by transfer
huffman@20754
   365
qed
paulson@14477
   366
paulson@15228
   367
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
paulson@14477
   368
by (blast intro: LIM_NSLIM NSLIM_LIM)
paulson@14477
   369
huffman@20755
   370
subsubsection {* Derived theorems about @{term LIM} *}
paulson@14477
   371
paulson@15228
   372
lemma LIM_mult2:
huffman@20552
   373
  fixes l m :: "'a::real_normed_algebra"
huffman@20552
   374
  shows "[| f -- x --> l; g -- x --> m |]
huffman@20552
   375
      ==> (%x. f(x) * g(x)) -- x --> (l * m)"
paulson@14477
   376
by (simp add: LIM_NSLIM_iff NSLIM_mult)
paulson@14477
   377
paulson@15228
   378
lemma LIM_add2:
paulson@15228
   379
     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
paulson@14477
   380
by (simp add: LIM_NSLIM_iff NSLIM_add)
paulson@14477
   381
paulson@14477
   382
lemma LIM_const2: "(%x. k) -- x --> k"
paulson@14477
   383
by (simp add: LIM_NSLIM_iff)
paulson@14477
   384
paulson@14477
   385
lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
paulson@14477
   386
by (simp add: LIM_NSLIM_iff NSLIM_minus)
paulson@14477
   387
paulson@14477
   388
lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
paulson@14477
   389
by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
paulson@14477
   390
huffman@20552
   391
lemma LIM_inverse:
huffman@20653
   392
  fixes L :: "'a::real_normed_div_algebra"
huffman@20552
   393
  shows "[| f -- a --> L; L \<noteq> 0 |]
huffman@20552
   394
      ==> (%x. inverse(f(x))) -- a --> (inverse L)"
paulson@14477
   395
by (simp add: LIM_NSLIM_iff NSLIM_inverse)
paulson@14477
   396
paulson@14477
   397
lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0"
paulson@14477
   398
by (simp add: LIM_NSLIM_iff NSLIM_zero)
paulson@14477
   399
paulson@14477
   400
lemma LIM_zero_cancel: "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l"
paulson@14477
   401
apply (drule_tac g = "%x. l" and M = l in LIM_add)
paulson@14477
   402
apply (auto simp add: diff_minus add_assoc)
paulson@14477
   403
done
paulson@14477
   404
huffman@20561
   405
lemma LIM_unique2:
huffman@20561
   406
  fixes a :: real
huffman@20561
   407
  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
paulson@14477
   408
by (simp add: LIM_NSLIM_iff NSLIM_unique)
paulson@14477
   409
paulson@14477
   410
(* we can use the corresponding thm LIM_mult2 *)
paulson@14477
   411
(* for standard definition of limit           *)
paulson@14477
   412
huffman@20552
   413
lemma LIM_mult_zero2:
huffman@20561
   414
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20552
   415
  shows "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
paulson@14477
   416
by (drule LIM_mult2, auto)
paulson@14477
   417
paulson@14477
   418
huffman@20755
   419
subsection {* Continuity *}
paulson@14477
   420
paulson@14477
   421
lemma isNSContD: "[| isNSCont f a; y \<approx> hypreal_of_real a |] ==> ( *f* f) y \<approx> hypreal_of_real (f a)"
paulson@14477
   422
by (simp add: isNSCont_def)
paulson@14477
   423
paulson@14477
   424
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
paulson@14477
   425
by (simp add: isNSCont_def NSLIM_def)
paulson@14477
   426
paulson@14477
   427
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
paulson@14477
   428
apply (simp add: isNSCont_def NSLIM_def, auto)
huffman@20561
   429
apply (case_tac "y = star_of a", auto)
paulson@14477
   430
done
paulson@14477
   431
paulson@15228
   432
text{*NS continuity can be defined using NS Limit in
paulson@15228
   433
    similar fashion to standard def of continuity*}
paulson@14477
   434
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
paulson@14477
   435
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
paulson@14477
   436
paulson@15228
   437
text{*Hence, NS continuity can be given
paulson@15228
   438
  in terms of standard limit*}
paulson@14477
   439
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
paulson@14477
   440
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
paulson@14477
   441
paulson@15228
   442
text{*Moreover, it's trivial now that NS continuity
paulson@15228
   443
  is equivalent to standard continuity*}
paulson@14477
   444
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
paulson@14477
   445
apply (simp add: isCont_def)
paulson@14477
   446
apply (rule isNSCont_LIM_iff)
paulson@14477
   447
done
paulson@14477
   448
paulson@15228
   449
text{*Standard continuity ==> NS continuity*}
paulson@14477
   450
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
paulson@14477
   451
by (erule isNSCont_isCont_iff [THEN iffD2])
paulson@14477
   452
paulson@15228
   453
text{*NS continuity ==> Standard continuity*}
paulson@14477
   454
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
paulson@14477
   455
by (erule isNSCont_isCont_iff [THEN iffD1])
paulson@14477
   456
paulson@14477
   457
text{*Alternative definition of continuity*}
paulson@14477
   458
(* Prove equivalence between NS limits - *)
paulson@14477
   459
(* seems easier than using standard def  *)
paulson@14477
   460
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
paulson@14477
   461
apply (simp add: NSLIM_def, auto)
huffman@20561
   462
apply (drule_tac x = "star_of a + x" in spec)
huffman@20561
   463
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
huffman@20561
   464
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
huffman@20561
   465
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
huffman@20561
   466
 prefer 2 apply (simp add: add_commute diff_def [symmetric])
huffman@20561
   467
apply (rule_tac x = x in star_cases)
huffman@17318
   468
apply (rule_tac [2] x = x in star_cases)
huffman@17318
   469
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
paulson@14477
   470
done
paulson@14477
   471
paulson@14477
   472
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
paulson@14477
   473
by (rule NSLIM_h_iff)
paulson@14477
   474
paulson@14477
   475
lemma LIM_isCont_iff: "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))"
paulson@14477
   476
by (simp add: LIM_NSLIM_iff NSLIM_isCont_iff)
paulson@14477
   477
paulson@14477
   478
lemma isCont_iff: "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))"
paulson@14477
   479
by (simp add: isCont_def LIM_isCont_iff)
paulson@14477
   480
paulson@15228
   481
text{*Immediate application of nonstandard criterion for continuity can offer
paulson@15228
   482
   very simple proofs of some standard property of continuous functions*}
paulson@14477
   483
text{*sum continuous*}
paulson@14477
   484
lemma isCont_add: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"
paulson@14477
   485
by (auto intro: approx_add simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
paulson@14477
   486
paulson@14477
   487
text{*mult continuous*}
huffman@20552
   488
lemma isCont_mult:
huffman@20561
   489
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
huffman@20552
   490
  shows "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
paulson@15228
   491
by (auto intro!: starfun_mult_HFinite_approx
paulson@15228
   492
            simp del: starfun_mult [symmetric]
paulson@14477
   493
            simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
paulson@14477
   494
paulson@15228
   495
text{*composition of continuous functions
paulson@15228
   496
     Note very short straightforard proof!*}
paulson@14477
   497
lemma isCont_o: "[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a"
paulson@14477
   498
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_def starfun_o [symmetric])
paulson@14477
   499
paulson@14477
   500
lemma isCont_o2: "[| isCont f a; isCont g (f a) |] ==> isCont (%x. g (f x)) a"
paulson@14477
   501
by (auto dest: isCont_o simp add: o_def)
paulson@14477
   502
paulson@14477
   503
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
paulson@14477
   504
by (simp add: isNSCont_def)
paulson@14477
   505
paulson@14477
   506
lemma isCont_minus: "isCont f a ==> isCont (%x. - f x) a"
paulson@14477
   507
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
paulson@14477
   508
paulson@14477
   509
lemma isCont_inverse:
huffman@20653
   510
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@20552
   511
  shows "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
paulson@14477
   512
apply (simp add: isCont_def)
paulson@14477
   513
apply (blast intro: LIM_inverse)
paulson@14477
   514
done
paulson@14477
   515
huffman@20552
   516
lemma isNSCont_inverse:
huffman@20653
   517
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
huffman@20552
   518
  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
paulson@14477
   519
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
paulson@14477
   520
paulson@14477
   521
lemma isCont_diff:
paulson@14477
   522
      "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"
paulson@14477
   523
apply (simp add: diff_minus)
paulson@14477
   524
apply (auto intro: isCont_add isCont_minus)
paulson@14477
   525
done
paulson@14477
   526
paulson@15228
   527
lemma isCont_const [simp]: "isCont (%x. k) a"
paulson@14477
   528
by (simp add: isCont_def)
paulson@14477
   529
paulson@15228
   530
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
paulson@14477
   531
by (simp add: isNSCont_def)
paulson@14477
   532
huffman@20561
   533
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
paulson@14477
   534
apply (simp add: isNSCont_def)
paulson@14477
   535
apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
paulson@14477
   536
done
paulson@14477
   537
huffman@20561
   538
lemma isCont_abs [simp]: "isCont abs (a::real)"
paulson@14477
   539
by (auto simp add: isNSCont_isCont_iff [symmetric])
paulson@15228
   540
paulson@14477
   541
paulson@14477
   542
(****************************************************************
paulson@14477
   543
(%* Leave as commented until I add topology theory or remove? *%)
paulson@14477
   544
(%*------------------------------------------------------------
paulson@14477
   545
  Elementary topology proof for a characterisation of
paulson@14477
   546
  continuity now: a function f is continuous if and only
paulson@14477
   547
  if the inverse image, {x. f(x) \<in> A}, of any open set A
paulson@14477
   548
  is always an open set
paulson@14477
   549
 ------------------------------------------------------------*%)
paulson@14477
   550
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
paulson@14477
   551
               ==> isNSopen {x. f x \<in> A}"
paulson@14477
   552
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
paulson@14477
   553
by (dtac (mem_monad_approx RS approx_sym);
paulson@14477
   554
by (dres_inst_tac [("x","a")] spec 1);
paulson@14477
   555
by (dtac isNSContD 1 THEN assume_tac 1)
paulson@14477
   556
by (dtac bspec 1 THEN assume_tac 1)
paulson@14477
   557
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
paulson@14477
   558
by (blast_tac (claset() addIs [starfun_mem_starset]);
paulson@14477
   559
qed "isNSCont_isNSopen";
paulson@14477
   560
paulson@14477
   561
Goalw [isNSCont_def]
paulson@14477
   562
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
paulson@14477
   563
\              ==> isNSCont f x";
paulson@14477
   564
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
paulson@14477
   565
     (approx_minus_iff RS iffD2)],simpset() addsimps
paulson@14477
   566
      [Infinitesimal_def,SReal_iff]));
paulson@14477
   567
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
paulson@14477
   568
by (etac (isNSopen_open_interval RSN (2,impE));
paulson@14477
   569
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
paulson@14477
   570
by (dres_inst_tac [("x","x")] spec 1);
paulson@14477
   571
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
paulson@14477
   572
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
paulson@14477
   573
qed "isNSopen_isNSCont";
paulson@14477
   574
paulson@14477
   575
Goal "(\<forall>x. isNSCont f x) = \
paulson@14477
   576
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
paulson@14477
   577
by (blast_tac (claset() addIs [isNSCont_isNSopen,
paulson@14477
   578
    isNSopen_isNSCont]);
paulson@14477
   579
qed "isNSCont_isNSopen_iff";
paulson@14477
   580
paulson@14477
   581
(%*------- Standard version of same theorem --------*%)
paulson@14477
   582
Goal "(\<forall>x. isCont f x) = \
paulson@14477
   583
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
paulson@14477
   584
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
paulson@14477
   585
              simpset() addsimps [isNSopen_isopen_iff RS sym,
paulson@14477
   586
              isNSCont_isCont_iff RS sym]));
paulson@14477
   587
qed "isCont_isopen_iff";
paulson@14477
   588
*******************************************************************)
paulson@14477
   589
huffman@20755
   590
subsection {* Uniform Continuity *}
huffman@20755
   591
paulson@14477
   592
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
paulson@14477
   593
by (simp add: isNSUCont_def)
paulson@14477
   594
paulson@14477
   595
lemma isUCont_isCont: "isUCont f ==> isCont f x"
paulson@14477
   596
by (simp add: isUCont_def isCont_def LIM_def, meson)
paulson@14477
   597
huffman@20754
   598
lemma isUCont_isNSUCont:
huffman@20754
   599
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   600
  assumes f: "isUCont f" shows "isNSUCont f"
huffman@20754
   601
proof (unfold isNSUCont_def, safe)
huffman@20754
   602
  fix x y :: "'a star"
huffman@20754
   603
  assume approx: "x \<approx> y"
huffman@20754
   604
  have "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   605
  proof (rule InfinitesimalI2)
huffman@20754
   606
    fix r::real assume r: "0 < r"
huffman@20754
   607
    with f obtain s where s: "0 < s" and
huffman@20754
   608
      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
huffman@20754
   609
      by (auto simp add: isUCont_def)
huffman@20754
   610
    from less_r have less_r':
huffman@20754
   611
       "\<And>x y. hnorm (x - y) < star_of s
huffman@20754
   612
        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   613
      by transfer
huffman@20754
   614
    from approx have "x - y \<in> Infinitesimal"
huffman@20754
   615
      by (unfold approx_def)
huffman@20754
   616
    hence "hnorm (x - y) < star_of s"
huffman@20754
   617
      using s by (rule InfinitesimalD2)
huffman@20754
   618
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   619
      by (rule less_r')
huffman@20754
   620
  qed
huffman@20754
   621
  thus "starfun f x \<approx> starfun f y"
huffman@20754
   622
    by (unfold approx_def)
huffman@20754
   623
qed
paulson@14477
   624
paulson@14477
   625
lemma isNSUCont_isUCont:
huffman@20754
   626
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
huffman@20754
   627
  assumes f: "isNSUCont f" shows "isUCont f"
huffman@20754
   628
proof (unfold isUCont_def, safe)
huffman@20754
   629
  fix r::real assume r: "0 < r"
huffman@20754
   630
  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
huffman@20754
   631
        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   632
  proof (rule exI, safe)
huffman@20754
   633
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
huffman@20754
   634
  next
huffman@20754
   635
    fix x y :: "'a star"
huffman@20754
   636
    assume "hnorm (x - y) < epsilon"
huffman@20754
   637
    with Infinitesimal_epsilon
huffman@20754
   638
    have "x - y \<in> Infinitesimal"
huffman@20754
   639
      by (rule hnorm_less_Infinitesimal)
huffman@20754
   640
    hence "x \<approx> y"
huffman@20754
   641
      by (unfold approx_def)
huffman@20754
   642
    with f have "starfun f x \<approx> starfun f y"
huffman@20754
   643
      by (simp add: isNSUCont_def)
huffman@20754
   644
    hence "starfun f x - starfun f y \<in> Infinitesimal"
huffman@20754
   645
      by (unfold approx_def)
huffman@20754
   646
    thus "hnorm (starfun f x - starfun f y) < star_of r"
huffman@20754
   647
      using r by (rule InfinitesimalD2)
huffman@20754
   648
  qed
huffman@20754
   649
  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
huffman@20754
   650
    by transfer
huffman@20754
   651
qed
paulson@14477
   652
huffman@20755
   653
subsection {* Derivatives *}
huffman@20755
   654
huffman@20563
   655
lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
paulson@14477
   656
by (simp add: deriv_def)
paulson@14477
   657
paulson@14477
   658
lemma DERIV_NS_iff:
huffman@20563
   659
      "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
paulson@14477
   660
by (simp add: deriv_def LIM_NSLIM_iff)
paulson@14477
   661
huffman@20563
   662
lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
paulson@14477
   663
by (simp add: deriv_def)
paulson@14477
   664
huffman@20563
   665
lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NS> D"
paulson@14477
   666
by (simp add: deriv_def LIM_NSLIM_iff)
paulson@14477
   667
paulson@14477
   668
subsubsection{*Uniqueness*}
paulson@14477
   669
paulson@14477
   670
lemma DERIV_unique:
paulson@14477
   671
      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
paulson@14477
   672
apply (simp add: deriv_def)
paulson@14477
   673
apply (blast intro: LIM_unique)
paulson@14477
   674
done
paulson@14477
   675
paulson@14477
   676
lemma NSDeriv_unique:
paulson@14477
   677
     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
paulson@14477
   678
apply (simp add: nsderiv_def)
paulson@14477
   679
apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
paulson@15228
   680
apply (auto dest!: bspec [where x=epsilon]
paulson@15228
   681
            intro!: inj_hypreal_of_real [THEN injD]
paulson@14477
   682
            dest: approx_trans3)
paulson@14477
   683
done
paulson@14477
   684
paulson@14477
   685
subsubsection{*Alternative definition for differentiability*}
paulson@14477
   686
paulson@14477
   687
lemma DERIV_LIM_iff:
huffman@20561
   688
     "((%h::real. (f(a + h) - f(a)) / h) -- 0 --> D) =
paulson@14477
   689
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
paulson@14477
   690
proof (intro iffI LIM_I)
paulson@14477
   691
  fix r::real
paulson@14477
   692
  assume r: "0<r"
paulson@14477
   693
  assume "(\<lambda>h. (f (a + h) - f a) / h) -- 0 --> D"
paulson@14477
   694
  from LIM_D [OF this r]
paulson@14477
   695
  obtain s
paulson@14477
   696
    where s:    "0 < s"
paulson@14477
   697
      and s_lt: "\<forall>x. x \<noteq> 0 & \<bar>x\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r"
paulson@14477
   698
  by auto
paulson@14477
   699
  show "\<exists>s. 0 < s \<and>
huffman@20561
   700
        (\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm ((f x - f a) / (x-a) - D) < r)"
paulson@14477
   701
  proof (intro exI conjI strip)
paulson@14477
   702
    show "0 < s"  by (rule s)
paulson@14477
   703
  next
paulson@14477
   704
    fix x::real
huffman@20561
   705
    assume "x \<noteq> a \<and> norm (x-a) < s"
paulson@14477
   706
    with s_lt [THEN spec [where x="x-a"]]
huffman@20552
   707
    show "norm ((f x - f a) / (x-a) - D) < r" by auto
paulson@14477
   708
  qed
paulson@14477
   709
next
paulson@14477
   710
  fix r::real
paulson@14477
   711
  assume r: "0<r"
paulson@14477
   712
  assume "(\<lambda>x. (f x - f a) / (x-a)) -- a --> D"
paulson@14477
   713
  from LIM_D [OF this r]
paulson@14477
   714
  obtain s
paulson@14477
   715
    where s:    "0 < s"
paulson@14477
   716
      and s_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>(f x - f a)/(x-a) - D\<bar> < r"
paulson@14477
   717
  by auto
paulson@14477
   718
  show "\<exists>s. 0 < s \<and>
huffman@20561
   719
        (\<forall>x. x \<noteq> 0 & norm (x - 0) < s --> norm ((f (a + x) - f a) / x - D) < r)"
paulson@14477
   720
  proof (intro exI conjI strip)
paulson@14477
   721
    show "0 < s"  by (rule s)
paulson@14477
   722
  next
paulson@14477
   723
    fix x::real
huffman@20561
   724
    assume "x \<noteq> 0 \<and> norm (x - 0) < s"
paulson@14477
   725
    with s_lt [THEN spec [where x="x+a"]]
huffman@20552
   726
    show "norm ((f (a + x) - f a) / x - D) < r" by (auto simp add: add_ac)
paulson@14477
   727
  qed
paulson@14477
   728
qed
paulson@14477
   729
paulson@14477
   730
lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
paulson@14477
   731
by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
paulson@14477
   732
paulson@14477
   733
huffman@20755
   734
subsubsection{*Equivalence of NS and standard definitions of differentiation*}
paulson@14477
   735
huffman@20755
   736
text {*First NSDERIV in terms of NSLIM*}
paulson@14477
   737
paulson@15228
   738
text{*first equivalence *}
paulson@14477
   739
lemma NSDERIV_NSLIM_iff:
huffman@20563
   740
      "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
paulson@14477
   741
apply (simp add: nsderiv_def NSLIM_def, auto)
paulson@14477
   742
apply (drule_tac x = xa in bspec)
paulson@14477
   743
apply (rule_tac [3] ccontr)
paulson@14477
   744
apply (drule_tac [3] x = h in spec)
paulson@14477
   745
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
paulson@14477
   746
done
paulson@14477
   747
paulson@15228
   748
text{*second equivalence *}
paulson@14477
   749
lemma NSDERIV_NSLIM_iff2:
paulson@14477
   750
     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
paulson@15228
   751
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
paulson@14477
   752
              LIM_NSLIM_iff [symmetric])
paulson@14477
   753
paulson@14477
   754
(* while we're at it! *)
paulson@14477
   755
lemma NSDERIV_iff2:
paulson@14477
   756
     "(NSDERIV f x :> D) =
paulson@14477
   757
      (\<forall>w.
paulson@14477
   758
        w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
paulson@14477
   759
        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
paulson@14477
   760
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
paulson@14477
   761
paulson@14477
   762
(*FIXME DELETE*)
huffman@20563
   763
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
paulson@14477
   764
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
paulson@14477
   765
paulson@14477
   766
lemma NSDERIVD5:
paulson@14477
   767
  "(NSDERIV f x :> D) ==>
paulson@14477
   768
   (\<forall>u. u \<approx> hypreal_of_real x -->
paulson@14477
   769
     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
paulson@14477
   770
apply (auto simp add: NSDERIV_iff2)
paulson@14477
   771
apply (case_tac "u = hypreal_of_real x", auto)
paulson@14477
   772
apply (drule_tac x = u in spec, auto)
paulson@14477
   773
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
paulson@14477
   774
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
paulson@14477
   775
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
huffman@20563
   776
apply (auto simp add:
kleing@19023
   777
         approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
kleing@19023
   778
         Infinitesimal_subset_HFinite [THEN subsetD])
paulson@14477
   779
done
paulson@14477
   780
paulson@14477
   781
lemma NSDERIVD4:
paulson@14477
   782
     "(NSDERIV f x :> D) ==>
paulson@14477
   783
      (\<forall>h \<in> Infinitesimal.
paulson@14477
   784
               (( *f* f)(hypreal_of_real x + h) -
paulson@14477
   785
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
paulson@14477
   786
apply (auto simp add: nsderiv_def)
paulson@14477
   787
apply (case_tac "h = (0::hypreal) ")
paulson@14477
   788
apply (auto simp add: diff_minus)
paulson@14477
   789
apply (drule_tac x = h in bspec)
paulson@14477
   790
apply (drule_tac [2] c = h in approx_mult1)
paulson@14477
   791
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
nipkow@15539
   792
            simp add: diff_minus)
paulson@14477
   793
done
paulson@14477
   794
paulson@14477
   795
lemma NSDERIVD3:
paulson@14477
   796
     "(NSDERIV f x :> D) ==>
paulson@14477
   797
      (\<forall>h \<in> Infinitesimal - {0}.
paulson@14477
   798
               (( *f* f)(hypreal_of_real x + h) -
paulson@14477
   799
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
paulson@14477
   800
apply (auto simp add: nsderiv_def)
paulson@14477
   801
apply (rule ccontr, drule_tac x = h in bspec)
paulson@14477
   802
apply (drule_tac [2] c = h in approx_mult1)
paulson@14477
   803
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
nipkow@15539
   804
            simp add: mult_assoc diff_minus)
paulson@14477
   805
done
paulson@14477
   806
paulson@14477
   807
text{*Now equivalence between NSDERIV and DERIV*}
paulson@14477
   808
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
paulson@14477
   809
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
paulson@14477
   810
paulson@15228
   811
text{*Differentiability implies continuity
paulson@15228
   812
         nice and simple "algebraic" proof*}
paulson@14477
   813
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
paulson@14477
   814
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
paulson@14477
   815
apply (drule approx_minus_iff [THEN iffD1])
paulson@14477
   816
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@20563
   817
apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
paulson@15228
   818
 prefer 2 apply (simp add: add_assoc [symmetric])
paulson@15234
   819
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
huffman@20563
   820
apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
paulson@14477
   821
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
paulson@14477
   822
            simp add: mult_assoc)
paulson@14477
   823
apply (drule_tac x3=D in
paulson@14477
   824
           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
paulson@14477
   825
             THEN mem_infmal_iff [THEN iffD1]])
nipkow@15539
   826
apply (auto simp add: mult_commute
paulson@14477
   827
            intro: approx_trans approx_minus_iff [THEN iffD2])
paulson@14477
   828
done
paulson@14477
   829
paulson@14477
   830
text{*Now Sandard proof*}
paulson@14477
   831
lemma DERIV_isCont: "DERIV f x :> D ==> isCont f x"
paulson@15228
   832
by (simp add: NSDERIV_DERIV_iff [symmetric] isNSCont_isCont_iff [symmetric]
paulson@14477
   833
              NSDERIV_isNSCont)
paulson@14477
   834
huffman@20755
   835
subsubsection {* Derivatives of various functions *}
paulson@14477
   836
paulson@15228
   837
text{*Differentiation rules for combinations of functions
paulson@14477
   838
      follow from clear, straightforard, algebraic
paulson@15228
   839
      manipulations*}
paulson@14477
   840
text{*Constant function*}
paulson@14477
   841
paulson@14477
   842
(* use simple constant nslimit theorem *)
paulson@15228
   843
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
paulson@14477
   844
by (simp add: NSDERIV_NSLIM_iff)
paulson@14477
   845
paulson@15228
   846
lemma DERIV_const [simp]: "(DERIV (%x. k) x :> 0)"
paulson@14477
   847
by (simp add: NSDERIV_DERIV_iff [symmetric])
paulson@14477
   848
paulson@15228
   849
text{*Sum of functions- proved easily*}
paulson@14477
   850
paulson@14477
   851
paulson@14477
   852
lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
paulson@14477
   853
      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
paulson@14477
   854
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@20563
   855
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
paulson@14477
   856
apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
huffman@20563
   857
apply (auto simp add: diff_def add_ac)
paulson@14477
   858
done
paulson@14477
   859
paulson@14477
   860
(* Standard theorem *)
paulson@14477
   861
lemma DERIV_add: "[| DERIV f x :> Da; DERIV g x :> Db |]
paulson@14477
   862
      ==> DERIV (%x. f x + g x) x :> Da + Db"
paulson@14477
   863
apply (simp add: NSDERIV_add NSDERIV_DERIV_iff [symmetric])
paulson@14477
   864
done
paulson@14477
   865
paulson@15228
   866
text{*Product of functions - Proof is trivial but tedious
paulson@15228
   867
  and long due to rearrangement of terms*}
paulson@14477
   868
huffman@20563
   869
lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
huffman@20563
   870
by (simp add: right_diff_distrib)
paulson@14477
   871
huffman@20563
   872
lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
paulson@14477
   873
         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
huffman@20563
   874
      ==> x - y \<approx> 0"
paulson@14477
   875
apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
huffman@20563
   876
apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
paulson@14477
   877
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
nipkow@15539
   878
            simp add: mult_assoc mem_infmal_iff [symmetric])
paulson@14477
   879
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
paulson@14477
   880
done
paulson@14477
   881
paulson@14477
   882
paulson@14477
   883
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
paulson@14477
   884
      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
paulson@14477
   885
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
paulson@14477
   886
apply (auto dest!: spec
kleing@19023
   887
      simp add: starfun_lambda_cancel lemma_nsderiv1)
huffman@20563
   888
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
paulson@14477
   889
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
paulson@15234
   890
apply (auto simp add: times_divide_eq_right [symmetric]
paulson@15234
   891
            simp del: times_divide_eq)
huffman@20563
   892
apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
huffman@20563
   893
apply (drule_tac
paulson@15228
   894
     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
paulson@15228
   895
apply (auto intro!: approx_add_mono1
paulson@14477
   896
            simp add: left_distrib right_distrib mult_commute add_assoc)
paulson@15228
   897
apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
paulson@14477
   898
         in add_commute [THEN subst])
paulson@15228
   899
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
paulson@15228
   900
                    Infinitesimal_add Infinitesimal_mult
paulson@15228
   901
                    Infinitesimal_hypreal_of_real_mult
paulson@14477
   902
                    Infinitesimal_hypreal_of_real_mult2
paulson@14477
   903
          simp add: add_assoc [symmetric])
paulson@14477
   904
done
paulson@14477
   905
paulson@14477
   906
lemma DERIV_mult:
paulson@15228
   907
     "[| DERIV f x :> Da; DERIV g x :> Db |]
paulson@14477
   908
      ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
paulson@14477
   909
by (simp add: NSDERIV_mult NSDERIV_DERIV_iff [symmetric])
paulson@14477
   910
paulson@14477
   911
text{*Multiplying by a constant*}
paulson@14477
   912
lemma NSDERIV_cmult: "NSDERIV f x :> D
paulson@14477
   913
      ==> NSDERIV (%x. c * f x) x :> c*D"
paulson@15228
   914
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
huffman@20563
   915
                  minus_mult_right right_diff_distrib [symmetric])
paulson@14477
   916
apply (erule NSLIM_const [THEN NSLIM_mult])
paulson@14477
   917
done
paulson@14477
   918
paulson@14477
   919
(* let's do the standard proof though theorem *)
paulson@14477
   920
(* LIM_mult2 follows from a NS proof          *)
paulson@14477
   921
paulson@14477
   922
lemma DERIV_cmult:
paulson@14477
   923
      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
paulson@15228
   924
apply (simp only: deriv_def times_divide_eq_right [symmetric]
huffman@20563
   925
                  NSDERIV_NSLIM_iff minus_mult_right right_diff_distrib [symmetric])
paulson@14477
   926
apply (erule LIM_const [THEN LIM_mult2])
paulson@14477
   927
done
paulson@14477
   928
paulson@14477
   929
text{*Negation of function*}
paulson@14477
   930
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
paulson@14477
   931
proof (simp add: NSDERIV_NSLIM_iff)
huffman@20563
   932
  assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
huffman@20563
   933
  hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
paulson@14477
   934
    by (rule NSLIM_minus)
huffman@20563
   935
  have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
paulson@15228
   936
    by (simp add: minus_divide_left)
paulson@14477
   937
  with deriv
paulson@14477
   938
  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
paulson@14477
   939
qed
paulson@14477
   940
paulson@14477
   941
paulson@14477
   942
lemma DERIV_minus: "DERIV f x :> D ==> DERIV (%x. -(f x)) x :> -D"
paulson@14477
   943
by (simp add: NSDERIV_minus NSDERIV_DERIV_iff [symmetric])
paulson@14477
   944
paulson@14477
   945
text{*Subtraction*}
paulson@14477
   946
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
paulson@14477
   947
by (blast dest: NSDERIV_add NSDERIV_minus)
paulson@14477
   948
paulson@14477
   949
lemma DERIV_add_minus: "[| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + -g x) x :> Da + -Db"
paulson@14477
   950
by (blast dest: DERIV_add DERIV_minus)
paulson@14477
   951
paulson@14477
   952
lemma NSDERIV_diff:
paulson@14477
   953
     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
paulson@14477
   954
      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
paulson@14477
   955
apply (simp add: diff_minus)
paulson@14477
   956
apply (blast intro: NSDERIV_add_minus)
paulson@14477
   957
done
paulson@14477
   958
paulson@14477
   959
lemma DERIV_diff:
paulson@14477
   960
     "[| DERIV f x :> Da; DERIV g x :> Db |]
paulson@14477
   961
       ==> DERIV (%x. f x - g x) x :> Da-Db"
paulson@14477
   962
apply (simp add: diff_minus)
paulson@14477
   963
apply (blast intro: DERIV_add_minus)
paulson@14477
   964
done
paulson@14477
   965
paulson@14477
   966
text{*  Similarly to the above, the chain rule admits an entirely
paulson@14477
   967
   straightforward derivation. Compare this with Harrison's
paulson@14477
   968
   HOL proof of the chain rule, which proved to be trickier and
paulson@14477
   969
   required an alternative characterisation of differentiability-
paulson@14477
   970
   the so-called Carathedory derivative. Our main problem is
paulson@14477
   971
   manipulation of terms.*}
paulson@14477
   972
paulson@14477
   973
paulson@14477
   974
(* lemmas *)
paulson@14477
   975
lemma NSDERIV_zero:
paulson@14477
   976
      "[| NSDERIV g x :> D;
paulson@14477
   977
               ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
paulson@14477
   978
               xa \<in> Infinitesimal;
paulson@14477
   979
               xa \<noteq> 0
paulson@14477
   980
            |] ==> D = 0"
paulson@14477
   981
apply (simp add: nsderiv_def)
paulson@14477
   982
apply (drule bspec, auto)
paulson@14477
   983
done
paulson@14477
   984
paulson@14477
   985
(* can be proved differently using NSLIM_isCont_iff *)
paulson@14477
   986
lemma NSDERIV_approx:
paulson@14477
   987
     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
huffman@20563
   988
      ==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
paulson@14477
   989
apply (simp add: nsderiv_def)
paulson@14477
   990
apply (simp add: mem_infmal_iff [symmetric])
paulson@14477
   991
apply (rule Infinitesimal_ratio)
paulson@14477
   992
apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
paulson@14477
   993
done
paulson@14477
   994
paulson@14477
   995
(*---------------------------------------------------------------
paulson@14477
   996
   from one version of differentiability
paulson@14477
   997
paulson@14477
   998
                f(x) - f(a)
paulson@14477
   999
              --------------- \<approx> Db
paulson@14477
  1000
                  x - a
paulson@14477
  1001
 ---------------------------------------------------------------*)
paulson@14477
  1002
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
paulson@14477
  1003
         ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
paulson@14477
  1004
         ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
paulson@14477
  1005
      |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
huffman@20563
  1006
                   - hypreal_of_real (f (g x)))
huffman@20563
  1007
              / (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
paulson@14477
  1008
             \<approx> hypreal_of_real(Da)"
paulson@14477
  1009
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
paulson@14477
  1010
paulson@14477
  1011
(*--------------------------------------------------------------
paulson@14477
  1012
   from other version of differentiability
paulson@14477
  1013
paulson@14477
  1014
                f(x + h) - f(x)
paulson@14477
  1015
               ----------------- \<approx> Db
paulson@14477
  1016
                       h
paulson@14477
  1017
 --------------------------------------------------------------*)
paulson@14477
  1018
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
huffman@20563
  1019
      ==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
paulson@14477
  1020
          \<approx> hypreal_of_real(Db)"
paulson@14477
  1021
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
paulson@14477
  1022
paulson@14477
  1023
lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
paulson@14477
  1024
by auto
paulson@14477
  1025
paulson@15228
  1026
text{*This proof uses both definitions of differentiability.*}
paulson@14477
  1027
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
paulson@14477
  1028
      ==> NSDERIV (f o g) x :> Da * Db"
paulson@14477
  1029
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
paulson@14477
  1030
                mem_infmal_iff [symmetric])
paulson@14477
  1031
apply clarify
paulson@14477
  1032
apply (frule_tac f = g in NSDERIV_approx)
paulson@14477
  1033
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
paulson@14477
  1034
apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
paulson@14477
  1035
apply (drule_tac g = g in NSDERIV_zero)
paulson@14477
  1036
apply (auto simp add: divide_inverse)
huffman@20563
  1037
apply (rule_tac z1 = "( *f* g) (hypreal_of_real (x) + xa) - hypreal_of_real (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
paulson@14477
  1038
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
paulson@14477
  1039
apply (rule approx_mult_hypreal_of_real)
paulson@14477
  1040
apply (simp_all add: divide_inverse [symmetric])
paulson@14477
  1041
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
paulson@14477
  1042
apply (blast intro: NSDERIVD2)
paulson@14477
  1043
done
paulson@14477
  1044
paulson@14477
  1045
(* standard version *)
paulson@14477
  1046
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
paulson@14477
  1047
by (simp add: NSDERIV_DERIV_iff [symmetric] NSDERIV_chain)
paulson@14477
  1048
paulson@14477
  1049
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
paulson@14477
  1050
by (auto dest: DERIV_chain simp add: o_def)
paulson@14477
  1051
paulson@14477
  1052
text{*Differentiation of natural number powers*}
paulson@15228
  1053
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
paulson@15228
  1054
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
paulson@14477
  1055
paulson@14477
  1056
(*derivative of the identity function*)
paulson@15228
  1057
lemma DERIV_Id [simp]: "DERIV (%x. x) x :> 1"
paulson@14477
  1058
by (simp add: NSDERIV_DERIV_iff [symmetric])
paulson@14477
  1059
paulson@14477
  1060
lemmas isCont_Id = DERIV_Id [THEN DERIV_isCont, standard]
paulson@14477
  1061
paulson@14477
  1062
(*derivative of linear multiplication*)
paulson@15228
  1063
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
paulson@14477
  1064
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
paulson@14477
  1065
paulson@15228
  1066
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
paulson@14477
  1067
by (simp add: NSDERIV_DERIV_iff)
paulson@14477
  1068
paulson@14477
  1069
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
paulson@15251
  1070
apply (induct "n")
paulson@14477
  1071
apply (drule_tac [2] DERIV_Id [THEN DERIV_mult])
paulson@14477
  1072
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@14477
  1073
apply (case_tac "0 < n")
paulson@14477
  1074
apply (drule_tac x = x in realpow_minus_mult)
paulson@15234
  1075
apply (auto simp add: mult_assoc add_commute)
paulson@14477
  1076
done
paulson@14477
  1077
paulson@14477
  1078
(* NS version *)
paulson@14477
  1079
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
paulson@14477
  1080
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
paulson@14477
  1081
paulson@15228
  1082
text{*Power of -1*}
paulson@14477
  1083
paulson@14477
  1084
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
paulson@14477
  1085
lemma NSDERIV_inverse:
paulson@14477
  1086
     "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
paulson@14477
  1087
apply (simp add: nsderiv_def)
paulson@15228
  1088
apply (rule ballI, simp, clarify)
huffman@20563
  1089
apply (frule (1) Infinitesimal_add_not_zero)
huffman@20563
  1090
apply (simp add: add_commute)
huffman@20563
  1091
(*apply (auto simp add: starfun_inverse_inverse realpow_two
huffman@20563
  1092
        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
paulson@14477
  1093
apply (simp add: inverse_add inverse_mult_distrib [symmetric]
huffman@20563
  1094
              inverse_minus_eq [symmetric] add_ac mult_ac diff_def
paulson@15228
  1095
            del: inverse_mult_distrib inverse_minus_eq
paulson@14477
  1096
                 minus_mult_left [symmetric] minus_mult_right [symmetric])
paulson@14477
  1097
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
paulson@14477
  1098
            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
paulson@15234
  1099
apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
paulson@14477
  1100
apply (rule inverse_add_Infinitesimal_approx2)
paulson@15228
  1101
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
paulson@14477
  1102
            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
paulson@14477
  1103
apply (rule Infinitesimal_HFinite_mult2, auto)
paulson@14477
  1104
done
paulson@14477
  1105
paulson@14477
  1106
paulson@14477
  1107
paulson@14477
  1108
paulson@14477
  1109
lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
paulson@14477
  1110
by (simp add: NSDERIV_inverse NSDERIV_DERIV_iff [symmetric] del: realpow_Suc)
paulson@14477
  1111
paulson@14477
  1112
text{*Derivative of inverse*}
paulson@14477
  1113
lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
paulson@14477
  1114
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
paulson@14477
  1115
apply (simp only: mult_commute [of d] minus_mult_left power_inverse)
paulson@14477
  1116
apply (fold o_def)
paulson@14477
  1117
apply (blast intro!: DERIV_chain DERIV_inverse)
paulson@14477
  1118
done
paulson@14477
  1119
paulson@14477
  1120
lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
paulson@14477
  1121
      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
paulson@14477
  1122
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
paulson@14477
  1123
paulson@14477
  1124
text{*Derivative of quotient*}
paulson@14477
  1125
lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@20563
  1126
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
paulson@14477
  1127
apply (drule_tac f = g in DERIV_inverse_fun)
paulson@14477
  1128
apply (drule_tac [2] DERIV_mult)
paulson@14477
  1129
apply (assumption+)
paulson@14477
  1130
apply (simp add: divide_inverse right_distrib power_inverse minus_mult_left
huffman@20563
  1131
                 mult_ac diff_def
paulson@14477
  1132
     del: realpow_Suc minus_mult_right [symmetric] minus_mult_left [symmetric])
paulson@14477
  1133
done
paulson@14477
  1134
paulson@14477
  1135
lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
paulson@14477
  1136
       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
huffman@20563
  1137
                            - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
paulson@14477
  1138
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
paulson@14477
  1139
paulson@14477
  1140
(* ------------------------------------------------------------------------ *)
paulson@14477
  1141
(* Caratheodory formulation of derivative at a point: standard proof        *)
paulson@14477
  1142
(* ------------------------------------------------------------------------ *)
paulson@14477
  1143
paulson@14477
  1144
lemma CARAT_DERIV:
paulson@14477
  1145
     "(DERIV f x :> l) =
paulson@14477
  1146
      (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
paulson@14477
  1147
      (is "?lhs = ?rhs")
paulson@14477
  1148
proof
paulson@14477
  1149
  assume der: "DERIV f x :> l"
paulson@14477
  1150
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
paulson@14477
  1151
  proof (intro exI conjI)
paulson@14477
  1152
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
nipkow@15539
  1153
    show "\<forall>z. f z - f x = ?g z * (z-x)" by (simp)
paulson@15228
  1154
    show "isCont ?g x" using der
paulson@15228
  1155
      by (simp add: isCont_iff DERIV_iff diff_minus
paulson@14477
  1156
               cong: LIM_equal [rule_format])
paulson@14477
  1157
    show "?g x = l" by simp
paulson@14477
  1158
  qed
paulson@14477
  1159
next
paulson@14477
  1160
  assume "?rhs"
paulson@15228
  1161
  then obtain g where
paulson@14477
  1162
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
paulson@15228
  1163
  thus "(DERIV f x :> l)"
paulson@15228
  1164
     by (auto simp add: isCont_iff DERIV_iff diff_minus
paulson@14477
  1165
               cong: LIM_equal [rule_format])
paulson@14477
  1166
qed
paulson@14477
  1167
paulson@14477
  1168
paulson@14477
  1169
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
paulson@14477
  1170
      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
paulson@14477
  1171
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV)
paulson@14477
  1172
paulson@14477
  1173
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
paulson@14477
  1174
by auto
paulson@14477
  1175
paulson@14477
  1176
lemma CARAT_DERIVD:
paulson@14477
  1177
  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
paulson@14477
  1178
      and nsc: "isNSCont g x"
paulson@14477
  1179
  shows "NSDERIV f x :> g x"
paulson@14477
  1180
proof -
paulson@14477
  1181
  from nsc
paulson@14477
  1182
  have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
paulson@14477
  1183
         ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
paulson@15228
  1184
         hypreal_of_real (g x)"
paulson@14477
  1185
    by (simp add: diff_minus isNSCont_def)
paulson@14477
  1186
  thus ?thesis using all
paulson@15228
  1187
    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
paulson@14477
  1188
qed
paulson@14477
  1189
huffman@20755
  1190
subsubsection {* Differentiability predicate *}
huffman@20755
  1191
huffman@20755
  1192
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
huffman@20755
  1193
by (simp add: differentiable_def)
huffman@20755
  1194
huffman@20755
  1195
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
huffman@20755
  1196
by (force simp add: differentiable_def)
huffman@20755
  1197
huffman@20755
  1198
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
huffman@20755
  1199
by (simp add: NSdifferentiable_def)
huffman@20755
  1200
huffman@20755
  1201
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
huffman@20755
  1202
by (force simp add: NSdifferentiable_def)
huffman@20755
  1203
huffman@20755
  1204
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
huffman@20755
  1205
  apply (unfold differentiable_def)
huffman@20755
  1206
  apply (rule_tac x=0 in exI)
huffman@20755
  1207
  apply simp
huffman@20755
  1208
  done
huffman@20755
  1209
huffman@20755
  1210
lemma differentiable_sum:
huffman@20755
  1211
  assumes "f differentiable x"
huffman@20755
  1212
  and "g differentiable x"
huffman@20755
  1213
  shows "(\<lambda>x. f x + g x) differentiable x"
huffman@20755
  1214
proof -
huffman@20755
  1215
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@20755
  1216
  then obtain df where "DERIV f x :> df" ..
huffman@20755
  1217
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@20755
  1218
  then obtain dg where "DERIV g x :> dg" ..
huffman@20755
  1219
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@20755
  1220
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
huffman@20755
  1221
  thus ?thesis by (fold differentiable_def)
huffman@20755
  1222
qed
huffman@20755
  1223
huffman@20755
  1224
lemma differentiable_diff:
huffman@20755
  1225
  assumes "f differentiable x"
huffman@20755
  1226
  and "g differentiable x"
huffman@20755
  1227
  shows "(\<lambda>x. f x - g x) differentiable x"
huffman@20755
  1228
proof -
huffman@20755
  1229
  from prems have "f differentiable x" by simp
huffman@20755
  1230
  moreover
huffman@20755
  1231
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@20755
  1232
  then obtain dg where "DERIV g x :> dg" ..
huffman@20755
  1233
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
huffman@20755
  1234
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
huffman@20755
  1235
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
huffman@20755
  1236
  ultimately 
huffman@20755
  1237
  show ?thesis
huffman@20755
  1238
    by (auto simp: real_diff_def dest: differentiable_sum)
huffman@20755
  1239
qed
huffman@20755
  1240
huffman@20755
  1241
lemma differentiable_mult:
huffman@20755
  1242
  assumes "f differentiable x"
huffman@20755
  1243
  and "g differentiable x"
huffman@20755
  1244
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@20755
  1245
proof -
huffman@20755
  1246
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@20755
  1247
  then obtain df where "DERIV f x :> df" ..
huffman@20755
  1248
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@20755
  1249
  then obtain dg where "DERIV g x :> dg" ..
huffman@20755
  1250
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
huffman@20755
  1251
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
huffman@20755
  1252
  thus ?thesis by (fold differentiable_def)
huffman@20755
  1253
qed
huffman@20755
  1254
huffman@20755
  1255
subsection {*(NS) Increment*}
huffman@20755
  1256
lemma incrementI:
huffman@20755
  1257
      "f NSdifferentiable x ==>
huffman@20755
  1258
      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@20755
  1259
      hypreal_of_real (f x)"
huffman@20755
  1260
by (simp add: increment_def)
huffman@20755
  1261
huffman@20755
  1262
lemma incrementI2: "NSDERIV f x :> D ==>
huffman@20755
  1263
     increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@20755
  1264
     hypreal_of_real (f x)"
huffman@20755
  1265
apply (erule NSdifferentiableI [THEN incrementI])
huffman@20755
  1266
done
huffman@20755
  1267
huffman@20755
  1268
(* The Increment theorem -- Keisler p. 65 *)
huffman@20755
  1269
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
huffman@20755
  1270
      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
huffman@20755
  1271
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
huffman@20755
  1272
apply (drule bspec, auto)
huffman@20755
  1273
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
huffman@20755
  1274
apply (frule_tac b1 = "hypreal_of_real (D) + y"
huffman@20755
  1275
        in hypreal_mult_right_cancel [THEN iffD2])
huffman@20755
  1276
apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl)
huffman@20755
  1277
apply assumption
huffman@20755
  1278
apply (simp add: times_divide_eq_right [symmetric])
huffman@20755
  1279
apply (auto simp add: left_distrib)
huffman@20755
  1280
done
huffman@20755
  1281
huffman@20755
  1282
lemma increment_thm2:
huffman@20755
  1283
     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@20755
  1284
      ==> \<exists>e \<in> Infinitesimal. increment f x h =
huffman@20755
  1285
              hypreal_of_real(D)*h + e*h"
huffman@20755
  1286
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
huffman@20755
  1287
huffman@20755
  1288
huffman@20755
  1289
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@20755
  1290
      ==> increment f x h \<approx> 0"
huffman@20755
  1291
apply (drule increment_thm2,
huffman@20755
  1292
       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
huffman@20755
  1293
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@20755
  1294
done
huffman@20755
  1295
huffman@20755
  1296
subsection {* Nested Intervals and Bisection *}
huffman@20755
  1297
paulson@15234
  1298
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
paulson@15234
  1299
     All considerably tidied by lcp.*}
paulson@14477
  1300
paulson@14477
  1301
lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
paulson@15251
  1302
apply (induct "no")
paulson@14477
  1303
apply (auto intro: order_trans)
paulson@14477
  1304
done
paulson@14477
  1305
paulson@14477
  1306
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
paulson@14477
  1307
         \<forall>n. g(Suc n) \<le> g(n);
paulson@14477
  1308
         \<forall>n. f(n) \<le> g(n) |]
huffman@20552
  1309
      ==> Bseq (f :: nat \<Rightarrow> real)"
paulson@14477
  1310
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
paulson@14477
  1311
apply (induct_tac "n")
paulson@14477
  1312
apply (auto intro: order_trans)
paulson@15234
  1313
apply (rule_tac y = "g (Suc na)" in order_trans)
paulson@14477
  1314
apply (induct_tac [2] "na")
paulson@14477
  1315
apply (auto intro: order_trans)
paulson@14477
  1316
done
paulson@14477
  1317
paulson@14477
  1318
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
paulson@14477
  1319
         \<forall>n. g(Suc n) \<le> g(n);
paulson@14477
  1320
         \<forall>n. f(n) \<le> g(n) |]
huffman@20552
  1321
      ==> Bseq (g :: nat \<Rightarrow> real)"
paulson@14477
  1322
apply (subst Bseq_minus_iff [symmetric])
paulson@15234
  1323
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
paulson@14477
  1324
apply auto
paulson@14477
  1325
done
paulson@14477
  1326
huffman@20693
  1327
lemma f_inc_imp_le_lim:
huffman@20693
  1328
  fixes f :: "nat \<Rightarrow> real"
huffman@20693
  1329
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
paulson@14477
  1330
apply (rule linorder_not_less [THEN iffD1])
paulson@14477
  1331
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
paulson@14477
  1332
apply (drule real_less_sum_gt_zero)
paulson@14477
  1333
apply (drule_tac x = "f n + - lim f" in spec, safe)
paulson@14477
  1334
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
paulson@14477
  1335
apply (subgoal_tac "lim f \<le> f (no + n) ")
paulson@14477
  1336
apply (drule_tac no=no and m=n in lemma_f_mono_add)
paulson@14477
  1337
apply (auto simp add: add_commute)
webertj@20254
  1338
apply (induct_tac "no")
webertj@20254
  1339
apply simp
webertj@20254
  1340
apply (auto intro: order_trans simp add: diff_minus abs_if)
paulson@14477
  1341
done
paulson@14477
  1342
paulson@14477
  1343
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
paulson@14477
  1344
apply (rule LIMSEQ_minus [THEN limI])
paulson@14477
  1345
apply (simp add: convergent_LIMSEQ_iff)
paulson@14477
  1346
done
paulson@14477
  1347
huffman@20693
  1348
lemma g_dec_imp_lim_le:
huffman@20693
  1349
  fixes g :: "nat \<Rightarrow> real"
huffman@20693
  1350
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
paulson@14477
  1351
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
paulson@15234
  1352
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
paulson@14477
  1353
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
paulson@14477
  1354
done
paulson@14477
  1355
paulson@14477
  1356
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
paulson@14477
  1357
         \<forall>n. g(Suc n) \<le> g(n);
paulson@14477
  1358
         \<forall>n. f(n) \<le> g(n) |]
huffman@20552
  1359
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
paulson@14477
  1360
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
paulson@14477
  1361
apply (subgoal_tac "monoseq f & monoseq g")
paulson@14477
  1362
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
paulson@14477
  1363
apply (subgoal_tac "Bseq f & Bseq g")
paulson@14477
  1364
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
paulson@14477
  1365
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
paulson@14477
  1366
apply (rule_tac x = "lim f" in exI)
paulson@14477
  1367
apply (rule_tac x = "lim g" in exI)
paulson@14477
  1368
apply (auto intro: LIMSEQ_le)
paulson@14477
  1369
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
paulson@14477
  1370
done
paulson@14477
  1371
paulson@14477
  1372
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
paulson@14477
  1373
         \<forall>n. g(Suc n) \<le> g(n);
paulson@14477
  1374
         \<forall>n. f(n) \<le> g(n);
paulson@14477
  1375
         (%n. f(n) - g(n)) ----> 0 |]
huffman@20552
  1376
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
paulson@14477
  1377
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
paulson@14477
  1378
apply (drule lemma_nest, auto)
paulson@14477
  1379
apply (subgoal_tac "l = m")
paulson@14477
  1380
apply (drule_tac [2] X = f in LIMSEQ_diff)
paulson@14477
  1381
apply (auto intro: LIMSEQ_unique)
paulson@14477
  1382
done
paulson@14477
  1383
paulson@14477
  1384
text{*The universal quantifiers below are required for the declaration
paulson@14477
  1385
  of @{text Bolzano_nest_unique} below.*}
paulson@14477
  1386
paulson@14477
  1387
lemma Bolzano_bisect_le:
paulson@14477
  1388
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
paulson@14477
  1389
apply (rule allI)
paulson@14477
  1390
apply (induct_tac "n")
paulson@14477
  1391
apply (auto simp add: Let_def split_def)
paulson@14477
  1392
done
paulson@14477
  1393
paulson@14477
  1394
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
paulson@14477
  1395
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
paulson@14477
  1396
apply (rule allI)
paulson@14477
  1397
apply (induct_tac "n")
paulson@14477
  1398
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
paulson@14477
  1399
done
paulson@14477
  1400
paulson@14477
  1401
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
paulson@14477
  1402
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
paulson@14477
  1403
apply (rule allI)
paulson@14477
  1404
apply (induct_tac "n")
nipkow@15539
  1405
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
paulson@14477
  1406
done
paulson@14477
  1407
kleing@19023
  1408
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
nipkow@15539
  1409
apply (auto)
paulson@14477
  1410
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
nipkow@15539
  1411
apply (simp)
paulson@14477
  1412
done
paulson@14477
  1413
paulson@14477
  1414
lemma Bolzano_bisect_diff:
paulson@14477
  1415
     "a \<le> b ==>
paulson@14477
  1416
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
paulson@14477
  1417
      (b-a) / (2 ^ n)"
paulson@15251
  1418
apply (induct "n")
paulson@14477
  1419
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
paulson@14477
  1420
done
paulson@14477
  1421
paulson@14477
  1422
lemmas Bolzano_nest_unique =
paulson@14477
  1423
    lemma_nest_unique
paulson@14477
  1424
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
paulson@14477
  1425
paulson@14477
  1426
paulson@14477
  1427
lemma not_P_Bolzano_bisect:
paulson@14477
  1428
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
paulson@14477
  1429
      and notP: "~ P(a,b)"
paulson@14477
  1430
      and le:   "a \<le> b"
paulson@14477
  1431
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
paulson@14477
  1432
proof (induct n)
paulson@14477
  1433
  case 0 thus ?case by simp
paulson@14477
  1434
 next
paulson@14477
  1435
  case (Suc n)
paulson@14477
  1436
  thus ?case
paulson@15228
  1437
 by (auto simp del: surjective_pairing [symmetric]
paulson@15228
  1438
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
paulson@15228
  1439
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
paulson@14477
  1440
qed
paulson@14477
  1441
paulson@14477
  1442
(*Now we re-package P_prem as a formula*)
paulson@14477
  1443
lemma not_P_Bolzano_bisect':
paulson@14477
  1444
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
paulson@14477
  1445
         ~ P(a,b);  a \<le> b |] ==>
paulson@14477
  1446
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
paulson@14477
  1447
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
paulson@14477
  1448
paulson@14477
  1449
paulson@14477
  1450
paulson@14477
  1451
lemma lemma_BOLZANO:
paulson@14477
  1452
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
paulson@14477
  1453
         \<forall>x. \<exists>d::real. 0 < d &
paulson@14477
  1454
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
paulson@14477
  1455
         a \<le> b |]
paulson@14477
  1456
      ==> P(a,b)"
paulson@14477
  1457
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
paulson@14477
  1458
apply (rule LIMSEQ_minus_cancel)
paulson@14477
  1459
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
paulson@14477
  1460
apply (rule ccontr)
paulson@14477
  1461
apply (drule not_P_Bolzano_bisect', assumption+)
paulson@14477
  1462
apply (rename_tac "l")
paulson@14477
  1463
apply (drule_tac x = l in spec, clarify)
paulson@14477
  1464
apply (simp add: LIMSEQ_def)
paulson@14477
  1465
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
paulson@14477
  1466
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
paulson@15228
  1467
apply (drule real_less_half_sum, auto)
paulson@14477
  1468
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
paulson@14477
  1469
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
paulson@14477
  1470
apply safe
paulson@14477
  1471
apply (simp_all (no_asm_simp))
paulson@15234
  1472
apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
paulson@14477
  1473
apply (simp (no_asm_simp) add: abs_if)
paulson@14477
  1474
apply (rule real_sum_of_halves [THEN subst])
paulson@14477
  1475
apply (rule add_strict_mono)
paulson@14477
  1476
apply (simp_all add: diff_minus [symmetric])
paulson@14477
  1477
done
paulson@14477
  1478
paulson@14477
  1479
paulson@14477
  1480
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
paulson@14477
  1481
       (\<forall>x. \<exists>d::real. 0 < d &
paulson@14477
  1482
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
paulson@14477
  1483
      --> (\<forall>a b. a \<le> b --> P(a,b))"
paulson@14477
  1484
apply clarify
paulson@14477
  1485
apply (blast intro: lemma_BOLZANO)
paulson@14477
  1486
done
paulson@14477
  1487
paulson@14477
  1488
huffman@20755
  1489
subsection {* Intermediate Value Theorem *}
huffman@20755
  1490
huffman@20755
  1491
text {*Prove Contrapositive by Bisection*}
paulson@14477
  1492
huffman@20561
  1493
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
paulson@14477
  1494
         a \<le> b;
paulson@14477
  1495
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
paulson@14477
  1496
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
paulson@14477
  1497
apply (rule contrapos_pp, assumption)
paulson@14477
  1498
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
paulson@14477
  1499
apply safe
paulson@14477
  1500
apply simp_all
paulson@14477
  1501
apply (simp add: isCont_iff LIM_def)
paulson@14477
  1502
apply (rule ccontr)
paulson@14477
  1503
apply (subgoal_tac "a \<le> x & x \<le> b")
paulson@14477
  1504
 prefer 2
paulson@15228
  1505
 apply simp
paulson@14477
  1506
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
paulson@14477
  1507
apply (drule_tac x = x in spec)+
paulson@14477
  1508
apply simp
nipkow@15360
  1509
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
paulson@14477
  1510
apply safe
paulson@14477
  1511
apply simp
paulson@14477
  1512
apply (drule_tac x = s in spec, clarify)
paulson@14477
  1513
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
paulson@14477
  1514
apply (drule_tac x = "ba-x" in spec)
paulson@14477
  1515
apply (simp_all add: abs_if)
paulson@14477
  1516
apply (drule_tac x = "aa-x" in spec)
paulson@14477
  1517
apply (case_tac "x \<le> aa", simp_all)
paulson@14477
  1518
done
paulson@14477
  1519
huffman@20561
  1520
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
paulson@14477
  1521
         a \<le> b;
paulson@14477
  1522
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
paulson@14477
  1523
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
paulson@15228
  1524
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
paulson@14477
  1525
apply (drule IVT [where f = "%x. - f x"], assumption)
paulson@14477
  1526
apply (auto intro: isCont_minus)
paulson@14477
  1527
done
paulson@14477
  1528
paulson@14477
  1529
(*HOL style here: object-level formulations*)
huffman@20561
  1530
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
paulson@14477
  1531
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
paulson@14477
  1532
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
paulson@14477
  1533
apply (blast intro: IVT)
paulson@14477
  1534
done
paulson@14477
  1535
huffman@20561
  1536
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
paulson@14477
  1537
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
paulson@14477
  1538
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
paulson@14477
  1539
apply (blast intro: IVT2)
paulson@14477
  1540
done
paulson@14477
  1541
huffman@20755
  1542
text{*By bisection, function continuous on closed interval is bounded above*}
paulson@14477
  1543
paulson@14477
  1544
lemma isCont_bounded:
paulson@14477
  1545
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@20561
  1546
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
paulson@15234
  1547
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
paulson@14477
  1548
apply safe
paulson@14477
  1549
apply simp_all
paulson@14477
  1550
apply (rename_tac x xa ya M Ma)
paulson@14477
  1551
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
paulson@14477
  1552
apply (rule_tac x = Ma in exI, clarify)
paulson@14477
  1553
apply (cut_tac x = xb and y = xa in linorder_linear, force)
paulson@14477
  1554
apply (rule_tac x = M in exI, clarify)
paulson@14477
  1555
apply (cut_tac x = xb and y = xa in linorder_linear, force)
paulson@14477
  1556
apply (case_tac "a \<le> x & x \<le> b")
paulson@14477
  1557
apply (rule_tac [2] x = 1 in exI)
paulson@14477
  1558
prefer 2 apply force
paulson@14477
  1559
apply (simp add: LIM_def isCont_iff)
paulson@14477
  1560
apply (drule_tac x = x in spec, auto)
paulson@14477
  1561
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
paulson@14477
  1562
apply (drule_tac x = 1 in spec, auto)
paulson@14477
  1563
apply (rule_tac x = s in exI, clarify)
paulson@14477
  1564
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
paulson@14477
  1565
apply (drule_tac x = "xa-x" in spec)
webertj@20217
  1566
apply (auto simp add: abs_ge_self)
paulson@14477
  1567
done
paulson@14477
  1568
paulson@15234
  1569
text{*Refine the above to existence of least upper bound*}
paulson@14477
  1570
paulson@14477
  1571
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
paulson@14477
  1572
      (\<exists>t. isLub UNIV S t)"
paulson@15234
  1573
by (blast intro: reals_complete)
paulson@14477
  1574
paulson@14477
  1575
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@20561
  1576
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
paulson@14477
  1577
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
kleing@19023
  1578
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
paulson@15234
  1579
        in lemma_reals_complete)
paulson@14477
  1580
apply auto
paulson@14477
  1581
apply (drule isCont_bounded, assumption)
paulson@14477
  1582
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
paulson@14477
  1583
apply (rule exI, auto)
paulson@15228
  1584
apply (auto dest!: spec simp add: linorder_not_less)
paulson@14477
  1585
done
paulson@14477
  1586
paulson@15234
  1587
text{*Now show that it attains its upper bound*}
paulson@14477
  1588
paulson@14477
  1589
lemma isCont_eq_Ub:
paulson@14477
  1590
  assumes le: "a \<le> b"
huffman@20561
  1591
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@20552
  1592
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
paulson@14477
  1593
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
paulson@14477
  1594
proof -
paulson@14477
  1595
  from isCont_has_Ub [OF le con]
paulson@14477
  1596
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
paulson@14477
  1597
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
paulson@14477
  1598
  show ?thesis
paulson@14477
  1599
  proof (intro exI, intro conjI)
paulson@14477
  1600
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
paulson@15228
  1601
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
paulson@14477
  1602
    proof (rule ccontr)
paulson@14477
  1603
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
paulson@14477
  1604
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
nipkow@15195
  1605
        by (fastsimp simp add: linorder_not_le [symmetric])
paulson@14477
  1606
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
paulson@14477
  1607
        by (auto simp add: isCont_inverse isCont_diff con)
paulson@14477
  1608
      from isCont_bounded [OF le this]
paulson@14477
  1609
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
paulson@14477
  1610
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
paulson@15228
  1611
        by (simp add: M3 compare_rls)
paulson@15228
  1612
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
paulson@15228
  1613
        by (auto intro: order_le_less_trans [of _ k])
paulson@15228
  1614
      with Minv
paulson@15228
  1615
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
paulson@14477
  1616
        by (intro strip less_imp_inverse_less, simp_all)
paulson@15228
  1617
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
paulson@14477
  1618
        by simp
paulson@15228
  1619
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
paulson@14477
  1620
        by (simp, arith)
paulson@14477
  1621
      from M2 [OF this]
paulson@14477
  1622
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
paulson@14477
  1623
      thus False using invlt [of x] by force
paulson@14477
  1624
    qed
paulson@14477
  1625
  qed
paulson@14477
  1626
qed
paulson@14477
  1627
paulson@14477
  1628
paulson@15234
  1629
text{*Same theorem for lower bound*}
paulson@14477
  1630
paulson@14477
  1631
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@20561
  1632
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
paulson@14477
  1633
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
paulson@14477
  1634
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
paulson@14477
  1635
prefer 2 apply (blast intro: isCont_minus)
paulson@15234
  1636
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
paulson@14477
  1637
apply safe
paulson@14477
  1638
apply auto
paulson@14477
  1639
done
paulson@14477
  1640
paulson@14477
  1641
paulson@15234
  1642
text{*Another version.*}
paulson@14477
  1643
paulson@14477
  1644
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@20561
  1645
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
paulson@14477
  1646
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
paulson@14477
  1647
apply (frule isCont_eq_Lb)
paulson@14477
  1648
apply (frule_tac [2] isCont_eq_Ub)
paulson@14477
  1649
apply (assumption+, safe)
paulson@14477
  1650
apply (rule_tac x = "f x" in exI)
paulson@14477
  1651
apply (rule_tac x = "f xa" in exI, simp, safe)
paulson@14477
  1652
apply (cut_tac x = x and y = xa in linorder_linear, safe)
paulson@14477
  1653
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
paulson@14477
  1654
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
paulson@14477
  1655
apply (rule_tac [2] x = xb in exI)
paulson@14477
  1656
apply (rule_tac [4] x = xb in exI, simp_all)
paulson@14477
  1657
done
paulson@14477
  1658
paulson@15003
  1659
huffman@20755
  1660
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
paulson@14477
  1661
paulson@14477
  1662
lemma DERIV_left_inc:
paulson@15003
  1663
  assumes der: "DERIV f x :> l"
paulson@15003
  1664
      and l:   "0 < l"
nipkow@15360
  1665
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
paulson@15003
  1666
proof -
paulson@15003
  1667
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
nipkow@15360
  1668
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
paulson@15003
  1669
    by (simp add: diff_minus)
paulson@15003
  1670
  then obtain s
paulson@15228
  1671
        where s:   "0 < s"
paulson@15003
  1672
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
paulson@15003
  1673
    by auto
paulson@15003
  1674
  thus ?thesis
paulson@15003
  1675
  proof (intro exI conjI strip)
paulson@15003
  1676
    show "0<s" .
paulson@15003
  1677
    fix h::real
nipkow@15360
  1678
    assume "0 < h" "h < s"
paulson@15228
  1679
    with all [of h] show "f x < f (x+h)"
paulson@15228
  1680
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
kleing@19023
  1681
    split add: split_if_asm)
paulson@15228
  1682
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
paulson@15228
  1683
      with l
paulson@15003
  1684
      have "0 < (f (x+h) - f x) / h" by arith
paulson@15003
  1685
      thus "f x < f (x+h)"
kleing@19023
  1686
  by (simp add: pos_less_divide_eq h)
paulson@15003
  1687
    qed
paulson@15003
  1688
  qed
paulson@15003
  1689
qed
paulson@14477
  1690
paulson@14477
  1691
lemma DERIV_left_dec:
paulson@14477
  1692
  assumes der: "DERIV f x :> l"
paulson@14477
  1693
      and l:   "l < 0"
nipkow@15360
  1694
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
paulson@14477
  1695
proof -
paulson@14477
  1696
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
nipkow@15360
  1697
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
paulson@14477
  1698
    by (simp add: diff_minus)
paulson@14477
  1699
  then obtain s
paulson@15228
  1700
        where s:   "0 < s"
paulson@14477
  1701
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
paulson@14477
  1702
    by auto
paulson@14477
  1703
  thus ?thesis
paulson@14477
  1704
  proof (intro exI conjI strip)
paulson@14477
  1705
    show "0<s" .
paulson@14477
  1706
    fix h::real
nipkow@15360
  1707
    assume "0 < h" "h < s"
paulson@15228
  1708
    with all [of "-h"] show "f x < f (x-h)"
paulson@15228
  1709
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
kleing@19023
  1710
    split add: split_if_asm)
paulson@15228
  1711
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
paulson@15228
  1712
      with l
paulson@14477
  1713
      have "0 < (f (x-h) - f x) / h" by arith
paulson@14477
  1714
      thus "f x < f (x-h)"
kleing@19023
  1715
  by (simp add: pos_less_divide_eq h)
paulson@14477
  1716
    qed
paulson@14477
  1717
  qed
paulson@14477
  1718
qed
paulson@14477
  1719
paulson@15228
  1720
lemma DERIV_local_max:
paulson@14477
  1721
  assumes der: "DERIV f x :> l"
paulson@14477
  1722
      and d:   "0 < d"
paulson@14477
  1723
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
paulson@14477
  1724
  shows "l = 0"
paulson@14477
  1725
proof (cases rule: linorder_cases [of l 0])
paulson@14477
  1726
  case equal show ?thesis .
paulson@14477
  1727
next
paulson@14477
  1728
  case less
paulson@14477
  1729
  from DERIV_left_dec [OF der less]
paulson@14477
  1730
  obtain d' where d': "0 < d'"
nipkow@15360
  1731
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
paulson@14477
  1732
  from real_lbound_gt_zero [OF d d']
paulson@14477
  1733
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
paulson@15228
  1734
  with lt le [THEN spec [where x="x-e"]]
paulson@14477
  1735
  show ?thesis by (auto simp add: abs_if)
paulson@14477
  1736
next
paulson@14477
  1737
  case greater
paulson@14477
  1738
  from DERIV_left_inc [OF der greater]
paulson@14477
  1739
  obtain d' where d': "0 < d'"
nipkow@15360
  1740
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
paulson@14477
  1741
  from real_lbound_gt_zero [OF d d']
paulson@14477
  1742
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
paulson@14477
  1743
  with lt le [THEN spec [where x="x+e"]]
paulson@14477
  1744
  show ?thesis by (auto simp add: abs_if)
paulson@14477
  1745
qed
paulson@14477
  1746
paulson@14477
  1747
paulson@14477
  1748
text{*Similar theorem for a local minimum*}
paulson@14477
  1749
lemma DERIV_local_min:
paulson@14477
  1750
     "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
paulson@14477
  1751
by (drule DERIV_minus [THEN DERIV_local_max], auto)
paulson@14477
  1752
paulson@14477
  1753
paulson@14477
  1754
text{*In particular, if a function is locally flat*}
paulson@14477
  1755
lemma DERIV_local_const:
paulson@14477
  1756
     "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
paulson@14477
  1757
by (auto dest!: DERIV_local_max)
paulson@14477
  1758
paulson@14477
  1759
text{*Lemma about introducing open ball in open interval*}
paulson@14477
  1760
lemma lemma_interval_lt:
paulson@15228
  1761
     "[| a < x;  x < b |]
paulson@14477
  1762
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
paulson@14477
  1763
apply (simp add: abs_interval_iff)
paulson@14477
  1764
apply (insert linorder_linear [of "x-a" "b-x"], safe)
paulson@14477
  1765
apply (rule_tac x = "x-a" in exI)
paulson@14477
  1766
apply (rule_tac [2] x = "b-x" in exI, auto)
paulson@14477
  1767
done
paulson@14477
  1768
paulson@14477
  1769
lemma lemma_interval: "[| a < x;  x < b |] ==>
paulson@14477
  1770
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
paulson@14477
  1771
apply (drule lemma_interval_lt, auto)
paulson@14477
  1772
apply (auto intro!: exI)
paulson@14477
  1773
done
paulson@14477
  1774
paulson@14477
  1775
text{*Rolle's Theorem.
paulson@15228
  1776
   If @{term f} is defined and continuous on the closed interval
paulson@15228
  1777
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
paulson@14477
  1778
   and @{term "f(a) = f(b)"},
paulson@14477
  1779
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
paulson@15228
  1780
theorem Rolle:
paulson@14477
  1781
  assumes lt: "a < b"
paulson@14477
  1782
      and eq: "f(a) = f(b)"
paulson@14477
  1783
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
paulson@14477
  1784
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
paulson@14477
  1785
  shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
paulson@14477
  1786
proof -
paulson@14477
  1787
  have le: "a \<le> b" using lt by simp
paulson@14477
  1788
  from isCont_eq_Ub [OF le con]
paulson@15228
  1789
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
paulson@15228
  1790
             and alex: "a \<le> x" and xleb: "x \<le> b"
paulson@14477
  1791
    by blast
paulson@14477
  1792
  from isCont_eq_Lb [OF le con]
paulson@15228
  1793
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
paulson@15228
  1794
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
paulson@14477
  1795
    by blast
paulson@14477
  1796
  show ?thesis
paulson@14477
  1797
  proof cases
paulson@14477
  1798
    assume axb: "a < x & x < b"
paulson@14477
  1799
        --{*@{term f} attains its maximum within the interval*}
paulson@14477
  1800
    hence ax: "a<x" and xb: "x<b" by auto
paulson@14477
  1801
    from lemma_interval [OF ax xb]
paulson@14477
  1802
    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
paulson@14477
  1803
      by blast
paulson@14477
  1804
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
paulson@14477
  1805
      by blast
paulson@14477
  1806
    from differentiableD [OF dif [OF axb]]
paulson@14477
  1807
    obtain l where der: "DERIV f x :> l" ..
paulson@15228
  1808
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
paulson@14477
  1809
        --{*the derivative at a local maximum is zero*}
paulson@14477
  1810
    thus ?thesis using ax xb der by auto
paulson@14477
  1811
  next
paulson@14477
  1812
    assume notaxb: "~ (a < x & x < b)"
paulson@14477
  1813
    hence xeqab: "x=a | x=b" using alex xleb by arith
paulson@15228
  1814
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
paulson@14477
  1815
    show ?thesis
paulson@14477
  1816
    proof cases
paulson@14477
  1817
      assume ax'b: "a < x' & x' < b"
paulson@14477
  1818
        --{*@{term f} attains its minimum within the interval*}
paulson@14477
  1819
      hence ax': "a<x'" and x'b: "x'<b" by auto
paulson@14477
  1820
      from lemma_interval [OF ax' x'b]
paulson@14477
  1821
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
kleing@19023
  1822
  by blast
paulson@14477
  1823
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
kleing@19023
  1824
  by blast
paulson@14477
  1825
      from differentiableD [OF dif [OF ax'b]]
paulson@14477
  1826
      obtain l where der: "DERIV f x' :> l" ..
paulson@15228
  1827
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
paulson@14477
  1828
        --{*the derivative at a local minimum is zero*}
paulson@14477
  1829
      thus ?thesis using ax' x'b der by auto
paulson@14477
  1830
    next
paulson@14477
  1831
      assume notax'b: "~ (a < x' & x' < b)"
paulson@14477
  1832
        --{*@{term f} is constant througout the interval*}
paulson@14477
  1833
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
paulson@15228
  1834
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
paulson@14477
  1835
      from dense [OF lt]
paulson@14477
  1836
      obtain r where ar: "a < r" and rb: "r < b" by blast
paulson@14477
  1837
      from lemma_interval [OF ar rb]
paulson@14477
  1838
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
kleing@19023
  1839
  by blast
paulson@15228
  1840
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
paulson@15228
  1841
      proof (clarify)
paulson@14477
  1842
        fix z::real
paulson@14477
  1843
        assume az: "a \<le> z" and zb: "z \<le> b"
paulson@14477
  1844
        show "f z = f b"
paulson@14477
  1845
        proof (rule order_antisym)
nipkow@15195
  1846
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
nipkow@15195
  1847
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
paulson@14477
  1848
        qed
paulson@14477
  1849
      qed
paulson@14477
  1850
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
paulson@14477
  1851
      proof (intro strip)
paulson@14477
  1852
        fix y::real
paulson@14477
  1853
        assume lt: "\<bar>r-y\<bar> < d"
paulson@15228
  1854
        hence "f y = f b" by (simp add: eq_fb bound)
paulson@14477
  1855
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
paulson@14477
  1856
      qed
paulson@14477
  1857
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
paulson@14477
  1858
      obtain l where der: "DERIV f r :> l" ..
paulson@15228
  1859
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
paulson@14477
  1860
        --{*the derivative of a constant function is zero*}
paulson@14477
  1861
      thus ?thesis using ar rb der by auto
paulson@14477
  1862
    qed
paulson@14477
  1863
  qed
paulson@14477
  1864
qed
paulson@14477
  1865
paulson@14477
  1866
paulson@14477
  1867
subsection{*Mean Value Theorem*}
paulson@14477
  1868
paulson@14477
  1869
lemma lemma_MVT:
paulson@14477
  1870
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
paulson@14477
  1871
proof cases
paulson@14477
  1872
  assume "a=b" thus ?thesis by simp
paulson@14477
  1873
next
paulson@15228
  1874
  assume "a\<noteq>b"
paulson@14477
  1875
  hence ba: "b-a \<noteq> 0" by arith
paulson@14477
  1876
  show ?thesis
paulson@14477
  1877
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
kleing@19023
  1878
        simp add: right_diff_distrib,
paulson@15234
  1879
        simp add: left_diff_distrib)
paulson@14477
  1880
qed
paulson@14477
  1881
paulson@15228
  1882
theorem MVT:
paulson@14477
  1883
  assumes lt:  "a < b"
paulson@14477
  1884
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
paulson@14477
  1885
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
paulson@14477
  1886
  shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
paulson@14477
  1887
                   (f(b) - f(a) = (b-a) * l)"
paulson@14477
  1888
proof -
paulson@14477
  1889
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
paulson@14477
  1890
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
paulson@15228
  1891
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
paulson@14477
  1892
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
paulson@14477
  1893
  proof (clarify)
paulson@14477
  1894
    fix x::real
paulson@14477
  1895
    assume ax: "a < x" and xb: "x < b"
paulson@14477
  1896
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
paulson@14477
  1897
    obtain l where der: "DERIV f x :> l" ..
paulson@14477
  1898
    show "?F differentiable x"
paulson@14477
  1899
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
paulson@15228
  1900
          blast intro: DERIV_diff DERIV_cmult_Id der)
paulson@15228
  1901
  qed
paulson@14477
  1902
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
paulson@15228
  1903
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
paulson@14477
  1904
    by blast
paulson@14477
  1905
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
paulson@14477
  1906
    by (rule DERIV_cmult_Id)
paulson@15228
  1907
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
paulson@14477
  1908
                   :> 0 + (f b - f a) / (b - a)"
paulson@14477
  1909
    by (rule DERIV_add [OF der])
paulson@15228
  1910
  show ?thesis
paulson@14477
  1911
  proof (intro exI conjI)
paulson@14477
  1912
    show "a < z" .
paulson@14477
  1913
    show "z < b" .
nipkow@15539
  1914
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
paulson@14477
  1915
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
paulson@14477
  1916
  qed
paulson@14477
  1917
qed
paulson@14477
  1918
paulson@14477
  1919
paulson@14477
  1920
text{*A function is constant if its derivative is 0 over an interval.*}
paulson@14477
  1921
paulson@14477
  1922
lemma DERIV_isconst_end: "[| a < b;
paulson@14477
  1923
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
paulson@14477
  1924
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
nipkow@15360
  1925
        ==> f b = f a"
paulson@14477
  1926
apply (drule MVT, assumption)
paulson@14477
  1927
apply (blast intro: differentiableI)
paulson@14477
  1928
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
paulson@14477
  1929
done
paulson@14477
  1930
paulson@14477
  1931
lemma DERIV_isconst1: "[| a < b;
paulson@14477
  1932
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
paulson@14477
  1933
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
paulson@14477
  1934
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
paulson@14477
  1935
apply safe
paulson@14477
  1936
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
paulson@14477
  1937
apply (drule_tac b = x in DERIV_isconst_end, auto)
paulson@14477
  1938
done
paulson@14477
  1939
paulson@14477
  1940
lemma DERIV_isconst2: "[| a < b;
paulson@14477
  1941
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
paulson@14477
  1942
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
paulson@14477
  1943
         a \<le> x; x \<le> b |]
paulson@14477
  1944
        ==> f x = f a"
paulson@14477
  1945
apply (blast dest: DERIV_isconst1)
paulson@14477
  1946
done
paulson@14477
  1947
paulson@14477
  1948
lemma DERIV_isconst_all: "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
paulson@14477
  1949
apply (rule linorder_cases [of x y])
paulson@14477
  1950
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
paulson@14477
  1951
done
paulson@14477
  1952
paulson@14477
  1953
lemma DERIV_const_ratio_const:
paulson@14477
  1954
     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
paulson@14477
  1955
apply (rule linorder_cases [of a b], auto)
paulson@14477
  1956
apply (drule_tac [!] f = f in MVT)
paulson@14477
  1957
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
paulson@14477
  1958
apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
paulson@14477
  1959
done
paulson@14477
  1960
paulson@14477
  1961
lemma DERIV_const_ratio_const2:
paulson@14477
  1962
     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
paulson@14477
  1963
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
nipkow@15539
  1964
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
paulson@14477
  1965
done
paulson@14477
  1966
paulson@15228
  1967
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
kleing@19023
  1968
by (simp)
paulson@14477
  1969
paulson@15228
  1970
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
kleing@19023
  1971
by (simp)
paulson@14477
  1972
paulson@14477
  1973
text{*Gallileo's "trick": average velocity = av. of end velocities*}
paulson@14477
  1974
paulson@14477
  1975
lemma DERIV_const_average:
paulson@14477
  1976
  assumes neq: "a \<noteq> (b::real)"
paulson@14477
  1977
      and der: "\<forall>x. DERIV v x :> k"
paulson@14477
  1978
  shows "v ((a + b)/2) = (v a + v b)/2"
paulson@14477
  1979
proof (cases rule: linorder_cases [of a b])
paulson@14477
  1980
  case equal with neq show ?thesis by simp
paulson@14477
  1981
next
paulson@14477
  1982
  case less
paulson@14477
  1983
  have "(v b - v a) / (b - a) = k"
paulson@14477
  1984
    by (rule DERIV_const_ratio_const2 [OF neq der])
paulson@15228
  1985
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
paulson@14477
  1986
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
paulson@14477
  1987
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
paulson@14477
  1988
  ultimately show ?thesis using neq by force
paulson@14477
  1989
next
paulson@14477
  1990
  case greater
paulson@14477
  1991
  have "(v b - v a) / (b - a) = k"
paulson@14477
  1992
    by (rule DERIV_const_ratio_const2 [OF neq der])
paulson@15228
  1993
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
paulson@14477
  1994
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
paulson@14477
  1995
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
paulson@15228
  1996
  ultimately show ?thesis using neq by (force simp add: add_commute)
paulson@14477
  1997
qed
paulson@14477
  1998
paulson@14477
  1999
paulson@14477
  2000
text{*Dull lemma: an continuous injection on an interval must have a
paulson@14477
  2001
strict maximum at an end point, not in the middle.*}
paulson@14477
  2002
paulson@14477
  2003
lemma lemma_isCont_inj:
huffman@20552
  2004
  fixes f :: "real \<Rightarrow> real"
paulson@14477
  2005
  assumes d: "0 < d"
paulson@14477
  2006
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
paulson@14477
  2007
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
paulson@14477
  2008
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
paulson@14477
  2009
proof (rule ccontr)
paulson@14477
  2010
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
paulson@15228
  2011
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
paulson@14477
  2012
  show False
paulson@14477
  2013
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
paulson@14477
  2014
    case le
paulson@14477
  2015
    from d cont all [of "x+d"]
paulson@15228
  2016
    have flef: "f(x+d) \<le> f x"
paulson@15228
  2017
     and xlex: "x - d \<le> x"
paulson@15228
  2018
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
paulson@14477
  2019
       by (auto simp add: abs_if)
paulson@14477
  2020
    from IVT [OF le flef xlex cont']
paulson@14477
  2021
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
paulson@14477
  2022
    moreover
paulson@14477
  2023
    hence "g(f x') = g (f(x+d))" by simp
paulson@14477
  2024
    ultimately show False using d inj [of x'] inj [of "x+d"]
paulson@14477
  2025
      by (simp add: abs_le_interval_iff)
paulson@14477
  2026
  next
paulson@14477
  2027
    case ge
paulson@14477
  2028
    from d cont all [of "x-d"]
paulson@15228
  2029
    have flef: "f(x-d) \<le> f x"
paulson@15228
  2030
     and xlex: "x \<le> x+d"
paulson@15228
  2031
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
paulson@14477
  2032
       by (auto simp add: abs_if)
paulson@14477
  2033
    from IVT2 [OF ge flef xlex cont']
paulson@14477
  2034
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
paulson@14477
  2035
    moreover
paulson@14477
  2036
    hence "g(f x') = g (f(x-d))" by simp
paulson@14477
  2037
    ultimately show False using d inj [of x'] inj [of "x-d"]
paulson@14477
  2038
      by (simp add: abs_le_interval_iff)
paulson@14477
  2039
  qed
paulson@14477
  2040
qed
paulson@14477
  2041
paulson@14477
  2042
paulson@14477
  2043
text{*Similar version for lower bound.*}
paulson@14477
  2044
paulson@14477
  2045
lemma lemma_isCont_inj2:
huffman@20552
  2046
  fixes f g :: "real \<Rightarrow> real"
huffman@20552
  2047
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
paulson@14477
  2048
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
paulson@14477
  2049
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
paulson@14477
  2050
apply (insert lemma_isCont_inj
paulson@14477
  2051
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
paulson@15228
  2052
apply (simp add: isCont_minus linorder_not_le)
paulson@14477
  2053
done
paulson@14477
  2054
paulson@15228
  2055
text{*Show there's an interval surrounding @{term "f(x)"} in
paulson@14477
  2056
@{text "f[[x - d, x + d]]"} .*}
paulson@14477
  2057
paulson@15228
  2058
lemma isCont_inj_range:
huffman@20552
  2059
  fixes f :: "real \<Rightarrow> real"
paulson@14477
  2060
  assumes d: "0 < d"
paulson@14477
  2061
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
paulson@14477
  2062
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
nipkow@15360
  2063
  shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
paulson@14477
  2064
proof -
paulson@14477
  2065
  have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
paulson@14477
  2066
    by (auto simp add: abs_le_interval_iff)
paulson@14477
  2067
  from isCont_Lb_Ub [OF this]
paulson@15228
  2068
  obtain L M
paulson@14477
  2069
  where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
paulson@14477
  2070
    and all2 [rule_format]:
paulson@14477
  2071
           "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
paulson@14477
  2072
    by auto
paulson@14477
  2073
  with d have "L \<le> f x & f x \<le> M" by simp
paulson@14477
  2074
  moreover have "L \<noteq> f x"
paulson@14477
  2075
  proof -
paulson@14477
  2076
    from lemma_isCont_inj2 [OF d inj cont]
paulson@14477
  2077
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
paulson@14477
  2078
    thus ?thesis using all1 [of u] by arith
paulson@14477
  2079
  qed
paulson@14477
  2080
  moreover have "f x \<noteq> M"
paulson@14477
  2081
  proof -
paulson@14477
  2082
    from lemma_isCont_inj [OF d inj cont]
paulson@14477
  2083
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
paulson@14477
  2084
    thus ?thesis using all1 [of u] by arith
paulson@14477
  2085
  qed
paulson@14477
  2086
  ultimately have "L < f x & f x < M" by arith
paulson@14477
  2087
  hence "0 < f x - L" "0 < M - f x" by arith+
paulson@14477
  2088
  from real_lbound_gt_zero [OF this]
paulson@14477
  2089
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
paulson@14477
  2090
  thus ?thesis
paulson@14477
  2091
  proof (intro exI conjI)
paulson@14477
  2092
    show "0<e" .
paulson@14477
  2093
    show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
paulson@14477
  2094
    proof (intro strip)
paulson@14477
  2095
      fix y::real
paulson@14477
  2096
      assume "\<bar>y - f x\<bar> \<le> e"
paulson@14477
  2097
      with e have "L \<le> y \<and> y \<le> M" by arith
paulson@14477
  2098
      from all2 [OF this]
paulson@14477
  2099
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
paulson@15228
  2100
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
paulson@14477
  2101
        by (force simp add: abs_le_interval_iff)
paulson@14477
  2102
    qed
paulson@14477
  2103
  qed
paulson@14477
  2104
qed
paulson@14477
  2105
paulson@14477
  2106
paulson@14477
  2107
text{*Continuity of inverse function*}
paulson@14477
  2108
paulson@14477
  2109
lemma isCont_inverse_function:
huffman@20561
  2110
  fixes f g :: "real \<Rightarrow> real"
paulson@14477
  2111
  assumes d: "0 < d"
paulson@14477
  2112
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
paulson@14477
  2113
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
paulson@14477
  2114
  shows "isCont g (f x)"
paulson@14477
  2115
proof (simp add: isCont_iff LIM_eq)
paulson@14477
  2116
  show "\<forall>r. 0 < r \<longrightarrow>
nipkow@15360
  2117
         (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
paulson@14477
  2118
  proof (intro strip)
paulson@14477
  2119
    fix r::real
paulson@14477
  2120
    assume r: "0<r"
paulson@14477
  2121
    from real_lbound_gt_zero [OF r d]
paulson@14477
  2122
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
paulson@14477
  2123
    with inj cont
paulson@15228
  2124
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
paulson@14477
  2125
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
paulson@14477
  2126
    from isCont_inj_range [OF e this]
paulson@15228
  2127
    obtain e' where e': "0 < e'"
paulson@14477
  2128
        and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
paulson@14477
  2129
          by blast
nipkow@15360
  2130
    show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
paulson@14477
  2131
    proof (intro exI conjI)
paulson@14477
  2132
      show "0<e'" .
paulson@14477
  2133
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
paulson@14477
  2134
      proof (intro strip)
paulson@14477
  2135
        fix z::real
paulson@14477
  2136
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
paulson@14477
  2137
        with e e_lt e_simps all [rule_format, of "f x + z"]
paulson@14477
  2138
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
paulson@14477
  2139
      qed
paulson@14477
  2140
    qed
paulson@14477
  2141
  qed
paulson@15228
  2142
qed
paulson@14477
  2143
kleing@19023
  2144
theorem GMVT:
kleing@19023
  2145
  assumes alb: "a < b"
kleing@19023
  2146
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
kleing@19023
  2147
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
kleing@19023
  2148
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
kleing@19023
  2149
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
kleing@19023
  2150
  shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
kleing@19023
  2151
proof -
kleing@19023
  2152
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
kleing@19023
  2153
  from prems have "a < b" by simp
kleing@19023
  2154
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
kleing@19023
  2155
  proof -
kleing@19023
  2156
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
kleing@19023
  2157
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
kleing@19023
  2158
      by (auto intro: isCont_mult)
kleing@19023
  2159
    moreover
kleing@19023
  2160
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
kleing@19023
  2161
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
kleing@19023
  2162
      by (auto intro: isCont_mult)
kleing@19023
  2163
    ultimately show ?thesis
kleing@19023
  2164
      by (fastsimp intro: isCont_diff)
kleing@19023
  2165
  qed
kleing@19023
  2166
  moreover
kleing@19023
  2167
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
kleing@19023
  2168
  proof -
kleing@19023
  2169
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
kleing@19023
  2170
    with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
kleing@19023
  2171
    moreover
kleing@19023
  2172
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
kleing@19023
  2173
    with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
kleing@19023
  2174
    ultimately show ?thesis by (simp add: differentiable_diff)
kleing@19023
  2175
  qed
kleing@19023
  2176
  ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
kleing@19023
  2177
  then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
kleing@19023
  2178
  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
kleing@19023
  2179
kleing@19023
  2180
  from cdef have cint: "a < c \<and> c < b" by auto
kleing@19023
  2181
  with gd have "g differentiable c" by simp
kleing@19023
  2182
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
kleing@19023
  2183
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
kleing@19023
  2184
kleing@19023
  2185
  from cdef have "a < c \<and> c < b" by auto
kleing@19023
  2186
  with fd have "f differentiable c" by simp
kleing@19023
  2187
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
kleing@19023
  2188
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
kleing@19023
  2189
kleing@19023
  2190
  from cdef have "DERIV ?h c :> l" by auto
kleing@19023
  2191
  moreover
kleing@19023
  2192
  {
kleing@19023
  2193
    from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
kleing@19023
  2194
      apply (insert DERIV_const [where k="f b - f a"])
kleing@19023
  2195
      apply (drule meta_spec [of _ c])
kleing@19023
  2196
      apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
kleing@19023
  2197
      by simp_all
kleing@19023
  2198
    moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
kleing@19023
  2199
      apply (insert DERIV_const [where k="g b - g a"])
kleing@19023
  2200
      apply (drule meta_spec [of _ c])
kleing@19023
  2201
      apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
kleing@19023
  2202
      by simp_all
kleing@19023
  2203
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
kleing@19023
  2204
      by (simp add: DERIV_diff)
kleing@19023
  2205
  }
kleing@19023
  2206
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
kleing@19023
  2207
kleing@19023
  2208
  {
kleing@19023
  2209
    from cdef have "?h b - ?h a = (b - a) * l" by auto
kleing@19023
  2210
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
kleing@19023
  2211
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
kleing@19023
  2212
  }
kleing@19023
  2213
  moreover
kleing@19023
  2214
  {
kleing@19023
  2215
    have "?h b - ?h a =
kleing@19023
  2216
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
kleing@19023
  2217
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
kleing@19023
  2218
      by (simp add: mult_ac add_ac real_diff_mult_distrib)
kleing@19023
  2219
    hence "?h b - ?h a = 0" by auto
kleing@19023
  2220
  }
kleing@19023
  2221
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
kleing@19023
  2222
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
kleing@19023
  2223
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
kleing@19023
  2224
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
kleing@19023
  2225
kleing@19023
  2226
  with g'cdef f'cdef cint show ?thesis by auto
kleing@19023
  2227
qed
kleing@19023
  2228
kleing@19023
  2229
kleing@19023
  2230
lemma LIMSEQ_SEQ_conv1:
huffman@20561
  2231
  fixes a :: real
kleing@19023
  2232
  assumes "X -- a --> L"
kleing@19023
  2233
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
  2234
proof -
kleing@19023
  2235
  {
huffman@20563
  2236
    from prems have Xdef: "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (X x - L) < r" by (unfold LIM_def)
kleing@19023
  2237
    
kleing@19023
  2238
    fix S
kleing@19023
  2239
    assume as: "(\<forall>n. S n \<noteq> a) \<and> S ----> a"
kleing@19023
  2240
    then have "S ----> a" by auto
huffman@20563
  2241
    then have Sdef: "(\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (S n - a) < r))" by (unfold LIMSEQ_def)
kleing@19023
  2242
    {
kleing@19023
  2243
      fix r
huffman@20563
  2244
      from Xdef have Xdef2: "0 < r --> (\<exists>s > 0. \<forall>x. x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r)" by simp
kleing@19023
  2245
      {
kleing@19023
  2246
        assume rgz: "0 < r"
kleing@19023
  2247
huffman@20563
  2248
        from Xdef2 rgz have "\<exists>s > 0. \<forall>x. x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r" by simp 
huffman@20563
  2249
        then obtain s where sdef: "s > 0 \<and> (\<forall>x. x\<noteq>a \<and> \<bar>x - a\<bar> < s \<longrightarrow> norm (X x - L) < r)" by auto
huffman@20563
  2250
        then have aux: "\<forall>x. x\<noteq>a \<and> \<bar>x - a\<bar> < s \<longrightarrow> norm (X x - L) < r" by auto
kleing@19023
  2251
        {
kleing@19023
  2252
          fix n
huffman@20563
  2253
          from aux have "S n \<noteq> a \<and> \<bar>S n - a\<bar> < s \<longrightarrow> norm (X (S n) - L) < r" by simp
huffman@20563
  2254
          with as have imp2: "\<bar>S n - a\<bar> < s --> norm (X (S n) - L) < r" by auto
kleing@19023
  2255
        }
huffman@20563
  2256
        hence "\<forall>n. \<bar>S n - a\<bar> < s --> norm (X (S n) - L) < r" ..
kleing@19023
  2257
        moreover
huffman@20563
  2258
        from Sdef sdef have imp1: "\<exists>no. \<forall>n. no \<le> n --> \<bar>S n - a\<bar> < s" by auto  
huffman@20563
  2259
        then obtain no where "\<forall>n. no \<le> n --> \<bar>S n - a\<bar> < s" by auto
huffman@20563
  2260
        ultimately have "\<forall>n. no \<le> n \<longrightarrow> norm (X (S n) - L) < r" by simp
huffman@20563
  2261
        hence "\<exists>no. \<forall>n. no \<le> n \<longrightarrow> norm (X (S n) - L) < r" by auto
kleing@19023
  2262
      }
kleing@19023
  2263
    }
huffman@20563
  2264
    hence "(\<forall>r. 0 < r --> (\<exists>no. \<forall>n. no \<le> n --> norm (X (S n) - L) < r))" by simp
kleing@19023
  2265
    hence "(\<lambda>n. X (S n)) ----> L" by (fold LIMSEQ_def)
kleing@19023
  2266
  }
kleing@19023
  2267
  thus ?thesis by simp
kleing@19023
  2268
qed
kleing@19023
  2269
webertj@20432
  2270
ML {* fast_arith_split_limit := 0; *}  (* FIXME *)
webertj@20217
  2271
kleing@19023
  2272
lemma LIMSEQ_SEQ_conv2:
huffman@20561
  2273
  fixes a :: real
kleing@19023
  2274
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
  2275
  shows "X -- a --> L"
kleing@19023
  2276
proof (rule ccontr)
kleing@19023
  2277
  assume "\<not> (X -- a --> L)"
huffman@20563
  2278
  hence "\<not> (\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (X x - L) < r)" by (unfold LIM_def)
huffman@20563
  2279
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. \<not>(x \<noteq> a \<and> \<bar>x - a\<bar> < s --> norm (X x - L) < r)" by simp
huffman@20563
  2280
  hence "\<exists>r > 0. \<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r)" by (simp add: linorder_not_less)
huffman@20563
  2281
  then obtain r where rdef: "r > 0 \<and> (\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r))" by auto
kleing@19023
  2282
huffman@20563
  2283
  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"
kleing@19023
  2284
  have "?F ----> a"
kleing@19023
  2285
  proof -
kleing@19023
  2286
    {
kleing@19023
  2287
      fix e::real
kleing@19023
  2288
      assume "0 < e"
kleing@19023
  2289
        (* choose no such that inverse (real (Suc n)) < e *)
kleing@19023
  2290
      have "\<exists>no. inverse (real (Suc no)) < e" by (rule reals_Archimedean)
kleing@19023
  2291
      then obtain m where nodef: "inverse (real (Suc m)) < e" by auto
kleing@19023
  2292
      {
kleing@19023
  2293
        fix n
kleing@19023
  2294
        assume mlen: "m \<le> n"
kleing@19023
  2295
        then have
kleing@19023
  2296
          "inverse (real (Suc n)) \<le> inverse (real (Suc m))"
kleing@19023
  2297
          by auto
kleing@19023
  2298
        moreover have
huffman@20563
  2299
          "\<bar>?F n - a\<bar> < inverse (real (Suc n))"
kleing@19023
  2300
        proof -
kleing@19023
  2301
          from rdef have
huffman@20563
  2302
            "\<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
kleing@19023
  2303
            by simp
kleing@19023
  2304
          hence
huffman@20563
  2305
            "(?F n)\<noteq>a \<and> \<bar>(?F n) - a\<bar> < inverse (real (Suc n)) \<and> norm (X (?F n) - L) \<ge> r"
kleing@19023
  2306
            by (simp add: some_eq_ex [symmetric])
kleing@19023
  2307
          thus ?thesis by simp
kleing@19023
  2308
        qed
kleing@19023
  2309
        moreover from nodef have
kleing@19023
  2310
          "inverse (real (Suc m)) < e" .
huffman@20563
  2311
        ultimately have "\<bar>?F n - a\<bar> < e" by arith
kleing@19023
  2312
      }
huffman@20563
  2313
      then have "\<exists>no. \<forall>n. no \<le> n --> \<bar>?F n - a\<bar> < e" by auto
kleing@19023
  2314
    }
kleing@19023
  2315
    thus ?thesis by (unfold LIMSEQ_def, simp)
kleing@19023
  2316
  qed
kleing@19023
  2317
  
kleing@19023
  2318
  moreover have "\<forall>n. ?F n \<noteq> a"
kleing@19023
  2319
  proof -
kleing@19023
  2320
    {
kleing@19023
  2321
      fix n
kleing@19023
  2322
      from rdef have
huffman@20563
  2323
        "\<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r"
kleing@19023
  2324
        by simp
kleing@19023
  2325
      hence "?F n \<noteq> a" by (simp add: some_eq_ex [symmetric])
kleing@19023
  2326
    }
kleing@19023
  2327
    thus ?thesis ..
kleing@19023
  2328
  qed
kleing@19023
  2329
  moreover from prems have "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by simp
kleing@19023
  2330
  ultimately have "(\<lambda>n. X (?F n)) ----> L" by simp
kleing@19023
  2331
  
kleing@19023
  2332
  moreover have "\<not> ((\<lambda>n. X (?F n)) ----> L)"
kleing@19023
  2333
  proof -
kleing@19023
  2334
    {
kleing@19023
  2335
      fix no::nat
kleing@19023
  2336
      obtain n where "n = no + 1" by simp
kleing@19023
  2337
      then have nolen: "no \<le> n" by simp
kleing@19023
  2338
        (* We prove this by showing that for any m there is an n\<ge>m such that |X (?F n) - L| \<ge> r *)
huffman@20563
  2339
      from rdef have "\<forall>s > 0. \<exists>x. (x \<noteq> a \<and> \<bar>x - a\<bar> < s \<and> norm (X x - L) \<ge> r)" ..
kleing@19023
  2340
huffman@20563
  2341
      then have "\<exists>x. x\<noteq>a \<and> \<bar>x - a\<bar> < inverse (real (Suc n)) \<and> norm (X x - L) \<ge> r" by simp
kleing@19023
  2342
      
huffman@20563
  2343
      hence "norm (X (?F n) - L) \<ge> r" by (simp add: some_eq_ex [symmetric])
huffman@20563
  2344
      with nolen have "\<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r" by auto
kleing@19023
  2345
    }
huffman@20563
  2346
    then have "(\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> r)" by simp
huffman@20563
  2347
    with rdef have "\<exists>e>0. (\<forall>no. \<exists>n. no \<le> n \<and> norm (X (?F n) - L) \<ge> e)" by auto
kleing@19023
  2348
    thus ?thesis by (unfold LIMSEQ_def, auto simp add: linorder_not_less)
kleing@19023
  2349
  qed
kleing@19023
  2350
  ultimately show False by simp
kleing@19023
  2351
qed
kleing@19023
  2352
webertj@20432
  2353
ML {* fast_arith_split_limit := 9; *}  (* FIXME *)
kleing@19023
  2354
kleing@19023
  2355
lemma LIMSEQ_SEQ_conv:
huffman@20561
  2356
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::real) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
huffman@20561
  2357
   (X -- a --> L)"
kleing@19023
  2358
proof
kleing@19023
  2359
  assume "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
kleing@19023
  2360
  show "X -- a --> L" by (rule LIMSEQ_SEQ_conv2)
kleing@19023
  2361
next
kleing@19023
  2362
  assume "(X -- a --> L)"
kleing@19023
  2363
  show "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L" by (rule LIMSEQ_SEQ_conv1)
kleing@19023
  2364
qed
kleing@19023
  2365
kleing@19023
  2366
lemma real_sqz:
kleing@19023
  2367
  fixes a::real
kleing@19023
  2368
  assumes "a < c"
kleing@19023
  2369
  shows "\<exists>b. a < b \<and> b < c"
huffman@20563
  2370
by (rule dense)
kleing@19023
  2371
kleing@19023
  2372
lemma LIM_offset:
kleing@19023
  2373
  assumes "(\<lambda>x. f x) -- a --> L"
kleing@19023
  2374
  shows "(\<lambda>x. f (x+c)) -- (a-c) --> L"
kleing@19023
  2375
proof -
kleing@19023
  2376
  have "f -- a --> L" .
kleing@19023
  2377
  hence
huffman@20563
  2378
    fd: "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> a & norm (x - a) < s --> norm (f x - L) < r"
kleing@19023
  2379
    by (unfold LIM_def)
kleing@19023
  2380
  {
kleing@19023
  2381
    fix r::real
kleing@19023
  2382
    assume rgz: "0 < r"
huffman@20563
  2383
    with fd have "\<exists>s > 0. \<forall>x. x\<noteq>a \<and> norm (x - a) < s --> norm (f x - L) < r" by simp
huffman@20563
  2384
    then obtain s where sgz: "s > 0" and ax: "\<forall>x. x\<noteq>a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r" by auto
huffman@20563
  2385
    from ax have ax2: "\<forall>x. (x+c)\<noteq>a \<and> norm ((x+c) - a) < s \<longrightarrow> norm (f (x+c) - L) < r" by auto
kleing@19023
  2386
    {
huffman@20561
  2387
      fix x
huffman@20563
  2388
      from ax2 have nt: "(x+c)\<noteq>a \<and> norm ((x+c) - a) < s \<longrightarrow> norm (f (x+c) - L) < r" ..
kleing@19023
  2389
      moreover have "((x+c)\<noteq>a) = (x\<noteq>(a-c))" by auto
huffman@20563
  2390
      moreover have "((x+c) - a) = (x - (a-c))" by simp
huffman@20563
  2391
      ultimately have "x\<noteq>(a-c) \<and> norm (x - (a-c)) < s \<longrightarrow> norm (f (x+c) - L) < r" by simp
kleing@19023
  2392
    }
huffman@20563
  2393
    then have "\<forall>x. x\<noteq>(a-c) \<and> norm (x - (a-c)) < s \<longrightarrow> norm (f (x+c) - L) < r" ..
huffman@20563
  2394
    with sgz have "\<exists>s > 0. \<forall>x. x\<noteq>(a-c) \<and> norm (x - (a-c)) < s \<longrightarrow> norm (f (x+c) - L) < r" by auto
kleing@19023
  2395
  }
kleing@19023
  2396
  then have
huffman@20563
  2397
    "\<forall>r > 0. \<exists>s > 0. \<forall>x. x \<noteq> (a-c) & norm (x - (a-c)) < s --> norm (f (x+c) - L) < r" by simp
kleing@19023
  2398
  thus ?thesis by (fold LIM_def)
kleing@19023
  2399
qed
kleing@19023
  2400
paulson@10751
  2401
end