src/HOL/Hyperreal/Lim.thy
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(*  Title       : Lim.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    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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   228
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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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   234
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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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   239
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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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   242
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   243
lemma NSLIM_minus: "f -- a --NS> L ==> (%x. -f(x)) -- a --NS> -L"
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   244
by (simp add: NSLIM_def)
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   245
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   246
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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   248
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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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   254
apply (drule spec)
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   255
apply (auto simp add: star_of_approx_inverse)
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   256
done
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   257
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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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   260
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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   265
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lemma NSLIM_zero_cancel: "(%x. f(x) - l) -- x --NS> 0 ==> f -- x --NS> l"
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   267
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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   269
done
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   270
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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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   273
  shows "k \<noteq> L ==> ~ ((%x. k) -- a --NS> L)"
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   274
apply (simp add: NSLIM_def)
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   275
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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   278
done
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   279
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lemma NSLIM_not_zero:
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  fixes a :: real
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   282
  shows "k \<noteq> 0 ==> ~ ((%x. k) -- a --NS> 0)"
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   283
by (rule NSLIM_const_not_eq)
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   284
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lemma NSLIM_const_eq:
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  fixes a :: real
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   287
  shows "(%x. k) -- a --NS> L ==> k = L"
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   288
apply (rule ccontr)
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   289
apply (blast dest: NSLIM_const_not_eq)
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   290
done
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   291
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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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   295
  shows "[| f -- a --NS> L; f -- a --NS> M |] ==> L = M"
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parents: 20754
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   296
apply (drule NSLIM_minus)
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diff changeset
   297
apply (drule NSLIM_add, assumption)
956a0377a408 reorganize sections
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   298
apply (auto dest!: NSLIM_const_eq [symmetric])
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   299
apply (simp add: diff_def [symmetric])
956a0377a408 reorganize sections
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diff changeset
   300
done
956a0377a408 reorganize sections
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diff changeset
   301
956a0377a408 reorganize sections
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   302
lemma NSLIM_mult_zero:
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   303
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
956a0377a408 reorganize sections
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parents: 20754
diff changeset
   304
  shows "[| f -- x --NS> 0; g -- x --NS> 0 |] ==> (%x. f(x)*g(x)) -- x --NS> 0"
956a0377a408 reorganize sections
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parents: 20754
diff changeset
   305
by (drule NSLIM_mult, auto)
956a0377a408 reorganize sections
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diff changeset
   306
956a0377a408 reorganize sections
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   307
lemma NSLIM_self: "(%x. x) -- a --NS> a"
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diff changeset
   308
by (simp add: NSLIM_def)
956a0377a408 reorganize sections
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parents: 20754
diff changeset
   309
956a0377a408 reorganize sections
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   310
subsubsection {* Equivalence of @{term LIM} and @{term NSLIM} *}
956a0377a408 reorganize sections
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diff changeset
   311
20754
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   312
lemma LIM_NSLIM:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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   313
  assumes f: "f -- a --> L" shows "f -- a --NS> L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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   314
proof (rule NSLIM_I)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   315
  fix x
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   316
  assume neq: "x \<noteq> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   317
  assume approx: "x \<approx> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   318
  have "starfun f x - star_of L \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   319
  proof (rule InfinitesimalI2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   320
    fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   321
    from LIM_D [OF f r]
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   322
    obtain s where s: "0 < s" and
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   323
      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   324
      by fast
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   325
    from less_r have less_r':
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   326
       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   327
        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   328
      by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   329
    from approx have "x - star_of a \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   330
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   331
    hence "hnorm (x - star_of a) < star_of s"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   332
      using s by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   333
    with neq show "hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   334
      by (rule less_r')
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   335
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   336
  thus "starfun f x \<approx> star_of L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
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   337
    by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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   338
qed
20552
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   339
20754
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   340
lemma NSLIM_LIM:
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parents: 20752
diff changeset
   341
  assumes f: "f -- a --NS> L" shows "f -- a --> L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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diff changeset
   342
proof (rule LIM_I)
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parents: 20752
diff changeset
   343
  fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   344
  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   345
        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   346
  proof (rule exI, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
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parents: 20752
diff changeset
   347
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   348
  next
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   349
    fix x assume neq: "x \<noteq> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   350
    assume "hnorm (x - star_of a) < epsilon"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   351
    with Infinitesimal_epsilon
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   352
    have "x - star_of a \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   353
      by (rule hnorm_less_Infinitesimal)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   354
    hence "x \<approx> star_of a"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   355
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   356
    with f neq have "starfun f x \<approx> star_of L"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   357
      by (rule NSLIM_D)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   358
    hence "starfun f x - star_of L \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   359
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   360
    thus "hnorm (starfun f x - star_of L) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   361
      using r by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   362
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   363
  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   364
    by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   365
qed
14477
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paulson
parents: 14387
diff changeset
   366
15228
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parents: 15195
diff changeset
   367
theorem LIM_NSLIM_iff: "(f -- x --> L) = (f -- x --NS> L)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   368
by (blast intro: LIM_NSLIM NSLIM_LIM)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   369
20755
956a0377a408 reorganize sections
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parents: 20754
diff changeset
   370
subsubsection {* Derived theorems about @{term LIM} *}
14477
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paulson
parents: 14387
diff changeset
   371
15228
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diff changeset
   372
lemma LIM_mult2:
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   373
  fixes l m :: "'a::real_normed_algebra"
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   374
  shows "[| f -- x --> l; g -- x --> m |]
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   375
      ==> (%x. f(x) * g(x)) -- x --> (l * m)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   376
by (simp add: LIM_NSLIM_iff NSLIM_mult)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   377
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   378
lemma LIM_add2:
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   379
     "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + g(x)) -- x --> (l + m)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   380
by (simp add: LIM_NSLIM_iff NSLIM_add)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   381
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   382
lemma LIM_const2: "(%x. k) -- x --> k"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   383
by (simp add: LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   384
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   385
lemma LIM_minus2: "f -- a --> L ==> (%x. -f(x)) -- a --> -L"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   386
by (simp add: LIM_NSLIM_iff NSLIM_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   387
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   388
lemma LIM_add_minus2: "[| f -- x --> l; g -- x --> m |] ==> (%x. f(x) + -g(x)) -- x --> (l + -m)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   389
by (simp add: LIM_NSLIM_iff NSLIM_add_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   390
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   391
lemma LIM_inverse:
20653
24cda2c5fd40 removed division_by_zero class requirements from several lemmas
huffman
parents: 20635
diff changeset
   392
  fixes L :: "'a::real_normed_div_algebra"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   393
  shows "[| f -- a --> L; L \<noteq> 0 |]
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   394
      ==> (%x. inverse(f(x))) -- a --> (inverse L)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   395
by (simp add: LIM_NSLIM_iff NSLIM_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   396
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   397
lemma LIM_zero2: "f -- a --> l ==> (%x. f(x) + -l) -- a --> 0"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   398
by (simp add: LIM_NSLIM_iff NSLIM_zero)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   399
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   400
lemma LIM_zero_cancel: "(%x. f(x) - l) -- x --> 0 ==> f -- x --> l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   401
apply (drule_tac g = "%x. l" and M = l in LIM_add)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   402
apply (auto simp add: diff_minus add_assoc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   403
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   404
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   405
lemma LIM_unique2:
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   406
  fixes a :: real
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   407
  shows "[| f -- a --> L; f -- a --> M |] ==> L = M"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   408
by (simp add: LIM_NSLIM_iff NSLIM_unique)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   409
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   410
(* we can use the corresponding thm LIM_mult2 *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   411
(* for standard definition of limit           *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   412
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   413
lemma LIM_mult_zero2:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   414
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   415
  shows "[| f -- x --> 0; g -- x --> 0 |] ==> (%x. f(x)*g(x)) -- x --> 0"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   416
by (drule LIM_mult2, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   417
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   418
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   419
subsection {* Continuity *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   420
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   421
lemma isNSContD: "[| isNSCont f a; y \<approx> hypreal_of_real a |] ==> ( *f* f) y \<approx> hypreal_of_real (f a)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   422
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   423
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   424
lemma isNSCont_NSLIM: "isNSCont f a ==> f -- a --NS> (f a) "
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   425
by (simp add: isNSCont_def NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   426
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   427
lemma NSLIM_isNSCont: "f -- a --NS> (f a) ==> isNSCont f a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   428
apply (simp add: isNSCont_def NSLIM_def, auto)
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   429
apply (case_tac "y = star_of a", auto)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   430
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   431
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   432
text{*NS continuity can be defined using NS Limit in
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   433
    similar fashion to standard def of continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   434
lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f -- a --NS> (f a))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   435
by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   436
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   437
text{*Hence, NS continuity can be given
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   438
  in terms of standard limit*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   439
lemma isNSCont_LIM_iff: "(isNSCont f a) = (f -- a --> (f a))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   440
by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   441
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   442
text{*Moreover, it's trivial now that NS continuity
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   443
  is equivalent to standard continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   444
lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   445
apply (simp add: isCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   446
apply (rule isNSCont_LIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   447
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   448
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   449
text{*Standard continuity ==> NS continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   450
lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   451
by (erule isNSCont_isCont_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   452
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   453
text{*NS continuity ==> Standard continuity*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   454
lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   455
by (erule isNSCont_isCont_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   456
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   457
text{*Alternative definition of continuity*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   458
(* Prove equivalence between NS limits - *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   459
(* seems easier than using standard def  *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   460
lemma NSLIM_h_iff: "(f -- a --NS> L) = ((%h. f(a + h)) -- 0 --NS> L)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   461
apply (simp add: NSLIM_def, auto)
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   462
apply (drule_tac x = "star_of a + x" in spec)
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   463
apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   464
apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   465
apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   466
 prefer 2 apply (simp add: add_commute diff_def [symmetric])
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   467
apply (rule_tac x = x in star_cases)
17318
bc1c75855f3d starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents: 17298
diff changeset
   468
apply (rule_tac [2] x = x in star_cases)
bc1c75855f3d starfun, starset, and other functions on NS types are now polymorphic;
huffman
parents: 17298
diff changeset
   469
apply (auto simp add: starfun star_of_def star_n_minus star_n_add add_assoc approx_refl star_n_zero_num)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   470
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   471
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   472
lemma NSLIM_isCont_iff: "(f -- a --NS> f a) = ((%h. f(a + h)) -- 0 --NS> f a)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   473
by (rule NSLIM_h_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   474
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   475
lemma LIM_isCont_iff: "(f -- a --> f a) = ((%h. f(a + h)) -- 0 --> f(a))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   476
by (simp add: LIM_NSLIM_iff NSLIM_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   478
lemma isCont_iff: "(isCont f x) = ((%h. f(x + h)) -- 0 --> f(x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   479
by (simp add: isCont_def LIM_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   480
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   481
text{*Immediate application of nonstandard criterion for continuity can offer
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   482
   very simple proofs of some standard property of continuous functions*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   483
text{*sum continuous*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   484
lemma isCont_add: "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) + g(x)) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   485
by (auto intro: approx_add simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   486
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   487
text{*mult continuous*}
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   488
lemma isCont_mult:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   489
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   490
  shows "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) * g(x)) a"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   491
by (auto intro!: starfun_mult_HFinite_approx
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   492
            simp del: starfun_mult [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   493
            simp add: isNSCont_isCont_iff [symmetric] isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   494
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   495
text{*composition of continuous functions
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   496
     Note very short straightforard proof!*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   497
lemma isCont_o: "[| isCont f a; isCont g (f a) |] ==> isCont (g o f) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   498
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_def starfun_o [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   499
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   500
lemma isCont_o2: "[| isCont f a; isCont g (f a) |] ==> isCont (%x. g (f x)) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   501
by (auto dest: isCont_o simp add: o_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   502
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   503
lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   504
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   505
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   506
lemma isCont_minus: "isCont f a ==> isCont (%x. - f x) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   507
by (auto simp add: isNSCont_isCont_iff [symmetric] isNSCont_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   508
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   509
lemma isCont_inverse:
20653
24cda2c5fd40 removed division_by_zero class requirements from several lemmas
huffman
parents: 20635
diff changeset
   510
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   511
  shows "[| isCont f x; f x \<noteq> 0 |] ==> isCont (%x. inverse (f x)) x"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   512
apply (simp add: isCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   513
apply (blast intro: LIM_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   514
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   515
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   516
lemma isNSCont_inverse:
20653
24cda2c5fd40 removed division_by_zero class requirements from several lemmas
huffman
parents: 20635
diff changeset
   517
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   518
  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   519
by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   520
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   521
lemma isCont_diff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   522
      "[| isCont f a; isCont g a |] ==> isCont (%x. f(x) - g(x)) a"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   523
apply (simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   524
apply (auto intro: isCont_add isCont_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   525
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   526
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   527
lemma isCont_const [simp]: "isCont (%x. k) a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   528
by (simp add: isCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   529
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   530
lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   531
by (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   532
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   533
lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   534
apply (simp add: isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   535
apply (auto intro: approx_hrabs simp add: hypreal_of_real_hrabs [symmetric] starfun_rabs_hrabs)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   536
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   537
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   538
lemma isCont_abs [simp]: "isCont abs (a::real)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   539
by (auto simp add: isNSCont_isCont_iff [symmetric])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   540
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   541
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   542
(****************************************************************
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   543
(%* Leave as commented until I add topology theory or remove? *%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   544
(%*------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   545
  Elementary topology proof for a characterisation of
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   546
  continuity now: a function f is continuous if and only
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   547
  if the inverse image, {x. f(x) \<in> A}, of any open set A
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   548
  is always an open set
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   549
 ------------------------------------------------------------*%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   550
Goal "[| isNSopen A; \<forall>x. isNSCont f x |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   551
               ==> isNSopen {x. f x \<in> A}"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   552
by (auto_tac (claset(),simpset() addsimps [isNSopen_iff1]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   553
by (dtac (mem_monad_approx RS approx_sym);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   554
by (dres_inst_tac [("x","a")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   555
by (dtac isNSContD 1 THEN assume_tac 1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   556
by (dtac bspec 1 THEN assume_tac 1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   557
by (dres_inst_tac [("x","( *f* f) x")] approx_mem_monad2 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   558
by (blast_tac (claset() addIs [starfun_mem_starset]);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   559
qed "isNSCont_isNSopen";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   560
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   561
Goalw [isNSCont_def]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   562
          "\<forall>A. isNSopen A --> isNSopen {x. f x \<in> A} \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   563
\              ==> isNSCont f x";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   564
by (auto_tac (claset() addSIs [(mem_infmal_iff RS iffD1) RS
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   565
     (approx_minus_iff RS iffD2)],simpset() addsimps
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   566
      [Infinitesimal_def,SReal_iff]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   567
by (dres_inst_tac [("x","{z. abs(z + -f(x)) < ya}")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   568
by (etac (isNSopen_open_interval RSN (2,impE));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   569
by (auto_tac (claset(),simpset() addsimps [isNSopen_def,isNSnbhd_def]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   570
by (dres_inst_tac [("x","x")] spec 1);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   571
by (auto_tac (claset() addDs [approx_sym RS approx_mem_monad],
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   572
    simpset() addsimps [hypreal_of_real_zero RS sym,STAR_starfun_rabs_add_minus]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   573
qed "isNSopen_isNSCont";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   574
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   575
Goal "(\<forall>x. isNSCont f x) = \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   576
\     (\<forall>A. isNSopen A --> isNSopen {x. f(x) \<in> A})";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   577
by (blast_tac (claset() addIs [isNSCont_isNSopen,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   578
    isNSopen_isNSCont]);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   579
qed "isNSCont_isNSopen_iff";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   580
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   581
(%*------- Standard version of same theorem --------*%)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   582
Goal "(\<forall>x. isCont f x) = \
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   583
\         (\<forall>A. isopen A --> isopen {x. f(x) \<in> A})";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   584
by (auto_tac (claset() addSIs [isNSCont_isNSopen_iff],
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   585
              simpset() addsimps [isNSopen_isopen_iff RS sym,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   586
              isNSCont_isCont_iff RS sym]));
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   587
qed "isCont_isopen_iff";
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   588
*******************************************************************)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   589
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   590
subsection {* Uniform Continuity *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   591
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   592
lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   593
by (simp add: isNSUCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   594
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   595
lemma isUCont_isCont: "isUCont f ==> isCont f x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   596
by (simp add: isUCont_def isCont_def LIM_def, meson)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   597
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   598
lemma isUCont_isNSUCont:
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   599
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   600
  assumes f: "isUCont f" shows "isNSUCont f"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   601
proof (unfold isNSUCont_def, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   602
  fix x y :: "'a star"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   603
  assume approx: "x \<approx> y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   604
  have "starfun f x - starfun f y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   605
  proof (rule InfinitesimalI2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   606
    fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   607
    with f obtain s where s: "0 < s" and
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   608
      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   609
      by (auto simp add: isUCont_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   610
    from less_r have less_r':
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   611
       "\<And>x y. hnorm (x - y) < star_of s
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   612
        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   613
      by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   614
    from approx have "x - y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   615
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   616
    hence "hnorm (x - y) < star_of s"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   617
      using s by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   618
    thus "hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   619
      by (rule less_r')
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   620
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   621
  thus "starfun f x \<approx> starfun f y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   622
    by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   623
qed
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   624
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   625
lemma isNSUCont_isUCont:
20754
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   626
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   627
  assumes f: "isNSUCont f" shows "isUCont f"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   628
proof (unfold isUCont_def, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   629
  fix r::real assume r: "0 < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   630
  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   631
        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   632
  proof (rule exI, safe)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   633
    show "0 < epsilon" by (rule hypreal_epsilon_gt_zero)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   634
  next
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   635
    fix x y :: "'a star"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   636
    assume "hnorm (x - y) < epsilon"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   637
    with Infinitesimal_epsilon
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   638
    have "x - y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   639
      by (rule hnorm_less_Infinitesimal)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   640
    hence "x \<approx> y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   641
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   642
    with f have "starfun f x \<approx> starfun f y"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   643
      by (simp add: isNSUCont_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   644
    hence "starfun f x - starfun f y \<in> Infinitesimal"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   645
      by (unfold approx_def)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   646
    thus "hnorm (starfun f x - starfun f y) < star_of r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   647
      using r by (rule InfinitesimalD2)
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   648
  qed
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   649
  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   650
    by transfer
9c053a494dc6 add intro/dest rules for NSLIM; rewrite equivalence proofs using transfer
huffman
parents: 20752
diff changeset
   651
qed
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   652
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   653
subsection {* Derivatives *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   654
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   655
lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --> D)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   656
by (simp add: deriv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   657
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   658
lemma DERIV_NS_iff:
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   659
      "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   660
by (simp add: deriv_def LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   661
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   662
lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --> D"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   663
by (simp add: deriv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   664
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   665
lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) -- 0 --NS> D"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   666
by (simp add: deriv_def LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   667
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   668
subsubsection{*Uniqueness*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   669
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   670
lemma DERIV_unique:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   671
      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   672
apply (simp add: deriv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   673
apply (blast intro: LIM_unique)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   674
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   675
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   676
lemma NSDeriv_unique:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   677
     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   678
apply (simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   679
apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   680
apply (auto dest!: bspec [where x=epsilon]
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   681
            intro!: inj_hypreal_of_real [THEN injD]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   682
            dest: approx_trans3)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   683
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   684
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   685
subsubsection{*Alternative definition for differentiability*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   686
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   687
lemma DERIV_LIM_iff:
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   688
     "((%h::real. (f(a + h) - f(a)) / h) -- 0 --> D) =
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   689
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   690
proof (intro iffI LIM_I)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   691
  fix r::real
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   692
  assume r: "0<r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   693
  assume "(\<lambda>h. (f (a + h) - f a) / h) -- 0 --> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   694
  from LIM_D [OF this r]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   695
  obtain s
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   696
    where s:    "0 < s"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   697
      and s_lt: "\<forall>x. x \<noteq> 0 & \<bar>x\<bar> < s --> \<bar>(f (a + x) - f a) / x - D\<bar> < r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   698
  by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   699
  show "\<exists>s. 0 < s \<and>
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   700
        (\<forall>x. x \<noteq> a \<and> norm (x-a) < s \<longrightarrow> norm ((f x - f a) / (x-a) - D) < r)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   701
  proof (intro exI conjI strip)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   702
    show "0 < s"  by (rule s)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   703
  next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   704
    fix x::real
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   705
    assume "x \<noteq> a \<and> norm (x-a) < s"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   706
    with s_lt [THEN spec [where x="x-a"]]
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   707
    show "norm ((f x - f a) / (x-a) - D) < r" by auto
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   708
  qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   709
next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   710
  fix r::real
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   711
  assume r: "0<r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   712
  assume "(\<lambda>x. (f x - f a) / (x-a)) -- a --> D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   713
  from LIM_D [OF this r]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   714
  obtain s
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   715
    where s:    "0 < s"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   716
      and s_lt: "\<forall>x. x \<noteq> a & \<bar>x-a\<bar> < s --> \<bar>(f x - f a)/(x-a) - D\<bar> < r"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   717
  by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   718
  show "\<exists>s. 0 < s \<and>
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   719
        (\<forall>x. x \<noteq> 0 & norm (x - 0) < s --> norm ((f (a + x) - f a) / x - D) < r)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   720
  proof (intro exI conjI strip)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   721
    show "0 < s"  by (rule s)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   722
  next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   723
    fix x::real
20561
6a6d8004322f generalize type of (NS)LIM to work on functions with vector space domain types
huffman
parents: 20552
diff changeset
   724
    assume "x \<noteq> 0 \<and> norm (x - 0) < s"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   725
    with s_lt [THEN spec [where x="x+a"]]
20552
2c31dd358c21 generalized types of many constants to work over arbitrary vector spaces;
huffman
parents: 20432
diff changeset
   726
    show "norm ((f (a + x) - f a) / x - D) < r" by (auto simp add: add_ac)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   727
  qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   728
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   729
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   730
lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   731
by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   732
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   733
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   734
subsubsection{*Equivalence of NS and standard definitions of differentiation*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   735
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   736
text {*First NSDERIV in terms of NSLIM*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   737
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   738
text{*first equivalence *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   739
lemma NSDERIV_NSLIM_iff:
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   740
      "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   741
apply (simp add: nsderiv_def NSLIM_def, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   742
apply (drule_tac x = xa in bspec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   743
apply (rule_tac [3] ccontr)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   744
apply (drule_tac [3] x = h in spec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   745
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   746
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   747
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   748
text{*second equivalence *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   749
lemma NSDERIV_NSLIM_iff2:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   750
     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   751
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   752
              LIM_NSLIM_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   753
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   754
(* while we're at it! *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   755
lemma NSDERIV_iff2:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   756
     "(NSDERIV f x :> D) =
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   757
      (\<forall>w.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   758
        w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   759
        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   760
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   761
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   762
(*FIXME DELETE*)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   763
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   764
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   765
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   766
lemma NSDERIVD5:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   767
  "(NSDERIV f x :> D) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   768
   (\<forall>u. u \<approx> hypreal_of_real x -->
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   769
     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   770
apply (auto simp add: NSDERIV_iff2)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   771
apply (case_tac "u = hypreal_of_real x", auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   772
apply (drule_tac x = u in spec, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   773
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   774
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   775
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   776
apply (auto simp add:
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   777
         approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   778
         Infinitesimal_subset_HFinite [THEN subsetD])
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   779
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   780
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   781
lemma NSDERIVD4:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   782
     "(NSDERIV f x :> D) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   783
      (\<forall>h \<in> Infinitesimal.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   784
               (( *f* f)(hypreal_of_real x + h) -
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   785
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   786
apply (auto simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   787
apply (case_tac "h = (0::hypreal) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   788
apply (auto simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   789
apply (drule_tac x = h in bspec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   790
apply (drule_tac [2] c = h in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   791
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15360
diff changeset
   792
            simp add: diff_minus)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   793
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   794
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   795
lemma NSDERIVD3:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   796
     "(NSDERIV f x :> D) ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   797
      (\<forall>h \<in> Infinitesimal - {0}.
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   798
               (( *f* f)(hypreal_of_real x + h) -
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   799
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   800
apply (auto simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   801
apply (rule ccontr, drule_tac x = h in bspec)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   802
apply (drule_tac [2] c = h in approx_mult1)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   803
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15360
diff changeset
   804
            simp add: mult_assoc diff_minus)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   805
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   806
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   807
text{*Now equivalence between NSDERIV and DERIV*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   808
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   809
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   810
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   811
text{*Differentiability implies continuity
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   812
         nice and simple "algebraic" proof*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   813
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   814
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   815
apply (drule approx_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   816
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   817
apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   818
 prefer 2 apply (simp add: add_assoc [symmetric])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15228
diff changeset
   819
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   820
apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   821
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   822
            simp add: mult_assoc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   823
apply (drule_tac x3=D in
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   824
           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   825
             THEN mem_infmal_iff [THEN iffD1]])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15360
diff changeset
   826
apply (auto simp add: mult_commute
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   827
            intro: approx_trans approx_minus_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   828
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   829
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   830
text{*Now Sandard proof*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   831
lemma DERIV_isCont: "DERIV f x :> D ==> isCont f x"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   832
by (simp add: NSDERIV_DERIV_iff [symmetric] isNSCont_isCont_iff [symmetric]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   833
              NSDERIV_isNSCont)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   834
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
   835
subsubsection {* Derivatives of various functions *}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   836
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   837
text{*Differentiation rules for combinations of functions
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   838
      follow from clear, straightforard, algebraic
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   839
      manipulations*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   840
text{*Constant function*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   841
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   842
(* use simple constant nslimit theorem *)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   843
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   844
by (simp add: NSDERIV_NSLIM_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   845
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   846
lemma DERIV_const [simp]: "(DERIV (%x. k) x :> 0)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   847
by (simp add: NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   848
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   849
text{*Sum of functions- proved easily*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   850
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   851
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   852
lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   853
      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   854
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   855
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   856
apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   857
apply (auto simp add: diff_def add_ac)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   858
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   859
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   860
(* Standard theorem *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   861
lemma DERIV_add: "[| DERIV f x :> Da; DERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   862
      ==> DERIV (%x. f x + g x) x :> Da + Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   863
apply (simp add: NSDERIV_add NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   864
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   865
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   866
text{*Product of functions - Proof is trivial but tedious
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   867
  and long due to rearrangement of terms*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   868
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   869
lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   870
by (simp add: right_diff_distrib)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   871
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   872
lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   873
         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   874
      ==> x - y \<approx> 0"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   875
apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   876
apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   877
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15360
diff changeset
   878
            simp add: mult_assoc mem_infmal_iff [symmetric])
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   879
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   880
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   881
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   882
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   883
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   884
      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   885
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   886
apply (auto dest!: spec
19023
5652a536b7e8 * include generalised MVT in HyperReal (contributed by Benjamin Porter)
kleing
parents: 17318
diff changeset
   887
      simp add: starfun_lambda_cancel lemma_nsderiv1)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   888
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   889
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15228
diff changeset
   890
apply (auto simp add: times_divide_eq_right [symmetric]
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15228
diff changeset
   891
            simp del: times_divide_eq)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   892
apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   893
apply (drule_tac
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   894
     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   895
apply (auto intro!: approx_add_mono1
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   896
            simp add: left_distrib right_distrib mult_commute add_assoc)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   897
apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   898
         in add_commute [THEN subst])
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   899
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   900
                    Infinitesimal_add Infinitesimal_mult
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   901
                    Infinitesimal_hypreal_of_real_mult
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   902
                    Infinitesimal_hypreal_of_real_mult2
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   903
          simp add: add_assoc [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   904
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   905
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   906
lemma DERIV_mult:
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   907
     "[| DERIV f x :> Da; DERIV g x :> Db |]
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   908
      ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   909
by (simp add: NSDERIV_mult NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   910
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   911
text{*Multiplying by a constant*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   912
lemma NSDERIV_cmult: "NSDERIV f x :> D
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   913
      ==> NSDERIV (%x. c * f x) x :> c*D"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   914
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   915
                  minus_mult_right right_diff_distrib [symmetric])
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   916
apply (erule NSLIM_const [THEN NSLIM_mult])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   917
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   918
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   919
(* let's do the standard proof though theorem *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   920
(* LIM_mult2 follows from a NS proof          *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   921
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   922
lemma DERIV_cmult:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   923
      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   924
apply (simp only: deriv_def times_divide_eq_right [symmetric]
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   925
                  NSDERIV_NSLIM_iff minus_mult_right right_diff_distrib [symmetric])
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   926
apply (erule LIM_const [THEN LIM_mult2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   927
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   928
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   929
text{*Negation of function*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   930
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   931
proof (simp add: NSDERIV_NSLIM_iff)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   932
  assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   933
  hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   934
    by (rule NSLIM_minus)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   935
  have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
   936
    by (simp add: minus_divide_left)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   937
  with deriv
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   938
  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   939
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   940
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   941
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   942
lemma DERIV_minus: "DERIV f x :> D ==> DERIV (%x. -(f x)) x :> -D"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   943
by (simp add: NSDERIV_minus NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   944
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   945
text{*Subtraction*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   946
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   947
by (blast dest: NSDERIV_add NSDERIV_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   948
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   949
lemma DERIV_add_minus: "[| DERIV f x :> Da; DERIV g x :> Db |] ==> DERIV (%x. f x + -g x) x :> Da + -Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   950
by (blast dest: DERIV_add DERIV_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   951
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   952
lemma NSDERIV_diff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   953
     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   954
      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   955
apply (simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   956
apply (blast intro: NSDERIV_add_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   957
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   958
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   959
lemma DERIV_diff:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   960
     "[| DERIV f x :> Da; DERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   961
       ==> DERIV (%x. f x - g x) x :> Da-Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   962
apply (simp add: diff_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   963
apply (blast intro: DERIV_add_minus)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   964
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   965
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   966
text{*  Similarly to the above, the chain rule admits an entirely
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   967
   straightforward derivation. Compare this with Harrison's
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   968
   HOL proof of the chain rule, which proved to be trickier and
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   969
   required an alternative characterisation of differentiability-
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   970
   the so-called Carathedory derivative. Our main problem is
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   971
   manipulation of terms.*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   972
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   973
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   974
(* lemmas *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   975
lemma NSDERIV_zero:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   976
      "[| NSDERIV g x :> D;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   977
               ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   978
               xa \<in> Infinitesimal;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   979
               xa \<noteq> 0
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   980
            |] ==> D = 0"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   981
apply (simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   982
apply (drule bspec, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   983
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   984
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   985
(* can be proved differently using NSLIM_isCont_iff *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   986
lemma NSDERIV_approx:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   987
     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
   988
      ==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   989
apply (simp add: nsderiv_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   990
apply (simp add: mem_infmal_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   991
apply (rule Infinitesimal_ratio)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   992
apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   993
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   994
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   995
(*---------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   996
   from one version of differentiability
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   997
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   998
                f(x) - f(a)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
   999
              --------------- \<approx> Db
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1000
                  x - a
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1001
 ---------------------------------------------------------------*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1002
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1003
         ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1004
         ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1005
      |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1006
                   - hypreal_of_real (f (g x)))
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1007
              / (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1008
             \<approx> hypreal_of_real(Da)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1009
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1010
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1011
(*--------------------------------------------------------------
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1012
   from other version of differentiability
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1013
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1014
                f(x + h) - f(x)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1015
               ----------------- \<approx> Db
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1016
                       h
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1017
 --------------------------------------------------------------*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1018
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1019
      ==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1020
          \<approx> hypreal_of_real(Db)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1021
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1022
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1023
lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1024
by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1025
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1026
text{*This proof uses both definitions of differentiability.*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1027
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1028
      ==> NSDERIV (f o g) x :> Da * Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1029
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1030
                mem_infmal_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1031
apply clarify
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1032
apply (frule_tac f = g in NSDERIV_approx)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1033
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1034
apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1035
apply (drule_tac g = g in NSDERIV_zero)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1036
apply (auto simp add: divide_inverse)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  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])
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1038
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1039
apply (rule approx_mult_hypreal_of_real)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1040
apply (simp_all add: divide_inverse [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1041
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1042
apply (blast intro: NSDERIVD2)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1043
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1044
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1045
(* standard version *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1046
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1047
by (simp add: NSDERIV_DERIV_iff [symmetric] NSDERIV_chain)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1048
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1049
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1050
by (auto dest: DERIV_chain simp add: o_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1051
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1052
text{*Differentiation of natural number powers*}
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1053
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1054
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1055
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1056
(*derivative of the identity function*)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1057
lemma DERIV_Id [simp]: "DERIV (%x. x) x :> 1"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1058
by (simp add: NSDERIV_DERIV_iff [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1059
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1060
lemmas isCont_Id = DERIV_Id [THEN DERIV_isCont, standard]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1061
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1062
(*derivative of linear multiplication*)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1063
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1064
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1065
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1066
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1067
by (simp add: NSDERIV_DERIV_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1068
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1069
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15234
diff changeset
  1070
apply (induct "n")
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1071
apply (drule_tac [2] DERIV_Id [THEN DERIV_mult])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1072
apply (auto simp add: real_of_nat_Suc left_distrib)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1073
apply (case_tac "0 < n")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1074
apply (drule_tac x = x in realpow_minus_mult)
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15228
diff changeset
  1075
apply (auto simp add: mult_assoc add_commute)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1076
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1077
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1078
(* NS version *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1079
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1080
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1081
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1082
text{*Power of -1*}
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1083
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1084
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1085
lemma NSDERIV_inverse:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1086
     "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1087
apply (simp add: nsderiv_def)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1088
apply (rule ballI, simp, clarify)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1089
apply (frule (1) Infinitesimal_add_not_zero)
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1090
apply (simp add: add_commute)
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1091
(*apply (auto simp add: starfun_inverse_inverse realpow_two
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1092
        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1093
apply (simp add: inverse_add inverse_mult_distrib [symmetric]
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1094
              inverse_minus_eq [symmetric] add_ac mult_ac diff_def
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1095
            del: inverse_mult_distrib inverse_minus_eq
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1096
                 minus_mult_left [symmetric] minus_mult_right [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1097
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1098
            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15228
diff changeset
  1099
apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1100
apply (rule inverse_add_Infinitesimal_approx2)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1101
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1102
            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1103
apply (rule Infinitesimal_HFinite_mult2, auto)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1104
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1105
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1106
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1107
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1108
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1109
lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1110
by (simp add: NSDERIV_inverse NSDERIV_DERIV_iff [symmetric] del: realpow_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1111
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1112
text{*Derivative of inverse*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1113
lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1114
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1115
apply (simp only: mult_commute [of d] minus_mult_left power_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1116
apply (fold o_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1117
apply (blast intro!: DERIV_chain DERIV_inverse)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1118
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1119
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1120
lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1121
      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1122
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1123
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1124
text{*Derivative of quotient*}
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1125
lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1126
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1127
apply (drule_tac f = g in DERIV_inverse_fun)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1128
apply (drule_tac [2] DERIV_mult)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1129
apply (assumption+)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1130
apply (simp add: divide_inverse right_distrib power_inverse minus_mult_left
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1131
                 mult_ac diff_def
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1132
     del: realpow_Suc minus_mult_right [symmetric] minus_mult_left [symmetric])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1133
done
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1134
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1135
lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1136
       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
20563
44eda2314aab replace (x + - y) with (x - y)
huffman
parents: 20561
diff changeset
  1137
                            - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1138
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1139
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1140
(* ------------------------------------------------------------------------ *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1141
(* Caratheodory formulation of derivative at a point: standard proof        *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1142
(* ------------------------------------------------------------------------ *)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1143
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1144
lemma CARAT_DERIV:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1145
     "(DERIV f x :> l) =
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1146
      (\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isCont g x & g x = l)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1147
      (is "?lhs = ?rhs")
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1148
proof
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1149
  assume der: "DERIV f x :> l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1150
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1151
  proof (intro exI conjI)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1152
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15360
diff changeset
  1153
    show "\<forall>z. f z - f x = ?g z * (z-x)" by (simp)
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1154
    show "isCont ?g x" using der
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1155
      by (simp add: isCont_iff DERIV_iff diff_minus
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1156
               cong: LIM_equal [rule_format])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1157
    show "?g x = l" by simp
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1158
  qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1159
next
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1160
  assume "?rhs"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1161
  then obtain g where
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1162
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1163
  thus "(DERIV f x :> l)"
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1164
     by (auto simp add: isCont_iff DERIV_iff diff_minus
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1165
               cong: LIM_equal [rule_format])
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1166
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1167
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1168
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1169
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1170
      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1171
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1172
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1173
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1174
by auto
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1175
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1176
lemma CARAT_DERIVD:
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1177
  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1178
      and nsc: "isNSCont g x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1179
  shows "NSDERIV f x :> g x"
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1180
proof -
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1181
  from nsc
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1182
  have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1183
         ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1184
         hypreal_of_real (g x)"
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1185
    by (simp add: diff_minus isNSCont_def)
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1186
  thus ?thesis using all
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15195
diff changeset
  1187
    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
14477
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1188
qed
cc61fd03e589 conversion of Hyperreal/Lim to new-style
paulson
parents: 14387
diff changeset
  1189
20755
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1190
subsubsection {* Differentiability predicate *}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1191
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1192
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1193
by (simp add: differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1194
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1195
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1196
by (force simp add: differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1197
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1198
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1199
by (simp add: NSdifferentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1200
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1201
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1202
by (force simp add: NSdifferentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1203
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1204
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1205
  apply (unfold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1206
  apply (rule_tac x=0 in exI)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1207
  apply simp
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1208
  done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1209
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1210
lemma differentiable_sum:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1211
  assumes "f differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1212
  and "g differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1213
  shows "(\<lambda>x. f x + g x) differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1214
proof -
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1215
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1216
  then obtain df where "DERIV f x :> df" ..
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1217
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1218
  then obtain dg where "DERIV g x :> dg" ..
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1219
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1220
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1221
  thus ?thesis by (fold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1222
qed
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1223
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1224
lemma differentiable_diff:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1225
  assumes "f differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1226
  and "g differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1227
  shows "(\<lambda>x. f x - g x) differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1228
proof -
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1229
  from prems have "f differentiable x" by simp
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1230
  moreover
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1231
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1232
  then obtain dg where "DERIV g x :> dg" ..
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1233
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1234
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1235
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1236
  ultimately 
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1237
  show ?thesis
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1238
    by (auto simp: real_diff_def dest: differentiable_sum)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1239
qed
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1240
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1241
lemma differentiable_mult:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1242
  assumes "f differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1243
  and "g differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1244
  shows "(\<lambda>x. f x * g x) differentiable x"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1245
proof -
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1246
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1247
  then obtain df where "DERIV f x :> df" ..
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1248
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1249
  then obtain dg where "DERIV g x :> dg" ..
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1250
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1251
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1252
  thus ?thesis by (fold differentiable_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1253
qed
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1254
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1255
subsection {*(NS) Increment*}
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1256
lemma incrementI:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1257
      "f NSdifferentiable x ==>
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1258
      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1259
      hypreal_of_real (f x)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1260
by (simp add: increment_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1261
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1262
lemma incrementI2: "NSDERIV f x :> D ==>
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1263
     increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1264
     hypreal_of_real (f x)"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1265
apply (erule NSdifferentiableI [THEN incrementI])
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1266
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1267
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1268
(* The Increment theorem -- Keisler p. 65 *)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1269
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1270
      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1271
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1272
apply (drule bspec, auto)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1273
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1274
apply (frule_tac b1 = "hypreal_of_real (D) + y"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1275
        in hypreal_mult_right_cancel [THEN iffD2])
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  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)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1277
apply assumption
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1278
apply (simp add: times_divide_eq_right [symmetric])
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1279
apply (auto simp add: left_distrib)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1280
done
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1281
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1282
lemma increment_thm2:
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1283
     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1284
      ==> \<exists>e \<in> Infinitesimal. increment f x h =
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1285
              hypreal_of_real(D)*h + e*h"
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1286
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
956a0377a408 reorganize sections
huffman
parents: 20754
diff changeset
  1287
956a0377a408 reorganize sections
huffman
parents: 20754
diff