src/HOL/Hyperreal/Deriv.thy
author huffman
Wed Nov 08 02:13:02 2006 +0100 (2006-11-08)
changeset 21239 d4fbe2c87ef1
parent 21199 2d83f93c3580
child 21404 eb85850d3eb7
permissions -rw-r--r--
LIM_compose -> isCont_LIM_compose;
cleaned up and reorganized sections
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(*  Title       : Deriv.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{* Differentiation *}
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theory Deriv
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imports Lim
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begin
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text{*Standard and Nonstandard Definitions*}
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definition
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  deriv :: "[real \<Rightarrow> 'a::real_normed_vector, real, 'a] \<Rightarrow> 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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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 {* Derivatives *}
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subsubsection {* Purely standard proofs *}
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lemma DERIV_iff: "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --> D)"
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by (simp add: deriv_def)
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lemma DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --> D"
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by (simp add: deriv_def)
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lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x :> 0"
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by (simp add: deriv_def)
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lemma DERIV_Id [simp]: "DERIV (\<lambda>x. x) x :> 1"
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by (simp add: deriv_def real_scaleR_def cong: LIM_cong)
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lemma add_diff_add:
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  fixes a b c d :: "'a::ab_group_add"
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  shows "(a + c) - (b + d) = (a - b) + (c - d)"
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by simp
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lemma DERIV_add:
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  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x :> D + E"
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by (simp only: deriv_def add_diff_add scaleR_right_distrib LIM_add)
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lemma DERIV_minus:
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  "DERIV f x :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x :> - D"
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by (simp only: deriv_def minus_diff_minus scaleR_minus_right LIM_minus)
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lemma DERIV_diff:
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  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x :> D - E"
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by (simp only: diff_def DERIV_add DERIV_minus)
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lemma DERIV_add_minus:
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  "\<lbrakk>DERIV f x :> D; DERIV g x :> E\<rbrakk> \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x :> D + - E"
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by (simp only: DERIV_add DERIV_minus)
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lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
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proof (unfold isCont_iff)
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  assume "DERIV f x :> D"
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  hence "(\<lambda>h. (f(x+h) - f(x)) /# h) -- 0 --> D"
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    by (rule DERIV_D)
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  hence "(\<lambda>h. h *# ((f(x+h) - f(x)) /# h)) -- 0 --> 0 *# D"
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    by (intro LIM_scaleR LIM_self)
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  hence "(\<lambda>h. (f(x+h) - f(x))) -- 0 --> 0"
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    by (simp cong: LIM_cong)
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  thus "(\<lambda>h. f(x+h)) -- 0 --> f(x)"
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    by (simp add: LIM_def)
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qed
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lemma DERIV_mult_lemma:
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  fixes a b c d :: "'a::real_algebra"
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  shows "(a * b - c * d) /# h = a * ((b - d) /# h) + ((a - c) /# h) * d"
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by (simp add: diff_minus scaleR_right_distrib [symmetric] ring_distrib)
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lemma DERIV_mult':
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  fixes f g :: "real \<Rightarrow> 'a::real_normed_algebra"
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  assumes f: "DERIV f x :> D"
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  assumes g: "DERIV g x :> E"
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  shows "DERIV (\<lambda>x. f x * g x) x :> f x * E + D * g x"
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proof (unfold deriv_def)
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  from f have "isCont f x"
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    by (rule DERIV_isCont)
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  hence "(\<lambda>h. f(x+h)) -- 0 --> f x"
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    by (simp only: isCont_iff)
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  hence "(\<lambda>h. f(x+h) * ((g(x+h) - g x) /# h) +
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              ((f(x+h) - f x) /# h) * g x)
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          -- 0 --> f x * E + D * g x"
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    by (intro LIM_add LIM_mult2 LIM_const DERIV_D f g)
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  thus "(\<lambda>h. (f(x+h) * g(x+h) - f x * g x) /# h)
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         -- 0 --> f x * E + D * g x"
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    by (simp only: DERIV_mult_lemma)
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qed
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lemma DERIV_mult:
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  fixes f g :: "real \<Rightarrow> 'a::{real_normed_algebra,comm_ring}" shows
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     "[| DERIV f x :> Da; DERIV g x :> Db |]
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      ==> DERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
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by (drule (1) DERIV_mult', simp only: mult_commute add_commute)
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lemma DERIV_unique:
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      "[| DERIV f x :> D; DERIV f x :> E |] ==> D = E"
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apply (simp add: deriv_def)
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apply (blast intro: LIM_unique)
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done
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text{*Differentiation of finite sum*}
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lemma DERIV_sumr [rule_format (no_asm)]:
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     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x :> (f' r x))
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      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x :> (\<Sum>r=m..<n. f' r x)"
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apply (induct "n")
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apply (auto intro: DERIV_add)
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done
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text{*Alternative definition for differentiability*}
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lemma DERIV_LIM_iff:
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     "((%h::real. (f(a + h) - f(a)) / h) -- 0 --> D) =
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      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
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apply (rule iffI)
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apply (drule_tac k="- a" in LIM_offset)
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apply (simp add: diff_minus)
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apply (drule_tac k="a" in LIM_offset)
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apply (simp add: add_commute)
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done
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lemma DERIV_LIM_iff':
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     "((%h::real. (f(a + h) - f(a)) /# h) -- 0 --> D) =
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      ((%x. (f(x)-f(a)) /# (x-a)) -- a --> D)"
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apply (rule iffI)
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apply (drule_tac k="- a" in LIM_offset)
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apply (simp add: diff_minus)
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apply (drule_tac k="a" in LIM_offset)
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apply (simp add: add_commute)
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done
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lemma DERIV_iff2: "(DERIV f x :> D) = ((%z. (f(z) - f(x)) /# (z-x)) -- x --> D)"
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by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff')
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lemma inverse_diff_inverse:
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  "\<lbrakk>(a::'a::division_ring) \<noteq> 0; b \<noteq> 0\<rbrakk>
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   \<Longrightarrow> inverse a - inverse b = - (inverse a * (a - b) * inverse b)"
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by (simp add: right_diff_distrib left_diff_distrib mult_assoc)
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lemma DERIV_inverse_lemma:
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  "\<lbrakk>a \<noteq> 0; b \<noteq> (0::'a::real_normed_div_algebra)\<rbrakk>
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   \<Longrightarrow> (inverse a - inverse b) /# h
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     = - (inverse a * ((a - b) /# h) * inverse b)"
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by (simp add: inverse_diff_inverse)
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lemma LIM_equal2:
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  assumes 1: "0 < R"
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  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
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  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
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apply (unfold LIM_def, safe)
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apply (drule_tac x="r" in spec, safe)
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apply (rule_tac x="min s R" in exI, safe)
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apply (simp add: 1)
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apply (simp add: 2)
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done
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lemma DERIV_inverse':
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  fixes f :: "real \<Rightarrow> 'a::real_normed_div_algebra"
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  assumes der: "DERIV f x :> D"
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  assumes neq: "f x \<noteq> 0"
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  shows "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * D * inverse (f x))"
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    (is "DERIV _ _ :> ?E")
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proof (unfold DERIV_iff2)
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  from der have lim_f: "f -- x --> f x"
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    by (rule DERIV_isCont [unfolded isCont_def])
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  from neq have "0 < norm (f x)" by simp
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  with LIM_D [OF lim_f] obtain s
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    where s: "0 < s"
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    and less_fx: "\<And>z. \<lbrakk>z \<noteq> x; norm (z - x) < s\<rbrakk>
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                  \<Longrightarrow> norm (f z - f x) < norm (f x)"
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    by fast
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  show "(\<lambda>z. (inverse (f z) - inverse (f x)) /# (z - x)) -- x --> ?E"
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  proof (rule LIM_equal2 [OF s])
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    fix z :: real
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    assume "z \<noteq> x" "norm (z - x) < s"
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    hence "norm (f z - f x) < norm (f x)" by (rule less_fx)
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    hence "f z \<noteq> 0" by auto
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    thus "(inverse (f z) - inverse (f x)) /# (z - x) =
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          - (inverse (f z) * ((f z - f x) /# (z - x)) * inverse (f x))"
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      using neq by (rule DERIV_inverse_lemma)
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  next
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    from der have "(\<lambda>z. (f z - f x) /# (z - x)) -- x --> D"
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      by (unfold DERIV_iff2)
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    thus "(\<lambda>z. - (inverse (f z) * ((f z - f x) /# (z - x)) * inverse (f x)))
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          -- x --> ?E"
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      by (intro LIM_mult2 LIM_inverse LIM_minus LIM_const lim_f neq)
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  qed
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qed
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lemma DERIV_divide:
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  fixes D E :: "'a::{real_normed_div_algebra,field}"
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  shows "\<lbrakk>DERIV f x :> D; DERIV g x :> E; g x \<noteq> 0\<rbrakk>
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         \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x :> (D * g x - f x * E) / (g x * g x)"
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apply (subgoal_tac "f x * - (inverse (g x) * E * inverse (g x)) +
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          D * inverse (g x) = (D * g x - f x * E) / (g x * g x)")
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apply (erule subst)
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apply (unfold divide_inverse)
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apply (erule DERIV_mult')
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apply (erule (1) DERIV_inverse')
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apply (simp add: left_diff_distrib nonzero_inverse_mult_distrib)
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apply (simp add: mult_ac)
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done
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lemma DERIV_power_Suc:
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  fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower,comm_monoid_mult}"
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  assumes f: "DERIV f x :> D"
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  shows "DERIV (\<lambda>x. f x ^ Suc n) x :> (of_nat n + 1) *# (D * f x ^ n)"
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proof (induct n)
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case 0
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  show ?case by (simp add: power_Suc f)
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case (Suc k)
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  from DERIV_mult' [OF f Suc] show ?case
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    apply (simp only: of_nat_Suc scaleR_left_distrib scaleR_one)
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    apply (simp only: power_Suc right_distrib mult_scaleR_right mult_ac)
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    done
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qed
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lemma DERIV_power:
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  fixes f :: "real \<Rightarrow> 'a::{real_normed_algebra,recpower,comm_monoid_mult}"
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  assumes f: "DERIV f x :> D"
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  shows "DERIV (\<lambda>x. f x ^ n) x :> of_nat n *# (D * f x ^ (n - Suc 0))"
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by (cases "n", simp, simp add: DERIV_power_Suc f)
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(* ------------------------------------------------------------------------ *)
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(* Caratheodory formulation of derivative at a point: standard proof        *)
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(* ------------------------------------------------------------------------ *)
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lemma CARAT_DERIV:
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     "(DERIV f x :> l) =
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      (\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) & isCont g x & g x = l)"
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      (is "?lhs = ?rhs")
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proof
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  assume der: "DERIV f x :> l"
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  show "\<exists>g. (\<forall>z. f z - f x = (z-x) *# g z) \<and> isCont g x \<and> g x = l"
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  proof (intro exI conjI)
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    let ?g = "(%z. if z = x then l else (f z - f x) /# (z-x))"
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    show "\<forall>z. f z - f x = (z-x) *# ?g z" by (simp)
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    show "isCont ?g x" using der
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      by (simp add: isCont_iff DERIV_iff diff_minus
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               cong: LIM_equal [rule_format])
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    show "?g x = l" by simp
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  qed
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next
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  assume "?rhs"
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  then obtain g where
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    "(\<forall>z. f z - f x = (z-x) *# g z)" and "isCont g x" and "g x = l" by blast
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  thus "(DERIV f x :> l)"
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     by (auto simp add: isCont_iff DERIV_iff diff_minus
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   289
               cong: LIM_equal [rule_format])
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   290
qed
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   291
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   292
lemma DERIV_chain':
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   293
  assumes f: "DERIV f x :> D"
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   294
  assumes g: "DERIV g (f x) :> E"
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   295
  shows "DERIV (\<lambda>x. g (f x)) x :> D *# E"
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   296
proof (unfold DERIV_iff2)
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   297
  obtain d where d: "\<forall>y. g y - g (f x) = (y - f x) *# d y"
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   298
    and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
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   299
    using CARAT_DERIV [THEN iffD1, OF g] by fast
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   300
  from f have "f -- x --> f x"
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   301
    by (rule DERIV_isCont [unfolded isCont_def])
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   302
  with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
huffman@21239
   303
    by (rule isCont_LIM_compose)
huffman@21164
   304
  hence "(\<lambda>z. ((f z - f x) /# (z - x)) *# d (f z))
huffman@21164
   305
          -- x --> D *# d (f x)"
huffman@21164
   306
    by (rule LIM_scaleR [OF f [unfolded DERIV_iff2]])
huffman@21164
   307
  thus "(\<lambda>z. (g (f z) - g (f x)) /# (z - x)) -- x --> D *# E"
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   308
    by (simp add: d dfx real_scaleR_def)
huffman@21164
   309
qed
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   310
huffman@21164
   311
huffman@21164
   312
subsubsection {* Nonstandard proofs *}
huffman@21164
   313
huffman@21164
   314
lemma DERIV_NS_iff:
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   315
      "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/#h) -- 0 --NS> D)"
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   316
by (simp add: deriv_def LIM_NSLIM_iff)
huffman@21164
   317
huffman@21164
   318
lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --NS> D"
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   319
by (simp add: deriv_def LIM_NSLIM_iff)
huffman@21164
   320
huffman@21164
   321
lemma NSDeriv_unique:
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   322
     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
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   323
apply (simp add: nsderiv_def)
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   324
apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
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   325
apply (auto dest!: bspec [where x=epsilon]
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   326
            intro!: inj_hypreal_of_real [THEN injD]
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   327
            dest: approx_trans3)
huffman@21164
   328
done
huffman@21164
   329
huffman@21164
   330
text {*First NSDERIV in terms of NSLIM*}
huffman@21164
   331
huffman@21164
   332
text{*first equivalence *}
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   333
lemma NSDERIV_NSLIM_iff:
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   334
      "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
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   335
apply (simp add: nsderiv_def NSLIM_def, auto)
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   336
apply (drule_tac x = xa in bspec)
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   337
apply (rule_tac [3] ccontr)
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   338
apply (drule_tac [3] x = h in spec)
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   339
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
huffman@21164
   340
done
huffman@21164
   341
huffman@21164
   342
text{*second equivalence *}
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   343
lemma NSDERIV_NSLIM_iff2:
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   344
     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
huffman@21164
   345
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
huffman@21164
   346
              LIM_NSLIM_iff [symmetric])
huffman@21164
   347
huffman@21164
   348
(* while we're at it! *)
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   349
lemma NSDERIV_iff2:
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   350
     "(NSDERIV f x :> D) =
huffman@21164
   351
      (\<forall>w.
huffman@21164
   352
        w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
huffman@21164
   353
        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
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   354
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
huffman@21164
   355
huffman@21164
   356
(*FIXME DELETE*)
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   357
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
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   358
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
huffman@21164
   359
huffman@21164
   360
lemma NSDERIVD5:
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   361
  "(NSDERIV f x :> D) ==>
huffman@21164
   362
   (\<forall>u. u \<approx> hypreal_of_real x -->
huffman@21164
   363
     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
huffman@21164
   364
apply (auto simp add: NSDERIV_iff2)
huffman@21164
   365
apply (case_tac "u = hypreal_of_real x", auto)
huffman@21164
   366
apply (drule_tac x = u in spec, auto)
huffman@21164
   367
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
huffman@21164
   368
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164
   369
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
huffman@21164
   370
apply (auto simp add:
huffman@21164
   371
         approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
huffman@21164
   372
         Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164
   373
done
huffman@21164
   374
huffman@21164
   375
lemma NSDERIVD4:
huffman@21164
   376
     "(NSDERIV f x :> D) ==>
huffman@21164
   377
      (\<forall>h \<in> Infinitesimal.
huffman@21164
   378
               (( *f* f)(hypreal_of_real x + h) -
huffman@21164
   379
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
huffman@21164
   380
apply (auto simp add: nsderiv_def)
huffman@21164
   381
apply (case_tac "h = (0::hypreal) ")
huffman@21164
   382
apply (auto simp add: diff_minus)
huffman@21164
   383
apply (drule_tac x = h in bspec)
huffman@21164
   384
apply (drule_tac [2] c = h in approx_mult1)
huffman@21164
   385
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164
   386
            simp add: diff_minus)
huffman@21164
   387
done
huffman@21164
   388
huffman@21164
   389
lemma NSDERIVD3:
huffman@21164
   390
     "(NSDERIV f x :> D) ==>
huffman@21164
   391
      (\<forall>h \<in> Infinitesimal - {0}.
huffman@21164
   392
               (( *f* f)(hypreal_of_real x + h) -
huffman@21164
   393
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
huffman@21164
   394
apply (auto simp add: nsderiv_def)
huffman@21164
   395
apply (rule ccontr, drule_tac x = h in bspec)
huffman@21164
   396
apply (drule_tac [2] c = h in approx_mult1)
huffman@21164
   397
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164
   398
            simp add: mult_assoc diff_minus)
huffman@21164
   399
done
huffman@21164
   400
huffman@21164
   401
text{*Differentiability implies continuity
huffman@21164
   402
         nice and simple "algebraic" proof*}
huffman@21164
   403
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
huffman@21164
   404
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
huffman@21164
   405
apply (drule approx_minus_iff [THEN iffD1])
huffman@21164
   406
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164
   407
apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
huffman@21164
   408
 prefer 2 apply (simp add: add_assoc [symmetric])
huffman@21164
   409
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
huffman@21164
   410
apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
huffman@21164
   411
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164
   412
            simp add: mult_assoc)
huffman@21164
   413
apply (drule_tac x3=D in
huffman@21164
   414
           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
huffman@21164
   415
             THEN mem_infmal_iff [THEN iffD1]])
huffman@21164
   416
apply (auto simp add: mult_commute
huffman@21164
   417
            intro: approx_trans approx_minus_iff [THEN iffD2])
huffman@21164
   418
done
huffman@21164
   419
huffman@21164
   420
text{*Differentiation rules for combinations of functions
huffman@21164
   421
      follow from clear, straightforard, algebraic
huffman@21164
   422
      manipulations*}
huffman@21164
   423
text{*Constant function*}
huffman@21164
   424
huffman@21164
   425
(* use simple constant nslimit theorem *)
huffman@21164
   426
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
huffman@21164
   427
by (simp add: NSDERIV_NSLIM_iff)
huffman@21164
   428
huffman@21164
   429
text{*Sum of functions- proved easily*}
huffman@21164
   430
huffman@21164
   431
lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
huffman@21164
   432
      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
huffman@21164
   433
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@21164
   434
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
huffman@21164
   435
apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
huffman@21164
   436
apply (auto simp add: diff_def add_ac)
huffman@21164
   437
done
huffman@21164
   438
huffman@21164
   439
text{*Product of functions - Proof is trivial but tedious
huffman@21164
   440
  and long due to rearrangement of terms*}
huffman@21164
   441
huffman@21164
   442
lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
huffman@21164
   443
by (simp add: right_diff_distrib)
huffman@21164
   444
huffman@21164
   445
lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
huffman@21164
   446
         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
huffman@21164
   447
      ==> x - y \<approx> 0"
huffman@21164
   448
apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
huffman@21164
   449
apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
huffman@21164
   450
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
huffman@21164
   451
            simp add: mult_assoc mem_infmal_iff [symmetric])
huffman@21164
   452
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164
   453
done
huffman@21164
   454
huffman@21164
   455
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
huffman@21164
   456
      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
huffman@21164
   457
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@21164
   458
apply (auto dest!: spec
huffman@21164
   459
      simp add: starfun_lambda_cancel lemma_nsderiv1)
huffman@21164
   460
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
huffman@21164
   461
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
huffman@21164
   462
apply (auto simp add: times_divide_eq_right [symmetric]
huffman@21164
   463
            simp del: times_divide_eq)
huffman@21164
   464
apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
huffman@21164
   465
apply (drule_tac
huffman@21164
   466
     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
huffman@21164
   467
apply (auto intro!: approx_add_mono1
huffman@21164
   468
            simp add: left_distrib right_distrib mult_commute add_assoc)
huffman@21164
   469
apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
huffman@21164
   470
         in add_commute [THEN subst])
huffman@21164
   471
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
huffman@21164
   472
                    Infinitesimal_add Infinitesimal_mult
huffman@21164
   473
                    Infinitesimal_hypreal_of_real_mult
huffman@21164
   474
                    Infinitesimal_hypreal_of_real_mult2
huffman@21164
   475
          simp add: add_assoc [symmetric])
huffman@21164
   476
done
huffman@21164
   477
huffman@21164
   478
text{*Multiplying by a constant*}
huffman@21164
   479
lemma NSDERIV_cmult: "NSDERIV f x :> D
huffman@21164
   480
      ==> NSDERIV (%x. c * f x) x :> c*D"
huffman@21164
   481
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
huffman@21164
   482
                  minus_mult_right right_diff_distrib [symmetric])
huffman@21164
   483
apply (erule NSLIM_const [THEN NSLIM_mult])
huffman@21164
   484
done
huffman@21164
   485
huffman@21164
   486
text{*Negation of function*}
huffman@21164
   487
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
huffman@21164
   488
proof (simp add: NSDERIV_NSLIM_iff)
huffman@21164
   489
  assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
huffman@21164
   490
  hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
huffman@21164
   491
    by (rule NSLIM_minus)
huffman@21164
   492
  have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
huffman@21164
   493
    by (simp add: minus_divide_left)
huffman@21164
   494
  with deriv
huffman@21164
   495
  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
huffman@21164
   496
qed
huffman@21164
   497
huffman@21164
   498
text{*Subtraction*}
huffman@21164
   499
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
huffman@21164
   500
by (blast dest: NSDERIV_add NSDERIV_minus)
huffman@21164
   501
huffman@21164
   502
lemma NSDERIV_diff:
huffman@21164
   503
     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
huffman@21164
   504
      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
huffman@21164
   505
apply (simp add: diff_minus)
huffman@21164
   506
apply (blast intro: NSDERIV_add_minus)
huffman@21164
   507
done
huffman@21164
   508
huffman@21164
   509
text{*  Similarly to the above, the chain rule admits an entirely
huffman@21164
   510
   straightforward derivation. Compare this with Harrison's
huffman@21164
   511
   HOL proof of the chain rule, which proved to be trickier and
huffman@21164
   512
   required an alternative characterisation of differentiability-
huffman@21164
   513
   the so-called Carathedory derivative. Our main problem is
huffman@21164
   514
   manipulation of terms.*}
huffman@21164
   515
huffman@21164
   516
huffman@21164
   517
(* lemmas *)
huffman@21164
   518
lemma NSDERIV_zero:
huffman@21164
   519
      "[| NSDERIV g x :> D;
huffman@21164
   520
               ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
huffman@21164
   521
               xa \<in> Infinitesimal;
huffman@21164
   522
               xa \<noteq> 0
huffman@21164
   523
            |] ==> D = 0"
huffman@21164
   524
apply (simp add: nsderiv_def)
huffman@21164
   525
apply (drule bspec, auto)
huffman@21164
   526
done
huffman@21164
   527
huffman@21164
   528
(* can be proved differently using NSLIM_isCont_iff *)
huffman@21164
   529
lemma NSDERIV_approx:
huffman@21164
   530
     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
huffman@21164
   531
      ==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
huffman@21164
   532
apply (simp add: nsderiv_def)
huffman@21164
   533
apply (simp add: mem_infmal_iff [symmetric])
huffman@21164
   534
apply (rule Infinitesimal_ratio)
huffman@21164
   535
apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
huffman@21164
   536
done
huffman@21164
   537
huffman@21164
   538
(*---------------------------------------------------------------
huffman@21164
   539
   from one version of differentiability
huffman@21164
   540
huffman@21164
   541
                f(x) - f(a)
huffman@21164
   542
              --------------- \<approx> Db
huffman@21164
   543
                  x - a
huffman@21164
   544
 ---------------------------------------------------------------*)
huffman@21164
   545
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
huffman@21164
   546
         ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
huffman@21164
   547
         ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
huffman@21164
   548
      |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
huffman@21164
   549
                   - hypreal_of_real (f (g x)))
huffman@21164
   550
              / (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
huffman@21164
   551
             \<approx> hypreal_of_real(Da)"
huffman@21164
   552
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
huffman@21164
   553
huffman@21164
   554
(*--------------------------------------------------------------
huffman@21164
   555
   from other version of differentiability
huffman@21164
   556
huffman@21164
   557
                f(x + h) - f(x)
huffman@21164
   558
               ----------------- \<approx> Db
huffman@21164
   559
                       h
huffman@21164
   560
 --------------------------------------------------------------*)
huffman@21164
   561
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
huffman@21164
   562
      ==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
huffman@21164
   563
          \<approx> hypreal_of_real(Db)"
huffman@21164
   564
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
huffman@21164
   565
huffman@21164
   566
lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
huffman@21164
   567
by auto
huffman@21164
   568
huffman@21164
   569
text{*This proof uses both definitions of differentiability.*}
huffman@21164
   570
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
huffman@21164
   571
      ==> NSDERIV (f o g) x :> Da * Db"
huffman@21164
   572
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
huffman@21164
   573
                mem_infmal_iff [symmetric])
huffman@21164
   574
apply clarify
huffman@21164
   575
apply (frule_tac f = g in NSDERIV_approx)
huffman@21164
   576
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
huffman@21164
   577
apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
huffman@21164
   578
apply (drule_tac g = g in NSDERIV_zero)
huffman@21164
   579
apply (auto simp add: divide_inverse)
huffman@21164
   580
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])
huffman@21164
   581
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164
   582
apply (rule approx_mult_hypreal_of_real)
huffman@21164
   583
apply (simp_all add: divide_inverse [symmetric])
huffman@21164
   584
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
huffman@21164
   585
apply (blast intro: NSDERIVD2)
huffman@21164
   586
done
huffman@21164
   587
huffman@21164
   588
text{*Differentiation of natural number powers*}
huffman@21164
   589
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
huffman@21164
   590
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
huffman@21164
   591
huffman@21164
   592
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
huffman@21164
   593
by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
huffman@21164
   594
huffman@21164
   595
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
huffman@21164
   596
lemma NSDERIV_inverse:
huffman@21164
   597
     "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
huffman@21164
   598
apply (simp add: nsderiv_def)
huffman@21164
   599
apply (rule ballI, simp, clarify)
huffman@21164
   600
apply (frule (1) Infinitesimal_add_not_zero)
huffman@21164
   601
apply (simp add: add_commute)
huffman@21164
   602
(*apply (auto simp add: starfun_inverse_inverse realpow_two
huffman@21164
   603
        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
huffman@21164
   604
apply (simp add: inverse_add inverse_mult_distrib [symmetric]
huffman@21164
   605
              inverse_minus_eq [symmetric] add_ac mult_ac diff_def
huffman@21164
   606
            del: inverse_mult_distrib inverse_minus_eq
huffman@21164
   607
                 minus_mult_left [symmetric] minus_mult_right [symmetric])
huffman@21164
   608
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
huffman@21164
   609
            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
huffman@21164
   610
apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
huffman@21164
   611
apply (rule inverse_add_Infinitesimal_approx2)
huffman@21164
   612
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
huffman@21164
   613
            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
huffman@21164
   614
apply (rule Infinitesimal_HFinite_mult2, auto)
huffman@21164
   615
done
huffman@21164
   616
huffman@21164
   617
subsubsection {* Equivalence of NS and Standard definitions *}
huffman@21164
   618
huffman@21164
   619
lemma divideR_eq_divide [simp]: "x /# y = x / y"
huffman@21164
   620
by (simp add: real_scaleR_def divide_inverse mult_commute)
huffman@21164
   621
huffman@21164
   622
text{*Now equivalence between NSDERIV and DERIV*}
huffman@21164
   623
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
huffman@21164
   624
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
huffman@21164
   625
huffman@21164
   626
(* let's do the standard proof though theorem *)
huffman@21164
   627
(* LIM_mult2 follows from a NS proof          *)
huffman@21164
   628
huffman@21164
   629
lemma DERIV_cmult:
huffman@21164
   630
  fixes f :: "real \<Rightarrow> 'a::real_normed_algebra" shows
huffman@21164
   631
      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
huffman@21164
   632
by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164
   633
huffman@21164
   634
(* standard version *)
huffman@21164
   635
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
huffman@21164
   636
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
huffman@21164
   637
huffman@21164
   638
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
huffman@21164
   639
by (auto dest: DERIV_chain simp add: o_def)
huffman@21164
   640
huffman@21164
   641
(*derivative of linear multiplication*)
huffman@21164
   642
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
huffman@21164
   643
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
huffman@21164
   644
huffman@21164
   645
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@21164
   646
apply (cut_tac DERIV_power [OF DERIV_Id])
huffman@21164
   647
apply (simp add: real_scaleR_def real_of_nat_def)
huffman@21164
   648
done
huffman@21164
   649
huffman@21164
   650
(* NS version *)
huffman@21164
   651
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@21164
   652
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
huffman@21164
   653
huffman@21164
   654
text{*Power of -1*}
huffman@21164
   655
huffman@21164
   656
lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
huffman@21164
   657
by (drule DERIV_inverse' [OF DERIV_Id], simp)
huffman@21164
   658
huffman@21164
   659
text{*Derivative of inverse*}
huffman@21164
   660
lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21164
   661
      ==> DERIV (%x. inverse(f x)::real) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21164
   662
by (drule (1) DERIV_inverse', simp add: mult_ac)
huffman@21164
   663
huffman@21164
   664
lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21164
   665
      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21164
   666
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
huffman@21164
   667
huffman@21164
   668
text{*Derivative of quotient*}
huffman@21164
   669
lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21164
   670
       ==> DERIV (%y. f(y) / (g y) :: real) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21164
   671
by (drule (2) DERIV_divide, simp add: mult_commute)
huffman@21164
   672
huffman@21164
   673
lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21164
   674
       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
huffman@21164
   675
                            - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21164
   676
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
huffman@21164
   677
huffman@21164
   678
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
huffman@21164
   679
      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
huffman@21164
   680
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
huffman@21164
   681
                   real_scaleR_def mult_commute)
huffman@21164
   682
huffman@21164
   683
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
huffman@21164
   684
by auto
huffman@21164
   685
huffman@21164
   686
lemma CARAT_DERIVD:
huffman@21164
   687
  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
huffman@21164
   688
      and nsc: "isNSCont g x"
huffman@21164
   689
  shows "NSDERIV f x :> g x"
huffman@21164
   690
proof -
huffman@21164
   691
  from nsc
huffman@21164
   692
  have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
huffman@21164
   693
         ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
huffman@21164
   694
         hypreal_of_real (g x)"
huffman@21164
   695
    by (simp add: diff_minus isNSCont_def)
huffman@21164
   696
  thus ?thesis using all
huffman@21164
   697
    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
huffman@21164
   698
qed
huffman@21164
   699
huffman@21164
   700
subsubsection {* Differentiability predicate *}
huffman@21164
   701
huffman@21164
   702
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
huffman@21164
   703
by (simp add: differentiable_def)
huffman@21164
   704
huffman@21164
   705
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
huffman@21164
   706
by (force simp add: differentiable_def)
huffman@21164
   707
huffman@21164
   708
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
huffman@21164
   709
by (simp add: NSdifferentiable_def)
huffman@21164
   710
huffman@21164
   711
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
huffman@21164
   712
by (force simp add: NSdifferentiable_def)
huffman@21164
   713
huffman@21164
   714
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
huffman@21164
   715
  apply (unfold differentiable_def)
huffman@21164
   716
  apply (rule_tac x=0 in exI)
huffman@21164
   717
  apply simp
huffman@21164
   718
  done
huffman@21164
   719
huffman@21164
   720
lemma differentiable_sum:
huffman@21164
   721
  assumes "f differentiable x"
huffman@21164
   722
  and "g differentiable x"
huffman@21164
   723
  shows "(\<lambda>x. f x + g x) differentiable x"
huffman@21164
   724
proof -
huffman@21164
   725
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   726
  then obtain df where "DERIV f x :> df" ..
huffman@21164
   727
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   728
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   729
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@21164
   730
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
huffman@21164
   731
  thus ?thesis by (fold differentiable_def)
huffman@21164
   732
qed
huffman@21164
   733
huffman@21164
   734
lemma differentiable_diff:
huffman@21164
   735
  assumes "f differentiable x"
huffman@21164
   736
  and "g differentiable x"
huffman@21164
   737
  shows "(\<lambda>x. f x - g x) differentiable x"
huffman@21164
   738
proof -
huffman@21164
   739
  from prems have "f differentiable x" by simp
huffman@21164
   740
  moreover
huffman@21164
   741
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   742
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   743
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
huffman@21164
   744
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
huffman@21164
   745
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
huffman@21164
   746
  ultimately 
huffman@21164
   747
  show ?thesis
huffman@21164
   748
    by (auto simp: real_diff_def dest: differentiable_sum)
huffman@21164
   749
qed
huffman@21164
   750
huffman@21164
   751
lemma differentiable_mult:
huffman@21164
   752
  assumes "f differentiable x"
huffman@21164
   753
  and "g differentiable x"
huffman@21164
   754
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164
   755
proof -
huffman@21164
   756
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   757
  then obtain df where "DERIV f x :> df" ..
huffman@21164
   758
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   759
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   760
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
huffman@21164
   761
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
huffman@21164
   762
  thus ?thesis by (fold differentiable_def)
huffman@21164
   763
qed
huffman@21164
   764
huffman@21164
   765
subsection {*(NS) Increment*}
huffman@21164
   766
lemma incrementI:
huffman@21164
   767
      "f NSdifferentiable x ==>
huffman@21164
   768
      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@21164
   769
      hypreal_of_real (f x)"
huffman@21164
   770
by (simp add: increment_def)
huffman@21164
   771
huffman@21164
   772
lemma incrementI2: "NSDERIV f x :> D ==>
huffman@21164
   773
     increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@21164
   774
     hypreal_of_real (f x)"
huffman@21164
   775
apply (erule NSdifferentiableI [THEN incrementI])
huffman@21164
   776
done
huffman@21164
   777
huffman@21164
   778
(* The Increment theorem -- Keisler p. 65 *)
huffman@21164
   779
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
huffman@21164
   780
      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
huffman@21164
   781
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
huffman@21164
   782
apply (drule bspec, auto)
huffman@21164
   783
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
huffman@21164
   784
apply (frule_tac b1 = "hypreal_of_real (D) + y"
huffman@21164
   785
        in hypreal_mult_right_cancel [THEN iffD2])
huffman@21164
   786
apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl)
huffman@21164
   787
apply assumption
huffman@21164
   788
apply (simp add: times_divide_eq_right [symmetric])
huffman@21164
   789
apply (auto simp add: left_distrib)
huffman@21164
   790
done
huffman@21164
   791
huffman@21164
   792
lemma increment_thm2:
huffman@21164
   793
     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@21164
   794
      ==> \<exists>e \<in> Infinitesimal. increment f x h =
huffman@21164
   795
              hypreal_of_real(D)*h + e*h"
huffman@21164
   796
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
huffman@21164
   797
huffman@21164
   798
huffman@21164
   799
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@21164
   800
      ==> increment f x h \<approx> 0"
huffman@21164
   801
apply (drule increment_thm2,
huffman@21164
   802
       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
huffman@21164
   803
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164
   804
done
huffman@21164
   805
huffman@21164
   806
subsection {* Nested Intervals and Bisection *}
huffman@21164
   807
huffman@21164
   808
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164
   809
     All considerably tidied by lcp.*}
huffman@21164
   810
huffman@21164
   811
lemma lemma_f_mono_add [rule_format (no_asm)]: "(\<forall>n. (f::nat=>real) n \<le> f (Suc n)) --> f m \<le> f(m + no)"
huffman@21164
   812
apply (induct "no")
huffman@21164
   813
apply (auto intro: order_trans)
huffman@21164
   814
done
huffman@21164
   815
huffman@21164
   816
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   817
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   818
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   819
      ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164
   820
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@21164
   821
apply (induct_tac "n")
huffman@21164
   822
apply (auto intro: order_trans)
huffman@21164
   823
apply (rule_tac y = "g (Suc na)" in order_trans)
huffman@21164
   824
apply (induct_tac [2] "na")
huffman@21164
   825
apply (auto intro: order_trans)
huffman@21164
   826
done
huffman@21164
   827
huffman@21164
   828
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   829
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   830
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   831
      ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164
   832
apply (subst Bseq_minus_iff [symmetric])
huffman@21164
   833
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164
   834
apply auto
huffman@21164
   835
done
huffman@21164
   836
huffman@21164
   837
lemma f_inc_imp_le_lim:
huffman@21164
   838
  fixes f :: "nat \<Rightarrow> real"
huffman@21164
   839
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@21164
   840
apply (rule linorder_not_less [THEN iffD1])
huffman@21164
   841
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
huffman@21164
   842
apply (drule real_less_sum_gt_zero)
huffman@21164
   843
apply (drule_tac x = "f n + - lim f" in spec, safe)
huffman@21164
   844
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
huffman@21164
   845
apply (subgoal_tac "lim f \<le> f (no + n) ")
huffman@21164
   846
apply (drule_tac no=no and m=n in lemma_f_mono_add)
huffman@21164
   847
apply (auto simp add: add_commute)
huffman@21164
   848
apply (induct_tac "no")
huffman@21164
   849
apply simp
huffman@21164
   850
apply (auto intro: order_trans simp add: diff_minus abs_if)
huffman@21164
   851
done
huffman@21164
   852
huffman@21164
   853
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@21164
   854
apply (rule LIMSEQ_minus [THEN limI])
huffman@21164
   855
apply (simp add: convergent_LIMSEQ_iff)
huffman@21164
   856
done
huffman@21164
   857
huffman@21164
   858
lemma g_dec_imp_lim_le:
huffman@21164
   859
  fixes g :: "nat \<Rightarrow> real"
huffman@21164
   860
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@21164
   861
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
huffman@21164
   862
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
huffman@21164
   863
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
huffman@21164
   864
done
huffman@21164
   865
huffman@21164
   866
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   867
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   868
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   869
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   870
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164
   871
apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164
   872
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164
   873
apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164
   874
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164
   875
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164
   876
apply (rule_tac x = "lim f" in exI)
huffman@21164
   877
apply (rule_tac x = "lim g" in exI)
huffman@21164
   878
apply (auto intro: LIMSEQ_le)
huffman@21164
   879
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164
   880
done
huffman@21164
   881
huffman@21164
   882
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   883
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   884
         \<forall>n. f(n) \<le> g(n);
huffman@21164
   885
         (%n. f(n) - g(n)) ----> 0 |]
huffman@21164
   886
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   887
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164
   888
apply (drule lemma_nest, auto)
huffman@21164
   889
apply (subgoal_tac "l = m")
huffman@21164
   890
apply (drule_tac [2] X = f in LIMSEQ_diff)
huffman@21164
   891
apply (auto intro: LIMSEQ_unique)
huffman@21164
   892
done
huffman@21164
   893
huffman@21164
   894
text{*The universal quantifiers below are required for the declaration
huffman@21164
   895
  of @{text Bolzano_nest_unique} below.*}
huffman@21164
   896
huffman@21164
   897
lemma Bolzano_bisect_le:
huffman@21164
   898
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   899
apply (rule allI)
huffman@21164
   900
apply (induct_tac "n")
huffman@21164
   901
apply (auto simp add: Let_def split_def)
huffman@21164
   902
done
huffman@21164
   903
huffman@21164
   904
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164
   905
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164
   906
apply (rule allI)
huffman@21164
   907
apply (induct_tac "n")
huffman@21164
   908
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   909
done
huffman@21164
   910
huffman@21164
   911
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164
   912
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   913
apply (rule allI)
huffman@21164
   914
apply (induct_tac "n")
huffman@21164
   915
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   916
done
huffman@21164
   917
huffman@21164
   918
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164
   919
apply (auto)
huffman@21164
   920
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164
   921
apply (simp)
huffman@21164
   922
done
huffman@21164
   923
huffman@21164
   924
lemma Bolzano_bisect_diff:
huffman@21164
   925
     "a \<le> b ==>
huffman@21164
   926
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164
   927
      (b-a) / (2 ^ n)"
huffman@21164
   928
apply (induct "n")
huffman@21164
   929
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164
   930
done
huffman@21164
   931
huffman@21164
   932
lemmas Bolzano_nest_unique =
huffman@21164
   933
    lemma_nest_unique
huffman@21164
   934
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164
   935
huffman@21164
   936
huffman@21164
   937
lemma not_P_Bolzano_bisect:
huffman@21164
   938
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164
   939
      and notP: "~ P(a,b)"
huffman@21164
   940
      and le:   "a \<le> b"
huffman@21164
   941
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   942
proof (induct n)
huffman@21164
   943
  case 0 thus ?case by simp
huffman@21164
   944
 next
huffman@21164
   945
  case (Suc n)
huffman@21164
   946
  thus ?case
huffman@21164
   947
 by (auto simp del: surjective_pairing [symmetric]
huffman@21164
   948
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164
   949
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164
   950
qed
huffman@21164
   951
huffman@21164
   952
(*Now we re-package P_prem as a formula*)
huffman@21164
   953
lemma not_P_Bolzano_bisect':
huffman@21164
   954
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   955
         ~ P(a,b);  a \<le> b |] ==>
huffman@21164
   956
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   957
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164
   958
huffman@21164
   959
huffman@21164
   960
huffman@21164
   961
lemma lemma_BOLZANO:
huffman@21164
   962
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   963
         \<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   964
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164
   965
         a \<le> b |]
huffman@21164
   966
      ==> P(a,b)"
huffman@21164
   967
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
huffman@21164
   968
apply (rule LIMSEQ_minus_cancel)
huffman@21164
   969
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164
   970
apply (rule ccontr)
huffman@21164
   971
apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164
   972
apply (rename_tac "l")
huffman@21164
   973
apply (drule_tac x = l in spec, clarify)
huffman@21164
   974
apply (simp add: LIMSEQ_def)
huffman@21164
   975
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   976
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   977
apply (drule real_less_half_sum, auto)
huffman@21164
   978
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   979
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   980
apply safe
huffman@21164
   981
apply (simp_all (no_asm_simp))
huffman@21164
   982
apply (rule_tac y = "abs (fst (Bolzano_bisect P a b (no + noa)) - l) + abs (snd (Bolzano_bisect P a b (no + noa)) - l)" in order_le_less_trans)
huffman@21164
   983
apply (simp (no_asm_simp) add: abs_if)
huffman@21164
   984
apply (rule real_sum_of_halves [THEN subst])
huffman@21164
   985
apply (rule add_strict_mono)
huffman@21164
   986
apply (simp_all add: diff_minus [symmetric])
huffman@21164
   987
done
huffman@21164
   988
huffman@21164
   989
huffman@21164
   990
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164
   991
       (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   992
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164
   993
      --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164
   994
apply clarify
huffman@21164
   995
apply (blast intro: lemma_BOLZANO)
huffman@21164
   996
done
huffman@21164
   997
huffman@21164
   998
huffman@21164
   999
subsection {* Intermediate Value Theorem *}
huffman@21164
  1000
huffman@21164
  1001
text {*Prove Contrapositive by Bisection*}
huffman@21164
  1002
huffman@21164
  1003
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164
  1004
         a \<le> b;
huffman@21164
  1005
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164
  1006
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
  1007
apply (rule contrapos_pp, assumption)
huffman@21164
  1008
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> ~ (f (u) \<le> y & y \<le> f (v))" in lemma_BOLZANO2)
huffman@21164
  1009
apply safe
huffman@21164
  1010
apply simp_all
huffman@21164
  1011
apply (simp add: isCont_iff LIM_def)
huffman@21164
  1012
apply (rule ccontr)
huffman@21164
  1013
apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164
  1014
 prefer 2
huffman@21164
  1015
 apply simp
huffman@21164
  1016
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164
  1017
apply (drule_tac x = x in spec)+
huffman@21164
  1018
apply simp
huffman@21164
  1019
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164
  1020
apply safe
huffman@21164
  1021
apply simp
huffman@21164
  1022
apply (drule_tac x = s in spec, clarify)
huffman@21164
  1023
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164
  1024
apply (drule_tac x = "ba-x" in spec)
huffman@21164
  1025
apply (simp_all add: abs_if)
huffman@21164
  1026
apply (drule_tac x = "aa-x" in spec)
huffman@21164
  1027
apply (case_tac "x \<le> aa", simp_all)
huffman@21164
  1028
done
huffman@21164
  1029
huffman@21164
  1030
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164
  1031
         a \<le> b;
huffman@21164
  1032
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164
  1033
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
  1034
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164
  1035
apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@21164
  1036
apply (auto intro: isCont_minus)
huffman@21164
  1037
done
huffman@21164
  1038
huffman@21164
  1039
(*HOL style here: object-level formulations*)
huffman@21164
  1040
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164
  1041
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
  1042
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
  1043
apply (blast intro: IVT)
huffman@21164
  1044
done
huffman@21164
  1045
huffman@21164
  1046
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164
  1047
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
  1048
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
  1049
apply (blast intro: IVT2)
huffman@21164
  1050
done
huffman@21164
  1051
huffman@21164
  1052
text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164
  1053
huffman@21164
  1054
lemma isCont_bounded:
huffman@21164
  1055
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1056
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164
  1057
apply (cut_tac P = "% (u,v) . a \<le> u & u \<le> v & v \<le> b --> (\<exists>M. \<forall>x. u \<le> x & x \<le> v --> f x \<le> M)" in lemma_BOLZANO2)
huffman@21164
  1058
apply safe
huffman@21164
  1059
apply simp_all
huffman@21164
  1060
apply (rename_tac x xa ya M Ma)
huffman@21164
  1061
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
huffman@21164
  1062
apply (rule_tac x = Ma in exI, clarify)
huffman@21164
  1063
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
  1064
apply (rule_tac x = M in exI, clarify)
huffman@21164
  1065
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
  1066
apply (case_tac "a \<le> x & x \<le> b")
huffman@21164
  1067
apply (rule_tac [2] x = 1 in exI)
huffman@21164
  1068
prefer 2 apply force
huffman@21164
  1069
apply (simp add: LIM_def isCont_iff)
huffman@21164
  1070
apply (drule_tac x = x in spec, auto)
huffman@21164
  1071
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164
  1072
apply (drule_tac x = 1 in spec, auto)
huffman@21164
  1073
apply (rule_tac x = s in exI, clarify)
huffman@21164
  1074
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164
  1075
apply (drule_tac x = "xa-x" in spec)
huffman@21164
  1076
apply (auto simp add: abs_ge_self)
huffman@21164
  1077
done
huffman@21164
  1078
huffman@21164
  1079
text{*Refine the above to existence of least upper bound*}
huffman@21164
  1080
huffman@21164
  1081
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164
  1082
      (\<exists>t. isLub UNIV S t)"
huffman@21164
  1083
by (blast intro: reals_complete)
huffman@21164
  1084
huffman@21164
  1085
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1086
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
  1087
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164
  1088
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164
  1089
        in lemma_reals_complete)
huffman@21164
  1090
apply auto
huffman@21164
  1091
apply (drule isCont_bounded, assumption)
huffman@21164
  1092
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164
  1093
apply (rule exI, auto)
huffman@21164
  1094
apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164
  1095
done
huffman@21164
  1096
huffman@21164
  1097
text{*Now show that it attains its upper bound*}
huffman@21164
  1098
huffman@21164
  1099
lemma isCont_eq_Ub:
huffman@21164
  1100
  assumes le: "a \<le> b"
huffman@21164
  1101
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1102
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
  1103
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
  1104
proof -
huffman@21164
  1105
  from isCont_has_Ub [OF le con]
huffman@21164
  1106
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164
  1107
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164
  1108
  show ?thesis
huffman@21164
  1109
  proof (intro exI, intro conjI)
huffman@21164
  1110
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164
  1111
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164
  1112
    proof (rule ccontr)
huffman@21164
  1113
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164
  1114
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
huffman@21164
  1115
        by (fastsimp simp add: linorder_not_le [symmetric])
huffman@21164
  1116
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@21164
  1117
        by (auto simp add: isCont_inverse isCont_diff con)
huffman@21164
  1118
      from isCont_bounded [OF le this]
huffman@21164
  1119
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164
  1120
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
huffman@21164
  1121
        by (simp add: M3 compare_rls)
huffman@21164
  1122
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164
  1123
        by (auto intro: order_le_less_trans [of _ k])
huffman@21164
  1124
      with Minv
huffman@21164
  1125
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164
  1126
        by (intro strip less_imp_inverse_less, simp_all)
huffman@21164
  1127
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164
  1128
        by simp
huffman@21164
  1129
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164
  1130
        by (simp, arith)
huffman@21164
  1131
      from M2 [OF this]
huffman@21164
  1132
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164
  1133
      thus False using invlt [of x] by force
huffman@21164
  1134
    qed
huffman@21164
  1135
  qed
huffman@21164
  1136
qed
huffman@21164
  1137
huffman@21164
  1138
huffman@21164
  1139
text{*Same theorem for lower bound*}
huffman@21164
  1140
huffman@21164
  1141
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1142
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164
  1143
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
  1144
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164
  1145
prefer 2 apply (blast intro: isCont_minus)
huffman@21164
  1146
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164
  1147
apply safe
huffman@21164
  1148
apply auto
huffman@21164
  1149
done
huffman@21164
  1150
huffman@21164
  1151
huffman@21164
  1152
text{*Another version.*}
huffman@21164
  1153
huffman@21164
  1154
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1155
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164
  1156
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164
  1157
apply (frule isCont_eq_Lb)
huffman@21164
  1158
apply (frule_tac [2] isCont_eq_Ub)
huffman@21164
  1159
apply (assumption+, safe)
huffman@21164
  1160
apply (rule_tac x = "f x" in exI)
huffman@21164
  1161
apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164
  1162
apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164
  1163
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164
  1164
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164
  1165
apply (rule_tac [2] x = xb in exI)
huffman@21164
  1166
apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164
  1167
done
huffman@21164
  1168
huffman@21164
  1169
huffman@21164
  1170
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
  1171
huffman@21164
  1172
lemma DERIV_left_inc:
huffman@21164
  1173
  fixes f :: "real => real"
huffman@21164
  1174
  assumes der: "DERIV f x :> l"
huffman@21164
  1175
      and l:   "0 < l"
huffman@21164
  1176
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
  1177
proof -
huffman@21164
  1178
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
  1179
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
huffman@21164
  1180
    by (simp add: diff_minus)
huffman@21164
  1181
  then obtain s
huffman@21164
  1182
        where s:   "0 < s"
huffman@21164
  1183
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
  1184
    by auto
huffman@21164
  1185
  thus ?thesis
huffman@21164
  1186
  proof (intro exI conjI strip)
huffman@21164
  1187
    show "0<s" .
huffman@21164
  1188
    fix h::real
huffman@21164
  1189
    assume "0 < h" "h < s"
huffman@21164
  1190
    with all [of h] show "f x < f (x+h)"
huffman@21164
  1191
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
  1192
    split add: split_if_asm)
huffman@21164
  1193
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
  1194
      with l
huffman@21164
  1195
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
  1196
      thus "f x < f (x+h)"
huffman@21164
  1197
  by (simp add: pos_less_divide_eq h)
huffman@21164
  1198
    qed
huffman@21164
  1199
  qed
huffman@21164
  1200
qed
huffman@21164
  1201
huffman@21164
  1202
lemma DERIV_left_dec:
huffman@21164
  1203
  fixes f :: "real => real"
huffman@21164
  1204
  assumes der: "DERIV f x :> l"
huffman@21164
  1205
      and l:   "l < 0"
huffman@21164
  1206
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
  1207
proof -
huffman@21164
  1208
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
  1209
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
huffman@21164
  1210
    by (simp add: diff_minus)
huffman@21164
  1211
  then obtain s
huffman@21164
  1212
        where s:   "0 < s"
huffman@21164
  1213
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
  1214
    by auto
huffman@21164
  1215
  thus ?thesis
huffman@21164
  1216
  proof (intro exI conjI strip)
huffman@21164
  1217
    show "0<s" .
huffman@21164
  1218
    fix h::real
huffman@21164
  1219
    assume "0 < h" "h < s"
huffman@21164
  1220
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
  1221
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
  1222
    split add: split_if_asm)
huffman@21164
  1223
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
  1224
      with l
huffman@21164
  1225
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
  1226
      thus "f x < f (x-h)"
huffman@21164
  1227
  by (simp add: pos_less_divide_eq h)
huffman@21164
  1228
    qed
huffman@21164
  1229
  qed
huffman@21164
  1230
qed
huffman@21164
  1231
huffman@21164
  1232
lemma DERIV_local_max:
huffman@21164
  1233
  fixes f :: "real => real"
huffman@21164
  1234
  assumes der: "DERIV f x :> l"
huffman@21164
  1235
      and d:   "0 < d"
huffman@21164
  1236
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
  1237
  shows "l = 0"
huffman@21164
  1238
proof (cases rule: linorder_cases [of l 0])
huffman@21164
  1239
  case equal show ?thesis .
huffman@21164
  1240
next
huffman@21164
  1241
  case less
huffman@21164
  1242
  from DERIV_left_dec [OF der less]
huffman@21164
  1243
  obtain d' where d': "0 < d'"
huffman@21164
  1244
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
  1245
  from real_lbound_gt_zero [OF d d']
huffman@21164
  1246
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
  1247
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
  1248
  show ?thesis by (auto simp add: abs_if)
huffman@21164
  1249
next
huffman@21164
  1250
  case greater
huffman@21164
  1251
  from DERIV_left_inc [OF der greater]
huffman@21164
  1252
  obtain d' where d': "0 < d'"
huffman@21164
  1253
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
  1254
  from real_lbound_gt_zero [OF d d']
huffman@21164
  1255
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
  1256
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
  1257
  show ?thesis by (auto simp add: abs_if)
huffman@21164
  1258
qed
huffman@21164
  1259
huffman@21164
  1260
huffman@21164
  1261
text{*Similar theorem for a local minimum*}
huffman@21164
  1262
lemma DERIV_local_min:
huffman@21164
  1263
  fixes f :: "real => real"
huffman@21164
  1264
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
  1265
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
  1266
huffman@21164
  1267
huffman@21164
  1268
text{*In particular, if a function is locally flat*}
huffman@21164
  1269
lemma DERIV_local_const:
huffman@21164
  1270
  fixes f :: "real => real"
huffman@21164
  1271
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
  1272
by (auto dest!: DERIV_local_max)
huffman@21164
  1273
huffman@21164
  1274
text{*Lemma about introducing open ball in open interval*}
huffman@21164
  1275
lemma lemma_interval_lt:
huffman@21164
  1276
     "[| a < x;  x < b |]
huffman@21164
  1277
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
huffman@21164
  1278
apply (simp add: abs_interval_iff)
huffman@21164
  1279
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
  1280
apply (rule_tac x = "x-a" in exI)
huffman@21164
  1281
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
  1282
done
huffman@21164
  1283
huffman@21164
  1284
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
  1285
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
  1286
apply (drule lemma_interval_lt, auto)
huffman@21164
  1287
apply (auto intro!: exI)
huffman@21164
  1288
done
huffman@21164
  1289
huffman@21164
  1290
text{*Rolle's Theorem.
huffman@21164
  1291
   If @{term f} is defined and continuous on the closed interval
huffman@21164
  1292
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
  1293
   and @{term "f(a) = f(b)"},
huffman@21164
  1294
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
  1295
theorem Rolle:
huffman@21164
  1296
  assumes lt: "a < b"
huffman@21164
  1297
      and eq: "f(a) = f(b)"
huffman@21164
  1298
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1299
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21164
  1300
  shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
huffman@21164
  1301
proof -
huffman@21164
  1302
  have le: "a \<le> b" using lt by simp
huffman@21164
  1303
  from isCont_eq_Ub [OF le con]
huffman@21164
  1304
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
  1305
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
  1306
    by blast
huffman@21164
  1307
  from isCont_eq_Lb [OF le con]
huffman@21164
  1308
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
  1309
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
  1310
    by blast
huffman@21164
  1311
  show ?thesis
huffman@21164
  1312
  proof cases
huffman@21164
  1313
    assume axb: "a < x & x < b"
huffman@21164
  1314
        --{*@{term f} attains its maximum within the interval*}
huffman@21164
  1315
    hence ax: "a<x" and xb: "x<b" by auto
huffman@21164
  1316
    from lemma_interval [OF ax xb]
huffman@21164
  1317
    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164
  1318
      by blast
huffman@21164
  1319
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
  1320
      by blast
huffman@21164
  1321
    from differentiableD [OF dif [OF axb]]
huffman@21164
  1322
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1323
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
  1324
        --{*the derivative at a local maximum is zero*}
huffman@21164
  1325
    thus ?thesis using ax xb der by auto
huffman@21164
  1326
  next
huffman@21164
  1327
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
  1328
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
  1329
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
  1330
    show ?thesis
huffman@21164
  1331
    proof cases
huffman@21164
  1332
      assume ax'b: "a < x' & x' < b"
huffman@21164
  1333
        --{*@{term f} attains its minimum within the interval*}
huffman@21164
  1334
      hence ax': "a<x'" and x'b: "x'<b" by auto
huffman@21164
  1335
      from lemma_interval [OF ax' x'b]
huffman@21164
  1336
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164
  1337
  by blast
huffman@21164
  1338
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
  1339
  by blast
huffman@21164
  1340
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
  1341
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
  1342
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
  1343
        --{*the derivative at a local minimum is zero*}
huffman@21164
  1344
      thus ?thesis using ax' x'b der by auto
huffman@21164
  1345
    next
huffman@21164
  1346
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
  1347
        --{*@{term f} is constant througout the interval*}
huffman@21164
  1348
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
  1349
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
  1350
      from dense [OF lt]
huffman@21164
  1351
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
  1352
      from lemma_interval [OF ar rb]
huffman@21164
  1353
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
huffman@21164
  1354
  by blast
huffman@21164
  1355
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
  1356
      proof (clarify)
huffman@21164
  1357
        fix z::real
huffman@21164
  1358
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1359
        show "f z = f b"
huffman@21164
  1360
        proof (rule order_antisym)
huffman@21164
  1361
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1362
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1363
        qed
huffman@21164
  1364
      qed
huffman@21164
  1365
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1366
      proof (intro strip)
huffman@21164
  1367
        fix y::real
huffman@21164
  1368
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1369
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1370
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1371
      qed
huffman@21164
  1372
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1373
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1374
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1375
        --{*the derivative of a constant function is zero*}
huffman@21164
  1376
      thus ?thesis using ar rb der by auto
huffman@21164
  1377
    qed
huffman@21164
  1378
  qed
huffman@21164
  1379
qed
huffman@21164
  1380
huffman@21164
  1381
huffman@21164
  1382
subsection{*Mean Value Theorem*}
huffman@21164
  1383
huffman@21164
  1384
lemma lemma_MVT:
huffman@21164
  1385
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164
  1386
proof cases
huffman@21164
  1387
  assume "a=b" thus ?thesis by simp
huffman@21164
  1388
next
huffman@21164
  1389
  assume "a\<noteq>b"
huffman@21164
  1390
  hence ba: "b-a \<noteq> 0" by arith
huffman@21164
  1391
  show ?thesis
huffman@21164
  1392
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164
  1393
        simp add: right_diff_distrib,
huffman@21164
  1394
        simp add: left_diff_distrib)
huffman@21164
  1395
qed
huffman@21164
  1396
huffman@21164
  1397
theorem MVT:
huffman@21164
  1398
  assumes lt:  "a < b"
huffman@21164
  1399
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1400
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21164
  1401
  shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
huffman@21164
  1402
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1403
proof -
huffman@21164
  1404
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@21164
  1405
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
huffman@21164
  1406
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
huffman@21164
  1407
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
  1408
  proof (clarify)
huffman@21164
  1409
    fix x::real
huffman@21164
  1410
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1411
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1412
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1413
    show "?F differentiable x"
huffman@21164
  1414
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1415
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1416
  qed
huffman@21164
  1417
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1418
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1419
    by blast
huffman@21164
  1420
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1421
    by (rule DERIV_cmult_Id)
huffman@21164
  1422
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1423
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1424
    by (rule DERIV_add [OF der])
huffman@21164
  1425
  show ?thesis
huffman@21164
  1426
  proof (intro exI conjI)
huffman@21164
  1427
    show "a < z" .
huffman@21164
  1428
    show "z < b" .
huffman@21164
  1429
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1430
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1431
  qed
huffman@21164
  1432
qed
huffman@21164
  1433
huffman@21164
  1434
huffman@21164
  1435
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1436
huffman@21164
  1437
lemma DERIV_isconst_end:
huffman@21164
  1438
  fixes f :: "real => real"
huffman@21164
  1439
  shows "[| a < b;
huffman@21164
  1440
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1441
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1442
        ==> f b = f a"
huffman@21164
  1443
apply (drule MVT, assumption)
huffman@21164
  1444
apply (blast intro: differentiableI)
huffman@21164
  1445
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1446
done
huffman@21164
  1447
huffman@21164
  1448
lemma DERIV_isconst1:
huffman@21164
  1449
  fixes f :: "real => real"
huffman@21164
  1450
  shows "[| a < b;
huffman@21164
  1451
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1452
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1453
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1454
apply safe
huffman@21164
  1455
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1456
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1457
done
huffman@21164
  1458
huffman@21164
  1459
lemma DERIV_isconst2:
huffman@21164
  1460
  fixes f :: "real => real"
huffman@21164
  1461
  shows "[| a < b;
huffman@21164
  1462
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1463
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1464
         a \<le> x; x \<le> b |]
huffman@21164
  1465
        ==> f x = f a"
huffman@21164
  1466
apply (blast dest: DERIV_isconst1)
huffman@21164
  1467
done
huffman@21164
  1468
huffman@21164
  1469
lemma DERIV_isconst_all:
huffman@21164
  1470
  fixes f :: "real => real"
huffman@21164
  1471
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1472
apply (rule linorder_cases [of x y])
huffman@21164
  1473
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1474
done
huffman@21164
  1475
huffman@21164
  1476
lemma DERIV_const_ratio_const:
huffman@21164
  1477
     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1478
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1479
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1480
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
huffman@21164
  1481
apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
huffman@21164
  1482
done
huffman@21164
  1483
huffman@21164
  1484
lemma DERIV_const_ratio_const2:
huffman@21164
  1485
     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1486
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1487
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1488
done
huffman@21164
  1489
huffman@21164
  1490
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1491
by (simp)
huffman@21164
  1492
huffman@21164
  1493
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1494
by (simp)
huffman@21164
  1495
huffman@21164
  1496
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1497
huffman@21164
  1498
lemma DERIV_const_average:
huffman@21164
  1499
  fixes v :: "real => real"
huffman@21164
  1500
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1501
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1502
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1503
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1504
  case equal with neq show ?thesis by simp
huffman@21164
  1505
next
huffman@21164
  1506
  case less
huffman@21164
  1507
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1508
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1509
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1510
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1511
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1512
  ultimately show ?thesis using neq by force
huffman@21164
  1513
next
huffman@21164
  1514
  case greater
huffman@21164
  1515
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1516
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1517
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1518
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1519
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1520
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1521
qed
huffman@21164
  1522
huffman@21164
  1523
huffman@21164
  1524
text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164
  1525
strict maximum at an end point, not in the middle.*}
huffman@21164
  1526
huffman@21164
  1527
lemma lemma_isCont_inj:
huffman@21164
  1528
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1529
  assumes d: "0 < d"
huffman@21164
  1530
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1531
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1532
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164
  1533
proof (rule ccontr)
huffman@21164
  1534
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164
  1535
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164
  1536
  show False
huffman@21164
  1537
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164
  1538
    case le
huffman@21164
  1539
    from d cont all [of "x+d"]
huffman@21164
  1540
    have flef: "f(x+d) \<le> f x"
huffman@21164
  1541
     and xlex: "x - d \<le> x"
huffman@21164
  1542
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164
  1543
       by (auto simp add: abs_if)
huffman@21164
  1544
    from IVT [OF le flef xlex cont']
huffman@21164
  1545
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164
  1546
    moreover
huffman@21164
  1547
    hence "g(f x') = g (f(x+d))" by simp
huffman@21164
  1548
    ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@21164
  1549
      by (simp add: abs_le_interval_iff)
huffman@21164
  1550
  next
huffman@21164
  1551
    case ge
huffman@21164
  1552
    from d cont all [of "x-d"]
huffman@21164
  1553
    have flef: "f(x-d) \<le> f x"
huffman@21164
  1554
     and xlex: "x \<le> x+d"
huffman@21164
  1555
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164
  1556
       by (auto simp add: abs_if)
huffman@21164
  1557
    from IVT2 [OF ge flef xlex cont']
huffman@21164
  1558
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164
  1559
    moreover
huffman@21164
  1560
    hence "g(f x') = g (f(x-d))" by simp
huffman@21164
  1561
    ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@21164
  1562
      by (simp add: abs_le_interval_iff)
huffman@21164
  1563
  qed
huffman@21164
  1564
qed
huffman@21164
  1565
huffman@21164
  1566
huffman@21164
  1567
text{*Similar version for lower bound.*}
huffman@21164
  1568
huffman@21164
  1569
lemma lemma_isCont_inj2:
huffman@21164
  1570
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1571
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164
  1572
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164
  1573
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164
  1574
apply (insert lemma_isCont_inj
huffman@21164
  1575
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@21164
  1576
apply (simp add: isCont_minus linorder_not_le)
huffman@21164
  1577
done
huffman@21164
  1578
huffman@21164
  1579
text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164
  1580
@{text "f[[x - d, x + d]]"} .*}
huffman@21164
  1581
huffman@21164
  1582
lemma isCont_inj_range:
huffman@21164
  1583
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1584
  assumes d: "0 < d"
huffman@21164
  1585
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1586
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1587
  shows "\<exists>e>0. \<forall>y. \<bar>y - f x\<bar> \<le> e --> (\<exists>z. \<bar>z-x\<bar> \<le> d & f z = y)"
huffman@21164
  1588
proof -
huffman@21164
  1589
  have "x-d \<le> x+d" "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z" using cont d
huffman@21164
  1590
    by (auto simp add: abs_le_interval_iff)
huffman@21164
  1591
  from isCont_Lb_Ub [OF this]
huffman@21164
  1592
  obtain L M
huffman@21164
  1593
  where all1 [rule_format]: "\<forall>z. x-d \<le> z \<and> z \<le> x+d \<longrightarrow> L \<le> f z \<and> f z \<le> M"
huffman@21164
  1594
    and all2 [rule_format]:
huffman@21164
  1595
           "\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>z. x-d \<le> z \<and> z \<le> x+d \<and> f z = y)"
huffman@21164
  1596
    by auto
huffman@21164
  1597
  with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164
  1598
  moreover have "L \<noteq> f x"
huffman@21164
  1599
  proof -
huffman@21164
  1600
    from lemma_isCont_inj2 [OF d inj cont]
huffman@21164
  1601
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164
  1602
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1603
  qed
huffman@21164
  1604
  moreover have "f x \<noteq> M"
huffman@21164
  1605
  proof -
huffman@21164
  1606
    from lemma_isCont_inj [OF d inj cont]
huffman@21164
  1607
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164
  1608
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1609
  qed
huffman@21164
  1610
  ultimately have "L < f x & f x < M" by arith
huffman@21164
  1611
  hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164
  1612
  from real_lbound_gt_zero [OF this]
huffman@21164
  1613
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164
  1614
  thus ?thesis
huffman@21164
  1615
  proof (intro exI conjI)
huffman@21164
  1616
    show "0<e" .
huffman@21164
  1617
    show "\<forall>y. \<bar>y - f x\<bar> \<le> e \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y)"
huffman@21164
  1618
    proof (intro strip)
huffman@21164
  1619
      fix y::real
huffman@21164
  1620
      assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164
  1621
      with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164
  1622
      from all2 [OF this]
huffman@21164
  1623
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
huffman@21164
  1624
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
huffman@21164
  1625
        by (force simp add: abs_le_interval_iff)
huffman@21164
  1626
    qed
huffman@21164
  1627
  qed
huffman@21164
  1628
qed
huffman@21164
  1629
huffman@21164
  1630
huffman@21164
  1631
text{*Continuity of inverse function*}
huffman@21164
  1632
huffman@21164
  1633
lemma isCont_inverse_function:
huffman@21164
  1634
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1635
  assumes d: "0 < d"
huffman@21164
  1636
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1637
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1638
  shows "isCont g (f x)"
huffman@21164
  1639
proof (simp add: isCont_iff LIM_eq)
huffman@21164
  1640
  show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164
  1641
         (\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r)"
huffman@21164
  1642
  proof (intro strip)
huffman@21164
  1643
    fix r::real
huffman@21164
  1644
    assume r: "0<r"
huffman@21164
  1645
    from real_lbound_gt_zero [OF r d]
huffman@21164
  1646
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164
  1647
    with inj cont
huffman@21164
  1648
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164
  1649
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164
  1650
    from isCont_inj_range [OF e this]
huffman@21164
  1651
    obtain e' where e': "0 < e'"
huffman@21164
  1652
        and all: "\<forall>y. \<bar>y - f x\<bar> \<le> e' \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)"
huffman@21164
  1653
          by blast
huffman@21164
  1654
    show "\<exists>s>0. \<forall>z. z\<noteq>0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>g(f x + z) - g(f x)\<bar> < r"
huffman@21164
  1655
    proof (intro exI conjI)
huffman@21164
  1656
      show "0<e'" .
huffman@21164
  1657
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164
  1658
      proof (intro strip)
huffman@21164
  1659
        fix z::real
huffman@21164
  1660
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164
  1661
        with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164
  1662
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164
  1663
      qed
huffman@21164
  1664
    qed
huffman@21164
  1665
  qed
huffman@21164
  1666
qed
huffman@21164
  1667
huffman@21164
  1668
theorem GMVT:
huffman@21164
  1669
  assumes alb: "a < b"
huffman@21164
  1670
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
huffman@21164
  1671
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
huffman@21164
  1672
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
huffman@21164
  1673
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1674
  shows "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
huffman@21164
  1675
proof -
huffman@21164
  1676
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
huffman@21164
  1677
  from prems have "a < b" by simp
huffman@21164
  1678
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@21164
  1679
  proof -
huffman@21164
  1680
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
huffman@21164
  1681
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
huffman@21164
  1682
      by (auto intro: isCont_mult)
huffman@21164
  1683
    moreover
huffman@21164
  1684
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
huffman@21164
  1685
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
huffman@21164
  1686
      by (auto intro: isCont_mult)
huffman@21164
  1687
    ultimately show ?thesis
huffman@21164
  1688
      by (fastsimp intro: isCont_diff)
huffman@21164
  1689
  qed
huffman@21164
  1690
  moreover
huffman@21164
  1691
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@21164
  1692
  proof -
huffman@21164
  1693
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1694
    with gd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (f b - f a) * g x) differentiable x" by (simp add: differentiable_mult)
huffman@21164
  1695
    moreover
huffman@21164
  1696
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1697
    with fd have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. (g b - g a) * f x) differentiable x" by (simp add: differentiable_mult)
huffman@21164
  1698
    ultimately show ?thesis by (simp add: differentiable_diff)
huffman@21164
  1699
  qed
huffman@21164
  1700
  ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
huffman@21164
  1701
  then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164
  1702
  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164
  1703
huffman@21164
  1704
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1705
  with gd have "g differentiable c" by simp
huffman@21164
  1706
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1707
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1708
huffman@21164
  1709
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1710
  with fd have "f differentiable c" by simp
huffman@21164
  1711
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1712
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1713
huffman@21164
  1714
  from cdef have "DERIV ?h c :> l" by auto
huffman@21164
  1715
  moreover
huffman@21164
  1716
  {
huffman@21164
  1717
    from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
huffman@21164
  1718
      apply (insert DERIV_const [where k="f b - f a"])
huffman@21164
  1719
      apply (drule meta_spec [of _ c])
huffman@21164
  1720
      apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
huffman@21164
  1721
      by simp_all
huffman@21164
  1722
    moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
huffman@21164
  1723
      apply (insert DERIV_const [where k="g b - g a"])
huffman@21164
  1724
      apply (drule meta_spec [of _ c])
huffman@21164
  1725
      apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
huffman@21164
  1726
      by simp_all
huffman@21164
  1727
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
huffman@21164
  1728
      by (simp add: DERIV_diff)
huffman@21164
  1729
  }
huffman@21164
  1730
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1731
huffman@21164
  1732
  {
huffman@21164
  1733
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1734
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1735
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1736
  }
huffman@21164
  1737
  moreover
huffman@21164
  1738
  {
huffman@21164
  1739
    have "?h b - ?h a =
huffman@21164
  1740
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1741
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
huffman@21164
  1742
      by (simp add: mult_ac add_ac real_diff_mult_distrib)
huffman@21164
  1743
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1744
  }
huffman@21164
  1745
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1746
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1747
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1748
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1749
huffman@21164
  1750
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1751
qed
huffman@21164
  1752
huffman@21164
  1753
end