src/HOL/Hyperreal/Deriv.thy
author wenzelm
Fri Nov 17 02:20:03 2006 +0100 (2006-11-17)
changeset 21404 eb85850d3eb7
parent 21239 d4fbe2c87ef1
child 21784 e76faa6e65fd
permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
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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) where
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  "DERIV f x :> D = ((%h. (f(x + h) - f x) /# h) -- 0 --> D)"
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definition
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  nsderiv :: "[real=>real,real,real] => bool"
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          ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
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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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definition
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  differentiable :: "[real=>real,real] => bool"   (infixl "differentiable" 60) where
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  "f differentiable x = (\<exists>D. DERIV f x :> D)"
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definition
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  NSdifferentiable :: "[real=>real,real] => bool"
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                       (infixl "NSdifferentiable" 60) where
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  "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
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definition
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  increment :: "[real=>real,real,hypreal] => hypreal" where
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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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   290
  thus "(DERIV f x :> l)"
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   291
     by (auto simp add: isCont_iff DERIV_iff diff_minus
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   292
               cong: LIM_equal [rule_format])
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   293
qed
huffman@21164
   294
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   295
lemma DERIV_chain':
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   296
  assumes f: "DERIV f x :> D"
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   297
  assumes g: "DERIV g (f x) :> E"
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   298
  shows "DERIV (\<lambda>x. g (f x)) x :> D *# E"
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   299
proof (unfold DERIV_iff2)
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   300
  obtain d where d: "\<forall>y. g y - g (f x) = (y - f x) *# d y"
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   301
    and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
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   302
    using CARAT_DERIV [THEN iffD1, OF g] by fast
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   303
  from f have "f -- x --> f x"
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   304
    by (rule DERIV_isCont [unfolded isCont_def])
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   305
  with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
huffman@21239
   306
    by (rule isCont_LIM_compose)
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   307
  hence "(\<lambda>z. ((f z - f x) /# (z - x)) *# d (f z))
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   308
          -- x --> D *# d (f x)"
huffman@21164
   309
    by (rule LIM_scaleR [OF f [unfolded DERIV_iff2]])
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   310
  thus "(\<lambda>z. (g (f z) - g (f x)) /# (z - x)) -- x --> D *# E"
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   311
    by (simp add: d dfx real_scaleR_def)
huffman@21164
   312
qed
huffman@21164
   313
huffman@21164
   314
huffman@21164
   315
subsubsection {* Nonstandard proofs *}
huffman@21164
   316
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   317
lemma DERIV_NS_iff:
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   318
      "(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 NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/#h) -- 0 --NS> D"
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   322
by (simp add: deriv_def LIM_NSLIM_iff)
huffman@21164
   323
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   324
lemma NSDeriv_unique:
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   325
     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
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   326
apply (simp add: nsderiv_def)
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   327
apply (cut_tac Infinitesimal_epsilon hypreal_epsilon_not_zero)
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   328
apply (auto dest!: bspec [where x=epsilon]
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   329
            intro!: inj_hypreal_of_real [THEN injD]
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   330
            dest: approx_trans3)
huffman@21164
   331
done
huffman@21164
   332
huffman@21164
   333
text {*First NSDERIV in terms of NSLIM*}
huffman@21164
   334
huffman@21164
   335
text{*first equivalence *}
huffman@21164
   336
lemma NSDERIV_NSLIM_iff:
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   337
      "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) -- 0 --NS> D)"
huffman@21164
   338
apply (simp add: nsderiv_def NSLIM_def, auto)
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   339
apply (drule_tac x = xa in bspec)
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   340
apply (rule_tac [3] ccontr)
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   341
apply (drule_tac [3] x = h in spec)
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   342
apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
huffman@21164
   343
done
huffman@21164
   344
huffman@21164
   345
text{*second equivalence *}
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   346
lemma NSDERIV_NSLIM_iff2:
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   347
     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) -- x --NS> D)"
huffman@21164
   348
by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff  diff_minus [symmetric]
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   349
              LIM_NSLIM_iff [symmetric])
huffman@21164
   350
huffman@21164
   351
(* while we're at it! *)
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   352
lemma NSDERIV_iff2:
huffman@21164
   353
     "(NSDERIV f x :> D) =
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   354
      (\<forall>w.
huffman@21164
   355
        w \<noteq> hypreal_of_real x & w \<approx> hypreal_of_real x -->
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   356
        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> hypreal_of_real D)"
huffman@21164
   357
by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
huffman@21164
   358
huffman@21164
   359
(*FIXME DELETE*)
huffman@21164
   360
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x - a \<noteq> (0::hypreal))"
huffman@21164
   361
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
huffman@21164
   362
huffman@21164
   363
lemma NSDERIVD5:
huffman@21164
   364
  "(NSDERIV f x :> D) ==>
huffman@21164
   365
   (\<forall>u. u \<approx> hypreal_of_real x -->
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   366
     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
huffman@21164
   367
apply (auto simp add: NSDERIV_iff2)
huffman@21164
   368
apply (case_tac "u = hypreal_of_real x", auto)
huffman@21164
   369
apply (drule_tac x = u in spec, auto)
huffman@21164
   370
apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
huffman@21164
   371
apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164
   372
apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
huffman@21164
   373
apply (auto simp add:
huffman@21164
   374
         approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
huffman@21164
   375
         Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164
   376
done
huffman@21164
   377
huffman@21164
   378
lemma NSDERIVD4:
huffman@21164
   379
     "(NSDERIV f x :> D) ==>
huffman@21164
   380
      (\<forall>h \<in> Infinitesimal.
huffman@21164
   381
               (( *f* f)(hypreal_of_real x + h) -
huffman@21164
   382
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
huffman@21164
   383
apply (auto simp add: nsderiv_def)
huffman@21164
   384
apply (case_tac "h = (0::hypreal) ")
huffman@21164
   385
apply (auto simp add: diff_minus)
huffman@21164
   386
apply (drule_tac x = h in bspec)
huffman@21164
   387
apply (drule_tac [2] c = h in approx_mult1)
huffman@21164
   388
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164
   389
            simp add: diff_minus)
huffman@21164
   390
done
huffman@21164
   391
huffman@21164
   392
lemma NSDERIVD3:
huffman@21164
   393
     "(NSDERIV f x :> D) ==>
huffman@21164
   394
      (\<forall>h \<in> Infinitesimal - {0}.
huffman@21164
   395
               (( *f* f)(hypreal_of_real x + h) -
huffman@21164
   396
                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
huffman@21164
   397
apply (auto simp add: nsderiv_def)
huffman@21164
   398
apply (rule ccontr, drule_tac x = h in bspec)
huffman@21164
   399
apply (drule_tac [2] c = h in approx_mult1)
huffman@21164
   400
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164
   401
            simp add: mult_assoc diff_minus)
huffman@21164
   402
done
huffman@21164
   403
huffman@21164
   404
text{*Differentiability implies continuity
huffman@21164
   405
         nice and simple "algebraic" proof*}
huffman@21164
   406
lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
huffman@21164
   407
apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
huffman@21164
   408
apply (drule approx_minus_iff [THEN iffD1])
huffman@21164
   409
apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164
   410
apply (drule_tac x = "xa - hypreal_of_real x" in bspec)
huffman@21164
   411
 prefer 2 apply (simp add: add_assoc [symmetric])
huffman@21164
   412
apply (auto simp add: mem_infmal_iff [symmetric] add_commute)
huffman@21164
   413
apply (drule_tac c = "xa - hypreal_of_real x" in approx_mult1)
huffman@21164
   414
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
huffman@21164
   415
            simp add: mult_assoc)
huffman@21164
   416
apply (drule_tac x3=D in
huffman@21164
   417
           HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult,
huffman@21164
   418
             THEN mem_infmal_iff [THEN iffD1]])
huffman@21164
   419
apply (auto simp add: mult_commute
huffman@21164
   420
            intro: approx_trans approx_minus_iff [THEN iffD2])
huffman@21164
   421
done
huffman@21164
   422
huffman@21164
   423
text{*Differentiation rules for combinations of functions
huffman@21164
   424
      follow from clear, straightforard, algebraic
huffman@21164
   425
      manipulations*}
huffman@21164
   426
text{*Constant function*}
huffman@21164
   427
huffman@21164
   428
(* use simple constant nslimit theorem *)
huffman@21164
   429
lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
huffman@21164
   430
by (simp add: NSDERIV_NSLIM_iff)
huffman@21164
   431
huffman@21164
   432
text{*Sum of functions- proved easily*}
huffman@21164
   433
huffman@21164
   434
lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
huffman@21164
   435
      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
huffman@21164
   436
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@21164
   437
apply (auto simp add: add_divide_distrib diff_divide_distrib dest!: spec)
huffman@21164
   438
apply (drule_tac b = "hypreal_of_real Da" and d = "hypreal_of_real Db" in approx_add)
huffman@21164
   439
apply (auto simp add: diff_def add_ac)
huffman@21164
   440
done
huffman@21164
   441
huffman@21164
   442
text{*Product of functions - Proof is trivial but tedious
huffman@21164
   443
  and long due to rearrangement of terms*}
huffman@21164
   444
huffman@21164
   445
lemma lemma_nsderiv1: "((a::hypreal)*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
huffman@21164
   446
by (simp add: right_diff_distrib)
huffman@21164
   447
huffman@21164
   448
lemma lemma_nsderiv2: "[| (x - y) / z = hypreal_of_real D + yb; z \<noteq> 0;
huffman@21164
   449
         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
huffman@21164
   450
      ==> x - y \<approx> 0"
huffman@21164
   451
apply (frule_tac c1 = z in hypreal_mult_right_cancel [THEN iffD2], assumption)
huffman@21164
   452
apply (erule_tac V = "(x - y) / z = hypreal_of_real D + yb" in thin_rl)
huffman@21164
   453
apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
huffman@21164
   454
            simp add: mult_assoc mem_infmal_iff [symmetric])
huffman@21164
   455
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164
   456
done
huffman@21164
   457
huffman@21164
   458
lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
huffman@21164
   459
      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
huffman@21164
   460
apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
huffman@21164
   461
apply (auto dest!: spec
huffman@21164
   462
      simp add: starfun_lambda_cancel lemma_nsderiv1)
huffman@21164
   463
apply (simp (no_asm) add: add_divide_distrib diff_divide_distrib)
huffman@21164
   464
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
huffman@21164
   465
apply (auto simp add: times_divide_eq_right [symmetric]
huffman@21164
   466
            simp del: times_divide_eq)
huffman@21164
   467
apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
huffman@21164
   468
apply (drule_tac
huffman@21164
   469
     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
huffman@21164
   470
apply (auto intro!: approx_add_mono1
huffman@21164
   471
            simp add: left_distrib right_distrib mult_commute add_assoc)
huffman@21164
   472
apply (rule_tac b1 = "hypreal_of_real Db * hypreal_of_real (f x)"
huffman@21164
   473
         in add_commute [THEN subst])
huffman@21164
   474
apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
huffman@21164
   475
                    Infinitesimal_add Infinitesimal_mult
huffman@21164
   476
                    Infinitesimal_hypreal_of_real_mult
huffman@21164
   477
                    Infinitesimal_hypreal_of_real_mult2
huffman@21164
   478
          simp add: add_assoc [symmetric])
huffman@21164
   479
done
huffman@21164
   480
huffman@21164
   481
text{*Multiplying by a constant*}
huffman@21164
   482
lemma NSDERIV_cmult: "NSDERIV f x :> D
huffman@21164
   483
      ==> NSDERIV (%x. c * f x) x :> c*D"
huffman@21164
   484
apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
huffman@21164
   485
                  minus_mult_right right_diff_distrib [symmetric])
huffman@21164
   486
apply (erule NSLIM_const [THEN NSLIM_mult])
huffman@21164
   487
done
huffman@21164
   488
huffman@21164
   489
text{*Negation of function*}
huffman@21164
   490
lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
huffman@21164
   491
proof (simp add: NSDERIV_NSLIM_iff)
huffman@21164
   492
  assume "(\<lambda>h. (f (x + h) - f x) / h) -- 0 --NS> D"
huffman@21164
   493
  hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) -- 0 --NS> - D"
huffman@21164
   494
    by (rule NSLIM_minus)
huffman@21164
   495
  have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
huffman@21164
   496
    by (simp add: minus_divide_left)
huffman@21164
   497
  with deriv
huffman@21164
   498
  show "(\<lambda>h. (- f (x + h) + f x) / h) -- 0 --NS> - D" by simp
huffman@21164
   499
qed
huffman@21164
   500
huffman@21164
   501
text{*Subtraction*}
huffman@21164
   502
lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
huffman@21164
   503
by (blast dest: NSDERIV_add NSDERIV_minus)
huffman@21164
   504
huffman@21164
   505
lemma NSDERIV_diff:
huffman@21164
   506
     "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
huffman@21164
   507
      ==> NSDERIV (%x. f x - g x) x :> Da-Db"
huffman@21164
   508
apply (simp add: diff_minus)
huffman@21164
   509
apply (blast intro: NSDERIV_add_minus)
huffman@21164
   510
done
huffman@21164
   511
huffman@21164
   512
text{*  Similarly to the above, the chain rule admits an entirely
huffman@21164
   513
   straightforward derivation. Compare this with Harrison's
huffman@21164
   514
   HOL proof of the chain rule, which proved to be trickier and
huffman@21164
   515
   required an alternative characterisation of differentiability-
huffman@21164
   516
   the so-called Carathedory derivative. Our main problem is
huffman@21164
   517
   manipulation of terms.*}
huffman@21164
   518
huffman@21164
   519
huffman@21164
   520
(* lemmas *)
huffman@21164
   521
lemma NSDERIV_zero:
huffman@21164
   522
      "[| NSDERIV g x :> D;
huffman@21164
   523
               ( *f* g) (hypreal_of_real(x) + xa) = hypreal_of_real(g x);
huffman@21164
   524
               xa \<in> Infinitesimal;
huffman@21164
   525
               xa \<noteq> 0
huffman@21164
   526
            |] ==> D = 0"
huffman@21164
   527
apply (simp add: nsderiv_def)
huffman@21164
   528
apply (drule bspec, auto)
huffman@21164
   529
done
huffman@21164
   530
huffman@21164
   531
(* can be proved differently using NSLIM_isCont_iff *)
huffman@21164
   532
lemma NSDERIV_approx:
huffman@21164
   533
     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
huffman@21164
   534
      ==> ( *f* f) (hypreal_of_real(x) + h) - hypreal_of_real(f x) \<approx> 0"
huffman@21164
   535
apply (simp add: nsderiv_def)
huffman@21164
   536
apply (simp add: mem_infmal_iff [symmetric])
huffman@21164
   537
apply (rule Infinitesimal_ratio)
huffman@21164
   538
apply (rule_tac [3] approx_hypreal_of_real_HFinite, auto)
huffman@21164
   539
done
huffman@21164
   540
huffman@21164
   541
(*---------------------------------------------------------------
huffman@21164
   542
   from one version of differentiability
huffman@21164
   543
huffman@21164
   544
                f(x) - f(a)
huffman@21164
   545
              --------------- \<approx> Db
huffman@21164
   546
                  x - a
huffman@21164
   547
 ---------------------------------------------------------------*)
huffman@21164
   548
lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
huffman@21164
   549
         ( *f* g) (hypreal_of_real(x) + xa) \<noteq> hypreal_of_real (g x);
huffman@21164
   550
         ( *f* g) (hypreal_of_real(x) + xa) \<approx> hypreal_of_real (g x)
huffman@21164
   551
      |] ==> (( *f* f) (( *f* g) (hypreal_of_real(x) + xa))
huffman@21164
   552
                   - hypreal_of_real (f (g x)))
huffman@21164
   553
              / (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real (g x))
huffman@21164
   554
             \<approx> hypreal_of_real(Da)"
huffman@21164
   555
by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def diff_minus [symmetric])
huffman@21164
   556
huffman@21164
   557
(*--------------------------------------------------------------
huffman@21164
   558
   from other version of differentiability
huffman@21164
   559
huffman@21164
   560
                f(x + h) - f(x)
huffman@21164
   561
               ----------------- \<approx> Db
huffman@21164
   562
                       h
huffman@21164
   563
 --------------------------------------------------------------*)
huffman@21164
   564
lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
huffman@21164
   565
      ==> (( *f* g) (hypreal_of_real(x) + xa) - hypreal_of_real(g x)) / xa
huffman@21164
   566
          \<approx> hypreal_of_real(Db)"
huffman@21164
   567
by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
huffman@21164
   568
huffman@21164
   569
lemma lemma_chain: "(z::hypreal) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
huffman@21164
   570
by auto
huffman@21164
   571
huffman@21164
   572
text{*This proof uses both definitions of differentiability.*}
huffman@21164
   573
lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
huffman@21164
   574
      ==> NSDERIV (f o g) x :> Da * Db"
huffman@21164
   575
apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
huffman@21164
   576
                mem_infmal_iff [symmetric])
huffman@21164
   577
apply clarify
huffman@21164
   578
apply (frule_tac f = g in NSDERIV_approx)
huffman@21164
   579
apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
huffman@21164
   580
apply (case_tac "( *f* g) (hypreal_of_real (x) + xa) = hypreal_of_real (g x) ")
huffman@21164
   581
apply (drule_tac g = g in NSDERIV_zero)
huffman@21164
   582
apply (auto simp add: divide_inverse)
huffman@21164
   583
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
   584
apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
huffman@21164
   585
apply (rule approx_mult_hypreal_of_real)
huffman@21164
   586
apply (simp_all add: divide_inverse [symmetric])
huffman@21164
   587
apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
huffman@21164
   588
apply (blast intro: NSDERIVD2)
huffman@21164
   589
done
huffman@21164
   590
huffman@21164
   591
text{*Differentiation of natural number powers*}
huffman@21164
   592
lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
huffman@21164
   593
by (simp add: NSDERIV_NSLIM_iff NSLIM_def divide_self del: divide_self_if)
huffman@21164
   594
huffman@21164
   595
lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
huffman@21164
   596
by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
huffman@21164
   597
huffman@21164
   598
(*Can't get rid of x \<noteq> 0 because it isn't continuous at zero*)
huffman@21164
   599
lemma NSDERIV_inverse:
huffman@21164
   600
     "x \<noteq> 0 ==> NSDERIV (%x. inverse(x)) x :> (- (inverse x ^ Suc (Suc 0)))"
huffman@21164
   601
apply (simp add: nsderiv_def)
huffman@21164
   602
apply (rule ballI, simp, clarify)
huffman@21164
   603
apply (frule (1) Infinitesimal_add_not_zero)
huffman@21164
   604
apply (simp add: add_commute)
huffman@21164
   605
(*apply (auto simp add: starfun_inverse_inverse realpow_two
huffman@21164
   606
        simp del: minus_mult_left [symmetric] minus_mult_right [symmetric])*)
huffman@21164
   607
apply (simp add: inverse_add inverse_mult_distrib [symmetric]
huffman@21164
   608
              inverse_minus_eq [symmetric] add_ac mult_ac diff_def
huffman@21164
   609
            del: inverse_mult_distrib inverse_minus_eq
huffman@21164
   610
                 minus_mult_left [symmetric] minus_mult_right [symmetric])
huffman@21164
   611
apply (simp (no_asm_simp) add: mult_assoc [symmetric] right_distrib
huffman@21164
   612
            del: minus_mult_left [symmetric] minus_mult_right [symmetric])
huffman@21164
   613
apply (rule_tac y = "inverse (- hypreal_of_real x * hypreal_of_real x)" in approx_trans)
huffman@21164
   614
apply (rule inverse_add_Infinitesimal_approx2)
huffman@21164
   615
apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
huffman@21164
   616
            simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
huffman@21164
   617
apply (rule Infinitesimal_HFinite_mult2, auto)
huffman@21164
   618
done
huffman@21164
   619
huffman@21164
   620
subsubsection {* Equivalence of NS and Standard definitions *}
huffman@21164
   621
huffman@21164
   622
lemma divideR_eq_divide [simp]: "x /# y = x / y"
huffman@21164
   623
by (simp add: real_scaleR_def divide_inverse mult_commute)
huffman@21164
   624
huffman@21164
   625
text{*Now equivalence between NSDERIV and DERIV*}
huffman@21164
   626
lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
huffman@21164
   627
by (simp add: deriv_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
huffman@21164
   628
huffman@21164
   629
(* let's do the standard proof though theorem *)
huffman@21164
   630
(* LIM_mult2 follows from a NS proof          *)
huffman@21164
   631
huffman@21164
   632
lemma DERIV_cmult:
huffman@21164
   633
  fixes f :: "real \<Rightarrow> 'a::real_normed_algebra" shows
huffman@21164
   634
      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
huffman@21164
   635
by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164
   636
huffman@21164
   637
(* standard version *)
huffman@21164
   638
lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
huffman@21164
   639
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
huffman@21164
   640
huffman@21164
   641
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
huffman@21164
   642
by (auto dest: DERIV_chain simp add: o_def)
huffman@21164
   643
huffman@21164
   644
(*derivative of linear multiplication*)
huffman@21164
   645
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
huffman@21164
   646
by (cut_tac c = c and x = x in DERIV_Id [THEN DERIV_cmult], simp)
huffman@21164
   647
huffman@21164
   648
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@21164
   649
apply (cut_tac DERIV_power [OF DERIV_Id])
huffman@21164
   650
apply (simp add: real_scaleR_def real_of_nat_def)
huffman@21164
   651
done
huffman@21164
   652
huffman@21164
   653
(* NS version *)
huffman@21164
   654
lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@21164
   655
by (simp add: NSDERIV_DERIV_iff DERIV_pow)
huffman@21164
   656
huffman@21164
   657
text{*Power of -1*}
huffman@21164
   658
huffman@21164
   659
lemma DERIV_inverse: "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
huffman@21164
   660
by (drule DERIV_inverse' [OF DERIV_Id], simp)
huffman@21164
   661
huffman@21164
   662
text{*Derivative of inverse*}
huffman@21164
   663
lemma DERIV_inverse_fun: "[| DERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21164
   664
      ==> DERIV (%x. inverse(f x)::real) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21164
   665
by (drule (1) DERIV_inverse', simp add: mult_ac)
huffman@21164
   666
huffman@21164
   667
lemma NSDERIV_inverse_fun: "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21164
   668
      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21164
   669
by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: realpow_Suc)
huffman@21164
   670
huffman@21164
   671
text{*Derivative of quotient*}
huffman@21164
   672
lemma DERIV_quotient: "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21164
   673
       ==> DERIV (%y. f(y) / (g y) :: real) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21164
   674
by (drule (2) DERIV_divide, simp add: mult_commute)
huffman@21164
   675
huffman@21164
   676
lemma NSDERIV_quotient: "[| NSDERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21164
   677
       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
huffman@21164
   678
                            - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21164
   679
by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: realpow_Suc)
huffman@21164
   680
huffman@21164
   681
lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
huffman@21164
   682
      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
huffman@21164
   683
by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
huffman@21164
   684
                   real_scaleR_def mult_commute)
huffman@21164
   685
huffman@21164
   686
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
huffman@21164
   687
by auto
huffman@21164
   688
huffman@21164
   689
lemma CARAT_DERIVD:
huffman@21164
   690
  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
huffman@21164
   691
      and nsc: "isNSCont g x"
huffman@21164
   692
  shows "NSDERIV f x :> g x"
huffman@21164
   693
proof -
huffman@21164
   694
  from nsc
huffman@21164
   695
  have "\<forall>w. w \<noteq> hypreal_of_real x \<and> w \<approx> hypreal_of_real x \<longrightarrow>
huffman@21164
   696
         ( *f* g) w * (w - hypreal_of_real x) / (w - hypreal_of_real x) \<approx>
huffman@21164
   697
         hypreal_of_real (g x)"
huffman@21164
   698
    by (simp add: diff_minus isNSCont_def)
huffman@21164
   699
  thus ?thesis using all
huffman@21164
   700
    by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
huffman@21164
   701
qed
huffman@21164
   702
huffman@21164
   703
subsubsection {* Differentiability predicate *}
huffman@21164
   704
huffman@21164
   705
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
huffman@21164
   706
by (simp add: differentiable_def)
huffman@21164
   707
huffman@21164
   708
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
huffman@21164
   709
by (force simp add: differentiable_def)
huffman@21164
   710
huffman@21164
   711
lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
huffman@21164
   712
by (simp add: NSdifferentiable_def)
huffman@21164
   713
huffman@21164
   714
lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
huffman@21164
   715
by (force simp add: NSdifferentiable_def)
huffman@21164
   716
huffman@21164
   717
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
huffman@21164
   718
  apply (unfold differentiable_def)
huffman@21164
   719
  apply (rule_tac x=0 in exI)
huffman@21164
   720
  apply simp
huffman@21164
   721
  done
huffman@21164
   722
huffman@21164
   723
lemma differentiable_sum:
huffman@21164
   724
  assumes "f differentiable x"
huffman@21164
   725
  and "g differentiable x"
huffman@21164
   726
  shows "(\<lambda>x. f x + g x) differentiable x"
huffman@21164
   727
proof -
huffman@21164
   728
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   729
  then obtain df where "DERIV f x :> df" ..
huffman@21164
   730
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   731
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   732
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@21164
   733
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
huffman@21164
   734
  thus ?thesis by (fold differentiable_def)
huffman@21164
   735
qed
huffman@21164
   736
huffman@21164
   737
lemma differentiable_diff:
huffman@21164
   738
  assumes "f differentiable x"
huffman@21164
   739
  and "g differentiable x"
huffman@21164
   740
  shows "(\<lambda>x. f x - g x) differentiable x"
huffman@21164
   741
proof -
huffman@21164
   742
  from prems have "f differentiable x" by simp
huffman@21164
   743
  moreover
huffman@21164
   744
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   745
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   746
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
huffman@21164
   747
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
huffman@21164
   748
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
huffman@21164
   749
  ultimately 
huffman@21164
   750
  show ?thesis
huffman@21164
   751
    by (auto simp: real_diff_def dest: differentiable_sum)
huffman@21164
   752
qed
huffman@21164
   753
huffman@21164
   754
lemma differentiable_mult:
huffman@21164
   755
  assumes "f differentiable x"
huffman@21164
   756
  and "g differentiable x"
huffman@21164
   757
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164
   758
proof -
huffman@21164
   759
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   760
  then obtain df where "DERIV f x :> df" ..
huffman@21164
   761
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   762
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   763
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
huffman@21164
   764
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
huffman@21164
   765
  thus ?thesis by (fold differentiable_def)
huffman@21164
   766
qed
huffman@21164
   767
huffman@21164
   768
subsection {*(NS) Increment*}
huffman@21164
   769
lemma incrementI:
huffman@21164
   770
      "f NSdifferentiable x ==>
huffman@21164
   771
      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@21164
   772
      hypreal_of_real (f x)"
huffman@21164
   773
by (simp add: increment_def)
huffman@21164
   774
huffman@21164
   775
lemma incrementI2: "NSDERIV f x :> D ==>
huffman@21164
   776
     increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
huffman@21164
   777
     hypreal_of_real (f x)"
huffman@21164
   778
apply (erule NSdifferentiableI [THEN incrementI])
huffman@21164
   779
done
huffman@21164
   780
huffman@21164
   781
(* The Increment theorem -- Keisler p. 65 *)
huffman@21164
   782
lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
huffman@21164
   783
      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
huffman@21164
   784
apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
huffman@21164
   785
apply (drule bspec, auto)
huffman@21164
   786
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
huffman@21164
   787
apply (frule_tac b1 = "hypreal_of_real (D) + y"
huffman@21164
   788
        in hypreal_mult_right_cancel [THEN iffD2])
huffman@21164
   789
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
   790
apply assumption
huffman@21164
   791
apply (simp add: times_divide_eq_right [symmetric])
huffman@21164
   792
apply (auto simp add: left_distrib)
huffman@21164
   793
done
huffman@21164
   794
huffman@21164
   795
lemma increment_thm2:
huffman@21164
   796
     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@21164
   797
      ==> \<exists>e \<in> Infinitesimal. increment f x h =
huffman@21164
   798
              hypreal_of_real(D)*h + e*h"
huffman@21164
   799
by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
huffman@21164
   800
huffman@21164
   801
huffman@21164
   802
lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
huffman@21164
   803
      ==> increment f x h \<approx> 0"
huffman@21164
   804
apply (drule increment_thm2,
huffman@21164
   805
       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: left_distrib [symmetric] mem_infmal_iff [symmetric])
huffman@21164
   806
apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
huffman@21164
   807
done
huffman@21164
   808
huffman@21164
   809
subsection {* Nested Intervals and Bisection *}
huffman@21164
   810
huffman@21164
   811
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164
   812
     All considerably tidied by lcp.*}
huffman@21164
   813
huffman@21164
   814
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
   815
apply (induct "no")
huffman@21164
   816
apply (auto intro: order_trans)
huffman@21164
   817
done
huffman@21164
   818
huffman@21164
   819
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   820
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   821
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   822
      ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164
   823
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@21164
   824
apply (induct_tac "n")
huffman@21164
   825
apply (auto intro: order_trans)
huffman@21164
   826
apply (rule_tac y = "g (Suc na)" in order_trans)
huffman@21164
   827
apply (induct_tac [2] "na")
huffman@21164
   828
apply (auto intro: order_trans)
huffman@21164
   829
done
huffman@21164
   830
huffman@21164
   831
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   832
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   833
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   834
      ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164
   835
apply (subst Bseq_minus_iff [symmetric])
huffman@21164
   836
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164
   837
apply auto
huffman@21164
   838
done
huffman@21164
   839
huffman@21164
   840
lemma f_inc_imp_le_lim:
huffman@21164
   841
  fixes f :: "nat \<Rightarrow> real"
huffman@21164
   842
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@21164
   843
apply (rule linorder_not_less [THEN iffD1])
huffman@21164
   844
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
huffman@21164
   845
apply (drule real_less_sum_gt_zero)
huffman@21164
   846
apply (drule_tac x = "f n + - lim f" in spec, safe)
huffman@21164
   847
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
huffman@21164
   848
apply (subgoal_tac "lim f \<le> f (no + n) ")
huffman@21164
   849
apply (drule_tac no=no and m=n in lemma_f_mono_add)
huffman@21164
   850
apply (auto simp add: add_commute)
huffman@21164
   851
apply (induct_tac "no")
huffman@21164
   852
apply simp
huffman@21164
   853
apply (auto intro: order_trans simp add: diff_minus abs_if)
huffman@21164
   854
done
huffman@21164
   855
huffman@21164
   856
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@21164
   857
apply (rule LIMSEQ_minus [THEN limI])
huffman@21164
   858
apply (simp add: convergent_LIMSEQ_iff)
huffman@21164
   859
done
huffman@21164
   860
huffman@21164
   861
lemma g_dec_imp_lim_le:
huffman@21164
   862
  fixes g :: "nat \<Rightarrow> real"
huffman@21164
   863
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@21164
   864
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
huffman@21164
   865
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
huffman@21164
   866
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
huffman@21164
   867
done
huffman@21164
   868
huffman@21164
   869
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   870
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   871
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   872
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   873
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164
   874
apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164
   875
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164
   876
apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164
   877
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164
   878
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164
   879
apply (rule_tac x = "lim f" in exI)
huffman@21164
   880
apply (rule_tac x = "lim g" in exI)
huffman@21164
   881
apply (auto intro: LIMSEQ_le)
huffman@21164
   882
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164
   883
done
huffman@21164
   884
huffman@21164
   885
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   886
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   887
         \<forall>n. f(n) \<le> g(n);
huffman@21164
   888
         (%n. f(n) - g(n)) ----> 0 |]
huffman@21164
   889
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   890
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164
   891
apply (drule lemma_nest, auto)
huffman@21164
   892
apply (subgoal_tac "l = m")
huffman@21164
   893
apply (drule_tac [2] X = f in LIMSEQ_diff)
huffman@21164
   894
apply (auto intro: LIMSEQ_unique)
huffman@21164
   895
done
huffman@21164
   896
huffman@21164
   897
text{*The universal quantifiers below are required for the declaration
huffman@21164
   898
  of @{text Bolzano_nest_unique} below.*}
huffman@21164
   899
huffman@21164
   900
lemma Bolzano_bisect_le:
huffman@21164
   901
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   902
apply (rule allI)
huffman@21164
   903
apply (induct_tac "n")
huffman@21164
   904
apply (auto simp add: Let_def split_def)
huffman@21164
   905
done
huffman@21164
   906
huffman@21164
   907
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164
   908
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164
   909
apply (rule allI)
huffman@21164
   910
apply (induct_tac "n")
huffman@21164
   911
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   912
done
huffman@21164
   913
huffman@21164
   914
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164
   915
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   916
apply (rule allI)
huffman@21164
   917
apply (induct_tac "n")
huffman@21164
   918
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   919
done
huffman@21164
   920
huffman@21164
   921
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164
   922
apply (auto)
huffman@21164
   923
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164
   924
apply (simp)
huffman@21164
   925
done
huffman@21164
   926
huffman@21164
   927
lemma Bolzano_bisect_diff:
huffman@21164
   928
     "a \<le> b ==>
huffman@21164
   929
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164
   930
      (b-a) / (2 ^ n)"
huffman@21164
   931
apply (induct "n")
huffman@21164
   932
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164
   933
done
huffman@21164
   934
huffman@21164
   935
lemmas Bolzano_nest_unique =
huffman@21164
   936
    lemma_nest_unique
huffman@21164
   937
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164
   938
huffman@21164
   939
huffman@21164
   940
lemma not_P_Bolzano_bisect:
huffman@21164
   941
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164
   942
      and notP: "~ P(a,b)"
huffman@21164
   943
      and le:   "a \<le> b"
huffman@21164
   944
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   945
proof (induct n)
huffman@21164
   946
  case 0 thus ?case by simp
huffman@21164
   947
 next
huffman@21164
   948
  case (Suc n)
huffman@21164
   949
  thus ?case
huffman@21164
   950
 by (auto simp del: surjective_pairing [symmetric]
huffman@21164
   951
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164
   952
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164
   953
qed
huffman@21164
   954
huffman@21164
   955
(*Now we re-package P_prem as a formula*)
huffman@21164
   956
lemma not_P_Bolzano_bisect':
huffman@21164
   957
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   958
         ~ P(a,b);  a \<le> b |] ==>
huffman@21164
   959
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   960
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164
   961
huffman@21164
   962
huffman@21164
   963
huffman@21164
   964
lemma lemma_BOLZANO:
huffman@21164
   965
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   966
         \<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   967
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164
   968
         a \<le> b |]
huffman@21164
   969
      ==> P(a,b)"
huffman@21164
   970
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
huffman@21164
   971
apply (rule LIMSEQ_minus_cancel)
huffman@21164
   972
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164
   973
apply (rule ccontr)
huffman@21164
   974
apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164
   975
apply (rename_tac "l")
huffman@21164
   976
apply (drule_tac x = l in spec, clarify)
huffman@21164
   977
apply (simp add: LIMSEQ_def)
huffman@21164
   978
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   979
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   980
apply (drule real_less_half_sum, auto)
huffman@21164
   981
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   982
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   983
apply safe
huffman@21164
   984
apply (simp_all (no_asm_simp))
huffman@21164
   985
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
   986
apply (simp (no_asm_simp) add: abs_if)
huffman@21164
   987
apply (rule real_sum_of_halves [THEN subst])
huffman@21164
   988
apply (rule add_strict_mono)
huffman@21164
   989
apply (simp_all add: diff_minus [symmetric])
huffman@21164
   990
done
huffman@21164
   991
huffman@21164
   992
huffman@21164
   993
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164
   994
       (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   995
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164
   996
      --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164
   997
apply clarify
huffman@21164
   998
apply (blast intro: lemma_BOLZANO)
huffman@21164
   999
done
huffman@21164
  1000
huffman@21164
  1001
huffman@21164
  1002
subsection {* Intermediate Value Theorem *}
huffman@21164
  1003
huffman@21164
  1004
text {*Prove Contrapositive by Bisection*}
huffman@21164
  1005
huffman@21164
  1006
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164
  1007
         a \<le> b;
huffman@21164
  1008
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164
  1009
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
  1010
apply (rule contrapos_pp, assumption)
huffman@21164
  1011
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
  1012
apply safe
huffman@21164
  1013
apply simp_all
huffman@21164
  1014
apply (simp add: isCont_iff LIM_def)
huffman@21164
  1015
apply (rule ccontr)
huffman@21164
  1016
apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164
  1017
 prefer 2
huffman@21164
  1018
 apply simp
huffman@21164
  1019
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164
  1020
apply (drule_tac x = x in spec)+
huffman@21164
  1021
apply simp
huffman@21164
  1022
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164
  1023
apply safe
huffman@21164
  1024
apply simp
huffman@21164
  1025
apply (drule_tac x = s in spec, clarify)
huffman@21164
  1026
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164
  1027
apply (drule_tac x = "ba-x" in spec)
huffman@21164
  1028
apply (simp_all add: abs_if)
huffman@21164
  1029
apply (drule_tac x = "aa-x" in spec)
huffman@21164
  1030
apply (case_tac "x \<le> aa", simp_all)
huffman@21164
  1031
done
huffman@21164
  1032
huffman@21164
  1033
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164
  1034
         a \<le> b;
huffman@21164
  1035
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164
  1036
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
  1037
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164
  1038
apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@21164
  1039
apply (auto intro: isCont_minus)
huffman@21164
  1040
done
huffman@21164
  1041
huffman@21164
  1042
(*HOL style here: object-level formulations*)
huffman@21164
  1043
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164
  1044
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
  1045
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
  1046
apply (blast intro: IVT)
huffman@21164
  1047
done
huffman@21164
  1048
huffman@21164
  1049
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164
  1050
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
  1051
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
  1052
apply (blast intro: IVT2)
huffman@21164
  1053
done
huffman@21164
  1054
huffman@21164
  1055
text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164
  1056
huffman@21164
  1057
lemma isCont_bounded:
huffman@21164
  1058
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1059
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164
  1060
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
  1061
apply safe
huffman@21164
  1062
apply simp_all
huffman@21164
  1063
apply (rename_tac x xa ya M Ma)
huffman@21164
  1064
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
huffman@21164
  1065
apply (rule_tac x = Ma in exI, clarify)
huffman@21164
  1066
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
  1067
apply (rule_tac x = M in exI, clarify)
huffman@21164
  1068
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
  1069
apply (case_tac "a \<le> x & x \<le> b")
huffman@21164
  1070
apply (rule_tac [2] x = 1 in exI)
huffman@21164
  1071
prefer 2 apply force
huffman@21164
  1072
apply (simp add: LIM_def isCont_iff)
huffman@21164
  1073
apply (drule_tac x = x in spec, auto)
huffman@21164
  1074
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164
  1075
apply (drule_tac x = 1 in spec, auto)
huffman@21164
  1076
apply (rule_tac x = s in exI, clarify)
huffman@21164
  1077
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164
  1078
apply (drule_tac x = "xa-x" in spec)
huffman@21164
  1079
apply (auto simp add: abs_ge_self)
huffman@21164
  1080
done
huffman@21164
  1081
huffman@21164
  1082
text{*Refine the above to existence of least upper bound*}
huffman@21164
  1083
huffman@21164
  1084
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164
  1085
      (\<exists>t. isLub UNIV S t)"
huffman@21164
  1086
by (blast intro: reals_complete)
huffman@21164
  1087
huffman@21164
  1088
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1089
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
  1090
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164
  1091
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164
  1092
        in lemma_reals_complete)
huffman@21164
  1093
apply auto
huffman@21164
  1094
apply (drule isCont_bounded, assumption)
huffman@21164
  1095
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164
  1096
apply (rule exI, auto)
huffman@21164
  1097
apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164
  1098
done
huffman@21164
  1099
huffman@21164
  1100
text{*Now show that it attains its upper bound*}
huffman@21164
  1101
huffman@21164
  1102
lemma isCont_eq_Ub:
huffman@21164
  1103
  assumes le: "a \<le> b"
huffman@21164
  1104
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1105
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
  1106
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
  1107
proof -
huffman@21164
  1108
  from isCont_has_Ub [OF le con]
huffman@21164
  1109
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164
  1110
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164
  1111
  show ?thesis
huffman@21164
  1112
  proof (intro exI, intro conjI)
huffman@21164
  1113
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164
  1114
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164
  1115
    proof (rule ccontr)
huffman@21164
  1116
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164
  1117
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
huffman@21164
  1118
        by (fastsimp simp add: linorder_not_le [symmetric])
huffman@21164
  1119
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@21164
  1120
        by (auto simp add: isCont_inverse isCont_diff con)
huffman@21164
  1121
      from isCont_bounded [OF le this]
huffman@21164
  1122
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164
  1123
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
huffman@21164
  1124
        by (simp add: M3 compare_rls)
huffman@21164
  1125
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164
  1126
        by (auto intro: order_le_less_trans [of _ k])
huffman@21164
  1127
      with Minv
huffman@21164
  1128
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164
  1129
        by (intro strip less_imp_inverse_less, simp_all)
huffman@21164
  1130
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164
  1131
        by simp
huffman@21164
  1132
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164
  1133
        by (simp, arith)
huffman@21164
  1134
      from M2 [OF this]
huffman@21164
  1135
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164
  1136
      thus False using invlt [of x] by force
huffman@21164
  1137
    qed
huffman@21164
  1138
  qed
huffman@21164
  1139
qed
huffman@21164
  1140
huffman@21164
  1141
huffman@21164
  1142
text{*Same theorem for lower bound*}
huffman@21164
  1143
huffman@21164
  1144
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1145
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164
  1146
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
  1147
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164
  1148
prefer 2 apply (blast intro: isCont_minus)
huffman@21164
  1149
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164
  1150
apply safe
huffman@21164
  1151
apply auto
huffman@21164
  1152
done
huffman@21164
  1153
huffman@21164
  1154
huffman@21164
  1155
text{*Another version.*}
huffman@21164
  1156
huffman@21164
  1157
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
  1158
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164
  1159
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164
  1160
apply (frule isCont_eq_Lb)
huffman@21164
  1161
apply (frule_tac [2] isCont_eq_Ub)
huffman@21164
  1162
apply (assumption+, safe)
huffman@21164
  1163
apply (rule_tac x = "f x" in exI)
huffman@21164
  1164
apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164
  1165
apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164
  1166
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164
  1167
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164
  1168
apply (rule_tac [2] x = xb in exI)
huffman@21164
  1169
apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164
  1170
done
huffman@21164
  1171
huffman@21164
  1172
huffman@21164
  1173
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
  1174
huffman@21164
  1175
lemma DERIV_left_inc:
huffman@21164
  1176
  fixes f :: "real => real"
huffman@21164
  1177
  assumes der: "DERIV f x :> l"
huffman@21164
  1178
      and l:   "0 < l"
huffman@21164
  1179
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
  1180
proof -
huffman@21164
  1181
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
  1182
  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
  1183
    by (simp add: diff_minus)
huffman@21164
  1184
  then obtain s
huffman@21164
  1185
        where s:   "0 < s"
huffman@21164
  1186
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
  1187
    by auto
huffman@21164
  1188
  thus ?thesis
huffman@21164
  1189
  proof (intro exI conjI strip)
huffman@21164
  1190
    show "0<s" .
huffman@21164
  1191
    fix h::real
huffman@21164
  1192
    assume "0 < h" "h < s"
huffman@21164
  1193
    with all [of h] show "f x < f (x+h)"
huffman@21164
  1194
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
  1195
    split add: split_if_asm)
huffman@21164
  1196
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
  1197
      with l
huffman@21164
  1198
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
  1199
      thus "f x < f (x+h)"
huffman@21164
  1200
  by (simp add: pos_less_divide_eq h)
huffman@21164
  1201
    qed
huffman@21164
  1202
  qed
huffman@21164
  1203
qed
huffman@21164
  1204
huffman@21164
  1205
lemma DERIV_left_dec:
huffman@21164
  1206
  fixes f :: "real => real"
huffman@21164
  1207
  assumes der: "DERIV f x :> l"
huffman@21164
  1208
      and l:   "l < 0"
huffman@21164
  1209
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
  1210
proof -
huffman@21164
  1211
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
  1212
  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
  1213
    by (simp add: diff_minus)
huffman@21164
  1214
  then obtain s
huffman@21164
  1215
        where s:   "0 < s"
huffman@21164
  1216
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
  1217
    by auto
huffman@21164
  1218
  thus ?thesis
huffman@21164
  1219
  proof (intro exI conjI strip)
huffman@21164
  1220
    show "0<s" .
huffman@21164
  1221
    fix h::real
huffman@21164
  1222
    assume "0 < h" "h < s"
huffman@21164
  1223
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
  1224
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
  1225
    split add: split_if_asm)
huffman@21164
  1226
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
  1227
      with l
huffman@21164
  1228
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
  1229
      thus "f x < f (x-h)"
huffman@21164
  1230
  by (simp add: pos_less_divide_eq h)
huffman@21164
  1231
    qed
huffman@21164
  1232
  qed
huffman@21164
  1233
qed
huffman@21164
  1234
huffman@21164
  1235
lemma DERIV_local_max:
huffman@21164
  1236
  fixes f :: "real => real"
huffman@21164
  1237
  assumes der: "DERIV f x :> l"
huffman@21164
  1238
      and d:   "0 < d"
huffman@21164
  1239
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
  1240
  shows "l = 0"
huffman@21164
  1241
proof (cases rule: linorder_cases [of l 0])
huffman@21164
  1242
  case equal show ?thesis .
huffman@21164
  1243
next
huffman@21164
  1244
  case less
huffman@21164
  1245
  from DERIV_left_dec [OF der less]
huffman@21164
  1246
  obtain d' where d': "0 < d'"
huffman@21164
  1247
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
  1248
  from real_lbound_gt_zero [OF d d']
huffman@21164
  1249
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
  1250
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
  1251
  show ?thesis by (auto simp add: abs_if)
huffman@21164
  1252
next
huffman@21164
  1253
  case greater
huffman@21164
  1254
  from DERIV_left_inc [OF der greater]
huffman@21164
  1255
  obtain d' where d': "0 < d'"
huffman@21164
  1256
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
  1257
  from real_lbound_gt_zero [OF d d']
huffman@21164
  1258
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
  1259
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
  1260
  show ?thesis by (auto simp add: abs_if)
huffman@21164
  1261
qed
huffman@21164
  1262
huffman@21164
  1263
huffman@21164
  1264
text{*Similar theorem for a local minimum*}
huffman@21164
  1265
lemma DERIV_local_min:
huffman@21164
  1266
  fixes f :: "real => real"
huffman@21164
  1267
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
  1268
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
  1269
huffman@21164
  1270
huffman@21164
  1271
text{*In particular, if a function is locally flat*}
huffman@21164
  1272
lemma DERIV_local_const:
huffman@21164
  1273
  fixes f :: "real => real"
huffman@21164
  1274
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
  1275
by (auto dest!: DERIV_local_max)
huffman@21164
  1276
huffman@21164
  1277
text{*Lemma about introducing open ball in open interval*}
huffman@21164
  1278
lemma lemma_interval_lt:
huffman@21164
  1279
     "[| a < x;  x < b |]
huffman@21164
  1280
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
huffman@21164
  1281
apply (simp add: abs_interval_iff)
huffman@21164
  1282
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
  1283
apply (rule_tac x = "x-a" in exI)
huffman@21164
  1284
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
  1285
done
huffman@21164
  1286
huffman@21164
  1287
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
  1288
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
  1289
apply (drule lemma_interval_lt, auto)
huffman@21164
  1290
apply (auto intro!: exI)
huffman@21164
  1291
done
huffman@21164
  1292
huffman@21164
  1293
text{*Rolle's Theorem.
huffman@21164
  1294
   If @{term f} is defined and continuous on the closed interval
huffman@21164
  1295
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
  1296
   and @{term "f(a) = f(b)"},
huffman@21164
  1297
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
  1298
theorem Rolle:
huffman@21164
  1299
  assumes lt: "a < b"
huffman@21164
  1300
      and eq: "f(a) = f(b)"
huffman@21164
  1301
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1302
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21164
  1303
  shows "\<exists>z. a < z & z < b & DERIV f z :> 0"
huffman@21164
  1304
proof -
huffman@21164
  1305
  have le: "a \<le> b" using lt by simp
huffman@21164
  1306
  from isCont_eq_Ub [OF le con]
huffman@21164
  1307
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
  1308
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
  1309
    by blast
huffman@21164
  1310
  from isCont_eq_Lb [OF le con]
huffman@21164
  1311
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
  1312
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
  1313
    by blast
huffman@21164
  1314
  show ?thesis
huffman@21164
  1315
  proof cases
huffman@21164
  1316
    assume axb: "a < x & x < b"
huffman@21164
  1317
        --{*@{term f} attains its maximum within the interval*}
huffman@21164
  1318
    hence ax: "a<x" and xb: "x<b" by auto
huffman@21164
  1319
    from lemma_interval [OF ax xb]
huffman@21164
  1320
    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
  1321
      by blast
huffman@21164
  1322
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
  1323
      by blast
huffman@21164
  1324
    from differentiableD [OF dif [OF axb]]
huffman@21164
  1325
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1326
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
  1327
        --{*the derivative at a local maximum is zero*}
huffman@21164
  1328
    thus ?thesis using ax xb der by auto
huffman@21164
  1329
  next
huffman@21164
  1330
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
  1331
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
  1332
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
  1333
    show ?thesis
huffman@21164
  1334
    proof cases
huffman@21164
  1335
      assume ax'b: "a < x' & x' < b"
huffman@21164
  1336
        --{*@{term f} attains its minimum within the interval*}
huffman@21164
  1337
      hence ax': "a<x'" and x'b: "x'<b" by auto
huffman@21164
  1338
      from lemma_interval [OF ax' x'b]
huffman@21164
  1339
      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
  1340
  by blast
huffman@21164
  1341
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
  1342
  by blast
huffman@21164
  1343
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
  1344
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
  1345
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
  1346
        --{*the derivative at a local minimum is zero*}
huffman@21164
  1347
      thus ?thesis using ax' x'b der by auto
huffman@21164
  1348
    next
huffman@21164
  1349
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
  1350
        --{*@{term f} is constant througout the interval*}
huffman@21164
  1351
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
  1352
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
  1353
      from dense [OF lt]
huffman@21164
  1354
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
  1355
      from lemma_interval [OF ar rb]
huffman@21164
  1356
      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
  1357
  by blast
huffman@21164
  1358
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
  1359
      proof (clarify)
huffman@21164
  1360
        fix z::real
huffman@21164
  1361
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1362
        show "f z = f b"
huffman@21164
  1363
        proof (rule order_antisym)
huffman@21164
  1364
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1365
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1366
        qed
huffman@21164
  1367
      qed
huffman@21164
  1368
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1369
      proof (intro strip)
huffman@21164
  1370
        fix y::real
huffman@21164
  1371
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1372
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1373
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1374
      qed
huffman@21164
  1375
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1376
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1377
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1378
        --{*the derivative of a constant function is zero*}
huffman@21164
  1379
      thus ?thesis using ar rb der by auto
huffman@21164
  1380
    qed
huffman@21164
  1381
  qed
huffman@21164
  1382
qed
huffman@21164
  1383
huffman@21164
  1384
huffman@21164
  1385
subsection{*Mean Value Theorem*}
huffman@21164
  1386
huffman@21164
  1387
lemma lemma_MVT:
huffman@21164
  1388
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164
  1389
proof cases
huffman@21164
  1390
  assume "a=b" thus ?thesis by simp
huffman@21164
  1391
next
huffman@21164
  1392
  assume "a\<noteq>b"
huffman@21164
  1393
  hence ba: "b-a \<noteq> 0" by arith
huffman@21164
  1394
  show ?thesis
huffman@21164
  1395
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164
  1396
        simp add: right_diff_distrib,
huffman@21164
  1397
        simp add: left_diff_distrib)
huffman@21164
  1398
qed
huffman@21164
  1399
huffman@21164
  1400
theorem MVT:
huffman@21164
  1401
  assumes lt:  "a < b"
huffman@21164
  1402
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1403
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21164
  1404
  shows "\<exists>l z. a < z & z < b & DERIV f z :> l &
huffman@21164
  1405
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1406
proof -
huffman@21164
  1407
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@21164
  1408
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
huffman@21164
  1409
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_Id)
huffman@21164
  1410
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
  1411
  proof (clarify)
huffman@21164
  1412
    fix x::real
huffman@21164
  1413
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1414
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1415
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1416
    show "?F differentiable x"
huffman@21164
  1417
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1418
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1419
  qed
huffman@21164
  1420
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1421
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1422
    by blast
huffman@21164
  1423
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1424
    by (rule DERIV_cmult_Id)
huffman@21164
  1425
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1426
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1427
    by (rule DERIV_add [OF der])
huffman@21164
  1428
  show ?thesis
huffman@21164
  1429
  proof (intro exI conjI)
huffman@21164
  1430
    show "a < z" .
huffman@21164
  1431
    show "z < b" .
huffman@21164
  1432
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1433
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1434
  qed
huffman@21164
  1435
qed
huffman@21164
  1436
huffman@21164
  1437
huffman@21164
  1438
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1439
huffman@21164
  1440
lemma DERIV_isconst_end:
huffman@21164
  1441
  fixes f :: "real => real"
huffman@21164
  1442
  shows "[| a < b;
huffman@21164
  1443
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1444
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1445
        ==> f b = f a"
huffman@21164
  1446
apply (drule MVT, assumption)
huffman@21164
  1447
apply (blast intro: differentiableI)
huffman@21164
  1448
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1449
done
huffman@21164
  1450
huffman@21164
  1451
lemma DERIV_isconst1:
huffman@21164
  1452
  fixes f :: "real => real"
huffman@21164
  1453
  shows "[| a < b;
huffman@21164
  1454
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1455
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1456
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1457
apply safe
huffman@21164
  1458
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1459
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1460
done
huffman@21164
  1461
huffman@21164
  1462
lemma DERIV_isconst2:
huffman@21164
  1463
  fixes f :: "real => real"
huffman@21164
  1464
  shows "[| a < b;
huffman@21164
  1465
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1466
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1467
         a \<le> x; x \<le> b |]
huffman@21164
  1468
        ==> f x = f a"
huffman@21164
  1469
apply (blast dest: DERIV_isconst1)
huffman@21164
  1470
done
huffman@21164
  1471
huffman@21164
  1472
lemma DERIV_isconst_all:
huffman@21164
  1473
  fixes f :: "real => real"
huffman@21164
  1474
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1475
apply (rule linorder_cases [of x y])
huffman@21164
  1476
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1477
done
huffman@21164
  1478
huffman@21164
  1479
lemma DERIV_const_ratio_const:
huffman@21164
  1480
     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1481
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1482
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1483
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
huffman@21164
  1484
apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
huffman@21164
  1485
done
huffman@21164
  1486
huffman@21164
  1487
lemma DERIV_const_ratio_const2:
huffman@21164
  1488
     "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1489
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1490
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1491
done
huffman@21164
  1492
huffman@21164
  1493
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1494
by (simp)
huffman@21164
  1495
huffman@21164
  1496
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1497
by (simp)
huffman@21164
  1498
huffman@21164
  1499
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1500
huffman@21164
  1501
lemma DERIV_const_average:
huffman@21164
  1502
  fixes v :: "real => real"
huffman@21164
  1503
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1504
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1505
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1506
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1507
  case equal with neq show ?thesis by simp
huffman@21164
  1508
next
huffman@21164
  1509
  case less
huffman@21164
  1510
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1511
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1512
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1513
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1514
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1515
  ultimately show ?thesis using neq by force
huffman@21164
  1516
next
huffman@21164
  1517
  case greater
huffman@21164
  1518
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1519
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1520
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1521
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1522
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1523
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1524
qed
huffman@21164
  1525
huffman@21164
  1526
huffman@21164
  1527
text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164
  1528
strict maximum at an end point, not in the middle.*}
huffman@21164
  1529
huffman@21164
  1530
lemma lemma_isCont_inj:
huffman@21164
  1531
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1532
  assumes d: "0 < d"
huffman@21164
  1533
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1534
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1535
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164
  1536
proof (rule ccontr)
huffman@21164
  1537
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164
  1538
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164
  1539
  show False
huffman@21164
  1540
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164
  1541
    case le
huffman@21164
  1542
    from d cont all [of "x+d"]
huffman@21164
  1543
    have flef: "f(x+d) \<le> f x"
huffman@21164
  1544
     and xlex: "x - d \<le> x"
huffman@21164
  1545
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164
  1546
       by (auto simp add: abs_if)
huffman@21164
  1547
    from IVT [OF le flef xlex cont']
huffman@21164
  1548
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164
  1549
    moreover
huffman@21164
  1550
    hence "g(f x') = g (f(x+d))" by simp
huffman@21164
  1551
    ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@21164
  1552
      by (simp add: abs_le_interval_iff)
huffman@21164
  1553
  next
huffman@21164
  1554
    case ge
huffman@21164
  1555
    from d cont all [of "x-d"]
huffman@21164
  1556
    have flef: "f(x-d) \<le> f x"
huffman@21164
  1557
     and xlex: "x \<le> x+d"
huffman@21164
  1558
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164
  1559
       by (auto simp add: abs_if)
huffman@21164
  1560
    from IVT2 [OF ge flef xlex cont']
huffman@21164
  1561
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164
  1562
    moreover
huffman@21164
  1563
    hence "g(f x') = g (f(x-d))" by simp
huffman@21164
  1564
    ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@21164
  1565
      by (simp add: abs_le_interval_iff)
huffman@21164
  1566
  qed
huffman@21164
  1567
qed
huffman@21164
  1568
huffman@21164
  1569
huffman@21164
  1570
text{*Similar version for lower bound.*}
huffman@21164
  1571
huffman@21164
  1572
lemma lemma_isCont_inj2:
huffman@21164
  1573
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1574
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164
  1575
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164
  1576
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164
  1577
apply (insert lemma_isCont_inj
huffman@21164
  1578
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@21164
  1579
apply (simp add: isCont_minus linorder_not_le)
huffman@21164
  1580
done
huffman@21164
  1581
huffman@21164
  1582
text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164
  1583
@{text "f[[x - d, x + d]]"} .*}
huffman@21164
  1584
huffman@21164
  1585
lemma isCont_inj_range:
huffman@21164
  1586
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1587
  assumes d: "0 < d"
huffman@21164
  1588
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1589
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1590
  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
  1591
proof -
huffman@21164
  1592
  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
  1593
    by (auto simp add: abs_le_interval_iff)
huffman@21164
  1594
  from isCont_Lb_Ub [OF this]
huffman@21164
  1595
  obtain L M
huffman@21164
  1596
  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
  1597
    and all2 [rule_format]:
huffman@21164
  1598
           "\<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
  1599
    by auto
huffman@21164
  1600
  with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164
  1601
  moreover have "L \<noteq> f x"
huffman@21164
  1602
  proof -
huffman@21164
  1603
    from lemma_isCont_inj2 [OF d inj cont]
huffman@21164
  1604
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164
  1605
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1606
  qed
huffman@21164
  1607
  moreover have "f x \<noteq> M"
huffman@21164
  1608
  proof -
huffman@21164
  1609
    from lemma_isCont_inj [OF d inj cont]
huffman@21164
  1610
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164
  1611
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1612
  qed
huffman@21164
  1613
  ultimately have "L < f x & f x < M" by arith
huffman@21164
  1614
  hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164
  1615
  from real_lbound_gt_zero [OF this]
huffman@21164
  1616
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164
  1617
  thus ?thesis
huffman@21164
  1618
  proof (intro exI conjI)
huffman@21164
  1619
    show "0<e" .
huffman@21164
  1620
    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
  1621
    proof (intro strip)
huffman@21164
  1622
      fix y::real
huffman@21164
  1623
      assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164
  1624
      with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164
  1625
      from all2 [OF this]
huffman@21164
  1626
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
huffman@21164
  1627
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
huffman@21164
  1628
        by (force simp add: abs_le_interval_iff)
huffman@21164
  1629
    qed
huffman@21164
  1630
  qed
huffman@21164
  1631
qed
huffman@21164
  1632
huffman@21164
  1633
huffman@21164
  1634
text{*Continuity of inverse function*}
huffman@21164
  1635
huffman@21164
  1636
lemma isCont_inverse_function:
huffman@21164
  1637
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1638
  assumes d: "0 < d"
huffman@21164
  1639
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1640
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1641
  shows "isCont g (f x)"
huffman@21164
  1642
proof (simp add: isCont_iff LIM_eq)
huffman@21164
  1643
  show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164
  1644
         (\<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
  1645
  proof (intro strip)
huffman@21164
  1646
    fix r::real
huffman@21164
  1647
    assume r: "0<r"
huffman@21164
  1648
    from real_lbound_gt_zero [OF r d]
huffman@21164
  1649
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164
  1650
    with inj cont
huffman@21164
  1651
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164
  1652
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164
  1653
    from isCont_inj_range [OF e this]
huffman@21164
  1654
    obtain e' where e': "0 < e'"
huffman@21164
  1655
        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
  1656
          by blast
huffman@21164
  1657
    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
  1658
    proof (intro exI conjI)
huffman@21164
  1659
      show "0<e'" .
huffman@21164
  1660
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164
  1661
      proof (intro strip)
huffman@21164
  1662
        fix z::real
huffman@21164
  1663
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164
  1664
        with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164
  1665
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164
  1666
      qed
huffman@21164
  1667
    qed
huffman@21164
  1668
  qed
huffman@21164
  1669
qed
huffman@21164
  1670
huffman@21164
  1671
theorem GMVT:
huffman@21164
  1672
  assumes alb: "a < b"
huffman@21164
  1673
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
huffman@21164
  1674
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
huffman@21164
  1675
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
huffman@21164
  1676
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1677
  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
  1678
proof -
huffman@21164
  1679
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
huffman@21164
  1680
  from prems have "a < b" by simp
huffman@21164
  1681
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@21164
  1682
  proof -
huffman@21164
  1683
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
huffman@21164
  1684
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
huffman@21164
  1685
      by (auto intro: isCont_mult)
huffman@21164
  1686
    moreover
huffman@21164
  1687
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
huffman@21164
  1688
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
huffman@21164
  1689
      by (auto intro: isCont_mult)
huffman@21164
  1690
    ultimately show ?thesis
huffman@21164
  1691
      by (fastsimp intro: isCont_diff)
huffman@21164
  1692
  qed
huffman@21164
  1693
  moreover
huffman@21164
  1694
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@21164
  1695
  proof -
huffman@21164
  1696
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1697
    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
  1698
    moreover
huffman@21164
  1699
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1700
    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
  1701
    ultimately show ?thesis by (simp add: differentiable_diff)
huffman@21164
  1702
  qed
huffman@21164
  1703
  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
  1704
  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
  1705
  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
  1706
huffman@21164
  1707
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1708
  with gd have "g differentiable c" by simp
huffman@21164
  1709
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1710
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1711
huffman@21164
  1712
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1713
  with fd have "f differentiable c" by simp
huffman@21164
  1714
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1715
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1716
huffman@21164
  1717
  from cdef have "DERIV ?h c :> l" by auto
huffman@21164
  1718
  moreover
huffman@21164
  1719
  {
huffman@21164
  1720
    from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
huffman@21164
  1721
      apply (insert DERIV_const [where k="f b - f a"])
huffman@21164
  1722
      apply (drule meta_spec [of _ c])
huffman@21164
  1723
      apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
huffman@21164
  1724
      by simp_all
huffman@21164
  1725
    moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
huffman@21164
  1726
      apply (insert DERIV_const [where k="g b - g a"])
huffman@21164
  1727
      apply (drule meta_spec [of _ c])
huffman@21164
  1728
      apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
huffman@21164
  1729
      by simp_all
huffman@21164
  1730
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
huffman@21164
  1731
      by (simp add: DERIV_diff)
huffman@21164
  1732
  }
huffman@21164
  1733
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1734
huffman@21164
  1735
  {
huffman@21164
  1736
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1737
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1738
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1739
  }
huffman@21164
  1740
  moreover
huffman@21164
  1741
  {
huffman@21164
  1742
    have "?h b - ?h a =
huffman@21164
  1743
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1744
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
huffman@21164
  1745
      by (simp add: mult_ac add_ac real_diff_mult_distrib)
huffman@21164
  1746
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1747
  }
huffman@21164
  1748
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1749
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1750
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1751
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1752
huffman@21164
  1753
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1754
qed
huffman@21164
  1755
huffman@21164
  1756
end