src/HOL/Hyperreal/Deriv.thy
author huffman
Tue May 22 07:29:49 2007 +0200 (2007-05-22)
changeset 23069 cdfff0241c12
parent 23044 2ad82c359175
child 23255 631bd424fd72
permissions -rw-r--r--
rename lemmas LIM_ident, isCont_ident, DERIV_ident
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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 Definitions*}
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definition
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  deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, '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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  differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
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    (infixl "differentiable" 60) where
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  "f differentiable x = (\<exists>D. DERIV f x :> D)"
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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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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_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
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by (simp add: deriv_def divide_self 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 add_divide_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 divide_minus_left 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. (f(x+h) - f(x)) / h * h) -- 0 --> D * 0"
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    by (intro LIM_mult LIM_ident)
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  hence "(\<lambda>h. (f(x+h) - f(x)) * (h / h)) -- 0 --> 0"
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    by simp
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  hence "(\<lambda>h. f(x+h) - f(x)) -- 0 --> 0"
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    by (simp cong: LIM_cong add: divide_self)
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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_field"
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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 add_divide_distrib [symmetric] ring_distrib)
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lemma DERIV_mult':
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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_mult 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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     "[| 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. (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_field)\<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 DERIV_inverse':
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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
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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_mult 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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  "\<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 :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
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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 left_distrib mult_1_left)
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    apply (simp only: power_Suc right_distrib 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 :: "'a \<Rightarrow> 'a::{real_normed_field,recpower}"
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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 nonzero_mult_divide_cancel_right:
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  "b \<noteq> 0 \<Longrightarrow> a * b / b = (a::'a::field)"
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proof -
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  assume b: "b \<noteq> 0"
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  have "a * b / b = a * (b / b)" by simp
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  also have "\<dots> = a" by (simp add: divide_self b)
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  finally show "a * b / b = a" .
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qed
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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 = g z * (z-x)) & 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 = g z * (z-x)) \<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 = ?g z * (z-x)"
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      by (simp add: nonzero_mult_divide_cancel_right)
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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 = g z * (z-x))" and "isCont g x" and "g x = l" by blast
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  thus "(DERIV f x :> l)"
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     by (auto simp add: isCont_iff DERIV_iff nonzero_mult_divide_cancel_right
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              cong: LIM_cong)
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qed
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lemma DERIV_chain':
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  assumes f: "DERIV f x :> D"
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  assumes g: "DERIV g (f x) :> E"
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  shows "DERIV (\<lambda>x. g (f x)) x :> E * D"
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proof (unfold DERIV_iff2)
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  obtain d where d: "\<forall>y. g y - g (f x) = d y * (y - f x)"
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    and cont_d: "isCont d (f x)" and dfx: "d (f x) = E"
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    using CARAT_DERIV [THEN iffD1, OF g] by fast
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  from f have "f -- x --> f x"
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    by (rule DERIV_isCont [unfolded isCont_def])
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  with cont_d have "(\<lambda>z. d (f z)) -- x --> d (f x)"
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    by (rule isCont_LIM_compose)
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  hence "(\<lambda>z. d (f z) * ((f z - f x) / (z - x)))
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          -- x --> d (f x) * D"
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    by (rule LIM_mult [OF _ f [unfolded DERIV_iff2]])
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  thus "(\<lambda>z. (g (f z) - g (f x)) / (z - x)) -- x --> E * D"
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    by (simp add: d dfx real_scaleR_def)
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qed
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(* let's do the standard proof though theorem *)
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(* LIM_mult2 follows from a NS proof          *)
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lemma DERIV_cmult:
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      "DERIV f x :> D ==> DERIV (%x. c * f x) x :> c*D"
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by (drule DERIV_mult' [OF DERIV_const], simp)
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(* standard version *)
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lemma DERIV_chain: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (f o g) x :> Da * Db"
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   292
by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
huffman@21164
   293
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   294
lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
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   295
by (auto dest: DERIV_chain simp add: o_def)
huffman@21164
   296
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   297
(*derivative of linear multiplication*)
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   298
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
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   299
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
huffman@21164
   300
huffman@21164
   301
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
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   302
apply (cut_tac DERIV_power [OF DERIV_ident])
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   303
apply (simp add: real_scaleR_def real_of_nat_def)
huffman@21164
   304
done
huffman@21164
   305
huffman@21164
   306
text{*Power of -1*}
huffman@21164
   307
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   308
lemma DERIV_inverse:
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   309
  fixes x :: "'a::{real_normed_field,recpower}"
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   310
  shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
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   311
by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)
huffman@21164
   312
huffman@21164
   313
text{*Derivative of inverse*}
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   314
lemma DERIV_inverse_fun:
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   315
  fixes x :: "'a::{real_normed_field,recpower}"
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   316
  shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
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   317
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
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   318
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)
huffman@21164
   319
huffman@21164
   320
text{*Derivative of quotient*}
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   321
lemma DERIV_quotient:
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   322
  fixes x :: "'a::{real_normed_field,recpower}"
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   323
  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21784
   324
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21784
   325
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
huffman@21164
   326
huffman@22984
   327
huffman@22984
   328
subsection {* Differentiability predicate *}
huffman@21164
   329
huffman@21164
   330
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
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   331
by (simp add: differentiable_def)
huffman@21164
   332
huffman@21164
   333
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
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   334
by (force simp add: differentiable_def)
huffman@21164
   335
huffman@21164
   336
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
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   337
  apply (unfold differentiable_def)
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   338
  apply (rule_tac x=0 in exI)
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   339
  apply simp
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   340
  done
huffman@21164
   341
huffman@21164
   342
lemma differentiable_sum:
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   343
  assumes "f differentiable x"
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   344
  and "g differentiable x"
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   345
  shows "(\<lambda>x. f x + g x) differentiable x"
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   346
proof -
huffman@21164
   347
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   348
  then obtain df where "DERIV f x :> df" ..
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   349
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   350
  then obtain dg where "DERIV g x :> dg" ..
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   351
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@21164
   352
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
huffman@21164
   353
  thus ?thesis by (fold differentiable_def)
huffman@21164
   354
qed
huffman@21164
   355
huffman@21164
   356
lemma differentiable_diff:
huffman@21164
   357
  assumes "f differentiable x"
huffman@21164
   358
  and "g differentiable x"
huffman@21164
   359
  shows "(\<lambda>x. f x - g x) differentiable x"
huffman@21164
   360
proof -
huffman@21164
   361
  from prems have "f differentiable x" by simp
huffman@21164
   362
  moreover
huffman@21164
   363
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   364
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   365
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
huffman@21164
   366
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
huffman@21164
   367
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
huffman@21164
   368
  ultimately 
huffman@21164
   369
  show ?thesis
huffman@21784
   370
    by (auto simp: diff_def dest: differentiable_sum)
huffman@21164
   371
qed
huffman@21164
   372
huffman@21164
   373
lemma differentiable_mult:
huffman@21164
   374
  assumes "f differentiable x"
huffman@21164
   375
  and "g differentiable x"
huffman@21164
   376
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164
   377
proof -
huffman@21164
   378
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   379
  then obtain df where "DERIV f x :> df" ..
huffman@21164
   380
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   381
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   382
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
huffman@21164
   383
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
huffman@21164
   384
  thus ?thesis by (fold differentiable_def)
huffman@21164
   385
qed
huffman@21164
   386
huffman@22984
   387
huffman@21164
   388
subsection {* Nested Intervals and Bisection *}
huffman@21164
   389
huffman@21164
   390
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164
   391
     All considerably tidied by lcp.*}
huffman@21164
   392
huffman@21164
   393
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
   394
apply (induct "no")
huffman@21164
   395
apply (auto intro: order_trans)
huffman@21164
   396
done
huffman@21164
   397
huffman@21164
   398
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   399
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   400
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   401
      ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164
   402
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@21164
   403
apply (induct_tac "n")
huffman@21164
   404
apply (auto intro: order_trans)
huffman@21164
   405
apply (rule_tac y = "g (Suc na)" in order_trans)
huffman@21164
   406
apply (induct_tac [2] "na")
huffman@21164
   407
apply (auto intro: order_trans)
huffman@21164
   408
done
huffman@21164
   409
huffman@21164
   410
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   411
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   412
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   413
      ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164
   414
apply (subst Bseq_minus_iff [symmetric])
huffman@21164
   415
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164
   416
apply auto
huffman@21164
   417
done
huffman@21164
   418
huffman@21164
   419
lemma f_inc_imp_le_lim:
huffman@21164
   420
  fixes f :: "nat \<Rightarrow> real"
huffman@21164
   421
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@21164
   422
apply (rule linorder_not_less [THEN iffD1])
huffman@21164
   423
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
huffman@21164
   424
apply (drule real_less_sum_gt_zero)
huffman@21164
   425
apply (drule_tac x = "f n + - lim f" in spec, safe)
huffman@21164
   426
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
huffman@21164
   427
apply (subgoal_tac "lim f \<le> f (no + n) ")
huffman@21164
   428
apply (drule_tac no=no and m=n in lemma_f_mono_add)
huffman@21164
   429
apply (auto simp add: add_commute)
huffman@21164
   430
apply (induct_tac "no")
huffman@21164
   431
apply simp
huffman@21164
   432
apply (auto intro: order_trans simp add: diff_minus abs_if)
huffman@21164
   433
done
huffman@21164
   434
huffman@21164
   435
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@21164
   436
apply (rule LIMSEQ_minus [THEN limI])
huffman@21164
   437
apply (simp add: convergent_LIMSEQ_iff)
huffman@21164
   438
done
huffman@21164
   439
huffman@21164
   440
lemma g_dec_imp_lim_le:
huffman@21164
   441
  fixes g :: "nat \<Rightarrow> real"
huffman@21164
   442
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@21164
   443
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
huffman@21164
   444
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
huffman@21164
   445
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
huffman@21164
   446
done
huffman@21164
   447
huffman@21164
   448
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   449
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   450
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   451
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   452
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164
   453
apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164
   454
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164
   455
apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164
   456
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164
   457
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164
   458
apply (rule_tac x = "lim f" in exI)
huffman@21164
   459
apply (rule_tac x = "lim g" in exI)
huffman@21164
   460
apply (auto intro: LIMSEQ_le)
huffman@21164
   461
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164
   462
done
huffman@21164
   463
huffman@21164
   464
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   465
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   466
         \<forall>n. f(n) \<le> g(n);
huffman@21164
   467
         (%n. f(n) - g(n)) ----> 0 |]
huffman@21164
   468
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   469
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164
   470
apply (drule lemma_nest, auto)
huffman@21164
   471
apply (subgoal_tac "l = m")
huffman@21164
   472
apply (drule_tac [2] X = f in LIMSEQ_diff)
huffman@21164
   473
apply (auto intro: LIMSEQ_unique)
huffman@21164
   474
done
huffman@21164
   475
huffman@21164
   476
text{*The universal quantifiers below are required for the declaration
huffman@21164
   477
  of @{text Bolzano_nest_unique} below.*}
huffman@21164
   478
huffman@21164
   479
lemma Bolzano_bisect_le:
huffman@21164
   480
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   481
apply (rule allI)
huffman@21164
   482
apply (induct_tac "n")
huffman@21164
   483
apply (auto simp add: Let_def split_def)
huffman@21164
   484
done
huffman@21164
   485
huffman@21164
   486
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164
   487
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164
   488
apply (rule allI)
huffman@21164
   489
apply (induct_tac "n")
huffman@21164
   490
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   491
done
huffman@21164
   492
huffman@21164
   493
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164
   494
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   495
apply (rule allI)
huffman@21164
   496
apply (induct_tac "n")
huffman@21164
   497
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   498
done
huffman@21164
   499
huffman@21164
   500
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164
   501
apply (auto)
huffman@21164
   502
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164
   503
apply (simp)
huffman@21164
   504
done
huffman@21164
   505
huffman@21164
   506
lemma Bolzano_bisect_diff:
huffman@21164
   507
     "a \<le> b ==>
huffman@21164
   508
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164
   509
      (b-a) / (2 ^ n)"
huffman@21164
   510
apply (induct "n")
huffman@21164
   511
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164
   512
done
huffman@21164
   513
huffman@21164
   514
lemmas Bolzano_nest_unique =
huffman@21164
   515
    lemma_nest_unique
huffman@21164
   516
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164
   517
huffman@21164
   518
huffman@21164
   519
lemma not_P_Bolzano_bisect:
huffman@21164
   520
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164
   521
      and notP: "~ P(a,b)"
huffman@21164
   522
      and le:   "a \<le> b"
huffman@21164
   523
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   524
proof (induct n)
huffman@21164
   525
  case 0 thus ?case by simp
huffman@21164
   526
 next
huffman@21164
   527
  case (Suc n)
huffman@21164
   528
  thus ?case
huffman@21164
   529
 by (auto simp del: surjective_pairing [symmetric]
huffman@21164
   530
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164
   531
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164
   532
qed
huffman@21164
   533
huffman@21164
   534
(*Now we re-package P_prem as a formula*)
huffman@21164
   535
lemma not_P_Bolzano_bisect':
huffman@21164
   536
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   537
         ~ P(a,b);  a \<le> b |] ==>
huffman@21164
   538
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   539
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164
   540
huffman@21164
   541
huffman@21164
   542
huffman@21164
   543
lemma lemma_BOLZANO:
huffman@21164
   544
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   545
         \<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   546
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164
   547
         a \<le> b |]
huffman@21164
   548
      ==> P(a,b)"
huffman@21164
   549
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
huffman@21164
   550
apply (rule LIMSEQ_minus_cancel)
huffman@21164
   551
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164
   552
apply (rule ccontr)
huffman@21164
   553
apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164
   554
apply (rename_tac "l")
huffman@21164
   555
apply (drule_tac x = l in spec, clarify)
huffman@21164
   556
apply (simp add: LIMSEQ_def)
huffman@21164
   557
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   558
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   559
apply (drule real_less_half_sum, auto)
huffman@21164
   560
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   561
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   562
apply safe
huffman@21164
   563
apply (simp_all (no_asm_simp))
huffman@21164
   564
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
   565
apply (simp (no_asm_simp) add: abs_if)
huffman@21164
   566
apply (rule real_sum_of_halves [THEN subst])
huffman@21164
   567
apply (rule add_strict_mono)
huffman@21164
   568
apply (simp_all add: diff_minus [symmetric])
huffman@21164
   569
done
huffman@21164
   570
huffman@21164
   571
huffman@21164
   572
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164
   573
       (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   574
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164
   575
      --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164
   576
apply clarify
huffman@21164
   577
apply (blast intro: lemma_BOLZANO)
huffman@21164
   578
done
huffman@21164
   579
huffman@21164
   580
huffman@21164
   581
subsection {* Intermediate Value Theorem *}
huffman@21164
   582
huffman@21164
   583
text {*Prove Contrapositive by Bisection*}
huffman@21164
   584
huffman@21164
   585
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164
   586
         a \<le> b;
huffman@21164
   587
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164
   588
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   589
apply (rule contrapos_pp, assumption)
huffman@21164
   590
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
   591
apply safe
huffman@21164
   592
apply simp_all
huffman@21164
   593
apply (simp add: isCont_iff LIM_def)
huffman@21164
   594
apply (rule ccontr)
huffman@21164
   595
apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164
   596
 prefer 2
huffman@21164
   597
 apply simp
huffman@21164
   598
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164
   599
apply (drule_tac x = x in spec)+
huffman@21164
   600
apply simp
huffman@21164
   601
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164
   602
apply safe
huffman@21164
   603
apply simp
huffman@21164
   604
apply (drule_tac x = s in spec, clarify)
huffman@21164
   605
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164
   606
apply (drule_tac x = "ba-x" in spec)
huffman@21164
   607
apply (simp_all add: abs_if)
huffman@21164
   608
apply (drule_tac x = "aa-x" in spec)
huffman@21164
   609
apply (case_tac "x \<le> aa", simp_all)
huffman@21164
   610
done
huffman@21164
   611
huffman@21164
   612
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164
   613
         a \<le> b;
huffman@21164
   614
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164
   615
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   616
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164
   617
apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@21164
   618
apply (auto intro: isCont_minus)
huffman@21164
   619
done
huffman@21164
   620
huffman@21164
   621
(*HOL style here: object-level formulations*)
huffman@21164
   622
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164
   623
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   624
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   625
apply (blast intro: IVT)
huffman@21164
   626
done
huffman@21164
   627
huffman@21164
   628
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164
   629
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   630
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   631
apply (blast intro: IVT2)
huffman@21164
   632
done
huffman@21164
   633
huffman@21164
   634
text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164
   635
huffman@21164
   636
lemma isCont_bounded:
huffman@21164
   637
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   638
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164
   639
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
   640
apply safe
huffman@21164
   641
apply simp_all
huffman@21164
   642
apply (rename_tac x xa ya M Ma)
huffman@21164
   643
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
huffman@21164
   644
apply (rule_tac x = Ma in exI, clarify)
huffman@21164
   645
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
   646
apply (rule_tac x = M in exI, clarify)
huffman@21164
   647
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
   648
apply (case_tac "a \<le> x & x \<le> b")
huffman@21164
   649
apply (rule_tac [2] x = 1 in exI)
huffman@21164
   650
prefer 2 apply force
huffman@21164
   651
apply (simp add: LIM_def isCont_iff)
huffman@21164
   652
apply (drule_tac x = x in spec, auto)
huffman@21164
   653
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164
   654
apply (drule_tac x = 1 in spec, auto)
huffman@21164
   655
apply (rule_tac x = s in exI, clarify)
huffman@21164
   656
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164
   657
apply (drule_tac x = "xa-x" in spec)
huffman@21164
   658
apply (auto simp add: abs_ge_self)
huffman@21164
   659
done
huffman@21164
   660
huffman@21164
   661
text{*Refine the above to existence of least upper bound*}
huffman@21164
   662
huffman@21164
   663
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164
   664
      (\<exists>t. isLub UNIV S t)"
huffman@21164
   665
by (blast intro: reals_complete)
huffman@21164
   666
huffman@21164
   667
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   668
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   669
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164
   670
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164
   671
        in lemma_reals_complete)
huffman@21164
   672
apply auto
huffman@21164
   673
apply (drule isCont_bounded, assumption)
huffman@21164
   674
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164
   675
apply (rule exI, auto)
huffman@21164
   676
apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164
   677
done
huffman@21164
   678
huffman@21164
   679
text{*Now show that it attains its upper bound*}
huffman@21164
   680
huffman@21164
   681
lemma isCont_eq_Ub:
huffman@21164
   682
  assumes le: "a \<le> b"
huffman@21164
   683
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   684
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   685
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   686
proof -
huffman@21164
   687
  from isCont_has_Ub [OF le con]
huffman@21164
   688
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164
   689
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164
   690
  show ?thesis
huffman@21164
   691
  proof (intro exI, intro conjI)
huffman@21164
   692
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164
   693
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164
   694
    proof (rule ccontr)
huffman@21164
   695
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164
   696
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
huffman@21164
   697
        by (fastsimp simp add: linorder_not_le [symmetric])
huffman@21164
   698
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@21164
   699
        by (auto simp add: isCont_inverse isCont_diff con)
huffman@21164
   700
      from isCont_bounded [OF le this]
huffman@21164
   701
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164
   702
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
huffman@21164
   703
        by (simp add: M3 compare_rls)
huffman@21164
   704
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164
   705
        by (auto intro: order_le_less_trans [of _ k])
huffman@21164
   706
      with Minv
huffman@21164
   707
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164
   708
        by (intro strip less_imp_inverse_less, simp_all)
huffman@21164
   709
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164
   710
        by simp
huffman@21164
   711
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164
   712
        by (simp, arith)
huffman@21164
   713
      from M2 [OF this]
huffman@21164
   714
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164
   715
      thus False using invlt [of x] by force
huffman@21164
   716
    qed
huffman@21164
   717
  qed
huffman@21164
   718
qed
huffman@21164
   719
huffman@21164
   720
huffman@21164
   721
text{*Same theorem for lower bound*}
huffman@21164
   722
huffman@21164
   723
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   724
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164
   725
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   726
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164
   727
prefer 2 apply (blast intro: isCont_minus)
huffman@21164
   728
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164
   729
apply safe
huffman@21164
   730
apply auto
huffman@21164
   731
done
huffman@21164
   732
huffman@21164
   733
huffman@21164
   734
text{*Another version.*}
huffman@21164
   735
huffman@21164
   736
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   737
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164
   738
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164
   739
apply (frule isCont_eq_Lb)
huffman@21164
   740
apply (frule_tac [2] isCont_eq_Ub)
huffman@21164
   741
apply (assumption+, safe)
huffman@21164
   742
apply (rule_tac x = "f x" in exI)
huffman@21164
   743
apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164
   744
apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164
   745
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164
   746
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164
   747
apply (rule_tac [2] x = xb in exI)
huffman@21164
   748
apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164
   749
done
huffman@21164
   750
huffman@21164
   751
huffman@21164
   752
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   753
huffman@21164
   754
lemma DERIV_left_inc:
huffman@21164
   755
  fixes f :: "real => real"
huffman@21164
   756
  assumes der: "DERIV f x :> l"
huffman@21164
   757
      and l:   "0 < l"
huffman@21164
   758
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   759
proof -
huffman@21164
   760
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   761
  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
   762
    by (simp add: diff_minus)
huffman@21164
   763
  then obtain s
huffman@21164
   764
        where s:   "0 < s"
huffman@21164
   765
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   766
    by auto
huffman@21164
   767
  thus ?thesis
huffman@21164
   768
  proof (intro exI conjI strip)
huffman@21164
   769
    show "0<s" .
huffman@21164
   770
    fix h::real
huffman@21164
   771
    assume "0 < h" "h < s"
huffman@21164
   772
    with all [of h] show "f x < f (x+h)"
huffman@21164
   773
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   774
    split add: split_if_asm)
huffman@21164
   775
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   776
      with l
huffman@21164
   777
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   778
      thus "f x < f (x+h)"
huffman@21164
   779
  by (simp add: pos_less_divide_eq h)
huffman@21164
   780
    qed
huffman@21164
   781
  qed
huffman@21164
   782
qed
huffman@21164
   783
huffman@21164
   784
lemma DERIV_left_dec:
huffman@21164
   785
  fixes f :: "real => real"
huffman@21164
   786
  assumes der: "DERIV f x :> l"
huffman@21164
   787
      and l:   "l < 0"
huffman@21164
   788
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   789
proof -
huffman@21164
   790
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   791
  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
   792
    by (simp add: diff_minus)
huffman@21164
   793
  then obtain s
huffman@21164
   794
        where s:   "0 < s"
huffman@21164
   795
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   796
    by auto
huffman@21164
   797
  thus ?thesis
huffman@21164
   798
  proof (intro exI conjI strip)
huffman@21164
   799
    show "0<s" .
huffman@21164
   800
    fix h::real
huffman@21164
   801
    assume "0 < h" "h < s"
huffman@21164
   802
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
   803
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   804
    split add: split_if_asm)
huffman@21164
   805
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   806
      with l
huffman@21164
   807
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   808
      thus "f x < f (x-h)"
huffman@21164
   809
  by (simp add: pos_less_divide_eq h)
huffman@21164
   810
    qed
huffman@21164
   811
  qed
huffman@21164
   812
qed
huffman@21164
   813
huffman@21164
   814
lemma DERIV_local_max:
huffman@21164
   815
  fixes f :: "real => real"
huffman@21164
   816
  assumes der: "DERIV f x :> l"
huffman@21164
   817
      and d:   "0 < d"
huffman@21164
   818
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   819
  shows "l = 0"
huffman@21164
   820
proof (cases rule: linorder_cases [of l 0])
huffman@21164
   821
  case equal show ?thesis .
huffman@21164
   822
next
huffman@21164
   823
  case less
huffman@21164
   824
  from DERIV_left_dec [OF der less]
huffman@21164
   825
  obtain d' where d': "0 < d'"
huffman@21164
   826
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   827
  from real_lbound_gt_zero [OF d d']
huffman@21164
   828
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   829
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   830
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   831
next
huffman@21164
   832
  case greater
huffman@21164
   833
  from DERIV_left_inc [OF der greater]
huffman@21164
   834
  obtain d' where d': "0 < d'"
huffman@21164
   835
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   836
  from real_lbound_gt_zero [OF d d']
huffman@21164
   837
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   838
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   839
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   840
qed
huffman@21164
   841
huffman@21164
   842
huffman@21164
   843
text{*Similar theorem for a local minimum*}
huffman@21164
   844
lemma DERIV_local_min:
huffman@21164
   845
  fixes f :: "real => real"
huffman@21164
   846
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   847
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   848
huffman@21164
   849
huffman@21164
   850
text{*In particular, if a function is locally flat*}
huffman@21164
   851
lemma DERIV_local_const:
huffman@21164
   852
  fixes f :: "real => real"
huffman@21164
   853
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   854
by (auto dest!: DERIV_local_max)
huffman@21164
   855
huffman@21164
   856
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   857
lemma lemma_interval_lt:
huffman@21164
   858
     "[| a < x;  x < b |]
huffman@21164
   859
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
huffman@22998
   860
apply (simp add: abs_less_iff)
huffman@21164
   861
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   862
apply (rule_tac x = "x-a" in exI)
huffman@21164
   863
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   864
done
huffman@21164
   865
huffman@21164
   866
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   867
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   868
apply (drule lemma_interval_lt, auto)
huffman@21164
   869
apply (auto intro!: exI)
huffman@21164
   870
done
huffman@21164
   871
huffman@21164
   872
text{*Rolle's Theorem.
huffman@21164
   873
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   874
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
   875
   and @{term "f(a) = f(b)"},
huffman@21164
   876
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
   877
theorem Rolle:
huffman@21164
   878
  assumes lt: "a < b"
huffman@21164
   879
      and eq: "f(a) = f(b)"
huffman@21164
   880
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   881
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   882
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
   883
proof -
huffman@21164
   884
  have le: "a \<le> b" using lt by simp
huffman@21164
   885
  from isCont_eq_Ub [OF le con]
huffman@21164
   886
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
   887
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
   888
    by blast
huffman@21164
   889
  from isCont_eq_Lb [OF le con]
huffman@21164
   890
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
   891
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
   892
    by blast
huffman@21164
   893
  show ?thesis
huffman@21164
   894
  proof cases
huffman@21164
   895
    assume axb: "a < x & x < b"
huffman@21164
   896
        --{*@{term f} attains its maximum within the interval*}
huffman@21164
   897
    hence ax: "a<x" and xb: "x<b" by auto
huffman@21164
   898
    from lemma_interval [OF ax xb]
huffman@21164
   899
    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
   900
      by blast
huffman@21164
   901
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
   902
      by blast
huffman@21164
   903
    from differentiableD [OF dif [OF axb]]
huffman@21164
   904
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   905
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
   906
        --{*the derivative at a local maximum is zero*}
huffman@21164
   907
    thus ?thesis using ax xb der by auto
huffman@21164
   908
  next
huffman@21164
   909
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
   910
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
   911
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
   912
    show ?thesis
huffman@21164
   913
    proof cases
huffman@21164
   914
      assume ax'b: "a < x' & x' < b"
huffman@21164
   915
        --{*@{term f} attains its minimum within the interval*}
huffman@21164
   916
      hence ax': "a<x'" and x'b: "x'<b" by auto
huffman@21164
   917
      from lemma_interval [OF ax' x'b]
huffman@21164
   918
      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
   919
  by blast
huffman@21164
   920
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
   921
  by blast
huffman@21164
   922
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
   923
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
   924
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
   925
        --{*the derivative at a local minimum is zero*}
huffman@21164
   926
      thus ?thesis using ax' x'b der by auto
huffman@21164
   927
    next
huffman@21164
   928
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
   929
        --{*@{term f} is constant througout the interval*}
huffman@21164
   930
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
   931
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
   932
      from dense [OF lt]
huffman@21164
   933
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
   934
      from lemma_interval [OF ar rb]
huffman@21164
   935
      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
   936
  by blast
huffman@21164
   937
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
   938
      proof (clarify)
huffman@21164
   939
        fix z::real
huffman@21164
   940
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
   941
        show "f z = f b"
huffman@21164
   942
        proof (rule order_antisym)
huffman@21164
   943
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
   944
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
   945
        qed
huffman@21164
   946
      qed
huffman@21164
   947
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
   948
      proof (intro strip)
huffman@21164
   949
        fix y::real
huffman@21164
   950
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
   951
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
   952
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
   953
      qed
huffman@21164
   954
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
   955
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
   956
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
   957
        --{*the derivative of a constant function is zero*}
huffman@21164
   958
      thus ?thesis using ar rb der by auto
huffman@21164
   959
    qed
huffman@21164
   960
  qed
huffman@21164
   961
qed
huffman@21164
   962
huffman@21164
   963
huffman@21164
   964
subsection{*Mean Value Theorem*}
huffman@21164
   965
huffman@21164
   966
lemma lemma_MVT:
huffman@21164
   967
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164
   968
proof cases
huffman@21164
   969
  assume "a=b" thus ?thesis by simp
huffman@21164
   970
next
huffman@21164
   971
  assume "a\<noteq>b"
huffman@21164
   972
  hence ba: "b-a \<noteq> 0" by arith
huffman@21164
   973
  show ?thesis
huffman@21164
   974
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164
   975
        simp add: right_diff_distrib,
huffman@21164
   976
        simp add: left_diff_distrib)
huffman@21164
   977
qed
huffman@21164
   978
huffman@21164
   979
theorem MVT:
huffman@21164
   980
  assumes lt:  "a < b"
huffman@21164
   981
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   982
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   983
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
   984
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
   985
proof -
huffman@21164
   986
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@21164
   987
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
huffman@23069
   988
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
huffman@21164
   989
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
   990
  proof (clarify)
huffman@21164
   991
    fix x::real
huffman@21164
   992
    assume ax: "a < x" and xb: "x < b"
huffman@21164
   993
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
   994
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   995
    show "?F differentiable x"
huffman@21164
   996
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
   997
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
   998
  qed
huffman@21164
   999
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1000
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1001
    by blast
huffman@21164
  1002
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1003
    by (rule DERIV_cmult_Id)
huffman@21164
  1004
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1005
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1006
    by (rule DERIV_add [OF der])
huffman@21164
  1007
  show ?thesis
huffman@21164
  1008
  proof (intro exI conjI)
huffman@21164
  1009
    show "a < z" .
huffman@21164
  1010
    show "z < b" .
huffman@21164
  1011
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1012
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1013
  qed
huffman@21164
  1014
qed
huffman@21164
  1015
huffman@21164
  1016
huffman@21164
  1017
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1018
huffman@21164
  1019
lemma DERIV_isconst_end:
huffman@21164
  1020
  fixes f :: "real => real"
huffman@21164
  1021
  shows "[| a < b;
huffman@21164
  1022
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1023
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1024
        ==> f b = f a"
huffman@21164
  1025
apply (drule MVT, assumption)
huffman@21164
  1026
apply (blast intro: differentiableI)
huffman@21164
  1027
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1028
done
huffman@21164
  1029
huffman@21164
  1030
lemma DERIV_isconst1:
huffman@21164
  1031
  fixes f :: "real => real"
huffman@21164
  1032
  shows "[| a < b;
huffman@21164
  1033
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1034
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1035
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1036
apply safe
huffman@21164
  1037
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1038
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1039
done
huffman@21164
  1040
huffman@21164
  1041
lemma DERIV_isconst2:
huffman@21164
  1042
  fixes f :: "real => real"
huffman@21164
  1043
  shows "[| a < b;
huffman@21164
  1044
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1045
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1046
         a \<le> x; x \<le> b |]
huffman@21164
  1047
        ==> f x = f a"
huffman@21164
  1048
apply (blast dest: DERIV_isconst1)
huffman@21164
  1049
done
huffman@21164
  1050
huffman@21164
  1051
lemma DERIV_isconst_all:
huffman@21164
  1052
  fixes f :: "real => real"
huffman@21164
  1053
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1054
apply (rule linorder_cases [of x y])
huffman@21164
  1055
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1056
done
huffman@21164
  1057
huffman@21164
  1058
lemma DERIV_const_ratio_const:
huffman@21784
  1059
  fixes f :: "real => real"
huffman@21784
  1060
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1061
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1062
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1063
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
huffman@21164
  1064
apply (auto dest: DERIV_unique simp add: left_distrib diff_minus)
huffman@21164
  1065
done
huffman@21164
  1066
huffman@21164
  1067
lemma DERIV_const_ratio_const2:
huffman@21784
  1068
  fixes f :: "real => real"
huffman@21784
  1069
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1070
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1071
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1072
done
huffman@21164
  1073
huffman@21164
  1074
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1075
by (simp)
huffman@21164
  1076
huffman@21164
  1077
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1078
by (simp)
huffman@21164
  1079
huffman@21164
  1080
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1081
huffman@21164
  1082
lemma DERIV_const_average:
huffman@21164
  1083
  fixes v :: "real => real"
huffman@21164
  1084
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1085
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1086
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1087
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1088
  case equal with neq show ?thesis by simp
huffman@21164
  1089
next
huffman@21164
  1090
  case less
huffman@21164
  1091
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1092
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1093
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1094
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1095
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1096
  ultimately show ?thesis using neq by force
huffman@21164
  1097
next
huffman@21164
  1098
  case greater
huffman@21164
  1099
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1100
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1101
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1102
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1103
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1104
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1105
qed
huffman@21164
  1106
huffman@21164
  1107
huffman@21164
  1108
text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164
  1109
strict maximum at an end point, not in the middle.*}
huffman@21164
  1110
huffman@21164
  1111
lemma lemma_isCont_inj:
huffman@21164
  1112
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1113
  assumes d: "0 < d"
huffman@21164
  1114
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1115
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1116
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164
  1117
proof (rule ccontr)
huffman@21164
  1118
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164
  1119
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164
  1120
  show False
huffman@21164
  1121
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164
  1122
    case le
huffman@21164
  1123
    from d cont all [of "x+d"]
huffman@21164
  1124
    have flef: "f(x+d) \<le> f x"
huffman@21164
  1125
     and xlex: "x - d \<le> x"
huffman@21164
  1126
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164
  1127
       by (auto simp add: abs_if)
huffman@21164
  1128
    from IVT [OF le flef xlex cont']
huffman@21164
  1129
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164
  1130
    moreover
huffman@21164
  1131
    hence "g(f x') = g (f(x+d))" by simp
huffman@21164
  1132
    ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@22998
  1133
      by (simp add: abs_le_iff)
huffman@21164
  1134
  next
huffman@21164
  1135
    case ge
huffman@21164
  1136
    from d cont all [of "x-d"]
huffman@21164
  1137
    have flef: "f(x-d) \<le> f x"
huffman@21164
  1138
     and xlex: "x \<le> x+d"
huffman@21164
  1139
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164
  1140
       by (auto simp add: abs_if)
huffman@21164
  1141
    from IVT2 [OF ge flef xlex cont']
huffman@21164
  1142
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164
  1143
    moreover
huffman@21164
  1144
    hence "g(f x') = g (f(x-d))" by simp
huffman@21164
  1145
    ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@22998
  1146
      by (simp add: abs_le_iff)
huffman@21164
  1147
  qed
huffman@21164
  1148
qed
huffman@21164
  1149
huffman@21164
  1150
huffman@21164
  1151
text{*Similar version for lower bound.*}
huffman@21164
  1152
huffman@21164
  1153
lemma lemma_isCont_inj2:
huffman@21164
  1154
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1155
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164
  1156
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164
  1157
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164
  1158
apply (insert lemma_isCont_inj
huffman@21164
  1159
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@21164
  1160
apply (simp add: isCont_minus linorder_not_le)
huffman@21164
  1161
done
huffman@21164
  1162
huffman@21164
  1163
text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164
  1164
@{text "f[[x - d, x + d]]"} .*}
huffman@21164
  1165
huffman@21164
  1166
lemma isCont_inj_range:
huffman@21164
  1167
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1168
  assumes d: "0 < d"
huffman@21164
  1169
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1170
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1171
  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
  1172
proof -
huffman@21164
  1173
  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@22998
  1174
    by (auto simp add: abs_le_iff)
huffman@21164
  1175
  from isCont_Lb_Ub [OF this]
huffman@21164
  1176
  obtain L M
huffman@21164
  1177
  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
  1178
    and all2 [rule_format]:
huffman@21164
  1179
           "\<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
  1180
    by auto
huffman@21164
  1181
  with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164
  1182
  moreover have "L \<noteq> f x"
huffman@21164
  1183
  proof -
huffman@21164
  1184
    from lemma_isCont_inj2 [OF d inj cont]
huffman@21164
  1185
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164
  1186
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1187
  qed
huffman@21164
  1188
  moreover have "f x \<noteq> M"
huffman@21164
  1189
  proof -
huffman@21164
  1190
    from lemma_isCont_inj [OF d inj cont]
huffman@21164
  1191
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164
  1192
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1193
  qed
huffman@21164
  1194
  ultimately have "L < f x & f x < M" by arith
huffman@21164
  1195
  hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164
  1196
  from real_lbound_gt_zero [OF this]
huffman@21164
  1197
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164
  1198
  thus ?thesis
huffman@21164
  1199
  proof (intro exI conjI)
huffman@21164
  1200
    show "0<e" .
huffman@21164
  1201
    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
  1202
    proof (intro strip)
huffman@21164
  1203
      fix y::real
huffman@21164
  1204
      assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164
  1205
      with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164
  1206
      from all2 [OF this]
huffman@21164
  1207
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
huffman@21164
  1208
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y"
huffman@22998
  1209
        by (force simp add: abs_le_iff)
huffman@21164
  1210
    qed
huffman@21164
  1211
  qed
huffman@21164
  1212
qed
huffman@21164
  1213
huffman@21164
  1214
huffman@21164
  1215
text{*Continuity of inverse function*}
huffman@21164
  1216
huffman@21164
  1217
lemma isCont_inverse_function:
huffman@21164
  1218
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1219
  assumes d: "0 < d"
huffman@21164
  1220
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1221
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1222
  shows "isCont g (f x)"
huffman@21164
  1223
proof (simp add: isCont_iff LIM_eq)
huffman@21164
  1224
  show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164
  1225
         (\<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
  1226
  proof (intro strip)
huffman@21164
  1227
    fix r::real
huffman@21164
  1228
    assume r: "0<r"
huffman@21164
  1229
    from real_lbound_gt_zero [OF r d]
huffman@21164
  1230
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164
  1231
    with inj cont
huffman@21164
  1232
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164
  1233
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164
  1234
    from isCont_inj_range [OF e this]
huffman@21164
  1235
    obtain e' where e': "0 < e'"
huffman@21164
  1236
        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
  1237
          by blast
huffman@21164
  1238
    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
  1239
    proof (intro exI conjI)
huffman@21164
  1240
      show "0<e'" .
huffman@21164
  1241
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164
  1242
      proof (intro strip)
huffman@21164
  1243
        fix z::real
huffman@21164
  1244
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164
  1245
        with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164
  1246
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164
  1247
      qed
huffman@21164
  1248
    qed
huffman@21164
  1249
  qed
huffman@21164
  1250
qed
huffman@21164
  1251
huffman@23041
  1252
text {* Derivative of inverse function *}
huffman@23041
  1253
huffman@23041
  1254
lemma DERIV_inverse_function:
huffman@23041
  1255
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1256
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1257
  assumes neq: "D \<noteq> 0"
huffman@23044
  1258
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1259
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1260
  assumes cont: "isCont g x"
huffman@23041
  1261
  shows "DERIV g x :> inverse D"
huffman@23041
  1262
unfolding DERIV_iff2
huffman@23044
  1263
proof (rule LIM_equal2)
huffman@23044
  1264
  show "0 < min (x - a) (b - x)"
huffman@23044
  1265
    using a b by simp
huffman@23044
  1266
next
huffman@23041
  1267
  fix y
huffman@23044
  1268
  assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1269
  hence "a < y" and "y < b"
huffman@23044
  1270
    by (simp_all add: abs_less_iff)
huffman@23041
  1271
  thus "(g y - g x) / (y - x) =
huffman@23041
  1272
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1273
    by (simp add: inj)
huffman@23041
  1274
next
huffman@23041
  1275
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1276
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1277
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1278
    using inj a b by simp
huffman@23041
  1279
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
huffman@23041
  1280
  proof (safe intro!: exI)
huffman@23044
  1281
    show "0 < min (x - a) (b - x)"
huffman@23044
  1282
      using a b by simp
huffman@23041
  1283
  next
huffman@23041
  1284
    fix y
huffman@23044
  1285
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1286
    hence y: "a < y" "y < b"
huffman@23044
  1287
      by (simp_all add: abs_less_iff)
huffman@23041
  1288
    assume "g y = g x"
huffman@23041
  1289
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1290
    hence "y = x" using inj y a b by simp
huffman@23041
  1291
    also assume "y \<noteq> x"
huffman@23041
  1292
    finally show False by simp
huffman@23041
  1293
  qed
huffman@23041
  1294
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1295
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1296
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1297
        -- x --> inverse D"
huffman@23041
  1298
    using neq by (rule LIM_inverse)
huffman@23041
  1299
qed
huffman@23041
  1300
huffman@21164
  1301
theorem GMVT:
huffman@21784
  1302
  fixes a b :: real
huffman@21164
  1303
  assumes alb: "a < b"
huffman@21164
  1304
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
huffman@21164
  1305
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
huffman@21164
  1306
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
huffman@21164
  1307
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1308
  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
  1309
proof -
huffman@21164
  1310
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
huffman@21164
  1311
  from prems have "a < b" by simp
huffman@21164
  1312
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@21164
  1313
  proof -
huffman@21164
  1314
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
huffman@21164
  1315
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
huffman@21164
  1316
      by (auto intro: isCont_mult)
huffman@21164
  1317
    moreover
huffman@21164
  1318
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
huffman@21164
  1319
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
huffman@21164
  1320
      by (auto intro: isCont_mult)
huffman@21164
  1321
    ultimately show ?thesis
huffman@21164
  1322
      by (fastsimp intro: isCont_diff)
huffman@21164
  1323
  qed
huffman@21164
  1324
  moreover
huffman@21164
  1325
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@21164
  1326
  proof -
huffman@21164
  1327
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1328
    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
  1329
    moreover
huffman@21164
  1330
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1331
    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
  1332
    ultimately show ?thesis by (simp add: differentiable_diff)
huffman@21164
  1333
  qed
huffman@21164
  1334
  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
  1335
  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
  1336
  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
  1337
huffman@21164
  1338
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1339
  with gd have "g differentiable c" by simp
huffman@21164
  1340
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1341
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1342
huffman@21164
  1343
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1344
  with fd have "f differentiable c" by simp
huffman@21164
  1345
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1346
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1347
huffman@21164
  1348
  from cdef have "DERIV ?h c :> l" by auto
huffman@21164
  1349
  moreover
huffman@21164
  1350
  {
huffman@21164
  1351
    from g'cdef have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
huffman@21164
  1352
      apply (insert DERIV_const [where k="f b - f a"])
huffman@21164
  1353
      apply (drule meta_spec [of _ c])
huffman@21164
  1354
      apply (drule DERIV_mult [where f="(\<lambda>x. f b - f a)" and g=g])
huffman@21164
  1355
      by simp_all
huffman@21164
  1356
    moreover from f'cdef have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
huffman@21164
  1357
      apply (insert DERIV_const [where k="g b - g a"])
huffman@21164
  1358
      apply (drule meta_spec [of _ c])
huffman@21164
  1359
      apply (drule DERIV_mult [where f="(\<lambda>x. g b - g a)" and g=f])
huffman@21164
  1360
      by simp_all
huffman@21164
  1361
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
huffman@21164
  1362
      by (simp add: DERIV_diff)
huffman@21164
  1363
  }
huffman@21164
  1364
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1365
huffman@21164
  1366
  {
huffman@21164
  1367
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1368
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1369
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1370
  }
huffman@21164
  1371
  moreover
huffman@21164
  1372
  {
huffman@21164
  1373
    have "?h b - ?h a =
huffman@21164
  1374
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1375
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
huffman@22998
  1376
      by (simp add: mult_ac add_ac right_diff_distrib)
huffman@21164
  1377
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1378
  }
huffman@21164
  1379
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1380
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1381
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1382
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1383
huffman@21164
  1384
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1385
qed
huffman@21164
  1386
huffman@21164
  1387
end