src/HOL/Deriv.thy
author haftmann
Wed Dec 03 15:58:44 2008 +0100 (2008-12-03)
changeset 28952 15a4b2cf8c34
parent 27668 src/HOL/Hyperreal/Deriv.thy@6eb20b2cecf8
child 29166 c23b2d108612
permissions -rw-r--r--
made repository layout more coherent with logical distribution structure; stripped some $Id$s
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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 Univ_Poly
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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 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)
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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_distribs)
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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: ring_simps)
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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: ring_distribs 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 :> (1 + of_nat n) * (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 ring_distribs mult_1_left)
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    apply (simp only: power_Suc right_distrib mult_ac add_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 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)" 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 = 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 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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by (drule (1) DERIV_chain', simp add: o_def real_scaleR_def mult_commute)
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lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
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by (auto dest: DERIV_chain simp add: o_def)
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(*derivative of linear multiplication*)
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lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
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by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
huffman@21164
   289
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   290
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
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   291
apply (cut_tac DERIV_power [OF DERIV_ident])
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   292
apply (simp add: real_scaleR_def real_of_nat_def)
huffman@21164
   293
done
huffman@21164
   294
huffman@21164
   295
text{*Power of -1*}
huffman@21164
   296
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   297
lemma DERIV_inverse:
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   298
  fixes x :: "'a::{real_normed_field,recpower}"
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   299
  shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
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   300
by (drule DERIV_inverse' [OF DERIV_ident]) (simp add: power_Suc)
huffman@21164
   301
huffman@21164
   302
text{*Derivative of inverse*}
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   303
lemma DERIV_inverse_fun:
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   304
  fixes x :: "'a::{real_normed_field,recpower}"
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   305
  shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21784
   306
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@21784
   307
by (drule (1) DERIV_inverse') (simp add: mult_ac power_Suc nonzero_inverse_mult_distrib)
huffman@21164
   308
huffman@21164
   309
text{*Derivative of quotient*}
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   310
lemma DERIV_quotient:
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   311
  fixes x :: "'a::{real_normed_field,recpower}"
huffman@21784
   312
  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21784
   313
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@21784
   314
by (drule (2) DERIV_divide) (simp add: mult_commute power_Suc)
huffman@21164
   315
huffman@22984
   316
huffman@22984
   317
subsection {* Differentiability predicate *}
huffman@21164
   318
huffman@21164
   319
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
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   320
by (simp add: differentiable_def)
huffman@21164
   321
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   322
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
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   323
by (force simp add: differentiable_def)
huffman@21164
   324
huffman@21164
   325
lemma differentiable_const: "(\<lambda>z. a) differentiable x"
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   326
  apply (unfold differentiable_def)
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   327
  apply (rule_tac x=0 in exI)
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   328
  apply simp
huffman@21164
   329
  done
huffman@21164
   330
huffman@21164
   331
lemma differentiable_sum:
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   332
  assumes "f differentiable x"
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   333
  and "g differentiable x"
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   334
  shows "(\<lambda>x. f x + g x) differentiable x"
huffman@21164
   335
proof -
huffman@21164
   336
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   337
  then obtain df where "DERIV f x :> df" ..
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   338
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   339
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   340
  ultimately have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@21164
   341
  hence "\<exists>D. DERIV (\<lambda>x. f x + g x) x :> D" by auto
huffman@21164
   342
  thus ?thesis by (fold differentiable_def)
huffman@21164
   343
qed
huffman@21164
   344
huffman@21164
   345
lemma differentiable_diff:
huffman@21164
   346
  assumes "f differentiable x"
huffman@21164
   347
  and "g differentiable x"
huffman@21164
   348
  shows "(\<lambda>x. f x - g x) differentiable x"
huffman@21164
   349
proof -
huffman@21164
   350
  from prems have "f differentiable x" by simp
huffman@21164
   351
  moreover
huffman@21164
   352
  from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   353
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   354
  then have "DERIV (\<lambda>x. - g x) x :> -dg" by (rule DERIV_minus)
huffman@21164
   355
  hence "\<exists>D. DERIV (\<lambda>x. - g x) x :> D" by auto
huffman@21164
   356
  hence "(\<lambda>x. - g x) differentiable x" by (fold differentiable_def)
huffman@21164
   357
  ultimately 
huffman@21164
   358
  show ?thesis
huffman@21784
   359
    by (auto simp: diff_def dest: differentiable_sum)
huffman@21164
   360
qed
huffman@21164
   361
huffman@21164
   362
lemma differentiable_mult:
huffman@21164
   363
  assumes "f differentiable x"
huffman@21164
   364
  and "g differentiable x"
huffman@21164
   365
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164
   366
proof -
huffman@21164
   367
  from prems have "\<exists>D. DERIV f x :> D" by (unfold differentiable_def)
huffman@21164
   368
  then obtain df where "DERIV f x :> df" ..
huffman@21164
   369
  moreover from prems have "\<exists>D. DERIV g x :> D" by (unfold differentiable_def)
huffman@21164
   370
  then obtain dg where "DERIV g x :> dg" ..
huffman@21164
   371
  ultimately have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (simp add: DERIV_mult)
huffman@21164
   372
  hence "\<exists>D. DERIV (\<lambda>x. f x * g x) x :> D" by auto
huffman@21164
   373
  thus ?thesis by (fold differentiable_def)
huffman@21164
   374
qed
huffman@21164
   375
huffman@22984
   376
huffman@21164
   377
subsection {* Nested Intervals and Bisection *}
huffman@21164
   378
huffman@21164
   379
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164
   380
     All considerably tidied by lcp.*}
huffman@21164
   381
huffman@21164
   382
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
   383
apply (induct "no")
huffman@21164
   384
apply (auto intro: order_trans)
huffman@21164
   385
done
huffman@21164
   386
huffman@21164
   387
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   388
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   389
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   390
      ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164
   391
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@21164
   392
apply (induct_tac "n")
huffman@21164
   393
apply (auto intro: order_trans)
huffman@21164
   394
apply (rule_tac y = "g (Suc na)" in order_trans)
huffman@21164
   395
apply (induct_tac [2] "na")
huffman@21164
   396
apply (auto intro: order_trans)
huffman@21164
   397
done
huffman@21164
   398
huffman@21164
   399
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   400
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   401
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   402
      ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164
   403
apply (subst Bseq_minus_iff [symmetric])
huffman@21164
   404
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164
   405
apply auto
huffman@21164
   406
done
huffman@21164
   407
huffman@21164
   408
lemma f_inc_imp_le_lim:
huffman@21164
   409
  fixes f :: "nat \<Rightarrow> real"
huffman@21164
   410
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@21164
   411
apply (rule linorder_not_less [THEN iffD1])
huffman@21164
   412
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
huffman@21164
   413
apply (drule real_less_sum_gt_zero)
huffman@21164
   414
apply (drule_tac x = "f n + - lim f" in spec, safe)
huffman@21164
   415
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
huffman@21164
   416
apply (subgoal_tac "lim f \<le> f (no + n) ")
huffman@21164
   417
apply (drule_tac no=no and m=n in lemma_f_mono_add)
huffman@21164
   418
apply (auto simp add: add_commute)
huffman@21164
   419
apply (induct_tac "no")
huffman@21164
   420
apply simp
huffman@21164
   421
apply (auto intro: order_trans simp add: diff_minus abs_if)
huffman@21164
   422
done
huffman@21164
   423
huffman@21164
   424
lemma lim_uminus: "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@21164
   425
apply (rule LIMSEQ_minus [THEN limI])
huffman@21164
   426
apply (simp add: convergent_LIMSEQ_iff)
huffman@21164
   427
done
huffman@21164
   428
huffman@21164
   429
lemma g_dec_imp_lim_le:
huffman@21164
   430
  fixes g :: "nat \<Rightarrow> real"
huffman@21164
   431
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@21164
   432
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
huffman@21164
   433
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
huffman@21164
   434
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
huffman@21164
   435
done
huffman@21164
   436
huffman@21164
   437
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   438
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   439
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   440
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   441
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164
   442
apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164
   443
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164
   444
apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164
   445
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164
   446
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164
   447
apply (rule_tac x = "lim f" in exI)
huffman@21164
   448
apply (rule_tac x = "lim g" in exI)
huffman@21164
   449
apply (auto intro: LIMSEQ_le)
huffman@21164
   450
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164
   451
done
huffman@21164
   452
huffman@21164
   453
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   454
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   455
         \<forall>n. f(n) \<le> g(n);
huffman@21164
   456
         (%n. f(n) - g(n)) ----> 0 |]
huffman@21164
   457
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   458
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164
   459
apply (drule lemma_nest, auto)
huffman@21164
   460
apply (subgoal_tac "l = m")
huffman@21164
   461
apply (drule_tac [2] X = f in LIMSEQ_diff)
huffman@21164
   462
apply (auto intro: LIMSEQ_unique)
huffman@21164
   463
done
huffman@21164
   464
huffman@21164
   465
text{*The universal quantifiers below are required for the declaration
huffman@21164
   466
  of @{text Bolzano_nest_unique} below.*}
huffman@21164
   467
huffman@21164
   468
lemma Bolzano_bisect_le:
huffman@21164
   469
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   470
apply (rule allI)
huffman@21164
   471
apply (induct_tac "n")
huffman@21164
   472
apply (auto simp add: Let_def split_def)
huffman@21164
   473
done
huffman@21164
   474
huffman@21164
   475
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164
   476
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164
   477
apply (rule allI)
huffman@21164
   478
apply (induct_tac "n")
huffman@21164
   479
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   480
done
huffman@21164
   481
huffman@21164
   482
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164
   483
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   484
apply (rule allI)
huffman@21164
   485
apply (induct_tac "n")
huffman@21164
   486
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   487
done
huffman@21164
   488
huffman@21164
   489
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164
   490
apply (auto)
huffman@21164
   491
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164
   492
apply (simp)
huffman@21164
   493
done
huffman@21164
   494
huffman@21164
   495
lemma Bolzano_bisect_diff:
huffman@21164
   496
     "a \<le> b ==>
huffman@21164
   497
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164
   498
      (b-a) / (2 ^ n)"
huffman@21164
   499
apply (induct "n")
huffman@21164
   500
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164
   501
done
huffman@21164
   502
huffman@21164
   503
lemmas Bolzano_nest_unique =
huffman@21164
   504
    lemma_nest_unique
huffman@21164
   505
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164
   506
huffman@21164
   507
huffman@21164
   508
lemma not_P_Bolzano_bisect:
huffman@21164
   509
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164
   510
      and notP: "~ P(a,b)"
huffman@21164
   511
      and le:   "a \<le> b"
huffman@21164
   512
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   513
proof (induct n)
huffman@23441
   514
  case 0 show ?case using notP by simp
huffman@21164
   515
 next
huffman@21164
   516
  case (Suc n)
huffman@21164
   517
  thus ?case
huffman@21164
   518
 by (auto simp del: surjective_pairing [symmetric]
huffman@21164
   519
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164
   520
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164
   521
qed
huffman@21164
   522
huffman@21164
   523
(*Now we re-package P_prem as a formula*)
huffman@21164
   524
lemma not_P_Bolzano_bisect':
huffman@21164
   525
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   526
         ~ P(a,b);  a \<le> b |] ==>
huffman@21164
   527
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   528
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164
   529
huffman@21164
   530
huffman@21164
   531
huffman@21164
   532
lemma lemma_BOLZANO:
huffman@21164
   533
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   534
         \<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   535
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164
   536
         a \<le> b |]
huffman@21164
   537
      ==> P(a,b)"
huffman@21164
   538
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
huffman@21164
   539
apply (rule LIMSEQ_minus_cancel)
huffman@21164
   540
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164
   541
apply (rule ccontr)
huffman@21164
   542
apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164
   543
apply (rename_tac "l")
huffman@21164
   544
apply (drule_tac x = l in spec, clarify)
huffman@21164
   545
apply (simp add: LIMSEQ_def)
huffman@21164
   546
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   547
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   548
apply (drule real_less_half_sum, auto)
huffman@21164
   549
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   550
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   551
apply safe
huffman@21164
   552
apply (simp_all (no_asm_simp))
huffman@21164
   553
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
   554
apply (simp (no_asm_simp) add: abs_if)
huffman@21164
   555
apply (rule real_sum_of_halves [THEN subst])
huffman@21164
   556
apply (rule add_strict_mono)
huffman@21164
   557
apply (simp_all add: diff_minus [symmetric])
huffman@21164
   558
done
huffman@21164
   559
huffman@21164
   560
huffman@21164
   561
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164
   562
       (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   563
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164
   564
      --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164
   565
apply clarify
huffman@21164
   566
apply (blast intro: lemma_BOLZANO)
huffman@21164
   567
done
huffman@21164
   568
huffman@21164
   569
huffman@21164
   570
subsection {* Intermediate Value Theorem *}
huffman@21164
   571
huffman@21164
   572
text {*Prove Contrapositive by Bisection*}
huffman@21164
   573
huffman@21164
   574
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164
   575
         a \<le> b;
huffman@21164
   576
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164
   577
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   578
apply (rule contrapos_pp, assumption)
huffman@21164
   579
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
   580
apply safe
huffman@21164
   581
apply simp_all
huffman@21164
   582
apply (simp add: isCont_iff LIM_def)
huffman@21164
   583
apply (rule ccontr)
huffman@21164
   584
apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164
   585
 prefer 2
huffman@21164
   586
 apply simp
huffman@21164
   587
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164
   588
apply (drule_tac x = x in spec)+
huffman@21164
   589
apply simp
huffman@21164
   590
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164
   591
apply safe
huffman@21164
   592
apply simp
huffman@21164
   593
apply (drule_tac x = s in spec, clarify)
huffman@21164
   594
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164
   595
apply (drule_tac x = "ba-x" in spec)
huffman@21164
   596
apply (simp_all add: abs_if)
huffman@21164
   597
apply (drule_tac x = "aa-x" in spec)
huffman@21164
   598
apply (case_tac "x \<le> aa", simp_all)
huffman@21164
   599
done
huffman@21164
   600
huffman@21164
   601
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164
   602
         a \<le> b;
huffman@21164
   603
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164
   604
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   605
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164
   606
apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@21164
   607
apply (auto intro: isCont_minus)
huffman@21164
   608
done
huffman@21164
   609
huffman@21164
   610
(*HOL style here: object-level formulations*)
huffman@21164
   611
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164
   612
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   613
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   614
apply (blast intro: IVT)
huffman@21164
   615
done
huffman@21164
   616
huffman@21164
   617
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164
   618
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   619
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   620
apply (blast intro: IVT2)
huffman@21164
   621
done
huffman@21164
   622
huffman@21164
   623
text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164
   624
huffman@21164
   625
lemma isCont_bounded:
huffman@21164
   626
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   627
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164
   628
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
   629
apply safe
huffman@21164
   630
apply simp_all
huffman@21164
   631
apply (rename_tac x xa ya M Ma)
huffman@21164
   632
apply (cut_tac x = M and y = Ma in linorder_linear, safe)
huffman@21164
   633
apply (rule_tac x = Ma in exI, clarify)
huffman@21164
   634
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
   635
apply (rule_tac x = M in exI, clarify)
huffman@21164
   636
apply (cut_tac x = xb and y = xa in linorder_linear, force)
huffman@21164
   637
apply (case_tac "a \<le> x & x \<le> b")
huffman@21164
   638
apply (rule_tac [2] x = 1 in exI)
huffman@21164
   639
prefer 2 apply force
huffman@21164
   640
apply (simp add: LIM_def isCont_iff)
huffman@21164
   641
apply (drule_tac x = x in spec, auto)
huffman@21164
   642
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164
   643
apply (drule_tac x = 1 in spec, auto)
huffman@21164
   644
apply (rule_tac x = s in exI, clarify)
huffman@21164
   645
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164
   646
apply (drule_tac x = "xa-x" in spec)
huffman@21164
   647
apply (auto simp add: abs_ge_self)
huffman@21164
   648
done
huffman@21164
   649
huffman@21164
   650
text{*Refine the above to existence of least upper bound*}
huffman@21164
   651
huffman@21164
   652
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164
   653
      (\<exists>t. isLub UNIV S t)"
huffman@21164
   654
by (blast intro: reals_complete)
huffman@21164
   655
huffman@21164
   656
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   657
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   658
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164
   659
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164
   660
        in lemma_reals_complete)
huffman@21164
   661
apply auto
huffman@21164
   662
apply (drule isCont_bounded, assumption)
huffman@21164
   663
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164
   664
apply (rule exI, auto)
huffman@21164
   665
apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164
   666
done
huffman@21164
   667
huffman@21164
   668
text{*Now show that it attains its upper bound*}
huffman@21164
   669
huffman@21164
   670
lemma isCont_eq_Ub:
huffman@21164
   671
  assumes le: "a \<le> b"
huffman@21164
   672
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   673
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   674
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   675
proof -
huffman@21164
   676
  from isCont_has_Ub [OF le con]
huffman@21164
   677
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164
   678
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164
   679
  show ?thesis
huffman@21164
   680
  proof (intro exI, intro conjI)
huffman@21164
   681
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164
   682
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164
   683
    proof (rule ccontr)
huffman@21164
   684
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164
   685
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
huffman@21164
   686
        by (fastsimp simp add: linorder_not_le [symmetric])
huffman@21164
   687
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@21164
   688
        by (auto simp add: isCont_inverse isCont_diff con)
huffman@21164
   689
      from isCont_bounded [OF le this]
huffman@21164
   690
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164
   691
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
huffman@21164
   692
        by (simp add: M3 compare_rls)
huffman@21164
   693
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164
   694
        by (auto intro: order_le_less_trans [of _ k])
huffman@21164
   695
      with Minv
huffman@21164
   696
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164
   697
        by (intro strip less_imp_inverse_less, simp_all)
huffman@21164
   698
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164
   699
        by simp
huffman@21164
   700
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164
   701
        by (simp, arith)
huffman@21164
   702
      from M2 [OF this]
huffman@21164
   703
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164
   704
      thus False using invlt [of x] by force
huffman@21164
   705
    qed
huffman@21164
   706
  qed
huffman@21164
   707
qed
huffman@21164
   708
huffman@21164
   709
huffman@21164
   710
text{*Same theorem for lower bound*}
huffman@21164
   711
huffman@21164
   712
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   713
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164
   714
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   715
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164
   716
prefer 2 apply (blast intro: isCont_minus)
huffman@21164
   717
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164
   718
apply safe
huffman@21164
   719
apply auto
huffman@21164
   720
done
huffman@21164
   721
huffman@21164
   722
huffman@21164
   723
text{*Another version.*}
huffman@21164
   724
huffman@21164
   725
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   726
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164
   727
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164
   728
apply (frule isCont_eq_Lb)
huffman@21164
   729
apply (frule_tac [2] isCont_eq_Ub)
huffman@21164
   730
apply (assumption+, safe)
huffman@21164
   731
apply (rule_tac x = "f x" in exI)
huffman@21164
   732
apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164
   733
apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164
   734
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164
   735
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164
   736
apply (rule_tac [2] x = xb in exI)
huffman@21164
   737
apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164
   738
done
huffman@21164
   739
huffman@21164
   740
huffman@21164
   741
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   742
huffman@21164
   743
lemma DERIV_left_inc:
huffman@21164
   744
  fixes f :: "real => real"
huffman@21164
   745
  assumes der: "DERIV f x :> l"
huffman@21164
   746
      and l:   "0 < l"
huffman@21164
   747
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   748
proof -
huffman@21164
   749
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   750
  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
   751
    by (simp add: diff_minus)
huffman@21164
   752
  then obtain s
huffman@21164
   753
        where s:   "0 < s"
huffman@21164
   754
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   755
    by auto
huffman@21164
   756
  thus ?thesis
huffman@21164
   757
  proof (intro exI conjI strip)
huffman@23441
   758
    show "0<s" using s .
huffman@21164
   759
    fix h::real
huffman@21164
   760
    assume "0 < h" "h < s"
huffman@21164
   761
    with all [of h] show "f x < f (x+h)"
huffman@21164
   762
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   763
    split add: split_if_asm)
huffman@21164
   764
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   765
      with l
huffman@21164
   766
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   767
      thus "f x < f (x+h)"
huffman@21164
   768
  by (simp add: pos_less_divide_eq h)
huffman@21164
   769
    qed
huffman@21164
   770
  qed
huffman@21164
   771
qed
huffman@21164
   772
huffman@21164
   773
lemma DERIV_left_dec:
huffman@21164
   774
  fixes f :: "real => real"
huffman@21164
   775
  assumes der: "DERIV f x :> l"
huffman@21164
   776
      and l:   "l < 0"
huffman@21164
   777
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   778
proof -
huffman@21164
   779
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   780
  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
   781
    by (simp add: diff_minus)
huffman@21164
   782
  then obtain s
huffman@21164
   783
        where s:   "0 < s"
huffman@21164
   784
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   785
    by auto
huffman@21164
   786
  thus ?thesis
huffman@21164
   787
  proof (intro exI conjI strip)
huffman@23441
   788
    show "0<s" using s .
huffman@21164
   789
    fix h::real
huffman@21164
   790
    assume "0 < h" "h < s"
huffman@21164
   791
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
   792
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   793
    split add: split_if_asm)
huffman@21164
   794
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   795
      with l
huffman@21164
   796
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   797
      thus "f x < f (x-h)"
huffman@21164
   798
  by (simp add: pos_less_divide_eq h)
huffman@21164
   799
    qed
huffman@21164
   800
  qed
huffman@21164
   801
qed
huffman@21164
   802
huffman@21164
   803
lemma DERIV_local_max:
huffman@21164
   804
  fixes f :: "real => real"
huffman@21164
   805
  assumes der: "DERIV f x :> l"
huffman@21164
   806
      and d:   "0 < d"
huffman@21164
   807
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   808
  shows "l = 0"
huffman@21164
   809
proof (cases rule: linorder_cases [of l 0])
huffman@23441
   810
  case equal thus ?thesis .
huffman@21164
   811
next
huffman@21164
   812
  case less
huffman@21164
   813
  from DERIV_left_dec [OF der less]
huffman@21164
   814
  obtain d' where d': "0 < d'"
huffman@21164
   815
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   816
  from real_lbound_gt_zero [OF d d']
huffman@21164
   817
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   818
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   819
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   820
next
huffman@21164
   821
  case greater
huffman@21164
   822
  from DERIV_left_inc [OF der greater]
huffman@21164
   823
  obtain d' where d': "0 < d'"
huffman@21164
   824
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   825
  from real_lbound_gt_zero [OF d d']
huffman@21164
   826
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   827
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   828
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   829
qed
huffman@21164
   830
huffman@21164
   831
huffman@21164
   832
text{*Similar theorem for a local minimum*}
huffman@21164
   833
lemma DERIV_local_min:
huffman@21164
   834
  fixes f :: "real => real"
huffman@21164
   835
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   836
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   837
huffman@21164
   838
huffman@21164
   839
text{*In particular, if a function is locally flat*}
huffman@21164
   840
lemma DERIV_local_const:
huffman@21164
   841
  fixes f :: "real => real"
huffman@21164
   842
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   843
by (auto dest!: DERIV_local_max)
huffman@21164
   844
huffman@21164
   845
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   846
lemma lemma_interval_lt:
huffman@21164
   847
     "[| a < x;  x < b |]
huffman@21164
   848
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
   849
huffman@22998
   850
apply (simp add: abs_less_iff)
huffman@21164
   851
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   852
apply (rule_tac x = "x-a" in exI)
huffman@21164
   853
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   854
done
huffman@21164
   855
huffman@21164
   856
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   857
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   858
apply (drule lemma_interval_lt, auto)
huffman@21164
   859
apply (auto intro!: exI)
huffman@21164
   860
done
huffman@21164
   861
huffman@21164
   862
text{*Rolle's Theorem.
huffman@21164
   863
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   864
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
   865
   and @{term "f(a) = f(b)"},
huffman@21164
   866
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
   867
theorem Rolle:
huffman@21164
   868
  assumes lt: "a < b"
huffman@21164
   869
      and eq: "f(a) = f(b)"
huffman@21164
   870
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   871
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   872
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
   873
proof -
huffman@21164
   874
  have le: "a \<le> b" using lt by simp
huffman@21164
   875
  from isCont_eq_Ub [OF le con]
huffman@21164
   876
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
   877
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
   878
    by blast
huffman@21164
   879
  from isCont_eq_Lb [OF le con]
huffman@21164
   880
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
   881
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
   882
    by blast
huffman@21164
   883
  show ?thesis
huffman@21164
   884
  proof cases
huffman@21164
   885
    assume axb: "a < x & x < b"
huffman@21164
   886
        --{*@{term f} attains its maximum within the interval*}
chaieb@27668
   887
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
   888
    from lemma_interval [OF ax xb]
huffman@21164
   889
    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
   890
      by blast
huffman@21164
   891
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
   892
      by blast
huffman@21164
   893
    from differentiableD [OF dif [OF axb]]
huffman@21164
   894
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   895
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
   896
        --{*the derivative at a local maximum is zero*}
huffman@21164
   897
    thus ?thesis using ax xb der by auto
huffman@21164
   898
  next
huffman@21164
   899
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
   900
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
   901
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
   902
    show ?thesis
huffman@21164
   903
    proof cases
huffman@21164
   904
      assume ax'b: "a < x' & x' < b"
huffman@21164
   905
        --{*@{term f} attains its minimum within the interval*}
chaieb@27668
   906
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
   907
      from lemma_interval [OF ax' x'b]
huffman@21164
   908
      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
   909
  by blast
huffman@21164
   910
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
   911
  by blast
huffman@21164
   912
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
   913
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
   914
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
   915
        --{*the derivative at a local minimum is zero*}
huffman@21164
   916
      thus ?thesis using ax' x'b der by auto
huffman@21164
   917
    next
huffman@21164
   918
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
   919
        --{*@{term f} is constant througout the interval*}
huffman@21164
   920
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
   921
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
   922
      from dense [OF lt]
huffman@21164
   923
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
   924
      from lemma_interval [OF ar rb]
huffman@21164
   925
      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
   926
  by blast
huffman@21164
   927
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
   928
      proof (clarify)
huffman@21164
   929
        fix z::real
huffman@21164
   930
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
   931
        show "f z = f b"
huffman@21164
   932
        proof (rule order_antisym)
huffman@21164
   933
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
   934
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
   935
        qed
huffman@21164
   936
      qed
huffman@21164
   937
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
   938
      proof (intro strip)
huffman@21164
   939
        fix y::real
huffman@21164
   940
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
   941
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
   942
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
   943
      qed
huffman@21164
   944
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
   945
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
   946
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
   947
        --{*the derivative of a constant function is zero*}
huffman@21164
   948
      thus ?thesis using ar rb der by auto
huffman@21164
   949
    qed
huffman@21164
   950
  qed
huffman@21164
   951
qed
huffman@21164
   952
huffman@21164
   953
huffman@21164
   954
subsection{*Mean Value Theorem*}
huffman@21164
   955
huffman@21164
   956
lemma lemma_MVT:
huffman@21164
   957
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164
   958
proof cases
huffman@21164
   959
  assume "a=b" thus ?thesis by simp
huffman@21164
   960
next
huffman@21164
   961
  assume "a\<noteq>b"
huffman@21164
   962
  hence ba: "b-a \<noteq> 0" by arith
huffman@21164
   963
  show ?thesis
huffman@21164
   964
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164
   965
        simp add: right_diff_distrib,
huffman@21164
   966
        simp add: left_diff_distrib)
huffman@21164
   967
qed
huffman@21164
   968
huffman@21164
   969
theorem MVT:
huffman@21164
   970
  assumes lt:  "a < b"
huffman@21164
   971
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   972
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   973
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
   974
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
   975
proof -
huffman@21164
   976
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@21164
   977
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
huffman@23069
   978
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
huffman@21164
   979
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
   980
  proof (clarify)
huffman@21164
   981
    fix x::real
huffman@21164
   982
    assume ax: "a < x" and xb: "x < b"
huffman@21164
   983
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
   984
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   985
    show "?F differentiable x"
huffman@21164
   986
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
   987
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
   988
  qed
huffman@21164
   989
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
   990
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
   991
    by blast
huffman@21164
   992
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
   993
    by (rule DERIV_cmult_Id)
huffman@21164
   994
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
   995
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
   996
    by (rule DERIV_add [OF der])
huffman@21164
   997
  show ?thesis
huffman@21164
   998
  proof (intro exI conjI)
huffman@23441
   999
    show "a < z" using az .
huffman@23441
  1000
    show "z < b" using zb .
huffman@21164
  1001
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1002
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1003
  qed
huffman@21164
  1004
qed
huffman@21164
  1005
huffman@21164
  1006
huffman@21164
  1007
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1008
huffman@21164
  1009
lemma DERIV_isconst_end:
huffman@21164
  1010
  fixes f :: "real => real"
huffman@21164
  1011
  shows "[| a < b;
huffman@21164
  1012
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1013
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1014
        ==> f b = f a"
huffman@21164
  1015
apply (drule MVT, assumption)
huffman@21164
  1016
apply (blast intro: differentiableI)
huffman@21164
  1017
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1018
done
huffman@21164
  1019
huffman@21164
  1020
lemma DERIV_isconst1:
huffman@21164
  1021
  fixes f :: "real => real"
huffman@21164
  1022
  shows "[| a < b;
huffman@21164
  1023
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1024
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1025
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1026
apply safe
huffman@21164
  1027
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1028
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1029
done
huffman@21164
  1030
huffman@21164
  1031
lemma DERIV_isconst2:
huffman@21164
  1032
  fixes f :: "real => real"
huffman@21164
  1033
  shows "[| a < b;
huffman@21164
  1034
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1035
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1036
         a \<le> x; x \<le> b |]
huffman@21164
  1037
        ==> f x = f a"
huffman@21164
  1038
apply (blast dest: DERIV_isconst1)
huffman@21164
  1039
done
huffman@21164
  1040
huffman@21164
  1041
lemma DERIV_isconst_all:
huffman@21164
  1042
  fixes f :: "real => real"
huffman@21164
  1043
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1044
apply (rule linorder_cases [of x y])
huffman@21164
  1045
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1046
done
huffman@21164
  1047
huffman@21164
  1048
lemma DERIV_const_ratio_const:
huffman@21784
  1049
  fixes f :: "real => real"
huffman@21784
  1050
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1051
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1052
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1053
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
nipkow@23477
  1054
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
huffman@21164
  1055
done
huffman@21164
  1056
huffman@21164
  1057
lemma DERIV_const_ratio_const2:
huffman@21784
  1058
  fixes f :: "real => real"
huffman@21784
  1059
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1060
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1061
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1062
done
huffman@21164
  1063
huffman@21164
  1064
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1065
by (simp)
huffman@21164
  1066
huffman@21164
  1067
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1068
by (simp)
huffman@21164
  1069
huffman@21164
  1070
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1071
huffman@21164
  1072
lemma DERIV_const_average:
huffman@21164
  1073
  fixes v :: "real => real"
huffman@21164
  1074
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1075
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1076
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1077
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1078
  case equal with neq show ?thesis by simp
huffman@21164
  1079
next
huffman@21164
  1080
  case less
huffman@21164
  1081
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1082
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1083
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1084
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1085
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1086
  ultimately show ?thesis using neq by force
huffman@21164
  1087
next
huffman@21164
  1088
  case greater
huffman@21164
  1089
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1090
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1091
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1092
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1093
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1094
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1095
qed
huffman@21164
  1096
huffman@21164
  1097
huffman@21164
  1098
text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164
  1099
strict maximum at an end point, not in the middle.*}
huffman@21164
  1100
huffman@21164
  1101
lemma lemma_isCont_inj:
huffman@21164
  1102
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1103
  assumes d: "0 < d"
huffman@21164
  1104
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1105
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1106
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164
  1107
proof (rule ccontr)
huffman@21164
  1108
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164
  1109
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164
  1110
  show False
huffman@21164
  1111
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164
  1112
    case le
huffman@21164
  1113
    from d cont all [of "x+d"]
huffman@21164
  1114
    have flef: "f(x+d) \<le> f x"
huffman@21164
  1115
     and xlex: "x - d \<le> x"
huffman@21164
  1116
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164
  1117
       by (auto simp add: abs_if)
huffman@21164
  1118
    from IVT [OF le flef xlex cont']
huffman@21164
  1119
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164
  1120
    moreover
huffman@21164
  1121
    hence "g(f x') = g (f(x+d))" by simp
huffman@21164
  1122
    ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@22998
  1123
      by (simp add: abs_le_iff)
huffman@21164
  1124
  next
huffman@21164
  1125
    case ge
huffman@21164
  1126
    from d cont all [of "x-d"]
huffman@21164
  1127
    have flef: "f(x-d) \<le> f x"
huffman@21164
  1128
     and xlex: "x \<le> x+d"
huffman@21164
  1129
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164
  1130
       by (auto simp add: abs_if)
huffman@21164
  1131
    from IVT2 [OF ge flef xlex cont']
huffman@21164
  1132
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164
  1133
    moreover
huffman@21164
  1134
    hence "g(f x') = g (f(x-d))" by simp
huffman@21164
  1135
    ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@22998
  1136
      by (simp add: abs_le_iff)
huffman@21164
  1137
  qed
huffman@21164
  1138
qed
huffman@21164
  1139
huffman@21164
  1140
huffman@21164
  1141
text{*Similar version for lower bound.*}
huffman@21164
  1142
huffman@21164
  1143
lemma lemma_isCont_inj2:
huffman@21164
  1144
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1145
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164
  1146
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164
  1147
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164
  1148
apply (insert lemma_isCont_inj
huffman@21164
  1149
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@21164
  1150
apply (simp add: isCont_minus linorder_not_le)
huffman@21164
  1151
done
huffman@21164
  1152
huffman@21164
  1153
text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164
  1154
@{text "f[[x - d, x + d]]"} .*}
huffman@21164
  1155
huffman@21164
  1156
lemma isCont_inj_range:
huffman@21164
  1157
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1158
  assumes d: "0 < d"
huffman@21164
  1159
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1160
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1161
  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
  1162
proof -
huffman@21164
  1163
  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
  1164
    by (auto simp add: abs_le_iff)
huffman@21164
  1165
  from isCont_Lb_Ub [OF this]
huffman@21164
  1166
  obtain L M
huffman@21164
  1167
  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
  1168
    and all2 [rule_format]:
huffman@21164
  1169
           "\<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
  1170
    by auto
huffman@21164
  1171
  with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164
  1172
  moreover have "L \<noteq> f x"
huffman@21164
  1173
  proof -
huffman@21164
  1174
    from lemma_isCont_inj2 [OF d inj cont]
huffman@21164
  1175
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164
  1176
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1177
  qed
huffman@21164
  1178
  moreover have "f x \<noteq> M"
huffman@21164
  1179
  proof -
huffman@21164
  1180
    from lemma_isCont_inj [OF d inj cont]
huffman@21164
  1181
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164
  1182
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1183
  qed
huffman@21164
  1184
  ultimately have "L < f x & f x < M" by arith
huffman@21164
  1185
  hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164
  1186
  from real_lbound_gt_zero [OF this]
huffman@21164
  1187
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164
  1188
  thus ?thesis
huffman@21164
  1189
  proof (intro exI conjI)
huffman@23441
  1190
    show "0<e" using e(1) .
huffman@21164
  1191
    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
  1192
    proof (intro strip)
huffman@21164
  1193
      fix y::real
huffman@21164
  1194
      assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164
  1195
      with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164
  1196
      from all2 [OF this]
huffman@21164
  1197
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
chaieb@27668
  1198
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" 
huffman@22998
  1199
        by (force simp add: abs_le_iff)
huffman@21164
  1200
    qed
huffman@21164
  1201
  qed
huffman@21164
  1202
qed
huffman@21164
  1203
huffman@21164
  1204
huffman@21164
  1205
text{*Continuity of inverse function*}
huffman@21164
  1206
huffman@21164
  1207
lemma isCont_inverse_function:
huffman@21164
  1208
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1209
  assumes d: "0 < d"
huffman@21164
  1210
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1211
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1212
  shows "isCont g (f x)"
huffman@21164
  1213
proof (simp add: isCont_iff LIM_eq)
huffman@21164
  1214
  show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164
  1215
         (\<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
  1216
  proof (intro strip)
huffman@21164
  1217
    fix r::real
huffman@21164
  1218
    assume r: "0<r"
huffman@21164
  1219
    from real_lbound_gt_zero [OF r d]
huffman@21164
  1220
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164
  1221
    with inj cont
huffman@21164
  1222
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164
  1223
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164
  1224
    from isCont_inj_range [OF e this]
huffman@21164
  1225
    obtain e' where e': "0 < e'"
huffman@21164
  1226
        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
  1227
          by blast
huffman@21164
  1228
    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
  1229
    proof (intro exI conjI)
huffman@23441
  1230
      show "0<e'" using e' .
huffman@21164
  1231
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164
  1232
      proof (intro strip)
huffman@21164
  1233
        fix z::real
huffman@21164
  1234
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164
  1235
        with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164
  1236
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164
  1237
      qed
huffman@21164
  1238
    qed
huffman@21164
  1239
  qed
huffman@21164
  1240
qed
huffman@21164
  1241
huffman@23041
  1242
text {* Derivative of inverse function *}
huffman@23041
  1243
huffman@23041
  1244
lemma DERIV_inverse_function:
huffman@23041
  1245
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1246
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1247
  assumes neq: "D \<noteq> 0"
huffman@23044
  1248
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1249
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1250
  assumes cont: "isCont g x"
huffman@23041
  1251
  shows "DERIV g x :> inverse D"
huffman@23041
  1252
unfolding DERIV_iff2
huffman@23044
  1253
proof (rule LIM_equal2)
huffman@23044
  1254
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1255
    using a b by arith 
huffman@23044
  1256
next
huffman@23041
  1257
  fix y
huffman@23044
  1258
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1259
  hence "a < y" and "y < b" 
huffman@23044
  1260
    by (simp_all add: abs_less_iff)
huffman@23041
  1261
  thus "(g y - g x) / (y - x) =
huffman@23041
  1262
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1263
    by (simp add: inj)
huffman@23041
  1264
next
huffman@23041
  1265
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1266
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1267
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1268
    using inj a b by simp
huffman@23041
  1269
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
huffman@23041
  1270
  proof (safe intro!: exI)
huffman@23044
  1271
    show "0 < min (x - a) (b - x)"
huffman@23044
  1272
      using a b by simp
huffman@23041
  1273
  next
huffman@23041
  1274
    fix y
huffman@23044
  1275
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1276
    hence y: "a < y" "y < b"
huffman@23044
  1277
      by (simp_all add: abs_less_iff)
huffman@23041
  1278
    assume "g y = g x"
huffman@23041
  1279
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1280
    hence "y = x" using inj y a b by simp
huffman@23041
  1281
    also assume "y \<noteq> x"
huffman@23041
  1282
    finally show False by simp
huffman@23041
  1283
  qed
huffman@23041
  1284
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1285
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1286
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1287
        -- x --> inverse D"
huffman@23041
  1288
    using neq by (rule LIM_inverse)
huffman@23041
  1289
qed
huffman@23041
  1290
huffman@21164
  1291
theorem GMVT:
huffman@21784
  1292
  fixes a b :: real
huffman@21164
  1293
  assumes alb: "a < b"
huffman@21164
  1294
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
huffman@21164
  1295
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
huffman@21164
  1296
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
huffman@21164
  1297
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1298
  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
  1299
proof -
huffman@21164
  1300
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
huffman@21164
  1301
  from prems have "a < b" by simp
huffman@21164
  1302
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@21164
  1303
  proof -
huffman@21164
  1304
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
huffman@21164
  1305
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
huffman@21164
  1306
      by (auto intro: isCont_mult)
huffman@21164
  1307
    moreover
huffman@21164
  1308
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
huffman@21164
  1309
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
huffman@21164
  1310
      by (auto intro: isCont_mult)
huffman@21164
  1311
    ultimately show ?thesis
huffman@21164
  1312
      by (fastsimp intro: isCont_diff)
huffman@21164
  1313
  qed
huffman@21164
  1314
  moreover
huffman@21164
  1315
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@21164
  1316
  proof -
huffman@21164
  1317
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1318
    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
  1319
    moreover
huffman@21164
  1320
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1321
    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
  1322
    ultimately show ?thesis by (simp add: differentiable_diff)
huffman@21164
  1323
  qed
huffman@21164
  1324
  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
  1325
  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
  1326
  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
  1327
huffman@21164
  1328
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1329
  with gd have "g differentiable c" by simp
huffman@21164
  1330
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1331
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1332
huffman@21164
  1333
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1334
  with fd have "f differentiable c" by simp
huffman@21164
  1335
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1336
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1337
huffman@21164
  1338
  from cdef have "DERIV ?h c :> l" by auto
huffman@21164
  1339
  moreover
huffman@21164
  1340
  {
huffman@23441
  1341
    have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
huffman@21164
  1342
      apply (insert DERIV_const [where k="f b - f a"])
huffman@21164
  1343
      apply (drule meta_spec [of _ c])
huffman@23441
  1344
      apply (drule DERIV_mult [OF _ g'cdef])
huffman@23441
  1345
      by simp
huffman@23441
  1346
    moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
huffman@21164
  1347
      apply (insert DERIV_const [where k="g b - g a"])
huffman@21164
  1348
      apply (drule meta_spec [of _ c])
huffman@23441
  1349
      apply (drule DERIV_mult [OF _ f'cdef])
huffman@23441
  1350
      by simp
huffman@21164
  1351
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
huffman@21164
  1352
      by (simp add: DERIV_diff)
huffman@21164
  1353
  }
huffman@21164
  1354
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1355
huffman@21164
  1356
  {
huffman@21164
  1357
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1358
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1359
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1360
  }
huffman@21164
  1361
  moreover
huffman@21164
  1362
  {
huffman@21164
  1363
    have "?h b - ?h a =
huffman@21164
  1364
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1365
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
huffman@22998
  1366
      by (simp add: mult_ac add_ac right_diff_distrib)
huffman@21164
  1367
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1368
  }
huffman@21164
  1369
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1370
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1371
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1372
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1373
huffman@21164
  1374
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1375
qed
huffman@21164
  1376
chaieb@23255
  1377
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
chaieb@23255
  1378
by auto
chaieb@23255
  1379
chaieb@26120
  1380
subsection {* Derivatives of univariate polynomials *}
chaieb@26120
  1381
chaieb@26120
  1382
chaieb@26120
  1383
  
chaieb@26120
  1384
primrec pderiv_aux :: "nat => real list => real list" where
chaieb@26120
  1385
   pderiv_aux_Nil:  "pderiv_aux n [] = []"
chaieb@26120
  1386
|  pderiv_aux_Cons: "pderiv_aux n (h#t) =
chaieb@26120
  1387
                     (real n * h)#(pderiv_aux (Suc n) t)"
chaieb@26120
  1388
chaieb@26120
  1389
definition
chaieb@26120
  1390
  pderiv :: "real list => real list" where
chaieb@26120
  1391
  "pderiv p = (if p = [] then [] else pderiv_aux 1 (tl p))"
chaieb@26120
  1392
chaieb@26120
  1393
chaieb@26120
  1394
text{*The derivative*}
chaieb@26120
  1395
chaieb@26120
  1396
lemma pderiv_Nil: "pderiv [] = []"
chaieb@26120
  1397
chaieb@26120
  1398
apply (simp add: pderiv_def)
chaieb@26120
  1399
done
chaieb@26120
  1400
declare pderiv_Nil [simp]
chaieb@26120
  1401
chaieb@26120
  1402
lemma pderiv_singleton: "pderiv [c] = []"
chaieb@26120
  1403
by (simp add: pderiv_def)
chaieb@26120
  1404
declare pderiv_singleton [simp]
chaieb@26120
  1405
chaieb@26120
  1406
lemma pderiv_Cons: "pderiv (h#t) = pderiv_aux 1 t"
chaieb@26120
  1407
by (simp add: pderiv_def)
chaieb@26120
  1408
chaieb@26120
  1409
lemma DERIV_cmult2: "DERIV f x :> D ==> DERIV (%x. (f x) * c :: real) x :> D * c"
chaieb@26120
  1410
by (simp add: DERIV_cmult mult_commute [of _ c])
chaieb@26120
  1411
chaieb@26120
  1412
lemma DERIV_pow2: "DERIV (%x. x ^ Suc n) x :> real (Suc n) * (x ^ n)"
chaieb@26120
  1413
by (rule lemma_DERIV_subst, rule DERIV_pow, simp)
chaieb@26120
  1414
declare DERIV_pow2 [simp] DERIV_pow [simp]
chaieb@26120
  1415
chaieb@26120
  1416
lemma lemma_DERIV_poly1: "\<forall>n. DERIV (%x. (x ^ (Suc n) * poly p x)) x :>
chaieb@26120
  1417
        x ^ n * poly (pderiv_aux (Suc n) p) x "
chaieb@26120
  1418
apply (induct "p")
chaieb@26120
  1419
apply (auto intro!: DERIV_add DERIV_cmult2 
chaieb@26120
  1420
            simp add: pderiv_def right_distrib real_mult_assoc [symmetric] 
chaieb@26120
  1421
            simp del: realpow_Suc)
chaieb@26120
  1422
apply (subst mult_commute) 
chaieb@26120
  1423
apply (simp del: realpow_Suc) 
chaieb@26120
  1424
apply (simp add: mult_commute realpow_Suc [symmetric] del: realpow_Suc)
chaieb@26120
  1425
done
chaieb@26120
  1426
chaieb@26120
  1427
lemma lemma_DERIV_poly: "DERIV (%x. (x ^ (Suc n) * poly p x)) x :>
chaieb@26120
  1428
        x ^ n * poly (pderiv_aux (Suc n) p) x "
chaieb@26120
  1429
by (simp add: lemma_DERIV_poly1 del: realpow_Suc)
chaieb@26120
  1430
chaieb@26120
  1431
lemma DERIV_add_const: "DERIV f x :> D ==>  DERIV (%x. a + f x :: real) x :> D"
chaieb@26120
  1432
by (rule lemma_DERIV_subst, rule DERIV_add, auto)
chaieb@26120
  1433
chaieb@26120
  1434
lemma poly_DERIV[simp]: "DERIV (%x. poly p x) x :> poly (pderiv p) x"
chaieb@26120
  1435
apply (induct "p")
chaieb@26120
  1436
apply (auto simp add: pderiv_Cons)
chaieb@26120
  1437
apply (rule DERIV_add_const)
chaieb@26120
  1438
apply (rule lemma_DERIV_subst)
chaieb@26120
  1439
apply (rule lemma_DERIV_poly [where n=0, simplified], simp) 
chaieb@26120
  1440
done
chaieb@26120
  1441
chaieb@26120
  1442
chaieb@26120
  1443
text{* Consequences of the derivative theorem above*}
chaieb@26120
  1444
chaieb@26120
  1445
lemma poly_differentiable[simp]: "(%x. poly p x) differentiable (x::real)"
chaieb@26120
  1446
apply (simp add: differentiable_def)
chaieb@26120
  1447
apply (blast intro: poly_DERIV)
chaieb@26120
  1448
done
chaieb@26120
  1449
chaieb@26120
  1450
lemma poly_isCont[simp]: "isCont (%x. poly p x) (x::real)"
chaieb@26120
  1451
by (rule poly_DERIV [THEN DERIV_isCont])
chaieb@26120
  1452
chaieb@26120
  1453
lemma poly_IVT_pos: "[| a < b; poly p (a::real) < 0; 0 < poly p b |]
chaieb@26120
  1454
      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
chaieb@26120
  1455
apply (cut_tac f = "%x. poly p x" and a = a and b = b and y = 0 in IVT_objl)
chaieb@26120
  1456
apply (auto simp add: order_le_less)
chaieb@26120
  1457
done
chaieb@26120
  1458
chaieb@26120
  1459
lemma poly_IVT_neg: "[| (a::real) < b; 0 < poly p a; poly p b < 0 |]
chaieb@26120
  1460
      ==> \<exists>x. a < x & x < b & (poly p x = 0)"
chaieb@26120
  1461
apply (insert poly_IVT_pos [where p = "-- p" ]) 
chaieb@26120
  1462
apply (simp add: poly_minus neg_less_0_iff_less) 
chaieb@26120
  1463
done
chaieb@26120
  1464
chaieb@26120
  1465
lemma poly_MVT: "a < b ==>
chaieb@26120
  1466
     \<exists>x. a < x & x < b & (poly p b - poly p a = (b - a) * poly (pderiv p) x)"
chaieb@26120
  1467
apply (drule_tac f = "poly p" in MVT, auto)
chaieb@26120
  1468
apply (rule_tac x = z in exI)
chaieb@26120
  1469
apply (auto simp add: real_mult_left_cancel poly_DERIV [THEN DERIV_unique])
chaieb@26120
  1470
done
chaieb@26120
  1471
chaieb@26120
  1472
text{*Lemmas for Derivatives*}
chaieb@26120
  1473
chaieb@26120
  1474
lemma lemma_poly_pderiv_aux_add: "\<forall>p2 n. poly (pderiv_aux n (p1 +++ p2)) x =
chaieb@26120
  1475
                poly (pderiv_aux n p1 +++ pderiv_aux n p2) x"
chaieb@26120
  1476
apply (induct "p1", simp, clarify) 
chaieb@26120
  1477
apply (case_tac "p2")
chaieb@26120
  1478
apply (auto simp add: right_distrib)
chaieb@26120
  1479
done
chaieb@26120
  1480
chaieb@26120
  1481
lemma poly_pderiv_aux_add: "poly (pderiv_aux n (p1 +++ p2)) x =
chaieb@26120
  1482
      poly (pderiv_aux n p1 +++ pderiv_aux n p2) x"
chaieb@26120
  1483
apply (simp add: lemma_poly_pderiv_aux_add)
chaieb@26120
  1484
done
chaieb@26120
  1485
chaieb@26120
  1486
lemma lemma_poly_pderiv_aux_cmult: "\<forall>n. poly (pderiv_aux n (c %* p)) x = poly (c %* pderiv_aux n p) x"
chaieb@26120
  1487
apply (induct "p")
chaieb@26120
  1488
apply (auto simp add: poly_cmult mult_ac)
chaieb@26120
  1489
done
chaieb@26120
  1490
chaieb@26120
  1491
lemma poly_pderiv_aux_cmult: "poly (pderiv_aux n (c %* p)) x = poly (c %* pderiv_aux n p) x"
chaieb@26120
  1492
by (simp add: lemma_poly_pderiv_aux_cmult)
chaieb@26120
  1493
chaieb@26120
  1494
lemma poly_pderiv_aux_minus:
chaieb@26120
  1495
   "poly (pderiv_aux n (-- p)) x = poly (-- pderiv_aux n p) x"
chaieb@26120
  1496
apply (simp add: poly_minus_def poly_pderiv_aux_cmult)
chaieb@26120
  1497
done
chaieb@26120
  1498
chaieb@26120
  1499
lemma lemma_poly_pderiv_aux_mult1: "\<forall>n. poly (pderiv_aux (Suc n) p) x = poly ((pderiv_aux n p) +++ p) x"
chaieb@26120
  1500
apply (induct "p")
chaieb@26120
  1501
apply (auto simp add: real_of_nat_Suc left_distrib)
chaieb@26120
  1502
done
chaieb@26120
  1503
chaieb@26120
  1504
lemma lemma_poly_pderiv_aux_mult: "poly (pderiv_aux (Suc n) p) x = poly ((pderiv_aux n p) +++ p) x"
chaieb@26120
  1505
by (simp add: lemma_poly_pderiv_aux_mult1)
chaieb@26120
  1506
chaieb@26120
  1507
lemma lemma_poly_pderiv_add: "\<forall>q. poly (pderiv (p +++ q)) x = poly (pderiv p +++ pderiv q) x"
chaieb@26120
  1508
apply (induct "p", simp, clarify) 
chaieb@26120
  1509
apply (case_tac "q")
chaieb@26120
  1510
apply (auto simp add: poly_pderiv_aux_add poly_add pderiv_def)
chaieb@26120
  1511
done
chaieb@26120
  1512
chaieb@26120
  1513
lemma poly_pderiv_add: "poly (pderiv (p +++ q)) x = poly (pderiv p +++ pderiv q) x"
chaieb@26120
  1514
by (simp add: lemma_poly_pderiv_add)
chaieb@26120
  1515
chaieb@26120
  1516
lemma poly_pderiv_cmult: "poly (pderiv (c %* p)) x = poly (c %* (pderiv p)) x"
chaieb@26120
  1517
apply (induct "p")
chaieb@26120
  1518
apply (auto simp add: poly_pderiv_aux_cmult poly_cmult pderiv_def)
chaieb@26120
  1519
done
chaieb@26120
  1520
chaieb@26120
  1521
lemma poly_pderiv_minus: "poly (pderiv (--p)) x = poly (--(pderiv p)) x"
chaieb@26120
  1522
by (simp add: poly_minus_def poly_pderiv_cmult)
chaieb@26120
  1523
chaieb@26120
  1524
lemma lemma_poly_mult_pderiv:
chaieb@26120
  1525
   "poly (pderiv (h#t)) x = poly ((0 # (pderiv t)) +++ t) x"
chaieb@26120
  1526
apply (simp add: pderiv_def)
chaieb@26120
  1527
apply (induct "t")
chaieb@26120
  1528
apply (auto simp add: poly_add lemma_poly_pderiv_aux_mult)
chaieb@26120
  1529
done
chaieb@26120
  1530
chaieb@26120
  1531
lemma poly_pderiv_mult: "\<forall>q. poly (pderiv (p *** q)) x =
chaieb@26120
  1532
      poly (p *** (pderiv q) +++ q *** (pderiv p)) x"
chaieb@26120
  1533
apply (induct "p")
chaieb@26120
  1534
apply (auto simp add: poly_add poly_cmult poly_pderiv_cmult poly_pderiv_add poly_mult)
chaieb@26120
  1535
apply (rule lemma_poly_mult_pderiv [THEN ssubst])
chaieb@26120
  1536
apply (rule lemma_poly_mult_pderiv [THEN ssubst])
chaieb@26120
  1537
apply (rule poly_add [THEN ssubst])
chaieb@26120
  1538
apply (rule poly_add [THEN ssubst])
chaieb@26120
  1539
apply (simp (no_asm_simp) add: poly_mult right_distrib add_ac mult_ac)
chaieb@26120
  1540
done
chaieb@26120
  1541
chaieb@26120
  1542
lemma poly_pderiv_exp: "poly (pderiv (p %^ (Suc n))) x =
chaieb@26120
  1543
         poly ((real (Suc n)) %* (p %^ n) *** pderiv p) x"
chaieb@26120
  1544
apply (induct "n")
chaieb@26120
  1545
apply (auto simp add: poly_add poly_pderiv_cmult poly_cmult poly_pderiv_mult
chaieb@26120
  1546
                      real_of_nat_zero poly_mult real_of_nat_Suc 
chaieb@26120
  1547
                      right_distrib left_distrib mult_ac)
chaieb@26120
  1548
done
chaieb@26120
  1549
chaieb@26120
  1550
lemma poly_pderiv_exp_prime: "poly (pderiv ([-a, 1] %^ (Suc n))) x =
chaieb@26120
  1551
      poly (real (Suc n) %* ([-a, 1] %^ n)) x"
chaieb@26120
  1552
apply (simp add: poly_pderiv_exp poly_mult del: pexp_Suc)
chaieb@26120
  1553
apply (simp add: poly_cmult pderiv_def)
chaieb@26120
  1554
done
chaieb@26120
  1555
chaieb@26120
  1556
chaieb@26120
  1557
lemma real_mult_zero_disj_iff[simp]: "(x * y = 0) = (x = (0::real) | y = 0)"
chaieb@26120
  1558
by simp
chaieb@26120
  1559
chaieb@26120
  1560
lemma pderiv_aux_iszero [rule_format, simp]:
chaieb@26120
  1561
    "\<forall>n. list_all (%c. c = 0) (pderiv_aux (Suc n) p) = list_all (%c. c = 0) p"
chaieb@26120
  1562
by (induct "p", auto)
chaieb@26120
  1563
chaieb@26120
  1564
lemma pderiv_aux_iszero_num: "(number_of n :: nat) \<noteq> 0
chaieb@26120
  1565
      ==> (list_all (%c. c = 0) (pderiv_aux (number_of n) p) =
chaieb@26120
  1566
      list_all (%c. c = 0) p)"
chaieb@26120
  1567
unfolding neq0_conv
chaieb@26120
  1568
apply (rule_tac n1 = "number_of n" and m1 = 0 in less_imp_Suc_add [THEN exE], force)
chaieb@26120
  1569
apply (rule_tac n1 = "0 + x" in pderiv_aux_iszero [THEN subst])
chaieb@26120
  1570
apply (simp (no_asm_simp) del: pderiv_aux_iszero)
chaieb@26120
  1571
done
chaieb@26120
  1572
chaieb@26120
  1573
instance real:: idom_char_0
chaieb@26120
  1574
apply (intro_classes)
chaieb@26120
  1575
done
chaieb@26120
  1576
chaieb@26120
  1577
instance real:: recpower_idom_char_0
chaieb@26120
  1578
apply (intro_classes)
chaieb@26120
  1579
done
chaieb@26120
  1580
chaieb@26120
  1581
lemma pderiv_iszero [rule_format]:
chaieb@26120
  1582
     "poly (pderiv p) = poly [] --> (\<exists>h. poly p = poly [h])"
chaieb@26120
  1583
apply (simp add: poly_zero)
chaieb@26120
  1584
apply (induct "p", force)
chaieb@26120
  1585
apply (simp add: pderiv_Cons pderiv_aux_iszero_num del: poly_Cons)
chaieb@26120
  1586
apply (auto simp add: poly_zero [symmetric])
chaieb@26120
  1587
done
chaieb@26120
  1588
chaieb@26120
  1589
lemma pderiv_zero_obj: "poly p = poly [] --> (poly (pderiv p) = poly [])"
chaieb@26120
  1590
apply (simp add: poly_zero)
chaieb@26120
  1591
apply (induct "p", force)
chaieb@26120
  1592
apply (simp add: pderiv_Cons pderiv_aux_iszero_num del: poly_Cons)
chaieb@26120
  1593
done
chaieb@26120
  1594
chaieb@26120
  1595
lemma pderiv_zero: "poly p = poly [] ==> (poly (pderiv p) = poly [])"
chaieb@26120
  1596
by (blast elim: pderiv_zero_obj [THEN impE])
chaieb@26120
  1597
declare pderiv_zero [simp]
chaieb@26120
  1598
chaieb@26120
  1599
lemma poly_pderiv_welldef: "poly p = poly q ==> (poly (pderiv p) = poly (pderiv q))"
chaieb@26120
  1600
apply (cut_tac p = "p +++ --q" in pderiv_zero_obj)
chaieb@26120
  1601
apply (simp add: fun_eq poly_add poly_minus poly_pderiv_add poly_pderiv_minus del: pderiv_zero)
chaieb@26120
  1602
done
chaieb@26120
  1603
chaieb@26120
  1604
lemma lemma_order_pderiv [rule_format]:
chaieb@26120
  1605
     "\<forall>p q a. 0 < n &
chaieb@26120
  1606
       poly (pderiv p) \<noteq> poly [] &
chaieb@26120
  1607
       poly p = poly ([- a, 1] %^ n *** q) & ~ [- a, 1] divides q
chaieb@26120
  1608
       --> n = Suc (order a (pderiv p))"
chaieb@26120
  1609
apply (induct "n", safe)
chaieb@26120
  1610
apply (rule order_unique_lemma, rule conjI, assumption)
chaieb@26120
  1611
apply (subgoal_tac "\<forall>r. r divides (pderiv p) = r divides (pderiv ([-a, 1] %^ Suc n *** q))")
chaieb@26120
  1612
apply (drule_tac [2] poly_pderiv_welldef)
chaieb@26120
  1613
 prefer 2 apply (simp add: divides_def del: pmult_Cons pexp_Suc) 
chaieb@26120
  1614
apply (simp del: pmult_Cons pexp_Suc) 
chaieb@26120
  1615
apply (rule conjI)
chaieb@26120
  1616
apply (simp add: divides_def fun_eq del: pmult_Cons pexp_Suc)
chaieb@26120
  1617
apply (rule_tac x = "[-a, 1] *** (pderiv q) +++ real (Suc n) %* q" in exI)
chaieb@26120
  1618
apply (simp add: poly_pderiv_mult poly_pderiv_exp_prime poly_add poly_mult poly_cmult right_distrib mult_ac del: pmult_Cons pexp_Suc)
chaieb@26120
  1619
apply (simp add: poly_mult right_distrib left_distrib mult_ac del: pmult_Cons)
chaieb@26120
  1620
apply (erule_tac V = "\<forall>r. r divides pderiv p = r divides pderiv ([- a, 1] %^ Suc n *** q)" in thin_rl)
chaieb@26120
  1621
apply (unfold divides_def)
chaieb@26120
  1622
apply (simp (no_asm) add: poly_pderiv_mult poly_pderiv_exp_prime fun_eq poly_add poly_mult del: pmult_Cons pexp_Suc)
chaieb@26120
  1623
apply (rule contrapos_np, assumption)
chaieb@26120
  1624
apply (rotate_tac 3, erule contrapos_np)
chaieb@26120
  1625
apply (simp del: pmult_Cons pexp_Suc, safe)
chaieb@26120
  1626
apply (rule_tac x = "inverse (real (Suc n)) %* (qa +++ -- (pderiv q))" in exI)
chaieb@26120
  1627
apply (subgoal_tac "poly ([-a, 1] %^ n *** q) = poly ([-a, 1] %^ n *** ([-a, 1] *** (inverse (real (Suc n)) %* (qa +++ -- (pderiv q))))) ")
chaieb@26120
  1628
apply (drule poly_mult_left_cancel [THEN iffD1], simp)
chaieb@26120
  1629
apply (simp add: fun_eq poly_mult poly_add poly_cmult poly_minus del: pmult_Cons mult_cancel_left, safe)
chaieb@26120
  1630
apply (rule_tac c1 = "real (Suc n)" in real_mult_left_cancel [THEN iffD1])
chaieb@26120
  1631
apply (simp (no_asm))
chaieb@26120
  1632
apply (subgoal_tac "real (Suc n) * (poly ([- a, 1] %^ n) xa * poly q xa) =
chaieb@26120
  1633
          (poly qa xa + - poly (pderiv q) xa) *
chaieb@26120
  1634
          (poly ([- a, 1] %^ n) xa *
chaieb@26120
  1635
           ((- a + xa) * (inverse (real (Suc n)) * real (Suc n))))")
chaieb@26120
  1636
apply (simp only: mult_ac)  
chaieb@26120
  1637
apply (rotate_tac 2)
chaieb@26120
  1638
apply (drule_tac x = xa in spec)
chaieb@26120
  1639
apply (simp add: left_distrib mult_ac del: pmult_Cons)
chaieb@26120
  1640
done
chaieb@26120
  1641
chaieb@26120
  1642
lemma order_pderiv: "[| poly (pderiv p) \<noteq> poly []; order a p \<noteq> 0 |]
chaieb@26120
  1643
      ==> (order a p = Suc (order a (pderiv p)))"
chaieb@26120
  1644
apply (case_tac "poly p = poly []")
chaieb@26120
  1645
apply (auto dest: pderiv_zero)
chaieb@26120
  1646
apply (drule_tac a = a and p = p in order_decomp)
chaieb@26120
  1647
using neq0_conv
chaieb@26120
  1648
apply (blast intro: lemma_order_pderiv)
chaieb@26120
  1649
done
chaieb@26120
  1650
chaieb@26120
  1651
text{*Now justify the standard squarefree decomposition, i.e. f / gcd(f,f'). *}
chaieb@26120
  1652
chaieb@26120
  1653
lemma poly_squarefree_decomp_order: "[| poly (pderiv p) \<noteq> poly [];
chaieb@26120
  1654
         poly p = poly (q *** d);
chaieb@26120
  1655
         poly (pderiv p) = poly (e *** d);
chaieb@26120
  1656
         poly d = poly (r *** p +++ s *** pderiv p)
chaieb@26120
  1657
      |] ==> order a q = (if order a p = 0 then 0 else 1)"
chaieb@26120
  1658
apply (subgoal_tac "order a p = order a q + order a d")
chaieb@26120
  1659
apply (rule_tac [2] s = "order a (q *** d)" in trans)
chaieb@26120
  1660
prefer 2 apply (blast intro: order_poly)
chaieb@26120
  1661
apply (rule_tac [2] order_mult)
chaieb@26120
  1662
 prefer 2 apply force
chaieb@26120
  1663
apply (case_tac "order a p = 0", simp)
chaieb@26120
  1664
apply (subgoal_tac "order a (pderiv p) = order a e + order a d")
chaieb@26120
  1665
apply (rule_tac [2] s = "order a (e *** d)" in trans)
chaieb@26120
  1666
prefer 2 apply (blast intro: order_poly)
chaieb@26120
  1667
apply (rule_tac [2] order_mult)
chaieb@26120
  1668
 prefer 2 apply force
chaieb@26120
  1669
apply (case_tac "poly p = poly []")
chaieb@26120
  1670
apply (drule_tac p = p in pderiv_zero, simp)
chaieb@26120
  1671
apply (drule order_pderiv, assumption)
chaieb@26120
  1672
apply (subgoal_tac "order a (pderiv p) \<le> order a d")
chaieb@26120
  1673
apply (subgoal_tac [2] " ([-a, 1] %^ (order a (pderiv p))) divides d")
chaieb@26120
  1674
 prefer 2 apply (simp add: poly_entire order_divides)
chaieb@26120
  1675
apply (subgoal_tac [2] " ([-a, 1] %^ (order a (pderiv p))) divides p & ([-a, 1] %^ (order a (pderiv p))) divides (pderiv p) ")
chaieb@26120
  1676
 prefer 3 apply (simp add: order_divides)
chaieb@26120
  1677
 prefer 2 apply (simp add: divides_def del: pexp_Suc pmult_Cons, safe)
chaieb@26120
  1678
apply (rule_tac x = "r *** qa +++ s *** qaa" in exI)
chaieb@26120
  1679
apply (simp add: fun_eq poly_add poly_mult left_distrib right_distrib mult_ac del: pexp_Suc pmult_Cons, auto)
chaieb@26120
  1680
done
chaieb@26120
  1681
chaieb@26120
  1682
chaieb@26120
  1683
lemma poly_squarefree_decomp_order2: "[| poly (pderiv p) \<noteq> poly [];
chaieb@26120
  1684
         poly p = poly (q *** d);
chaieb@26120
  1685
         poly (pderiv p) = poly (e *** d);
chaieb@26120
  1686
         poly d = poly (r *** p +++ s *** pderiv p)
chaieb@26120
  1687
      |] ==> \<forall>a. order a q = (if order a p = 0 then 0 else 1)"
chaieb@26120
  1688
apply (blast intro: poly_squarefree_decomp_order)
chaieb@26120
  1689
done
chaieb@26120
  1690
chaieb@26120
  1691
lemma order_pderiv2: "[| poly (pderiv p) \<noteq> poly []; order a p \<noteq> 0 |]
chaieb@26120
  1692
      ==> (order a (pderiv p) = n) = (order a p = Suc n)"
chaieb@26120
  1693
apply (auto dest: order_pderiv)
chaieb@26120
  1694
done
chaieb@26120
  1695
chaieb@26120
  1696
lemma rsquarefree_roots:
chaieb@26120
  1697
  "rsquarefree p = (\<forall>a. ~(poly p a = 0 & poly (pderiv p) a = 0))"
chaieb@26120
  1698
apply (simp add: rsquarefree_def)
chaieb@26120
  1699
apply (case_tac "poly p = poly []", simp, simp)
chaieb@26120
  1700
apply (case_tac "poly (pderiv p) = poly []")
chaieb@26120
  1701
apply simp
chaieb@26120
  1702
apply (drule pderiv_iszero, clarify)
chaieb@26120
  1703
apply (subgoal_tac "\<forall>a. order a p = order a [h]")
chaieb@26120
  1704
apply (simp add: fun_eq)
chaieb@26120
  1705
apply (rule allI)
chaieb@26120
  1706
apply (cut_tac p = "[h]" and a = a in order_root)
chaieb@26120
  1707
apply (simp add: fun_eq)
chaieb@26120
  1708
apply (blast intro: order_poly)
chaieb@26120
  1709
apply (auto simp add: order_root order_pderiv2)
chaieb@26120
  1710
apply (erule_tac x="a" in allE, simp)
chaieb@26120
  1711
done
chaieb@26120
  1712
chaieb@26120
  1713
lemma poly_squarefree_decomp: "[| poly (pderiv p) \<noteq> poly [];
chaieb@26120
  1714
         poly p = poly (q *** d);
chaieb@26120
  1715
         poly (pderiv p) = poly (e *** d);
chaieb@26120
  1716
         poly d = poly (r *** p +++ s *** pderiv p)
chaieb@26120
  1717
      |] ==> rsquarefree q & (\<forall>a. (poly q a = 0) = (poly p a = 0))"
chaieb@26120
  1718
apply (frule poly_squarefree_decomp_order2, assumption+) 
chaieb@26120
  1719
apply (case_tac "poly p = poly []")
chaieb@26120
  1720
apply (blast dest: pderiv_zero)
chaieb@26120
  1721
apply (simp (no_asm) add: rsquarefree_def order_root del: pmult_Cons)
chaieb@26120
  1722
apply (simp add: poly_entire del: pmult_Cons)
chaieb@26120
  1723
done
chaieb@26120
  1724
huffman@21164
  1725
end