src/HOL/Deriv.thy
author wenzelm
Fri Nov 13 15:38:45 2009 +0100 (2009-11-13)
changeset 33659 2d7ab9458518
parent 33654 abf780db30ea
child 33690 889d06128608
permissions -rw-r--r--
more "anti_sym" -> "antisym" (cf. a4179bf442d1);
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(*  Title       : Deriv.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    GMVT by Benjamin Porter, 2005
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*)
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header{* Differentiation *}
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theory Deriv
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imports Lim
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begin
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text{*Standard Definitions*}
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definition
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  deriv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
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    --{*Differentiation: D is derivative of function f at x*}
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          ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
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  "DERIV f x :> D = ((%h. (f(x + h) - f x) / h) -- 0 --> D)"
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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 dist_norm)
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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_setsum:
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  assumes "finite S"
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  and "\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x :> (f' x n)"
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  shows "DERIV (%x. setsum (f x) S) x :> setsum (f' x) S"
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  using assms by induct (auto intro!: DERIV_add)
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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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  by (auto intro: DERIV_setsum)
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text{*Alternative definition for differentiability*}
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lemma DERIV_LIM_iff:
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  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
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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: algebra_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}"
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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: 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 algebra_simps)
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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}"
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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 del: power_Suc)
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text {* Caratheodory formulation of derivative at a point *}
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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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text {*
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 Let's do the standard proof, though theorem
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 @{text "LIM_mult2"} follows from a NS proof
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*}
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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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lemma DERIV_cdivide: "DERIV f x :> D ==> DERIV (%x. f x / c) x :> D / c"
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  apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x :> (1 / c) * D", force)
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  apply (erule DERIV_cmult)
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  done
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text {* 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"
huffman@21164
   286
by (auto dest: DERIV_chain simp add: o_def)
huffman@21164
   287
wenzelm@31899
   288
text {* Derivative of linear multiplication *}
huffman@21164
   289
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
huffman@23069
   290
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
huffman@21164
   291
huffman@21164
   292
lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
huffman@23069
   293
apply (cut_tac DERIV_power [OF DERIV_ident])
huffman@21164
   294
apply (simp add: real_scaleR_def real_of_nat_def)
huffman@21164
   295
done
huffman@21164
   296
wenzelm@31899
   297
text {* Power of @{text "-1"} *}
huffman@21164
   298
huffman@21784
   299
lemma DERIV_inverse:
haftmann@31017
   300
  fixes x :: "'a::{real_normed_field}"
huffman@21784
   301
  shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
huffman@30273
   302
by (drule DERIV_inverse' [OF DERIV_ident]) simp
huffman@21164
   303
wenzelm@31899
   304
text {* Derivative of inverse *}
huffman@21784
   305
lemma DERIV_inverse_fun:
haftmann@31017
   306
  fixes x :: "'a::{real_normed_field}"
huffman@21784
   307
  shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21784
   308
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@30273
   309
by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
huffman@21164
   310
wenzelm@31899
   311
text {* Derivative of quotient *}
huffman@21784
   312
lemma DERIV_quotient:
haftmann@31017
   313
  fixes x :: "'a::{real_normed_field}"
huffman@21784
   314
  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21784
   315
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@30273
   316
by (drule (2) DERIV_divide) (simp add: mult_commute)
huffman@21164
   317
huffman@29975
   318
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
huffman@29975
   319
by auto
huffman@29975
   320
wenzelm@31899
   321
text {* @{text "DERIV_intros"} *}
wenzelm@31899
   322
ML {*
wenzelm@31902
   323
structure Deriv_Intros = Named_Thms
wenzelm@31899
   324
(
wenzelm@31899
   325
  val name = "DERIV_intros"
wenzelm@31899
   326
  val description = "DERIV introduction rules"
wenzelm@31899
   327
)
wenzelm@31899
   328
*}
hoelzl@31880
   329
wenzelm@31902
   330
setup Deriv_Intros.setup
hoelzl@31880
   331
hoelzl@31880
   332
lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"
hoelzl@31880
   333
  by simp
hoelzl@31880
   334
hoelzl@31880
   335
declare
hoelzl@31880
   336
  DERIV_const[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   337
  DERIV_ident[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   338
  DERIV_add[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   339
  DERIV_minus[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   340
  DERIV_mult[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   341
  DERIV_diff[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   342
  DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   343
  DERIV_divide[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   344
  DERIV_power[where 'a=real, THEN DERIV_cong,
hoelzl@31880
   345
              unfolded real_of_nat_def[symmetric], DERIV_intros]
hoelzl@31880
   346
  DERIV_setsum[THEN DERIV_cong, DERIV_intros]
huffman@22984
   347
wenzelm@31899
   348
huffman@22984
   349
subsection {* Differentiability predicate *}
huffman@21164
   350
huffman@29169
   351
definition
huffman@29169
   352
  differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
huffman@29169
   353
    (infixl "differentiable" 60) where
huffman@29169
   354
  "f differentiable x = (\<exists>D. DERIV f x :> D)"
huffman@29169
   355
huffman@29169
   356
lemma differentiableE [elim?]:
huffman@29169
   357
  assumes "f differentiable x"
huffman@29169
   358
  obtains df where "DERIV f x :> df"
huffman@29169
   359
  using prems unfolding differentiable_def ..
huffman@29169
   360
huffman@21164
   361
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
huffman@21164
   362
by (simp add: differentiable_def)
huffman@21164
   363
huffman@21164
   364
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
huffman@21164
   365
by (force simp add: differentiable_def)
huffman@21164
   366
huffman@29169
   367
lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
huffman@29169
   368
  by (rule DERIV_ident [THEN differentiableI])
huffman@29169
   369
huffman@29169
   370
lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
huffman@29169
   371
  by (rule DERIV_const [THEN differentiableI])
huffman@21164
   372
huffman@29169
   373
lemma differentiable_compose:
huffman@29169
   374
  assumes f: "f differentiable (g x)"
huffman@29169
   375
  assumes g: "g differentiable x"
huffman@29169
   376
  shows "(\<lambda>x. f (g x)) differentiable x"
huffman@29169
   377
proof -
huffman@29169
   378
  from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
huffman@29169
   379
  moreover
huffman@29169
   380
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
huffman@29169
   381
  ultimately
huffman@29169
   382
  have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
huffman@29169
   383
  thus ?thesis by (rule differentiableI)
huffman@29169
   384
qed
huffman@29169
   385
huffman@29169
   386
lemma differentiable_sum [simp]:
huffman@21164
   387
  assumes "f differentiable x"
huffman@21164
   388
  and "g differentiable x"
huffman@21164
   389
  shows "(\<lambda>x. f x + g x) differentiable x"
huffman@21164
   390
proof -
huffman@29169
   391
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@29169
   392
  moreover
huffman@29169
   393
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
huffman@29169
   394
  ultimately
huffman@29169
   395
  have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@29169
   396
  thus ?thesis by (rule differentiableI)
huffman@29169
   397
qed
huffman@29169
   398
huffman@29169
   399
lemma differentiable_minus [simp]:
huffman@29169
   400
  assumes "f differentiable x"
huffman@29169
   401
  shows "(\<lambda>x. - f x) differentiable x"
huffman@29169
   402
proof -
huffman@29169
   403
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@29169
   404
  hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
huffman@29169
   405
  thus ?thesis by (rule differentiableI)
huffman@21164
   406
qed
huffman@21164
   407
huffman@29169
   408
lemma differentiable_diff [simp]:
huffman@21164
   409
  assumes "f differentiable x"
huffman@29169
   410
  assumes "g differentiable x"
huffman@21164
   411
  shows "(\<lambda>x. f x - g x) differentiable x"
huffman@29169
   412
  unfolding diff_minus using prems by simp
huffman@29169
   413
huffman@29169
   414
lemma differentiable_mult [simp]:
huffman@29169
   415
  assumes "f differentiable x"
huffman@29169
   416
  assumes "g differentiable x"
huffman@29169
   417
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164
   418
proof -
huffman@29169
   419
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@21164
   420
  moreover
huffman@29169
   421
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
huffman@29169
   422
  ultimately
huffman@29169
   423
  have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
huffman@29169
   424
  thus ?thesis by (rule differentiableI)
huffman@21164
   425
qed
huffman@21164
   426
huffman@29169
   427
lemma differentiable_inverse [simp]:
huffman@29169
   428
  assumes "f differentiable x" and "f x \<noteq> 0"
huffman@29169
   429
  shows "(\<lambda>x. inverse (f x)) differentiable x"
huffman@21164
   430
proof -
huffman@29169
   431
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@29169
   432
  hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
huffman@29169
   433
    using `f x \<noteq> 0` by (rule DERIV_inverse')
huffman@29169
   434
  thus ?thesis by (rule differentiableI)
huffman@21164
   435
qed
huffman@21164
   436
huffman@29169
   437
lemma differentiable_divide [simp]:
huffman@29169
   438
  assumes "f differentiable x"
huffman@29169
   439
  assumes "g differentiable x" and "g x \<noteq> 0"
huffman@29169
   440
  shows "(\<lambda>x. f x / g x) differentiable x"
huffman@29169
   441
  unfolding divide_inverse using prems by simp
huffman@29169
   442
huffman@29169
   443
lemma differentiable_power [simp]:
haftmann@31017
   444
  fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
huffman@29169
   445
  assumes "f differentiable x"
huffman@29169
   446
  shows "(\<lambda>x. f x ^ n) differentiable x"
huffman@30273
   447
  by (induct n, simp, simp add: prems)
huffman@29169
   448
huffman@22984
   449
huffman@21164
   450
subsection {* Nested Intervals and Bisection *}
huffman@21164
   451
huffman@21164
   452
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164
   453
     All considerably tidied by lcp.*}
huffman@21164
   454
huffman@21164
   455
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
   456
apply (induct "no")
huffman@21164
   457
apply (auto intro: order_trans)
huffman@21164
   458
done
huffman@21164
   459
huffman@21164
   460
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   461
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   462
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   463
      ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164
   464
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@21164
   465
apply (induct_tac "n")
huffman@21164
   466
apply (auto intro: order_trans)
huffman@21164
   467
apply (rule_tac y = "g (Suc na)" in order_trans)
huffman@21164
   468
apply (induct_tac [2] "na")
huffman@21164
   469
apply (auto intro: order_trans)
huffman@21164
   470
done
huffman@21164
   471
huffman@21164
   472
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   473
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   474
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   475
      ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164
   476
apply (subst Bseq_minus_iff [symmetric])
huffman@21164
   477
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164
   478
apply auto
huffman@21164
   479
done
huffman@21164
   480
huffman@21164
   481
lemma f_inc_imp_le_lim:
huffman@21164
   482
  fixes f :: "nat \<Rightarrow> real"
huffman@21164
   483
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@21164
   484
apply (rule linorder_not_less [THEN iffD1])
huffman@21164
   485
apply (auto simp add: convergent_LIMSEQ_iff LIMSEQ_iff monoseq_Suc)
huffman@21164
   486
apply (drule real_less_sum_gt_zero)
huffman@21164
   487
apply (drule_tac x = "f n + - lim f" in spec, safe)
huffman@21164
   488
apply (drule_tac P = "%na. no\<le>na --> ?Q na" and x = "no + n" in spec, auto)
huffman@21164
   489
apply (subgoal_tac "lim f \<le> f (no + n) ")
huffman@21164
   490
apply (drule_tac no=no and m=n in lemma_f_mono_add)
huffman@21164
   491
apply (auto simp add: add_commute)
huffman@21164
   492
apply (induct_tac "no")
huffman@21164
   493
apply simp
huffman@21164
   494
apply (auto intro: order_trans simp add: diff_minus abs_if)
huffman@21164
   495
done
huffman@21164
   496
huffman@31404
   497
lemma lim_uminus:
huffman@31404
   498
  fixes g :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31404
   499
  shows "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@21164
   500
apply (rule LIMSEQ_minus [THEN limI])
huffman@21164
   501
apply (simp add: convergent_LIMSEQ_iff)
huffman@21164
   502
done
huffman@21164
   503
huffman@21164
   504
lemma g_dec_imp_lim_le:
huffman@21164
   505
  fixes g :: "nat \<Rightarrow> real"
huffman@21164
   506
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@21164
   507
apply (subgoal_tac "- (g n) \<le> - (lim g) ")
huffman@21164
   508
apply (cut_tac [2] f = "%x. - (g x)" in f_inc_imp_le_lim)
huffman@21164
   509
apply (auto simp add: lim_uminus convergent_minus_iff [symmetric])
huffman@21164
   510
done
huffman@21164
   511
huffman@21164
   512
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   513
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   514
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   515
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   516
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164
   517
apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164
   518
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164
   519
apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164
   520
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164
   521
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164
   522
apply (rule_tac x = "lim f" in exI)
huffman@21164
   523
apply (rule_tac x = "lim g" in exI)
huffman@21164
   524
apply (auto intro: LIMSEQ_le)
huffman@21164
   525
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164
   526
done
huffman@21164
   527
huffman@21164
   528
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   529
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   530
         \<forall>n. f(n) \<le> g(n);
huffman@21164
   531
         (%n. f(n) - g(n)) ----> 0 |]
huffman@21164
   532
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   533
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164
   534
apply (drule lemma_nest, auto)
huffman@21164
   535
apply (subgoal_tac "l = m")
huffman@21164
   536
apply (drule_tac [2] X = f in LIMSEQ_diff)
huffman@21164
   537
apply (auto intro: LIMSEQ_unique)
huffman@21164
   538
done
huffman@21164
   539
huffman@21164
   540
text{*The universal quantifiers below are required for the declaration
huffman@21164
   541
  of @{text Bolzano_nest_unique} below.*}
huffman@21164
   542
huffman@21164
   543
lemma Bolzano_bisect_le:
huffman@21164
   544
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   545
apply (rule allI)
huffman@21164
   546
apply (induct_tac "n")
huffman@21164
   547
apply (auto simp add: Let_def split_def)
huffman@21164
   548
done
huffman@21164
   549
huffman@21164
   550
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164
   551
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164
   552
apply (rule allI)
huffman@21164
   553
apply (induct_tac "n")
huffman@21164
   554
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   555
done
huffman@21164
   556
huffman@21164
   557
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164
   558
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   559
apply (rule allI)
huffman@21164
   560
apply (induct_tac "n")
huffman@21164
   561
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   562
done
huffman@21164
   563
huffman@21164
   564
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164
   565
apply (auto)
huffman@21164
   566
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164
   567
apply (simp)
huffman@21164
   568
done
huffman@21164
   569
huffman@21164
   570
lemma Bolzano_bisect_diff:
huffman@21164
   571
     "a \<le> b ==>
huffman@21164
   572
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164
   573
      (b-a) / (2 ^ n)"
huffman@21164
   574
apply (induct "n")
huffman@21164
   575
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164
   576
done
huffman@21164
   577
huffman@21164
   578
lemmas Bolzano_nest_unique =
huffman@21164
   579
    lemma_nest_unique
huffman@21164
   580
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164
   581
huffman@21164
   582
huffman@21164
   583
lemma not_P_Bolzano_bisect:
huffman@21164
   584
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164
   585
      and notP: "~ P(a,b)"
huffman@21164
   586
      and le:   "a \<le> b"
huffman@21164
   587
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   588
proof (induct n)
huffman@23441
   589
  case 0 show ?case using notP by simp
huffman@21164
   590
 next
huffman@21164
   591
  case (Suc n)
huffman@21164
   592
  thus ?case
huffman@21164
   593
 by (auto simp del: surjective_pairing [symmetric]
huffman@21164
   594
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164
   595
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164
   596
qed
huffman@21164
   597
huffman@21164
   598
(*Now we re-package P_prem as a formula*)
huffman@21164
   599
lemma not_P_Bolzano_bisect':
huffman@21164
   600
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   601
         ~ P(a,b);  a \<le> b |] ==>
huffman@21164
   602
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   603
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164
   604
huffman@21164
   605
huffman@21164
   606
huffman@21164
   607
lemma lemma_BOLZANO:
huffman@21164
   608
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   609
         \<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   610
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164
   611
         a \<le> b |]
huffman@21164
   612
      ==> P(a,b)"
huffman@21164
   613
apply (rule Bolzano_nest_unique [where P1=P, THEN exE], assumption+)
huffman@21164
   614
apply (rule LIMSEQ_minus_cancel)
huffman@21164
   615
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164
   616
apply (rule ccontr)
huffman@21164
   617
apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164
   618
apply (rename_tac "l")
huffman@21164
   619
apply (drule_tac x = l in spec, clarify)
huffman@31336
   620
apply (simp add: LIMSEQ_iff)
huffman@21164
   621
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   622
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   623
apply (drule real_less_half_sum, auto)
huffman@21164
   624
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   625
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   626
apply safe
huffman@21164
   627
apply (simp_all (no_asm_simp))
huffman@21164
   628
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
   629
apply (simp (no_asm_simp) add: abs_if)
huffman@21164
   630
apply (rule real_sum_of_halves [THEN subst])
huffman@21164
   631
apply (rule add_strict_mono)
huffman@21164
   632
apply (simp_all add: diff_minus [symmetric])
huffman@21164
   633
done
huffman@21164
   634
huffman@21164
   635
huffman@21164
   636
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164
   637
       (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   638
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164
   639
      --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164
   640
apply clarify
huffman@21164
   641
apply (blast intro: lemma_BOLZANO)
huffman@21164
   642
done
huffman@21164
   643
huffman@21164
   644
huffman@21164
   645
subsection {* Intermediate Value Theorem *}
huffman@21164
   646
huffman@21164
   647
text {*Prove Contrapositive by Bisection*}
huffman@21164
   648
huffman@21164
   649
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164
   650
         a \<le> b;
huffman@21164
   651
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164
   652
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   653
apply (rule contrapos_pp, assumption)
huffman@21164
   654
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
   655
apply safe
huffman@21164
   656
apply simp_all
huffman@31338
   657
apply (simp add: isCont_iff LIM_eq)
huffman@21164
   658
apply (rule ccontr)
huffman@21164
   659
apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164
   660
 prefer 2
huffman@21164
   661
 apply simp
huffman@21164
   662
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164
   663
apply (drule_tac x = x in spec)+
huffman@21164
   664
apply simp
huffman@21164
   665
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164
   666
apply safe
huffman@21164
   667
apply simp
huffman@21164
   668
apply (drule_tac x = s in spec, clarify)
huffman@21164
   669
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164
   670
apply (drule_tac x = "ba-x" in spec)
huffman@21164
   671
apply (simp_all add: abs_if)
huffman@21164
   672
apply (drule_tac x = "aa-x" in spec)
huffman@21164
   673
apply (case_tac "x \<le> aa", simp_all)
huffman@21164
   674
done
huffman@21164
   675
huffman@21164
   676
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164
   677
         a \<le> b;
huffman@21164
   678
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164
   679
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   680
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164
   681
apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@21164
   682
apply (auto intro: isCont_minus)
huffman@21164
   683
done
huffman@21164
   684
huffman@21164
   685
(*HOL style here: object-level formulations*)
huffman@21164
   686
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164
   687
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   688
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   689
apply (blast intro: IVT)
huffman@21164
   690
done
huffman@21164
   691
huffman@21164
   692
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164
   693
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   694
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   695
apply (blast intro: IVT2)
huffman@21164
   696
done
huffman@21164
   697
huffman@29975
   698
huffman@29975
   699
subsection {* Boundedness of continuous functions *}
huffman@29975
   700
huffman@21164
   701
text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164
   702
huffman@21164
   703
lemma isCont_bounded:
huffman@21164
   704
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   705
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164
   706
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
   707
apply safe
huffman@21164
   708
apply simp_all
huffman@21164
   709
apply (rename_tac x xa ya M Ma)
paulson@33654
   710
apply (metis linorder_not_less order_le_less real_le_trans)
huffman@21164
   711
apply (case_tac "a \<le> x & x \<le> b")
paulson@33654
   712
 prefer 2
paulson@33654
   713
 apply (rule_tac x = 1 in exI, force)
huffman@31338
   714
apply (simp add: LIM_eq isCont_iff)
huffman@21164
   715
apply (drule_tac x = x in spec, auto)
huffman@21164
   716
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164
   717
apply (drule_tac x = 1 in spec, auto)
huffman@21164
   718
apply (rule_tac x = s in exI, clarify)
huffman@21164
   719
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164
   720
apply (drule_tac x = "xa-x" in spec)
huffman@21164
   721
apply (auto simp add: abs_ge_self)
huffman@21164
   722
done
huffman@21164
   723
huffman@21164
   724
text{*Refine the above to existence of least upper bound*}
huffman@21164
   725
huffman@21164
   726
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164
   727
      (\<exists>t. isLub UNIV S t)"
huffman@21164
   728
by (blast intro: reals_complete)
huffman@21164
   729
huffman@21164
   730
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   731
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   732
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164
   733
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164
   734
        in lemma_reals_complete)
huffman@21164
   735
apply auto
huffman@21164
   736
apply (drule isCont_bounded, assumption)
huffman@21164
   737
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164
   738
apply (rule exI, auto)
huffman@21164
   739
apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164
   740
done
huffman@21164
   741
huffman@21164
   742
text{*Now show that it attains its upper bound*}
huffman@21164
   743
huffman@21164
   744
lemma isCont_eq_Ub:
huffman@21164
   745
  assumes le: "a \<le> b"
huffman@21164
   746
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   747
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   748
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   749
proof -
huffman@21164
   750
  from isCont_has_Ub [OF le con]
huffman@21164
   751
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164
   752
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164
   753
  show ?thesis
huffman@21164
   754
  proof (intro exI, intro conjI)
huffman@21164
   755
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164
   756
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164
   757
    proof (rule ccontr)
huffman@21164
   758
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164
   759
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
huffman@21164
   760
        by (fastsimp simp add: linorder_not_le [symmetric])
huffman@21164
   761
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@21164
   762
        by (auto simp add: isCont_inverse isCont_diff con)
huffman@21164
   763
      from isCont_bounded [OF le this]
huffman@21164
   764
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164
   765
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
nipkow@29667
   766
        by (simp add: M3 algebra_simps)
huffman@21164
   767
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164
   768
        by (auto intro: order_le_less_trans [of _ k])
huffman@21164
   769
      with Minv
huffman@21164
   770
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164
   771
        by (intro strip less_imp_inverse_less, simp_all)
huffman@21164
   772
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164
   773
        by simp
huffman@21164
   774
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164
   775
        by (simp, arith)
huffman@21164
   776
      from M2 [OF this]
huffman@21164
   777
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164
   778
      thus False using invlt [of x] by force
huffman@21164
   779
    qed
huffman@21164
   780
  qed
huffman@21164
   781
qed
huffman@21164
   782
huffman@21164
   783
huffman@21164
   784
text{*Same theorem for lower bound*}
huffman@21164
   785
huffman@21164
   786
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   787
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164
   788
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   789
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164
   790
prefer 2 apply (blast intro: isCont_minus)
huffman@21164
   791
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164
   792
apply safe
huffman@21164
   793
apply auto
huffman@21164
   794
done
huffman@21164
   795
huffman@21164
   796
huffman@21164
   797
text{*Another version.*}
huffman@21164
   798
huffman@21164
   799
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   800
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164
   801
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164
   802
apply (frule isCont_eq_Lb)
huffman@21164
   803
apply (frule_tac [2] isCont_eq_Ub)
huffman@21164
   804
apply (assumption+, safe)
huffman@21164
   805
apply (rule_tac x = "f x" in exI)
huffman@21164
   806
apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164
   807
apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164
   808
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164
   809
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164
   810
apply (rule_tac [2] x = xb in exI)
huffman@21164
   811
apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164
   812
done
huffman@21164
   813
huffman@21164
   814
huffman@29975
   815
subsection {* Local extrema *}
huffman@29975
   816
huffman@21164
   817
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   818
paulson@33654
   819
lemma DERIV_pos_inc_right:
huffman@21164
   820
  fixes f :: "real => real"
huffman@21164
   821
  assumes der: "DERIV f x :> l"
huffman@21164
   822
      and l:   "0 < l"
huffman@21164
   823
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   824
proof -
huffman@21164
   825
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   826
  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
   827
    by (simp add: diff_minus)
huffman@21164
   828
  then obtain s
huffman@21164
   829
        where s:   "0 < s"
huffman@21164
   830
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   831
    by auto
huffman@21164
   832
  thus ?thesis
huffman@21164
   833
  proof (intro exI conjI strip)
huffman@23441
   834
    show "0<s" using s .
huffman@21164
   835
    fix h::real
huffman@21164
   836
    assume "0 < h" "h < s"
huffman@21164
   837
    with all [of h] show "f x < f (x+h)"
huffman@21164
   838
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   839
    split add: split_if_asm)
huffman@21164
   840
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   841
      with l
huffman@21164
   842
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   843
      thus "f x < f (x+h)"
huffman@21164
   844
  by (simp add: pos_less_divide_eq h)
huffman@21164
   845
    qed
huffman@21164
   846
  qed
huffman@21164
   847
qed
huffman@21164
   848
paulson@33654
   849
lemma DERIV_neg_dec_left:
huffman@21164
   850
  fixes f :: "real => real"
huffman@21164
   851
  assumes der: "DERIV f x :> l"
huffman@21164
   852
      and l:   "l < 0"
huffman@21164
   853
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   854
proof -
huffman@21164
   855
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   856
  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
   857
    by (simp add: diff_minus)
huffman@21164
   858
  then obtain s
huffman@21164
   859
        where s:   "0 < s"
huffman@21164
   860
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   861
    by auto
huffman@21164
   862
  thus ?thesis
huffman@21164
   863
  proof (intro exI conjI strip)
huffman@23441
   864
    show "0<s" using s .
huffman@21164
   865
    fix h::real
huffman@21164
   866
    assume "0 < h" "h < s"
huffman@21164
   867
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
   868
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   869
    split add: split_if_asm)
huffman@21164
   870
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   871
      with l
huffman@21164
   872
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   873
      thus "f x < f (x-h)"
huffman@21164
   874
  by (simp add: pos_less_divide_eq h)
huffman@21164
   875
    qed
huffman@21164
   876
  qed
huffman@21164
   877
qed
huffman@21164
   878
paulson@33654
   879
paulson@33654
   880
lemma DERIV_pos_inc_left:
paulson@33654
   881
  fixes f :: "real => real"
paulson@33654
   882
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
paulson@33654
   883
  apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
paulson@33654
   884
  apply (auto simp add: DERIV_minus) 
paulson@33654
   885
  done
paulson@33654
   886
paulson@33654
   887
lemma DERIV_neg_dec_right:
paulson@33654
   888
  fixes f :: "real => real"
paulson@33654
   889
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
paulson@33654
   890
  apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
paulson@33654
   891
  apply (auto simp add: DERIV_minus) 
paulson@33654
   892
  done
paulson@33654
   893
huffman@21164
   894
lemma DERIV_local_max:
huffman@21164
   895
  fixes f :: "real => real"
huffman@21164
   896
  assumes der: "DERIV f x :> l"
huffman@21164
   897
      and d:   "0 < d"
huffman@21164
   898
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   899
  shows "l = 0"
huffman@21164
   900
proof (cases rule: linorder_cases [of l 0])
huffman@23441
   901
  case equal thus ?thesis .
huffman@21164
   902
next
huffman@21164
   903
  case less
paulson@33654
   904
  from DERIV_neg_dec_left [OF der less]
huffman@21164
   905
  obtain d' where d': "0 < d'"
huffman@21164
   906
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   907
  from real_lbound_gt_zero [OF d d']
huffman@21164
   908
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   909
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   910
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   911
next
huffman@21164
   912
  case greater
paulson@33654
   913
  from DERIV_pos_inc_right [OF der greater]
huffman@21164
   914
  obtain d' where d': "0 < d'"
huffman@21164
   915
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   916
  from real_lbound_gt_zero [OF d d']
huffman@21164
   917
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   918
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   919
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   920
qed
huffman@21164
   921
huffman@21164
   922
huffman@21164
   923
text{*Similar theorem for a local minimum*}
huffman@21164
   924
lemma DERIV_local_min:
huffman@21164
   925
  fixes f :: "real => real"
huffman@21164
   926
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   927
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   928
huffman@21164
   929
huffman@21164
   930
text{*In particular, if a function is locally flat*}
huffman@21164
   931
lemma DERIV_local_const:
huffman@21164
   932
  fixes f :: "real => real"
huffman@21164
   933
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   934
by (auto dest!: DERIV_local_max)
huffman@21164
   935
huffman@29975
   936
huffman@29975
   937
subsection {* Rolle's Theorem *}
huffman@29975
   938
huffman@21164
   939
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   940
lemma lemma_interval_lt:
huffman@21164
   941
     "[| a < x;  x < b |]
huffman@21164
   942
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
   943
huffman@22998
   944
apply (simp add: abs_less_iff)
huffman@21164
   945
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   946
apply (rule_tac x = "x-a" in exI)
huffman@21164
   947
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   948
done
huffman@21164
   949
huffman@21164
   950
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   951
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   952
apply (drule lemma_interval_lt, auto)
huffman@21164
   953
apply (auto intro!: exI)
huffman@21164
   954
done
huffman@21164
   955
huffman@21164
   956
text{*Rolle's Theorem.
huffman@21164
   957
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   958
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
   959
   and @{term "f(a) = f(b)"},
huffman@21164
   960
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
   961
theorem Rolle:
huffman@21164
   962
  assumes lt: "a < b"
huffman@21164
   963
      and eq: "f(a) = f(b)"
huffman@21164
   964
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   965
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   966
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
   967
proof -
huffman@21164
   968
  have le: "a \<le> b" using lt by simp
huffman@21164
   969
  from isCont_eq_Ub [OF le con]
huffman@21164
   970
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
   971
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
   972
    by blast
huffman@21164
   973
  from isCont_eq_Lb [OF le con]
huffman@21164
   974
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
   975
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
   976
    by blast
huffman@21164
   977
  show ?thesis
huffman@21164
   978
  proof cases
huffman@21164
   979
    assume axb: "a < x & x < b"
huffman@21164
   980
        --{*@{term f} attains its maximum within the interval*}
chaieb@27668
   981
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
   982
    from lemma_interval [OF ax xb]
huffman@21164
   983
    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
   984
      by blast
huffman@21164
   985
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
   986
      by blast
huffman@21164
   987
    from differentiableD [OF dif [OF axb]]
huffman@21164
   988
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   989
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
   990
        --{*the derivative at a local maximum is zero*}
huffman@21164
   991
    thus ?thesis using ax xb der by auto
huffman@21164
   992
  next
huffman@21164
   993
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
   994
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
   995
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
   996
    show ?thesis
huffman@21164
   997
    proof cases
huffman@21164
   998
      assume ax'b: "a < x' & x' < b"
huffman@21164
   999
        --{*@{term f} attains its minimum within the interval*}
chaieb@27668
  1000
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
  1001
      from lemma_interval [OF ax' x'b]
huffman@21164
  1002
      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
  1003
  by blast
huffman@21164
  1004
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
  1005
  by blast
huffman@21164
  1006
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
  1007
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
  1008
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
  1009
        --{*the derivative at a local minimum is zero*}
huffman@21164
  1010
      thus ?thesis using ax' x'b der by auto
huffman@21164
  1011
    next
huffman@21164
  1012
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
  1013
        --{*@{term f} is constant througout the interval*}
huffman@21164
  1014
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
  1015
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
  1016
      from dense [OF lt]
huffman@21164
  1017
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
  1018
      from lemma_interval [OF ar rb]
huffman@21164
  1019
      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
  1020
  by blast
huffman@21164
  1021
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
  1022
      proof (clarify)
huffman@21164
  1023
        fix z::real
huffman@21164
  1024
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1025
        show "f z = f b"
huffman@21164
  1026
        proof (rule order_antisym)
huffman@21164
  1027
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1028
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1029
        qed
huffman@21164
  1030
      qed
huffman@21164
  1031
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1032
      proof (intro strip)
huffman@21164
  1033
        fix y::real
huffman@21164
  1034
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1035
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1036
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1037
      qed
huffman@21164
  1038
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1039
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1040
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1041
        --{*the derivative of a constant function is zero*}
huffman@21164
  1042
      thus ?thesis using ar rb der by auto
huffman@21164
  1043
    qed
huffman@21164
  1044
  qed
huffman@21164
  1045
qed
huffman@21164
  1046
huffman@21164
  1047
huffman@21164
  1048
subsection{*Mean Value Theorem*}
huffman@21164
  1049
huffman@21164
  1050
lemma lemma_MVT:
huffman@21164
  1051
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164
  1052
proof cases
huffman@21164
  1053
  assume "a=b" thus ?thesis by simp
huffman@21164
  1054
next
huffman@21164
  1055
  assume "a\<noteq>b"
huffman@21164
  1056
  hence ba: "b-a \<noteq> 0" by arith
huffman@21164
  1057
  show ?thesis
huffman@21164
  1058
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164
  1059
        simp add: right_diff_distrib,
huffman@21164
  1060
        simp add: left_diff_distrib)
huffman@21164
  1061
qed
huffman@21164
  1062
huffman@21164
  1063
theorem MVT:
huffman@21164
  1064
  assumes lt:  "a < b"
huffman@21164
  1065
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1066
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
  1067
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
  1068
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1069
proof -
huffman@21164
  1070
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@21164
  1071
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x" using con
huffman@23069
  1072
    by (fast intro: isCont_diff isCont_const isCont_mult isCont_ident)
huffman@21164
  1073
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
  1074
  proof (clarify)
huffman@21164
  1075
    fix x::real
huffman@21164
  1076
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1077
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1078
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1079
    show "?F differentiable x"
huffman@21164
  1080
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1081
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1082
  qed
huffman@21164
  1083
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1084
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1085
    by blast
huffman@21164
  1086
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1087
    by (rule DERIV_cmult_Id)
huffman@21164
  1088
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1089
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1090
    by (rule DERIV_add [OF der])
huffman@21164
  1091
  show ?thesis
huffman@21164
  1092
  proof (intro exI conjI)
huffman@23441
  1093
    show "a < z" using az .
huffman@23441
  1094
    show "z < b" using zb .
huffman@21164
  1095
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1096
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1097
  qed
huffman@21164
  1098
qed
huffman@21164
  1099
hoelzl@29803
  1100
lemma MVT2:
hoelzl@29803
  1101
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803
  1102
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803
  1103
apply (drule MVT)
hoelzl@29803
  1104
apply (blast intro: DERIV_isCont)
hoelzl@29803
  1105
apply (force dest: order_less_imp_le simp add: differentiable_def)
hoelzl@29803
  1106
apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803
  1107
done
hoelzl@29803
  1108
huffman@21164
  1109
huffman@21164
  1110
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1111
huffman@21164
  1112
lemma DERIV_isconst_end:
huffman@21164
  1113
  fixes f :: "real => real"
huffman@21164
  1114
  shows "[| a < b;
huffman@21164
  1115
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1116
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1117
        ==> f b = f a"
huffman@21164
  1118
apply (drule MVT, assumption)
huffman@21164
  1119
apply (blast intro: differentiableI)
huffman@21164
  1120
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1121
done
huffman@21164
  1122
huffman@21164
  1123
lemma DERIV_isconst1:
huffman@21164
  1124
  fixes f :: "real => real"
huffman@21164
  1125
  shows "[| a < b;
huffman@21164
  1126
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1127
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1128
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1129
apply safe
huffman@21164
  1130
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1131
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1132
done
huffman@21164
  1133
huffman@21164
  1134
lemma DERIV_isconst2:
huffman@21164
  1135
  fixes f :: "real => real"
huffman@21164
  1136
  shows "[| a < b;
huffman@21164
  1137
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1138
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1139
         a \<le> x; x \<le> b |]
huffman@21164
  1140
        ==> f x = f a"
huffman@21164
  1141
apply (blast dest: DERIV_isconst1)
huffman@21164
  1142
done
huffman@21164
  1143
hoelzl@29803
  1144
lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803
  1145
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803
  1146
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1147
  shows "f x = f y"
hoelzl@29803
  1148
proof (cases "x = y")
hoelzl@29803
  1149
  case False
hoelzl@29803
  1150
  let ?a = "min x y"
hoelzl@29803
  1151
  let ?b = "max x y"
hoelzl@29803
  1152
  
hoelzl@29803
  1153
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1154
  proof (rule allI, rule impI)
hoelzl@29803
  1155
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hoelzl@29803
  1156
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hoelzl@29803
  1157
    hence "z \<in> {a<..<b}" by auto
hoelzl@29803
  1158
    thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1159
  qed
hoelzl@29803
  1160
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803
  1161
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803
  1162
hoelzl@29803
  1163
  have "?a < ?b" using `x \<noteq> y` by auto
hoelzl@29803
  1164
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1165
  show ?thesis by auto
hoelzl@29803
  1166
qed auto
hoelzl@29803
  1167
huffman@21164
  1168
lemma DERIV_isconst_all:
huffman@21164
  1169
  fixes f :: "real => real"
huffman@21164
  1170
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1171
apply (rule linorder_cases [of x y])
huffman@21164
  1172
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1173
done
huffman@21164
  1174
huffman@21164
  1175
lemma DERIV_const_ratio_const:
huffman@21784
  1176
  fixes f :: "real => real"
huffman@21784
  1177
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1178
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1179
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1180
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
nipkow@23477
  1181
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
huffman@21164
  1182
done
huffman@21164
  1183
huffman@21164
  1184
lemma DERIV_const_ratio_const2:
huffman@21784
  1185
  fixes f :: "real => real"
huffman@21784
  1186
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1187
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1188
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1189
done
huffman@21164
  1190
huffman@21164
  1191
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1192
by (simp)
huffman@21164
  1193
huffman@21164
  1194
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1195
by (simp)
huffman@21164
  1196
huffman@21164
  1197
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1198
huffman@21164
  1199
lemma DERIV_const_average:
huffman@21164
  1200
  fixes v :: "real => real"
huffman@21164
  1201
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1202
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1203
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1204
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1205
  case equal with neq show ?thesis by simp
huffman@21164
  1206
next
huffman@21164
  1207
  case less
huffman@21164
  1208
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1209
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1210
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1211
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1212
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1213
  ultimately show ?thesis using neq by force
huffman@21164
  1214
next
huffman@21164
  1215
  case greater
huffman@21164
  1216
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1217
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1218
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1219
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1220
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1221
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1222
qed
huffman@21164
  1223
paulson@33654
  1224
(* A function with positive derivative is increasing. 
paulson@33654
  1225
   A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654
  1226
*)
paulson@33654
  1227
paulson@33654
  1228
lemma DERIV_pos_imp_increasing:
paulson@33654
  1229
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1230
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
paulson@33654
  1231
  shows "f a < f b"
paulson@33654
  1232
proof (rule ccontr)
paulson@33654
  1233
  assume "~ f a < f b"
paulson@33654
  1234
  from assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1235
      & f b - f a = (b - a) * l"
paulson@33654
  1236
    by (metis MVT DERIV_isCont differentiableI real_less_def)
paulson@33654
  1237
  then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
paulson@33654
  1238
      and "f b - f a = (b - a) * l"
paulson@33654
  1239
    by auto
paulson@33654
  1240
  
paulson@33654
  1241
  from prems have "~(l > 0)"
paulson@33654
  1242
    by (metis assms(1) linorder_not_le mult_le_0_iff real_le_eq_diff)
paulson@33654
  1243
  with prems show False
paulson@33654
  1244
    by (metis DERIV_unique real_less_def)
paulson@33654
  1245
qed
paulson@33654
  1246
paulson@33654
  1247
paulson@33654
  1248
lemma DERIV_nonneg_imp_nonincreasing:
paulson@33654
  1249
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1250
  assumes "a \<le> b" and
paulson@33654
  1251
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654
  1252
  shows "f a \<le> f b"
paulson@33654
  1253
proof (rule ccontr, cases "a = b")
paulson@33654
  1254
  assume "~ f a \<le> f b"
paulson@33654
  1255
  assume "a = b"
paulson@33654
  1256
  with prems show False by auto
paulson@33654
  1257
  next assume "~ f a \<le> f b"
paulson@33654
  1258
  assume "a ~= b"
paulson@33654
  1259
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1260
      & f b - f a = (b - a) * l"
paulson@33654
  1261
    apply (intro MVT)
paulson@33654
  1262
    apply auto
paulson@33654
  1263
    apply (metis DERIV_isCont)
paulson@33654
  1264
    apply (metis differentiableI real_less_def)
paulson@33654
  1265
    done
paulson@33654
  1266
  then obtain l z where "a < z" and "z < b" and "DERIV f z :> l"
paulson@33654
  1267
      and "f b - f a = (b - a) * l"
paulson@33654
  1268
    by auto
paulson@33654
  1269
  from prems have "~(l >= 0)"
wenzelm@33659
  1270
    by (metis diff_self le_eqI le_iff_diff_le_0 real_le_antisym real_le_linear
paulson@33654
  1271
              split_mult_pos_le)
paulson@33654
  1272
  with prems show False
paulson@33654
  1273
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1274
qed
paulson@33654
  1275
paulson@33654
  1276
lemma DERIV_neg_imp_decreasing:
paulson@33654
  1277
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1278
  assumes "a < b" and
paulson@33654
  1279
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654
  1280
  shows "f a > f b"
paulson@33654
  1281
proof -
paulson@33654
  1282
  have "(%x. -f x) a < (%x. -f x) b"
paulson@33654
  1283
    apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
paulson@33654
  1284
    apply (insert prems, auto)
paulson@33654
  1285
    apply (metis DERIV_minus neg_0_less_iff_less)
paulson@33654
  1286
    done
paulson@33654
  1287
  thus ?thesis
paulson@33654
  1288
    by simp
paulson@33654
  1289
qed
paulson@33654
  1290
paulson@33654
  1291
lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654
  1292
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1293
  assumes "a \<le> b" and
paulson@33654
  1294
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654
  1295
  shows "f a \<ge> f b"
paulson@33654
  1296
proof -
paulson@33654
  1297
  have "(%x. -f x) a \<le> (%x. -f x) b"
paulson@33654
  1298
    apply (rule DERIV_nonneg_imp_nonincreasing [of a b "%x. -f x"])
paulson@33654
  1299
    apply (insert prems, auto)
paulson@33654
  1300
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1301
    done
paulson@33654
  1302
  thus ?thesis
paulson@33654
  1303
    by simp
paulson@33654
  1304
qed
huffman@21164
  1305
huffman@29975
  1306
subsection {* Continuous injective functions *}
huffman@29975
  1307
huffman@21164
  1308
text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164
  1309
strict maximum at an end point, not in the middle.*}
huffman@21164
  1310
huffman@21164
  1311
lemma lemma_isCont_inj:
huffman@21164
  1312
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1313
  assumes d: "0 < d"
huffman@21164
  1314
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1315
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1316
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164
  1317
proof (rule ccontr)
huffman@21164
  1318
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164
  1319
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164
  1320
  show False
huffman@21164
  1321
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164
  1322
    case le
huffman@21164
  1323
    from d cont all [of "x+d"]
huffman@21164
  1324
    have flef: "f(x+d) \<le> f x"
huffman@21164
  1325
     and xlex: "x - d \<le> x"
huffman@21164
  1326
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164
  1327
       by (auto simp add: abs_if)
huffman@21164
  1328
    from IVT [OF le flef xlex cont']
huffman@21164
  1329
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164
  1330
    moreover
huffman@21164
  1331
    hence "g(f x') = g (f(x+d))" by simp
huffman@21164
  1332
    ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@22998
  1333
      by (simp add: abs_le_iff)
huffman@21164
  1334
  next
huffman@21164
  1335
    case ge
huffman@21164
  1336
    from d cont all [of "x-d"]
huffman@21164
  1337
    have flef: "f(x-d) \<le> f x"
huffman@21164
  1338
     and xlex: "x \<le> x+d"
huffman@21164
  1339
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164
  1340
       by (auto simp add: abs_if)
huffman@21164
  1341
    from IVT2 [OF ge flef xlex cont']
huffman@21164
  1342
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164
  1343
    moreover
huffman@21164
  1344
    hence "g(f x') = g (f(x-d))" by simp
huffman@21164
  1345
    ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@22998
  1346
      by (simp add: abs_le_iff)
huffman@21164
  1347
  qed
huffman@21164
  1348
qed
huffman@21164
  1349
huffman@21164
  1350
huffman@21164
  1351
text{*Similar version for lower bound.*}
huffman@21164
  1352
huffman@21164
  1353
lemma lemma_isCont_inj2:
huffman@21164
  1354
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1355
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164
  1356
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164
  1357
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164
  1358
apply (insert lemma_isCont_inj
huffman@21164
  1359
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@21164
  1360
apply (simp add: isCont_minus linorder_not_le)
huffman@21164
  1361
done
huffman@21164
  1362
huffman@21164
  1363
text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164
  1364
@{text "f[[x - d, x + d]]"} .*}
huffman@21164
  1365
huffman@21164
  1366
lemma isCont_inj_range:
huffman@21164
  1367
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1368
  assumes d: "0 < d"
huffman@21164
  1369
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1370
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1371
  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
  1372
proof -
huffman@21164
  1373
  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
  1374
    by (auto simp add: abs_le_iff)
huffman@21164
  1375
  from isCont_Lb_Ub [OF this]
huffman@21164
  1376
  obtain L M
huffman@21164
  1377
  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
  1378
    and all2 [rule_format]:
huffman@21164
  1379
           "\<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
  1380
    by auto
huffman@21164
  1381
  with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164
  1382
  moreover have "L \<noteq> f x"
huffman@21164
  1383
  proof -
huffman@21164
  1384
    from lemma_isCont_inj2 [OF d inj cont]
huffman@21164
  1385
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164
  1386
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1387
  qed
huffman@21164
  1388
  moreover have "f x \<noteq> M"
huffman@21164
  1389
  proof -
huffman@21164
  1390
    from lemma_isCont_inj [OF d inj cont]
huffman@21164
  1391
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164
  1392
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1393
  qed
huffman@21164
  1394
  ultimately have "L < f x & f x < M" by arith
huffman@21164
  1395
  hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164
  1396
  from real_lbound_gt_zero [OF this]
huffman@21164
  1397
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164
  1398
  thus ?thesis
huffman@21164
  1399
  proof (intro exI conjI)
huffman@23441
  1400
    show "0<e" using e(1) .
huffman@21164
  1401
    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
  1402
    proof (intro strip)
huffman@21164
  1403
      fix y::real
huffman@21164
  1404
      assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164
  1405
      with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164
  1406
      from all2 [OF this]
huffman@21164
  1407
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
chaieb@27668
  1408
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" 
huffman@22998
  1409
        by (force simp add: abs_le_iff)
huffman@21164
  1410
    qed
huffman@21164
  1411
  qed
huffman@21164
  1412
qed
huffman@21164
  1413
huffman@21164
  1414
huffman@21164
  1415
text{*Continuity of inverse function*}
huffman@21164
  1416
huffman@21164
  1417
lemma isCont_inverse_function:
huffman@21164
  1418
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1419
  assumes d: "0 < d"
huffman@21164
  1420
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1421
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1422
  shows "isCont g (f x)"
huffman@21164
  1423
proof (simp add: isCont_iff LIM_eq)
huffman@21164
  1424
  show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164
  1425
         (\<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
  1426
  proof (intro strip)
huffman@21164
  1427
    fix r::real
huffman@21164
  1428
    assume r: "0<r"
huffman@21164
  1429
    from real_lbound_gt_zero [OF r d]
huffman@21164
  1430
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164
  1431
    with inj cont
huffman@21164
  1432
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164
  1433
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164
  1434
    from isCont_inj_range [OF e this]
huffman@21164
  1435
    obtain e' where e': "0 < e'"
huffman@21164
  1436
        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
  1437
          by blast
huffman@21164
  1438
    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
  1439
    proof (intro exI conjI)
huffman@23441
  1440
      show "0<e'" using e' .
huffman@21164
  1441
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164
  1442
      proof (intro strip)
huffman@21164
  1443
        fix z::real
huffman@21164
  1444
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164
  1445
        with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164
  1446
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164
  1447
      qed
huffman@21164
  1448
    qed
huffman@21164
  1449
  qed
huffman@21164
  1450
qed
huffman@21164
  1451
huffman@23041
  1452
text {* Derivative of inverse function *}
huffman@23041
  1453
huffman@23041
  1454
lemma DERIV_inverse_function:
huffman@23041
  1455
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1456
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1457
  assumes neq: "D \<noteq> 0"
huffman@23044
  1458
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1459
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1460
  assumes cont: "isCont g x"
huffman@23041
  1461
  shows "DERIV g x :> inverse D"
huffman@23041
  1462
unfolding DERIV_iff2
huffman@23044
  1463
proof (rule LIM_equal2)
huffman@23044
  1464
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1465
    using a b by arith 
huffman@23044
  1466
next
huffman@23041
  1467
  fix y
huffman@23044
  1468
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1469
  hence "a < y" and "y < b" 
huffman@23044
  1470
    by (simp_all add: abs_less_iff)
huffman@23041
  1471
  thus "(g y - g x) / (y - x) =
huffman@23041
  1472
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1473
    by (simp add: inj)
huffman@23041
  1474
next
huffman@23041
  1475
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1476
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1477
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1478
    using inj a b by simp
huffman@23041
  1479
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
huffman@23041
  1480
  proof (safe intro!: exI)
huffman@23044
  1481
    show "0 < min (x - a) (b - x)"
huffman@23044
  1482
      using a b by simp
huffman@23041
  1483
  next
huffman@23041
  1484
    fix y
huffman@23044
  1485
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1486
    hence y: "a < y" "y < b"
huffman@23044
  1487
      by (simp_all add: abs_less_iff)
huffman@23041
  1488
    assume "g y = g x"
huffman@23041
  1489
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1490
    hence "y = x" using inj y a b by simp
huffman@23041
  1491
    also assume "y \<noteq> x"
huffman@23041
  1492
    finally show False by simp
huffman@23041
  1493
  qed
huffman@23041
  1494
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1495
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1496
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1497
        -- x --> inverse D"
huffman@23041
  1498
    using neq by (rule LIM_inverse)
huffman@23041
  1499
qed
huffman@23041
  1500
huffman@29975
  1501
huffman@29975
  1502
subsection {* Generalized Mean Value Theorem *}
huffman@29975
  1503
huffman@21164
  1504
theorem GMVT:
huffman@21784
  1505
  fixes a b :: real
huffman@21164
  1506
  assumes alb: "a < b"
huffman@21164
  1507
  and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
huffman@21164
  1508
  and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
huffman@21164
  1509
  and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
huffman@21164
  1510
  and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1511
  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
  1512
proof -
huffman@21164
  1513
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
huffman@21164
  1514
  from prems have "a < b" by simp
huffman@21164
  1515
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@21164
  1516
  proof -
huffman@21164
  1517
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. f b - f a) x" by simp
huffman@21164
  1518
    with gc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (f b - f a) * g x) x"
huffman@21164
  1519
      by (auto intro: isCont_mult)
huffman@21164
  1520
    moreover
huffman@21164
  1521
    have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. g b - g a) x" by simp
huffman@21164
  1522
    with fc have "\<forall>x. a <= x \<and> x <= b \<longrightarrow> isCont (\<lambda>x. (g b - g a) * f x) x"
huffman@21164
  1523
      by (auto intro: isCont_mult)
huffman@21164
  1524
    ultimately show ?thesis
huffman@21164
  1525
      by (fastsimp intro: isCont_diff)
huffman@21164
  1526
  qed
huffman@21164
  1527
  moreover
huffman@21164
  1528
  have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@21164
  1529
  proof -
huffman@21164
  1530
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. f b - f a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1531
    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
  1532
    moreover
huffman@21164
  1533
    have "\<forall>x. a < x \<and> x < b \<longrightarrow> (\<lambda>x. g b - g a) differentiable x" by (simp add: differentiable_const)
huffman@21164
  1534
    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
  1535
    ultimately show ?thesis by (simp add: differentiable_diff)
huffman@21164
  1536
  qed
huffman@21164
  1537
  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
  1538
  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
  1539
  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
  1540
huffman@21164
  1541
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1542
  with gd have "g differentiable c" by simp
huffman@21164
  1543
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1544
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1545
huffman@21164
  1546
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1547
  with fd have "f differentiable c" by simp
huffman@21164
  1548
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1549
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1550
huffman@21164
  1551
  from cdef have "DERIV ?h c :> l" by auto
huffman@21164
  1552
  moreover
huffman@21164
  1553
  {
huffman@23441
  1554
    have "DERIV (\<lambda>x. (f b - f a) * g x) c :> g'c * (f b - f a)"
huffman@21164
  1555
      apply (insert DERIV_const [where k="f b - f a"])
huffman@21164
  1556
      apply (drule meta_spec [of _ c])
huffman@23441
  1557
      apply (drule DERIV_mult [OF _ g'cdef])
huffman@23441
  1558
      by simp
huffman@23441
  1559
    moreover have "DERIV (\<lambda>x. (g b - g a) * f x) c :> f'c * (g b - g a)"
huffman@21164
  1560
      apply (insert DERIV_const [where k="g b - g a"])
huffman@21164
  1561
      apply (drule meta_spec [of _ c])
huffman@23441
  1562
      apply (drule DERIV_mult [OF _ f'cdef])
huffman@23441
  1563
      by simp
huffman@21164
  1564
    ultimately have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
huffman@21164
  1565
      by (simp add: DERIV_diff)
huffman@21164
  1566
  }
huffman@21164
  1567
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1568
huffman@21164
  1569
  {
huffman@21164
  1570
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1571
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1572
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1573
  }
huffman@21164
  1574
  moreover
huffman@21164
  1575
  {
huffman@21164
  1576
    have "?h b - ?h a =
huffman@21164
  1577
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1578
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1579
      by (simp add: algebra_simps)
huffman@21164
  1580
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1581
  }
huffman@21164
  1582
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1583
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1584
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1585
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1586
huffman@21164
  1587
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1588
qed
huffman@21164
  1589
huffman@29470
  1590
huffman@29166
  1591
subsection {* Theorems about Limits *}
huffman@29166
  1592
huffman@29166
  1593
(* need to rename second isCont_inverse *)
huffman@29166
  1594
huffman@29166
  1595
lemma isCont_inv_fun:
huffman@29166
  1596
  fixes f g :: "real \<Rightarrow> real"
huffman@29166
  1597
  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
huffman@29166
  1598
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
huffman@29166
  1599
      ==> isCont g (f x)"
huffman@29166
  1600
by (rule isCont_inverse_function)
huffman@29166
  1601
huffman@29166
  1602
lemma isCont_inv_fun_inv:
huffman@29166
  1603
  fixes f g :: "real \<Rightarrow> real"
huffman@29166
  1604
  shows "[| 0 < d;  
huffman@29166
  1605
         \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
huffman@29166
  1606
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
huffman@29166
  1607
       ==> \<exists>e. 0 < e &  
huffman@29166
  1608
             (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
huffman@29166
  1609
apply (drule isCont_inj_range)
huffman@29166
  1610
prefer 2 apply (assumption, assumption, auto)
huffman@29166
  1611
apply (rule_tac x = e in exI, auto)
huffman@29166
  1612
apply (rotate_tac 2)
huffman@29166
  1613
apply (drule_tac x = y in spec, auto)
huffman@29166
  1614
done
huffman@29166
  1615
huffman@29166
  1616
huffman@29166
  1617
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
huffman@29166
  1618
lemma LIM_fun_gt_zero:
huffman@29166
  1619
     "[| f -- c --> (l::real); 0 < l |]  
huffman@29166
  1620
         ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
huffman@31338
  1621
apply (auto simp add: LIM_eq)
huffman@29166
  1622
apply (drule_tac x = "l/2" in spec, safe, force)
huffman@29166
  1623
apply (rule_tac x = s in exI)
huffman@29166
  1624
apply (auto simp only: abs_less_iff)
huffman@29166
  1625
done
huffman@29166
  1626
huffman@29166
  1627
lemma LIM_fun_less_zero:
huffman@29166
  1628
     "[| f -- c --> (l::real); l < 0 |]  
huffman@29166
  1629
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
huffman@31338
  1630
apply (auto simp add: LIM_eq)
huffman@29166
  1631
apply (drule_tac x = "-l/2" in spec, safe, force)
huffman@29166
  1632
apply (rule_tac x = s in exI)
huffman@29166
  1633
apply (auto simp only: abs_less_iff)
huffman@29166
  1634
done
huffman@29166
  1635
huffman@29166
  1636
huffman@29166
  1637
lemma LIM_fun_not_zero:
huffman@29166
  1638
     "[| f -- c --> (l::real); l \<noteq> 0 |] 
huffman@29166
  1639
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
huffman@29166
  1640
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
huffman@29166
  1641
apply (drule LIM_fun_less_zero)
huffman@29166
  1642
apply (drule_tac [3] LIM_fun_gt_zero)
huffman@29166
  1643
apply force+
huffman@29166
  1644
done
huffman@29166
  1645
huffman@21164
  1646
end