src/HOL/Deriv.thy
author hoelzl
Mon Dec 03 18:19:11 2012 +0100 (2012-12-03)
changeset 50330 d0b12171118e
parent 50329 9bd6b6b8a554
child 50331 4b6dc5077e98
permissions -rw-r--r--
conversion rules for at, at_left and at_right; applied to l'Hopital's rules.
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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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primrec
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  Bolzano_bisect :: "(real \<times> real \<Rightarrow> bool) \<Rightarrow> real \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real \<times> real" where
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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 tendsto_const)
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lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x :> 1"
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  by (simp add: deriv_def tendsto_const cong: LIM_cong)
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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 tendsto_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 tendsto_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_minus 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 tendsto_mult tendsto_ident_at)
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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 tendsto_intros 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 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 tendsto_intros 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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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_tendsto_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 tendsto_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)
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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 mult_commute)
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lemma DERIV_chain2: "[| DERIV f (g x) :> Da; DERIV g x :> Db |] ==> DERIV (%x. f (g x)) x :> Da * Db"
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by (auto dest: DERIV_chain simp add: o_def)
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text {* Derivative of linear multiplication *}
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lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x :> c"
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by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
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lemma DERIV_pow: "DERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
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apply (cut_tac DERIV_power [OF DERIV_ident])
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apply (simp add: real_of_nat_def)
huffman@21164
   283
done
huffman@21164
   284
wenzelm@31899
   285
text {* Power of @{text "-1"} *}
huffman@21164
   286
huffman@21784
   287
lemma DERIV_inverse:
haftmann@31017
   288
  fixes x :: "'a::{real_normed_field}"
huffman@21784
   289
  shows "x \<noteq> 0 ==> DERIV (%x. inverse(x)) x :> (-(inverse x ^ Suc (Suc 0)))"
huffman@30273
   290
by (drule DERIV_inverse' [OF DERIV_ident]) simp
huffman@21164
   291
wenzelm@31899
   292
text {* Derivative of inverse *}
huffman@21784
   293
lemma DERIV_inverse_fun:
haftmann@31017
   294
  fixes x :: "'a::{real_normed_field}"
huffman@21784
   295
  shows "[| DERIV f x :> d; f(x) \<noteq> 0 |]
huffman@21784
   296
      ==> DERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
huffman@30273
   297
by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
huffman@21164
   298
wenzelm@31899
   299
text {* Derivative of quotient *}
huffman@21784
   300
lemma DERIV_quotient:
haftmann@31017
   301
  fixes x :: "'a::{real_normed_field}"
huffman@21784
   302
  shows "[| DERIV f x :> d; DERIV g x :> e; g(x) \<noteq> 0 |]
huffman@21784
   303
       ==> DERIV (%y. f(y) / (g y)) x :> (d*g(x) - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
huffman@30273
   304
by (drule (2) DERIV_divide) (simp add: mult_commute)
huffman@21164
   305
wenzelm@31899
   306
text {* @{text "DERIV_intros"} *}
wenzelm@31899
   307
ML {*
wenzelm@31902
   308
structure Deriv_Intros = Named_Thms
wenzelm@31899
   309
(
wenzelm@45294
   310
  val name = @{binding DERIV_intros}
wenzelm@31899
   311
  val description = "DERIV introduction rules"
wenzelm@31899
   312
)
wenzelm@31899
   313
*}
hoelzl@31880
   314
wenzelm@31902
   315
setup Deriv_Intros.setup
hoelzl@31880
   316
hoelzl@31880
   317
lemma DERIV_cong: "\<lbrakk> DERIV f x :> X ; X = Y \<rbrakk> \<Longrightarrow> DERIV f x :> Y"
hoelzl@31880
   318
  by simp
hoelzl@31880
   319
hoelzl@31880
   320
declare
hoelzl@31880
   321
  DERIV_const[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   322
  DERIV_ident[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   323
  DERIV_add[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   324
  DERIV_minus[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   325
  DERIV_mult[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   326
  DERIV_diff[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   327
  DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   328
  DERIV_divide[THEN DERIV_cong, DERIV_intros]
hoelzl@31880
   329
  DERIV_power[where 'a=real, THEN DERIV_cong,
hoelzl@31880
   330
              unfolded real_of_nat_def[symmetric], DERIV_intros]
hoelzl@31880
   331
  DERIV_setsum[THEN DERIV_cong, DERIV_intros]
huffman@22984
   332
wenzelm@31899
   333
huffman@22984
   334
subsection {* Differentiability predicate *}
huffman@21164
   335
huffman@29169
   336
definition
huffman@29169
   337
  differentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
huffman@29169
   338
    (infixl "differentiable" 60) where
huffman@29169
   339
  "f differentiable x = (\<exists>D. DERIV f x :> D)"
huffman@29169
   340
huffman@29169
   341
lemma differentiableE [elim?]:
huffman@29169
   342
  assumes "f differentiable x"
huffman@29169
   343
  obtains df where "DERIV f x :> df"
wenzelm@41550
   344
  using assms unfolding differentiable_def ..
huffman@29169
   345
huffman@21164
   346
lemma differentiableD: "f differentiable x ==> \<exists>D. DERIV f x :> D"
huffman@21164
   347
by (simp add: differentiable_def)
huffman@21164
   348
huffman@21164
   349
lemma differentiableI: "DERIV f x :> D ==> f differentiable x"
huffman@21164
   350
by (force simp add: differentiable_def)
huffman@21164
   351
huffman@29169
   352
lemma differentiable_ident [simp]: "(\<lambda>x. x) differentiable x"
huffman@29169
   353
  by (rule DERIV_ident [THEN differentiableI])
huffman@29169
   354
huffman@29169
   355
lemma differentiable_const [simp]: "(\<lambda>z. a) differentiable x"
huffman@29169
   356
  by (rule DERIV_const [THEN differentiableI])
huffman@21164
   357
huffman@29169
   358
lemma differentiable_compose:
huffman@29169
   359
  assumes f: "f differentiable (g x)"
huffman@29169
   360
  assumes g: "g differentiable x"
huffman@29169
   361
  shows "(\<lambda>x. f (g x)) differentiable x"
huffman@29169
   362
proof -
huffman@29169
   363
  from `f differentiable (g x)` obtain df where "DERIV f (g x) :> df" ..
huffman@29169
   364
  moreover
huffman@29169
   365
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
huffman@29169
   366
  ultimately
huffman@29169
   367
  have "DERIV (\<lambda>x. f (g x)) x :> df * dg" by (rule DERIV_chain2)
huffman@29169
   368
  thus ?thesis by (rule differentiableI)
huffman@29169
   369
qed
huffman@29169
   370
huffman@29169
   371
lemma differentiable_sum [simp]:
huffman@21164
   372
  assumes "f differentiable x"
huffman@21164
   373
  and "g differentiable x"
huffman@21164
   374
  shows "(\<lambda>x. f x + g x) differentiable x"
huffman@21164
   375
proof -
huffman@29169
   376
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@29169
   377
  moreover
huffman@29169
   378
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
huffman@29169
   379
  ultimately
huffman@29169
   380
  have "DERIV (\<lambda>x. f x + g x) x :> df + dg" by (rule DERIV_add)
huffman@29169
   381
  thus ?thesis by (rule differentiableI)
huffman@29169
   382
qed
huffman@29169
   383
huffman@29169
   384
lemma differentiable_minus [simp]:
huffman@29169
   385
  assumes "f differentiable x"
huffman@29169
   386
  shows "(\<lambda>x. - f x) differentiable x"
huffman@29169
   387
proof -
huffman@29169
   388
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@29169
   389
  hence "DERIV (\<lambda>x. - f x) x :> - df" by (rule DERIV_minus)
huffman@29169
   390
  thus ?thesis by (rule differentiableI)
huffman@21164
   391
qed
huffman@21164
   392
huffman@29169
   393
lemma differentiable_diff [simp]:
huffman@21164
   394
  assumes "f differentiable x"
huffman@29169
   395
  assumes "g differentiable x"
huffman@21164
   396
  shows "(\<lambda>x. f x - g x) differentiable x"
wenzelm@41550
   397
  unfolding diff_minus using assms by simp
huffman@29169
   398
huffman@29169
   399
lemma differentiable_mult [simp]:
huffman@29169
   400
  assumes "f differentiable x"
huffman@29169
   401
  assumes "g differentiable x"
huffman@29169
   402
  shows "(\<lambda>x. f x * g x) differentiable x"
huffman@21164
   403
proof -
huffman@29169
   404
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@21164
   405
  moreover
huffman@29169
   406
  from `g differentiable x` obtain dg where "DERIV g x :> dg" ..
huffman@29169
   407
  ultimately
huffman@29169
   408
  have "DERIV (\<lambda>x. f x * g x) x :> df * g x + dg * f x" by (rule DERIV_mult)
huffman@29169
   409
  thus ?thesis by (rule differentiableI)
huffman@21164
   410
qed
huffman@21164
   411
huffman@29169
   412
lemma differentiable_inverse [simp]:
huffman@29169
   413
  assumes "f differentiable x" and "f x \<noteq> 0"
huffman@29169
   414
  shows "(\<lambda>x. inverse (f x)) differentiable x"
huffman@21164
   415
proof -
huffman@29169
   416
  from `f differentiable x` obtain df where "DERIV f x :> df" ..
huffman@29169
   417
  hence "DERIV (\<lambda>x. inverse (f x)) x :> - (inverse (f x) * df * inverse (f x))"
huffman@29169
   418
    using `f x \<noteq> 0` by (rule DERIV_inverse')
huffman@29169
   419
  thus ?thesis by (rule differentiableI)
huffman@21164
   420
qed
huffman@21164
   421
huffman@29169
   422
lemma differentiable_divide [simp]:
huffman@29169
   423
  assumes "f differentiable x"
huffman@29169
   424
  assumes "g differentiable x" and "g x \<noteq> 0"
huffman@29169
   425
  shows "(\<lambda>x. f x / g x) differentiable x"
wenzelm@41550
   426
  unfolding divide_inverse using assms by simp
huffman@29169
   427
huffman@29169
   428
lemma differentiable_power [simp]:
haftmann@31017
   429
  fixes f :: "'a::{real_normed_field} \<Rightarrow> 'a"
huffman@29169
   430
  assumes "f differentiable x"
huffman@29169
   431
  shows "(\<lambda>x. f x ^ n) differentiable x"
wenzelm@41550
   432
  apply (induct n)
wenzelm@41550
   433
  apply simp
wenzelm@41550
   434
  apply (simp add: assms)
wenzelm@41550
   435
  done
huffman@29169
   436
huffman@22984
   437
huffman@21164
   438
subsection {* Nested Intervals and Bisection *}
huffman@21164
   439
huffman@21164
   440
text{*Lemmas about nested intervals and proof by bisection (cf.Harrison).
huffman@21164
   441
     All considerably tidied by lcp.*}
huffman@21164
   442
huffman@21164
   443
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
   444
apply (induct "no")
huffman@21164
   445
apply (auto intro: order_trans)
huffman@21164
   446
done
huffman@21164
   447
huffman@21164
   448
lemma f_inc_g_dec_Beq_f: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   449
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   450
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   451
      ==> Bseq (f :: nat \<Rightarrow> real)"
huffman@21164
   452
apply (rule_tac k = "f 0" and K = "g 0" in BseqI2, rule allI)
huffman@44921
   453
apply (rule conjI)
huffman@21164
   454
apply (induct_tac "n")
huffman@21164
   455
apply (auto intro: order_trans)
huffman@44921
   456
apply (rule_tac y = "g n" in order_trans)
huffman@44921
   457
apply (induct_tac [2] "n")
huffman@21164
   458
apply (auto intro: order_trans)
huffman@21164
   459
done
huffman@21164
   460
huffman@21164
   461
lemma f_inc_g_dec_Beq_g: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   462
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   463
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   464
      ==> Bseq (g :: nat \<Rightarrow> real)"
huffman@21164
   465
apply (subst Bseq_minus_iff [symmetric])
huffman@21164
   466
apply (rule_tac g = "%x. - (f x)" in f_inc_g_dec_Beq_f)
huffman@21164
   467
apply auto
huffman@21164
   468
done
huffman@21164
   469
huffman@21164
   470
lemma f_inc_imp_le_lim:
huffman@21164
   471
  fixes f :: "nat \<Rightarrow> real"
huffman@21164
   472
  shows "\<lbrakk>\<forall>n. f n \<le> f (Suc n); convergent f\<rbrakk> \<Longrightarrow> f n \<le> lim f"
huffman@44921
   473
  by (rule incseq_le, simp add: incseq_SucI, simp add: convergent_LIMSEQ_iff)
huffman@21164
   474
huffman@31404
   475
lemma lim_uminus:
huffman@31404
   476
  fixes g :: "nat \<Rightarrow> 'a::real_normed_vector"
huffman@31404
   477
  shows "convergent g ==> lim (%x. - g x) = - (lim g)"
huffman@44568
   478
apply (rule tendsto_minus [THEN limI])
huffman@21164
   479
apply (simp add: convergent_LIMSEQ_iff)
huffman@21164
   480
done
huffman@21164
   481
huffman@21164
   482
lemma g_dec_imp_lim_le:
huffman@21164
   483
  fixes g :: "nat \<Rightarrow> real"
huffman@21164
   484
  shows "\<lbrakk>\<forall>n. g (Suc n) \<le> g(n); convergent g\<rbrakk> \<Longrightarrow> lim g \<le> g n"
huffman@44921
   485
  by (rule decseq_le, simp add: decseq_SucI, simp add: convergent_LIMSEQ_iff)
huffman@21164
   486
huffman@21164
   487
lemma lemma_nest: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   488
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   489
         \<forall>n. f(n) \<le> g(n) |]
huffman@21164
   490
      ==> \<exists>l m :: real. l \<le> m &  ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   491
                            ((\<forall>n. m \<le> g(n)) & g ----> m)"
huffman@21164
   492
apply (subgoal_tac "monoseq f & monoseq g")
huffman@21164
   493
prefer 2 apply (force simp add: LIMSEQ_iff monoseq_Suc)
huffman@21164
   494
apply (subgoal_tac "Bseq f & Bseq g")
huffman@21164
   495
prefer 2 apply (blast intro: f_inc_g_dec_Beq_f f_inc_g_dec_Beq_g)
huffman@21164
   496
apply (auto dest!: Bseq_monoseq_convergent simp add: convergent_LIMSEQ_iff)
huffman@21164
   497
apply (rule_tac x = "lim f" in exI)
huffman@21164
   498
apply (rule_tac x = "lim g" in exI)
huffman@21164
   499
apply (auto intro: LIMSEQ_le)
huffman@21164
   500
apply (auto simp add: f_inc_imp_le_lim g_dec_imp_lim_le convergent_LIMSEQ_iff)
huffman@21164
   501
done
huffman@21164
   502
huffman@21164
   503
lemma lemma_nest_unique: "[| \<forall>n. f(n) \<le> f(Suc n);
huffman@21164
   504
         \<forall>n. g(Suc n) \<le> g(n);
huffman@21164
   505
         \<forall>n. f(n) \<le> g(n);
huffman@21164
   506
         (%n. f(n) - g(n)) ----> 0 |]
huffman@21164
   507
      ==> \<exists>l::real. ((\<forall>n. f(n) \<le> l) & f ----> l) &
huffman@21164
   508
                ((\<forall>n. l \<le> g(n)) & g ----> l)"
huffman@21164
   509
apply (drule lemma_nest, auto)
huffman@21164
   510
apply (subgoal_tac "l = m")
huffman@44568
   511
apply (drule_tac [2] f = f in tendsto_diff)
huffman@21164
   512
apply (auto intro: LIMSEQ_unique)
huffman@21164
   513
done
huffman@21164
   514
huffman@21164
   515
text{*The universal quantifiers below are required for the declaration
huffman@21164
   516
  of @{text Bolzano_nest_unique} below.*}
huffman@21164
   517
huffman@21164
   518
lemma Bolzano_bisect_le:
huffman@21164
   519
 "a \<le> b ==> \<forall>n. fst (Bolzano_bisect P a b n) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   520
apply (rule allI)
huffman@21164
   521
apply (induct_tac "n")
huffman@21164
   522
apply (auto simp add: Let_def split_def)
huffman@21164
   523
done
huffman@21164
   524
huffman@21164
   525
lemma Bolzano_bisect_fst_le_Suc: "a \<le> b ==>
huffman@21164
   526
   \<forall>n. fst(Bolzano_bisect P a b n) \<le> fst (Bolzano_bisect P a b (Suc n))"
huffman@21164
   527
apply (rule allI)
huffman@21164
   528
apply (induct_tac "n")
huffman@21164
   529
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   530
done
huffman@21164
   531
huffman@21164
   532
lemma Bolzano_bisect_Suc_le_snd: "a \<le> b ==>
huffman@21164
   533
   \<forall>n. snd(Bolzano_bisect P a b (Suc n)) \<le> snd (Bolzano_bisect P a b n)"
huffman@21164
   534
apply (rule allI)
huffman@21164
   535
apply (induct_tac "n")
huffman@21164
   536
apply (auto simp add: Bolzano_bisect_le Let_def split_def)
huffman@21164
   537
done
huffman@21164
   538
huffman@21164
   539
lemma eq_divide_2_times_iff: "((x::real) = y / (2 * z)) = (2 * x = y/z)"
huffman@21164
   540
apply (auto)
huffman@21164
   541
apply (drule_tac f = "%u. (1/2) *u" in arg_cong)
huffman@21164
   542
apply (simp)
huffman@21164
   543
done
huffman@21164
   544
huffman@21164
   545
lemma Bolzano_bisect_diff:
huffman@21164
   546
     "a \<le> b ==>
huffman@21164
   547
      snd(Bolzano_bisect P a b n) - fst(Bolzano_bisect P a b n) =
huffman@21164
   548
      (b-a) / (2 ^ n)"
huffman@21164
   549
apply (induct "n")
huffman@21164
   550
apply (auto simp add: eq_divide_2_times_iff add_divide_distrib Let_def split_def)
huffman@21164
   551
done
huffman@21164
   552
huffman@21164
   553
lemmas Bolzano_nest_unique =
huffman@21164
   554
    lemma_nest_unique
huffman@21164
   555
    [OF Bolzano_bisect_fst_le_Suc Bolzano_bisect_Suc_le_snd Bolzano_bisect_le]
huffman@21164
   556
huffman@21164
   557
huffman@21164
   558
lemma not_P_Bolzano_bisect:
huffman@21164
   559
  assumes P:    "!!a b c. [| P(a,b); P(b,c); a \<le> b; b \<le> c|] ==> P(a,c)"
huffman@21164
   560
      and notP: "~ P(a,b)"
huffman@21164
   561
      and le:   "a \<le> b"
huffman@21164
   562
  shows "~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   563
proof (induct n)
huffman@23441
   564
  case 0 show ?case using notP by simp
huffman@21164
   565
 next
huffman@21164
   566
  case (Suc n)
huffman@21164
   567
  thus ?case
huffman@21164
   568
 by (auto simp del: surjective_pairing [symmetric]
huffman@21164
   569
             simp add: Let_def split_def Bolzano_bisect_le [OF le]
huffman@21164
   570
     P [of "fst (Bolzano_bisect P a b n)" _ "snd (Bolzano_bisect P a b n)"])
huffman@21164
   571
qed
huffman@21164
   572
huffman@21164
   573
(*Now we re-package P_prem as a formula*)
huffman@21164
   574
lemma not_P_Bolzano_bisect':
huffman@21164
   575
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   576
         ~ P(a,b);  a \<le> b |] ==>
huffman@21164
   577
      \<forall>n. ~ P(fst(Bolzano_bisect P a b n), snd(Bolzano_bisect P a b n))"
huffman@21164
   578
by (blast elim!: not_P_Bolzano_bisect [THEN [2] rev_notE])
huffman@21164
   579
huffman@21164
   580
huffman@21164
   581
huffman@21164
   582
lemma lemma_BOLZANO:
huffman@21164
   583
     "[| \<forall>a b c. P(a,b) & P(b,c) & a \<le> b & b \<le> c --> P(a,c);
huffman@21164
   584
         \<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   585
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b));
huffman@21164
   586
         a \<le> b |]
huffman@21164
   587
      ==> P(a,b)"
wenzelm@45600
   588
apply (rule Bolzano_nest_unique [where P=P, THEN exE], assumption+)
huffman@44568
   589
apply (rule tendsto_minus_cancel)
huffman@21164
   590
apply (simp (no_asm_simp) add: Bolzano_bisect_diff LIMSEQ_divide_realpow_zero)
huffman@21164
   591
apply (rule ccontr)
huffman@21164
   592
apply (drule not_P_Bolzano_bisect', assumption+)
huffman@21164
   593
apply (rename_tac "l")
huffman@21164
   594
apply (drule_tac x = l in spec, clarify)
huffman@31336
   595
apply (simp add: LIMSEQ_iff)
huffman@21164
   596
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   597
apply (drule_tac P = "%r. 0<r --> ?Q r" and x = "d/2" in spec)
huffman@21164
   598
apply (drule real_less_half_sum, auto)
huffman@21164
   599
apply (drule_tac x = "fst (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   600
apply (drule_tac x = "snd (Bolzano_bisect P a b (no + noa))" in spec)
huffman@21164
   601
apply safe
huffman@21164
   602
apply (simp_all (no_asm_simp))
huffman@21164
   603
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
   604
apply (simp (no_asm_simp) add: abs_if)
huffman@21164
   605
apply (rule real_sum_of_halves [THEN subst])
huffman@21164
   606
apply (rule add_strict_mono)
huffman@21164
   607
apply (simp_all add: diff_minus [symmetric])
huffman@21164
   608
done
huffman@21164
   609
huffman@21164
   610
huffman@21164
   611
lemma lemma_BOLZANO2: "((\<forall>a b c. (a \<le> b & b \<le> c & P(a,b) & P(b,c)) --> P(a,c)) &
huffman@21164
   612
       (\<forall>x. \<exists>d::real. 0 < d &
huffman@21164
   613
                (\<forall>a b. a \<le> x & x \<le> b & (b-a) < d --> P(a,b))))
huffman@21164
   614
      --> (\<forall>a b. a \<le> b --> P(a,b))"
huffman@21164
   615
apply clarify
huffman@21164
   616
apply (blast intro: lemma_BOLZANO)
huffman@21164
   617
done
huffman@21164
   618
huffman@21164
   619
huffman@21164
   620
subsection {* Intermediate Value Theorem *}
huffman@21164
   621
huffman@21164
   622
text {*Prove Contrapositive by Bisection*}
huffman@21164
   623
huffman@21164
   624
lemma IVT: "[| f(a::real) \<le> (y::real); y \<le> f(b);
huffman@21164
   625
         a \<le> b;
huffman@21164
   626
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x) |]
huffman@21164
   627
      ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   628
apply (rule contrapos_pp, assumption)
huffman@21164
   629
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
   630
apply safe
huffman@21164
   631
apply simp_all
huffman@31338
   632
apply (simp add: isCont_iff LIM_eq)
huffman@21164
   633
apply (rule ccontr)
huffman@21164
   634
apply (subgoal_tac "a \<le> x & x \<le> b")
huffman@21164
   635
 prefer 2
huffman@21164
   636
 apply simp
huffman@21164
   637
 apply (drule_tac P = "%d. 0<d --> ?P d" and x = 1 in spec, arith)
huffman@21164
   638
apply (drule_tac x = x in spec)+
huffman@21164
   639
apply simp
huffman@21164
   640
apply (drule_tac P = "%r. ?P r --> (\<exists>s>0. ?Q r s) " and x = "\<bar>y - f x\<bar>" in spec)
huffman@21164
   641
apply safe
huffman@21164
   642
apply simp
huffman@21164
   643
apply (drule_tac x = s in spec, clarify)
huffman@21164
   644
apply (cut_tac x = "f x" and y = y in linorder_less_linear, safe)
huffman@21164
   645
apply (drule_tac x = "ba-x" in spec)
huffman@21164
   646
apply (simp_all add: abs_if)
huffman@21164
   647
apply (drule_tac x = "aa-x" in spec)
huffman@21164
   648
apply (case_tac "x \<le> aa", simp_all)
huffman@21164
   649
done
huffman@21164
   650
huffman@21164
   651
lemma IVT2: "[| f(b::real) \<le> (y::real); y \<le> f(a);
huffman@21164
   652
         a \<le> b;
huffman@21164
   653
         (\<forall>x. a \<le> x & x \<le> b --> isCont f x)
huffman@21164
   654
      |] ==> \<exists>x. a \<le> x & x \<le> b & f(x) = y"
huffman@21164
   655
apply (subgoal_tac "- f a \<le> -y & -y \<le> - f b", clarify)
huffman@21164
   656
apply (drule IVT [where f = "%x. - f x"], assumption)
huffman@44233
   657
apply simp_all
huffman@21164
   658
done
huffman@21164
   659
huffman@21164
   660
(*HOL style here: object-level formulations*)
huffman@21164
   661
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
huffman@21164
   662
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   663
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   664
apply (blast intro: IVT)
huffman@21164
   665
done
huffman@21164
   666
huffman@21164
   667
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
huffman@21164
   668
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
huffman@21164
   669
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
huffman@21164
   670
apply (blast intro: IVT2)
huffman@21164
   671
done
huffman@21164
   672
huffman@29975
   673
huffman@29975
   674
subsection {* Boundedness of continuous functions *}
huffman@29975
   675
huffman@21164
   676
text{*By bisection, function continuous on closed interval is bounded above*}
huffman@21164
   677
huffman@21164
   678
lemma isCont_bounded:
huffman@21164
   679
     "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   680
      ==> \<exists>M::real. \<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M"
huffman@21164
   681
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
   682
apply safe
huffman@21164
   683
apply simp_all
huffman@21164
   684
apply (rename_tac x xa ya M Ma)
huffman@36777
   685
apply (metis linorder_not_less order_le_less order_trans)
huffman@21164
   686
apply (case_tac "a \<le> x & x \<le> b")
paulson@33654
   687
 prefer 2
paulson@33654
   688
 apply (rule_tac x = 1 in exI, force)
huffman@31338
   689
apply (simp add: LIM_eq isCont_iff)
huffman@21164
   690
apply (drule_tac x = x in spec, auto)
huffman@21164
   691
apply (erule_tac V = "\<forall>M. \<exists>x. a \<le> x & x \<le> b & ~ f x \<le> M" in thin_rl)
huffman@21164
   692
apply (drule_tac x = 1 in spec, auto)
huffman@21164
   693
apply (rule_tac x = s in exI, clarify)
huffman@21164
   694
apply (rule_tac x = "\<bar>f x\<bar> + 1" in exI, clarify)
huffman@21164
   695
apply (drule_tac x = "xa-x" in spec)
huffman@21164
   696
apply (auto simp add: abs_ge_self)
huffman@21164
   697
done
huffman@21164
   698
huffman@21164
   699
text{*Refine the above to existence of least upper bound*}
huffman@21164
   700
huffman@21164
   701
lemma lemma_reals_complete: "((\<exists>x. x \<in> S) & (\<exists>y. isUb UNIV S (y::real))) -->
huffman@21164
   702
      (\<exists>t. isLub UNIV S t)"
huffman@21164
   703
by (blast intro: reals_complete)
huffman@21164
   704
huffman@21164
   705
lemma isCont_has_Ub: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   706
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   707
                   (\<forall>N. N < M --> (\<exists>x. a \<le> x & x \<le> b & N < f(x)))"
huffman@21164
   708
apply (cut_tac S = "Collect (%y. \<exists>x. a \<le> x & x \<le> b & y = f x)"
huffman@21164
   709
        in lemma_reals_complete)
huffman@21164
   710
apply auto
huffman@21164
   711
apply (drule isCont_bounded, assumption)
huffman@21164
   712
apply (auto simp add: isUb_def leastP_def isLub_def setge_def setle_def)
huffman@21164
   713
apply (rule exI, auto)
huffman@21164
   714
apply (auto dest!: spec simp add: linorder_not_less)
huffman@21164
   715
done
huffman@21164
   716
huffman@21164
   717
text{*Now show that it attains its upper bound*}
huffman@21164
   718
huffman@21164
   719
lemma isCont_eq_Ub:
huffman@21164
   720
  assumes le: "a \<le> b"
huffman@21164
   721
      and con: "\<forall>x::real. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   722
  shows "\<exists>M::real. (\<forall>x. a \<le> x & x \<le> b --> f(x) \<le> M) &
huffman@21164
   723
             (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   724
proof -
huffman@21164
   725
  from isCont_has_Ub [OF le con]
huffman@21164
   726
  obtain M where M1: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
huffman@21164
   727
             and M2: "!!N. N<M ==> \<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x"  by blast
huffman@21164
   728
  show ?thesis
huffman@21164
   729
  proof (intro exI, intro conjI)
huffman@21164
   730
    show " \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M" by (rule M1)
huffman@21164
   731
    show "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M"
huffman@21164
   732
    proof (rule ccontr)
huffman@21164
   733
      assume "\<not> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
huffman@21164
   734
      with M1 have M3: "\<forall>x. a \<le> x & x \<le> b --> f x < M"
nipkow@44890
   735
        by (fastforce simp add: linorder_not_le [symmetric])
huffman@21164
   736
      hence "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. inverse (M - f x)) x"
huffman@44233
   737
        by (auto simp add: con)
huffman@21164
   738
      from isCont_bounded [OF le this]
huffman@21164
   739
      obtain k where k: "!!x. a \<le> x & x \<le> b --> inverse (M - f x) \<le> k" by auto
huffman@21164
   740
      have Minv: "!!x. a \<le> x & x \<le> b --> 0 < inverse (M - f (x))"
nipkow@29667
   741
        by (simp add: M3 algebra_simps)
huffman@21164
   742
      have "!!x. a \<le> x & x \<le> b --> inverse (M - f x) < k+1" using k
huffman@21164
   743
        by (auto intro: order_le_less_trans [of _ k])
huffman@21164
   744
      with Minv
huffman@21164
   745
      have "!!x. a \<le> x & x \<le> b --> inverse(k+1) < inverse(inverse(M - f x))"
huffman@21164
   746
        by (intro strip less_imp_inverse_less, simp_all)
huffman@21164
   747
      hence invlt: "!!x. a \<le> x & x \<le> b --> inverse(k+1) < M - f x"
huffman@21164
   748
        by simp
huffman@21164
   749
      have "M - inverse (k+1) < M" using k [of a] Minv [of a] le
huffman@21164
   750
        by (simp, arith)
huffman@21164
   751
      from M2 [OF this]
huffman@21164
   752
      obtain x where ax: "a \<le> x & x \<le> b & M - inverse(k+1) < f x" ..
huffman@21164
   753
      thus False using invlt [of x] by force
huffman@21164
   754
    qed
huffman@21164
   755
  qed
huffman@21164
   756
qed
huffman@21164
   757
huffman@21164
   758
huffman@21164
   759
text{*Same theorem for lower bound*}
huffman@21164
   760
huffman@21164
   761
lemma isCont_eq_Lb: "[| a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   762
         ==> \<exists>M::real. (\<forall>x::real. a \<le> x & x \<le> b --> M \<le> f(x)) &
huffman@21164
   763
                   (\<exists>x. a \<le> x & x \<le> b & f(x) = M)"
huffman@21164
   764
apply (subgoal_tac "\<forall>x. a \<le> x & x \<le> b --> isCont (%x. - (f x)) x")
huffman@21164
   765
prefer 2 apply (blast intro: isCont_minus)
huffman@21164
   766
apply (drule_tac f = "(%x. - (f x))" in isCont_eq_Ub)
huffman@21164
   767
apply safe
huffman@21164
   768
apply auto
huffman@21164
   769
done
huffman@21164
   770
huffman@21164
   771
huffman@21164
   772
text{*Another version.*}
huffman@21164
   773
huffman@21164
   774
lemma isCont_Lb_Ub: "[|a \<le> b; \<forall>x. a \<le> x & x \<le> b --> isCont f x |]
huffman@21164
   775
      ==> \<exists>L M::real. (\<forall>x::real. a \<le> x & x \<le> b --> L \<le> f(x) & f(x) \<le> M) &
huffman@21164
   776
          (\<forall>y. L \<le> y & y \<le> M --> (\<exists>x. a \<le> x & x \<le> b & (f(x) = y)))"
huffman@21164
   777
apply (frule isCont_eq_Lb)
huffman@21164
   778
apply (frule_tac [2] isCont_eq_Ub)
huffman@21164
   779
apply (assumption+, safe)
huffman@21164
   780
apply (rule_tac x = "f x" in exI)
huffman@21164
   781
apply (rule_tac x = "f xa" in exI, simp, safe)
huffman@21164
   782
apply (cut_tac x = x and y = xa in linorder_linear, safe)
huffman@21164
   783
apply (cut_tac f = f and a = x and b = xa and y = y in IVT_objl)
huffman@21164
   784
apply (cut_tac [2] f = f and a = xa and b = x and y = y in IVT2_objl, safe)
huffman@21164
   785
apply (rule_tac [2] x = xb in exI)
huffman@21164
   786
apply (rule_tac [4] x = xb in exI, simp_all)
huffman@21164
   787
done
huffman@21164
   788
huffman@21164
   789
huffman@29975
   790
subsection {* Local extrema *}
huffman@29975
   791
huffman@21164
   792
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   793
paulson@33654
   794
lemma DERIV_pos_inc_right:
huffman@21164
   795
  fixes f :: "real => real"
huffman@21164
   796
  assumes der: "DERIV f x :> l"
huffman@21164
   797
      and l:   "0 < l"
huffman@21164
   798
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   799
proof -
huffman@21164
   800
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   801
  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
   802
    by (simp add: diff_minus)
huffman@21164
   803
  then obtain s
huffman@21164
   804
        where s:   "0 < s"
huffman@21164
   805
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   806
    by auto
huffman@21164
   807
  thus ?thesis
huffman@21164
   808
  proof (intro exI conjI strip)
huffman@23441
   809
    show "0<s" using s .
huffman@21164
   810
    fix h::real
huffman@21164
   811
    assume "0 < h" "h < s"
huffman@21164
   812
    with all [of h] show "f x < f (x+h)"
huffman@21164
   813
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   814
    split add: split_if_asm)
huffman@21164
   815
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   816
      with l
huffman@21164
   817
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   818
      thus "f x < f (x+h)"
huffman@21164
   819
  by (simp add: pos_less_divide_eq h)
huffman@21164
   820
    qed
huffman@21164
   821
  qed
huffman@21164
   822
qed
huffman@21164
   823
paulson@33654
   824
lemma DERIV_neg_dec_left:
huffman@21164
   825
  fixes f :: "real => real"
huffman@21164
   826
  assumes der: "DERIV f x :> l"
huffman@21164
   827
      and l:   "l < 0"
huffman@21164
   828
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   829
proof -
huffman@21164
   830
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   831
  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
   832
    by (simp add: diff_minus)
huffman@21164
   833
  then obtain s
huffman@21164
   834
        where s:   "0 < s"
huffman@21164
   835
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   836
    by auto
huffman@21164
   837
  thus ?thesis
huffman@21164
   838
  proof (intro exI conjI strip)
huffman@23441
   839
    show "0<s" using s .
huffman@21164
   840
    fix h::real
huffman@21164
   841
    assume "0 < h" "h < s"
huffman@21164
   842
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
   843
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   844
    split add: split_if_asm)
huffman@21164
   845
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   846
      with l
huffman@21164
   847
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   848
      thus "f x < f (x-h)"
huffman@21164
   849
  by (simp add: pos_less_divide_eq h)
huffman@21164
   850
    qed
huffman@21164
   851
  qed
huffman@21164
   852
qed
huffman@21164
   853
paulson@33654
   854
paulson@33654
   855
lemma DERIV_pos_inc_left:
paulson@33654
   856
  fixes f :: "real => real"
paulson@33654
   857
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
paulson@33654
   858
  apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
hoelzl@41368
   859
  apply (auto simp add: DERIV_minus)
paulson@33654
   860
  done
paulson@33654
   861
paulson@33654
   862
lemma DERIV_neg_dec_right:
paulson@33654
   863
  fixes f :: "real => real"
paulson@33654
   864
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
paulson@33654
   865
  apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
hoelzl@41368
   866
  apply (auto simp add: DERIV_minus)
paulson@33654
   867
  done
paulson@33654
   868
huffman@21164
   869
lemma DERIV_local_max:
huffman@21164
   870
  fixes f :: "real => real"
huffman@21164
   871
  assumes der: "DERIV f x :> l"
huffman@21164
   872
      and d:   "0 < d"
huffman@21164
   873
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   874
  shows "l = 0"
huffman@21164
   875
proof (cases rule: linorder_cases [of l 0])
huffman@23441
   876
  case equal thus ?thesis .
huffman@21164
   877
next
huffman@21164
   878
  case less
paulson@33654
   879
  from DERIV_neg_dec_left [OF der less]
huffman@21164
   880
  obtain d' where d': "0 < d'"
huffman@21164
   881
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   882
  from real_lbound_gt_zero [OF d d']
huffman@21164
   883
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   884
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   885
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   886
next
huffman@21164
   887
  case greater
paulson@33654
   888
  from DERIV_pos_inc_right [OF der greater]
huffman@21164
   889
  obtain d' where d': "0 < d'"
huffman@21164
   890
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   891
  from real_lbound_gt_zero [OF d d']
huffman@21164
   892
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   893
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   894
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   895
qed
huffman@21164
   896
huffman@21164
   897
huffman@21164
   898
text{*Similar theorem for a local minimum*}
huffman@21164
   899
lemma DERIV_local_min:
huffman@21164
   900
  fixes f :: "real => real"
huffman@21164
   901
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   902
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   903
huffman@21164
   904
huffman@21164
   905
text{*In particular, if a function is locally flat*}
huffman@21164
   906
lemma DERIV_local_const:
huffman@21164
   907
  fixes f :: "real => real"
huffman@21164
   908
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   909
by (auto dest!: DERIV_local_max)
huffman@21164
   910
huffman@29975
   911
huffman@29975
   912
subsection {* Rolle's Theorem *}
huffman@29975
   913
huffman@21164
   914
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   915
lemma lemma_interval_lt:
huffman@21164
   916
     "[| a < x;  x < b |]
huffman@21164
   917
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
   918
huffman@22998
   919
apply (simp add: abs_less_iff)
huffman@21164
   920
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   921
apply (rule_tac x = "x-a" in exI)
huffman@21164
   922
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   923
done
huffman@21164
   924
huffman@21164
   925
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   926
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   927
apply (drule lemma_interval_lt, auto)
huffman@44921
   928
apply force
huffman@21164
   929
done
huffman@21164
   930
huffman@21164
   931
text{*Rolle's Theorem.
huffman@21164
   932
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   933
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
   934
   and @{term "f(a) = f(b)"},
huffman@21164
   935
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
   936
theorem Rolle:
huffman@21164
   937
  assumes lt: "a < b"
huffman@21164
   938
      and eq: "f(a) = f(b)"
huffman@21164
   939
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   940
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   941
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
   942
proof -
huffman@21164
   943
  have le: "a \<le> b" using lt by simp
huffman@21164
   944
  from isCont_eq_Ub [OF le con]
huffman@21164
   945
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
   946
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
   947
    by blast
huffman@21164
   948
  from isCont_eq_Lb [OF le con]
huffman@21164
   949
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
   950
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
   951
    by blast
huffman@21164
   952
  show ?thesis
huffman@21164
   953
  proof cases
huffman@21164
   954
    assume axb: "a < x & x < b"
huffman@21164
   955
        --{*@{term f} attains its maximum within the interval*}
chaieb@27668
   956
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
   957
    from lemma_interval [OF ax xb]
huffman@21164
   958
    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
   959
      by blast
huffman@21164
   960
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
   961
      by blast
huffman@21164
   962
    from differentiableD [OF dif [OF axb]]
huffman@21164
   963
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   964
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
   965
        --{*the derivative at a local maximum is zero*}
huffman@21164
   966
    thus ?thesis using ax xb der by auto
huffman@21164
   967
  next
huffman@21164
   968
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
   969
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
   970
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
   971
    show ?thesis
huffman@21164
   972
    proof cases
huffman@21164
   973
      assume ax'b: "a < x' & x' < b"
huffman@21164
   974
        --{*@{term f} attains its minimum within the interval*}
chaieb@27668
   975
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
   976
      from lemma_interval [OF ax' x'b]
huffman@21164
   977
      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
   978
  by blast
huffman@21164
   979
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
   980
  by blast
huffman@21164
   981
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
   982
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
   983
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
   984
        --{*the derivative at a local minimum is zero*}
huffman@21164
   985
      thus ?thesis using ax' x'b der by auto
huffman@21164
   986
    next
huffman@21164
   987
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
   988
        --{*@{term f} is constant througout the interval*}
huffman@21164
   989
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
   990
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
   991
      from dense [OF lt]
huffman@21164
   992
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
   993
      from lemma_interval [OF ar rb]
huffman@21164
   994
      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
   995
  by blast
huffman@21164
   996
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
   997
      proof (clarify)
huffman@21164
   998
        fix z::real
huffman@21164
   999
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1000
        show "f z = f b"
huffman@21164
  1001
        proof (rule order_antisym)
huffman@21164
  1002
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1003
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1004
        qed
huffman@21164
  1005
      qed
huffman@21164
  1006
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1007
      proof (intro strip)
huffman@21164
  1008
        fix y::real
huffman@21164
  1009
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1010
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1011
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1012
      qed
huffman@21164
  1013
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1014
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1015
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1016
        --{*the derivative of a constant function is zero*}
huffman@21164
  1017
      thus ?thesis using ar rb der by auto
huffman@21164
  1018
    qed
huffman@21164
  1019
  qed
huffman@21164
  1020
qed
huffman@21164
  1021
huffman@21164
  1022
huffman@21164
  1023
subsection{*Mean Value Theorem*}
huffman@21164
  1024
huffman@21164
  1025
lemma lemma_MVT:
huffman@21164
  1026
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
huffman@21164
  1027
proof cases
huffman@21164
  1028
  assume "a=b" thus ?thesis by simp
huffman@21164
  1029
next
huffman@21164
  1030
  assume "a\<noteq>b"
huffman@21164
  1031
  hence ba: "b-a \<noteq> 0" by arith
huffman@21164
  1032
  show ?thesis
huffman@21164
  1033
    by (rule real_mult_left_cancel [OF ba, THEN iffD1],
huffman@21164
  1034
        simp add: right_diff_distrib,
huffman@21164
  1035
        simp add: left_diff_distrib)
huffman@21164
  1036
qed
huffman@21164
  1037
huffman@21164
  1038
theorem MVT:
huffman@21164
  1039
  assumes lt:  "a < b"
huffman@21164
  1040
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1041
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
  1042
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
  1043
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1044
proof -
huffman@21164
  1045
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@44233
  1046
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
huffman@44233
  1047
    using con by (fast intro: isCont_intros)
huffman@21164
  1048
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
  1049
  proof (clarify)
huffman@21164
  1050
    fix x::real
huffman@21164
  1051
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1052
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1053
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1054
    show "?F differentiable x"
huffman@21164
  1055
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1056
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1057
  qed
huffman@21164
  1058
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1059
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1060
    by blast
huffman@21164
  1061
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1062
    by (rule DERIV_cmult_Id)
huffman@21164
  1063
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1064
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1065
    by (rule DERIV_add [OF der])
huffman@21164
  1066
  show ?thesis
huffman@21164
  1067
  proof (intro exI conjI)
huffman@23441
  1068
    show "a < z" using az .
huffman@23441
  1069
    show "z < b" using zb .
huffman@21164
  1070
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1071
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1072
  qed
huffman@21164
  1073
qed
huffman@21164
  1074
hoelzl@29803
  1075
lemma MVT2:
hoelzl@29803
  1076
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803
  1077
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803
  1078
apply (drule MVT)
hoelzl@29803
  1079
apply (blast intro: DERIV_isCont)
hoelzl@29803
  1080
apply (force dest: order_less_imp_le simp add: differentiable_def)
hoelzl@29803
  1081
apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803
  1082
done
hoelzl@29803
  1083
huffman@21164
  1084
huffman@21164
  1085
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1086
huffman@21164
  1087
lemma DERIV_isconst_end:
huffman@21164
  1088
  fixes f :: "real => real"
huffman@21164
  1089
  shows "[| a < b;
huffman@21164
  1090
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1091
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1092
        ==> f b = f a"
huffman@21164
  1093
apply (drule MVT, assumption)
huffman@21164
  1094
apply (blast intro: differentiableI)
huffman@21164
  1095
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1096
done
huffman@21164
  1097
huffman@21164
  1098
lemma DERIV_isconst1:
huffman@21164
  1099
  fixes f :: "real => real"
huffman@21164
  1100
  shows "[| a < b;
huffman@21164
  1101
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1102
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1103
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1104
apply safe
huffman@21164
  1105
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1106
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1107
done
huffman@21164
  1108
huffman@21164
  1109
lemma DERIV_isconst2:
huffman@21164
  1110
  fixes f :: "real => real"
huffman@21164
  1111
  shows "[| a < b;
huffman@21164
  1112
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1113
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1114
         a \<le> x; x \<le> b |]
huffman@21164
  1115
        ==> f x = f a"
huffman@21164
  1116
apply (blast dest: DERIV_isconst1)
huffman@21164
  1117
done
huffman@21164
  1118
hoelzl@29803
  1119
lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803
  1120
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803
  1121
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1122
  shows "f x = f y"
hoelzl@29803
  1123
proof (cases "x = y")
hoelzl@29803
  1124
  case False
hoelzl@29803
  1125
  let ?a = "min x y"
hoelzl@29803
  1126
  let ?b = "max x y"
hoelzl@29803
  1127
  
hoelzl@29803
  1128
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1129
  proof (rule allI, rule impI)
hoelzl@29803
  1130
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hoelzl@29803
  1131
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hoelzl@29803
  1132
    hence "z \<in> {a<..<b}" by auto
hoelzl@29803
  1133
    thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1134
  qed
hoelzl@29803
  1135
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803
  1136
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803
  1137
hoelzl@29803
  1138
  have "?a < ?b" using `x \<noteq> y` by auto
hoelzl@29803
  1139
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1140
  show ?thesis by auto
hoelzl@29803
  1141
qed auto
hoelzl@29803
  1142
huffman@21164
  1143
lemma DERIV_isconst_all:
huffman@21164
  1144
  fixes f :: "real => real"
huffman@21164
  1145
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1146
apply (rule linorder_cases [of x y])
huffman@21164
  1147
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1148
done
huffman@21164
  1149
huffman@21164
  1150
lemma DERIV_const_ratio_const:
huffman@21784
  1151
  fixes f :: "real => real"
huffman@21784
  1152
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1153
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1154
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1155
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
nipkow@23477
  1156
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
huffman@21164
  1157
done
huffman@21164
  1158
huffman@21164
  1159
lemma DERIV_const_ratio_const2:
huffman@21784
  1160
  fixes f :: "real => real"
huffman@21784
  1161
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1162
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1163
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1164
done
huffman@21164
  1165
huffman@21164
  1166
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1167
by (simp)
huffman@21164
  1168
huffman@21164
  1169
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1170
by (simp)
huffman@21164
  1171
huffman@21164
  1172
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1173
huffman@21164
  1174
lemma DERIV_const_average:
huffman@21164
  1175
  fixes v :: "real => real"
huffman@21164
  1176
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1177
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1178
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1179
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1180
  case equal with neq show ?thesis by simp
huffman@21164
  1181
next
huffman@21164
  1182
  case less
huffman@21164
  1183
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1184
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1185
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1186
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1187
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1188
  ultimately show ?thesis using neq by force
huffman@21164
  1189
next
huffman@21164
  1190
  case greater
huffman@21164
  1191
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1192
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1193
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1194
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1195
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1196
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1197
qed
huffman@21164
  1198
paulson@33654
  1199
(* A function with positive derivative is increasing. 
paulson@33654
  1200
   A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654
  1201
*)
paulson@33654
  1202
lemma DERIV_pos_imp_increasing:
paulson@33654
  1203
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1204
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
paulson@33654
  1205
  shows "f a < f b"
paulson@33654
  1206
proof (rule ccontr)
wenzelm@41550
  1207
  assume f: "~ f a < f b"
wenzelm@33690
  1208
  have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1209
      & f b - f a = (b - a) * l"
wenzelm@33690
  1210
    apply (rule MVT)
wenzelm@33690
  1211
      using assms
wenzelm@33690
  1212
      apply auto
wenzelm@33690
  1213
      apply (metis DERIV_isCont)
huffman@36777
  1214
     apply (metis differentiableI less_le)
wenzelm@33690
  1215
    done
wenzelm@41550
  1216
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
paulson@33654
  1217
      and "f b - f a = (b - a) * l"
paulson@33654
  1218
    by auto
wenzelm@41550
  1219
  with assms f have "~(l > 0)"
huffman@36777
  1220
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550
  1221
  with assms z show False
huffman@36777
  1222
    by (metis DERIV_unique less_le)
paulson@33654
  1223
qed
paulson@33654
  1224
noschinl@45791
  1225
lemma DERIV_nonneg_imp_nondecreasing:
paulson@33654
  1226
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1227
  assumes "a \<le> b" and
paulson@33654
  1228
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654
  1229
  shows "f a \<le> f b"
paulson@33654
  1230
proof (rule ccontr, cases "a = b")
wenzelm@41550
  1231
  assume "~ f a \<le> f b" and "a = b"
wenzelm@41550
  1232
  then show False by auto
haftmann@37891
  1233
next
haftmann@37891
  1234
  assume A: "~ f a \<le> f b"
haftmann@37891
  1235
  assume B: "a ~= b"
paulson@33654
  1236
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1237
      & f b - f a = (b - a) * l"
wenzelm@33690
  1238
    apply -
wenzelm@33690
  1239
    apply (rule MVT)
wenzelm@33690
  1240
      apply auto
wenzelm@33690
  1241
      apply (metis DERIV_isCont)
huffman@36777
  1242
     apply (metis differentiableI less_le)
paulson@33654
  1243
    done
wenzelm@41550
  1244
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
haftmann@37891
  1245
      and C: "f b - f a = (b - a) * l"
paulson@33654
  1246
    by auto
haftmann@37891
  1247
  with A have "a < b" "f b < f a" by auto
haftmann@37891
  1248
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
huffman@45051
  1249
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@41550
  1250
  with assms z show False
paulson@33654
  1251
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1252
qed
paulson@33654
  1253
paulson@33654
  1254
lemma DERIV_neg_imp_decreasing:
paulson@33654
  1255
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1256
  assumes "a < b" and
paulson@33654
  1257
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654
  1258
  shows "f a > f b"
paulson@33654
  1259
proof -
paulson@33654
  1260
  have "(%x. -f x) a < (%x. -f x) b"
paulson@33654
  1261
    apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
wenzelm@33690
  1262
    using assms
wenzelm@33690
  1263
    apply auto
paulson@33654
  1264
    apply (metis DERIV_minus neg_0_less_iff_less)
paulson@33654
  1265
    done
paulson@33654
  1266
  thus ?thesis
paulson@33654
  1267
    by simp
paulson@33654
  1268
qed
paulson@33654
  1269
paulson@33654
  1270
lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654
  1271
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1272
  assumes "a \<le> b" and
paulson@33654
  1273
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654
  1274
  shows "f a \<ge> f b"
paulson@33654
  1275
proof -
paulson@33654
  1276
  have "(%x. -f x) a \<le> (%x. -f x) b"
noschinl@45791
  1277
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
wenzelm@33690
  1278
    using assms
wenzelm@33690
  1279
    apply auto
paulson@33654
  1280
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1281
    done
paulson@33654
  1282
  thus ?thesis
paulson@33654
  1283
    by simp
paulson@33654
  1284
qed
huffman@21164
  1285
huffman@29975
  1286
subsection {* Continuous injective functions *}
huffman@29975
  1287
huffman@21164
  1288
text{*Dull lemma: an continuous injection on an interval must have a
huffman@21164
  1289
strict maximum at an end point, not in the middle.*}
huffman@21164
  1290
huffman@21164
  1291
lemma lemma_isCont_inj:
huffman@21164
  1292
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1293
  assumes d: "0 < d"
huffman@21164
  1294
      and inj [rule_format]: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1295
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1296
  shows "\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z"
huffman@21164
  1297
proof (rule ccontr)
huffman@21164
  1298
  assume  "~ (\<exists>z. \<bar>z-x\<bar> \<le> d & f x < f z)"
huffman@21164
  1299
  hence all [rule_format]: "\<forall>z. \<bar>z - x\<bar> \<le> d --> f z \<le> f x" by auto
huffman@21164
  1300
  show False
huffman@21164
  1301
  proof (cases rule: linorder_le_cases [of "f(x-d)" "f(x+d)"])
huffman@21164
  1302
    case le
huffman@21164
  1303
    from d cont all [of "x+d"]
huffman@21164
  1304
    have flef: "f(x+d) \<le> f x"
huffman@21164
  1305
     and xlex: "x - d \<le> x"
huffman@21164
  1306
     and cont': "\<forall>z. x - d \<le> z \<and> z \<le> x \<longrightarrow> isCont f z"
huffman@21164
  1307
       by (auto simp add: abs_if)
huffman@21164
  1308
    from IVT [OF le flef xlex cont']
huffman@21164
  1309
    obtain x' where "x-d \<le> x'" "x' \<le> x" "f x' = f(x+d)" by blast
huffman@21164
  1310
    moreover
huffman@21164
  1311
    hence "g(f x') = g (f(x+d))" by simp
huffman@21164
  1312
    ultimately show False using d inj [of x'] inj [of "x+d"]
huffman@22998
  1313
      by (simp add: abs_le_iff)
huffman@21164
  1314
  next
huffman@21164
  1315
    case ge
huffman@21164
  1316
    from d cont all [of "x-d"]
huffman@21164
  1317
    have flef: "f(x-d) \<le> f x"
huffman@21164
  1318
     and xlex: "x \<le> x+d"
huffman@21164
  1319
     and cont': "\<forall>z. x \<le> z \<and> z \<le> x+d \<longrightarrow> isCont f z"
huffman@21164
  1320
       by (auto simp add: abs_if)
huffman@21164
  1321
    from IVT2 [OF ge flef xlex cont']
huffman@21164
  1322
    obtain x' where "x \<le> x'" "x' \<le> x+d" "f x' = f(x-d)" by blast
huffman@21164
  1323
    moreover
huffman@21164
  1324
    hence "g(f x') = g (f(x-d))" by simp
huffman@21164
  1325
    ultimately show False using d inj [of x'] inj [of "x-d"]
huffman@22998
  1326
      by (simp add: abs_le_iff)
huffman@21164
  1327
  qed
huffman@21164
  1328
qed
huffman@21164
  1329
huffman@21164
  1330
huffman@21164
  1331
text{*Similar version for lower bound.*}
huffman@21164
  1332
huffman@21164
  1333
lemma lemma_isCont_inj2:
huffman@21164
  1334
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1335
  shows "[|0 < d; \<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z;
huffman@21164
  1336
        \<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z |]
huffman@21164
  1337
      ==> \<exists>z. \<bar>z-x\<bar> \<le> d & f z < f x"
huffman@21164
  1338
apply (insert lemma_isCont_inj
huffman@21164
  1339
          [where f = "%x. - f x" and g = "%y. g(-y)" and x = x and d = d])
huffman@44233
  1340
apply (simp add: linorder_not_le)
huffman@21164
  1341
done
huffman@21164
  1342
huffman@21164
  1343
text{*Show there's an interval surrounding @{term "f(x)"} in
huffman@21164
  1344
@{text "f[[x - d, x + d]]"} .*}
huffman@21164
  1345
huffman@21164
  1346
lemma isCont_inj_range:
huffman@21164
  1347
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1348
  assumes d: "0 < d"
huffman@21164
  1349
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1350
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1351
  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
  1352
proof -
huffman@21164
  1353
  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
  1354
    by (auto simp add: abs_le_iff)
huffman@21164
  1355
  from isCont_Lb_Ub [OF this]
huffman@21164
  1356
  obtain L M
huffman@21164
  1357
  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
  1358
    and all2 [rule_format]:
huffman@21164
  1359
           "\<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
  1360
    by auto
huffman@21164
  1361
  with d have "L \<le> f x & f x \<le> M" by simp
huffman@21164
  1362
  moreover have "L \<noteq> f x"
huffman@21164
  1363
  proof -
huffman@21164
  1364
    from lemma_isCont_inj2 [OF d inj cont]
huffman@21164
  1365
    obtain u where "\<bar>u - x\<bar> \<le> d" "f u < f x"  by auto
huffman@21164
  1366
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1367
  qed
huffman@21164
  1368
  moreover have "f x \<noteq> M"
huffman@21164
  1369
  proof -
huffman@21164
  1370
    from lemma_isCont_inj [OF d inj cont]
huffman@21164
  1371
    obtain u where "\<bar>u - x\<bar> \<le> d" "f x < f u"  by auto
huffman@21164
  1372
    thus ?thesis using all1 [of u] by arith
huffman@21164
  1373
  qed
huffman@21164
  1374
  ultimately have "L < f x & f x < M" by arith
huffman@21164
  1375
  hence "0 < f x - L" "0 < M - f x" by arith+
huffman@21164
  1376
  from real_lbound_gt_zero [OF this]
huffman@21164
  1377
  obtain e where e: "0 < e" "e < f x - L" "e < M - f x" by auto
huffman@21164
  1378
  thus ?thesis
huffman@21164
  1379
  proof (intro exI conjI)
huffman@23441
  1380
    show "0<e" using e(1) .
huffman@21164
  1381
    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
  1382
    proof (intro strip)
huffman@21164
  1383
      fix y::real
huffman@21164
  1384
      assume "\<bar>y - f x\<bar> \<le> e"
huffman@21164
  1385
      with e have "L \<le> y \<and> y \<le> M" by arith
huffman@21164
  1386
      from all2 [OF this]
huffman@21164
  1387
      obtain z where "x - d \<le> z" "z \<le> x + d" "f z = y" by blast
chaieb@27668
  1388
      thus "\<exists>z. \<bar>z - x\<bar> \<le> d \<and> f z = y" 
huffman@22998
  1389
        by (force simp add: abs_le_iff)
huffman@21164
  1390
    qed
huffman@21164
  1391
  qed
huffman@21164
  1392
qed
huffman@21164
  1393
huffman@21164
  1394
huffman@21164
  1395
text{*Continuity of inverse function*}
huffman@21164
  1396
huffman@21164
  1397
lemma isCont_inverse_function:
huffman@21164
  1398
  fixes f g :: "real \<Rightarrow> real"
huffman@21164
  1399
  assumes d: "0 < d"
huffman@21164
  1400
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d --> g(f z) = z"
huffman@21164
  1401
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d --> isCont f z"
huffman@21164
  1402
  shows "isCont g (f x)"
huffman@21164
  1403
proof (simp add: isCont_iff LIM_eq)
huffman@21164
  1404
  show "\<forall>r. 0 < r \<longrightarrow>
huffman@21164
  1405
         (\<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
  1406
  proof (intro strip)
huffman@21164
  1407
    fix r::real
huffman@21164
  1408
    assume r: "0<r"
huffman@21164
  1409
    from real_lbound_gt_zero [OF r d]
huffman@21164
  1410
    obtain e where e: "0 < e" and e_lt: "e < r \<and> e < d" by blast
huffman@21164
  1411
    with inj cont
huffman@21164
  1412
    have e_simps: "\<forall>z. \<bar>z-x\<bar> \<le> e --> g (f z) = z"
huffman@21164
  1413
                  "\<forall>z. \<bar>z-x\<bar> \<le> e --> isCont f z"   by auto
huffman@21164
  1414
    from isCont_inj_range [OF e this]
huffman@21164
  1415
    obtain e' where e': "0 < e'"
huffman@21164
  1416
        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
  1417
          by blast
huffman@21164
  1418
    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
  1419
    proof (intro exI conjI)
huffman@23441
  1420
      show "0<e'" using e' .
huffman@21164
  1421
      show "\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < e' \<longrightarrow> \<bar>g (f x + z) - g (f x)\<bar> < r"
huffman@21164
  1422
      proof (intro strip)
huffman@21164
  1423
        fix z::real
huffman@21164
  1424
        assume z: "z \<noteq> 0 \<and> \<bar>z\<bar> < e'"
huffman@21164
  1425
        with e e_lt e_simps all [rule_format, of "f x + z"]
huffman@21164
  1426
        show "\<bar>g (f x + z) - g (f x)\<bar> < r" by force
huffman@21164
  1427
      qed
huffman@21164
  1428
    qed
huffman@21164
  1429
  qed
huffman@21164
  1430
qed
huffman@21164
  1431
huffman@23041
  1432
text {* Derivative of inverse function *}
huffman@23041
  1433
huffman@23041
  1434
lemma DERIV_inverse_function:
huffman@23041
  1435
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1436
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1437
  assumes neq: "D \<noteq> 0"
huffman@23044
  1438
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1439
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1440
  assumes cont: "isCont g x"
huffman@23041
  1441
  shows "DERIV g x :> inverse D"
huffman@23041
  1442
unfolding DERIV_iff2
huffman@23044
  1443
proof (rule LIM_equal2)
huffman@23044
  1444
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1445
    using a b by arith 
huffman@23044
  1446
next
huffman@23041
  1447
  fix y
huffman@23044
  1448
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1449
  hence "a < y" and "y < b" 
huffman@23044
  1450
    by (simp_all add: abs_less_iff)
huffman@23041
  1451
  thus "(g y - g x) / (y - x) =
huffman@23041
  1452
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1453
    by (simp add: inj)
huffman@23041
  1454
next
huffman@23041
  1455
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1456
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1457
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1458
    using inj a b by simp
huffman@23041
  1459
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
huffman@23041
  1460
  proof (safe intro!: exI)
huffman@23044
  1461
    show "0 < min (x - a) (b - x)"
huffman@23044
  1462
      using a b by simp
huffman@23041
  1463
  next
huffman@23041
  1464
    fix y
huffman@23044
  1465
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1466
    hence y: "a < y" "y < b"
huffman@23044
  1467
      by (simp_all add: abs_less_iff)
huffman@23041
  1468
    assume "g y = g x"
huffman@23041
  1469
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1470
    hence "y = x" using inj y a b by simp
huffman@23041
  1471
    also assume "y \<noteq> x"
huffman@23041
  1472
    finally show False by simp
huffman@23041
  1473
  qed
huffman@23041
  1474
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1475
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1476
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1477
        -- x --> inverse D"
huffman@44568
  1478
    using neq by (rule tendsto_inverse)
huffman@23041
  1479
qed
huffman@23041
  1480
huffman@29975
  1481
huffman@29975
  1482
subsection {* Generalized Mean Value Theorem *}
huffman@29975
  1483
huffman@21164
  1484
theorem GMVT:
huffman@21784
  1485
  fixes a b :: real
huffman@21164
  1486
  assumes alb: "a < b"
wenzelm@41550
  1487
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
wenzelm@41550
  1488
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
wenzelm@41550
  1489
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
wenzelm@41550
  1490
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1491
  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
  1492
proof -
huffman@21164
  1493
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
wenzelm@41550
  1494
  from assms have "a < b" by simp
huffman@21164
  1495
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@44233
  1496
    using fc gc by simp
huffman@44233
  1497
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@44233
  1498
    using fd gd by simp
huffman@21164
  1499
  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
  1500
  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
  1501
  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
  1502
huffman@21164
  1503
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1504
  with gd have "g differentiable c" by simp
huffman@21164
  1505
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1506
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1507
huffman@21164
  1508
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1509
  with fd have "f differentiable c" by simp
huffman@21164
  1510
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1511
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1512
huffman@21164
  1513
  from cdef have "DERIV ?h c :> l" by auto
hoelzl@41368
  1514
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
hoelzl@41368
  1515
    using g'cdef f'cdef by (auto intro!: DERIV_intros)
huffman@21164
  1516
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1517
huffman@21164
  1518
  {
huffman@21164
  1519
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1520
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1521
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1522
  }
huffman@21164
  1523
  moreover
huffman@21164
  1524
  {
huffman@21164
  1525
    have "?h b - ?h a =
huffman@21164
  1526
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1527
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1528
      by (simp add: algebra_simps)
huffman@21164
  1529
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1530
  }
huffman@21164
  1531
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1532
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1533
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1534
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1535
huffman@21164
  1536
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1537
qed
huffman@21164
  1538
huffman@29470
  1539
huffman@29166
  1540
subsection {* Theorems about Limits *}
huffman@29166
  1541
huffman@29166
  1542
(* need to rename second isCont_inverse *)
huffman@29166
  1543
huffman@29166
  1544
lemma isCont_inv_fun:
huffman@29166
  1545
  fixes f g :: "real \<Rightarrow> real"
huffman@29166
  1546
  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
huffman@29166
  1547
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
huffman@29166
  1548
      ==> isCont g (f x)"
huffman@29166
  1549
by (rule isCont_inverse_function)
huffman@29166
  1550
huffman@29166
  1551
lemma isCont_inv_fun_inv:
huffman@29166
  1552
  fixes f g :: "real \<Rightarrow> real"
huffman@29166
  1553
  shows "[| 0 < d;  
huffman@29166
  1554
         \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
huffman@29166
  1555
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
huffman@29166
  1556
       ==> \<exists>e. 0 < e &  
huffman@29166
  1557
             (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
huffman@29166
  1558
apply (drule isCont_inj_range)
huffman@29166
  1559
prefer 2 apply (assumption, assumption, auto)
huffman@29166
  1560
apply (rule_tac x = e in exI, auto)
huffman@29166
  1561
apply (rotate_tac 2)
huffman@29166
  1562
apply (drule_tac x = y in spec, auto)
huffman@29166
  1563
done
huffman@29166
  1564
huffman@29166
  1565
huffman@29166
  1566
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
huffman@29166
  1567
lemma LIM_fun_gt_zero:
huffman@29166
  1568
     "[| f -- c --> (l::real); 0 < l |]  
huffman@29166
  1569
         ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
huffman@44209
  1570
apply (drule (1) LIM_D, clarify)
huffman@29166
  1571
apply (rule_tac x = s in exI)
huffman@44209
  1572
apply (simp add: abs_less_iff)
huffman@29166
  1573
done
huffman@29166
  1574
huffman@29166
  1575
lemma LIM_fun_less_zero:
huffman@29166
  1576
     "[| f -- c --> (l::real); l < 0 |]  
huffman@29166
  1577
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
huffman@44209
  1578
apply (drule LIM_D [where r="-l"], simp, clarify)
huffman@29166
  1579
apply (rule_tac x = s in exI)
huffman@44209
  1580
apply (simp add: abs_less_iff)
huffman@29166
  1581
done
huffman@29166
  1582
huffman@29166
  1583
lemma LIM_fun_not_zero:
huffman@29166
  1584
     "[| f -- c --> (l::real); l \<noteq> 0 |] 
huffman@29166
  1585
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
huffman@44209
  1586
apply (rule linorder_cases [of l 0])
huffman@44209
  1587
apply (drule (1) LIM_fun_less_zero, force)
huffman@44209
  1588
apply simp
huffman@44209
  1589
apply (drule (1) LIM_fun_gt_zero, force)
huffman@29166
  1590
done
huffman@29166
  1591
hoelzl@50327
  1592
lemma GMVT':
hoelzl@50327
  1593
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1594
  assumes "a < b"
hoelzl@50327
  1595
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
hoelzl@50327
  1596
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
hoelzl@50327
  1597
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
hoelzl@50327
  1598
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327
  1599
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1600
proof -
hoelzl@50327
  1601
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
hoelzl@50327
  1602
    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@50327
  1603
    using assms by (intro GMVT) (force simp: differentiable_def)+
hoelzl@50327
  1604
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1605
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327
  1606
  then show ?thesis
hoelzl@50327
  1607
    by auto
hoelzl@50327
  1608
qed
hoelzl@50327
  1609
hoelzl@50329
  1610
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@50329
  1611
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
hoelzl@50329
  1612
  unfolding DERIV_iff2
hoelzl@50329
  1613
proof (rule filterlim_cong)
hoelzl@50329
  1614
  assume "eventually (\<lambda>x. f x = g x) (nhds x)"
hoelzl@50329
  1615
  moreover then have "f x = g x" by (auto simp: eventually_nhds)
hoelzl@50329
  1616
  moreover assume "x = y" "u = v"
hoelzl@50329
  1617
  ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
hoelzl@50329
  1618
    by (auto simp: eventually_within at_def elim: eventually_elim1)
hoelzl@50329
  1619
qed simp_all
hoelzl@50329
  1620
hoelzl@50330
  1621
lemma DERIV_shift:
hoelzl@50330
  1622
  "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
hoelzl@50330
  1623
  by (simp add: DERIV_iff field_simps)
hoelzl@50329
  1624
hoelzl@50330
  1625
lemma DERIV_mirror:
hoelzl@50330
  1626
  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
hoelzl@50330
  1627
  by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right
hoelzl@50330
  1628
                tendsto_minus_cancel_left field_simps conj_commute)
hoelzl@50329
  1629
hoelzl@50327
  1630
lemma lhopital_right_0:
hoelzl@50329
  1631
  fixes f0 g0 :: "real \<Rightarrow> real"
hoelzl@50329
  1632
  assumes f_0: "(f0 ---> 0) (at_right 0)"
hoelzl@50329
  1633
  assumes g_0: "(g0 ---> 0) (at_right 0)"
hoelzl@50327
  1634
  assumes ev:
hoelzl@50329
  1635
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1636
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50329
  1637
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
hoelzl@50329
  1638
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
hoelzl@50327
  1639
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50329
  1640
  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
hoelzl@50327
  1641
proof -
hoelzl@50329
  1642
  def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
hoelzl@50329
  1643
  then have "f 0 = 0" by simp
hoelzl@50329
  1644
hoelzl@50329
  1645
  def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
hoelzl@50329
  1646
  then have "g 0 = 0" by simp
hoelzl@50329
  1647
hoelzl@50329
  1648
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329
  1649
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329
  1650
    using ev by eventually_elim auto
hoelzl@50329
  1651
  then obtain a where [arith]: "0 < a"
hoelzl@50329
  1652
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327
  1653
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329
  1654
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329
  1655
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
hoelzl@50327
  1656
    unfolding eventually_within eventually_at by (auto simp: dist_real_def)
hoelzl@50327
  1657
hoelzl@50329
  1658
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329
  1659
    using g0_neq_0 by (simp add: g_def)
hoelzl@50329
  1660
hoelzl@50329
  1661
  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
hoelzl@50329
  1662
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
hoelzl@50329
  1663
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1664
  note f = this
hoelzl@50329
  1665
hoelzl@50329
  1666
  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
hoelzl@50329
  1667
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
hoelzl@50329
  1668
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1669
  note g = this
hoelzl@50329
  1670
hoelzl@50329
  1671
  have "isCont f 0"
hoelzl@50329
  1672
    using tendsto_const[of "0::real" "at 0"] f_0
hoelzl@50329
  1673
    unfolding isCont_def f_def
hoelzl@50329
  1674
    by (intro filterlim_split_at_real)
hoelzl@50329
  1675
       (auto elim: eventually_elim1
hoelzl@50329
  1676
             simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)
hoelzl@50329
  1677
    
hoelzl@50329
  1678
  have "isCont g 0"
hoelzl@50329
  1679
    using tendsto_const[of "0::real" "at 0"] g_0
hoelzl@50329
  1680
    unfolding isCont_def g_def
hoelzl@50329
  1681
    by (intro filterlim_split_at_real)
hoelzl@50329
  1682
       (auto elim: eventually_elim1
hoelzl@50329
  1683
             simp add: filterlim_def le_filter_def eventually_within eventually_filtermap)
hoelzl@50329
  1684
hoelzl@50327
  1685
  have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
hoelzl@50327
  1686
  proof (rule bchoice, rule)
hoelzl@50327
  1687
    fix x assume "x \<in> {0 <..< a}"
hoelzl@50327
  1688
    then have x[arith]: "0 < x" "x < a" by auto
hoelzl@50327
  1689
    with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
hoelzl@50327
  1690
      by auto
hoelzl@50328
  1691
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
hoelzl@50328
  1692
      using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1693
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
hoelzl@50328
  1694
      using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1695
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
hoelzl@50328
  1696
      using f g `x < a` by (intro GMVT') auto
hoelzl@50327
  1697
    then guess c ..
hoelzl@50327
  1698
    moreover
hoelzl@50327
  1699
    with g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327
  1700
      by (simp add: field_simps)
hoelzl@50327
  1701
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
hoelzl@50327
  1702
      using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
hoelzl@50327
  1703
  qed
hoelzl@50327
  1704
  then guess \<zeta> ..
hoelzl@50327
  1705
  then have \<zeta>: "eventually (\<lambda>x. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)) (at_right 0)"
hoelzl@50327
  1706
    unfolding eventually_within eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327
  1707
  moreover
hoelzl@50327
  1708
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327
  1709
    by eventually_elim auto
hoelzl@50327
  1710
  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
hoelzl@50327
  1711
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
hoelzl@50327
  1712
       (auto intro: tendsto_const tendsto_ident_at_within)
hoelzl@50327
  1713
  then have "(\<zeta> ---> 0) (at_right 0)"
hoelzl@50327
  1714
    by (rule tendsto_norm_zero_cancel)
hoelzl@50327
  1715
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
hoelzl@50327
  1716
    by (auto elim!: eventually_elim1 simp: filterlim_within filterlim_at)
hoelzl@50327
  1717
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
hoelzl@50327
  1718
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
hoelzl@50329
  1719
  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
hoelzl@50328
  1720
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
hoelzl@50328
  1721
       (auto elim: eventually_elim1)
hoelzl@50329
  1722
  also have "?P \<longleftrightarrow> ?thesis"
hoelzl@50329
  1723
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_within)
hoelzl@50329
  1724
  finally show ?thesis .
hoelzl@50327
  1725
qed
hoelzl@50327
  1726
hoelzl@50330
  1727
lemma lhopital_right:
hoelzl@50330
  1728
  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1729
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1730
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1731
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1732
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1733
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1734
  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1735
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1736
  by (rule lhopital_right_0)
hoelzl@50330
  1737
hoelzl@50330
  1738
lemma lhopital_left:
hoelzl@50330
  1739
  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1740
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1741
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1742
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1743
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1744
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1745
  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1746
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1747
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1748
hoelzl@50330
  1749
lemma lhopital:
hoelzl@50330
  1750
  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1751
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1752
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1753
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1754
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1755
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1756
  ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1757
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1758
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330
  1759
hoelzl@50327
  1760
lemma lhopital_right_0_at_top:
hoelzl@50327
  1761
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1762
  assumes g_0: "LIM x at_right 0. g x :> at_top"
hoelzl@50327
  1763
  assumes ev:
hoelzl@50327
  1764
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1765
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
hoelzl@50327
  1766
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
hoelzl@50327
  1767
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50327
  1768
  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
hoelzl@50327
  1769
  unfolding tendsto_iff
hoelzl@50327
  1770
proof safe
hoelzl@50327
  1771
  fix e :: real assume "0 < e"
hoelzl@50327
  1772
hoelzl@50327
  1773
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
hoelzl@50327
  1774
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
hoelzl@50327
  1775
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327
  1776
  obtain a where [arith]: "0 < a"
hoelzl@50327
  1777
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327
  1778
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327
  1779
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327
  1780
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@50327
  1781
    unfolding eventually_within_le by (auto simp: dist_real_def)
hoelzl@50327
  1782
hoelzl@50327
  1783
  from Df have
hoelzl@50328
  1784
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@50327
  1785
    unfolding eventually_within eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327
  1786
hoelzl@50327
  1787
  moreover
hoelzl@50328
  1788
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
hoelzl@50328
  1789
    using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top)
hoelzl@50327
  1790
hoelzl@50327
  1791
  moreover
hoelzl@50327
  1792
  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
hoelzl@50327
  1793
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327
  1794
    by (rule filterlim_compose)
hoelzl@50327
  1795
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327
  1796
    by (intro tendsto_intros)
hoelzl@50327
  1797
  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
hoelzl@50327
  1798
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1799
  from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327
  1800
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
hoelzl@50327
  1801
    by (auto elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50327
  1802
hoelzl@50327
  1803
  moreover
hoelzl@50327
  1804
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
hoelzl@50327
  1805
    by (intro tendsto_intros)
hoelzl@50327
  1806
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
hoelzl@50327
  1807
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1808
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
hoelzl@50327
  1809
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327
  1810
    by (auto simp: dist_real_def)
hoelzl@50327
  1811
hoelzl@50327
  1812
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327
  1813
  proof eventually_elim
hoelzl@50327
  1814
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327
  1815
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327
  1816
hoelzl@50327
  1817
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327
  1818
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
hoelzl@50327
  1819
    then guess y ..
hoelzl@50327
  1820
    from this
hoelzl@50327
  1821
    have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
hoelzl@50327
  1822
      using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327
  1823
hoelzl@50327
  1824
    have *: "f t / g t - x = ((f t - f a) / (g t - g a) - x) * (1 - g a / g t) + (f a - x * g a) / g t"
hoelzl@50327
  1825
      by (simp add: field_simps)
hoelzl@50327
  1826
    have "norm (f t / g t - x) \<le>
hoelzl@50327
  1827
        norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
hoelzl@50327
  1828
      unfolding * by (rule norm_triangle_ineq)
hoelzl@50327
  1829
    also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
hoelzl@50327
  1830
      by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327
  1831
    also have "\<dots> < (e / 4) * 2 + e / 2"
hoelzl@50327
  1832
      using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
hoelzl@50327
  1833
    finally show "dist (f t / g t) x < e"
hoelzl@50327
  1834
      by (simp add: dist_real_def)
hoelzl@50327
  1835
  qed
hoelzl@50327
  1836
qed
hoelzl@50327
  1837
hoelzl@50330
  1838
lemma lhopital_right_at_top:
hoelzl@50330
  1839
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1840
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1841
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1842
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1843
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1844
    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1845
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1846
  by (rule lhopital_right_0_at_top)
hoelzl@50330
  1847
hoelzl@50330
  1848
lemma lhopital_left_at_top:
hoelzl@50330
  1849
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1850
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1851
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1852
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1853
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1854
    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1855
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1856
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1857
hoelzl@50330
  1858
lemma lhopital_at_top:
hoelzl@50330
  1859
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1860
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1861
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1862
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1863
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1864
    ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1865
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1866
  by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])
hoelzl@50330
  1867
huffman@21164
  1868
end