src/HOL/Deriv.thy
author hoelzl
Tue Apr 09 14:04:47 2013 +0200 (2013-04-09)
changeset 51642 400ec5ae7f8f
parent 51641 cd05e9fcc63d
child 53374 a14d2a854c02
permissions -rw-r--r--
move FrechetDeriv from the Library to HOL/Deriv; base DERIV on FDERIV and both derivatives allow a restricted support set; FDERIV is now an abbreviation of has_derivative
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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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    Author      : Brian Huffman
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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 Limits
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begin
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definition
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  -- {* Frechet derivative: D is derivative of function f at x within s *}
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  has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"
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  (infixl "(has'_derivative)" 12)
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where
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  "(f has_derivative f') F \<longleftrightarrow>
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    (bounded_linear f' \<and>
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     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 0) F)"
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lemma FDERIV_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
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  by simp
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ML {*
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structure FDERIV_Intros = Named_Thms
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(
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  val name = @{binding FDERIV_intros}
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  val description = "introduction rules for FDERIV"
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)
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*}
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setup {*
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  FDERIV_Intros.setup #>
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  Global_Theory.add_thms_dynamic (@{binding FDERIV_eq_intros},
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    map_filter (try (fn thm => @{thm FDERIV_eq_rhs} OF [thm])) o FDERIV_Intros.get o Context.proof_of);
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*}
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lemma FDERIV_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
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  by (simp add: has_derivative_def)
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lemma FDERIV_ident[FDERIV_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
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  by (simp add: has_derivative_def tendsto_const)
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lemma FDERIV_const[FDERIV_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
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  by (simp add: has_derivative_def tendsto_const)
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lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
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lemma (in bounded_linear) FDERIV:
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  "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
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  using assms unfolding has_derivative_def
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  apply safe
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  apply (erule bounded_linear_compose [OF local.bounded_linear])
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  apply (drule local.tendsto)
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  apply (simp add: local.scaleR local.diff local.add local.zero)
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  done
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lemmas FDERIV_scaleR_right [FDERIV_intros] =
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  bounded_linear.FDERIV [OF bounded_linear_scaleR_right]
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lemmas FDERIV_scaleR_left [FDERIV_intros] =
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  bounded_linear.FDERIV [OF bounded_linear_scaleR_left]
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lemmas FDERIV_mult_right [FDERIV_intros] =
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  bounded_linear.FDERIV [OF bounded_linear_mult_right]
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lemmas FDERIV_mult_left [FDERIV_intros] =
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  bounded_linear.FDERIV [OF bounded_linear_mult_left]
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lemma FDERIV_add[simp, FDERIV_intros]:
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  assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
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  shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
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  unfolding has_derivative_def
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proof safe
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  let ?x = "Lim F (\<lambda>x. x)"
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  let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
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  have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
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    using f g by (intro tendsto_add) (auto simp: has_derivative_def)
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  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
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    by (simp add: field_simps scaleR_add_right scaleR_diff_right)
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qed (blast intro: bounded_linear_add f g FDERIV_bounded_linear)
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lemma FDERIV_setsum[simp, FDERIV_intros]:
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  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
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  shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
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proof cases
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  assume "finite I" from this f show ?thesis
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    by induct (simp_all add: f)
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qed simp
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lemma FDERIV_minus[simp, FDERIV_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
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  using FDERIV_scaleR_right[of f f' F "-1"] by simp
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lemma FDERIV_diff[simp, FDERIV_intros]:
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  "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
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  by (simp only: diff_minus FDERIV_add FDERIV_minus)
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abbreviation
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  -- {* Frechet derivative: D is derivative of function f at x within s *}
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  FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
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  ("(FDERIV (_)/ (_)/ : (_)/ :> (_))" [1000, 1000, 1000, 60] 60)
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where
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  "FDERIV f x : s :> f' \<equiv> (f has_derivative f') (at x within s)"
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abbreviation
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  fderiv_at ::
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    "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
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    ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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where
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  "FDERIV f x :> D \<equiv> FDERIV f x : UNIV :> D"
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lemma FDERIV_def:
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  "FDERIV f x : s :> f' \<longleftrightarrow>
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    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"
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  by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
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lemma FDERIV_iff_norm:
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  "FDERIV f x : s :> f' \<longleftrightarrow>
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    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"
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  using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
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  by (simp add: FDERIV_def divide_inverse ac_simps)
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lemma fderiv_def:
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  "FDERIV f x :> D = (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"
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  unfolding FDERIV_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
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lemma field_fderiv_def:
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  fixes x :: "'a::real_normed_field"
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  shows "FDERIV f x :> (\<lambda>h. h * D) = (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
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  apply (unfold fderiv_def)
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  apply (simp add: bounded_linear_mult_left)
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  apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
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  apply (subst diff_divide_distrib)
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  apply (subst times_divide_eq_left [symmetric])
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  apply (simp cong: LIM_cong)
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  apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
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  done
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lemma FDERIV_I:
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  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
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  FDERIV f x : s :> f'"
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  by (simp add: FDERIV_def)
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lemma FDERIV_I_sandwich:
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  assumes e: "0 < e" and bounded: "bounded_linear f'"
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    and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
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    and "(H ---> 0) (at x within s)"
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  shows "FDERIV f x : s :> f'"
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  unfolding FDERIV_iff_norm
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proof safe
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  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
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  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
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    show "(H ---> 0) (at x within s)" by fact
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    show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
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      unfolding eventually_at using e sandwich by auto
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  qed (auto simp: le_divide_eq tendsto_const)
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qed fact
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lemma FDERIV_subset: "FDERIV f x : s :> f' \<Longrightarrow> t \<subseteq> s \<Longrightarrow> FDERIV f x : t :> f'"
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  by (auto simp add: FDERIV_iff_norm intro: tendsto_within_subset)
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subsection {* Continuity *}
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lemma FDERIV_continuous:
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  assumes f: "FDERIV f x : s :> f'"
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  shows "continuous (at x within s) f"
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proof -
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  from f interpret F: bounded_linear f' by (rule FDERIV_bounded_linear)
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  note F.tendsto[tendsto_intros]
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  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
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  have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
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    using f unfolding FDERIV_iff_norm by blast
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  then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
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    by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
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  also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
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    by (intro filterlim_cong) (simp_all add: eventually_at_filter)
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  finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
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    by (rule tendsto_norm_zero_cancel)
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  then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
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    by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
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  then have "?L (\<lambda>y. f y - f x)"
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    by simp
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  from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
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    by (simp add: continuous_within)
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qed
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subsection {* Composition *}
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lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"
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  unfolding tendsto_def eventually_inf_principal eventually_at_filter
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  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
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lemma FDERIV_in_compose:
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  assumes f: "FDERIV f x : s :> f'"
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  assumes g: "FDERIV g (f x) : (f`s) :> g'"
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  shows "FDERIV (\<lambda>x. g (f x)) x : s :> (\<lambda>x. g' (f' x))"
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proof -
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  from f interpret F: bounded_linear f' by (rule FDERIV_bounded_linear)
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  from g interpret G: bounded_linear g' by (rule FDERIV_bounded_linear)
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  from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
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  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
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  note G.tendsto[tendsto_intros]
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  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
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  let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
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  let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
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  let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
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  def Nf \<equiv> "?N f f' x"
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  def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
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  show ?thesis
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  proof (rule FDERIV_I_sandwich[of 1])
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    show "bounded_linear (\<lambda>x. g' (f' x))"
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      using f g by (blast intro: bounded_linear_compose FDERIV_bounded_linear)
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  next
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    fix y::'a assume neq: "y \<noteq> x"
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    have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
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      by (simp add: G.diff G.add field_simps)
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    also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
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      by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
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    also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
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    proof (intro add_mono mult_left_mono)
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      have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
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        by simp
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      also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
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        by (rule norm_triangle_ineq)
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      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
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        using kF by (intro add_mono) simp
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      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
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        by (simp add: neq Nf_def field_simps)
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    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
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    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
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  next
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    have [tendsto_intros]: "?L Nf"
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      using f unfolding FDERIV_iff_norm Nf_def ..
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    from f have "(f ---> f x) (at x within s)"
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      by (blast intro: FDERIV_continuous continuous_within[THEN iffD1])
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    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
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      unfolding filterlim_def
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      by (simp add: eventually_filtermap eventually_at_filter le_principal)
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    have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
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      using g unfolding FDERIV_iff_norm ..
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    then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
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      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
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   252
    have [tendsto_intros]: "?L Ng"
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   253
      unfolding Ng_def by (rule filterlim_compose[OF g' f'])
hoelzl@51642
   254
    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
hoelzl@51642
   255
      by (intro tendsto_eq_intros) auto
hoelzl@51642
   256
  qed simp
hoelzl@51642
   257
qed
hoelzl@51642
   258
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   259
lemma FDERIV_compose:
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   260
  "FDERIV f x : s :> f' \<Longrightarrow> FDERIV g (f x) :> g' \<Longrightarrow> FDERIV (\<lambda>x. g (f x)) x : s :> (\<lambda>x. g' (f' x))"
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   261
  by (blast intro: FDERIV_in_compose FDERIV_subset)
hoelzl@51642
   262
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   263
lemma (in bounded_bilinear) FDERIV:
hoelzl@51642
   264
  assumes f: "FDERIV f x : s :> f'" and g: "FDERIV g x : s :> g'"
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   265
  shows "FDERIV (\<lambda>x. f x ** g x) x : s :> (\<lambda>h. f x ** g' h + f' h ** g x)"
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   266
proof -
hoelzl@51642
   267
  from bounded_linear.bounded [OF FDERIV_bounded_linear [OF f]]
hoelzl@51642
   268
  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
hoelzl@51642
   269
hoelzl@51642
   270
  from pos_bounded obtain K where K: "0 < K" and norm_prod:
hoelzl@51642
   271
    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
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   272
  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
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   273
  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
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   274
  def Ng =="?N g g'" and Nf =="?N f f'"
hoelzl@51642
   275
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   276
  let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
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   277
  let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
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   278
  let ?F = "at x within s"
huffman@21164
   279
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   280
  show ?thesis
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   281
  proof (rule FDERIV_I_sandwich[of 1])
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   282
    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
hoelzl@51642
   283
      by (intro bounded_linear_add
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   284
        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
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   285
        FDERIV_bounded_linear [OF g] FDERIV_bounded_linear [OF f])
hoelzl@51642
   286
  next
hoelzl@51642
   287
    from g have "(g ---> g x) ?F"
hoelzl@51642
   288
      by (intro continuous_within[THEN iffD1] FDERIV_continuous)
hoelzl@51642
   289
    moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
hoelzl@51642
   290
      by (simp_all add: FDERIV_iff_norm Ng_def Nf_def)
hoelzl@51642
   291
    ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
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   292
      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
hoelzl@51642
   293
    then show "(?fun2 ---> 0) ?F"
hoelzl@51642
   294
      by simp
hoelzl@51642
   295
  next
hoelzl@51642
   296
    fix y::'d assume "y \<noteq> x"
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   297
    have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
hoelzl@51642
   298
      by (simp add: diff_left diff_right add_left add_right field_simps)
hoelzl@51642
   299
    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
hoelzl@51642
   300
        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
hoelzl@51642
   301
      by (intro divide_right_mono mult_mono'
hoelzl@51642
   302
                order_trans [OF norm_triangle_ineq add_mono]
hoelzl@51642
   303
                order_trans [OF norm_prod mult_right_mono]
hoelzl@51642
   304
                mult_nonneg_nonneg order_refl norm_ge_zero norm_F
hoelzl@51642
   305
                K [THEN order_less_imp_le])
hoelzl@51642
   306
    also have "\<dots> = ?fun2 y"
hoelzl@51642
   307
      by (simp add: add_divide_distrib Ng_def Nf_def)
hoelzl@51642
   308
    finally show "?fun1 y \<le> ?fun2 y" .
hoelzl@51642
   309
  qed simp
hoelzl@51642
   310
qed
hoelzl@51642
   311
hoelzl@51642
   312
lemmas FDERIV_mult[simp, FDERIV_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
hoelzl@51642
   313
lemmas FDERIV_scaleR[simp, FDERIV_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
hoelzl@51642
   314
hoelzl@51642
   315
lemma FDERIV_setprod[simp, FDERIV_intros]:
hoelzl@51642
   316
  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@51642
   317
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> FDERIV (f i) x : s :> f' i"
hoelzl@51642
   318
  shows "FDERIV (\<lambda>x. \<Prod>i\<in>I. f i x) x : s :> (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))"
hoelzl@51642
   319
proof cases
hoelzl@51642
   320
  assume "finite I" from this f show ?thesis
hoelzl@51642
   321
  proof induct
hoelzl@51642
   322
    case (insert i I)
hoelzl@51642
   323
    let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
hoelzl@51642
   324
    have "FDERIV (\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) x : s :> ?P"
hoelzl@51642
   325
      using insert by (intro FDERIV_mult) auto
hoelzl@51642
   326
    also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
hoelzl@51642
   327
      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum_cong)
hoelzl@51642
   328
    finally show ?case
hoelzl@51642
   329
      using insert by simp
hoelzl@51642
   330
  qed simp  
hoelzl@51642
   331
qed simp
hoelzl@51642
   332
hoelzl@51642
   333
lemma FDERIV_power[simp, FDERIV_intros]:
hoelzl@51642
   334
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@51642
   335
  assumes f: "FDERIV f x : s :> f'"
hoelzl@51642
   336
  shows "FDERIV  (\<lambda>x. f x^n) x : s :> (\<lambda>y. of_nat n * f' y * f x^(n - 1))"
hoelzl@51642
   337
  using FDERIV_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
hoelzl@51642
   338
hoelzl@51642
   339
lemma FDERIV_inverse':
hoelzl@51642
   340
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51642
   341
  assumes x: "x \<noteq> 0"
hoelzl@51642
   342
  shows "FDERIV inverse x : s :> (\<lambda>h. - (inverse x * h * inverse x))"
hoelzl@51642
   343
        (is "FDERIV ?inv x : s :> ?f")
hoelzl@51642
   344
proof (rule FDERIV_I_sandwich)
hoelzl@51642
   345
  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
hoelzl@51642
   346
    apply (rule bounded_linear_minus)
hoelzl@51642
   347
    apply (rule bounded_linear_mult_const)
hoelzl@51642
   348
    apply (rule bounded_linear_const_mult)
hoelzl@51642
   349
    apply (rule bounded_linear_ident)
hoelzl@51642
   350
    done
hoelzl@51642
   351
next
hoelzl@51642
   352
  show "0 < norm x" using x by simp
hoelzl@51642
   353
next
hoelzl@51642
   354
  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
hoelzl@51642
   355
    apply (rule tendsto_mult_left_zero)
hoelzl@51642
   356
    apply (rule tendsto_norm_zero)
hoelzl@51642
   357
    apply (rule LIM_zero)
hoelzl@51642
   358
    apply (rule tendsto_inverse)
hoelzl@51642
   359
    apply (rule tendsto_ident_at)
hoelzl@51642
   360
    apply (rule x)
hoelzl@51642
   361
    done
hoelzl@51642
   362
next
hoelzl@51642
   363
  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
hoelzl@51642
   364
  then have "y \<noteq> 0"
hoelzl@51642
   365
    by (auto simp: norm_conv_dist dist_commute)
hoelzl@51642
   366
  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
hoelzl@51642
   367
    apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])
hoelzl@51642
   368
    apply (subst minus_diff_minus)
hoelzl@51642
   369
    apply (subst norm_minus_cancel)
hoelzl@51642
   370
    apply (simp add: left_diff_distrib)
hoelzl@51642
   371
    done
hoelzl@51642
   372
  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
hoelzl@51642
   373
    apply (rule divide_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   374
    apply (rule order_trans [OF norm_mult_ineq])
hoelzl@51642
   375
    apply (rule mult_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   376
    apply (rule norm_mult_ineq)
hoelzl@51642
   377
    done
hoelzl@51642
   378
  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
hoelzl@51642
   379
    by simp
hoelzl@51642
   380
  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
hoelzl@51642
   381
      norm (?inv y - ?inv x) * norm (?inv x)" .
hoelzl@51642
   382
qed
hoelzl@51642
   383
hoelzl@51642
   384
lemma FDERIV_inverse[simp, FDERIV_intros]:
hoelzl@51642
   385
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@51642
   386
  assumes x:  "f x \<noteq> 0" and f: "FDERIV f x : s :> f'"
hoelzl@51642
   387
  shows "FDERIV (\<lambda>x. inverse (f x)) x : s :> (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))"
hoelzl@51642
   388
  using FDERIV_compose[OF f FDERIV_inverse', OF x] .
hoelzl@51642
   389
hoelzl@51642
   390
lemma FDERIV_divide[simp, FDERIV_intros]:
hoelzl@51642
   391
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@51642
   392
  assumes g: "FDERIV g x : s :> g'"
hoelzl@51642
   393
  assumes x:  "f x \<noteq> 0" and f: "FDERIV f x : s :> f'"
hoelzl@51642
   394
  shows "FDERIV (\<lambda>x. g x / f x) x : s :> (\<lambda>h. - g x * (inverse (f x) * f' h * inverse (f x)) + g' h / f x)"
hoelzl@51642
   395
  using FDERIV_mult[OF g FDERIV_inverse[OF x f]]
hoelzl@51642
   396
  by (simp add: divide_inverse)
hoelzl@51642
   397
hoelzl@51642
   398
subsection {* Uniqueness *}
hoelzl@51642
   399
hoelzl@51642
   400
text {*
hoelzl@51642
   401
hoelzl@51642
   402
This can not generally shown for @{const FDERIV}, as we need to approach the point from
hoelzl@51642
   403
all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.
hoelzl@51642
   404
hoelzl@51642
   405
*}
hoelzl@51642
   406
hoelzl@51642
   407
lemma FDERIV_zero_unique:
hoelzl@51642
   408
  assumes "FDERIV (\<lambda>x. 0) x :> F" shows "F = (\<lambda>h. 0)"
hoelzl@51642
   409
proof -
hoelzl@51642
   410
  interpret F: bounded_linear F
hoelzl@51642
   411
    using assms by (rule FDERIV_bounded_linear)
hoelzl@51642
   412
  let ?r = "\<lambda>h. norm (F h) / norm h"
hoelzl@51642
   413
  have *: "?r -- 0 --> 0"
hoelzl@51642
   414
    using assms unfolding fderiv_def by simp
hoelzl@51642
   415
  show "F = (\<lambda>h. 0)"
hoelzl@51642
   416
  proof
hoelzl@51642
   417
    fix h show "F h = 0"
hoelzl@51642
   418
    proof (rule ccontr)
hoelzl@51642
   419
      assume "F h \<noteq> 0"
hoelzl@51642
   420
      moreover from this have h: "h \<noteq> 0"
hoelzl@51642
   421
        by (clarsimp simp add: F.zero)
hoelzl@51642
   422
      ultimately have "0 < ?r h"
hoelzl@51642
   423
        by (simp add: divide_pos_pos)
hoelzl@51642
   424
      from LIM_D [OF * this] obtain s where s: "0 < s"
hoelzl@51642
   425
        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
hoelzl@51642
   426
      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
hoelzl@51642
   427
      let ?x = "scaleR (t / norm h) h"
hoelzl@51642
   428
      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
hoelzl@51642
   429
      hence "?r ?x < ?r h" by (rule r)
hoelzl@51642
   430
      thus "False" using t h by (simp add: F.scaleR)
hoelzl@51642
   431
    qed
hoelzl@51642
   432
  qed
hoelzl@51642
   433
qed
hoelzl@51642
   434
hoelzl@51642
   435
lemma FDERIV_unique:
hoelzl@51642
   436
  assumes "FDERIV f x :> F" and "FDERIV f x :> F'" shows "F = F'"
hoelzl@51642
   437
proof -
hoelzl@51642
   438
  have "FDERIV (\<lambda>x. 0) x :> (\<lambda>h. F h - F' h)"
hoelzl@51642
   439
    using FDERIV_diff [OF assms] by simp
hoelzl@51642
   440
  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
hoelzl@51642
   441
    by (rule FDERIV_zero_unique)
hoelzl@51642
   442
  thus "F = F'"
hoelzl@51642
   443
    unfolding fun_eq_iff right_minus_eq .
hoelzl@51642
   444
qed
hoelzl@51642
   445
hoelzl@51642
   446
subsection {* Differentiability predicate *}
hoelzl@51642
   447
hoelzl@51642
   448
definition isDiff :: "'a filter \<Rightarrow> ('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool" where
hoelzl@51642
   449
  isDiff_def: "isDiff F f \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
hoelzl@51642
   450
hoelzl@51642
   451
abbreviation differentiable_in :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool"
hoelzl@51642
   452
    ("(_) differentiable (_) in (_)"  [1000, 1000, 60] 60) where
hoelzl@51642
   453
  "f differentiable x in s \<equiv> isDiff (at x within s) f"
hoelzl@51642
   454
hoelzl@51642
   455
abbreviation differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@51642
   456
    (infixl "differentiable" 60) where
hoelzl@51642
   457
  "f differentiable x \<equiv> f differentiable x in UNIV"
hoelzl@51642
   458
hoelzl@51642
   459
lemma differentiable_subset: "f differentiable x in s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable x in t"
hoelzl@51642
   460
  unfolding isDiff_def by (blast intro: FDERIV_subset)
hoelzl@51642
   461
hoelzl@51642
   462
lemma differentiable_ident [simp]: "isDiff F (\<lambda>x. x)"
hoelzl@51642
   463
  unfolding isDiff_def by (blast intro: FDERIV_ident)
hoelzl@51642
   464
hoelzl@51642
   465
lemma differentiable_const [simp]: "isDiff F (\<lambda>z. a)"
hoelzl@51642
   466
  unfolding isDiff_def by (blast intro: FDERIV_const)
hoelzl@51642
   467
hoelzl@51642
   468
lemma differentiable_in_compose:
hoelzl@51642
   469
  "f differentiable (g x) in (g`s) \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f (g x)) differentiable x in s"
hoelzl@51642
   470
  unfolding isDiff_def by (blast intro: FDERIV_in_compose)
hoelzl@51642
   471
hoelzl@51642
   472
lemma differentiable_compose:
hoelzl@51642
   473
  "f differentiable (g x) \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f (g x)) differentiable x in s"
hoelzl@51642
   474
  by (blast intro: differentiable_in_compose differentiable_subset)
hoelzl@51642
   475
hoelzl@51642
   476
lemma differentiable_sum [simp]:
hoelzl@51642
   477
  "isDiff F f \<Longrightarrow> isDiff F g \<Longrightarrow> isDiff F (\<lambda>x. f x + g x)"
hoelzl@51642
   478
  unfolding isDiff_def by (blast intro: FDERIV_add)
hoelzl@51642
   479
hoelzl@51642
   480
lemma differentiable_minus [simp]:
hoelzl@51642
   481
  "isDiff F f \<Longrightarrow> isDiff F (\<lambda>x. - f x)"
hoelzl@51642
   482
  unfolding isDiff_def by (blast intro: FDERIV_minus)
hoelzl@51642
   483
hoelzl@51642
   484
lemma differentiable_diff [simp]:
hoelzl@51642
   485
  "isDiff F f \<Longrightarrow> isDiff F g \<Longrightarrow> isDiff F (\<lambda>x. f x - g x)"
hoelzl@51642
   486
  unfolding isDiff_def by (blast intro: FDERIV_diff)
hoelzl@51642
   487
hoelzl@51642
   488
lemma differentiable_mult [simp]:
hoelzl@51642
   489
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
hoelzl@51642
   490
  shows "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f x * g x) differentiable x in s"
hoelzl@51642
   491
  unfolding isDiff_def by (blast intro: FDERIV_mult)
hoelzl@51642
   492
hoelzl@51642
   493
lemma differentiable_inverse [simp]:
hoelzl@51642
   494
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@51642
   495
  shows "f differentiable x in s \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable x in s"
hoelzl@51642
   496
  unfolding isDiff_def by (blast intro: FDERIV_inverse)
hoelzl@51642
   497
hoelzl@51642
   498
lemma differentiable_divide [simp]:
hoelzl@51642
   499
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@51642
   500
  shows "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable x in s"
hoelzl@51642
   501
  unfolding divide_inverse using assms by simp
hoelzl@51642
   502
hoelzl@51642
   503
lemma differentiable_power [simp]:
hoelzl@51642
   504
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@51642
   505
  shows "f differentiable x in s \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable x in s"
hoelzl@51642
   506
  unfolding isDiff_def by (blast intro: FDERIV_power)
hoelzl@51642
   507
hoelzl@51642
   508
lemma differentiable_scaleR [simp]:
hoelzl@51642
   509
  "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable x in s"
hoelzl@51642
   510
  unfolding isDiff_def by (blast intro: FDERIV_scaleR)
hoelzl@51642
   511
hoelzl@51642
   512
definition 
hoelzl@51642
   513
  -- {*Differentiation: D is derivative of function f at x*}
hoelzl@51642
   514
  deriv ::
hoelzl@51642
   515
    "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@51642
   516
    ("(DERIV (_)/ (_)/ : (_)/ :> (_))" [1000, 1000, 1000, 60] 60)
hoelzl@51642
   517
where
hoelzl@51642
   518
  deriv_fderiv: "DERIV f x : s :> D = FDERIV f x : s :> (\<lambda>x. x * D)"
hoelzl@51642
   519
hoelzl@51642
   520
abbreviation
hoelzl@51642
   521
  -- {*Differentiation: D is derivative of function f at x*}
hoelzl@51642
   522
  deriv_at ::
hoelzl@51642
   523
    "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@51642
   524
    ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@51642
   525
where
hoelzl@51642
   526
  "DERIV f x :> D \<equiv> DERIV f x : UNIV :> D"
hoelzl@51642
   527
hoelzl@51642
   528
lemma differentiable_def: "(f::real \<Rightarrow> real) differentiable x in s \<longleftrightarrow> (\<exists>D. DERIV f x : s :> D)"
hoelzl@51642
   529
proof safe
hoelzl@51642
   530
  assume "f differentiable x in s"
hoelzl@51642
   531
  then obtain f' where "FDERIV f x : s :> f'"
hoelzl@51642
   532
    unfolding isDiff_def by auto
hoelzl@51642
   533
  moreover then obtain c where "f' = (\<lambda>x. x * c)"
hoelzl@51642
   534
    by (metis real_bounded_linear FDERIV_bounded_linear)
hoelzl@51642
   535
  ultimately show "\<exists>D. DERIV f x : s :> D"
hoelzl@51642
   536
    unfolding deriv_fderiv by auto
hoelzl@51642
   537
qed (auto simp: isDiff_def deriv_fderiv)
hoelzl@51642
   538
hoelzl@51642
   539
lemma differentiableE [elim?]:
hoelzl@51642
   540
  fixes f :: "real \<Rightarrow> real"
hoelzl@51642
   541
  assumes f: "f differentiable x in s" obtains df where "DERIV f x : s :> df"
hoelzl@51642
   542
  using assms by (auto simp: differentiable_def)
hoelzl@51642
   543
hoelzl@51642
   544
lemma differentiableD: "(f::real \<Rightarrow> real) differentiable x in s \<Longrightarrow> \<exists>D. DERIV f x : s :> D"
hoelzl@51642
   545
  by (auto elim: differentiableE)
hoelzl@51642
   546
hoelzl@51642
   547
lemma differentiableI: "DERIV f x : s :> D \<Longrightarrow> (f::real \<Rightarrow> real) differentiable x in s"
hoelzl@51642
   548
  by (force simp add: differentiable_def)
hoelzl@51642
   549
hoelzl@51642
   550
lemma DERIV_I_FDERIV: "FDERIV f x : s :> F \<Longrightarrow> (\<And>x. x * F' = F x) \<Longrightarrow> DERIV f x : s :> F'"
hoelzl@51642
   551
  by (simp add: deriv_fderiv)
hoelzl@51642
   552
hoelzl@51642
   553
lemma DERIV_D_FDERIV: "DERIV f x : s :> F \<Longrightarrow> FDERIV f x : s :> (\<lambda>x. x * F)"
hoelzl@51642
   554
  by (simp add: deriv_fderiv)
hoelzl@51642
   555
hoelzl@51642
   556
lemma deriv_def:
hoelzl@51642
   557
  "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@51642
   558
  apply (simp add: deriv_fderiv fderiv_def bounded_linear_mult_left LIM_zero_iff[symmetric, of _ D])
hoelzl@51642
   559
  apply (subst (2) tendsto_norm_zero_iff[symmetric])
hoelzl@51642
   560
  apply (rule filterlim_cong)
hoelzl@51642
   561
  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
hoelzl@51642
   562
  done
huffman@21164
   563
huffman@21164
   564
subsection {* Derivatives *}
huffman@21164
   565
hoelzl@51642
   566
lemma DERIV_iff: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@51642
   567
  by (simp add: deriv_def)
huffman@21164
   568
hoelzl@51642
   569
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@51642
   570
  by (simp add: deriv_def)
huffman@21164
   571
hoelzl@51642
   572
lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x : s :> 0"
hoelzl@51642
   573
  by (rule DERIV_I_FDERIV[OF FDERIV_const]) auto
huffman@21164
   574
hoelzl@51642
   575
lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x : s :> 1"
hoelzl@51642
   576
  by (rule DERIV_I_FDERIV[OF FDERIV_ident]) auto
huffman@21164
   577
hoelzl@51642
   578
lemma DERIV_add: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x : s :> D + E"
hoelzl@51642
   579
  by (rule DERIV_I_FDERIV[OF FDERIV_add]) (auto simp: field_simps dest: DERIV_D_FDERIV)
huffman@21164
   580
hoelzl@51642
   581
lemma DERIV_minus: "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x : s :> - D"
hoelzl@51642
   582
  by (rule DERIV_I_FDERIV[OF FDERIV_minus]) (auto simp: field_simps dest: DERIV_D_FDERIV)
huffman@21164
   583
hoelzl@51642
   584
lemma DERIV_diff: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x : s :> D - E"
hoelzl@51642
   585
  by (rule DERIV_I_FDERIV[OF FDERIV_diff]) (auto simp: field_simps dest: DERIV_D_FDERIV)
huffman@21164
   586
hoelzl@51642
   587
lemma DERIV_add_minus: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x : s :> D + - E"
hoelzl@51642
   588
  by (simp only: DERIV_add DERIV_minus)
hoelzl@51642
   589
hoelzl@51642
   590
lemma DERIV_continuous: "DERIV f x : s :> D \<Longrightarrow> continuous (at x within s) f"
hoelzl@51642
   591
  by (drule FDERIV_continuous[OF DERIV_D_FDERIV]) simp
huffman@21164
   592
huffman@21164
   593
lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
hoelzl@51642
   594
  by (auto dest!: DERIV_continuous)
hoelzl@51642
   595
hoelzl@51642
   596
lemma DERIV_mult': "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x : s :> f x * E + D * g x"
hoelzl@51642
   597
  by (rule DERIV_I_FDERIV[OF FDERIV_mult]) (auto simp: field_simps dest: DERIV_D_FDERIV)
huffman@21164
   598
hoelzl@51642
   599
lemma DERIV_mult: "DERIV f x : s :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x : s :> Da * g x + Db * f x"
hoelzl@51642
   600
  by (rule DERIV_I_FDERIV[OF FDERIV_mult]) (auto simp: field_simps dest: DERIV_D_FDERIV)
hoelzl@51642
   601
hoelzl@51642
   602
text {* Derivative of linear multiplication *}
huffman@21164
   603
hoelzl@51642
   604
lemma DERIV_cmult:
hoelzl@51642
   605
  "DERIV f x : s :> D ==> DERIV (%x. c * f x) x : s :> c*D"
hoelzl@51642
   606
  by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164
   607
hoelzl@51642
   608
lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x : s :> c"
hoelzl@51642
   609
  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
hoelzl@51642
   610
hoelzl@51642
   611
lemma DERIV_cdivide: "DERIV f x : s :> D ==> DERIV (%x. f x / c) x : s :> D / c"
hoelzl@51642
   612
  apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x : s :> (1 / c) * D", force)
hoelzl@51642
   613
  apply (erule DERIV_cmult)
hoelzl@51642
   614
  done
huffman@21164
   615
huffman@21164
   616
lemma DERIV_unique:
hoelzl@51642
   617
  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
hoelzl@50331
   618
  unfolding deriv_def by (rule LIM_unique) 
huffman@21164
   619
hoelzl@51642
   620
lemma DERIV_setsum':
hoelzl@51642
   621
  "(\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x : s :> (f' x n)) \<Longrightarrow> DERIV (\<lambda>x. setsum (f x) S) x : s :> setsum (f' x) S"
hoelzl@51642
   622
  by (rule DERIV_I_FDERIV[OF FDERIV_setsum]) (auto simp: setsum_right_distrib dest: DERIV_D_FDERIV)
huffman@21164
   623
hoelzl@31880
   624
lemma DERIV_setsum:
hoelzl@51642
   625
  "finite S \<Longrightarrow> (\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x : s :> (f' x n)) \<Longrightarrow> DERIV (\<lambda>x. setsum (f x) S) x : s :> setsum (f' x) S"
hoelzl@51642
   626
  by (rule DERIV_setsum')
hoelzl@51642
   627
hoelzl@51642
   628
lemma DERIV_sumr [rule_format (no_asm)]: (* REMOVE *)
hoelzl@51642
   629
     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x : s :> (f' r x))
hoelzl@51642
   630
      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x : s :> (\<Sum>r=m..<n. f' r x)"
hoelzl@51642
   631
  by (auto intro: DERIV_setsum)
hoelzl@51642
   632
hoelzl@51642
   633
lemma DERIV_inverse':
hoelzl@51642
   634
  "DERIV f x : s :> D \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse (f x)) x : s :> - (inverse (f x) * D * inverse (f x))"
hoelzl@51642
   635
  by (rule DERIV_I_FDERIV[OF FDERIV_inverse]) (auto dest: DERIV_D_FDERIV)
hoelzl@51642
   636
hoelzl@51642
   637
text {* Power of @{text "-1"} *}
hoelzl@51642
   638
hoelzl@51642
   639
lemma DERIV_inverse:
hoelzl@51642
   640
  "x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse(x)) x : s :> - (inverse x ^ Suc (Suc 0))"
hoelzl@51642
   641
  by (drule DERIV_inverse' [OF DERIV_ident]) simp
hoelzl@51642
   642
hoelzl@51642
   643
text {* Derivative of inverse *}
hoelzl@51642
   644
hoelzl@51642
   645
lemma DERIV_inverse_fun:
hoelzl@51642
   646
  "DERIV f x : s :> d \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse (f x)) x : s :> (- (d * inverse(f x ^ Suc (Suc 0))))"
hoelzl@51642
   647
  by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
hoelzl@51642
   648
hoelzl@51642
   649
text {* Derivative of quotient *}
hoelzl@51642
   650
hoelzl@51642
   651
lemma DERIV_divide:
hoelzl@51642
   652
  "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x : s :> (D * g x - f x * E) / (g x * g x)"
hoelzl@51642
   653
  by (rule DERIV_I_FDERIV[OF FDERIV_divide])
hoelzl@51642
   654
     (auto dest: DERIV_D_FDERIV simp: field_simps nonzero_inverse_mult_distrib divide_inverse)
hoelzl@51642
   655
hoelzl@51642
   656
lemma DERIV_quotient:
hoelzl@51642
   657
  "DERIV f x : s :> d \<Longrightarrow> DERIV g x : s :> e \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>y. f y / g y) x : s :> (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))"
hoelzl@51642
   658
  by (drule (2) DERIV_divide) (simp add: mult_commute)
hoelzl@51642
   659
hoelzl@51642
   660
lemma DERIV_power_Suc:
hoelzl@51642
   661
  "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. f x ^ Suc n) x : s :> (1 + of_nat n) * (D * f x ^ n)"
hoelzl@51642
   662
  by (rule DERIV_I_FDERIV[OF FDERIV_power]) (auto simp: deriv_fderiv)
hoelzl@51642
   663
hoelzl@51642
   664
lemma DERIV_power:
hoelzl@51642
   665
  "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. f x ^ n) x : s :> of_nat n * (D * f x ^ (n - Suc 0))"
hoelzl@51642
   666
  by (rule DERIV_I_FDERIV[OF FDERIV_power]) (auto simp: deriv_fderiv)
hoelzl@31880
   667
hoelzl@51642
   668
lemma DERIV_pow: "DERIV (%x. x ^ n) x : s :> real n * (x ^ (n - Suc 0))"
hoelzl@51642
   669
  apply (cut_tac DERIV_power [OF DERIV_ident])
hoelzl@51642
   670
  apply (simp add: real_of_nat_def)
hoelzl@51642
   671
  done
hoelzl@51642
   672
hoelzl@51642
   673
lemma DERIV_chain': "DERIV f x : s :> D \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> DERIV (\<lambda>x. g (f x)) x : s :> E * D"
hoelzl@51642
   674
  using FDERIV_compose[of f "\<lambda>x. x * D" x s g "\<lambda>x. x * E"]
hoelzl@51642
   675
  by (auto simp: deriv_fderiv ac_simps dest: FDERIV_subset)
hoelzl@51642
   676
hoelzl@51642
   677
text {* Standard version *}
hoelzl@51642
   678
hoelzl@51642
   679
lemma DERIV_chain: "DERIV f (g x) :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (f o g) x : s :> Da * Db"
hoelzl@51642
   680
  by (drule (1) DERIV_chain', simp add: o_def mult_commute)
hoelzl@51642
   681
hoelzl@51642
   682
lemma DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (%x. f (g x)) x : s :> Da * Db"
hoelzl@51642
   683
  by (auto dest: DERIV_chain simp add: o_def)
hoelzl@51642
   684
hoelzl@51642
   685
subsubsection {* @{text "DERIV_intros"} *}
hoelzl@51642
   686
hoelzl@51642
   687
ML {*
hoelzl@51642
   688
structure Deriv_Intros = Named_Thms
hoelzl@51642
   689
(
hoelzl@51642
   690
  val name = @{binding DERIV_intros}
hoelzl@51642
   691
  val description = "DERIV introduction rules"
hoelzl@51642
   692
)
hoelzl@51642
   693
*}
hoelzl@51642
   694
hoelzl@51642
   695
setup Deriv_Intros.setup
hoelzl@51642
   696
hoelzl@51642
   697
lemma DERIV_cong: "DERIV f x : s :> X \<Longrightarrow> X = Y \<Longrightarrow> DERIV f x : s :> Y"
hoelzl@51642
   698
  by simp
hoelzl@51642
   699
hoelzl@51642
   700
declare
hoelzl@51642
   701
  DERIV_const[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   702
  DERIV_ident[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   703
  DERIV_add[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   704
  DERIV_minus[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   705
  DERIV_mult[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   706
  DERIV_diff[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   707
  DERIV_inverse'[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   708
  DERIV_divide[THEN DERIV_cong, DERIV_intros]
hoelzl@51642
   709
  DERIV_power[where 'a=real, THEN DERIV_cong,
hoelzl@51642
   710
              unfolded real_of_nat_def[symmetric], DERIV_intros]
hoelzl@51642
   711
  DERIV_setsum[THEN DERIV_cong, DERIV_intros]
huffman@21164
   712
huffman@21164
   713
text{*Alternative definition for differentiability*}
huffman@21164
   714
huffman@21164
   715
lemma DERIV_LIM_iff:
huffman@31338
   716
  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
huffman@21784
   717
     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
huffman@21164
   718
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
huffman@21164
   719
apply (rule iffI)
huffman@21164
   720
apply (drule_tac k="- a" in LIM_offset)
huffman@21164
   721
apply (simp add: diff_minus)
huffman@21164
   722
apply (drule_tac k="a" in LIM_offset)
huffman@21164
   723
apply (simp add: add_commute)
huffman@21164
   724
done
huffman@21164
   725
hoelzl@51642
   726
lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
hoelzl@51642
   727
  by (simp add: deriv_def diff_minus [symmetric] DERIV_LIM_iff)
huffman@21164
   728
hoelzl@51642
   729
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@51642
   730
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
hoelzl@51642
   731
  unfolding DERIV_iff2
hoelzl@51642
   732
proof (rule filterlim_cong)
hoelzl@51642
   733
  assume "eventually (\<lambda>x. f x = g x) (nhds x)"
hoelzl@51642
   734
  moreover then have "f x = g x" by (auto simp: eventually_nhds)
hoelzl@51642
   735
  moreover assume "x = y" "u = v"
hoelzl@51642
   736
  ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
hoelzl@51642
   737
    by (auto simp: eventually_at_filter elim: eventually_elim1)
hoelzl@51642
   738
qed simp_all
huffman@21164
   739
hoelzl@51642
   740
lemma DERIV_shift:
hoelzl@51642
   741
  "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
hoelzl@51642
   742
  by (simp add: DERIV_iff field_simps)
huffman@21164
   743
hoelzl@51642
   744
lemma DERIV_mirror:
hoelzl@51642
   745
  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
hoelzl@51642
   746
  by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right
hoelzl@51642
   747
                tendsto_minus_cancel_left field_simps conj_commute)
huffman@21164
   748
huffman@29975
   749
text {* Caratheodory formulation of derivative at a point *}
huffman@21164
   750
huffman@21164
   751
lemma CARAT_DERIV:
hoelzl@51642
   752
  "(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
huffman@21164
   753
      (is "?lhs = ?rhs")
huffman@21164
   754
proof
huffman@21164
   755
  assume der: "DERIV f x :> l"
huffman@21784
   756
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
huffman@21164
   757
  proof (intro exI conjI)
huffman@21784
   758
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
nipkow@23413
   759
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
huffman@21164
   760
    show "isCont ?g x" using der
huffman@21164
   761
      by (simp add: isCont_iff DERIV_iff diff_minus
huffman@21164
   762
               cong: LIM_equal [rule_format])
huffman@21164
   763
    show "?g x = l" by simp
huffman@21164
   764
  qed
huffman@21164
   765
next
huffman@21164
   766
  assume "?rhs"
huffman@21164
   767
  then obtain g where
huffman@21784
   768
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
huffman@21164
   769
  thus "(DERIV f x :> l)"
nipkow@23413
   770
     by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)
huffman@21164
   771
qed
huffman@21164
   772
wenzelm@31899
   773
text {*
wenzelm@31899
   774
 Let's do the standard proof, though theorem
wenzelm@31899
   775
 @{text "LIM_mult2"} follows from a NS proof
wenzelm@31899
   776
*}
huffman@21164
   777
huffman@29975
   778
subsection {* Local extrema *}
huffman@29975
   779
huffman@21164
   780
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   781
paulson@33654
   782
lemma DERIV_pos_inc_right:
huffman@21164
   783
  fixes f :: "real => real"
huffman@21164
   784
  assumes der: "DERIV f x :> l"
huffman@21164
   785
      and l:   "0 < l"
huffman@21164
   786
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   787
proof -
huffman@21164
   788
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   789
  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
   790
    by (simp add: diff_minus)
huffman@21164
   791
  then obtain s
huffman@21164
   792
        where s:   "0 < s"
huffman@21164
   793
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   794
    by auto
huffman@21164
   795
  thus ?thesis
huffman@21164
   796
  proof (intro exI conjI strip)
huffman@23441
   797
    show "0<s" using s .
huffman@21164
   798
    fix h::real
huffman@21164
   799
    assume "0 < h" "h < s"
huffman@21164
   800
    with all [of h] show "f x < f (x+h)"
huffman@21164
   801
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   802
    split add: split_if_asm)
huffman@21164
   803
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   804
      with l
huffman@21164
   805
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   806
      thus "f x < f (x+h)"
huffman@21164
   807
  by (simp add: pos_less_divide_eq h)
huffman@21164
   808
    qed
huffman@21164
   809
  qed
huffman@21164
   810
qed
huffman@21164
   811
paulson@33654
   812
lemma DERIV_neg_dec_left:
huffman@21164
   813
  fixes f :: "real => real"
huffman@21164
   814
  assumes der: "DERIV f x :> l"
huffman@21164
   815
      and l:   "l < 0"
huffman@21164
   816
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   817
proof -
huffman@21164
   818
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   819
  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
   820
    by (simp add: diff_minus)
huffman@21164
   821
  then obtain s
huffman@21164
   822
        where s:   "0 < s"
huffman@21164
   823
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   824
    by auto
huffman@21164
   825
  thus ?thesis
huffman@21164
   826
  proof (intro exI conjI strip)
huffman@23441
   827
    show "0<s" using s .
huffman@21164
   828
    fix h::real
huffman@21164
   829
    assume "0 < h" "h < s"
huffman@21164
   830
    with all [of "-h"] show "f x < f (x-h)"
huffman@21164
   831
    proof (simp add: abs_if pos_less_divide_eq diff_minus [symmetric]
huffman@21164
   832
    split add: split_if_asm)
huffman@21164
   833
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   834
      with l
huffman@21164
   835
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   836
      thus "f x < f (x-h)"
huffman@21164
   837
  by (simp add: pos_less_divide_eq h)
huffman@21164
   838
    qed
huffman@21164
   839
  qed
huffman@21164
   840
qed
huffman@21164
   841
paulson@33654
   842
lemma DERIV_pos_inc_left:
paulson@33654
   843
  fixes f :: "real => real"
paulson@33654
   844
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
paulson@33654
   845
  apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])
hoelzl@41368
   846
  apply (auto simp add: DERIV_minus)
paulson@33654
   847
  done
paulson@33654
   848
paulson@33654
   849
lemma DERIV_neg_dec_right:
paulson@33654
   850
  fixes f :: "real => real"
paulson@33654
   851
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
paulson@33654
   852
  apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])
hoelzl@41368
   853
  apply (auto simp add: DERIV_minus)
paulson@33654
   854
  done
paulson@33654
   855
huffman@21164
   856
lemma DERIV_local_max:
huffman@21164
   857
  fixes f :: "real => real"
huffman@21164
   858
  assumes der: "DERIV f x :> l"
huffman@21164
   859
      and d:   "0 < d"
huffman@21164
   860
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   861
  shows "l = 0"
huffman@21164
   862
proof (cases rule: linorder_cases [of l 0])
huffman@23441
   863
  case equal thus ?thesis .
huffman@21164
   864
next
huffman@21164
   865
  case less
paulson@33654
   866
  from DERIV_neg_dec_left [OF der less]
huffman@21164
   867
  obtain d' where d': "0 < d'"
huffman@21164
   868
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   869
  from real_lbound_gt_zero [OF d d']
huffman@21164
   870
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   871
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   872
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   873
next
huffman@21164
   874
  case greater
paulson@33654
   875
  from DERIV_pos_inc_right [OF der greater]
huffman@21164
   876
  obtain d' where d': "0 < d'"
huffman@21164
   877
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   878
  from real_lbound_gt_zero [OF d d']
huffman@21164
   879
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   880
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   881
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   882
qed
huffman@21164
   883
huffman@21164
   884
huffman@21164
   885
text{*Similar theorem for a local minimum*}
huffman@21164
   886
lemma DERIV_local_min:
huffman@21164
   887
  fixes f :: "real => real"
huffman@21164
   888
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   889
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   890
huffman@21164
   891
huffman@21164
   892
text{*In particular, if a function is locally flat*}
huffman@21164
   893
lemma DERIV_local_const:
huffman@21164
   894
  fixes f :: "real => real"
huffman@21164
   895
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   896
by (auto dest!: DERIV_local_max)
huffman@21164
   897
huffman@29975
   898
huffman@29975
   899
subsection {* Rolle's Theorem *}
huffman@29975
   900
huffman@21164
   901
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   902
lemma lemma_interval_lt:
huffman@21164
   903
     "[| a < x;  x < b |]
huffman@21164
   904
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
   905
huffman@22998
   906
apply (simp add: abs_less_iff)
huffman@21164
   907
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   908
apply (rule_tac x = "x-a" in exI)
huffman@21164
   909
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   910
done
huffman@21164
   911
huffman@21164
   912
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   913
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   914
apply (drule lemma_interval_lt, auto)
huffman@44921
   915
apply force
huffman@21164
   916
done
huffman@21164
   917
huffman@21164
   918
text{*Rolle's Theorem.
huffman@21164
   919
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   920
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
   921
   and @{term "f(a) = f(b)"},
huffman@21164
   922
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
   923
theorem Rolle:
huffman@21164
   924
  assumes lt: "a < b"
huffman@21164
   925
      and eq: "f(a) = f(b)"
huffman@21164
   926
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
   927
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
   928
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
   929
proof -
huffman@21164
   930
  have le: "a \<le> b" using lt by simp
huffman@21164
   931
  from isCont_eq_Ub [OF le con]
huffman@21164
   932
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
   933
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
   934
    by blast
huffman@21164
   935
  from isCont_eq_Lb [OF le con]
huffman@21164
   936
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
   937
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
   938
    by blast
huffman@21164
   939
  show ?thesis
huffman@21164
   940
  proof cases
huffman@21164
   941
    assume axb: "a < x & x < b"
huffman@21164
   942
        --{*@{term f} attains its maximum within the interval*}
chaieb@27668
   943
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
   944
    from lemma_interval [OF ax xb]
huffman@21164
   945
    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
   946
      by blast
huffman@21164
   947
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
   948
      by blast
huffman@21164
   949
    from differentiableD [OF dif [OF axb]]
huffman@21164
   950
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
   951
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
   952
        --{*the derivative at a local maximum is zero*}
huffman@21164
   953
    thus ?thesis using ax xb der by auto
huffman@21164
   954
  next
huffman@21164
   955
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
   956
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
   957
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
   958
    show ?thesis
huffman@21164
   959
    proof cases
huffman@21164
   960
      assume ax'b: "a < x' & x' < b"
huffman@21164
   961
        --{*@{term f} attains its minimum within the interval*}
chaieb@27668
   962
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
   963
      from lemma_interval [OF ax' x'b]
huffman@21164
   964
      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
   965
  by blast
huffman@21164
   966
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
   967
  by blast
huffman@21164
   968
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
   969
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
   970
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
   971
        --{*the derivative at a local minimum is zero*}
huffman@21164
   972
      thus ?thesis using ax' x'b der by auto
huffman@21164
   973
    next
huffman@21164
   974
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
   975
        --{*@{term f} is constant througout the interval*}
huffman@21164
   976
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
   977
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
   978
      from dense [OF lt]
huffman@21164
   979
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
   980
      from lemma_interval [OF ar rb]
huffman@21164
   981
      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
   982
  by blast
huffman@21164
   983
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
   984
      proof (clarify)
huffman@21164
   985
        fix z::real
huffman@21164
   986
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
   987
        show "f z = f b"
huffman@21164
   988
        proof (rule order_antisym)
huffman@21164
   989
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
   990
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
   991
        qed
huffman@21164
   992
      qed
huffman@21164
   993
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
   994
      proof (intro strip)
huffman@21164
   995
        fix y::real
huffman@21164
   996
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
   997
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
   998
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
   999
      qed
huffman@21164
  1000
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1001
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1002
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1003
        --{*the derivative of a constant function is zero*}
huffman@21164
  1004
      thus ?thesis using ar rb der by auto
huffman@21164
  1005
    qed
huffman@21164
  1006
  qed
huffman@21164
  1007
qed
huffman@21164
  1008
huffman@21164
  1009
huffman@21164
  1010
subsection{*Mean Value Theorem*}
huffman@21164
  1011
huffman@21164
  1012
lemma lemma_MVT:
huffman@21164
  1013
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
hoelzl@51481
  1014
  by (cases "a = b") (simp_all add: field_simps)
huffman@21164
  1015
huffman@21164
  1016
theorem MVT:
huffman@21164
  1017
  assumes lt:  "a < b"
huffman@21164
  1018
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
huffman@21164
  1019
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"
huffman@21784
  1020
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
  1021
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1022
proof -
huffman@21164
  1023
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@44233
  1024
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
huffman@44233
  1025
    using con by (fast intro: isCont_intros)
huffman@21164
  1026
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
huffman@21164
  1027
  proof (clarify)
huffman@21164
  1028
    fix x::real
huffman@21164
  1029
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1030
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1031
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1032
    show "?F differentiable x"
huffman@21164
  1033
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1034
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1035
  qed
huffman@21164
  1036
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1037
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1038
    by blast
huffman@21164
  1039
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1040
    by (rule DERIV_cmult_Id)
huffman@21164
  1041
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1042
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1043
    by (rule DERIV_add [OF der])
huffman@21164
  1044
  show ?thesis
huffman@21164
  1045
  proof (intro exI conjI)
huffman@23441
  1046
    show "a < z" using az .
huffman@23441
  1047
    show "z < b" using zb .
huffman@21164
  1048
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1049
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1050
  qed
huffman@21164
  1051
qed
huffman@21164
  1052
hoelzl@29803
  1053
lemma MVT2:
hoelzl@29803
  1054
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803
  1055
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803
  1056
apply (drule MVT)
hoelzl@29803
  1057
apply (blast intro: DERIV_isCont)
hoelzl@29803
  1058
apply (force dest: order_less_imp_le simp add: differentiable_def)
hoelzl@29803
  1059
apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803
  1060
done
hoelzl@29803
  1061
huffman@21164
  1062
huffman@21164
  1063
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1064
huffman@21164
  1065
lemma DERIV_isconst_end:
huffman@21164
  1066
  fixes f :: "real => real"
huffman@21164
  1067
  shows "[| a < b;
huffman@21164
  1068
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1069
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1070
        ==> f b = f a"
huffman@21164
  1071
apply (drule MVT, assumption)
huffman@21164
  1072
apply (blast intro: differentiableI)
huffman@21164
  1073
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1074
done
huffman@21164
  1075
huffman@21164
  1076
lemma DERIV_isconst1:
huffman@21164
  1077
  fixes f :: "real => real"
huffman@21164
  1078
  shows "[| a < b;
huffman@21164
  1079
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1080
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1081
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1082
apply safe
huffman@21164
  1083
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1084
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1085
done
huffman@21164
  1086
huffman@21164
  1087
lemma DERIV_isconst2:
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
         a \<le> x; x \<le> b |]
huffman@21164
  1093
        ==> f x = f a"
huffman@21164
  1094
apply (blast dest: DERIV_isconst1)
huffman@21164
  1095
done
huffman@21164
  1096
hoelzl@29803
  1097
lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803
  1098
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803
  1099
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1100
  shows "f x = f y"
hoelzl@29803
  1101
proof (cases "x = y")
hoelzl@29803
  1102
  case False
hoelzl@29803
  1103
  let ?a = "min x y"
hoelzl@29803
  1104
  let ?b = "max x y"
hoelzl@29803
  1105
  
hoelzl@29803
  1106
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1107
  proof (rule allI, rule impI)
hoelzl@29803
  1108
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hoelzl@29803
  1109
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hoelzl@29803
  1110
    hence "z \<in> {a<..<b}" by auto
hoelzl@29803
  1111
    thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1112
  qed
hoelzl@29803
  1113
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803
  1114
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803
  1115
hoelzl@29803
  1116
  have "?a < ?b" using `x \<noteq> y` by auto
hoelzl@29803
  1117
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1118
  show ?thesis by auto
hoelzl@29803
  1119
qed auto
hoelzl@29803
  1120
huffman@21164
  1121
lemma DERIV_isconst_all:
huffman@21164
  1122
  fixes f :: "real => real"
huffman@21164
  1123
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1124
apply (rule linorder_cases [of x y])
huffman@21164
  1125
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1126
done
huffman@21164
  1127
huffman@21164
  1128
lemma DERIV_const_ratio_const:
huffman@21784
  1129
  fixes f :: "real => real"
huffman@21784
  1130
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1131
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1132
apply (drule_tac [!] f = f in MVT)
huffman@21164
  1133
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
nipkow@23477
  1134
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
huffman@21164
  1135
done
huffman@21164
  1136
huffman@21164
  1137
lemma DERIV_const_ratio_const2:
huffman@21784
  1138
  fixes f :: "real => real"
huffman@21784
  1139
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
huffman@21164
  1140
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
huffman@21164
  1141
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1142
done
huffman@21164
  1143
huffman@21164
  1144
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1145
by (simp)
huffman@21164
  1146
huffman@21164
  1147
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1148
by (simp)
huffman@21164
  1149
huffman@21164
  1150
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1151
huffman@21164
  1152
lemma DERIV_const_average:
huffman@21164
  1153
  fixes v :: "real => real"
huffman@21164
  1154
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1155
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1156
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1157
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1158
  case equal with neq show ?thesis by simp
huffman@21164
  1159
next
huffman@21164
  1160
  case less
huffman@21164
  1161
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1162
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1163
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1164
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1165
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1166
  ultimately show ?thesis using neq by force
huffman@21164
  1167
next
huffman@21164
  1168
  case greater
huffman@21164
  1169
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1170
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1171
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1172
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1173
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1174
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1175
qed
huffman@21164
  1176
paulson@33654
  1177
(* A function with positive derivative is increasing. 
paulson@33654
  1178
   A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654
  1179
*)
paulson@33654
  1180
lemma DERIV_pos_imp_increasing:
paulson@33654
  1181
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1182
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
paulson@33654
  1183
  shows "f a < f b"
paulson@33654
  1184
proof (rule ccontr)
wenzelm@41550
  1185
  assume f: "~ f a < f b"
wenzelm@33690
  1186
  have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1187
      & f b - f a = (b - a) * l"
wenzelm@33690
  1188
    apply (rule MVT)
wenzelm@33690
  1189
      using assms
wenzelm@33690
  1190
      apply auto
wenzelm@33690
  1191
      apply (metis DERIV_isCont)
huffman@36777
  1192
     apply (metis differentiableI less_le)
wenzelm@33690
  1193
    done
wenzelm@41550
  1194
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
paulson@33654
  1195
      and "f b - f a = (b - a) * l"
paulson@33654
  1196
    by auto
wenzelm@41550
  1197
  with assms f have "~(l > 0)"
huffman@36777
  1198
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550
  1199
  with assms z show False
huffman@36777
  1200
    by (metis DERIV_unique less_le)
paulson@33654
  1201
qed
paulson@33654
  1202
noschinl@45791
  1203
lemma DERIV_nonneg_imp_nondecreasing:
paulson@33654
  1204
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1205
  assumes "a \<le> b" and
paulson@33654
  1206
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654
  1207
  shows "f a \<le> f b"
paulson@33654
  1208
proof (rule ccontr, cases "a = b")
wenzelm@41550
  1209
  assume "~ f a \<le> f b" and "a = b"
wenzelm@41550
  1210
  then show False by auto
haftmann@37891
  1211
next
haftmann@37891
  1212
  assume A: "~ f a \<le> f b"
haftmann@37891
  1213
  assume B: "a ~= b"
paulson@33654
  1214
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1215
      & f b - f a = (b - a) * l"
wenzelm@33690
  1216
    apply -
wenzelm@33690
  1217
    apply (rule MVT)
wenzelm@33690
  1218
      apply auto
wenzelm@33690
  1219
      apply (metis DERIV_isCont)
huffman@36777
  1220
     apply (metis differentiableI less_le)
paulson@33654
  1221
    done
wenzelm@41550
  1222
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
haftmann@37891
  1223
      and C: "f b - f a = (b - a) * l"
paulson@33654
  1224
    by auto
haftmann@37891
  1225
  with A have "a < b" "f b < f a" by auto
haftmann@37891
  1226
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
huffman@45051
  1227
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@41550
  1228
  with assms z show False
paulson@33654
  1229
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1230
qed
paulson@33654
  1231
paulson@33654
  1232
lemma DERIV_neg_imp_decreasing:
paulson@33654
  1233
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1234
  assumes "a < b" and
paulson@33654
  1235
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654
  1236
  shows "f a > f b"
paulson@33654
  1237
proof -
paulson@33654
  1238
  have "(%x. -f x) a < (%x. -f x) b"
paulson@33654
  1239
    apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
wenzelm@33690
  1240
    using assms
wenzelm@33690
  1241
    apply auto
paulson@33654
  1242
    apply (metis DERIV_minus neg_0_less_iff_less)
paulson@33654
  1243
    done
paulson@33654
  1244
  thus ?thesis
paulson@33654
  1245
    by simp
paulson@33654
  1246
qed
paulson@33654
  1247
paulson@33654
  1248
lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654
  1249
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1250
  assumes "a \<le> b" and
paulson@33654
  1251
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654
  1252
  shows "f a \<ge> f b"
paulson@33654
  1253
proof -
paulson@33654
  1254
  have "(%x. -f x) a \<le> (%x. -f x) b"
noschinl@45791
  1255
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
wenzelm@33690
  1256
    using assms
wenzelm@33690
  1257
    apply auto
paulson@33654
  1258
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1259
    done
paulson@33654
  1260
  thus ?thesis
paulson@33654
  1261
    by simp
paulson@33654
  1262
qed
huffman@21164
  1263
huffman@23041
  1264
text {* Derivative of inverse function *}
huffman@23041
  1265
huffman@23041
  1266
lemma DERIV_inverse_function:
huffman@23041
  1267
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1268
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1269
  assumes neq: "D \<noteq> 0"
huffman@23044
  1270
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1271
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1272
  assumes cont: "isCont g x"
huffman@23041
  1273
  shows "DERIV g x :> inverse D"
huffman@23041
  1274
unfolding DERIV_iff2
huffman@23044
  1275
proof (rule LIM_equal2)
huffman@23044
  1276
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1277
    using a b by arith 
huffman@23044
  1278
next
huffman@23041
  1279
  fix y
huffman@23044
  1280
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1281
  hence "a < y" and "y < b" 
huffman@23044
  1282
    by (simp_all add: abs_less_iff)
huffman@23041
  1283
  thus "(g y - g x) / (y - x) =
huffman@23041
  1284
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1285
    by (simp add: inj)
huffman@23041
  1286
next
huffman@23041
  1287
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1288
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1289
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1290
    using inj a b by simp
huffman@23041
  1291
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
huffman@23041
  1292
  proof (safe intro!: exI)
huffman@23044
  1293
    show "0 < min (x - a) (b - x)"
huffman@23044
  1294
      using a b by simp
huffman@23041
  1295
  next
huffman@23041
  1296
    fix y
huffman@23044
  1297
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1298
    hence y: "a < y" "y < b"
huffman@23044
  1299
      by (simp_all add: abs_less_iff)
huffman@23041
  1300
    assume "g y = g x"
huffman@23041
  1301
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1302
    hence "y = x" using inj y a b by simp
huffman@23041
  1303
    also assume "y \<noteq> x"
huffman@23041
  1304
    finally show False by simp
huffman@23041
  1305
  qed
huffman@23041
  1306
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1307
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1308
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1309
        -- x --> inverse D"
huffman@44568
  1310
    using neq by (rule tendsto_inverse)
huffman@23041
  1311
qed
huffman@23041
  1312
huffman@29975
  1313
subsection {* Generalized Mean Value Theorem *}
huffman@29975
  1314
huffman@21164
  1315
theorem GMVT:
huffman@21784
  1316
  fixes a b :: real
huffman@21164
  1317
  assumes alb: "a < b"
wenzelm@41550
  1318
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
wenzelm@41550
  1319
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
wenzelm@41550
  1320
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
wenzelm@41550
  1321
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
huffman@21164
  1322
  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
  1323
proof -
huffman@21164
  1324
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
wenzelm@41550
  1325
  from assms have "a < b" by simp
huffman@21164
  1326
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@44233
  1327
    using fc gc by simp
huffman@44233
  1328
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
huffman@44233
  1329
    using fd gd by simp
huffman@21164
  1330
  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
  1331
  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
  1332
  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
  1333
huffman@21164
  1334
  from cdef have cint: "a < c \<and> c < b" by auto
huffman@21164
  1335
  with gd have "g differentiable c" by simp
huffman@21164
  1336
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1337
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1338
huffman@21164
  1339
  from cdef have "a < c \<and> c < b" by auto
huffman@21164
  1340
  with fd have "f differentiable c" by simp
huffman@21164
  1341
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1342
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1343
huffman@21164
  1344
  from cdef have "DERIV ?h c :> l" by auto
hoelzl@41368
  1345
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
hoelzl@41368
  1346
    using g'cdef f'cdef by (auto intro!: DERIV_intros)
huffman@21164
  1347
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1348
huffman@21164
  1349
  {
huffman@21164
  1350
    from cdef have "?h b - ?h a = (b - a) * l" by auto
huffman@21164
  1351
    also with leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1352
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1353
  }
huffman@21164
  1354
  moreover
huffman@21164
  1355
  {
huffman@21164
  1356
    have "?h b - ?h a =
huffman@21164
  1357
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1358
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1359
      by (simp add: algebra_simps)
huffman@21164
  1360
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1361
  }
huffman@21164
  1362
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1363
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1364
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1365
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1366
huffman@21164
  1367
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1368
qed
huffman@21164
  1369
hoelzl@50327
  1370
lemma GMVT':
hoelzl@50327
  1371
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1372
  assumes "a < b"
hoelzl@50327
  1373
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
hoelzl@50327
  1374
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
hoelzl@50327
  1375
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
hoelzl@50327
  1376
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327
  1377
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1378
proof -
hoelzl@50327
  1379
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
hoelzl@50327
  1380
    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@50327
  1381
    using assms by (intro GMVT) (force simp: differentiable_def)+
hoelzl@50327
  1382
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1383
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327
  1384
  then show ?thesis
hoelzl@50327
  1385
    by auto
hoelzl@50327
  1386
qed
hoelzl@50327
  1387
hoelzl@51529
  1388
hoelzl@51529
  1389
subsection {* L'Hopitals rule *}
hoelzl@51529
  1390
hoelzl@51641
  1391
lemma isCont_If_ge:
hoelzl@51641
  1392
  fixes a :: "'a :: linorder_topology"
hoelzl@51641
  1393
  shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
hoelzl@51641
  1394
  unfolding isCont_def continuous_within
hoelzl@51641
  1395
  apply (intro filterlim_split_at)
hoelzl@51641
  1396
  apply (subst filterlim_cong[OF refl refl, where g=g])
hoelzl@51641
  1397
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1398
  apply (subst filterlim_cong[OF refl refl, where g=f])
hoelzl@51641
  1399
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1400
  done
hoelzl@51641
  1401
hoelzl@50327
  1402
lemma lhopital_right_0:
hoelzl@50329
  1403
  fixes f0 g0 :: "real \<Rightarrow> real"
hoelzl@50329
  1404
  assumes f_0: "(f0 ---> 0) (at_right 0)"
hoelzl@50329
  1405
  assumes g_0: "(g0 ---> 0) (at_right 0)"
hoelzl@50327
  1406
  assumes ev:
hoelzl@50329
  1407
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1408
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50329
  1409
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
hoelzl@50329
  1410
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
hoelzl@50327
  1411
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50329
  1412
  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
hoelzl@50327
  1413
proof -
hoelzl@50329
  1414
  def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
hoelzl@50329
  1415
  then have "f 0 = 0" by simp
hoelzl@50329
  1416
hoelzl@50329
  1417
  def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
hoelzl@50329
  1418
  then have "g 0 = 0" by simp
hoelzl@50329
  1419
hoelzl@50329
  1420
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329
  1421
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329
  1422
    using ev by eventually_elim auto
hoelzl@50329
  1423
  then obtain a where [arith]: "0 < a"
hoelzl@50329
  1424
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327
  1425
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329
  1426
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329
  1427
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
hoelzl@51641
  1428
    unfolding eventually_at eventually_at by (auto simp: dist_real_def)
hoelzl@50327
  1429
hoelzl@50329
  1430
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329
  1431
    using g0_neq_0 by (simp add: g_def)
hoelzl@50329
  1432
hoelzl@50329
  1433
  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
hoelzl@50329
  1434
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
hoelzl@50329
  1435
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1436
  note f = this
hoelzl@50329
  1437
hoelzl@50329
  1438
  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
hoelzl@50329
  1439
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
hoelzl@50329
  1440
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1441
  note g = this
hoelzl@50329
  1442
hoelzl@50329
  1443
  have "isCont f 0"
hoelzl@51641
  1444
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
hoelzl@51641
  1445
hoelzl@50329
  1446
  have "isCont g 0"
hoelzl@51641
  1447
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
hoelzl@50329
  1448
hoelzl@50327
  1449
  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
  1450
  proof (rule bchoice, rule)
hoelzl@50327
  1451
    fix x assume "x \<in> {0 <..< a}"
hoelzl@50327
  1452
    then have x[arith]: "0 < x" "x < a" by auto
hoelzl@50327
  1453
    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
  1454
      by auto
hoelzl@50328
  1455
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
hoelzl@50328
  1456
      using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1457
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
hoelzl@50328
  1458
      using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1459
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
hoelzl@50328
  1460
      using f g `x < a` by (intro GMVT') auto
hoelzl@50327
  1461
    then guess c ..
hoelzl@50327
  1462
    moreover
hoelzl@50327
  1463
    with g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327
  1464
      by (simp add: field_simps)
hoelzl@50327
  1465
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
hoelzl@50327
  1466
      using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
hoelzl@50327
  1467
  qed
hoelzl@50327
  1468
  then guess \<zeta> ..
hoelzl@50327
  1469
  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@51641
  1470
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327
  1471
  moreover
hoelzl@50327
  1472
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327
  1473
    by eventually_elim auto
hoelzl@50327
  1474
  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
hoelzl@50327
  1475
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
hoelzl@51641
  1476
       (auto intro: tendsto_const tendsto_ident_at)
hoelzl@50327
  1477
  then have "(\<zeta> ---> 0) (at_right 0)"
hoelzl@50327
  1478
    by (rule tendsto_norm_zero_cancel)
hoelzl@50327
  1479
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
hoelzl@51641
  1480
    by (auto elim!: eventually_elim1 simp: filterlim_at)
hoelzl@50327
  1481
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
hoelzl@50327
  1482
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
hoelzl@50329
  1483
  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
hoelzl@50328
  1484
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
hoelzl@50328
  1485
       (auto elim: eventually_elim1)
hoelzl@50329
  1486
  also have "?P \<longleftrightarrow> ?thesis"
hoelzl@51641
  1487
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
hoelzl@50329
  1488
  finally show ?thesis .
hoelzl@50327
  1489
qed
hoelzl@50327
  1490
hoelzl@50330
  1491
lemma lhopital_right:
hoelzl@50330
  1492
  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1493
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1494
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1495
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1496
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1497
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1498
  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1499
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1500
  by (rule lhopital_right_0)
hoelzl@50330
  1501
hoelzl@50330
  1502
lemma lhopital_left:
hoelzl@50330
  1503
  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1504
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1505
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1506
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1507
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1508
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1509
  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1510
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1511
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1512
hoelzl@50330
  1513
lemma lhopital:
hoelzl@50330
  1514
  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1515
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1516
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1517
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1518
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1519
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1520
  ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1521
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1522
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330
  1523
hoelzl@50327
  1524
lemma lhopital_right_0_at_top:
hoelzl@50327
  1525
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1526
  assumes g_0: "LIM x at_right 0. g x :> at_top"
hoelzl@50327
  1527
  assumes ev:
hoelzl@50327
  1528
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1529
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
hoelzl@50327
  1530
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
hoelzl@50327
  1531
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50327
  1532
  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
hoelzl@50327
  1533
  unfolding tendsto_iff
hoelzl@50327
  1534
proof safe
hoelzl@50327
  1535
  fix e :: real assume "0 < e"
hoelzl@50327
  1536
hoelzl@50327
  1537
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
hoelzl@50327
  1538
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
hoelzl@50327
  1539
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327
  1540
  obtain a where [arith]: "0 < a"
hoelzl@50327
  1541
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327
  1542
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327
  1543
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327
  1544
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@51641
  1545
    unfolding eventually_at_le by (auto simp: dist_real_def)
hoelzl@51641
  1546
    
hoelzl@50327
  1547
hoelzl@50327
  1548
  from Df have
hoelzl@50328
  1549
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@51641
  1550
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327
  1551
hoelzl@50327
  1552
  moreover
hoelzl@50328
  1553
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
hoelzl@50346
  1554
    using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
hoelzl@50327
  1555
hoelzl@50327
  1556
  moreover
hoelzl@50327
  1557
  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
hoelzl@50327
  1558
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327
  1559
    by (rule filterlim_compose)
hoelzl@50327
  1560
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327
  1561
    by (intro tendsto_intros)
hoelzl@50327
  1562
  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
hoelzl@50327
  1563
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1564
  from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327
  1565
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
hoelzl@50327
  1566
    by (auto elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50327
  1567
hoelzl@50327
  1568
  moreover
hoelzl@50327
  1569
  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
  1570
    by (intro tendsto_intros)
hoelzl@50327
  1571
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
hoelzl@50327
  1572
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1573
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
hoelzl@50327
  1574
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327
  1575
    by (auto simp: dist_real_def)
hoelzl@50327
  1576
hoelzl@50327
  1577
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327
  1578
  proof eventually_elim
hoelzl@50327
  1579
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327
  1580
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327
  1581
hoelzl@50327
  1582
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327
  1583
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
hoelzl@50327
  1584
    then guess y ..
hoelzl@50327
  1585
    from this
hoelzl@50327
  1586
    have [arith]: "t < y" "y < a" and D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
hoelzl@50327
  1587
      using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327
  1588
hoelzl@50327
  1589
    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
  1590
      by (simp add: field_simps)
hoelzl@50327
  1591
    have "norm (f t / g t - x) \<le>
hoelzl@50327
  1592
        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
  1593
      unfolding * by (rule norm_triangle_ineq)
hoelzl@50327
  1594
    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
  1595
      by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327
  1596
    also have "\<dots> < (e / 4) * 2 + e / 2"
hoelzl@50327
  1597
      using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
hoelzl@50327
  1598
    finally show "dist (f t / g t) x < e"
hoelzl@50327
  1599
      by (simp add: dist_real_def)
hoelzl@50327
  1600
  qed
hoelzl@50327
  1601
qed
hoelzl@50327
  1602
hoelzl@50330
  1603
lemma lhopital_right_at_top:
hoelzl@50330
  1604
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1605
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1606
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1607
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1608
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1609
    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1610
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1611
  by (rule lhopital_right_0_at_top)
hoelzl@50330
  1612
hoelzl@50330
  1613
lemma lhopital_left_at_top:
hoelzl@50330
  1614
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1615
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1616
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1617
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1618
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1619
    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1620
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1621
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1622
hoelzl@50330
  1623
lemma lhopital_at_top:
hoelzl@50330
  1624
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1625
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1626
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1627
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1628
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1629
    ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1630
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1631
  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
  1632
hoelzl@50347
  1633
lemma lhospital_at_top_at_top:
hoelzl@50347
  1634
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50347
  1635
  assumes g_0: "LIM x at_top. g x :> at_top"
hoelzl@50347
  1636
  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
hoelzl@50347
  1637
  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
hoelzl@50347
  1638
  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
hoelzl@50347
  1639
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
hoelzl@50347
  1640
  shows "((\<lambda> x. f x / g x) ---> x) at_top"
hoelzl@50347
  1641
  unfolding filterlim_at_top_to_right
hoelzl@50347
  1642
proof (rule lhopital_right_0_at_top)
hoelzl@50347
  1643
  let ?F = "\<lambda>x. f (inverse x)"
hoelzl@50347
  1644
  let ?G = "\<lambda>x. g (inverse x)"
hoelzl@50347
  1645
  let ?R = "at_right (0::real)"
hoelzl@50347
  1646
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
hoelzl@50347
  1647
hoelzl@50347
  1648
  show "LIM x ?R. ?G x :> at_top"
hoelzl@50347
  1649
    using g_0 unfolding filterlim_at_top_to_right .
hoelzl@50347
  1650
hoelzl@50347
  1651
  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
hoelzl@50347
  1652
    unfolding eventually_at_right_to_top
hoelzl@50347
  1653
    using Dg eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1654
    apply eventually_elim
hoelzl@50347
  1655
    apply (rule DERIV_cong)
hoelzl@50347
  1656
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1657
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1658
    done
hoelzl@50347
  1659
hoelzl@50347
  1660
  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
hoelzl@50347
  1661
    unfolding eventually_at_right_to_top
hoelzl@50347
  1662
    using Df eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1663
    apply eventually_elim
hoelzl@50347
  1664
    apply (rule DERIV_cong)
hoelzl@50347
  1665
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1666
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1667
    done
hoelzl@50347
  1668
hoelzl@50347
  1669
  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
hoelzl@50347
  1670
    unfolding eventually_at_right_to_top
hoelzl@50347
  1671
    using g' eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1672
    by eventually_elim auto
hoelzl@50347
  1673
    
hoelzl@50347
  1674
  show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
hoelzl@50347
  1675
    unfolding filterlim_at_right_to_top
hoelzl@50347
  1676
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
hoelzl@50347
  1677
    using eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1678
    by eventually_elim simp
hoelzl@50347
  1679
qed
hoelzl@50347
  1680
huffman@21164
  1681
end