src/HOL/Deriv.thy
author paulson <lp15@cam.ac.uk>
Mon May 23 15:33:24 2016 +0100 (2016-05-23)
changeset 63114 27afe7af7379
parent 63092 a949b2a5f51d
child 63170 eae6549dbea2
permissions -rw-r--r--
Lots of new material for multivariate analysis
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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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section\<open>Differentiation\<close>
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theory Deriv
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imports Limits
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begin
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subsection \<open>Frechet derivative\<close>
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definition
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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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  (infix "(has'_derivative)" 50)
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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))) \<longlongrightarrow> 0) F)"
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text \<open>
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  Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
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  (at x within s)"} means: @{term D} is the derivative of function @{term f} at point @{term x}
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  within the set @{term s}. Where @{term s} is used to express left or right sided derivatives. In
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  most cases @{term s} is either a variable or @{term UNIV}.
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\<close>
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lemma has_derivative_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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definition 
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  has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
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  (infix "(has'_field'_derivative)" 50)
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where
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  "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
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lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
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  by simp
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definition
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  has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
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  (infix "has'_vector'_derivative" 50)
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where
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  "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
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lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
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  by simp
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named_theorems derivative_intros "structural introduction rules for derivatives"
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setup \<open>
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  let
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    val eq_thms = @{thms has_derivative_eq_rhs DERIV_cong has_vector_derivative_eq_rhs}
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    fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms
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  in
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    Global_Theory.add_thms_dynamic
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      (@{binding derivative_eq_intros},
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        fn context =>
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          Named_Theorems.get (Context.proof_of context) @{named_theorems derivative_intros}
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          |> map_filter eq_rule)
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  end;
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\<close>
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text \<open>
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  The following syntax is only used as a legacy syntax.
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\<close>
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abbreviation (input)
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  FDERIV :: "('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 :> f' \<equiv> (f has_derivative f') (at x)"
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lemma has_derivative_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 has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
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  using bounded_linear.linear[OF has_derivative_bounded_linear] .
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lemma has_derivative_ident[derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
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  by (simp add: has_derivative_def)
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lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
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  by (simp add: has_derivative_def)
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lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
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lemma (in bounded_linear) has_derivative:
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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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  unfolding has_derivative_def
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  apply safe
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  apply (erule bounded_linear_compose [OF bounded_linear])
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  apply (drule tendsto)
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  apply (simp add: scaleR diff add zero)
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  done
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lemmas has_derivative_scaleR_right [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
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lemmas has_derivative_scaleR_left [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
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lemmas has_derivative_mult_right [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_mult_right]
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lemmas has_derivative_mult_left [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_mult_left]
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lemma has_derivative_add[simp, derivative_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) \<longlongrightarrow> (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) \<longlongrightarrow> 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 has_derivative_bounded_linear)
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lemma has_derivative_setsum[simp, derivative_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 has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
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  using has_derivative_scaleR_right[of f f' F "-1"] by simp
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lemma has_derivative_diff[simp, derivative_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_conv_add_uminus has_derivative_add has_derivative_minus)
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lemma has_derivative_at_within:
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  "(f has_derivative f') (at x within s) \<longleftrightarrow>
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    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 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 has_derivative_iff_norm:
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  "(f has_derivative f') (at x within s) \<longleftrightarrow>
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    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 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: has_derivative_at_within divide_inverse ac_simps)
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lemma has_derivative_at:
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  "(f has_derivative D) (at x) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0\<rightarrow> 0)"
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  unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
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lemma field_has_derivative_at:
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  fixes x :: "'a::real_normed_field"
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  shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
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  apply (unfold has_derivative_at)
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  apply (simp add: bounded_linear_mult_right)
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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 has_derivativeI:
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  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s) \<Longrightarrow>
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  (f has_derivative f') (at x within s)"
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  by (simp add: has_derivative_at_within)
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lemma has_derivativeI_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 \<longlongrightarrow> 0) (at x within s)"
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  shows "(f has_derivative f') (at x within s)"
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  unfolding has_derivative_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)) \<longlongrightarrow> 0) (at x within s)"
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  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
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    show "(H \<longlongrightarrow> 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)
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qed fact
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lemma has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
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  by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
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lemmas has_derivative_within_subset = has_derivative_subset 
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subsection \<open>Continuity\<close>
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lemma has_derivative_continuous:
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  assumes f: "(f has_derivative f') (at x within s)"
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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 has_derivative_bounded_linear)
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  note F.tendsto[tendsto_intros]
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  let ?L = "\<lambda>f. (f \<longlongrightarrow> 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 has_derivative_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 \<open>Composition\<close>
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lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> 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_mono)
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lemma has_derivative_in_compose:
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  assumes f: "(f has_derivative f') (at x within s)"
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  assumes g: "(g has_derivative g') (at (f x) within (f`s))"
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  shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
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proof -
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  from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
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  from g interpret G: bounded_linear g' by (rule has_derivative_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 \<longlongrightarrow> 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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  define Nf where "Nf = ?N f f' x"
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  define Ng where [abs_def]: "Ng y = ?N g g' (f x) (f y)" for y
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  show ?thesis
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  proof (rule has_derivativeI_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 has_derivative_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)
hoelzl@51642
   257
      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
hoelzl@51642
   258
        using kF by (intro add_mono) simp
hoelzl@51642
   259
      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
hoelzl@51642
   260
        by (simp add: neq Nf_def field_simps)
hoelzl@51642
   261
    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
hoelzl@51642
   262
    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
hoelzl@51642
   263
  next
hoelzl@51642
   264
    have [tendsto_intros]: "?L Nf"
hoelzl@56181
   265
      using f unfolding has_derivative_iff_norm Nf_def ..
wenzelm@61973
   266
    from f have "(f \<longlongrightarrow> f x) (at x within s)"
hoelzl@56181
   267
      by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
hoelzl@51642
   268
    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
hoelzl@51642
   269
      unfolding filterlim_def
hoelzl@51642
   270
      by (simp add: eventually_filtermap eventually_at_filter le_principal)
hoelzl@51642
   271
wenzelm@61973
   272
    have "((?N g  g' (f x)) \<longlongrightarrow> 0) (at (f x) within f`s)"
hoelzl@56181
   273
      using g unfolding has_derivative_iff_norm ..
wenzelm@61973
   274
    then have g': "((?N g  g' (f x)) \<longlongrightarrow> 0) (inf (nhds (f x)) (principal (f`s)))"
hoelzl@51642
   275
      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
hoelzl@51642
   276
hoelzl@51642
   277
    have [tendsto_intros]: "?L Ng"
hoelzl@51642
   278
      unfolding Ng_def by (rule filterlim_compose[OF g' f'])
wenzelm@61973
   279
    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) \<longlongrightarrow> 0) (at x within s)"
hoelzl@51642
   280
      by (intro tendsto_eq_intros) auto
hoelzl@51642
   281
  qed simp
hoelzl@51642
   282
qed
hoelzl@51642
   283
hoelzl@56181
   284
lemma has_derivative_compose:
hoelzl@56181
   285
  "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
hoelzl@56181
   286
  ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@56181
   287
  by (blast intro: has_derivative_in_compose has_derivative_subset)
hoelzl@51642
   288
hoelzl@51642
   289
lemma (in bounded_bilinear) FDERIV:
hoelzl@56181
   290
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
hoelzl@56181
   291
  shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
hoelzl@51642
   292
proof -
hoelzl@56181
   293
  from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
hoelzl@51642
   294
  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
hoelzl@51642
   295
hoelzl@51642
   296
  from pos_bounded obtain K where K: "0 < K" and norm_prod:
hoelzl@51642
   297
    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
hoelzl@51642
   298
  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
hoelzl@51642
   299
  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
wenzelm@63040
   300
  define Ng where "Ng = ?N g g'"
wenzelm@63040
   301
  define Nf where "Nf = ?N f f'"
hoelzl@51642
   302
hoelzl@51642
   303
  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)"
hoelzl@51642
   304
  let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
hoelzl@51642
   305
  let ?F = "at x within s"
huffman@21164
   306
hoelzl@51642
   307
  show ?thesis
hoelzl@56181
   308
  proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642
   309
    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
hoelzl@51642
   310
      by (intro bounded_linear_add
hoelzl@51642
   311
        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
hoelzl@56181
   312
        has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
hoelzl@51642
   313
  next
wenzelm@61973
   314
    from g have "(g \<longlongrightarrow> g x) ?F"
hoelzl@56181
   315
      by (intro continuous_within[THEN iffD1] has_derivative_continuous)
wenzelm@61973
   316
    moreover from f g have "(Nf \<longlongrightarrow> 0) ?F" "(Ng \<longlongrightarrow> 0) ?F"
hoelzl@56181
   317
      by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
wenzelm@61973
   318
    ultimately have "(?fun2 \<longlongrightarrow> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
hoelzl@51642
   319
      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
wenzelm@61973
   320
    then show "(?fun2 \<longlongrightarrow> 0) ?F"
hoelzl@51642
   321
      by simp
hoelzl@51642
   322
  next
hoelzl@51642
   323
    fix y::'d assume "y \<noteq> x"
hoelzl@51642
   324
    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
   325
      by (simp add: diff_left diff_right add_left add_right field_simps)
hoelzl@51642
   326
    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
hoelzl@51642
   327
        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
hoelzl@51642
   328
      by (intro divide_right_mono mult_mono'
hoelzl@51642
   329
                order_trans [OF norm_triangle_ineq add_mono]
hoelzl@51642
   330
                order_trans [OF norm_prod mult_right_mono]
hoelzl@51642
   331
                mult_nonneg_nonneg order_refl norm_ge_zero norm_F
hoelzl@51642
   332
                K [THEN order_less_imp_le])
hoelzl@51642
   333
    also have "\<dots> = ?fun2 y"
hoelzl@51642
   334
      by (simp add: add_divide_distrib Ng_def Nf_def)
hoelzl@51642
   335
    finally show "?fun1 y \<le> ?fun2 y" .
hoelzl@51642
   336
  qed simp
hoelzl@51642
   337
qed
hoelzl@51642
   338
hoelzl@56381
   339
lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
hoelzl@56381
   340
lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
hoelzl@51642
   341
hoelzl@56381
   342
lemma has_derivative_setprod[simp, derivative_intros]:
hoelzl@51642
   343
  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   344
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
hoelzl@56181
   345
  shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within s)"
hoelzl@51642
   346
proof cases
hoelzl@51642
   347
  assume "finite I" from this f show ?thesis
hoelzl@51642
   348
  proof induct
hoelzl@51642
   349
    case (insert i I)
hoelzl@51642
   350
    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@56181
   351
    have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
hoelzl@56181
   352
      using insert by (intro has_derivative_mult) auto
hoelzl@51642
   353
    also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
haftmann@57418
   354
      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum.cong)
hoelzl@51642
   355
    finally show ?case
hoelzl@51642
   356
      using insert by simp
hoelzl@51642
   357
  qed simp  
hoelzl@51642
   358
qed simp
hoelzl@51642
   359
hoelzl@56381
   360
lemma has_derivative_power[simp, derivative_intros]:
hoelzl@51642
   361
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   362
  assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   363
  shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within s)"
hoelzl@56181
   364
  using has_derivative_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
hoelzl@51642
   365
hoelzl@56181
   366
lemma has_derivative_inverse':
hoelzl@51642
   367
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51642
   368
  assumes x: "x \<noteq> 0"
hoelzl@56181
   369
  shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
hoelzl@56181
   370
        (is "(?inv has_derivative ?f) _")
hoelzl@56181
   371
proof (rule has_derivativeI_sandwich)
hoelzl@51642
   372
  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
hoelzl@51642
   373
    apply (rule bounded_linear_minus)
hoelzl@51642
   374
    apply (rule bounded_linear_mult_const)
hoelzl@51642
   375
    apply (rule bounded_linear_const_mult)
hoelzl@51642
   376
    apply (rule bounded_linear_ident)
hoelzl@51642
   377
    done
hoelzl@51642
   378
next
hoelzl@51642
   379
  show "0 < norm x" using x by simp
hoelzl@51642
   380
next
wenzelm@61973
   381
  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) \<longlongrightarrow> 0) (at x within s)"
hoelzl@51642
   382
    apply (rule tendsto_mult_left_zero)
hoelzl@51642
   383
    apply (rule tendsto_norm_zero)
hoelzl@51642
   384
    apply (rule LIM_zero)
hoelzl@51642
   385
    apply (rule tendsto_inverse)
hoelzl@51642
   386
    apply (rule tendsto_ident_at)
hoelzl@51642
   387
    apply (rule x)
hoelzl@51642
   388
    done
hoelzl@51642
   389
next
hoelzl@51642
   390
  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
lp15@62397
   391
  then have "y \<noteq> 0" by auto
hoelzl@51642
   392
  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
wenzelm@60758
   393
    apply (subst inverse_diff_inverse [OF \<open>y \<noteq> 0\<close> x])
hoelzl@51642
   394
    apply (subst minus_diff_minus)
hoelzl@51642
   395
    apply (subst norm_minus_cancel)
hoelzl@51642
   396
    apply (simp add: left_diff_distrib)
hoelzl@51642
   397
    done
hoelzl@51642
   398
  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
hoelzl@51642
   399
    apply (rule divide_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   400
    apply (rule order_trans [OF norm_mult_ineq])
hoelzl@51642
   401
    apply (rule mult_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   402
    apply (rule norm_mult_ineq)
hoelzl@51642
   403
    done
hoelzl@51642
   404
  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
hoelzl@51642
   405
    by simp
hoelzl@51642
   406
  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
hoelzl@51642
   407
      norm (?inv y - ?inv x) * norm (?inv x)" .
hoelzl@51642
   408
qed
hoelzl@51642
   409
hoelzl@56381
   410
lemma has_derivative_inverse[simp, derivative_intros]:
hoelzl@51642
   411
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181
   412
  assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   413
  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
hoelzl@56181
   414
  using has_derivative_compose[OF f has_derivative_inverse', OF x] .
hoelzl@51642
   415
hoelzl@56381
   416
lemma has_derivative_divide[simp, derivative_intros]:
hoelzl@51642
   417
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181
   418
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 
lp15@55967
   419
  assumes x: "g x \<noteq> 0"
hoelzl@56181
   420
  shows "((\<lambda>x. f x / g x) has_derivative
hoelzl@56181
   421
                (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
hoelzl@56181
   422
  using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
hoelzl@56480
   423
  by (simp add: field_simps)
lp15@55967
   424
wenzelm@60758
   425
text\<open>Conventional form requires mult-AC laws. Types real and complex only.\<close>
hoelzl@56181
   426
hoelzl@56381
   427
lemma has_derivative_divide'[derivative_intros]: 
lp15@55967
   428
  fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
hoelzl@56181
   429
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"
hoelzl@56181
   430
  shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within s)"
lp15@55967
   431
proof -
lp15@55967
   432
  { fix h
lp15@55967
   433
    have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
lp15@55967
   434
          (f' h * g x - f x * g' h) / (g x * g x)"
hoelzl@56480
   435
      by (simp add: field_simps x)
lp15@55967
   436
   }
lp15@55967
   437
  then show ?thesis
hoelzl@56181
   438
    using has_derivative_divide [OF f g] x
lp15@55967
   439
    by simp
lp15@55967
   440
qed
hoelzl@51642
   441
wenzelm@60758
   442
subsection \<open>Uniqueness\<close>
hoelzl@51642
   443
wenzelm@60758
   444
text \<open>
hoelzl@51642
   445
hoelzl@56181
   446
This can not generally shown for @{const has_derivative}, as we need to approach the point from
wenzelm@61799
   447
all directions. There is a proof in \<open>Multivariate_Analysis\<close> for \<open>euclidean_space\<close>.
hoelzl@51642
   448
wenzelm@60758
   449
\<close>
hoelzl@51642
   450
hoelzl@56181
   451
lemma has_derivative_zero_unique:
hoelzl@56181
   452
  assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
hoelzl@51642
   453
proof -
hoelzl@51642
   454
  interpret F: bounded_linear F
hoelzl@56181
   455
    using assms by (rule has_derivative_bounded_linear)
hoelzl@51642
   456
  let ?r = "\<lambda>h. norm (F h) / norm h"
wenzelm@61976
   457
  have *: "?r \<midarrow>0\<rightarrow> 0"
hoelzl@56181
   458
    using assms unfolding has_derivative_at by simp
hoelzl@51642
   459
  show "F = (\<lambda>h. 0)"
hoelzl@51642
   460
  proof
hoelzl@51642
   461
    fix h show "F h = 0"
hoelzl@51642
   462
    proof (rule ccontr)
wenzelm@53374
   463
      assume **: "F h \<noteq> 0"
nipkow@56541
   464
      hence h: "h \<noteq> 0" by (clarsimp simp add: F.zero)
nipkow@56541
   465
      with ** have "0 < ?r h" by simp
hoelzl@51642
   466
      from LIM_D [OF * this] obtain s where s: "0 < s"
hoelzl@51642
   467
        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
hoelzl@51642
   468
      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
hoelzl@51642
   469
      let ?x = "scaleR (t / norm h) h"
hoelzl@51642
   470
      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
hoelzl@51642
   471
      hence "?r ?x < ?r h" by (rule r)
hoelzl@51642
   472
      thus "False" using t h by (simp add: F.scaleR)
hoelzl@51642
   473
    qed
hoelzl@51642
   474
  qed
hoelzl@51642
   475
qed
hoelzl@51642
   476
hoelzl@56181
   477
lemma has_derivative_unique:
hoelzl@56181
   478
  assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
hoelzl@51642
   479
proof -
hoelzl@56181
   480
  have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
hoelzl@56181
   481
    using has_derivative_diff [OF assms] by simp
hoelzl@51642
   482
  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
hoelzl@56181
   483
    by (rule has_derivative_zero_unique)
hoelzl@51642
   484
  thus "F = F'"
hoelzl@51642
   485
    unfolding fun_eq_iff right_minus_eq .
hoelzl@51642
   486
qed
hoelzl@51642
   487
wenzelm@60758
   488
subsection \<open>Differentiability predicate\<close>
hoelzl@51642
   489
hoelzl@56181
   490
definition
hoelzl@56181
   491
  differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@56182
   492
  (infix "differentiable" 50)
hoelzl@56181
   493
where
hoelzl@56181
   494
  "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
hoelzl@51642
   495
hoelzl@56181
   496
lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
hoelzl@56181
   497
  unfolding differentiable_def by (blast intro: has_derivative_subset)
hoelzl@51642
   498
lp15@56261
   499
lemmas differentiable_within_subset = differentiable_subset
lp15@56261
   500
hoelzl@56381
   501
lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
hoelzl@56181
   502
  unfolding differentiable_def by (blast intro: has_derivative_ident)
hoelzl@51642
   503
hoelzl@56381
   504
lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
hoelzl@56181
   505
  unfolding differentiable_def by (blast intro: has_derivative_const)
hoelzl@51642
   506
hoelzl@51642
   507
lemma differentiable_in_compose:
hoelzl@56181
   508
  "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@56181
   509
  unfolding differentiable_def by (blast intro: has_derivative_in_compose)
hoelzl@51642
   510
hoelzl@51642
   511
lemma differentiable_compose:
hoelzl@56181
   512
  "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@51642
   513
  by (blast intro: differentiable_in_compose differentiable_subset)
hoelzl@51642
   514
hoelzl@56381
   515
lemma differentiable_sum [simp, derivative_intros]:
hoelzl@56181
   516
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
hoelzl@56181
   517
  unfolding differentiable_def by (blast intro: has_derivative_add)
hoelzl@51642
   518
hoelzl@56381
   519
lemma differentiable_minus [simp, derivative_intros]:
hoelzl@56181
   520
  "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
hoelzl@56181
   521
  unfolding differentiable_def by (blast intro: has_derivative_minus)
hoelzl@51642
   522
hoelzl@56381
   523
lemma differentiable_diff [simp, derivative_intros]:
hoelzl@56181
   524
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
hoelzl@56181
   525
  unfolding differentiable_def by (blast intro: has_derivative_diff)
hoelzl@51642
   526
hoelzl@56381
   527
lemma differentiable_mult [simp, derivative_intros]:
hoelzl@51642
   528
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
hoelzl@56181
   529
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x * g x) differentiable (at x within s)"
hoelzl@56181
   530
  unfolding differentiable_def by (blast intro: has_derivative_mult)
hoelzl@51642
   531
hoelzl@56381
   532
lemma differentiable_inverse [simp, derivative_intros]:
hoelzl@51642
   533
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   534
  shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable (at x within s)"
hoelzl@56181
   535
  unfolding differentiable_def by (blast intro: has_derivative_inverse)
hoelzl@51642
   536
hoelzl@56381
   537
lemma differentiable_divide [simp, derivative_intros]:
hoelzl@51642
   538
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   539
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
wenzelm@63092
   540
  unfolding divide_inverse by simp
hoelzl@51642
   541
hoelzl@56381
   542
lemma differentiable_power [simp, derivative_intros]:
hoelzl@51642
   543
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   544
  shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
hoelzl@56181
   545
  unfolding differentiable_def by (blast intro: has_derivative_power)
hoelzl@51642
   546
hoelzl@56381
   547
lemma differentiable_scaleR [simp, derivative_intros]:
hoelzl@56181
   548
  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
hoelzl@56181
   549
  unfolding differentiable_def by (blast intro: has_derivative_scaleR)
hoelzl@51642
   550
hoelzl@56181
   551
lemma has_derivative_imp_has_field_derivative:
hoelzl@56181
   552
  "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
hoelzl@56181
   553
  unfolding has_field_derivative_def 
haftmann@57512
   554
  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
hoelzl@56181
   555
hoelzl@56181
   556
lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
hoelzl@56181
   557
  by (simp add: has_field_derivative_def)
hoelzl@51642
   558
lp15@56261
   559
lemma DERIV_subset: 
lp15@56261
   560
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
lp15@56261
   561
   \<Longrightarrow> (f has_field_derivative f') (at x within t)"
lp15@56261
   562
  by (simp add: has_field_derivative_def has_derivative_within_subset)
lp15@56261
   563
lp15@59862
   564
lemma has_field_derivative_at_within:
lp15@59862
   565
    "(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
lp15@59862
   566
  using DERIV_subset by blast
lp15@59862
   567
hoelzl@56181
   568
abbreviation (input)
hoelzl@56381
   569
  DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56181
   570
  ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@51642
   571
where
hoelzl@56181
   572
  "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
hoelzl@51642
   573
hoelzl@56181
   574
abbreviation 
hoelzl@56181
   575
  has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
hoelzl@56182
   576
  (infix "(has'_real'_derivative)" 50)
hoelzl@56181
   577
where
hoelzl@56181
   578
  "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
hoelzl@56181
   579
hoelzl@56181
   580
lemma real_differentiable_def:
hoelzl@56181
   581
  "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
hoelzl@51642
   582
proof safe
hoelzl@56181
   583
  assume "f differentiable at x within s"
hoelzl@56181
   584
  then obtain f' where *: "(f has_derivative f') (at x within s)"
hoelzl@56181
   585
    unfolding differentiable_def by auto
hoelzl@56181
   586
  then obtain c where "f' = (op * c)"
haftmann@57512
   587
    by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
hoelzl@56181
   588
  with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   589
    unfolding has_field_derivative_def by auto
hoelzl@56181
   590
qed (auto simp: differentiable_def has_field_derivative_def)
hoelzl@51642
   591
hoelzl@56181
   592
lemma real_differentiableE [elim?]:
hoelzl@56181
   593
  assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
hoelzl@56181
   594
  using assms by (auto simp: real_differentiable_def)
hoelzl@51642
   595
hoelzl@56181
   596
lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   597
  by (auto elim: real_differentiableE)
hoelzl@51642
   598
hoelzl@56181
   599
lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
hoelzl@56181
   600
  by (force simp add: real_differentiable_def)
hoelzl@51642
   601
immler@63079
   602
lemma has_field_derivative_iff:
immler@63079
   603
  "(f has_field_derivative D) (at x within S) \<longleftrightarrow>
immler@63079
   604
    ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
immler@63079
   605
  apply (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right
immler@63079
   606
    LIM_zero_iff[symmetric, of _ D])
hoelzl@51642
   607
  apply (subst (2) tendsto_norm_zero_iff[symmetric])
hoelzl@51642
   608
  apply (rule filterlim_cong)
hoelzl@51642
   609
  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
hoelzl@51642
   610
  done
huffman@21164
   611
immler@63079
   612
lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
immler@63079
   613
  unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..
immler@63079
   614
hoelzl@56181
   615
lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
haftmann@57512
   616
  by (simp add: fun_eq_iff mult.commute)
huffman@21164
   617
wenzelm@60758
   618
subsection \<open>Vector derivative\<close>
immler@60177
   619
immler@60177
   620
lemma has_field_derivative_iff_has_vector_derivative:
immler@60177
   621
  "(f has_field_derivative y) F \<longleftrightarrow> (f has_vector_derivative y) F"
immler@60177
   622
  unfolding has_vector_derivative_def has_field_derivative_def real_scaleR_def mult_commute_abs ..
immler@60177
   623
immler@60177
   624
lemma has_field_derivative_subset:
immler@60177
   625
  "(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_field_derivative y) (at x within t)"
immler@60177
   626
  unfolding has_field_derivative_def by (rule has_derivative_subset)
immler@60177
   627
immler@60177
   628
lemma has_vector_derivative_const[simp, derivative_intros]: "((\<lambda>x. c) has_vector_derivative 0) net"
immler@60177
   629
  by (auto simp: has_vector_derivative_def)
immler@60177
   630
immler@60177
   631
lemma has_vector_derivative_id[simp, derivative_intros]: "((\<lambda>x. x) has_vector_derivative 1) net"
immler@60177
   632
  by (auto simp: has_vector_derivative_def)
immler@60177
   633
immler@60177
   634
lemma has_vector_derivative_minus[derivative_intros]:
immler@60177
   635
  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. - f x) has_vector_derivative (- f')) net"
immler@60177
   636
  by (auto simp: has_vector_derivative_def)
immler@60177
   637
immler@60177
   638
lemma has_vector_derivative_add[derivative_intros]:
immler@60177
   639
  "(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
immler@60177
   640
    ((\<lambda>x. f x + g x) has_vector_derivative (f' + g')) net"
immler@60177
   641
  by (auto simp: has_vector_derivative_def scaleR_right_distrib)
immler@60177
   642
immler@60177
   643
lemma has_vector_derivative_setsum[derivative_intros]:
immler@60177
   644
  "(\<And>i. i \<in> I \<Longrightarrow> (f i has_vector_derivative f' i) net) \<Longrightarrow>
immler@60177
   645
    ((\<lambda>x. \<Sum>i\<in>I. f i x) has_vector_derivative (\<Sum>i\<in>I. f' i)) net"
immler@60177
   646
  by (auto simp: has_vector_derivative_def fun_eq_iff scaleR_setsum_right intro!: derivative_eq_intros)
immler@60177
   647
immler@60177
   648
lemma has_vector_derivative_diff[derivative_intros]:
immler@60177
   649
  "(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
immler@60177
   650
    ((\<lambda>x. f x - g x) has_vector_derivative (f' - g')) net"
immler@60177
   651
  by (auto simp: has_vector_derivative_def scaleR_diff_right)
immler@60177
   652
paulson@61204
   653
lemma has_vector_derivative_add_const:
immler@63079
   654
     "((\<lambda>t. g t + z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
paulson@61204
   655
apply (intro iffI)
paulson@61204
   656
apply (drule has_vector_derivative_diff [where g = "\<lambda>t. z", OF _ has_vector_derivative_const], simp)
paulson@61204
   657
apply (drule has_vector_derivative_add [OF _ has_vector_derivative_const], simp)
paulson@61204
   658
done
paulson@61204
   659
paulson@61204
   660
lemma has_vector_derivative_diff_const:
immler@63079
   661
     "((\<lambda>t. g t - z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
paulson@61204
   662
using has_vector_derivative_add_const [where z = "-z"]
paulson@61204
   663
by simp
paulson@61204
   664
immler@60177
   665
lemma (in bounded_linear) has_vector_derivative:
immler@60177
   666
  assumes "(g has_vector_derivative g') F"
immler@60177
   667
  shows "((\<lambda>x. f (g x)) has_vector_derivative f g') F"
immler@60177
   668
  using has_derivative[OF assms[unfolded has_vector_derivative_def]]
immler@60177
   669
  by (simp add: has_vector_derivative_def scaleR)
immler@60177
   670
immler@60177
   671
lemma (in bounded_bilinear) has_vector_derivative:
immler@60177
   672
  assumes "(f has_vector_derivative f') (at x within s)"
immler@60177
   673
    and "(g has_vector_derivative g') (at x within s)"
immler@60177
   674
  shows "((\<lambda>x. f x ** g x) has_vector_derivative (f x ** g' + f' ** g x)) (at x within s)"
immler@60177
   675
  using FDERIV[OF assms(1-2)[unfolded has_vector_derivative_def]]
immler@60177
   676
  by (simp add: has_vector_derivative_def scaleR_right scaleR_left scaleR_right_distrib)
immler@60177
   677
immler@60177
   678
lemma has_vector_derivative_scaleR[derivative_intros]:
immler@60177
   679
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
immler@60177
   680
    ((\<lambda>x. f x *\<^sub>R g x) has_vector_derivative (f x *\<^sub>R g' + f' *\<^sub>R g x)) (at x within s)"
immler@60177
   681
  unfolding has_field_derivative_iff_has_vector_derivative
immler@60177
   682
  by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])
immler@60177
   683
immler@60177
   684
lemma has_vector_derivative_mult[derivative_intros]:
immler@60177
   685
  "(f has_vector_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
immler@60177
   686
    ((\<lambda>x. f x * g x) has_vector_derivative (f x * g' + f' * g x :: 'a :: real_normed_algebra)) (at x within s)"
immler@60177
   687
  by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_mult])
immler@60177
   688
immler@60177
   689
lemma has_vector_derivative_of_real[derivative_intros]:
immler@60177
   690
  "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_vector_derivative (of_real D)) F"
immler@60177
   691
  by (rule bounded_linear.has_vector_derivative[OF bounded_linear_of_real])
immler@60177
   692
     (simp add: has_field_derivative_iff_has_vector_derivative)
immler@60177
   693
immler@60177
   694
lemma has_vector_derivative_continuous: "(f has_vector_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
immler@60177
   695
  by (auto intro: has_derivative_continuous simp: has_vector_derivative_def)
immler@60177
   696
immler@60177
   697
lemma has_vector_derivative_mult_right[derivative_intros]:
immler@60177
   698
  fixes a :: "'a :: real_normed_algebra"
immler@60177
   699
  shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. a * f x) has_vector_derivative (a * x)) F"
immler@60177
   700
  by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_right])
immler@60177
   701
immler@60177
   702
lemma has_vector_derivative_mult_left[derivative_intros]:
immler@60177
   703
  fixes a :: "'a :: real_normed_algebra"
immler@60177
   704
  shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. f x * a) has_vector_derivative (x * a)) F"
immler@60177
   705
  by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left])
immler@60177
   706
immler@60177
   707
wenzelm@60758
   708
subsection \<open>Derivatives\<close>
huffman@21164
   709
wenzelm@61976
   710
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
hoelzl@56381
   711
  by (simp add: DERIV_def)
huffman@21164
   712
immler@63079
   713
lemma has_field_derivativeD:
immler@63079
   714
  "(f has_field_derivative D) (at x within S) \<Longrightarrow>
immler@63079
   715
    ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
immler@63079
   716
  by (simp add: has_field_derivative_iff)
immler@63079
   717
hoelzl@56381
   718
lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
hoelzl@56181
   719
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
hoelzl@56181
   720
hoelzl@56381
   721
lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
hoelzl@56181
   722
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
huffman@21164
   723
hoelzl@56381
   724
lemma field_differentiable_add[derivative_intros]:
hoelzl@56381
   725
  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> 
hoelzl@56381
   726
    ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
hoelzl@56381
   727
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
hoelzl@56381
   728
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
lp15@56261
   729
lp15@56261
   730
corollary DERIV_add:
hoelzl@56181
   731
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   732
  ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
lp15@56261
   733
  by (rule field_differentiable_add)
lp15@56261
   734
hoelzl@56381
   735
lemma field_differentiable_minus[derivative_intros]:
hoelzl@56381
   736
  "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
hoelzl@56381
   737
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
hoelzl@56381
   738
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
huffman@21164
   739
lp15@56261
   740
corollary DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
lp15@56261
   741
  by (rule field_differentiable_minus)
huffman@21164
   742
hoelzl@56381
   743
lemma field_differentiable_diff[derivative_intros]:
hoelzl@56381
   744
  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
wenzelm@63092
   745
  by (simp only: diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
lp15@56261
   746
lp15@56261
   747
corollary DERIV_diff:
hoelzl@56181
   748
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   749
  ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
lp15@56261
   750
  by (rule field_differentiable_diff)
hoelzl@51642
   751
hoelzl@56181
   752
lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
hoelzl@56181
   753
  by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
huffman@21164
   754
lp15@56261
   755
corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
lp15@56261
   756
  by (rule DERIV_continuous)
lp15@56261
   757
lp15@56261
   758
lemma DERIV_continuous_on:
eberlm@61552
   759
  "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (D x)) (at x)) \<Longrightarrow> continuous_on s f"
lp15@56261
   760
  by (metis DERIV_continuous continuous_at_imp_continuous_on)
hoelzl@51642
   761
hoelzl@56181
   762
lemma DERIV_mult':
hoelzl@56181
   763
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   764
  ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
hoelzl@56181
   765
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   766
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164
   767
hoelzl@56381
   768
lemma DERIV_mult[derivative_intros]:
hoelzl@56181
   769
  "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   770
  ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
hoelzl@56181
   771
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   772
     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   773
wenzelm@60758
   774
text \<open>Derivative of linear multiplication\<close>
huffman@21164
   775
hoelzl@51642
   776
lemma DERIV_cmult:
hoelzl@56181
   777
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
hoelzl@51642
   778
  by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164
   779
lp15@55967
   780
lemma DERIV_cmult_right:
hoelzl@56181
   781
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
haftmann@57514
   782
  using DERIV_cmult by (force simp add: ac_simps)
lp15@55967
   783
hoelzl@56181
   784
lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
hoelzl@51642
   785
  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
hoelzl@51642
   786
hoelzl@56181
   787
lemma DERIV_cdivide:
hoelzl@56181
   788
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
hoelzl@56181
   789
  using DERIV_cmult_right[of f D x s "1 / c"] by simp
huffman@21164
   790
huffman@21164
   791
lemma DERIV_unique:
hoelzl@51642
   792
  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
hoelzl@56381
   793
  unfolding DERIV_def by (rule LIM_unique) 
huffman@21164
   794
hoelzl@56381
   795
lemma DERIV_setsum[derivative_intros]:
hoelzl@56181
   796
  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 
hoelzl@56181
   797
    ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
hoelzl@56181
   798
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
hoelzl@56181
   799
     (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   800
hoelzl@56381
   801
lemma DERIV_inverse'[derivative_intros]:
haftmann@59867
   802
  assumes "(f has_field_derivative D) (at x within s)"
haftmann@59867
   803
    and "f x \<noteq> 0"
haftmann@59867
   804
  shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
haftmann@59867
   805
proof -
haftmann@59867
   806
  have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative op * D)"
haftmann@59867
   807
    by (rule arg_cong [of "\<lambda>x. x * D"]) (simp add: fun_eq_iff)
haftmann@59867
   808
  with assms have "(f has_derivative (\<lambda>x. x * D)) (at x within s)"
haftmann@59867
   809
    by (auto dest!: has_field_derivative_imp_has_derivative)
wenzelm@60758
   810
  then show ?thesis using \<open>f x \<noteq> 0\<close>
haftmann@59867
   811
    by (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse)
haftmann@59867
   812
qed
hoelzl@51642
   813
wenzelm@61799
   814
text \<open>Power of \<open>-1\<close>\<close>
hoelzl@51642
   815
hoelzl@51642
   816
lemma DERIV_inverse:
hoelzl@56181
   817
  "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642
   818
  by (drule DERIV_inverse' [OF DERIV_ident]) simp
hoelzl@51642
   819
wenzelm@60758
   820
text \<open>Derivative of inverse\<close>
hoelzl@51642
   821
hoelzl@51642
   822
lemma DERIV_inverse_fun:
hoelzl@56181
   823
  "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   824
  ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
haftmann@57514
   825
  by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)
hoelzl@51642
   826
wenzelm@60758
   827
text \<open>Derivative of quotient\<close>
hoelzl@51642
   828
hoelzl@56381
   829
lemma DERIV_divide[derivative_intros]:
hoelzl@56181
   830
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   831
  (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   832
  ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
hoelzl@56181
   833
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
hoelzl@56480
   834
     (auto dest: has_field_derivative_imp_has_derivative simp: field_simps)
hoelzl@51642
   835
hoelzl@51642
   836
lemma DERIV_quotient:
hoelzl@56181
   837
  "(f has_field_derivative d) (at x within s) \<Longrightarrow>
hoelzl@56181
   838
  (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
hoelzl@56181
   839
  ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
haftmann@57512
   840
  by (drule (2) DERIV_divide) (simp add: mult.commute)
hoelzl@51642
   841
hoelzl@51642
   842
lemma DERIV_power_Suc:
hoelzl@56181
   843
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   844
  ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
hoelzl@56181
   845
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   846
     (auto simp: has_field_derivative_def)
hoelzl@51642
   847
hoelzl@56381
   848
lemma DERIV_power[derivative_intros]:
hoelzl@56181
   849
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   850
  ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
hoelzl@56181
   851
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   852
     (auto simp: has_field_derivative_def)
hoelzl@31880
   853
hoelzl@56181
   854
lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
lp15@61609
   855
  using DERIV_power [OF DERIV_ident] by simp
hoelzl@51642
   856
hoelzl@56181
   857
lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
hoelzl@56181
   858
  ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
hoelzl@56181
   859
  using has_derivative_compose[of f "op * D" x s g "op * E"]
hoelzl@56181
   860
  unfolding has_field_derivative_def mult_commute_abs ac_simps .
hoelzl@51642
   861
hoelzl@56181
   862
corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   863
  ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
lp15@55967
   864
  by (rule DERIV_chain')
lp15@55967
   865
wenzelm@60758
   866
text \<open>Standard version\<close>
hoelzl@51642
   867
hoelzl@56181
   868
lemma DERIV_chain:
hoelzl@56181
   869
  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
hoelzl@56181
   870
  (f o g has_field_derivative Da * Db) (at x within s)"
haftmann@57512
   871
  by (drule (1) DERIV_chain', simp add: o_def mult.commute)
hoelzl@51642
   872
lp15@55967
   873
lemma DERIV_image_chain: 
hoelzl@56181
   874
  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   875
  (f o g has_field_derivative Da * Db) (at x within s)"
hoelzl@56181
   876
  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
hoelzl@56181
   877
  by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
lp15@55967
   878
lp15@55967
   879
(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
lp15@55967
   880
lemma DERIV_chain_s:
lp15@55967
   881
  assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
lp15@55967
   882
      and "DERIV f x :> f'" 
lp15@55967
   883
      and "f x \<in> s"
lp15@55967
   884
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
haftmann@57512
   885
  by (metis (full_types) DERIV_chain' mult.commute assms)
lp15@55967
   886
lp15@55967
   887
lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
lp15@55967
   888
  assumes "(\<And>x. DERIV g x :> g'(x))"
lp15@55967
   889
      and "DERIV f x :> f'" 
lp15@55967
   890
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967
   891
  by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
lp15@55967
   892
wenzelm@60758
   893
text\<open>Alternative definition for differentiability\<close>
huffman@21164
   894
huffman@21164
   895
lemma DERIV_LIM_iff:
huffman@31338
   896
  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
wenzelm@61976
   897
     "((%h. (f(a + h) - f(a)) / h) \<midarrow>0\<rightarrow> D) =
wenzelm@61976
   898
      ((%x. (f(x)-f(a)) / (x-a)) \<midarrow>a\<rightarrow> D)"
huffman@21164
   899
apply (rule iffI)
huffman@21164
   900
apply (drule_tac k="- a" in LIM_offset)
haftmann@54230
   901
apply simp
huffman@21164
   902
apply (drule_tac k="a" in LIM_offset)
haftmann@57512
   903
apply (simp add: add.commute)
huffman@21164
   904
done
huffman@21164
   905
immler@63079
   906
lemmas DERIV_iff2 = has_field_derivative_iff
immler@63079
   907
immler@63079
   908
lemma has_field_derivative_cong_ev:
immler@63079
   909
  assumes "x = y"
immler@63079
   910
    and *: "eventually (\<lambda>x. x \<in> s \<longrightarrow> f x = g x) (nhds x)"
immler@63079
   911
    and "u = v" "s = t" "x \<in> s"
immler@63079
   912
    shows "(f has_field_derivative u) (at x within s) = (g has_field_derivative v) (at y within t)"
immler@63079
   913
  unfolding DERIV_iff2
immler@63079
   914
proof (rule filterlim_cong)
immler@63079
   915
  from assms have "f y = g y" by (auto simp: eventually_nhds)
immler@63079
   916
  with * show "\<forall>\<^sub>F xa in at x within s. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)"
immler@63079
   917
    unfolding eventually_at_filter
immler@63079
   918
    by eventually_elim (auto simp: assms \<open>f y = g y\<close>)
immler@63079
   919
qed (simp_all add: assms)
huffman@21164
   920
hoelzl@51642
   921
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@51642
   922
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
immler@63079
   923
  by (rule has_field_derivative_cong_ev) simp_all
huffman@21164
   924
hoelzl@51642
   925
lemma DERIV_shift:
immler@63079
   926
  "(f has_field_derivative y) (at (x + z)) = ((\<lambda>x. f (x + z)) has_field_derivative y) (at x)"
hoelzl@56381
   927
  by (simp add: DERIV_def field_simps)
huffman@21164
   928
hoelzl@51642
   929
lemma DERIV_mirror:
hoelzl@51642
   930
  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
hoelzl@56479
   931
  by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
hoelzl@51642
   932
                tendsto_minus_cancel_left field_simps conj_commute)
huffman@21164
   933
wenzelm@60758
   934
text \<open>Caratheodory formulation of derivative at a point\<close>
huffman@21164
   935
lp15@55970
   936
lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
hoelzl@51642
   937
  "(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
   938
      (is "?lhs = ?rhs")
huffman@21164
   939
proof
huffman@21164
   940
  assume der: "DERIV f x :> l"
huffman@21784
   941
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
huffman@21164
   942
  proof (intro exI conjI)
huffman@21784
   943
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
nipkow@23413
   944
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
huffman@21164
   945
    show "isCont ?g x" using der
hoelzl@56381
   946
      by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
huffman@21164
   947
    show "?g x = l" by simp
huffman@21164
   948
  qed
huffman@21164
   949
next
huffman@21164
   950
  assume "?rhs"
huffman@21164
   951
  then obtain g where
huffman@21784
   952
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
huffman@21164
   953
  thus "(DERIV f x :> l)"
hoelzl@56381
   954
     by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
huffman@21164
   955
qed
huffman@21164
   956
huffman@21164
   957
wenzelm@60758
   958
subsection \<open>Local extrema\<close>
huffman@29975
   959
wenzelm@60758
   960
text\<open>If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right\<close>
huffman@21164
   961
immler@63079
   962
lemma has_real_derivative_pos_inc_right:
immler@63079
   963
  fixes f :: "real => real"
immler@63079
   964
  assumes der: "(f has_real_derivative l) (at x within S)"
immler@63079
   965
      and l:   "0 < l"
immler@63079
   966
  shows "\<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x + h)"
immler@63079
   967
  using assms
immler@63079
   968
proof -
immler@63079
   969
  from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
immler@63079
   970
  obtain s where s:   "0 < s"
immler@63079
   971
    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> abs ((f xa - f x) / (xa - x) - l) < l"
immler@63079
   972
    by (auto simp: dist_real_def)
immler@63079
   973
  then show ?thesis
immler@63079
   974
  proof (intro exI conjI strip)
immler@63079
   975
    show "0<s" using s .
immler@63079
   976
    fix h::real
immler@63079
   977
    assume "0 < h" "h < s" "x + h \<in> S"
immler@63079
   978
    with all [of "x + h"] show "f x < f (x+h)"
immler@63079
   979
    proof (simp add: abs_if dist_real_def pos_less_divide_eq split: if_split_asm)
immler@63079
   980
      assume "\<not> (f (x+h) - f x) / h < l" and h: "0 < h"
immler@63079
   981
      with l
immler@63079
   982
      have "0 < (f (x+h) - f x) / h" by arith
immler@63079
   983
      thus "f x < f (x+h)"
immler@63079
   984
        by (simp add: pos_less_divide_eq h)
immler@63079
   985
    qed
immler@63079
   986
  qed
immler@63079
   987
qed
immler@63079
   988
paulson@33654
   989
lemma DERIV_pos_inc_right:
huffman@21164
   990
  fixes f :: "real => real"
huffman@21164
   991
  assumes der: "DERIV f x :> l"
huffman@21164
   992
      and l:   "0 < l"
huffman@21164
   993
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
immler@63079
   994
  using has_real_derivative_pos_inc_right[OF assms]
immler@63079
   995
  by auto
immler@63079
   996
immler@63079
   997
lemma has_real_derivative_neg_dec_left:
immler@63079
   998
  fixes f :: "real => real"
immler@63079
   999
  assumes der: "(f has_real_derivative l) (at x within S)"
immler@63079
  1000
      and "l < 0"
immler@63079
  1001
  shows "\<exists>d > 0. \<forall>h > 0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x - h)"
huffman@21164
  1002
proof -
immler@63079
  1003
  from \<open>l < 0\<close> have l: "- l > 0" by simp
immler@63079
  1004
  from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
immler@63079
  1005
  obtain s where s:   "0 < s"
immler@63079
  1006
    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> abs ((f xa - f x) / (xa - x) - l) < - l"
immler@63079
  1007
    by (auto simp: dist_real_def)
huffman@21164
  1008
  thus ?thesis
huffman@21164
  1009
  proof (intro exI conjI strip)
huffman@23441
  1010
    show "0<s" using s .
huffman@21164
  1011
    fix h::real
immler@63079
  1012
    assume "0 < h" "h < s" "x - h \<in> S"
immler@63079
  1013
    with all [of "x - h"] show "f x < f (x-h)"
immler@63079
  1014
    proof (simp add: abs_if pos_less_divide_eq dist_real_def split add: if_split_asm)
immler@63079
  1015
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
  1016
      with l
immler@63079
  1017
      have "0 < (f (x-h) - f x) / h" by arith
immler@63079
  1018
      thus "f x < f (x-h)"
immler@63079
  1019
        by (simp add: pos_less_divide_eq h)
huffman@21164
  1020
    qed
huffman@21164
  1021
  qed
huffman@21164
  1022
qed
huffman@21164
  1023
paulson@33654
  1024
lemma DERIV_neg_dec_left:
huffman@21164
  1025
  fixes f :: "real => real"
huffman@21164
  1026
  assumes der: "DERIV f x :> l"
huffman@21164
  1027
      and l:   "l < 0"
huffman@21164
  1028
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
immler@63079
  1029
  using has_real_derivative_neg_dec_left[OF assms]
immler@63079
  1030
  by auto
immler@63079
  1031
immler@63079
  1032
lemma has_real_derivative_pos_inc_left:
immler@63079
  1033
  fixes f :: "real => real"
immler@63079
  1034
  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
immler@63079
  1035
  by (rule has_real_derivative_neg_dec_left [of "%x. - f x" "-l" x S, simplified])
immler@63079
  1036
      (auto simp add: DERIV_minus)
huffman@21164
  1037
paulson@33654
  1038
lemma DERIV_pos_inc_left:
paulson@33654
  1039
  fixes f :: "real => real"
paulson@33654
  1040
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
immler@63079
  1041
  using has_real_derivative_pos_inc_left
immler@63079
  1042
  by blast
immler@63079
  1043
immler@63079
  1044
lemma has_real_derivative_neg_dec_right:
immler@63079
  1045
  fixes f :: "real => real"
immler@63079
  1046
  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f(x) > f(x + h)"
immler@63079
  1047
  by (rule has_real_derivative_pos_inc_right [of "%x. - f x" "-l" x S, simplified])
immler@63079
  1048
      (auto simp add: DERIV_minus)
paulson@33654
  1049
paulson@33654
  1050
lemma DERIV_neg_dec_right:
paulson@33654
  1051
  fixes f :: "real => real"
paulson@33654
  1052
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
immler@63079
  1053
  using has_real_derivative_neg_dec_right by blast
paulson@33654
  1054
huffman@21164
  1055
lemma DERIV_local_max:
huffman@21164
  1056
  fixes f :: "real => real"
huffman@21164
  1057
  assumes der: "DERIV f x :> l"
huffman@21164
  1058
      and d:   "0 < d"
huffman@21164
  1059
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
  1060
  shows "l = 0"
huffman@21164
  1061
proof (cases rule: linorder_cases [of l 0])
huffman@23441
  1062
  case equal thus ?thesis .
huffman@21164
  1063
next
huffman@21164
  1064
  case less
paulson@33654
  1065
  from DERIV_neg_dec_left [OF der less]
huffman@21164
  1066
  obtain d' where d': "0 < d'"
huffman@21164
  1067
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
  1068
  from real_lbound_gt_zero [OF d d']
huffman@21164
  1069
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
  1070
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
  1071
  show ?thesis by (auto simp add: abs_if)
huffman@21164
  1072
next
huffman@21164
  1073
  case greater
paulson@33654
  1074
  from DERIV_pos_inc_right [OF der greater]
huffman@21164
  1075
  obtain d' where d': "0 < d'"
huffman@21164
  1076
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
  1077
  from real_lbound_gt_zero [OF d d']
huffman@21164
  1078
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
  1079
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
  1080
  show ?thesis by (auto simp add: abs_if)
huffman@21164
  1081
qed
huffman@21164
  1082
huffman@21164
  1083
wenzelm@60758
  1084
text\<open>Similar theorem for a local minimum\<close>
huffman@21164
  1085
lemma DERIV_local_min:
huffman@21164
  1086
  fixes f :: "real => real"
huffman@21164
  1087
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
  1088
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
  1089
huffman@21164
  1090
wenzelm@60758
  1091
text\<open>In particular, if a function is locally flat\<close>
huffman@21164
  1092
lemma DERIV_local_const:
huffman@21164
  1093
  fixes f :: "real => real"
huffman@21164
  1094
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
  1095
by (auto dest!: DERIV_local_max)
huffman@21164
  1096
huffman@29975
  1097
wenzelm@60758
  1098
subsection \<open>Rolle's Theorem\<close>
huffman@29975
  1099
wenzelm@60758
  1100
text\<open>Lemma about introducing open ball in open interval\<close>
huffman@21164
  1101
lemma lemma_interval_lt:
huffman@21164
  1102
     "[| a < x;  x < b |]
huffman@21164
  1103
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
  1104
huffman@22998
  1105
apply (simp add: abs_less_iff)
huffman@21164
  1106
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
  1107
apply (rule_tac x = "x-a" in exI)
huffman@21164
  1108
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
  1109
done
huffman@21164
  1110
huffman@21164
  1111
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
  1112
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
  1113
apply (drule lemma_interval_lt, auto)
huffman@44921
  1114
apply force
huffman@21164
  1115
done
huffman@21164
  1116
wenzelm@60758
  1117
text\<open>Rolle's Theorem.
huffman@21164
  1118
   If @{term f} is defined and continuous on the closed interval
wenzelm@61799
  1119
   \<open>[a,b]\<close> and differentiable on the open interval \<open>(a,b)\<close>,
huffman@21164
  1120
   and @{term "f(a) = f(b)"},
wenzelm@61799
  1121
   then there exists \<open>x0 \<in> (a,b)\<close> such that @{term "f'(x0) = 0"}\<close>
huffman@21164
  1122
theorem Rolle:
huffman@21164
  1123
  assumes lt: "a < b"
huffman@21164
  1124
      and eq: "f(a) = f(b)"
huffman@21164
  1125
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181
  1126
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784
  1127
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
  1128
proof -
huffman@21164
  1129
  have le: "a \<le> b" using lt by simp
huffman@21164
  1130
  from isCont_eq_Ub [OF le con]
huffman@21164
  1131
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
  1132
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
  1133
    by blast
huffman@21164
  1134
  from isCont_eq_Lb [OF le con]
huffman@21164
  1135
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
  1136
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
  1137
    by blast
huffman@21164
  1138
  show ?thesis
huffman@21164
  1139
  proof cases
huffman@21164
  1140
    assume axb: "a < x & x < b"
wenzelm@61799
  1141
        \<comment>\<open>@{term f} attains its maximum within the interval\<close>
chaieb@27668
  1142
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
  1143
    from lemma_interval [OF ax xb]
huffman@21164
  1144
    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
  1145
      by blast
huffman@21164
  1146
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
  1147
      by blast
huffman@21164
  1148
    from differentiableD [OF dif [OF axb]]
huffman@21164
  1149
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1150
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
wenzelm@61799
  1151
        \<comment>\<open>the derivative at a local maximum is zero\<close>
huffman@21164
  1152
    thus ?thesis using ax xb der by auto
huffman@21164
  1153
  next
huffman@21164
  1154
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
  1155
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
  1156
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
  1157
    show ?thesis
huffman@21164
  1158
    proof cases
huffman@21164
  1159
      assume ax'b: "a < x' & x' < b"
wenzelm@61799
  1160
        \<comment>\<open>@{term f} attains its minimum within the interval\<close>
chaieb@27668
  1161
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
  1162
      from lemma_interval [OF ax' x'b]
huffman@21164
  1163
      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
  1164
  by blast
huffman@21164
  1165
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
  1166
  by blast
huffman@21164
  1167
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
  1168
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
  1169
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
wenzelm@61799
  1170
        \<comment>\<open>the derivative at a local minimum is zero\<close>
huffman@21164
  1171
      thus ?thesis using ax' x'b der by auto
huffman@21164
  1172
    next
huffman@21164
  1173
      assume notax'b: "~ (a < x' & x' < b)"
wenzelm@61799
  1174
        \<comment>\<open>@{term f} is constant througout the interval\<close>
huffman@21164
  1175
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
  1176
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
  1177
      from dense [OF lt]
huffman@21164
  1178
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
  1179
      from lemma_interval [OF ar rb]
huffman@21164
  1180
      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
  1181
  by blast
huffman@21164
  1182
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
  1183
      proof (clarify)
huffman@21164
  1184
        fix z::real
huffman@21164
  1185
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1186
        show "f z = f b"
huffman@21164
  1187
        proof (rule order_antisym)
huffman@21164
  1188
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1189
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1190
        qed
huffman@21164
  1191
      qed
huffman@21164
  1192
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1193
      proof (intro strip)
huffman@21164
  1194
        fix y::real
huffman@21164
  1195
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1196
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1197
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1198
      qed
huffman@21164
  1199
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1200
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1201
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
wenzelm@61799
  1202
        \<comment>\<open>the derivative of a constant function is zero\<close>
huffman@21164
  1203
      thus ?thesis using ar rb der by auto
huffman@21164
  1204
    qed
huffman@21164
  1205
  qed
huffman@21164
  1206
qed
huffman@21164
  1207
huffman@21164
  1208
wenzelm@60758
  1209
subsection\<open>Mean Value Theorem\<close>
huffman@21164
  1210
huffman@21164
  1211
lemma lemma_MVT:
huffman@21164
  1212
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
hoelzl@51481
  1213
  by (cases "a = b") (simp_all add: field_simps)
huffman@21164
  1214
huffman@21164
  1215
theorem MVT:
huffman@21164
  1216
  assumes lt:  "a < b"
huffman@21164
  1217
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181
  1218
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784
  1219
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
  1220
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1221
proof -
huffman@21164
  1222
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@44233
  1223
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
hoelzl@56371
  1224
    using con by (fast intro: continuous_intros)
hoelzl@56181
  1225
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
huffman@21164
  1226
  proof (clarify)
huffman@21164
  1227
    fix x::real
huffman@21164
  1228
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1229
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1230
    obtain l where der: "DERIV f x :> l" ..
hoelzl@56181
  1231
    show "?F differentiable (at x)"
huffman@21164
  1232
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1233
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1234
  qed
huffman@21164
  1235
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1236
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1237
    by blast
huffman@21164
  1238
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1239
    by (rule DERIV_cmult_Id)
huffman@21164
  1240
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1241
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1242
    by (rule DERIV_add [OF der])
huffman@21164
  1243
  show ?thesis
huffman@21164
  1244
  proof (intro exI conjI)
huffman@23441
  1245
    show "a < z" using az .
huffman@23441
  1246
    show "z < b" using zb .
huffman@21164
  1247
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1248
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1249
  qed
huffman@21164
  1250
qed
huffman@21164
  1251
hoelzl@29803
  1252
lemma MVT2:
hoelzl@29803
  1253
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803
  1254
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803
  1255
apply (drule MVT)
hoelzl@29803
  1256
apply (blast intro: DERIV_isCont)
hoelzl@56181
  1257
apply (force dest: order_less_imp_le simp add: real_differentiable_def)
hoelzl@29803
  1258
apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803
  1259
done
hoelzl@29803
  1260
huffman@21164
  1261
wenzelm@60758
  1262
text\<open>A function is constant if its derivative is 0 over an interval.\<close>
huffman@21164
  1263
huffman@21164
  1264
lemma DERIV_isconst_end:
huffman@21164
  1265
  fixes f :: "real => real"
huffman@21164
  1266
  shows "[| a < b;
huffman@21164
  1267
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1268
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1269
        ==> f b = f a"
huffman@21164
  1270
apply (drule MVT, assumption)
huffman@21164
  1271
apply (blast intro: differentiableI)
huffman@21164
  1272
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1273
done
huffman@21164
  1274
huffman@21164
  1275
lemma DERIV_isconst1:
huffman@21164
  1276
  fixes f :: "real => real"
huffman@21164
  1277
  shows "[| a < b;
huffman@21164
  1278
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1279
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1280
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1281
apply safe
huffman@21164
  1282
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1283
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1284
done
huffman@21164
  1285
huffman@21164
  1286
lemma DERIV_isconst2:
huffman@21164
  1287
  fixes f :: "real => real"
huffman@21164
  1288
  shows "[| a < b;
huffman@21164
  1289
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1290
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1291
         a \<le> x; x \<le> b |]
huffman@21164
  1292
        ==> f x = f a"
huffman@21164
  1293
apply (blast dest: DERIV_isconst1)
huffman@21164
  1294
done
huffman@21164
  1295
hoelzl@29803
  1296
lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803
  1297
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803
  1298
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1299
  shows "f x = f y"
hoelzl@29803
  1300
proof (cases "x = y")
hoelzl@29803
  1301
  case False
hoelzl@29803
  1302
  let ?a = "min x y"
hoelzl@29803
  1303
  let ?b = "max x y"
hoelzl@29803
  1304
  
hoelzl@29803
  1305
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1306
  proof (rule allI, rule impI)
hoelzl@29803
  1307
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
wenzelm@60758
  1308
    hence "a < z" and "z < b" using \<open>x \<in> {a <..< b}\<close> and \<open>y \<in> {a <..< b}\<close> by auto
hoelzl@29803
  1309
    hence "z \<in> {a<..<b}" by auto
hoelzl@29803
  1310
    thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1311
  qed
hoelzl@29803
  1312
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803
  1313
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803
  1314
wenzelm@60758
  1315
  have "?a < ?b" using \<open>x \<noteq> y\<close> by auto
hoelzl@29803
  1316
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1317
  show ?thesis by auto
hoelzl@29803
  1318
qed auto
hoelzl@29803
  1319
huffman@21164
  1320
lemma DERIV_isconst_all:
huffman@21164
  1321
  fixes f :: "real => real"
huffman@21164
  1322
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1323
apply (rule linorder_cases [of x y])
huffman@21164
  1324
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1325
done
huffman@21164
  1326
huffman@21164
  1327
lemma DERIV_const_ratio_const:
huffman@21784
  1328
  fixes f :: "real => real"
huffman@21784
  1329
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1330
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1331
apply (drule_tac [!] f = f in MVT)
hoelzl@56181
  1332
apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
haftmann@54230
  1333
apply (auto dest: DERIV_unique simp add: ring_distribs)
huffman@21164
  1334
done
huffman@21164
  1335
huffman@21164
  1336
lemma DERIV_const_ratio_const2:
huffman@21784
  1337
  fixes f :: "real => real"
huffman@21784
  1338
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
lp15@56217
  1339
apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
haftmann@57512
  1340
apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
huffman@21164
  1341
done
huffman@21164
  1342
huffman@21164
  1343
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1344
by (simp)
huffman@21164
  1345
huffman@21164
  1346
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1347
by (simp)
huffman@21164
  1348
wenzelm@60758
  1349
text\<open>Gallileo's "trick": average velocity = av. of end velocities\<close>
huffman@21164
  1350
huffman@21164
  1351
lemma DERIV_const_average:
huffman@21164
  1352
  fixes v :: "real => real"
huffman@21164
  1353
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1354
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1355
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1356
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1357
  case equal with neq show ?thesis by simp
huffman@21164
  1358
next
huffman@21164
  1359
  case less
huffman@21164
  1360
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1361
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1362
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1363
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1364
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1365
  ultimately show ?thesis using neq by force
huffman@21164
  1366
next
huffman@21164
  1367
  case greater
huffman@21164
  1368
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1369
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1370
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1371
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1372
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
haftmann@57512
  1373
  ultimately show ?thesis using neq by (force simp add: add.commute)
huffman@21164
  1374
qed
huffman@21164
  1375
paulson@33654
  1376
(* A function with positive derivative is increasing. 
paulson@33654
  1377
   A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654
  1378
*)
lp15@56261
  1379
lemma DERIV_pos_imp_increasing_open:
paulson@33654
  1380
  fixes a::real and b::real and f::"real => real"
lp15@56261
  1381
  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
lp15@56261
  1382
      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
paulson@33654
  1383
  shows "f a < f b"
paulson@33654
  1384
proof (rule ccontr)
wenzelm@41550
  1385
  assume f: "~ f a < f b"
wenzelm@33690
  1386
  have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1387
      & f b - f a = (b - a) * l"
wenzelm@33690
  1388
    apply (rule MVT)
lp15@56261
  1389
      using assms Deriv.differentiableI
lp15@56261
  1390
      apply force+
wenzelm@33690
  1391
    done
wenzelm@41550
  1392
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
paulson@33654
  1393
      and "f b - f a = (b - a) * l"
paulson@33654
  1394
    by auto
wenzelm@41550
  1395
  with assms f have "~(l > 0)"
huffman@36777
  1396
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550
  1397
  with assms z show False
lp15@56261
  1398
    by (metis DERIV_unique)
paulson@33654
  1399
qed
paulson@33654
  1400
lp15@56261
  1401
lemma DERIV_pos_imp_increasing:
lp15@56261
  1402
  fixes a::real and b::real and f::"real => real"
lp15@56261
  1403
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
lp15@56261
  1404
  shows "f a < f b"
lp15@56261
  1405
by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
lp15@56261
  1406
noschinl@45791
  1407
lemma DERIV_nonneg_imp_nondecreasing:
paulson@33654
  1408
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1409
  assumes "a \<le> b" and
paulson@33654
  1410
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654
  1411
  shows "f a \<le> f b"
paulson@33654
  1412
proof (rule ccontr, cases "a = b")
wenzelm@41550
  1413
  assume "~ f a \<le> f b" and "a = b"
wenzelm@41550
  1414
  then show False by auto
haftmann@37891
  1415
next
haftmann@37891
  1416
  assume A: "~ f a \<le> f b"
haftmann@37891
  1417
  assume B: "a ~= b"
paulson@33654
  1418
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1419
      & f b - f a = (b - a) * l"
wenzelm@33690
  1420
    apply -
wenzelm@33690
  1421
    apply (rule MVT)
wenzelm@33690
  1422
      apply auto
wenzelm@33690
  1423
      apply (metis DERIV_isCont)
huffman@36777
  1424
     apply (metis differentiableI less_le)
paulson@33654
  1425
    done
wenzelm@41550
  1426
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
haftmann@37891
  1427
      and C: "f b - f a = (b - a) * l"
paulson@33654
  1428
    by auto
haftmann@37891
  1429
  with A have "a < b" "f b < f a" by auto
haftmann@37891
  1430
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
huffman@45051
  1431
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@41550
  1432
  with assms z show False
paulson@33654
  1433
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1434
qed
paulson@33654
  1435
lp15@56261
  1436
lemma DERIV_neg_imp_decreasing_open:
lp15@56261
  1437
  fixes a::real and b::real and f::"real => real"
lp15@56261
  1438
  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
lp15@56261
  1439
      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
lp15@56261
  1440
  shows "f a > f b"
lp15@56261
  1441
proof -
lp15@56261
  1442
  have "(%x. -f x) a < (%x. -f x) b"
lp15@56261
  1443
    apply (rule DERIV_pos_imp_increasing_open [of a b "%x. -f x"])
lp15@56261
  1444
    using assms
lp15@56261
  1445
    apply auto
lp15@56261
  1446
    apply (metis field_differentiable_minus neg_0_less_iff_less)
lp15@56261
  1447
    done
lp15@56261
  1448
  thus ?thesis
lp15@56261
  1449
    by simp
lp15@56261
  1450
qed
lp15@56261
  1451
paulson@33654
  1452
lemma DERIV_neg_imp_decreasing:
paulson@33654
  1453
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1454
  assumes "a < b" and
paulson@33654
  1455
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654
  1456
  shows "f a > f b"
lp15@56261
  1457
by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
paulson@33654
  1458
paulson@33654
  1459
lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654
  1460
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1461
  assumes "a \<le> b" and
paulson@33654
  1462
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654
  1463
  shows "f a \<ge> f b"
paulson@33654
  1464
proof -
paulson@33654
  1465
  have "(%x. -f x) a \<le> (%x. -f x) b"
noschinl@45791
  1466
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
wenzelm@33690
  1467
    using assms
wenzelm@33690
  1468
    apply auto
paulson@33654
  1469
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1470
    done
paulson@33654
  1471
  thus ?thesis
paulson@33654
  1472
    by simp
paulson@33654
  1473
qed
huffman@21164
  1474
lp15@56289
  1475
lemma DERIV_pos_imp_increasing_at_bot:
lp15@56289
  1476
  fixes f :: "real => real"
lp15@56289
  1477
  assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
wenzelm@61973
  1478
      and lim: "(f \<longlongrightarrow> flim) at_bot"
lp15@56289
  1479
  shows "flim < f b"
lp15@56289
  1480
proof -
lp15@56289
  1481
  have "flim \<le> f (b - 1)"
lp15@56289
  1482
    apply (rule tendsto_ge_const [OF _ lim])
lp15@56289
  1483
    apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
lp15@56289
  1484
    apply (rule_tac x="b - 2" in exI)
lp15@56289
  1485
    apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289
  1486
    done
lp15@56289
  1487
  also have "... < f b"
lp15@56289
  1488
    by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289
  1489
  finally show ?thesis .
lp15@56289
  1490
qed
lp15@56289
  1491
lp15@56289
  1492
lemma DERIV_neg_imp_decreasing_at_top:
lp15@56289
  1493
  fixes f :: "real => real"
lp15@56289
  1494
  assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
wenzelm@61973
  1495
      and lim: "(f \<longlongrightarrow> flim) at_top"
lp15@56289
  1496
  shows "flim < f b"
lp15@56289
  1497
  apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
lp15@56289
  1498
  apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
lp15@56289
  1499
  apply (metis filterlim_at_top_mirror lim)
lp15@56289
  1500
  done
lp15@56289
  1501
wenzelm@60758
  1502
text \<open>Derivative of inverse function\<close>
huffman@23041
  1503
huffman@23041
  1504
lemma DERIV_inverse_function:
huffman@23041
  1505
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1506
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1507
  assumes neq: "D \<noteq> 0"
huffman@23044
  1508
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1509
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1510
  assumes cont: "isCont g x"
huffman@23041
  1511
  shows "DERIV g x :> inverse D"
huffman@23041
  1512
unfolding DERIV_iff2
huffman@23044
  1513
proof (rule LIM_equal2)
huffman@23044
  1514
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1515
    using a b by arith 
huffman@23044
  1516
next
huffman@23041
  1517
  fix y
huffman@23044
  1518
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1519
  hence "a < y" and "y < b" 
huffman@23044
  1520
    by (simp_all add: abs_less_iff)
huffman@23041
  1521
  thus "(g y - g x) / (y - x) =
huffman@23041
  1522
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1523
    by (simp add: inj)
huffman@23041
  1524
next
wenzelm@61976
  1525
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) \<midarrow>g x\<rightarrow> D"
huffman@23041
  1526
    by (rule der [unfolded DERIV_iff2])
wenzelm@61976
  1527
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
huffman@23044
  1528
    using inj a b by simp
huffman@23041
  1529
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
wenzelm@56219
  1530
  proof (rule exI, safe)
huffman@23044
  1531
    show "0 < min (x - a) (b - x)"
huffman@23044
  1532
      using a b by simp
huffman@23041
  1533
  next
huffman@23041
  1534
    fix y
huffman@23044
  1535
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1536
    hence y: "a < y" "y < b"
huffman@23044
  1537
      by (simp_all add: abs_less_iff)
huffman@23041
  1538
    assume "g y = g x"
huffman@23041
  1539
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1540
    hence "y = x" using inj y a b by simp
huffman@23041
  1541
    also assume "y \<noteq> x"
huffman@23041
  1542
    finally show False by simp
huffman@23041
  1543
  qed
wenzelm@61976
  1544
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) \<midarrow>x\<rightarrow> D"
huffman@23041
  1545
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1546
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
wenzelm@61976
  1547
        \<midarrow>x\<rightarrow> inverse D"
huffman@44568
  1548
    using neq by (rule tendsto_inverse)
huffman@23041
  1549
qed
huffman@23041
  1550
wenzelm@60758
  1551
subsection \<open>Generalized Mean Value Theorem\<close>
huffman@29975
  1552
huffman@21164
  1553
theorem GMVT:
huffman@21784
  1554
  fixes a b :: real
huffman@21164
  1555
  assumes alb: "a < b"
wenzelm@41550
  1556
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
hoelzl@56181
  1557
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@41550
  1558
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
hoelzl@56181
  1559
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
wenzelm@53381
  1560
  shows "\<exists>g'c f'c c.
wenzelm@53381
  1561
    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
  1562
proof -
huffman@21164
  1563
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
wenzelm@41550
  1564
  from assms have "a < b" by simp
huffman@21164
  1565
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@44233
  1566
    using fc gc by simp
hoelzl@56181
  1567
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
huffman@44233
  1568
    using fd gd by simp
huffman@21164
  1569
  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
  1570
  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
  1571
  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
  1572
huffman@21164
  1573
  from cdef have cint: "a < c \<and> c < b" by auto
hoelzl@56181
  1574
  with gd have "g differentiable (at c)" by simp
huffman@21164
  1575
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1576
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1577
huffman@21164
  1578
  from cdef have "a < c \<and> c < b" by auto
hoelzl@56181
  1579
  with fd have "f differentiable (at c)" by simp
huffman@21164
  1580
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1581
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1582
huffman@21164
  1583
  from cdef have "DERIV ?h c :> l" by auto
hoelzl@41368
  1584
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
hoelzl@56381
  1585
    using g'cdef f'cdef by (auto intro!: derivative_eq_intros)
huffman@21164
  1586
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1587
huffman@21164
  1588
  {
huffman@21164
  1589
    from cdef have "?h b - ?h a = (b - a) * l" by auto
wenzelm@53374
  1590
    also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1591
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1592
  }
huffman@21164
  1593
  moreover
huffman@21164
  1594
  {
huffman@21164
  1595
    have "?h b - ?h a =
huffman@21164
  1596
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1597
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1598
      by (simp add: algebra_simps)
huffman@21164
  1599
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1600
  }
huffman@21164
  1601
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1602
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1603
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
haftmann@57514
  1604
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
huffman@21164
  1605
huffman@21164
  1606
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1607
qed
huffman@21164
  1608
hoelzl@50327
  1609
lemma GMVT':
hoelzl@50327
  1610
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1611
  assumes "a < b"
hoelzl@50327
  1612
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
hoelzl@50327
  1613
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
hoelzl@50327
  1614
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
hoelzl@50327
  1615
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327
  1616
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1617
proof -
hoelzl@50327
  1618
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
hoelzl@50327
  1619
    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@56181
  1620
    using assms by (intro GMVT) (force simp: real_differentiable_def)+
hoelzl@50327
  1621
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1622
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327
  1623
  then show ?thesis
hoelzl@50327
  1624
    by auto
hoelzl@50327
  1625
qed
hoelzl@50327
  1626
hoelzl@51529
  1627
wenzelm@60758
  1628
subsection \<open>L'Hopitals rule\<close>
hoelzl@51529
  1629
hoelzl@51641
  1630
lemma isCont_If_ge:
hoelzl@51641
  1631
  fixes a :: "'a :: linorder_topology"
wenzelm@61973
  1632
  shows "continuous (at_left a) g \<Longrightarrow> (f \<longlongrightarrow> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
hoelzl@51641
  1633
  unfolding isCont_def continuous_within
hoelzl@51641
  1634
  apply (intro filterlim_split_at)
hoelzl@51641
  1635
  apply (subst filterlim_cong[OF refl refl, where g=g])
hoelzl@51641
  1636
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1637
  apply (subst filterlim_cong[OF refl refl, where g=f])
hoelzl@51641
  1638
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1639
  done
hoelzl@51641
  1640
hoelzl@50327
  1641
lemma lhopital_right_0:
hoelzl@50329
  1642
  fixes f0 g0 :: "real \<Rightarrow> real"
wenzelm@61973
  1643
  assumes f_0: "(f0 \<longlongrightarrow> 0) (at_right 0)"
wenzelm@61973
  1644
  assumes g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1645
  assumes ev:
hoelzl@50329
  1646
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1647
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50329
  1648
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
hoelzl@50329
  1649
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
wenzelm@61973
  1650
  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
wenzelm@61973
  1651
  shows "((\<lambda> x. f0 x / g0 x) \<longlongrightarrow> x) (at_right 0)"
hoelzl@50327
  1652
proof -
wenzelm@63040
  1653
  define f where [abs_def]: "f x = (if x \<le> 0 then 0 else f0 x)" for x
hoelzl@50329
  1654
  then have "f 0 = 0" by simp
hoelzl@50329
  1655
wenzelm@63040
  1656
  define g where [abs_def]: "g x = (if x \<le> 0 then 0 else g0 x)" for x
hoelzl@50329
  1657
  then have "g 0 = 0" by simp
hoelzl@50329
  1658
hoelzl@50329
  1659
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329
  1660
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329
  1661
    using ev by eventually_elim auto
hoelzl@50329
  1662
  then obtain a where [arith]: "0 < a"
hoelzl@50329
  1663
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327
  1664
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329
  1665
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329
  1666
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
wenzelm@56219
  1667
    unfolding eventually_at by (auto simp: dist_real_def)
hoelzl@50327
  1668
hoelzl@50329
  1669
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329
  1670
    using g0_neq_0 by (simp add: g_def)
hoelzl@50329
  1671
hoelzl@50329
  1672
  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
hoelzl@50329
  1673
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
hoelzl@50329
  1674
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1675
  note f = this
hoelzl@50329
  1676
hoelzl@50329
  1677
  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
hoelzl@50329
  1678
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
hoelzl@50329
  1679
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1680
  note g = this
hoelzl@50329
  1681
hoelzl@50329
  1682
  have "isCont f 0"
hoelzl@51641
  1683
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
hoelzl@51641
  1684
hoelzl@50329
  1685
  have "isCont g 0"
hoelzl@51641
  1686
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
hoelzl@50329
  1687
hoelzl@50327
  1688
  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
  1689
  proof (rule bchoice, rule)
hoelzl@50327
  1690
    fix x assume "x \<in> {0 <..< a}"
hoelzl@50327
  1691
    then have x[arith]: "0 < x" "x < a" by auto
wenzelm@60758
  1692
    with g'_neq_0 g_neq_0 \<open>g 0 = 0\<close> have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
hoelzl@50327
  1693
      by auto
hoelzl@50328
  1694
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
wenzelm@60758
  1695
      using \<open>isCont f 0\<close> f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1696
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
wenzelm@60758
  1697
      using \<open>isCont g 0\<close> g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1698
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@60758
  1699
      using f g \<open>x < a\<close> by (intro GMVT') auto
wenzelm@53374
  1700
    then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@53374
  1701
      by blast
hoelzl@50327
  1702
    moreover
wenzelm@53374
  1703
    from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327
  1704
      by (simp add: field_simps)
hoelzl@50327
  1705
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
wenzelm@60758
  1706
      using \<open>f 0 = 0\<close> \<open>g 0 = 0\<close> by (auto intro!: exI[of _ c])
hoelzl@50327
  1707
  qed
wenzelm@53381
  1708
  then obtain \<zeta> where "\<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
  1709
  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
  1710
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327
  1711
  moreover
hoelzl@50327
  1712
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327
  1713
    by eventually_elim auto
wenzelm@61973
  1714
  then have "((\<lambda>x. norm (\<zeta> x)) \<longlongrightarrow> 0) (at_right 0)"
hoelzl@58729
  1715
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"]) auto
wenzelm@61973
  1716
  then have "(\<zeta> \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1717
    by (rule tendsto_norm_zero_cancel)
hoelzl@50327
  1718
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
lp15@61810
  1719
    by (auto elim!: eventually_mono simp: filterlim_at)
wenzelm@61973
  1720
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) \<longlongrightarrow> x) (at_right 0)"
hoelzl@50327
  1721
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
wenzelm@61973
  1722
  ultimately have "((\<lambda>t. f t / g t) \<longlongrightarrow> x) (at_right 0)" (is ?P)
hoelzl@50328
  1723
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
lp15@61810
  1724
       (auto elim: eventually_mono)
hoelzl@50329
  1725
  also have "?P \<longleftrightarrow> ?thesis"
hoelzl@51641
  1726
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
hoelzl@50329
  1727
  finally show ?thesis .
hoelzl@50327
  1728
qed
hoelzl@50327
  1729
hoelzl@50330
  1730
lemma lhopital_right:
wenzelm@61973
  1731
  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at_right x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1732
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1733
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1734
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1735
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
wenzelm@61973
  1736
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
wenzelm@61973
  1737
  ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
hoelzl@50330
  1738
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1739
  by (rule lhopital_right_0)
hoelzl@50330
  1740
hoelzl@50330
  1741
lemma lhopital_left:
wenzelm@61973
  1742
  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at_left x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1743
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1744
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1745
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1746
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
wenzelm@61973
  1747
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
wenzelm@61973
  1748
  ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
hoelzl@50330
  1749
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@56479
  1750
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1751
hoelzl@50330
  1752
lemma lhopital:
wenzelm@61973
  1753
  "((f::real \<Rightarrow> real) \<longlongrightarrow> 0) (at x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1754
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1755
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1756
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1757
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
wenzelm@61973
  1758
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
wenzelm@61973
  1759
  ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
hoelzl@50330
  1760
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1761
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330
  1762
hoelzl@50327
  1763
lemma lhopital_right_0_at_top:
hoelzl@50327
  1764
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1765
  assumes g_0: "LIM x at_right 0. g x :> at_top"
hoelzl@50327
  1766
  assumes ev:
hoelzl@50327
  1767
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1768
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
hoelzl@50327
  1769
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
wenzelm@61973
  1770
  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
wenzelm@61973
  1771
  shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
hoelzl@50327
  1772
  unfolding tendsto_iff
hoelzl@50327
  1773
proof safe
hoelzl@50327
  1774
  fix e :: real assume "0 < e"
hoelzl@50327
  1775
hoelzl@50327
  1776
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
hoelzl@50327
  1777
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
hoelzl@50327
  1778
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327
  1779
  obtain a where [arith]: "0 < a"
hoelzl@50327
  1780
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327
  1781
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327
  1782
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327
  1783
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@51641
  1784
    unfolding eventually_at_le by (auto simp: dist_real_def)
hoelzl@51641
  1785
    
hoelzl@50327
  1786
hoelzl@50327
  1787
  from Df have
hoelzl@50328
  1788
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@51641
  1789
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327
  1790
hoelzl@50327
  1791
  moreover
hoelzl@50328
  1792
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
lp15@61810
  1793
    using g_0 by (auto elim: eventually_mono simp: filterlim_at_top_dense)
hoelzl@50327
  1794
hoelzl@50327
  1795
  moreover
wenzelm@61973
  1796
  have inv_g: "((\<lambda>x. inverse (g x)) \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1797
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327
  1798
    by (rule filterlim_compose)
wenzelm@61973
  1799
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) \<longlongrightarrow> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327
  1800
    by (intro tendsto_intros)
wenzelm@61973
  1801
  then have "((\<lambda>x. norm (1 - g a / g x)) \<longlongrightarrow> 1) (at_right 0)"
hoelzl@50327
  1802
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1803
  from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327
  1804
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
lp15@61810
  1805
    by (auto elim!: eventually_mono simp: dist_real_def)
hoelzl@50327
  1806
hoelzl@50327
  1807
  moreover
wenzelm@61973
  1808
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) \<longlongrightarrow> norm ((f a - x * g a) * 0)) (at_right 0)"
hoelzl@50327
  1809
    by (intro tendsto_intros)
wenzelm@61973
  1810
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1811
    by (simp add: inverse_eq_divide)
wenzelm@60758
  1812
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] \<open>0 < e\<close>
hoelzl@50327
  1813
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327
  1814
    by (auto simp: dist_real_def)
hoelzl@50327
  1815
hoelzl@50327
  1816
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327
  1817
  proof eventually_elim
hoelzl@50327
  1818
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327
  1819
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327
  1820
hoelzl@50327
  1821
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327
  1822
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
wenzelm@53381
  1823
    then obtain y where [arith]: "t < y" "y < a"
wenzelm@53381
  1824
      and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
wenzelm@53381
  1825
      by blast
wenzelm@53381
  1826
    from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
wenzelm@60758
  1827
      using \<open>g a < g t\<close> g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327
  1828
hoelzl@50327
  1829
    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
  1830
      by (simp add: field_simps)
hoelzl@50327
  1831
    have "norm (f t / g t - x) \<le>
hoelzl@50327
  1832
        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
  1833
      unfolding * by (rule norm_triangle_ineq)
hoelzl@50327
  1834
    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
  1835
      by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327
  1836
    also have "\<dots> < (e / 4) * 2 + e / 2"
wenzelm@60758
  1837
      using ineq Df[of y] \<open>0 < e\<close> by (intro add_le_less_mono mult_mono) auto
hoelzl@50327
  1838
    finally show "dist (f t / g t) x < e"
hoelzl@50327
  1839
      by (simp add: dist_real_def)
hoelzl@50327
  1840
  qed
hoelzl@50327
  1841
qed
hoelzl@50327
  1842
hoelzl@50330
  1843
lemma lhopital_right_at_top:
hoelzl@50330
  1844
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1845
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1846
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1847
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
wenzelm@61973
  1848
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
wenzelm@61973
  1849
    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
hoelzl@50330
  1850
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1851
  by (rule lhopital_right_0_at_top)
hoelzl@50330
  1852
hoelzl@50330
  1853
lemma lhopital_left_at_top:
hoelzl@50330
  1854
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1855
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1856
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1857
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
wenzelm@61973
  1858
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
wenzelm@61973
  1859
    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
hoelzl@50330
  1860
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@56479
  1861
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1862
hoelzl@50330
  1863
lemma lhopital_at_top:
hoelzl@50330
  1864
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1865
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1866
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1867
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
wenzelm@61973
  1868
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
wenzelm@61973
  1869
    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
hoelzl@50330
  1870
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1871
  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
  1872
hoelzl@50347
  1873
lemma lhospital_at_top_at_top:
hoelzl@50347
  1874
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50347
  1875
  assumes g_0: "LIM x at_top. g x :> at_top"
hoelzl@50347
  1876
  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
hoelzl@50347
  1877
  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
hoelzl@50347
  1878
  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
wenzelm@61973
  1879
  assumes lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
wenzelm@61973
  1880
  shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) at_top"
hoelzl@50347
  1881
  unfolding filterlim_at_top_to_right
hoelzl@50347
  1882
proof (rule lhopital_right_0_at_top)
hoelzl@50347
  1883
  let ?F = "\<lambda>x. f (inverse x)"
hoelzl@50347
  1884
  let ?G = "\<lambda>x. g (inverse x)"
hoelzl@50347
  1885
  let ?R = "at_right (0::real)"
hoelzl@50347
  1886
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
hoelzl@50347
  1887
hoelzl@50347
  1888
  show "LIM x ?R. ?G x :> at_top"
hoelzl@50347
  1889
    using g_0 unfolding filterlim_at_top_to_right .
hoelzl@50347
  1890
hoelzl@50347
  1891
  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
hoelzl@50347
  1892
    unfolding eventually_at_right_to_top
hoelzl@50347
  1893
    using Dg eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1894
    apply eventually_elim
hoelzl@50347
  1895
    apply (rule DERIV_cong)
hoelzl@50347
  1896
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1897
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1898
    done
hoelzl@50347
  1899
hoelzl@50347
  1900
  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
hoelzl@50347
  1901
    unfolding eventually_at_right_to_top
hoelzl@50347
  1902
    using Df eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1903
    apply eventually_elim
hoelzl@50347
  1904
    apply (rule DERIV_cong)
hoelzl@50347
  1905
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1906
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1907
    done
hoelzl@50347
  1908
hoelzl@50347
  1909
  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
hoelzl@50347
  1910
    unfolding eventually_at_right_to_top
hoelzl@50347
  1911
    using g' eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1912
    by eventually_elim auto
hoelzl@50347
  1913
    
wenzelm@61973
  1914
  show "((\<lambda>x. ?D f' x / ?D g' x) \<longlongrightarrow> x) ?R"
hoelzl@50347
  1915
    unfolding filterlim_at_right_to_top
hoelzl@50347
  1916
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
hoelzl@50347
  1917
    using eventually_ge_at_top[where c="1::real"]
hoelzl@56479
  1918
    by eventually_elim simp
hoelzl@50347
  1919
qed
hoelzl@50347
  1920
huffman@21164
  1921
end