src/HOL/Deriv.thy
author hoelzl
Thu Apr 03 17:56:08 2014 +0200 (2014-04-03)
changeset 56381 0556204bc230
parent 56371 fb9ae0727548
child 56409 36489d77c484
permissions -rw-r--r--
merged DERIV_intros, has_derivative_intros into derivative_intros
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(*  Title       : Deriv.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Author      : Brian Huffman
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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    GMVT by Benjamin Porter, 2005
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*)
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header{* Differentiation *}
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theory Deriv
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imports Limits
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begin
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subsection {* Frechet derivative *}
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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))) ---> 0) F)"
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text {*
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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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*}
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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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ML {*
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structure Derivative_Intros = Named_Thms
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(
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  val name = @{binding derivative_intros}
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  val description = "structural introduction rules for derivatives"
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)
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*}
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setup {*
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  let
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    val eq_thms = [@{thm has_derivative_eq_rhs}, @{thm DERIV_cong}, @{thm 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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    Derivative_Intros.setup #>
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    Global_Theory.add_thms_dynamic
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      (@{binding derivative_eq_intros}, map_filter eq_rule o Derivative_Intros.get o Context.proof_of)
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  end;
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*}
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text {*
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  The following syntax is only used as a legacy syntax.
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*}
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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 tendsto_const)
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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 tendsto_const)
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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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  using assms 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) ---> (0 + 0)) F"
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    using f g by (intro tendsto_add) (auto simp: has_derivative_def)
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  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
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    by (simp add: field_simps scaleR_add_right scaleR_diff_right)
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qed (blast intro: bounded_linear_add f g 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)) ---> 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)) ---> 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) -- 0 --> 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) -- 0 --> 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)) ---> 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 ---> 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)) ---> 0) (at x within s)"
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  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
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    show "(H ---> 0) (at x within s)" by fact
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    show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
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      unfolding eventually_at using e sandwich by auto
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  qed (auto simp: le_divide_eq tendsto_const)
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qed fact
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lemma 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 {* Continuity *}
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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 ---> 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 {* Composition *}
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lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"
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  unfolding tendsto_def eventually_inf_principal eventually_at_filter
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  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
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lemma 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 ---> 0) (at x within s)"
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  let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
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  let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
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  let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
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  def Nf \<equiv> "?N f f' x"
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  def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
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  show ?thesis
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  proof (rule 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))"
hoelzl@51642
   261
        by simp
hoelzl@51642
   262
      also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
hoelzl@51642
   263
        by (rule norm_triangle_ineq)
hoelzl@51642
   264
      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
hoelzl@51642
   265
        using kF by (intro add_mono) simp
hoelzl@51642
   266
      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
hoelzl@51642
   267
        by (simp add: neq Nf_def field_simps)
hoelzl@51642
   268
    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
hoelzl@51642
   269
    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
hoelzl@51642
   270
  next
hoelzl@51642
   271
    have [tendsto_intros]: "?L Nf"
hoelzl@56181
   272
      using f unfolding has_derivative_iff_norm Nf_def ..
hoelzl@51642
   273
    from f have "(f ---> f x) (at x within s)"
hoelzl@56181
   274
      by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
hoelzl@51642
   275
    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
hoelzl@51642
   276
      unfolding filterlim_def
hoelzl@51642
   277
      by (simp add: eventually_filtermap eventually_at_filter le_principal)
hoelzl@51642
   278
hoelzl@51642
   279
    have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"
hoelzl@56181
   280
      using g unfolding has_derivative_iff_norm ..
hoelzl@51642
   281
    then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"
hoelzl@51642
   282
      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
hoelzl@51642
   283
hoelzl@51642
   284
    have [tendsto_intros]: "?L Ng"
hoelzl@51642
   285
      unfolding Ng_def by (rule filterlim_compose[OF g' f'])
hoelzl@51642
   286
    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"
hoelzl@51642
   287
      by (intro tendsto_eq_intros) auto
hoelzl@51642
   288
  qed simp
hoelzl@51642
   289
qed
hoelzl@51642
   290
hoelzl@56181
   291
lemma has_derivative_compose:
hoelzl@56181
   292
  "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
hoelzl@56181
   293
  ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@56181
   294
  by (blast intro: has_derivative_in_compose has_derivative_subset)
hoelzl@51642
   295
hoelzl@51642
   296
lemma (in bounded_bilinear) FDERIV:
hoelzl@56181
   297
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
hoelzl@56181
   298
  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
   299
proof -
hoelzl@56181
   300
  from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
hoelzl@51642
   301
  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
hoelzl@51642
   302
hoelzl@51642
   303
  from pos_bounded obtain K where K: "0 < K" and norm_prod:
hoelzl@51642
   304
    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
hoelzl@51642
   305
  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
hoelzl@51642
   306
  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
hoelzl@51642
   307
  def Ng =="?N g g'" and Nf =="?N f f'"
hoelzl@51642
   308
hoelzl@51642
   309
  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
   310
  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
   311
  let ?F = "at x within s"
huffman@21164
   312
hoelzl@51642
   313
  show ?thesis
hoelzl@56181
   314
  proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642
   315
    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
hoelzl@51642
   316
      by (intro bounded_linear_add
hoelzl@51642
   317
        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
hoelzl@56181
   318
        has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
hoelzl@51642
   319
  next
hoelzl@51642
   320
    from g have "(g ---> g x) ?F"
hoelzl@56181
   321
      by (intro continuous_within[THEN iffD1] has_derivative_continuous)
hoelzl@51642
   322
    moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"
hoelzl@56181
   323
      by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
hoelzl@51642
   324
    ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
hoelzl@51642
   325
      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
hoelzl@51642
   326
    then show "(?fun2 ---> 0) ?F"
hoelzl@51642
   327
      by simp
hoelzl@51642
   328
  next
hoelzl@51642
   329
    fix y::'d assume "y \<noteq> x"
hoelzl@51642
   330
    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
   331
      by (simp add: diff_left diff_right add_left add_right field_simps)
hoelzl@51642
   332
    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
hoelzl@51642
   333
        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
hoelzl@51642
   334
      by (intro divide_right_mono mult_mono'
hoelzl@51642
   335
                order_trans [OF norm_triangle_ineq add_mono]
hoelzl@51642
   336
                order_trans [OF norm_prod mult_right_mono]
hoelzl@51642
   337
                mult_nonneg_nonneg order_refl norm_ge_zero norm_F
hoelzl@51642
   338
                K [THEN order_less_imp_le])
hoelzl@51642
   339
    also have "\<dots> = ?fun2 y"
hoelzl@51642
   340
      by (simp add: add_divide_distrib Ng_def Nf_def)
hoelzl@51642
   341
    finally show "?fun1 y \<le> ?fun2 y" .
hoelzl@51642
   342
  qed simp
hoelzl@51642
   343
qed
hoelzl@51642
   344
hoelzl@56381
   345
lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
hoelzl@56381
   346
lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
hoelzl@51642
   347
hoelzl@56381
   348
lemma has_derivative_setprod[simp, derivative_intros]:
hoelzl@51642
   349
  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   350
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
hoelzl@56181
   351
  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
   352
proof cases
hoelzl@51642
   353
  assume "finite I" from this f show ?thesis
hoelzl@51642
   354
  proof induct
hoelzl@51642
   355
    case (insert i I)
hoelzl@51642
   356
    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
   357
    have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
hoelzl@56181
   358
      using insert by (intro has_derivative_mult) auto
hoelzl@51642
   359
    also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
hoelzl@51642
   360
      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum_cong)
hoelzl@51642
   361
    finally show ?case
hoelzl@51642
   362
      using insert by simp
hoelzl@51642
   363
  qed simp  
hoelzl@51642
   364
qed simp
hoelzl@51642
   365
hoelzl@56381
   366
lemma has_derivative_power[simp, derivative_intros]:
hoelzl@51642
   367
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   368
  assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   369
  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
   370
  using has_derivative_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
hoelzl@51642
   371
hoelzl@56181
   372
lemma has_derivative_inverse':
hoelzl@51642
   373
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51642
   374
  assumes x: "x \<noteq> 0"
hoelzl@56181
   375
  shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
hoelzl@56181
   376
        (is "(?inv has_derivative ?f) _")
hoelzl@56181
   377
proof (rule has_derivativeI_sandwich)
hoelzl@51642
   378
  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
hoelzl@51642
   379
    apply (rule bounded_linear_minus)
hoelzl@51642
   380
    apply (rule bounded_linear_mult_const)
hoelzl@51642
   381
    apply (rule bounded_linear_const_mult)
hoelzl@51642
   382
    apply (rule bounded_linear_ident)
hoelzl@51642
   383
    done
hoelzl@51642
   384
next
hoelzl@51642
   385
  show "0 < norm x" using x by simp
hoelzl@51642
   386
next
hoelzl@51642
   387
  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"
hoelzl@51642
   388
    apply (rule tendsto_mult_left_zero)
hoelzl@51642
   389
    apply (rule tendsto_norm_zero)
hoelzl@51642
   390
    apply (rule LIM_zero)
hoelzl@51642
   391
    apply (rule tendsto_inverse)
hoelzl@51642
   392
    apply (rule tendsto_ident_at)
hoelzl@51642
   393
    apply (rule x)
hoelzl@51642
   394
    done
hoelzl@51642
   395
next
hoelzl@51642
   396
  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"
hoelzl@51642
   397
  then have "y \<noteq> 0"
hoelzl@51642
   398
    by (auto simp: norm_conv_dist dist_commute)
hoelzl@51642
   399
  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
hoelzl@51642
   400
    apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])
hoelzl@51642
   401
    apply (subst minus_diff_minus)
hoelzl@51642
   402
    apply (subst norm_minus_cancel)
hoelzl@51642
   403
    apply (simp add: left_diff_distrib)
hoelzl@51642
   404
    done
hoelzl@51642
   405
  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
hoelzl@51642
   406
    apply (rule divide_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   407
    apply (rule order_trans [OF norm_mult_ineq])
hoelzl@51642
   408
    apply (rule mult_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   409
    apply (rule norm_mult_ineq)
hoelzl@51642
   410
    done
hoelzl@51642
   411
  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
hoelzl@51642
   412
    by simp
hoelzl@51642
   413
  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
hoelzl@51642
   414
      norm (?inv y - ?inv x) * norm (?inv x)" .
hoelzl@51642
   415
qed
hoelzl@51642
   416
hoelzl@56381
   417
lemma has_derivative_inverse[simp, derivative_intros]:
hoelzl@51642
   418
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181
   419
  assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   420
  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
hoelzl@56181
   421
  using has_derivative_compose[OF f has_derivative_inverse', OF x] .
hoelzl@51642
   422
hoelzl@56381
   423
lemma has_derivative_divide[simp, derivative_intros]:
hoelzl@51642
   424
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
hoelzl@56181
   425
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 
lp15@55967
   426
  assumes x: "g x \<noteq> 0"
hoelzl@56181
   427
  shows "((\<lambda>x. f x / g x) has_derivative
hoelzl@56181
   428
                (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
hoelzl@56181
   429
  using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
lp15@55967
   430
  by (simp add: divide_inverse field_simps)
lp15@55967
   431
lp15@55967
   432
text{*Conventional form requires mult-AC laws. Types real and complex only.*}
hoelzl@56181
   433
hoelzl@56381
   434
lemma has_derivative_divide'[derivative_intros]: 
lp15@55967
   435
  fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
hoelzl@56181
   436
  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
   437
  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
   438
proof -
lp15@55967
   439
  { fix h
lp15@55967
   440
    have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
lp15@55967
   441
          (f' h * g x - f x * g' h) / (g x * g x)"
lp15@55967
   442
      by (simp add: divide_inverse field_simps nonzero_inverse_mult_distrib x)
lp15@55967
   443
   }
lp15@55967
   444
  then show ?thesis
hoelzl@56181
   445
    using has_derivative_divide [OF f g] x
lp15@55967
   446
    by simp
lp15@55967
   447
qed
hoelzl@51642
   448
hoelzl@51642
   449
subsection {* Uniqueness *}
hoelzl@51642
   450
hoelzl@51642
   451
text {*
hoelzl@51642
   452
hoelzl@56181
   453
This can not generally shown for @{const has_derivative}, as we need to approach the point from
hoelzl@51642
   454
all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.
hoelzl@51642
   455
hoelzl@51642
   456
*}
hoelzl@51642
   457
hoelzl@56181
   458
lemma has_derivative_zero_unique:
hoelzl@56181
   459
  assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
hoelzl@51642
   460
proof -
hoelzl@51642
   461
  interpret F: bounded_linear F
hoelzl@56181
   462
    using assms by (rule has_derivative_bounded_linear)
hoelzl@51642
   463
  let ?r = "\<lambda>h. norm (F h) / norm h"
hoelzl@51642
   464
  have *: "?r -- 0 --> 0"
hoelzl@56181
   465
    using assms unfolding has_derivative_at by simp
hoelzl@51642
   466
  show "F = (\<lambda>h. 0)"
hoelzl@51642
   467
  proof
hoelzl@51642
   468
    fix h show "F h = 0"
hoelzl@51642
   469
    proof (rule ccontr)
wenzelm@53374
   470
      assume **: "F h \<noteq> 0"
wenzelm@53374
   471
      then have h: "h \<noteq> 0"
hoelzl@51642
   472
        by (clarsimp simp add: F.zero)
wenzelm@53374
   473
      with ** have "0 < ?r h"
hoelzl@51642
   474
        by (simp add: divide_pos_pos)
hoelzl@51642
   475
      from LIM_D [OF * this] obtain s where s: "0 < s"
hoelzl@51642
   476
        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
hoelzl@51642
   477
      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
hoelzl@51642
   478
      let ?x = "scaleR (t / norm h) h"
hoelzl@51642
   479
      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
hoelzl@51642
   480
      hence "?r ?x < ?r h" by (rule r)
hoelzl@51642
   481
      thus "False" using t h by (simp add: F.scaleR)
hoelzl@51642
   482
    qed
hoelzl@51642
   483
  qed
hoelzl@51642
   484
qed
hoelzl@51642
   485
hoelzl@56181
   486
lemma has_derivative_unique:
hoelzl@56181
   487
  assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
hoelzl@51642
   488
proof -
hoelzl@56181
   489
  have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
hoelzl@56181
   490
    using has_derivative_diff [OF assms] by simp
hoelzl@51642
   491
  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
hoelzl@56181
   492
    by (rule has_derivative_zero_unique)
hoelzl@51642
   493
  thus "F = F'"
hoelzl@51642
   494
    unfolding fun_eq_iff right_minus_eq .
hoelzl@51642
   495
qed
hoelzl@51642
   496
hoelzl@51642
   497
subsection {* Differentiability predicate *}
hoelzl@51642
   498
hoelzl@56181
   499
definition
hoelzl@56181
   500
  differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@56182
   501
  (infix "differentiable" 50)
hoelzl@56181
   502
where
hoelzl@56181
   503
  "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
hoelzl@51642
   504
hoelzl@56181
   505
lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
hoelzl@56181
   506
  unfolding differentiable_def by (blast intro: has_derivative_subset)
hoelzl@51642
   507
lp15@56261
   508
lemmas differentiable_within_subset = differentiable_subset
lp15@56261
   509
hoelzl@56381
   510
lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
hoelzl@56181
   511
  unfolding differentiable_def by (blast intro: has_derivative_ident)
hoelzl@51642
   512
hoelzl@56381
   513
lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
hoelzl@56181
   514
  unfolding differentiable_def by (blast intro: has_derivative_const)
hoelzl@51642
   515
hoelzl@51642
   516
lemma differentiable_in_compose:
hoelzl@56181
   517
  "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
   518
  unfolding differentiable_def by (blast intro: has_derivative_in_compose)
hoelzl@51642
   519
hoelzl@51642
   520
lemma differentiable_compose:
hoelzl@56181
   521
  "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
   522
  by (blast intro: differentiable_in_compose differentiable_subset)
hoelzl@51642
   523
hoelzl@56381
   524
lemma differentiable_sum [simp, derivative_intros]:
hoelzl@56181
   525
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
hoelzl@56181
   526
  unfolding differentiable_def by (blast intro: has_derivative_add)
hoelzl@51642
   527
hoelzl@56381
   528
lemma differentiable_minus [simp, derivative_intros]:
hoelzl@56181
   529
  "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
hoelzl@56181
   530
  unfolding differentiable_def by (blast intro: has_derivative_minus)
hoelzl@51642
   531
hoelzl@56381
   532
lemma differentiable_diff [simp, derivative_intros]:
hoelzl@56181
   533
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
hoelzl@56181
   534
  unfolding differentiable_def by (blast intro: has_derivative_diff)
hoelzl@51642
   535
hoelzl@56381
   536
lemma differentiable_mult [simp, derivative_intros]:
hoelzl@51642
   537
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
hoelzl@56181
   538
  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
   539
  unfolding differentiable_def by (blast intro: has_derivative_mult)
hoelzl@51642
   540
hoelzl@56381
   541
lemma differentiable_inverse [simp, derivative_intros]:
hoelzl@51642
   542
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   543
  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
   544
  unfolding differentiable_def by (blast intro: has_derivative_inverse)
hoelzl@51642
   545
hoelzl@56381
   546
lemma differentiable_divide [simp, derivative_intros]:
hoelzl@51642
   547
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   548
  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)"
hoelzl@51642
   549
  unfolding divide_inverse using assms by simp
hoelzl@51642
   550
hoelzl@56381
   551
lemma differentiable_power [simp, derivative_intros]:
hoelzl@51642
   552
  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   553
  shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
hoelzl@56181
   554
  unfolding differentiable_def by (blast intro: has_derivative_power)
hoelzl@51642
   555
hoelzl@56381
   556
lemma differentiable_scaleR [simp, derivative_intros]:
hoelzl@56181
   557
  "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
   558
  unfolding differentiable_def by (blast intro: has_derivative_scaleR)
hoelzl@51642
   559
hoelzl@56181
   560
lemma has_derivative_imp_has_field_derivative:
hoelzl@56181
   561
  "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
hoelzl@56181
   562
  unfolding has_field_derivative_def 
hoelzl@56181
   563
  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult_commute)
hoelzl@56181
   564
hoelzl@56181
   565
lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
hoelzl@56181
   566
  by (simp add: has_field_derivative_def)
hoelzl@51642
   567
lp15@56261
   568
lemma DERIV_subset: 
lp15@56261
   569
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
lp15@56261
   570
   \<Longrightarrow> (f has_field_derivative f') (at x within t)"
lp15@56261
   571
  by (simp add: has_field_derivative_def has_derivative_within_subset)
lp15@56261
   572
hoelzl@56181
   573
abbreviation (input)
hoelzl@56381
   574
  DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
hoelzl@56181
   575
  ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
hoelzl@51642
   576
where
hoelzl@56181
   577
  "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
hoelzl@51642
   578
hoelzl@56181
   579
abbreviation 
hoelzl@56181
   580
  has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
hoelzl@56182
   581
  (infix "(has'_real'_derivative)" 50)
hoelzl@56181
   582
where
hoelzl@56181
   583
  "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
hoelzl@56181
   584
hoelzl@56181
   585
lemma real_differentiable_def:
hoelzl@56181
   586
  "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
hoelzl@51642
   587
proof safe
hoelzl@56181
   588
  assume "f differentiable at x within s"
hoelzl@56181
   589
  then obtain f' where *: "(f has_derivative f') (at x within s)"
hoelzl@56181
   590
    unfolding differentiable_def by auto
hoelzl@56181
   591
  then obtain c where "f' = (op * c)"
hoelzl@56181
   592
    by (metis real_bounded_linear has_derivative_bounded_linear mult_commute fun_eq_iff)
hoelzl@56181
   593
  with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   594
    unfolding has_field_derivative_def by auto
hoelzl@56181
   595
qed (auto simp: differentiable_def has_field_derivative_def)
hoelzl@51642
   596
hoelzl@56181
   597
lemma real_differentiableE [elim?]:
hoelzl@56181
   598
  assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
hoelzl@56181
   599
  using assms by (auto simp: real_differentiable_def)
hoelzl@51642
   600
hoelzl@56181
   601
lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   602
  by (auto elim: real_differentiableE)
hoelzl@51642
   603
hoelzl@56181
   604
lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
hoelzl@56181
   605
  by (force simp add: real_differentiable_def)
hoelzl@51642
   606
hoelzl@56381
   607
lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56181
   608
  apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right LIM_zero_iff[symmetric, of _ D])
hoelzl@51642
   609
  apply (subst (2) tendsto_norm_zero_iff[symmetric])
hoelzl@51642
   610
  apply (rule filterlim_cong)
hoelzl@51642
   611
  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
hoelzl@51642
   612
  done
huffman@21164
   613
hoelzl@56181
   614
lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
hoelzl@56181
   615
  by (simp add: fun_eq_iff mult_commute)
huffman@21164
   616
hoelzl@56181
   617
subsection {* Derivatives *}
huffman@21164
   618
hoelzl@51642
   619
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
hoelzl@56381
   620
  by (simp add: DERIV_def)
huffman@21164
   621
hoelzl@56381
   622
lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
hoelzl@56181
   623
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
hoelzl@56181
   624
hoelzl@56381
   625
lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
hoelzl@56181
   626
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
huffman@21164
   627
hoelzl@56381
   628
lemma field_differentiable_add[derivative_intros]:
hoelzl@56381
   629
  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> 
hoelzl@56381
   630
    ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
hoelzl@56381
   631
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
hoelzl@56381
   632
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
lp15@56261
   633
lp15@56261
   634
corollary DERIV_add:
hoelzl@56181
   635
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   636
  ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
lp15@56261
   637
  by (rule field_differentiable_add)
lp15@56261
   638
hoelzl@56381
   639
lemma field_differentiable_minus[derivative_intros]:
hoelzl@56381
   640
  "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
hoelzl@56381
   641
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
hoelzl@56381
   642
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
huffman@21164
   643
lp15@56261
   644
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
   645
  by (rule field_differentiable_minus)
huffman@21164
   646
hoelzl@56381
   647
lemma field_differentiable_diff[derivative_intros]:
hoelzl@56381
   648
  "(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"
hoelzl@56381
   649
  by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
lp15@56261
   650
lp15@56261
   651
corollary DERIV_diff:
hoelzl@56181
   652
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   653
  ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
lp15@56261
   654
  by (rule field_differentiable_diff)
hoelzl@51642
   655
hoelzl@56181
   656
lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
hoelzl@56181
   657
  by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
huffman@21164
   658
lp15@56261
   659
corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
lp15@56261
   660
  by (rule DERIV_continuous)
lp15@56261
   661
lp15@56261
   662
lemma DERIV_continuous_on:
lp15@56261
   663
  "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D) (at x)) \<Longrightarrow> continuous_on s f"
lp15@56261
   664
  by (metis DERIV_continuous continuous_at_imp_continuous_on)
hoelzl@51642
   665
hoelzl@56181
   666
lemma DERIV_mult':
hoelzl@56181
   667
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
hoelzl@56181
   668
  ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
hoelzl@56181
   669
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   670
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164
   671
hoelzl@56381
   672
lemma DERIV_mult[derivative_intros]:
hoelzl@56181
   673
  "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   674
  ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
hoelzl@56181
   675
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   676
     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   677
hoelzl@51642
   678
text {* Derivative of linear multiplication *}
huffman@21164
   679
hoelzl@51642
   680
lemma DERIV_cmult:
hoelzl@56181
   681
  "(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
   682
  by (drule DERIV_mult' [OF DERIV_const], simp)
huffman@21164
   683
lp15@55967
   684
lemma DERIV_cmult_right:
hoelzl@56181
   685
  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
hoelzl@56381
   686
  using DERIV_cmult by (force simp add: mult_ac)
lp15@55967
   687
hoelzl@56181
   688
lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
hoelzl@51642
   689
  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
hoelzl@51642
   690
hoelzl@56181
   691
lemma DERIV_cdivide:
hoelzl@56181
   692
  "(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
   693
  using DERIV_cmult_right[of f D x s "1 / c"] by simp
huffman@21164
   694
huffman@21164
   695
lemma DERIV_unique:
hoelzl@51642
   696
  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
hoelzl@56381
   697
  unfolding DERIV_def by (rule LIM_unique) 
huffman@21164
   698
hoelzl@56381
   699
lemma DERIV_setsum[derivative_intros]:
hoelzl@56181
   700
  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 
hoelzl@56181
   701
    ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
hoelzl@56181
   702
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
hoelzl@56181
   703
     (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   704
hoelzl@56381
   705
lemma DERIV_inverse'[derivative_intros]:
hoelzl@56181
   706
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   707
  ((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
hoelzl@56181
   708
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_inverse])
hoelzl@56181
   709
     (auto dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   710
hoelzl@51642
   711
text {* Power of @{text "-1"} *}
hoelzl@51642
   712
hoelzl@51642
   713
lemma DERIV_inverse:
hoelzl@56181
   714
  "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642
   715
  by (drule DERIV_inverse' [OF DERIV_ident]) simp
hoelzl@51642
   716
hoelzl@51642
   717
text {* Derivative of inverse *}
hoelzl@51642
   718
hoelzl@51642
   719
lemma DERIV_inverse_fun:
hoelzl@56181
   720
  "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   721
  ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
hoelzl@51642
   722
  by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)
hoelzl@51642
   723
hoelzl@51642
   724
text {* Derivative of quotient *}
hoelzl@51642
   725
hoelzl@56381
   726
lemma DERIV_divide[derivative_intros]:
hoelzl@56181
   727
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   728
  (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
hoelzl@56181
   729
  ((\<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
   730
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
hoelzl@56181
   731
     (auto dest: has_field_derivative_imp_has_derivative simp: field_simps nonzero_inverse_mult_distrib divide_inverse)
hoelzl@51642
   732
hoelzl@51642
   733
lemma DERIV_quotient:
hoelzl@56181
   734
  "(f has_field_derivative d) (at x within s) \<Longrightarrow>
hoelzl@56181
   735
  (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 
hoelzl@56181
   736
  ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642
   737
  by (drule (2) DERIV_divide) (simp add: mult_commute)
hoelzl@51642
   738
hoelzl@51642
   739
lemma DERIV_power_Suc:
hoelzl@56181
   740
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   741
  ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
hoelzl@56181
   742
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   743
     (auto simp: has_field_derivative_def)
hoelzl@51642
   744
hoelzl@56381
   745
lemma DERIV_power[derivative_intros]:
hoelzl@56181
   746
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
hoelzl@56181
   747
  ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
hoelzl@56181
   748
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   749
     (auto simp: has_field_derivative_def)
hoelzl@31880
   750
hoelzl@56181
   751
lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
hoelzl@51642
   752
  apply (cut_tac DERIV_power [OF DERIV_ident])
hoelzl@51642
   753
  apply (simp add: real_of_nat_def)
hoelzl@51642
   754
  done
hoelzl@51642
   755
hoelzl@56181
   756
lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 
hoelzl@56181
   757
  ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
hoelzl@56181
   758
  using has_derivative_compose[of f "op * D" x s g "op * E"]
hoelzl@56181
   759
  unfolding has_field_derivative_def mult_commute_abs ac_simps .
hoelzl@51642
   760
hoelzl@56181
   761
corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   762
  ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
lp15@55967
   763
  by (rule DERIV_chain')
lp15@55967
   764
hoelzl@51642
   765
text {* Standard version *}
hoelzl@51642
   766
hoelzl@56181
   767
lemma DERIV_chain:
hoelzl@56181
   768
  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 
hoelzl@56181
   769
  (f o g has_field_derivative Da * Db) (at x within s)"
hoelzl@51642
   770
  by (drule (1) DERIV_chain', simp add: o_def mult_commute)
hoelzl@51642
   771
lp15@55967
   772
lemma DERIV_image_chain: 
hoelzl@56181
   773
  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   774
  (f o g has_field_derivative Da * Db) (at x within s)"
hoelzl@56181
   775
  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
hoelzl@56181
   776
  by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
lp15@55967
   777
lp15@55967
   778
(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
lp15@55967
   779
lemma DERIV_chain_s:
lp15@55967
   780
  assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
lp15@55967
   781
      and "DERIV f x :> f'" 
lp15@55967
   782
      and "f x \<in> s"
lp15@55967
   783
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967
   784
  by (metis (full_types) DERIV_chain' mult_commute assms)
lp15@55967
   785
lp15@55967
   786
lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
lp15@55967
   787
  assumes "(\<And>x. DERIV g x :> g'(x))"
lp15@55967
   788
      and "DERIV f x :> f'" 
lp15@55967
   789
    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967
   790
  by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
lp15@55967
   791
hoelzl@51642
   792
declare
hoelzl@56381
   793
  DERIV_power[where 'a=real, unfolded real_of_nat_def[symmetric], derivative_intros]
huffman@21164
   794
huffman@21164
   795
text{*Alternative definition for differentiability*}
huffman@21164
   796
huffman@21164
   797
lemma DERIV_LIM_iff:
huffman@31338
   798
  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
huffman@21784
   799
     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
huffman@21164
   800
      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"
huffman@21164
   801
apply (rule iffI)
huffman@21164
   802
apply (drule_tac k="- a" in LIM_offset)
haftmann@54230
   803
apply simp
huffman@21164
   804
apply (drule_tac k="a" in LIM_offset)
huffman@21164
   805
apply (simp add: add_commute)
huffman@21164
   806
done
huffman@21164
   807
hoelzl@51642
   808
lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
hoelzl@56381
   809
  by (simp add: DERIV_def DERIV_LIM_iff)
huffman@21164
   810
hoelzl@51642
   811
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@51642
   812
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
hoelzl@51642
   813
  unfolding DERIV_iff2
hoelzl@51642
   814
proof (rule filterlim_cong)
wenzelm@53374
   815
  assume *: "eventually (\<lambda>x. f x = g x) (nhds x)"
wenzelm@53374
   816
  moreover from * have "f x = g x" by (auto simp: eventually_nhds)
hoelzl@51642
   817
  moreover assume "x = y" "u = v"
hoelzl@51642
   818
  ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
hoelzl@51642
   819
    by (auto simp: eventually_at_filter elim: eventually_elim1)
hoelzl@51642
   820
qed simp_all
huffman@21164
   821
hoelzl@51642
   822
lemma DERIV_shift:
hoelzl@51642
   823
  "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
hoelzl@56381
   824
  by (simp add: DERIV_def field_simps)
huffman@21164
   825
hoelzl@51642
   826
lemma DERIV_mirror:
hoelzl@51642
   827
  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
hoelzl@56381
   828
  by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
hoelzl@51642
   829
                tendsto_minus_cancel_left field_simps conj_commute)
huffman@21164
   830
huffman@29975
   831
text {* Caratheodory formulation of derivative at a point *}
huffman@21164
   832
lp15@55970
   833
lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
hoelzl@51642
   834
  "(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
   835
      (is "?lhs = ?rhs")
huffman@21164
   836
proof
huffman@21164
   837
  assume der: "DERIV f x :> l"
huffman@21784
   838
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
huffman@21164
   839
  proof (intro exI conjI)
huffman@21784
   840
    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
nipkow@23413
   841
    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
huffman@21164
   842
    show "isCont ?g x" using der
hoelzl@56381
   843
      by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
huffman@21164
   844
    show "?g x = l" by simp
huffman@21164
   845
  qed
huffman@21164
   846
next
huffman@21164
   847
  assume "?rhs"
huffman@21164
   848
  then obtain g where
huffman@21784
   849
    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
huffman@21164
   850
  thus "(DERIV f x :> l)"
hoelzl@56381
   851
     by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
huffman@21164
   852
qed
huffman@21164
   853
wenzelm@31899
   854
text {*
wenzelm@31899
   855
 Let's do the standard proof, though theorem
wenzelm@31899
   856
 @{text "LIM_mult2"} follows from a NS proof
wenzelm@31899
   857
*}
huffman@21164
   858
huffman@29975
   859
subsection {* Local extrema *}
huffman@29975
   860
huffman@21164
   861
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
huffman@21164
   862
paulson@33654
   863
lemma DERIV_pos_inc_right:
huffman@21164
   864
  fixes f :: "real => real"
huffman@21164
   865
  assumes der: "DERIV f x :> l"
huffman@21164
   866
      and l:   "0 < l"
huffman@21164
   867
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
huffman@21164
   868
proof -
huffman@21164
   869
  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
huffman@21164
   870
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
haftmann@54230
   871
    by simp
huffman@21164
   872
  then obtain s
huffman@21164
   873
        where s:   "0 < s"
huffman@21164
   874
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
huffman@21164
   875
    by auto
huffman@21164
   876
  thus ?thesis
huffman@21164
   877
  proof (intro exI conjI strip)
huffman@23441
   878
    show "0<s" using s .
huffman@21164
   879
    fix h::real
huffman@21164
   880
    assume "0 < h" "h < s"
huffman@21164
   881
    with all [of h] show "f x < f (x+h)"
haftmann@54230
   882
    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
huffman@21164
   883
      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"
huffman@21164
   884
      with l
huffman@21164
   885
      have "0 < (f (x+h) - f x) / h" by arith
huffman@21164
   886
      thus "f x < f (x+h)"
huffman@21164
   887
  by (simp add: pos_less_divide_eq h)
huffman@21164
   888
    qed
huffman@21164
   889
  qed
huffman@21164
   890
qed
huffman@21164
   891
paulson@33654
   892
lemma DERIV_neg_dec_left:
huffman@21164
   893
  fixes f :: "real => real"
huffman@21164
   894
  assumes der: "DERIV f x :> l"
huffman@21164
   895
      and l:   "l < 0"
huffman@21164
   896
  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
huffman@21164
   897
proof -
huffman@21164
   898
  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
huffman@21164
   899
  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
haftmann@54230
   900
    by simp
huffman@21164
   901
  then obtain s
huffman@21164
   902
        where s:   "0 < s"
huffman@21164
   903
          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
huffman@21164
   904
    by auto
huffman@21164
   905
  thus ?thesis
huffman@21164
   906
  proof (intro exI conjI strip)
huffman@23441
   907
    show "0<s" using s .
huffman@21164
   908
    fix h::real
huffman@21164
   909
    assume "0 < h" "h < s"
huffman@21164
   910
    with all [of "-h"] show "f x < f (x-h)"
haftmann@54230
   911
    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)
huffman@21164
   912
      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"
huffman@21164
   913
      with l
huffman@21164
   914
      have "0 < (f (x-h) - f x) / h" by arith
huffman@21164
   915
      thus "f x < f (x-h)"
huffman@21164
   916
  by (simp add: pos_less_divide_eq h)
huffman@21164
   917
    qed
huffman@21164
   918
  qed
huffman@21164
   919
qed
huffman@21164
   920
paulson@33654
   921
lemma DERIV_pos_inc_left:
paulson@33654
   922
  fixes f :: "real => real"
paulson@33654
   923
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
hoelzl@56181
   924
  apply (rule DERIV_neg_dec_left [of "%x. - f x" "-l" x, simplified])
hoelzl@41368
   925
  apply (auto simp add: DERIV_minus)
paulson@33654
   926
  done
paulson@33654
   927
paulson@33654
   928
lemma DERIV_neg_dec_right:
paulson@33654
   929
  fixes f :: "real => real"
paulson@33654
   930
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
hoelzl@56181
   931
  apply (rule DERIV_pos_inc_right [of "%x. - f x" "-l" x, simplified])
hoelzl@41368
   932
  apply (auto simp add: DERIV_minus)
paulson@33654
   933
  done
paulson@33654
   934
huffman@21164
   935
lemma DERIV_local_max:
huffman@21164
   936
  fixes f :: "real => real"
huffman@21164
   937
  assumes der: "DERIV f x :> l"
huffman@21164
   938
      and d:   "0 < d"
huffman@21164
   939
      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
huffman@21164
   940
  shows "l = 0"
huffman@21164
   941
proof (cases rule: linorder_cases [of l 0])
huffman@23441
   942
  case equal thus ?thesis .
huffman@21164
   943
next
huffman@21164
   944
  case less
paulson@33654
   945
  from DERIV_neg_dec_left [OF der less]
huffman@21164
   946
  obtain d' where d': "0 < d'"
huffman@21164
   947
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
huffman@21164
   948
  from real_lbound_gt_zero [OF d d']
huffman@21164
   949
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   950
  with lt le [THEN spec [where x="x-e"]]
huffman@21164
   951
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   952
next
huffman@21164
   953
  case greater
paulson@33654
   954
  from DERIV_pos_inc_right [OF der greater]
huffman@21164
   955
  obtain d' where d': "0 < d'"
huffman@21164
   956
             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
huffman@21164
   957
  from real_lbound_gt_zero [OF d d']
huffman@21164
   958
  obtain e where "0 < e \<and> e < d \<and> e < d'" ..
huffman@21164
   959
  with lt le [THEN spec [where x="x+e"]]
huffman@21164
   960
  show ?thesis by (auto simp add: abs_if)
huffman@21164
   961
qed
huffman@21164
   962
huffman@21164
   963
huffman@21164
   964
text{*Similar theorem for a local minimum*}
huffman@21164
   965
lemma DERIV_local_min:
huffman@21164
   966
  fixes f :: "real => real"
huffman@21164
   967
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"
huffman@21164
   968
by (drule DERIV_minus [THEN DERIV_local_max], auto)
huffman@21164
   969
huffman@21164
   970
huffman@21164
   971
text{*In particular, if a function is locally flat*}
huffman@21164
   972
lemma DERIV_local_const:
huffman@21164
   973
  fixes f :: "real => real"
huffman@21164
   974
  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"
huffman@21164
   975
by (auto dest!: DERIV_local_max)
huffman@21164
   976
huffman@29975
   977
huffman@29975
   978
subsection {* Rolle's Theorem *}
huffman@29975
   979
huffman@21164
   980
text{*Lemma about introducing open ball in open interval*}
huffman@21164
   981
lemma lemma_interval_lt:
huffman@21164
   982
     "[| a < x;  x < b |]
huffman@21164
   983
      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
chaieb@27668
   984
huffman@22998
   985
apply (simp add: abs_less_iff)
huffman@21164
   986
apply (insert linorder_linear [of "x-a" "b-x"], safe)
huffman@21164
   987
apply (rule_tac x = "x-a" in exI)
huffman@21164
   988
apply (rule_tac [2] x = "b-x" in exI, auto)
huffman@21164
   989
done
huffman@21164
   990
huffman@21164
   991
lemma lemma_interval: "[| a < x;  x < b |] ==>
huffman@21164
   992
        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
huffman@21164
   993
apply (drule lemma_interval_lt, auto)
huffman@44921
   994
apply force
huffman@21164
   995
done
huffman@21164
   996
huffman@21164
   997
text{*Rolle's Theorem.
huffman@21164
   998
   If @{term f} is defined and continuous on the closed interval
huffman@21164
   999
   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
huffman@21164
  1000
   and @{term "f(a) = f(b)"},
huffman@21164
  1001
   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
huffman@21164
  1002
theorem Rolle:
huffman@21164
  1003
  assumes lt: "a < b"
huffman@21164
  1004
      and eq: "f(a) = f(b)"
huffman@21164
  1005
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181
  1006
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784
  1007
  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
huffman@21164
  1008
proof -
huffman@21164
  1009
  have le: "a \<le> b" using lt by simp
huffman@21164
  1010
  from isCont_eq_Ub [OF le con]
huffman@21164
  1011
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
huffman@21164
  1012
             and alex: "a \<le> x" and xleb: "x \<le> b"
huffman@21164
  1013
    by blast
huffman@21164
  1014
  from isCont_eq_Lb [OF le con]
huffman@21164
  1015
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
huffman@21164
  1016
              and alex': "a \<le> x'" and x'leb: "x' \<le> b"
huffman@21164
  1017
    by blast
huffman@21164
  1018
  show ?thesis
huffman@21164
  1019
  proof cases
huffman@21164
  1020
    assume axb: "a < x & x < b"
huffman@21164
  1021
        --{*@{term f} attains its maximum within the interval*}
chaieb@27668
  1022
    hence ax: "a<x" and xb: "x<b" by arith + 
huffman@21164
  1023
    from lemma_interval [OF ax xb]
huffman@21164
  1024
    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
  1025
      by blast
huffman@21164
  1026
    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
huffman@21164
  1027
      by blast
huffman@21164
  1028
    from differentiableD [OF dif [OF axb]]
huffman@21164
  1029
    obtain l where der: "DERIV f x :> l" ..
huffman@21164
  1030
    have "l=0" by (rule DERIV_local_max [OF der d bound'])
huffman@21164
  1031
        --{*the derivative at a local maximum is zero*}
huffman@21164
  1032
    thus ?thesis using ax xb der by auto
huffman@21164
  1033
  next
huffman@21164
  1034
    assume notaxb: "~ (a < x & x < b)"
huffman@21164
  1035
    hence xeqab: "x=a | x=b" using alex xleb by arith
huffman@21164
  1036
    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
huffman@21164
  1037
    show ?thesis
huffman@21164
  1038
    proof cases
huffman@21164
  1039
      assume ax'b: "a < x' & x' < b"
huffman@21164
  1040
        --{*@{term f} attains its minimum within the interval*}
chaieb@27668
  1041
      hence ax': "a<x'" and x'b: "x'<b" by arith+ 
huffman@21164
  1042
      from lemma_interval [OF ax' x'b]
huffman@21164
  1043
      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
  1044
  by blast
huffman@21164
  1045
      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
huffman@21164
  1046
  by blast
huffman@21164
  1047
      from differentiableD [OF dif [OF ax'b]]
huffman@21164
  1048
      obtain l where der: "DERIV f x' :> l" ..
huffman@21164
  1049
      have "l=0" by (rule DERIV_local_min [OF der d bound'])
huffman@21164
  1050
        --{*the derivative at a local minimum is zero*}
huffman@21164
  1051
      thus ?thesis using ax' x'b der by auto
huffman@21164
  1052
    next
huffman@21164
  1053
      assume notax'b: "~ (a < x' & x' < b)"
huffman@21164
  1054
        --{*@{term f} is constant througout the interval*}
huffman@21164
  1055
      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
huffman@21164
  1056
      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
huffman@21164
  1057
      from dense [OF lt]
huffman@21164
  1058
      obtain r where ar: "a < r" and rb: "r < b" by blast
huffman@21164
  1059
      from lemma_interval [OF ar rb]
huffman@21164
  1060
      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
  1061
  by blast
huffman@21164
  1062
      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
huffman@21164
  1063
      proof (clarify)
huffman@21164
  1064
        fix z::real
huffman@21164
  1065
        assume az: "a \<le> z" and zb: "z \<le> b"
huffman@21164
  1066
        show "f z = f b"
huffman@21164
  1067
        proof (rule order_antisym)
huffman@21164
  1068
          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
huffman@21164
  1069
          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
huffman@21164
  1070
        qed
huffman@21164
  1071
      qed
huffman@21164
  1072
      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1073
      proof (intro strip)
huffman@21164
  1074
        fix y::real
huffman@21164
  1075
        assume lt: "\<bar>r-y\<bar> < d"
huffman@21164
  1076
        hence "f y = f b" by (simp add: eq_fb bound)
huffman@21164
  1077
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
huffman@21164
  1078
      qed
huffman@21164
  1079
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
huffman@21164
  1080
      obtain l where der: "DERIV f r :> l" ..
huffman@21164
  1081
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
huffman@21164
  1082
        --{*the derivative of a constant function is zero*}
huffman@21164
  1083
      thus ?thesis using ar rb der by auto
huffman@21164
  1084
    qed
huffman@21164
  1085
  qed
huffman@21164
  1086
qed
huffman@21164
  1087
huffman@21164
  1088
huffman@21164
  1089
subsection{*Mean Value Theorem*}
huffman@21164
  1090
huffman@21164
  1091
lemma lemma_MVT:
huffman@21164
  1092
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
hoelzl@51481
  1093
  by (cases "a = b") (simp_all add: field_simps)
huffman@21164
  1094
huffman@21164
  1095
theorem MVT:
huffman@21164
  1096
  assumes lt:  "a < b"
huffman@21164
  1097
      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
hoelzl@56181
  1098
      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
huffman@21784
  1099
  shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
huffman@21164
  1100
                   (f(b) - f(a) = (b-a) * l)"
huffman@21164
  1101
proof -
huffman@21164
  1102
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
huffman@44233
  1103
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
hoelzl@56371
  1104
    using con by (fast intro: continuous_intros)
hoelzl@56181
  1105
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
huffman@21164
  1106
  proof (clarify)
huffman@21164
  1107
    fix x::real
huffman@21164
  1108
    assume ax: "a < x" and xb: "x < b"
huffman@21164
  1109
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
huffman@21164
  1110
    obtain l where der: "DERIV f x :> l" ..
hoelzl@56181
  1111
    show "?F differentiable (at x)"
huffman@21164
  1112
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
huffman@21164
  1113
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1114
  qed
huffman@21164
  1115
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
huffman@21164
  1116
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1117
    by blast
huffman@21164
  1118
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
huffman@21164
  1119
    by (rule DERIV_cmult_Id)
huffman@21164
  1120
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
huffman@21164
  1121
                   :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1122
    by (rule DERIV_add [OF der])
huffman@21164
  1123
  show ?thesis
huffman@21164
  1124
  proof (intro exI conjI)
huffman@23441
  1125
    show "a < z" using az .
huffman@23441
  1126
    show "z < b" using zb .
huffman@21164
  1127
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
huffman@21164
  1128
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
huffman@21164
  1129
  qed
huffman@21164
  1130
qed
huffman@21164
  1131
hoelzl@29803
  1132
lemma MVT2:
hoelzl@29803
  1133
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
hoelzl@29803
  1134
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
hoelzl@29803
  1135
apply (drule MVT)
hoelzl@29803
  1136
apply (blast intro: DERIV_isCont)
hoelzl@56181
  1137
apply (force dest: order_less_imp_le simp add: real_differentiable_def)
hoelzl@29803
  1138
apply (blast dest: DERIV_unique order_less_imp_le)
hoelzl@29803
  1139
done
hoelzl@29803
  1140
huffman@21164
  1141
huffman@21164
  1142
text{*A function is constant if its derivative is 0 over an interval.*}
huffman@21164
  1143
huffman@21164
  1144
lemma DERIV_isconst_end:
huffman@21164
  1145
  fixes f :: "real => real"
huffman@21164
  1146
  shows "[| a < b;
huffman@21164
  1147
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1148
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1149
        ==> f b = f a"
huffman@21164
  1150
apply (drule MVT, assumption)
huffman@21164
  1151
apply (blast intro: differentiableI)
huffman@21164
  1152
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
huffman@21164
  1153
done
huffman@21164
  1154
huffman@21164
  1155
lemma DERIV_isconst1:
huffman@21164
  1156
  fixes f :: "real => real"
huffman@21164
  1157
  shows "[| a < b;
huffman@21164
  1158
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1159
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
huffman@21164
  1160
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
huffman@21164
  1161
apply safe
huffman@21164
  1162
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
huffman@21164
  1163
apply (drule_tac b = x in DERIV_isconst_end, auto)
huffman@21164
  1164
done
huffman@21164
  1165
huffman@21164
  1166
lemma DERIV_isconst2:
huffman@21164
  1167
  fixes f :: "real => real"
huffman@21164
  1168
  shows "[| a < b;
huffman@21164
  1169
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
huffman@21164
  1170
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
huffman@21164
  1171
         a \<le> x; x \<le> b |]
huffman@21164
  1172
        ==> f x = f a"
huffman@21164
  1173
apply (blast dest: DERIV_isconst1)
huffman@21164
  1174
done
huffman@21164
  1175
hoelzl@29803
  1176
lemma DERIV_isconst3: fixes a b x y :: real
hoelzl@29803
  1177
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
hoelzl@29803
  1178
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1179
  shows "f x = f y"
hoelzl@29803
  1180
proof (cases "x = y")
hoelzl@29803
  1181
  case False
hoelzl@29803
  1182
  let ?a = "min x y"
hoelzl@29803
  1183
  let ?b = "max x y"
hoelzl@29803
  1184
  
hoelzl@29803
  1185
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1186
  proof (rule allI, rule impI)
hoelzl@29803
  1187
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hoelzl@29803
  1188
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hoelzl@29803
  1189
    hence "z \<in> {a<..<b}" by auto
hoelzl@29803
  1190
    thus "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1191
  qed
hoelzl@29803
  1192
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
hoelzl@29803
  1193
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
hoelzl@29803
  1194
hoelzl@29803
  1195
  have "?a < ?b" using `x \<noteq> y` by auto
hoelzl@29803
  1196
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1197
  show ?thesis by auto
hoelzl@29803
  1198
qed auto
hoelzl@29803
  1199
huffman@21164
  1200
lemma DERIV_isconst_all:
huffman@21164
  1201
  fixes f :: "real => real"
huffman@21164
  1202
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
huffman@21164
  1203
apply (rule linorder_cases [of x y])
huffman@21164
  1204
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
huffman@21164
  1205
done
huffman@21164
  1206
huffman@21164
  1207
lemma DERIV_const_ratio_const:
huffman@21784
  1208
  fixes f :: "real => real"
huffman@21784
  1209
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
huffman@21164
  1210
apply (rule linorder_cases [of a b], auto)
huffman@21164
  1211
apply (drule_tac [!] f = f in MVT)
hoelzl@56181
  1212
apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
haftmann@54230
  1213
apply (auto dest: DERIV_unique simp add: ring_distribs)
huffman@21164
  1214
done
huffman@21164
  1215
huffman@21164
  1216
lemma DERIV_const_ratio_const2:
huffman@21784
  1217
  fixes f :: "real => real"
huffman@21784
  1218
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
lp15@56217
  1219
apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
huffman@21164
  1220
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
huffman@21164
  1221
done
huffman@21164
  1222
huffman@21164
  1223
lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
huffman@21164
  1224
by (simp)
huffman@21164
  1225
huffman@21164
  1226
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
huffman@21164
  1227
by (simp)
huffman@21164
  1228
huffman@21164
  1229
text{*Gallileo's "trick": average velocity = av. of end velocities*}
huffman@21164
  1230
huffman@21164
  1231
lemma DERIV_const_average:
huffman@21164
  1232
  fixes v :: "real => real"
huffman@21164
  1233
  assumes neq: "a \<noteq> (b::real)"
huffman@21164
  1234
      and der: "\<forall>x. DERIV v x :> k"
huffman@21164
  1235
  shows "v ((a + b)/2) = (v a + v b)/2"
huffman@21164
  1236
proof (cases rule: linorder_cases [of a b])
huffman@21164
  1237
  case equal with neq show ?thesis by simp
huffman@21164
  1238
next
huffman@21164
  1239
  case less
huffman@21164
  1240
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1241
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1242
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1243
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
huffman@21164
  1244
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1245
  ultimately show ?thesis using neq by force
huffman@21164
  1246
next
huffman@21164
  1247
  case greater
huffman@21164
  1248
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1249
    by (rule DERIV_const_ratio_const2 [OF neq der])
huffman@21164
  1250
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
huffman@21164
  1251
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
huffman@21164
  1252
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
huffman@21164
  1253
  ultimately show ?thesis using neq by (force simp add: add_commute)
huffman@21164
  1254
qed
huffman@21164
  1255
paulson@33654
  1256
(* A function with positive derivative is increasing. 
paulson@33654
  1257
   A simple proof using the MVT, by Jeremy Avigad. And variants.
paulson@33654
  1258
*)
lp15@56261
  1259
lemma DERIV_pos_imp_increasing_open:
paulson@33654
  1260
  fixes a::real and b::real and f::"real => real"
lp15@56261
  1261
  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
lp15@56261
  1262
      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
paulson@33654
  1263
  shows "f a < f b"
paulson@33654
  1264
proof (rule ccontr)
wenzelm@41550
  1265
  assume f: "~ f a < f b"
wenzelm@33690
  1266
  have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1267
      & f b - f a = (b - a) * l"
wenzelm@33690
  1268
    apply (rule MVT)
lp15@56261
  1269
      using assms Deriv.differentiableI
lp15@56261
  1270
      apply force+
wenzelm@33690
  1271
    done
wenzelm@41550
  1272
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
paulson@33654
  1273
      and "f b - f a = (b - a) * l"
paulson@33654
  1274
    by auto
wenzelm@41550
  1275
  with assms f have "~(l > 0)"
huffman@36777
  1276
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550
  1277
  with assms z show False
lp15@56261
  1278
    by (metis DERIV_unique)
paulson@33654
  1279
qed
paulson@33654
  1280
lp15@56261
  1281
lemma DERIV_pos_imp_increasing:
lp15@56261
  1282
  fixes a::real and b::real and f::"real => real"
lp15@56261
  1283
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
lp15@56261
  1284
  shows "f a < f b"
lp15@56261
  1285
by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
lp15@56261
  1286
noschinl@45791
  1287
lemma DERIV_nonneg_imp_nondecreasing:
paulson@33654
  1288
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1289
  assumes "a \<le> b" and
paulson@33654
  1290
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
paulson@33654
  1291
  shows "f a \<le> f b"
paulson@33654
  1292
proof (rule ccontr, cases "a = b")
wenzelm@41550
  1293
  assume "~ f a \<le> f b" and "a = b"
wenzelm@41550
  1294
  then show False by auto
haftmann@37891
  1295
next
haftmann@37891
  1296
  assume A: "~ f a \<le> f b"
haftmann@37891
  1297
  assume B: "a ~= b"
paulson@33654
  1298
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
paulson@33654
  1299
      & f b - f a = (b - a) * l"
wenzelm@33690
  1300
    apply -
wenzelm@33690
  1301
    apply (rule MVT)
wenzelm@33690
  1302
      apply auto
wenzelm@33690
  1303
      apply (metis DERIV_isCont)
huffman@36777
  1304
     apply (metis differentiableI less_le)
paulson@33654
  1305
    done
wenzelm@41550
  1306
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
haftmann@37891
  1307
      and C: "f b - f a = (b - a) * l"
paulson@33654
  1308
    by auto
haftmann@37891
  1309
  with A have "a < b" "f b < f a" by auto
haftmann@37891
  1310
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
huffman@45051
  1311
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@41550
  1312
  with assms z show False
paulson@33654
  1313
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1314
qed
paulson@33654
  1315
lp15@56261
  1316
lemma DERIV_neg_imp_decreasing_open:
lp15@56261
  1317
  fixes a::real and b::real and f::"real => real"
lp15@56261
  1318
  assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
lp15@56261
  1319
      and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
lp15@56261
  1320
  shows "f a > f b"
lp15@56261
  1321
proof -
lp15@56261
  1322
  have "(%x. -f x) a < (%x. -f x) b"
lp15@56261
  1323
    apply (rule DERIV_pos_imp_increasing_open [of a b "%x. -f x"])
lp15@56261
  1324
    using assms
lp15@56261
  1325
    apply auto
lp15@56261
  1326
    apply (metis field_differentiable_minus neg_0_less_iff_less)
lp15@56261
  1327
    done
lp15@56261
  1328
  thus ?thesis
lp15@56261
  1329
    by simp
lp15@56261
  1330
qed
lp15@56261
  1331
paulson@33654
  1332
lemma DERIV_neg_imp_decreasing:
paulson@33654
  1333
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1334
  assumes "a < b" and
paulson@33654
  1335
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
paulson@33654
  1336
  shows "f a > f b"
lp15@56261
  1337
by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
paulson@33654
  1338
paulson@33654
  1339
lemma DERIV_nonpos_imp_nonincreasing:
paulson@33654
  1340
  fixes a::real and b::real and f::"real => real"
paulson@33654
  1341
  assumes "a \<le> b" and
paulson@33654
  1342
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
paulson@33654
  1343
  shows "f a \<ge> f b"
paulson@33654
  1344
proof -
paulson@33654
  1345
  have "(%x. -f x) a \<le> (%x. -f x) b"
noschinl@45791
  1346
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
wenzelm@33690
  1347
    using assms
wenzelm@33690
  1348
    apply auto
paulson@33654
  1349
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1350
    done
paulson@33654
  1351
  thus ?thesis
paulson@33654
  1352
    by simp
paulson@33654
  1353
qed
huffman@21164
  1354
lp15@56289
  1355
lemma DERIV_pos_imp_increasing_at_bot:
lp15@56289
  1356
  fixes f :: "real => real"
lp15@56289
  1357
  assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
lp15@56289
  1358
      and lim: "(f ---> flim) at_bot"
lp15@56289
  1359
  shows "flim < f b"
lp15@56289
  1360
proof -
lp15@56289
  1361
  have "flim \<le> f (b - 1)"
lp15@56289
  1362
    apply (rule tendsto_ge_const [OF _ lim])
lp15@56289
  1363
    apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
lp15@56289
  1364
    apply (rule_tac x="b - 2" in exI)
lp15@56289
  1365
    apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289
  1366
    done
lp15@56289
  1367
  also have "... < f b"
lp15@56289
  1368
    by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289
  1369
  finally show ?thesis .
lp15@56289
  1370
qed
lp15@56289
  1371
lp15@56289
  1372
lemma DERIV_neg_imp_decreasing_at_top:
lp15@56289
  1373
  fixes f :: "real => real"
lp15@56289
  1374
  assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
lp15@56289
  1375
      and lim: "(f ---> flim) at_top"
lp15@56289
  1376
  shows "flim < f b"
lp15@56289
  1377
  apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
lp15@56289
  1378
  apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
lp15@56289
  1379
  apply (metis filterlim_at_top_mirror lim)
lp15@56289
  1380
  done
lp15@56289
  1381
huffman@23041
  1382
text {* Derivative of inverse function *}
huffman@23041
  1383
huffman@23041
  1384
lemma DERIV_inverse_function:
huffman@23041
  1385
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1386
  assumes der: "DERIV f (g x) :> D"
huffman@23041
  1387
  assumes neq: "D \<noteq> 0"
huffman@23044
  1388
  assumes a: "a < x" and b: "x < b"
huffman@23044
  1389
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
huffman@23041
  1390
  assumes cont: "isCont g x"
huffman@23041
  1391
  shows "DERIV g x :> inverse D"
huffman@23041
  1392
unfolding DERIV_iff2
huffman@23044
  1393
proof (rule LIM_equal2)
huffman@23044
  1394
  show "0 < min (x - a) (b - x)"
chaieb@27668
  1395
    using a b by arith 
huffman@23044
  1396
next
huffman@23041
  1397
  fix y
huffman@23044
  1398
  assume "norm (y - x) < min (x - a) (b - x)"
chaieb@27668
  1399
  hence "a < y" and "y < b" 
huffman@23044
  1400
    by (simp_all add: abs_less_iff)
huffman@23041
  1401
  thus "(g y - g x) / (y - x) =
huffman@23041
  1402
        inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1403
    by (simp add: inj)
huffman@23041
  1404
next
huffman@23041
  1405
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
huffman@23041
  1406
    by (rule der [unfolded DERIV_iff2])
huffman@23041
  1407
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
huffman@23044
  1408
    using inj a b by simp
huffman@23041
  1409
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
wenzelm@56219
  1410
  proof (rule exI, safe)
huffman@23044
  1411
    show "0 < min (x - a) (b - x)"
huffman@23044
  1412
      using a b by simp
huffman@23041
  1413
  next
huffman@23041
  1414
    fix y
huffman@23044
  1415
    assume "norm (y - x) < min (x - a) (b - x)"
huffman@23044
  1416
    hence y: "a < y" "y < b"
huffman@23044
  1417
      by (simp_all add: abs_less_iff)
huffman@23041
  1418
    assume "g y = g x"
huffman@23041
  1419
    hence "f (g y) = f (g x)" by simp
huffman@23044
  1420
    hence "y = x" using inj y a b by simp
huffman@23041
  1421
    also assume "y \<noteq> x"
huffman@23041
  1422
    finally show False by simp
huffman@23041
  1423
  qed
huffman@23041
  1424
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
huffman@23041
  1425
    using cont 1 2 by (rule isCont_LIM_compose2)
huffman@23041
  1426
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
huffman@23041
  1427
        -- x --> inverse D"
huffman@44568
  1428
    using neq by (rule tendsto_inverse)
huffman@23041
  1429
qed
huffman@23041
  1430
huffman@29975
  1431
subsection {* Generalized Mean Value Theorem *}
huffman@29975
  1432
huffman@21164
  1433
theorem GMVT:
huffman@21784
  1434
  fixes a b :: real
huffman@21164
  1435
  assumes alb: "a < b"
wenzelm@41550
  1436
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
hoelzl@56181
  1437
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@41550
  1438
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
hoelzl@56181
  1439
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
wenzelm@53381
  1440
  shows "\<exists>g'c f'c c.
wenzelm@53381
  1441
    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
  1442
proof -
huffman@21164
  1443
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
wenzelm@41550
  1444
  from assms have "a < b" by simp
huffman@21164
  1445
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
huffman@44233
  1446
    using fc gc by simp
hoelzl@56181
  1447
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
huffman@44233
  1448
    using fd gd by simp
huffman@21164
  1449
  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
  1450
  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
  1451
  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
  1452
huffman@21164
  1453
  from cdef have cint: "a < c \<and> c < b" by auto
hoelzl@56181
  1454
  with gd have "g differentiable (at c)" by simp
huffman@21164
  1455
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
huffman@21164
  1456
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
huffman@21164
  1457
huffman@21164
  1458
  from cdef have "a < c \<and> c < b" by auto
hoelzl@56181
  1459
  with fd have "f differentiable (at c)" by simp
huffman@21164
  1460
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
huffman@21164
  1461
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
huffman@21164
  1462
huffman@21164
  1463
  from cdef have "DERIV ?h c :> l" by auto
hoelzl@41368
  1464
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
hoelzl@56381
  1465
    using g'cdef f'cdef by (auto intro!: derivative_eq_intros)
huffman@21164
  1466
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1467
huffman@21164
  1468
  {
huffman@21164
  1469
    from cdef have "?h b - ?h a = (b - a) * l" by auto
wenzelm@53374
  1470
    also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1471
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
huffman@21164
  1472
  }
huffman@21164
  1473
  moreover
huffman@21164
  1474
  {
huffman@21164
  1475
    have "?h b - ?h a =
huffman@21164
  1476
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
huffman@21164
  1477
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1478
      by (simp add: algebra_simps)
huffman@21164
  1479
    hence "?h b - ?h a = 0" by auto
huffman@21164
  1480
  }
huffman@21164
  1481
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1482
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
huffman@21164
  1483
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
huffman@21164
  1484
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)
huffman@21164
  1485
huffman@21164
  1486
  with g'cdef f'cdef cint show ?thesis by auto
huffman@21164
  1487
qed
huffman@21164
  1488
hoelzl@50327
  1489
lemma GMVT':
hoelzl@50327
  1490
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1491
  assumes "a < b"
hoelzl@50327
  1492
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
hoelzl@50327
  1493
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
hoelzl@50327
  1494
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
hoelzl@50327
  1495
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327
  1496
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1497
proof -
hoelzl@50327
  1498
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
hoelzl@50327
  1499
    a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@56181
  1500
    using assms by (intro GMVT) (force simp: real_differentiable_def)+
hoelzl@50327
  1501
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1502
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327
  1503
  then show ?thesis
hoelzl@50327
  1504
    by auto
hoelzl@50327
  1505
qed
hoelzl@50327
  1506
hoelzl@51529
  1507
hoelzl@51529
  1508
subsection {* L'Hopitals rule *}
hoelzl@51529
  1509
hoelzl@51641
  1510
lemma isCont_If_ge:
hoelzl@51641
  1511
  fixes a :: "'a :: linorder_topology"
hoelzl@51641
  1512
  shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
hoelzl@51641
  1513
  unfolding isCont_def continuous_within
hoelzl@51641
  1514
  apply (intro filterlim_split_at)
hoelzl@51641
  1515
  apply (subst filterlim_cong[OF refl refl, where g=g])
hoelzl@51641
  1516
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1517
  apply (subst filterlim_cong[OF refl refl, where g=f])
hoelzl@51641
  1518
  apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1519
  done
hoelzl@51641
  1520
hoelzl@50327
  1521
lemma lhopital_right_0:
hoelzl@50329
  1522
  fixes f0 g0 :: "real \<Rightarrow> real"
hoelzl@50329
  1523
  assumes f_0: "(f0 ---> 0) (at_right 0)"
hoelzl@50329
  1524
  assumes g_0: "(g0 ---> 0) (at_right 0)"
hoelzl@50327
  1525
  assumes ev:
hoelzl@50329
  1526
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1527
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50329
  1528
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
hoelzl@50329
  1529
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
hoelzl@50327
  1530
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50329
  1531
  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
hoelzl@50327
  1532
proof -
hoelzl@50329
  1533
  def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
hoelzl@50329
  1534
  then have "f 0 = 0" by simp
hoelzl@50329
  1535
hoelzl@50329
  1536
  def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
hoelzl@50329
  1537
  then have "g 0 = 0" by simp
hoelzl@50329
  1538
hoelzl@50329
  1539
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329
  1540
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329
  1541
    using ev by eventually_elim auto
hoelzl@50329
  1542
  then obtain a where [arith]: "0 < a"
hoelzl@50329
  1543
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327
  1544
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329
  1545
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329
  1546
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
wenzelm@56219
  1547
    unfolding eventually_at by (auto simp: dist_real_def)
hoelzl@50327
  1548
hoelzl@50329
  1549
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329
  1550
    using g0_neq_0 by (simp add: g_def)
hoelzl@50329
  1551
hoelzl@50329
  1552
  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
hoelzl@50329
  1553
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
hoelzl@50329
  1554
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1555
  note f = this
hoelzl@50329
  1556
hoelzl@50329
  1557
  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
hoelzl@50329
  1558
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
hoelzl@50329
  1559
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
hoelzl@50329
  1560
  note g = this
hoelzl@50329
  1561
hoelzl@50329
  1562
  have "isCont f 0"
hoelzl@51641
  1563
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
hoelzl@51641
  1564
hoelzl@50329
  1565
  have "isCont g 0"
hoelzl@51641
  1566
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
hoelzl@50329
  1567
hoelzl@50327
  1568
  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
  1569
  proof (rule bchoice, rule)
hoelzl@50327
  1570
    fix x assume "x \<in> {0 <..< a}"
hoelzl@50327
  1571
    then have x[arith]: "0 < x" "x < a" by auto
hoelzl@50327
  1572
    with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
hoelzl@50327
  1573
      by auto
hoelzl@50328
  1574
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
hoelzl@50328
  1575
      using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1576
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
hoelzl@50328
  1577
      using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1578
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
hoelzl@50328
  1579
      using f g `x < a` by (intro GMVT') auto
wenzelm@53374
  1580
    then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@53374
  1581
      by blast
hoelzl@50327
  1582
    moreover
wenzelm@53374
  1583
    from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327
  1584
      by (simp add: field_simps)
hoelzl@50327
  1585
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
hoelzl@50327
  1586
      using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
hoelzl@50327
  1587
  qed
wenzelm@53381
  1588
  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
  1589
  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
  1590
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327
  1591
  moreover
hoelzl@50327
  1592
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327
  1593
    by eventually_elim auto
hoelzl@50327
  1594
  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
hoelzl@50327
  1595
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
hoelzl@51641
  1596
       (auto intro: tendsto_const tendsto_ident_at)
hoelzl@50327
  1597
  then have "(\<zeta> ---> 0) (at_right 0)"
hoelzl@50327
  1598
    by (rule tendsto_norm_zero_cancel)
hoelzl@50327
  1599
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
hoelzl@51641
  1600
    by (auto elim!: eventually_elim1 simp: filterlim_at)
hoelzl@50327
  1601
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
hoelzl@50327
  1602
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
hoelzl@50329
  1603
  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
hoelzl@50328
  1604
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
hoelzl@50328
  1605
       (auto elim: eventually_elim1)
hoelzl@50329
  1606
  also have "?P \<longleftrightarrow> ?thesis"
hoelzl@51641
  1607
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
hoelzl@50329
  1608
  finally show ?thesis .
hoelzl@50327
  1609
qed
hoelzl@50327
  1610
hoelzl@50330
  1611
lemma lhopital_right:
hoelzl@50330
  1612
  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1613
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1614
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1615
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1616
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1617
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1618
  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1619
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1620
  by (rule lhopital_right_0)
hoelzl@50330
  1621
hoelzl@50330
  1622
lemma lhopital_left:
hoelzl@50330
  1623
  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1624
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1625
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1626
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1627
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1628
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1629
  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1630
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1631
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1632
hoelzl@50330
  1633
lemma lhopital:
hoelzl@50330
  1634
  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1635
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1636
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1637
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1638
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1639
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1640
  ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1641
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1642
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330
  1643
hoelzl@50327
  1644
lemma lhopital_right_0_at_top:
hoelzl@50327
  1645
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1646
  assumes g_0: "LIM x at_right 0. g x :> at_top"
hoelzl@50327
  1647
  assumes ev:
hoelzl@50327
  1648
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
hoelzl@50327
  1649
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
hoelzl@50327
  1650
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
hoelzl@50327
  1651
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
hoelzl@50327
  1652
  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
hoelzl@50327
  1653
  unfolding tendsto_iff
hoelzl@50327
  1654
proof safe
hoelzl@50327
  1655
  fix e :: real assume "0 < e"
hoelzl@50327
  1656
hoelzl@50327
  1657
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
hoelzl@50327
  1658
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
hoelzl@50327
  1659
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327
  1660
  obtain a where [arith]: "0 < a"
hoelzl@50327
  1661
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327
  1662
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327
  1663
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327
  1664
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@51641
  1665
    unfolding eventually_at_le by (auto simp: dist_real_def)
hoelzl@51641
  1666
    
hoelzl@50327
  1667
hoelzl@50327
  1668
  from Df have
hoelzl@50328
  1669
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@51641
  1670
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327
  1671
hoelzl@50327
  1672
  moreover
hoelzl@50328
  1673
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
hoelzl@50346
  1674
    using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
hoelzl@50327
  1675
hoelzl@50327
  1676
  moreover
hoelzl@50327
  1677
  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
hoelzl@50327
  1678
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327
  1679
    by (rule filterlim_compose)
hoelzl@50327
  1680
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327
  1681
    by (intro tendsto_intros)
hoelzl@50327
  1682
  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
hoelzl@50327
  1683
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1684
  from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327
  1685
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
hoelzl@50327
  1686
    by (auto elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50327
  1687
hoelzl@50327
  1688
  moreover
hoelzl@50327
  1689
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
hoelzl@50327
  1690
    by (intro tendsto_intros)
hoelzl@50327
  1691
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
hoelzl@50327
  1692
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1693
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
hoelzl@50327
  1694
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327
  1695
    by (auto simp: dist_real_def)
hoelzl@50327
  1696
hoelzl@50327
  1697
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327
  1698
  proof eventually_elim
hoelzl@50327
  1699
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327
  1700
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327
  1701
hoelzl@50327
  1702
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327
  1703
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
wenzelm@53381
  1704
    then obtain y where [arith]: "t < y" "y < a"
wenzelm@53381
  1705
      and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
wenzelm@53381
  1706
      by blast
wenzelm@53381
  1707
    from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
hoelzl@50327
  1708
      using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327
  1709
hoelzl@50327
  1710
    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
  1711
      by (simp add: field_simps)
hoelzl@50327
  1712
    have "norm (f t / g t - x) \<le>
hoelzl@50327
  1713
        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
  1714
      unfolding * by (rule norm_triangle_ineq)
hoelzl@50327
  1715
    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
  1716
      by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327
  1717
    also have "\<dots> < (e / 4) * 2 + e / 2"
hoelzl@50327
  1718
      using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
hoelzl@50327
  1719
    finally show "dist (f t / g t) x < e"
hoelzl@50327
  1720
      by (simp add: dist_real_def)
hoelzl@50327
  1721
  qed
hoelzl@50327
  1722
qed
hoelzl@50327
  1723
hoelzl@50330
  1724
lemma lhopital_right_at_top:
hoelzl@50330
  1725
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1726
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1727
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1728
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1729
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
hoelzl@50330
  1730
    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
hoelzl@50330
  1731
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1732
  by (rule lhopital_right_0_at_top)
hoelzl@50330
  1733
hoelzl@50330
  1734
lemma lhopital_left_at_top:
hoelzl@50330
  1735
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1736
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1737
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1738
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1739
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
hoelzl@50330
  1740
    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
hoelzl@50330
  1741
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@50330
  1742
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1743
hoelzl@50330
  1744
lemma lhopital_at_top:
hoelzl@50330
  1745
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1746
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1747
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1748
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
hoelzl@50330
  1749
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
hoelzl@50330
  1750
    ((\<lambda> x. f x / g x) ---> y) (at x)"
hoelzl@50330
  1751
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1752
  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
  1753
hoelzl@50347
  1754
lemma lhospital_at_top_at_top:
hoelzl@50347
  1755
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50347
  1756
  assumes g_0: "LIM x at_top. g x :> at_top"
hoelzl@50347
  1757
  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
hoelzl@50347
  1758
  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
hoelzl@50347
  1759
  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
hoelzl@50347
  1760
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
hoelzl@50347
  1761
  shows "((\<lambda> x. f x / g x) ---> x) at_top"
hoelzl@50347
  1762
  unfolding filterlim_at_top_to_right
hoelzl@50347
  1763
proof (rule lhopital_right_0_at_top)
hoelzl@50347
  1764
  let ?F = "\<lambda>x. f (inverse x)"
hoelzl@50347
  1765
  let ?G = "\<lambda>x. g (inverse x)"
hoelzl@50347
  1766
  let ?R = "at_right (0::real)"
hoelzl@50347
  1767
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
hoelzl@50347
  1768
hoelzl@50347
  1769
  show "LIM x ?R. ?G x :> at_top"
hoelzl@50347
  1770
    using g_0 unfolding filterlim_at_top_to_right .
hoelzl@50347
  1771
hoelzl@50347
  1772
  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
hoelzl@50347
  1773
    unfolding eventually_at_right_to_top
hoelzl@50347
  1774
    using Dg eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1775
    apply eventually_elim
hoelzl@50347
  1776
    apply (rule DERIV_cong)
hoelzl@50347
  1777
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1778
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1779
    done
hoelzl@50347
  1780
hoelzl@50347
  1781
  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
hoelzl@50347
  1782
    unfolding eventually_at_right_to_top
hoelzl@50347
  1783
    using Df eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1784
    apply eventually_elim
hoelzl@50347
  1785
    apply (rule DERIV_cong)
hoelzl@50347
  1786
    apply (rule DERIV_chain'[where f=inverse])
hoelzl@50347
  1787
    apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1788
    done
hoelzl@50347
  1789
hoelzl@50347
  1790
  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
hoelzl@50347
  1791
    unfolding eventually_at_right_to_top
hoelzl@50347
  1792
    using g' eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1793
    by eventually_elim auto
hoelzl@50347
  1794
    
hoelzl@50347
  1795
  show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
hoelzl@50347
  1796
    unfolding filterlim_at_right_to_top
hoelzl@50347
  1797
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
hoelzl@50347
  1798
    using eventually_ge_at_top[where c="1::real"]
hoelzl@50347
  1799
    by eventually_elim simp
hoelzl@50347
  1800
qed
hoelzl@50347
  1801
huffman@21164
  1802
end