src/HOL/Deriv.thy
author wenzelm
Sat Nov 04 15:24:40 2017 +0100 (19 months ago)
changeset 67003 49850a679c2c
parent 64272 f76b6dda2e56
child 67149 e61557884799
permissions -rw-r--r--
more robust sorted_entries;
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(*  Title:      HOL/Deriv.thy
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    Author:     Jacques D. Fleuriot, University of Cambridge, 1998
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    Author:     Brian Huffman
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    Author:     Lawrence C Paulson, 2004
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    Author:     Benjamin Porter, 2005
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*)
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section \<open>Differentiation\<close>
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theory Deriv
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  imports Limits
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begin
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subsection \<open>Frechet derivative\<close>
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definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow>
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    ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool"  (infix "(has'_derivative)" 50)
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  where "(f has_derivative f') F \<longleftrightarrow>
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    bounded_linear f' \<and>
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    ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F"
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text \<open>
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  Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)
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  (at x within s)"} means: @{term D} is the derivative of function @{term f} at point @{term x}
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  within the set @{term s}. Where @{term s} is used to express left or right sided derivatives. In
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  most cases @{term s} is either a variable or @{term UNIV}.
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\<close>
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lemma has_derivative_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
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  by simp
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definition 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 "(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 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 "(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:
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  "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
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  by simp
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named_theorems derivative_intros "structural introduction rules for derivatives"
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setup \<open>
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  let
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    val eq_thms = @{thms has_derivative_eq_rhs DERIV_cong has_vector_derivative_eq_rhs}
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    fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms
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  in
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    Global_Theory.add_thms_dynamic
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      (@{binding derivative_eq_intros},
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        fn context =>
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          Named_Theorems.get (Context.proof_of context) @{named_theorems derivative_intros}
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          |> map_filter eq_rule)
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  end;
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\<close>
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text \<open>
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  The following syntax is only used as a legacy syntax.
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\<close>
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abbreviation (input)
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  FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
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  ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
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  where "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
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lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
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  by (simp add: has_derivative_def)
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lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
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  using bounded_linear.linear[OF has_derivative_bounded_linear] .
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lemma has_derivative_ident[derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
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  by (simp add: has_derivative_def)
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lemma has_derivative_id [derivative_intros, simp]: "(id has_derivative id) (at a)"
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  by (metis eq_id_iff has_derivative_ident)
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lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"
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  by (simp add: has_derivative_def)
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lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..
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lemma (in bounded_linear) has_derivative:
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  "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
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  unfolding has_derivative_def
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  apply safe
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   apply (erule bounded_linear_compose [OF bounded_linear])
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  apply (drule tendsto)
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  apply (simp add: scaleR diff add zero)
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  done
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lemmas has_derivative_scaleR_right [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_scaleR_right]
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lemmas has_derivative_scaleR_left [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_scaleR_left]
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lemmas has_derivative_mult_right [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_mult_right]
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lemmas has_derivative_mult_left [derivative_intros] =
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  bounded_linear.has_derivative [OF bounded_linear_mult_left]
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lemma has_derivative_add[simp, derivative_intros]:
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  assumes f: "(f has_derivative f') F"
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    and g: "(g has_derivative g') F"
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  shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
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  unfolding has_derivative_def
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proof safe
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  let ?x = "Lim F (\<lambda>x. x)"
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  let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
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  have "((\<lambda>x. ?D f f' x + ?D g g' x) \<longlongrightarrow> (0 + 0)) F"
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    using f g by (intro tendsto_add) (auto simp: has_derivative_def)
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  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) \<longlongrightarrow> 0) F"
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    by (simp add: field_simps scaleR_add_right scaleR_diff_right)
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qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
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lemma has_derivative_sum[simp, derivative_intros]:
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  "(\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F) \<Longrightarrow>
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    ((\<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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  by (induct I rule: infinite_finite_induct) simp_all
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lemma has_derivative_minus[simp, derivative_intros]:
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  "(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>
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    ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
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  by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
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lemma has_derivative_at_within:
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  "(f has_derivative f') (at x within s) \<longleftrightarrow>
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    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
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  by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)
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lemma has_derivative_iff_norm:
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  "(f has_derivative f') (at x within s) \<longleftrightarrow>
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    bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
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  using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
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  by (simp add: has_derivative_at_within divide_inverse ac_simps)
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lemma has_derivative_at:
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  "(f has_derivative D) (at x) \<longleftrightarrow>
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    (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) \<midarrow>0\<rightarrow> 0)"
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  unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp
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lemma field_has_derivative_at:
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  fixes x :: "'a::real_normed_field"
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  shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
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  apply (unfold has_derivative_at)
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  apply (simp add: bounded_linear_mult_right)
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  apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
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  apply (subst diff_divide_distrib)
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  apply (subst times_divide_eq_left [symmetric])
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  apply (simp cong: LIM_cong)
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  apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
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  done
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lemma has_derivativeI:
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  "bounded_linear f' \<Longrightarrow>
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    ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s) \<Longrightarrow>
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    (f has_derivative f') (at x within s)"
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  by (simp add: has_derivative_at_within)
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lemma has_derivativeI_sandwich:
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  assumes e: "0 < e"
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    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>
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      norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
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    and "(H \<longlongrightarrow> 0) (at x within s)"
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  shows "(f has_derivative f') (at x within s)"
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  unfolding has_derivative_iff_norm
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proof safe
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  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
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  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
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    show "(H \<longlongrightarrow> 0) (at x within s)" by fact
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    show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
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      unfolding eventually_at using e sandwich by auto
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  qed (auto simp: le_divide_eq)
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qed fact
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lemma has_derivative_subset:
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  "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
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  by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
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lemmas has_derivative_within_subset = has_derivative_subset
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subsection \<open>Continuity\<close>
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lemma has_derivative_continuous:
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  assumes f: "(f has_derivative f') (at x within s)"
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  shows "continuous (at x within s) f"
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proof -
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  from f interpret F: bounded_linear f'
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    by (rule has_derivative_bounded_linear)
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  note F.tendsto[tendsto_intros]
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  let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
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  have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
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    using f unfolding has_derivative_iff_norm by blast
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  then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
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    by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
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  also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
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    by (intro filterlim_cong) (simp_all add: eventually_at_filter)
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  finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
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    by (rule tendsto_norm_zero_cancel)
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  then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
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    by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
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  then have "?L (\<lambda>y. f y - f x)"
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    by simp
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  from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
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    by (simp add: continuous_within)
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qed
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subsection \<open>Composition\<close>
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lemma tendsto_at_iff_tendsto_nhds_within:
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  "f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
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  unfolding tendsto_def eventually_inf_principal eventually_at_filter
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  by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
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lemma has_derivative_in_compose:
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  assumes f: "(f has_derivative f') (at x within s)"
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    and 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'
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    by (rule has_derivative_bounded_linear)
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  from g interpret G: bounded_linear g'
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    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"
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    by fast
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  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG"
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    by fast
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  note G.tendsto[tendsto_intros]
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  let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
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  let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
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  let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
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  let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
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  define Nf where "Nf = ?N f f' x"
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  define Ng where [abs_def]: "Ng y = ?N g g' (f x) (f y)" for y
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  show ?thesis
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  proof (rule has_derivativeI_sandwich[of 1])
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    show "bounded_linear (\<lambda>x. g' (f' x))"
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      using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
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  next
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    fix y :: 'a
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    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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   262
      have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
hoelzl@51642
   263
        by simp
hoelzl@51642
   264
      also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
hoelzl@51642
   265
        by (rule norm_triangle_ineq)
hoelzl@51642
   266
      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
hoelzl@51642
   267
        using kF by (intro add_mono) simp
hoelzl@51642
   268
      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
hoelzl@51642
   269
        by (simp add: neq Nf_def field_simps)
wenzelm@63558
   270
    qed (use kG in \<open>simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps\<close>)
hoelzl@51642
   271
    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
hoelzl@51642
   272
  next
hoelzl@51642
   273
    have [tendsto_intros]: "?L Nf"
hoelzl@56181
   274
      using f unfolding has_derivative_iff_norm Nf_def ..
wenzelm@61973
   275
    from f have "(f \<longlongrightarrow> f x) (at x within s)"
hoelzl@56181
   276
      by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
hoelzl@51642
   277
    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
hoelzl@51642
   278
      unfolding filterlim_def
hoelzl@51642
   279
      by (simp add: eventually_filtermap eventually_at_filter le_principal)
hoelzl@51642
   280
wenzelm@61973
   281
    have "((?N g  g' (f x)) \<longlongrightarrow> 0) (at (f x) within f`s)"
hoelzl@56181
   282
      using g unfolding has_derivative_iff_norm ..
wenzelm@61973
   283
    then have g': "((?N g  g' (f x)) \<longlongrightarrow> 0) (inf (nhds (f x)) (principal (f`s)))"
hoelzl@51642
   284
      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp
hoelzl@51642
   285
hoelzl@51642
   286
    have [tendsto_intros]: "?L Ng"
hoelzl@51642
   287
      unfolding Ng_def by (rule filterlim_compose[OF g' f'])
wenzelm@61973
   288
    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) \<longlongrightarrow> 0) (at x within s)"
hoelzl@51642
   289
      by (intro tendsto_eq_intros) auto
hoelzl@51642
   290
  qed simp
hoelzl@51642
   291
qed
hoelzl@51642
   292
hoelzl@56181
   293
lemma has_derivative_compose:
hoelzl@56181
   294
  "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
hoelzl@56181
   295
  ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
hoelzl@56181
   296
  by (blast intro: has_derivative_in_compose has_derivative_subset)
hoelzl@51642
   297
hoelzl@51642
   298
lemma (in bounded_bilinear) FDERIV:
hoelzl@56181
   299
  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
hoelzl@56181
   300
  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
   301
proof -
hoelzl@56181
   302
  from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
hoelzl@51642
   303
  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
hoelzl@51642
   304
wenzelm@63558
   305
  from pos_bounded obtain K
wenzelm@63558
   306
    where K: "0 < K" and norm_prod: "\<And>a b. norm (a ** b) \<le> norm a * norm b * K"
wenzelm@63558
   307
    by fast
hoelzl@51642
   308
  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
hoelzl@51642
   309
  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
wenzelm@63040
   310
  define Ng where "Ng = ?N g g'"
wenzelm@63040
   311
  define Nf where "Nf = ?N f f'"
hoelzl@51642
   312
hoelzl@51642
   313
  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
   314
  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
   315
  let ?F = "at x within s"
huffman@21164
   316
hoelzl@51642
   317
  show ?thesis
hoelzl@56181
   318
  proof (rule has_derivativeI_sandwich[of 1])
hoelzl@51642
   319
    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
hoelzl@51642
   320
      by (intro bounded_linear_add
hoelzl@51642
   321
        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
hoelzl@56181
   322
        has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
hoelzl@51642
   323
  next
wenzelm@61973
   324
    from g have "(g \<longlongrightarrow> g x) ?F"
hoelzl@56181
   325
      by (intro continuous_within[THEN iffD1] has_derivative_continuous)
wenzelm@61973
   326
    moreover from f g have "(Nf \<longlongrightarrow> 0) ?F" "(Ng \<longlongrightarrow> 0) ?F"
hoelzl@56181
   327
      by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
wenzelm@61973
   328
    ultimately have "(?fun2 \<longlongrightarrow> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
hoelzl@51642
   329
      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
wenzelm@61973
   330
    then show "(?fun2 \<longlongrightarrow> 0) ?F"
hoelzl@51642
   331
      by simp
hoelzl@51642
   332
  next
wenzelm@63558
   333
    fix y :: 'd
wenzelm@63558
   334
    assume "y \<noteq> x"
wenzelm@63558
   335
    have "?fun1 y =
wenzelm@63558
   336
        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
   337
      by (simp add: diff_left diff_right add_left add_right field_simps)
hoelzl@51642
   338
    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
hoelzl@51642
   339
        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
hoelzl@51642
   340
      by (intro divide_right_mono mult_mono'
hoelzl@51642
   341
                order_trans [OF norm_triangle_ineq add_mono]
hoelzl@51642
   342
                order_trans [OF norm_prod mult_right_mono]
hoelzl@51642
   343
                mult_nonneg_nonneg order_refl norm_ge_zero norm_F
hoelzl@51642
   344
                K [THEN order_less_imp_le])
hoelzl@51642
   345
    also have "\<dots> = ?fun2 y"
hoelzl@51642
   346
      by (simp add: add_divide_distrib Ng_def Nf_def)
hoelzl@51642
   347
    finally show "?fun1 y \<le> ?fun2 y" .
hoelzl@51642
   348
  qed simp
hoelzl@51642
   349
qed
hoelzl@51642
   350
hoelzl@56381
   351
lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
hoelzl@56381
   352
lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
hoelzl@51642
   353
nipkow@64272
   354
lemma has_derivative_prod[simp, derivative_intros]:
wenzelm@63558
   355
  fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
wenzelm@63915
   356
  shows "(\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)) \<Longrightarrow>
wenzelm@63915
   357
    ((\<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)"
wenzelm@63915
   358
proof (induct I rule: infinite_finite_induct)
wenzelm@63915
   359
  case infinite
wenzelm@63915
   360
  then show ?case by simp
wenzelm@63915
   361
next
wenzelm@63915
   362
  case empty
wenzelm@63915
   363
  then show ?case by simp
wenzelm@63558
   364
next
wenzelm@63915
   365
  case (insert i I)
wenzelm@63915
   366
  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)"
wenzelm@63915
   367
  have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
wenzelm@63915
   368
    using insert by (intro has_derivative_mult) auto
wenzelm@63915
   369
  also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
wenzelm@63915
   370
    using insert(1,2)
nipkow@64267
   371
    by (auto simp add: sum_distrib_left insert_Diff_if intro!: ext sum.cong)
wenzelm@63915
   372
  finally show ?case
wenzelm@63915
   373
    using insert by simp
wenzelm@63558
   374
qed
hoelzl@51642
   375
hoelzl@56381
   376
lemma has_derivative_power[simp, derivative_intros]:
hoelzl@51642
   377
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
hoelzl@56181
   378
  assumes f: "(f has_derivative f') (at x within s)"
hoelzl@56181
   379
  shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within s)"
nipkow@64272
   380
  using has_derivative_prod[OF f, of "{..< n}"] by (simp add: prod_constant ac_simps)
hoelzl@51642
   381
hoelzl@56181
   382
lemma has_derivative_inverse':
hoelzl@51642
   383
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51642
   384
  assumes x: "x \<noteq> 0"
hoelzl@56181
   385
  shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
wenzelm@63558
   386
    (is "(?inv has_derivative ?f) _")
hoelzl@56181
   387
proof (rule has_derivativeI_sandwich)
hoelzl@51642
   388
  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"
hoelzl@51642
   389
    apply (rule bounded_linear_minus)
hoelzl@51642
   390
    apply (rule bounded_linear_mult_const)
hoelzl@51642
   391
    apply (rule bounded_linear_const_mult)
hoelzl@51642
   392
    apply (rule bounded_linear_ident)
hoelzl@51642
   393
    done
hoelzl@51642
   394
  show "0 < norm x" using x by simp
wenzelm@61973
   395
  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) \<longlongrightarrow> 0) (at x within s)"
hoelzl@51642
   396
    apply (rule tendsto_mult_left_zero)
hoelzl@51642
   397
    apply (rule tendsto_norm_zero)
hoelzl@51642
   398
    apply (rule LIM_zero)
hoelzl@51642
   399
    apply (rule tendsto_inverse)
wenzelm@63558
   400
     apply (rule tendsto_ident_at)
hoelzl@51642
   401
    apply (rule x)
hoelzl@51642
   402
    done
hoelzl@51642
   403
next
wenzelm@63558
   404
  fix y :: 'a
wenzelm@63558
   405
  assume h: "y \<noteq> x" "dist y x < norm x"
lp15@62397
   406
  then have "y \<noteq> 0" by auto
wenzelm@63558
   407
  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) =
wenzelm@63558
   408
      norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
wenzelm@60758
   409
    apply (subst inverse_diff_inverse [OF \<open>y \<noteq> 0\<close> x])
hoelzl@51642
   410
    apply (subst minus_diff_minus)
hoelzl@51642
   411
    apply (subst norm_minus_cancel)
hoelzl@51642
   412
    apply (simp add: left_diff_distrib)
hoelzl@51642
   413
    done
hoelzl@51642
   414
  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
hoelzl@51642
   415
    apply (rule divide_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   416
    apply (rule order_trans [OF norm_mult_ineq])
hoelzl@51642
   417
    apply (rule mult_right_mono [OF _ norm_ge_zero])
hoelzl@51642
   418
    apply (rule norm_mult_ineq)
hoelzl@51642
   419
    done
hoelzl@51642
   420
  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
hoelzl@51642
   421
    by simp
hoelzl@51642
   422
  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
wenzelm@63558
   423
    norm (?inv y - ?inv x) * norm (?inv x)" .
hoelzl@51642
   424
qed
hoelzl@51642
   425
hoelzl@56381
   426
lemma has_derivative_inverse[simp, derivative_intros]:
hoelzl@51642
   427
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
wenzelm@63558
   428
  assumes x:  "f x \<noteq> 0"
wenzelm@63558
   429
    and f: "(f has_derivative f') (at x within s)"
wenzelm@63558
   430
  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x))))
wenzelm@63558
   431
    (at x within s)"
hoelzl@56181
   432
  using has_derivative_compose[OF f has_derivative_inverse', OF x] .
hoelzl@51642
   433
hoelzl@56381
   434
lemma has_derivative_divide[simp, derivative_intros]:
hoelzl@51642
   435
  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
wenzelm@63558
   436
  assumes f: "(f has_derivative f') (at x within s)"
wenzelm@63558
   437
    and g: "(g has_derivative g') (at x within s)"
lp15@55967
   438
  assumes x: "g x \<noteq> 0"
hoelzl@56181
   439
  shows "((\<lambda>x. f x / g x) has_derivative
hoelzl@56181
   440
                (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
hoelzl@56181
   441
  using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
hoelzl@56480
   442
  by (simp add: field_simps)
lp15@55967
   443
wenzelm@63558
   444
wenzelm@63558
   445
text \<open>Conventional form requires mult-AC laws. Types real and complex only.\<close>
hoelzl@56181
   446
wenzelm@63558
   447
lemma has_derivative_divide'[derivative_intros]:
lp15@55967
   448
  fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
wenzelm@63558
   449
  assumes f: "(f has_derivative f') (at x within s)"
wenzelm@63558
   450
    and g: "(g has_derivative g') (at x within s)"
wenzelm@63558
   451
    and x: "g x \<noteq> 0"
hoelzl@56181
   452
  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
   453
proof -
wenzelm@63558
   454
  have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
wenzelm@63558
   455
      (f' h * g x - f x * g' h) / (g x * g x)" for h
wenzelm@63558
   456
    by (simp add: field_simps x)
lp15@55967
   457
  then show ?thesis
hoelzl@56181
   458
    using has_derivative_divide [OF f g] x
lp15@55967
   459
    by simp
lp15@55967
   460
qed
hoelzl@51642
   461
wenzelm@63558
   462
wenzelm@60758
   463
subsection \<open>Uniqueness\<close>
hoelzl@51642
   464
wenzelm@60758
   465
text \<open>
hoelzl@56181
   466
This can not generally shown for @{const has_derivative}, as we need to approach the point from
hoelzl@63627
   467
all directions. There is a proof in \<open>Analysis\<close> for \<open>euclidean_space\<close>.
wenzelm@60758
   468
\<close>
hoelzl@51642
   469
hoelzl@56181
   470
lemma has_derivative_zero_unique:
wenzelm@63558
   471
  assumes "((\<lambda>x. 0) has_derivative F) (at x)"
wenzelm@63558
   472
  shows "F = (\<lambda>h. 0)"
hoelzl@51642
   473
proof -
hoelzl@51642
   474
  interpret F: bounded_linear F
hoelzl@56181
   475
    using assms by (rule has_derivative_bounded_linear)
hoelzl@51642
   476
  let ?r = "\<lambda>h. norm (F h) / norm h"
wenzelm@61976
   477
  have *: "?r \<midarrow>0\<rightarrow> 0"
hoelzl@56181
   478
    using assms unfolding has_derivative_at by simp
hoelzl@51642
   479
  show "F = (\<lambda>h. 0)"
hoelzl@51642
   480
  proof
wenzelm@63558
   481
    show "F h = 0" for h
hoelzl@51642
   482
    proof (rule ccontr)
wenzelm@63558
   483
      assume **: "\<not> ?thesis"
wenzelm@63558
   484
      then have h: "h \<noteq> 0"
wenzelm@63558
   485
        by (auto simp add: F.zero)
wenzelm@63558
   486
      with ** have "0 < ?r h"
wenzelm@63558
   487
        by simp
wenzelm@63558
   488
      from LIM_D [OF * this] obtain s
wenzelm@63558
   489
        where s: "0 < s" and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h"
wenzelm@63558
   490
        by auto
hoelzl@51642
   491
      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
hoelzl@51642
   492
      let ?x = "scaleR (t / norm h) h"
wenzelm@63558
   493
      have "?x \<noteq> 0" and "norm ?x < s"
wenzelm@63558
   494
        using t h by simp_all
wenzelm@63558
   495
      then have "?r ?x < ?r h"
wenzelm@63558
   496
        by (rule r)
wenzelm@63558
   497
      then show False
wenzelm@63558
   498
        using t h by (simp add: F.scaleR)
hoelzl@51642
   499
    qed
hoelzl@51642
   500
  qed
hoelzl@51642
   501
qed
hoelzl@51642
   502
hoelzl@56181
   503
lemma has_derivative_unique:
wenzelm@63558
   504
  assumes "(f has_derivative F) (at x)"
wenzelm@63558
   505
    and "(f has_derivative F') (at x)"
wenzelm@63558
   506
  shows "F = F'"
hoelzl@51642
   507
proof -
hoelzl@56181
   508
  have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
hoelzl@56181
   509
    using has_derivative_diff [OF assms] by simp
wenzelm@63558
   510
  then have "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
hoelzl@56181
   511
    by (rule has_derivative_zero_unique)
wenzelm@63558
   512
  then show "F = F'"
hoelzl@51642
   513
    unfolding fun_eq_iff right_minus_eq .
hoelzl@51642
   514
qed
hoelzl@51642
   515
wenzelm@63558
   516
wenzelm@60758
   517
subsection \<open>Differentiability predicate\<close>
hoelzl@51642
   518
wenzelm@63558
   519
definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
wenzelm@63558
   520
    (infix "differentiable" 50)
wenzelm@63558
   521
  where "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
hoelzl@51642
   522
wenzelm@63558
   523
lemma differentiable_subset:
wenzelm@63558
   524
  "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
hoelzl@56181
   525
  unfolding differentiable_def by (blast intro: has_derivative_subset)
hoelzl@51642
   526
lp15@56261
   527
lemmas differentiable_within_subset = differentiable_subset
lp15@56261
   528
hoelzl@56381
   529
lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
hoelzl@56181
   530
  unfolding differentiable_def by (blast intro: has_derivative_ident)
hoelzl@51642
   531
hoelzl@56381
   532
lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
hoelzl@56181
   533
  unfolding differentiable_def by (blast intro: has_derivative_const)
hoelzl@51642
   534
hoelzl@51642
   535
lemma differentiable_in_compose:
wenzelm@63558
   536
  "f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
wenzelm@63558
   537
    (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@56181
   538
  unfolding differentiable_def by (blast intro: has_derivative_in_compose)
hoelzl@51642
   539
hoelzl@51642
   540
lemma differentiable_compose:
wenzelm@63558
   541
  "f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
wenzelm@63558
   542
    (\<lambda>x. f (g x)) differentiable (at x within s)"
hoelzl@51642
   543
  by (blast intro: differentiable_in_compose differentiable_subset)
hoelzl@51642
   544
hoelzl@56381
   545
lemma differentiable_sum [simp, derivative_intros]:
hoelzl@56181
   546
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
hoelzl@56181
   547
  unfolding differentiable_def by (blast intro: has_derivative_add)
hoelzl@51642
   548
hoelzl@56381
   549
lemma differentiable_minus [simp, derivative_intros]:
hoelzl@56181
   550
  "f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
hoelzl@56181
   551
  unfolding differentiable_def by (blast intro: has_derivative_minus)
hoelzl@51642
   552
hoelzl@56381
   553
lemma differentiable_diff [simp, derivative_intros]:
hoelzl@56181
   554
  "f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
hoelzl@56181
   555
  unfolding differentiable_def by (blast intro: has_derivative_diff)
hoelzl@51642
   556
hoelzl@56381
   557
lemma differentiable_mult [simp, derivative_intros]:
wenzelm@63558
   558
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
wenzelm@63558
   559
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
wenzelm@63558
   560
    (\<lambda>x. f x * g x) differentiable (at x within s)"
hoelzl@56181
   561
  unfolding differentiable_def by (blast intro: has_derivative_mult)
hoelzl@51642
   562
hoelzl@56381
   563
lemma differentiable_inverse [simp, derivative_intros]:
wenzelm@63558
   564
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
wenzelm@63558
   565
  shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
wenzelm@63558
   566
    (\<lambda>x. inverse (f x)) differentiable (at x within s)"
hoelzl@56181
   567
  unfolding differentiable_def by (blast intro: has_derivative_inverse)
hoelzl@51642
   568
hoelzl@56381
   569
lemma differentiable_divide [simp, derivative_intros]:
wenzelm@63558
   570
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
wenzelm@63558
   571
  shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
wenzelm@63558
   572
    g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
wenzelm@63092
   573
  unfolding divide_inverse by simp
hoelzl@51642
   574
hoelzl@56381
   575
lemma differentiable_power [simp, derivative_intros]:
wenzelm@63558
   576
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
hoelzl@56181
   577
  shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
hoelzl@56181
   578
  unfolding differentiable_def by (blast intro: has_derivative_power)
hoelzl@51642
   579
hoelzl@56381
   580
lemma differentiable_scaleR [simp, derivative_intros]:
wenzelm@63558
   581
  "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
wenzelm@63558
   582
    (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
hoelzl@56181
   583
  unfolding differentiable_def by (blast intro: has_derivative_scaleR)
hoelzl@51642
   584
hoelzl@56181
   585
lemma has_derivative_imp_has_field_derivative:
hoelzl@56181
   586
  "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
wenzelm@63558
   587
  unfolding has_field_derivative_def
haftmann@57512
   588
  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
hoelzl@56181
   589
wenzelm@63558
   590
lemma has_field_derivative_imp_has_derivative:
wenzelm@63558
   591
  "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
hoelzl@56181
   592
  by (simp add: has_field_derivative_def)
hoelzl@51642
   593
wenzelm@63558
   594
lemma DERIV_subset:
wenzelm@63558
   595
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
wenzelm@63558
   596
    (f has_field_derivative f') (at x within t)"
lp15@56261
   597
  by (simp add: has_field_derivative_def has_derivative_within_subset)
lp15@56261
   598
lp15@59862
   599
lemma has_field_derivative_at_within:
wenzelm@63558
   600
  "(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
lp15@59862
   601
  using DERIV_subset by blast
lp15@59862
   602
hoelzl@56181
   603
abbreviation (input)
hoelzl@56381
   604
  DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
wenzelm@63558
   605
    ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
wenzelm@63558
   606
  where "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
hoelzl@51642
   607
wenzelm@63558
   608
abbreviation has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
wenzelm@63558
   609
    (infix "(has'_real'_derivative)" 50)
wenzelm@63558
   610
  where "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
hoelzl@56181
   611
hoelzl@56181
   612
lemma real_differentiable_def:
hoelzl@56181
   613
  "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
hoelzl@51642
   614
proof safe
hoelzl@56181
   615
  assume "f differentiable at x within s"
hoelzl@56181
   616
  then obtain f' where *: "(f has_derivative f') (at x within s)"
hoelzl@56181
   617
    unfolding differentiable_def by auto
hoelzl@56181
   618
  then obtain c where "f' = (op * c)"
haftmann@57512
   619
    by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
hoelzl@56181
   620
  with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   621
    unfolding has_field_derivative_def by auto
hoelzl@56181
   622
qed (auto simp: differentiable_def has_field_derivative_def)
hoelzl@51642
   623
hoelzl@56181
   624
lemma real_differentiableE [elim?]:
wenzelm@63558
   625
  assumes f: "f differentiable (at x within s)"
wenzelm@63558
   626
  obtains df where "(f has_real_derivative df) (at x within s)"
hoelzl@56181
   627
  using assms by (auto simp: real_differentiable_def)
hoelzl@51642
   628
wenzelm@63558
   629
lemma differentiableD:
wenzelm@63558
   630
  "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"
hoelzl@56181
   631
  by (auto elim: real_differentiableE)
hoelzl@51642
   632
wenzelm@63558
   633
lemma differentiableI:
wenzelm@63558
   634
  "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"
hoelzl@56181
   635
  by (force simp add: real_differentiable_def)
hoelzl@51642
   636
immler@63079
   637
lemma has_field_derivative_iff:
immler@63079
   638
  "(f has_field_derivative D) (at x within S) \<longleftrightarrow>
immler@63079
   639
    ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
immler@63079
   640
  apply (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right
wenzelm@63558
   641
      LIM_zero_iff[symmetric, of _ D])
hoelzl@51642
   642
  apply (subst (2) tendsto_norm_zero_iff[symmetric])
hoelzl@51642
   643
  apply (rule filterlim_cong)
wenzelm@63558
   644
    apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
hoelzl@51642
   645
  done
huffman@21164
   646
immler@63079
   647
lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
immler@63079
   648
  unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..
immler@63079
   649
wenzelm@63558
   650
lemma mult_commute_abs: "(\<lambda>x. x * c) = op * c"
wenzelm@63558
   651
  for c :: "'a::ab_semigroup_mult"
haftmann@57512
   652
  by (simp add: fun_eq_iff mult.commute)
huffman@21164
   653
wenzelm@63558
   654
wenzelm@60758
   655
subsection \<open>Vector derivative\<close>
immler@60177
   656
immler@60177
   657
lemma has_field_derivative_iff_has_vector_derivative:
immler@60177
   658
  "(f has_field_derivative y) F \<longleftrightarrow> (f has_vector_derivative y) F"
immler@60177
   659
  unfolding has_vector_derivative_def has_field_derivative_def real_scaleR_def mult_commute_abs ..
immler@60177
   660
immler@60177
   661
lemma has_field_derivative_subset:
wenzelm@63558
   662
  "(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
wenzelm@63558
   663
    (f has_field_derivative y) (at x within t)"
immler@60177
   664
  unfolding has_field_derivative_def by (rule has_derivative_subset)
immler@60177
   665
immler@60177
   666
lemma has_vector_derivative_const[simp, derivative_intros]: "((\<lambda>x. c) has_vector_derivative 0) net"
immler@60177
   667
  by (auto simp: has_vector_derivative_def)
immler@60177
   668
immler@60177
   669
lemma has_vector_derivative_id[simp, derivative_intros]: "((\<lambda>x. x) has_vector_derivative 1) net"
immler@60177
   670
  by (auto simp: has_vector_derivative_def)
immler@60177
   671
immler@60177
   672
lemma has_vector_derivative_minus[derivative_intros]:
immler@60177
   673
  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. - f x) has_vector_derivative (- f')) net"
immler@60177
   674
  by (auto simp: has_vector_derivative_def)
immler@60177
   675
immler@60177
   676
lemma has_vector_derivative_add[derivative_intros]:
immler@60177
   677
  "(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
immler@60177
   678
    ((\<lambda>x. f x + g x) has_vector_derivative (f' + g')) net"
immler@60177
   679
  by (auto simp: has_vector_derivative_def scaleR_right_distrib)
immler@60177
   680
nipkow@64267
   681
lemma has_vector_derivative_sum[derivative_intros]:
immler@60177
   682
  "(\<And>i. i \<in> I \<Longrightarrow> (f i has_vector_derivative f' i) net) \<Longrightarrow>
immler@60177
   683
    ((\<lambda>x. \<Sum>i\<in>I. f i x) has_vector_derivative (\<Sum>i\<in>I. f' i)) net"
nipkow@64267
   684
  by (auto simp: has_vector_derivative_def fun_eq_iff scaleR_sum_right intro!: derivative_eq_intros)
immler@60177
   685
immler@60177
   686
lemma has_vector_derivative_diff[derivative_intros]:
immler@60177
   687
  "(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
immler@60177
   688
    ((\<lambda>x. f x - g x) has_vector_derivative (f' - g')) net"
immler@60177
   689
  by (auto simp: has_vector_derivative_def scaleR_diff_right)
immler@60177
   690
paulson@61204
   691
lemma has_vector_derivative_add_const:
wenzelm@63558
   692
  "((\<lambda>t. g t + z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
wenzelm@63558
   693
  apply (intro iffI)
wenzelm@63558
   694
   apply (drule has_vector_derivative_diff [where g = "\<lambda>t. z", OF _ has_vector_derivative_const])
wenzelm@63558
   695
   apply simp
wenzelm@63558
   696
  apply (drule has_vector_derivative_add [OF _ has_vector_derivative_const])
wenzelm@63558
   697
  apply simp
wenzelm@63558
   698
  done
paulson@61204
   699
paulson@61204
   700
lemma has_vector_derivative_diff_const:
wenzelm@63558
   701
  "((\<lambda>t. g t - z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
wenzelm@63558
   702
  using has_vector_derivative_add_const [where z = "-z"]
wenzelm@63558
   703
  by simp
paulson@61204
   704
immler@60177
   705
lemma (in bounded_linear) has_vector_derivative:
immler@60177
   706
  assumes "(g has_vector_derivative g') F"
immler@60177
   707
  shows "((\<lambda>x. f (g x)) has_vector_derivative f g') F"
immler@60177
   708
  using has_derivative[OF assms[unfolded has_vector_derivative_def]]
immler@60177
   709
  by (simp add: has_vector_derivative_def scaleR)
immler@60177
   710
immler@60177
   711
lemma (in bounded_bilinear) has_vector_derivative:
immler@60177
   712
  assumes "(f has_vector_derivative f') (at x within s)"
immler@60177
   713
    and "(g has_vector_derivative g') (at x within s)"
immler@60177
   714
  shows "((\<lambda>x. f x ** g x) has_vector_derivative (f x ** g' + f' ** g x)) (at x within s)"
immler@60177
   715
  using FDERIV[OF assms(1-2)[unfolded has_vector_derivative_def]]
immler@60177
   716
  by (simp add: has_vector_derivative_def scaleR_right scaleR_left scaleR_right_distrib)
immler@60177
   717
immler@60177
   718
lemma has_vector_derivative_scaleR[derivative_intros]:
immler@60177
   719
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
immler@60177
   720
    ((\<lambda>x. f x *\<^sub>R g x) has_vector_derivative (f x *\<^sub>R g' + f' *\<^sub>R g x)) (at x within s)"
immler@60177
   721
  unfolding has_field_derivative_iff_has_vector_derivative
immler@60177
   722
  by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])
immler@60177
   723
immler@60177
   724
lemma has_vector_derivative_mult[derivative_intros]:
immler@60177
   725
  "(f has_vector_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
wenzelm@63558
   726
    ((\<lambda>x. f x * g x) has_vector_derivative (f x * g' + f' * g x)) (at x within s)"
wenzelm@63558
   727
  for f g :: "real \<Rightarrow> 'a::real_normed_algebra"
immler@60177
   728
  by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_mult])
immler@60177
   729
immler@60177
   730
lemma has_vector_derivative_of_real[derivative_intros]:
immler@60177
   731
  "(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_vector_derivative (of_real D)) F"
immler@60177
   732
  by (rule bounded_linear.has_vector_derivative[OF bounded_linear_of_real])
wenzelm@63558
   733
    (simp add: has_field_derivative_iff_has_vector_derivative)
immler@60177
   734
wenzelm@63558
   735
lemma has_vector_derivative_continuous:
wenzelm@63558
   736
  "(f has_vector_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
immler@60177
   737
  by (auto intro: has_derivative_continuous simp: has_vector_derivative_def)
immler@60177
   738
immler@60177
   739
lemma has_vector_derivative_mult_right[derivative_intros]:
wenzelm@63558
   740
  fixes a :: "'a::real_normed_algebra"
immler@60177
   741
  shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. a * f x) has_vector_derivative (a * x)) F"
immler@60177
   742
  by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_right])
immler@60177
   743
immler@60177
   744
lemma has_vector_derivative_mult_left[derivative_intros]:
wenzelm@63558
   745
  fixes a :: "'a::real_normed_algebra"
immler@60177
   746
  shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. f x * a) has_vector_derivative (x * a)) F"
immler@60177
   747
  by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left])
immler@60177
   748
immler@60177
   749
wenzelm@60758
   750
subsection \<open>Derivatives\<close>
huffman@21164
   751
wenzelm@61976
   752
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
hoelzl@56381
   753
  by (simp add: DERIV_def)
huffman@21164
   754
immler@63079
   755
lemma has_field_derivativeD:
immler@63079
   756
  "(f has_field_derivative D) (at x within S) \<Longrightarrow>
immler@63079
   757
    ((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
immler@63079
   758
  by (simp add: has_field_derivative_iff)
immler@63079
   759
hoelzl@56381
   760
lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
hoelzl@56181
   761
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
hoelzl@56181
   762
hoelzl@56381
   763
lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
hoelzl@56181
   764
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
huffman@21164
   765
hoelzl@56381
   766
lemma field_differentiable_add[derivative_intros]:
wenzelm@63558
   767
  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow>
hoelzl@56381
   768
    ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
hoelzl@56381
   769
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
hoelzl@56381
   770
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
lp15@56261
   771
lp15@56261
   772
corollary DERIV_add:
hoelzl@56181
   773
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
wenzelm@63558
   774
    ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
lp15@56261
   775
  by (rule field_differentiable_add)
lp15@56261
   776
hoelzl@56381
   777
lemma field_differentiable_minus[derivative_intros]:
hoelzl@56381
   778
  "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
hoelzl@56381
   779
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
hoelzl@56381
   780
     (auto simp: has_field_derivative_def field_simps mult_commute_abs)
huffman@21164
   781
wenzelm@63558
   782
corollary DERIV_minus:
wenzelm@63558
   783
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   784
    ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
lp15@56261
   785
  by (rule field_differentiable_minus)
huffman@21164
   786
hoelzl@56381
   787
lemma field_differentiable_diff[derivative_intros]:
wenzelm@63558
   788
  "(f has_field_derivative f') F \<Longrightarrow>
wenzelm@63558
   789
    (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
wenzelm@63092
   790
  by (simp only: diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
lp15@56261
   791
lp15@56261
   792
corollary DERIV_diff:
wenzelm@63558
   793
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   794
    (g has_field_derivative E) (at x within s) \<Longrightarrow>
wenzelm@63558
   795
    ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
lp15@56261
   796
  by (rule field_differentiable_diff)
hoelzl@51642
   797
hoelzl@56181
   798
lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
hoelzl@56181
   799
  by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
huffman@21164
   800
lp15@56261
   801
corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
lp15@56261
   802
  by (rule DERIV_continuous)
lp15@56261
   803
lp15@56261
   804
lemma DERIV_continuous_on:
eberlm@63299
   805
  "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (D x)) (at x within s)) \<Longrightarrow> continuous_on s f"
eberlm@63299
   806
  unfolding continuous_on_eq_continuous_within
wenzelm@63558
   807
  by (intro continuous_at_imp_continuous_on ballI DERIV_continuous)
hoelzl@51642
   808
hoelzl@56181
   809
lemma DERIV_mult':
hoelzl@56181
   810
  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
wenzelm@63558
   811
    ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
hoelzl@56181
   812
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   813
     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
huffman@21164
   814
hoelzl@56381
   815
lemma DERIV_mult[derivative_intros]:
hoelzl@56181
   816
  "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
wenzelm@63558
   817
    ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
hoelzl@56181
   818
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
hoelzl@56181
   819
     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   820
wenzelm@60758
   821
text \<open>Derivative of linear multiplication\<close>
huffman@21164
   822
hoelzl@51642
   823
lemma DERIV_cmult:
wenzelm@63558
   824
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   825
    ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
wenzelm@63558
   826
  by (drule DERIV_mult' [OF DERIV_const]) simp
huffman@21164
   827
lp15@55967
   828
lemma DERIV_cmult_right:
wenzelm@63558
   829
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   830
    ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
wenzelm@63558
   831
  using DERIV_cmult by (auto simp add: ac_simps)
lp15@55967
   832
hoelzl@56181
   833
lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
wenzelm@63558
   834
  using DERIV_ident [THEN DERIV_cmult, where c = c and x = x] by simp
hoelzl@51642
   835
hoelzl@56181
   836
lemma DERIV_cdivide:
wenzelm@63558
   837
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   838
    ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
hoelzl@56181
   839
  using DERIV_cmult_right[of f D x s "1 / c"] by simp
huffman@21164
   840
wenzelm@63558
   841
lemma DERIV_unique: "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
wenzelm@63558
   842
  unfolding DERIV_def by (rule LIM_unique)
huffman@21164
   843
nipkow@64267
   844
lemma DERIV_sum[derivative_intros]:
wenzelm@63558
   845
  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow>
nipkow@64267
   846
    ((\<lambda>x. sum (f x) S) has_field_derivative sum (f' x) S) F"
nipkow@64267
   847
  by (rule has_derivative_imp_has_field_derivative [OF has_derivative_sum])
nipkow@64267
   848
     (auto simp: sum_distrib_left mult_commute_abs dest: has_field_derivative_imp_has_derivative)
hoelzl@51642
   849
hoelzl@56381
   850
lemma DERIV_inverse'[derivative_intros]:
haftmann@59867
   851
  assumes "(f has_field_derivative D) (at x within s)"
haftmann@59867
   852
    and "f x \<noteq> 0"
wenzelm@63558
   853
  shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))
wenzelm@63558
   854
    (at x within s)"
haftmann@59867
   855
proof -
haftmann@59867
   856
  have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative op * D)"
haftmann@59867
   857
    by (rule arg_cong [of "\<lambda>x. x * D"]) (simp add: fun_eq_iff)
haftmann@59867
   858
  with assms have "(f has_derivative (\<lambda>x. x * D)) (at x within s)"
haftmann@59867
   859
    by (auto dest!: has_field_derivative_imp_has_derivative)
wenzelm@60758
   860
  then show ?thesis using \<open>f x \<noteq> 0\<close>
haftmann@59867
   861
    by (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse)
haftmann@59867
   862
qed
hoelzl@51642
   863
wenzelm@61799
   864
text \<open>Power of \<open>-1\<close>\<close>
hoelzl@51642
   865
hoelzl@51642
   866
lemma DERIV_inverse:
hoelzl@56181
   867
  "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
hoelzl@51642
   868
  by (drule DERIV_inverse' [OF DERIV_ident]) simp
hoelzl@51642
   869
wenzelm@60758
   870
text \<open>Derivative of inverse\<close>
hoelzl@51642
   871
hoelzl@51642
   872
lemma DERIV_inverse_fun:
hoelzl@56181
   873
  "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
wenzelm@63558
   874
    ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
haftmann@57514
   875
  by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)
hoelzl@51642
   876
wenzelm@60758
   877
text \<open>Derivative of quotient\<close>
hoelzl@51642
   878
hoelzl@56381
   879
lemma DERIV_divide[derivative_intros]:
hoelzl@56181
   880
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   881
    (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
wenzelm@63558
   882
    ((\<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
   883
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
hoelzl@56480
   884
     (auto dest: has_field_derivative_imp_has_derivative simp: field_simps)
hoelzl@51642
   885
hoelzl@51642
   886
lemma DERIV_quotient:
hoelzl@56181
   887
  "(f has_field_derivative d) (at x within s) \<Longrightarrow>
wenzelm@63558
   888
    (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
wenzelm@63558
   889
    ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
haftmann@57512
   890
  by (drule (2) DERIV_divide) (simp add: mult.commute)
hoelzl@51642
   891
hoelzl@51642
   892
lemma DERIV_power_Suc:
hoelzl@56181
   893
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   894
    ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
hoelzl@56181
   895
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   896
     (auto simp: has_field_derivative_def)
hoelzl@51642
   897
hoelzl@56381
   898
lemma DERIV_power[derivative_intros]:
hoelzl@56181
   899
  "(f has_field_derivative D) (at x within s) \<Longrightarrow>
wenzelm@63558
   900
    ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
hoelzl@56181
   901
  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
hoelzl@56181
   902
     (auto simp: has_field_derivative_def)
hoelzl@31880
   903
hoelzl@56181
   904
lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
lp15@61609
   905
  using DERIV_power [OF DERIV_ident] by simp
hoelzl@51642
   906
wenzelm@63558
   907
lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow>
hoelzl@56181
   908
  ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
hoelzl@56181
   909
  using has_derivative_compose[of f "op * D" x s g "op * E"]
wenzelm@63170
   910
  by (simp only: has_field_derivative_def mult_commute_abs ac_simps)
hoelzl@51642
   911
hoelzl@56181
   912
corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
hoelzl@56181
   913
  ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
lp15@55967
   914
  by (rule DERIV_chain')
lp15@55967
   915
wenzelm@60758
   916
text \<open>Standard version\<close>
hoelzl@51642
   917
hoelzl@56181
   918
lemma DERIV_chain:
wenzelm@63558
   919
  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
wenzelm@63558
   920
    (f \<circ> g has_field_derivative Da * Db) (at x within s)"
haftmann@57512
   921
  by (drule (1) DERIV_chain', simp add: o_def mult.commute)
hoelzl@51642
   922
wenzelm@63558
   923
lemma DERIV_image_chain:
wenzelm@63558
   924
  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow>
wenzelm@63558
   925
    (g has_field_derivative Db) (at x within s) \<Longrightarrow>
wenzelm@63558
   926
    (f \<circ> g has_field_derivative Da * Db) (at x within s)"
hoelzl@56181
   927
  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]
hoelzl@56181
   928
  by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)
lp15@55967
   929
lp15@55967
   930
(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
lp15@55967
   931
lemma DERIV_chain_s:
lp15@55967
   932
  assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
wenzelm@63558
   933
    and "DERIV f x :> f'"
wenzelm@63558
   934
    and "f x \<in> s"
wenzelm@63558
   935
  shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
haftmann@57512
   936
  by (metis (full_types) DERIV_chain' mult.commute assms)
lp15@55967
   937
lp15@55967
   938
lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
lp15@55967
   939
  assumes "(\<And>x. DERIV g x :> g'(x))"
wenzelm@63558
   940
    and "DERIV f x :> f'"
wenzelm@63558
   941
  shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
lp15@55967
   942
  by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
lp15@55967
   943
wenzelm@63558
   944
text \<open>Alternative definition for differentiability\<close>
huffman@21164
   945
huffman@21164
   946
lemma DERIV_LIM_iff:
wenzelm@63558
   947
  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a"
wenzelm@63558
   948
  shows "((\<lambda>h. (f (a + h) - f a) / h) \<midarrow>0\<rightarrow> D) = ((\<lambda>x. (f x - f a) / (x - a)) \<midarrow>a\<rightarrow> D)"
wenzelm@63558
   949
  apply (rule iffI)
wenzelm@63558
   950
   apply (drule_tac k="- a" in LIM_offset)
wenzelm@63558
   951
   apply simp
wenzelm@63558
   952
  apply (drule_tac k="a" in LIM_offset)
wenzelm@63558
   953
  apply (simp add: add.commute)
wenzelm@63558
   954
  done
huffman@21164
   955
immler@63079
   956
lemmas DERIV_iff2 = has_field_derivative_iff
immler@63079
   957
immler@63079
   958
lemma has_field_derivative_cong_ev:
immler@63079
   959
  assumes "x = y"
immler@63079
   960
    and *: "eventually (\<lambda>x. x \<in> s \<longrightarrow> f x = g x) (nhds x)"
immler@63079
   961
    and "u = v" "s = t" "x \<in> s"
wenzelm@63558
   962
  shows "(f has_field_derivative u) (at x within s) = (g has_field_derivative v) (at y within t)"
immler@63079
   963
  unfolding DERIV_iff2
immler@63079
   964
proof (rule filterlim_cong)
wenzelm@63558
   965
  from assms have "f y = g y"
wenzelm@63558
   966
    by (auto simp: eventually_nhds)
immler@63079
   967
  with * show "\<forall>\<^sub>F xa in at x within s. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)"
immler@63079
   968
    unfolding eventually_at_filter
immler@63079
   969
    by eventually_elim (auto simp: assms \<open>f y = g y\<close>)
immler@63079
   970
qed (simp_all add: assms)
huffman@21164
   971
wenzelm@63558
   972
lemma DERIV_cong_ev:
wenzelm@63558
   973
  "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
hoelzl@51642
   974
    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
immler@63079
   975
  by (rule has_field_derivative_cong_ev) simp_all
huffman@21164
   976
hoelzl@51642
   977
lemma DERIV_shift:
immler@63079
   978
  "(f has_field_derivative y) (at (x + z)) = ((\<lambda>x. f (x + z)) has_field_derivative y) (at x)"
hoelzl@56381
   979
  by (simp add: DERIV_def field_simps)
huffman@21164
   980
wenzelm@63558
   981
lemma DERIV_mirror: "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x)) x :> - y)"
wenzelm@63558
   982
  for f :: "real \<Rightarrow> real" and x y :: real
hoelzl@56479
   983
  by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
wenzelm@63558
   984
      tendsto_minus_cancel_left field_simps conj_commute)
huffman@21164
   985
immler@63263
   986
lemma floor_has_real_derivative:
wenzelm@63558
   987
  fixes f :: "real \<Rightarrow> 'a::{floor_ceiling,order_topology}"
immler@63263
   988
  assumes "isCont f x"
wenzelm@63558
   989
    and "f x \<notin> \<int>"
immler@63263
   990
  shows "((\<lambda>x. floor (f x)) has_real_derivative 0) (at x)"
immler@63263
   991
proof (subst DERIV_cong_ev[OF refl _ refl])
wenzelm@63558
   992
  show "((\<lambda>_. floor (f x)) has_real_derivative 0) (at x)"
wenzelm@63558
   993
    by simp
immler@63263
   994
  have "\<forall>\<^sub>F y in at x. \<lfloor>f y\<rfloor> = \<lfloor>f x\<rfloor>"
immler@63263
   995
    by (rule eventually_floor_eq[OF assms[unfolded continuous_at]])
immler@63263
   996
  then show "\<forall>\<^sub>F y in nhds x. real_of_int \<lfloor>f y\<rfloor> = real_of_int \<lfloor>f x\<rfloor>"
immler@63263
   997
    unfolding eventually_at_filter
immler@63263
   998
    by eventually_elim auto
immler@63263
   999
qed
immler@63263
  1000
immler@63263
  1001
wenzelm@60758
  1002
text \<open>Caratheodory formulation of derivative at a point\<close>
huffman@21164
  1003
lp15@55970
  1004
lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
hoelzl@51642
  1005
  "(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)"
wenzelm@63558
  1006
  (is "?lhs = ?rhs")
huffman@21164
  1007
proof
wenzelm@63558
  1008
  assume ?lhs
wenzelm@63558
  1009
  show "\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l"
huffman@21164
  1010
  proof (intro exI conjI)
wenzelm@63558
  1011
    let ?g = "(\<lambda>z. if z = x then l else (f z - f x) / (z-x))"
wenzelm@63558
  1012
    show "\<forall>z. f z - f x = ?g z * (z - x)"
wenzelm@63558
  1013
      by simp
wenzelm@63558
  1014
    show "isCont ?g x"
wenzelm@63558
  1015
      using \<open>?lhs\<close> by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
wenzelm@63558
  1016
    show "?g x = l"
wenzelm@63558
  1017
      by simp
huffman@21164
  1018
  qed
huffman@21164
  1019
next
wenzelm@63558
  1020
  assume ?rhs
wenzelm@63558
  1021
  then obtain g where "(\<forall>z. f z - f x = g z * (z - x))" and "isCont g x" and "g x = l"
wenzelm@63558
  1022
    by blast
wenzelm@63558
  1023
  then show ?lhs
wenzelm@63558
  1024
    by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
huffman@21164
  1025
qed
huffman@21164
  1026
huffman@21164
  1027
wenzelm@60758
  1028
subsection \<open>Local extrema\<close>
huffman@29975
  1029
wenzelm@63558
  1030
text \<open>If @{term "0 < f' x"} then @{term x} is Locally Strictly Increasing At The Right.\<close>
huffman@21164
  1031
immler@63079
  1032
lemma has_real_derivative_pos_inc_right:
wenzelm@63558
  1033
  fixes f :: "real \<Rightarrow> real"
immler@63079
  1034
  assumes der: "(f has_real_derivative l) (at x within S)"
wenzelm@63558
  1035
    and l: "0 < l"
immler@63079
  1036
  shows "\<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x + h)"
immler@63079
  1037
  using assms
immler@63079
  1038
proof -
immler@63079
  1039
  from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
wenzelm@63558
  1040
  obtain s where s: "0 < s"
wenzelm@63558
  1041
    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < l"
immler@63079
  1042
    by (auto simp: dist_real_def)
immler@63079
  1043
  then show ?thesis
immler@63079
  1044
  proof (intro exI conjI strip)
wenzelm@63558
  1045
    show "0 < s" by (rule s)
wenzelm@63558
  1046
  next
wenzelm@63558
  1047
    fix h :: real
immler@63079
  1048
    assume "0 < h" "h < s" "x + h \<in> S"
immler@63079
  1049
    with all [of "x + h"] show "f x < f (x+h)"
immler@63079
  1050
    proof (simp add: abs_if dist_real_def pos_less_divide_eq split: if_split_asm)
wenzelm@63558
  1051
      assume "\<not> (f (x + h) - f x) / h < l" and h: "0 < h"
wenzelm@63558
  1052
      with l have "0 < (f (x + h) - f x) / h"
wenzelm@63558
  1053
        by arith
wenzelm@63558
  1054
      then show "f x < f (x + h)"
immler@63079
  1055
        by (simp add: pos_less_divide_eq h)
immler@63079
  1056
    qed
immler@63079
  1057
  qed
immler@63079
  1058
qed
immler@63079
  1059
paulson@33654
  1060
lemma DERIV_pos_inc_right:
wenzelm@63558
  1061
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1062
  assumes der: "DERIV f x :> l"
wenzelm@63558
  1063
    and l: "0 < l"
wenzelm@63558
  1064
  shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x + h)"
immler@63079
  1065
  using has_real_derivative_pos_inc_right[OF assms]
immler@63079
  1066
  by auto
immler@63079
  1067
immler@63079
  1068
lemma has_real_derivative_neg_dec_left:
wenzelm@63558
  1069
  fixes f :: "real \<Rightarrow> real"
immler@63079
  1070
  assumes der: "(f has_real_derivative l) (at x within S)"
wenzelm@63558
  1071
    and "l < 0"
immler@63079
  1072
  shows "\<exists>d > 0. \<forall>h > 0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x - h)"
huffman@21164
  1073
proof -
wenzelm@63558
  1074
  from \<open>l < 0\<close> have l: "- l > 0"
wenzelm@63558
  1075
    by simp
immler@63079
  1076
  from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
wenzelm@63558
  1077
  obtain s where s: "0 < s"
wenzelm@63558
  1078
    and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < - l"
immler@63079
  1079
    by (auto simp: dist_real_def)
wenzelm@63558
  1080
  then show ?thesis
huffman@21164
  1081
  proof (intro exI conjI strip)
wenzelm@63558
  1082
    show "0 < s" by (rule s)
wenzelm@63558
  1083
  next
wenzelm@63558
  1084
    fix h :: real
immler@63079
  1085
    assume "0 < h" "h < s" "x - h \<in> S"
immler@63079
  1086
    with all [of "x - h"] show "f x < f (x-h)"
nipkow@63648
  1087
    proof (simp add: abs_if pos_less_divide_eq dist_real_def split: if_split_asm)
wenzelm@63558
  1088
      assume "- ((f (x-h) - f x) / h) < l" and h: "0 < h"
wenzelm@63558
  1089
      with l have "0 < (f (x-h) - f x) / h"
wenzelm@63558
  1090
        by arith
wenzelm@63558
  1091
      then show "f x < f (x - h)"
immler@63079
  1092
        by (simp add: pos_less_divide_eq h)
huffman@21164
  1093
    qed
huffman@21164
  1094
  qed
huffman@21164
  1095
qed
huffman@21164
  1096
paulson@33654
  1097
lemma DERIV_neg_dec_left:
wenzelm@63558
  1098
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1099
  assumes der: "DERIV f x :> l"
wenzelm@63558
  1100
    and l: "l < 0"
wenzelm@63558
  1101
  shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x - h)"
immler@63079
  1102
  using has_real_derivative_neg_dec_left[OF assms]
immler@63079
  1103
  by auto
immler@63079
  1104
immler@63079
  1105
lemma has_real_derivative_pos_inc_left:
wenzelm@63558
  1106
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1107
  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> 0 < l \<Longrightarrow>
wenzelm@63558
  1108
    \<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
wenzelm@63558
  1109
  by (rule has_real_derivative_neg_dec_left [of "\<lambda>x. - f x" "-l" x S, simplified])
immler@63079
  1110
      (auto simp add: DERIV_minus)
huffman@21164
  1111
paulson@33654
  1112
lemma DERIV_pos_inc_left:
wenzelm@63558
  1113
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1114
  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f (x - h) < f x"
immler@63079
  1115
  using has_real_derivative_pos_inc_left
immler@63079
  1116
  by blast
immler@63079
  1117
immler@63079
  1118
lemma has_real_derivative_neg_dec_right:
wenzelm@63558
  1119
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1120
  shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> l < 0 \<Longrightarrow>
wenzelm@63558
  1121
    \<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x > f (x + h)"
wenzelm@63558
  1122
  by (rule has_real_derivative_pos_inc_right [of "\<lambda>x. - f x" "-l" x S, simplified])
immler@63079
  1123
      (auto simp add: DERIV_minus)
paulson@33654
  1124
paulson@33654
  1125
lemma DERIV_neg_dec_right:
wenzelm@63558
  1126
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1127
  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x > f (x + h)"
immler@63079
  1128
  using has_real_derivative_neg_dec_right by blast
paulson@33654
  1129
huffman@21164
  1130
lemma DERIV_local_max:
wenzelm@63558
  1131
  fixes f :: "real \<Rightarrow> real"
huffman@21164
  1132
  assumes der: "DERIV f x :> l"
wenzelm@63558
  1133
    and d: "0 < d"
wenzelm@63558
  1134
    and le: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
huffman@21164
  1135
  shows "l = 0"
huffman@21164
  1136
proof (cases rule: linorder_cases [of l 0])
wenzelm@63558
  1137
  case equal
wenzelm@63558
  1138
  then show ?thesis .
huffman@21164
  1139
next
huffman@21164
  1140
  case less
paulson@33654
  1141
  from DERIV_neg_dec_left [OF der less]
wenzelm@63558
  1142
  obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x - h)"
wenzelm@63558
  1143
    by blast
wenzelm@63558
  1144
  obtain e where "0 < e \<and> e < d \<and> e < d'"
wenzelm@63558
  1145
    using real_lbound_gt_zero [OF d d']  ..
wenzelm@63558
  1146
  with lt le [THEN spec [where x="x - e"]] show ?thesis
wenzelm@63558
  1147
    by (auto simp add: abs_if)
huffman@21164
  1148
next
huffman@21164
  1149
  case greater
paulson@33654
  1150
  from DERIV_pos_inc_right [OF der greater]
wenzelm@63558
  1151
  obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)"
wenzelm@63558
  1152
    by blast
wenzelm@63558
  1153
  obtain e where "0 < e \<and> e < d \<and> e < d'"
wenzelm@63558
  1154
    using real_lbound_gt_zero [OF d d'] ..
wenzelm@63558
  1155
  with lt le [THEN spec [where x="x + e"]] show ?thesis
wenzelm@63558
  1156
    by (auto simp add: abs_if)
huffman@21164
  1157
qed
huffman@21164
  1158
wenzelm@63558
  1159
text \<open>Similar theorem for a local minimum\<close>
huffman@21164
  1160
lemma DERIV_local_min:
wenzelm@63558
  1161
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1162
  shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x \<le> f y \<Longrightarrow> l = 0"
wenzelm@63558
  1163
  by (drule DERIV_minus [THEN DERIV_local_max]) auto
huffman@21164
  1164
huffman@21164
  1165
wenzelm@60758
  1166
text\<open>In particular, if a function is locally flat\<close>
huffman@21164
  1167
lemma DERIV_local_const:
wenzelm@63558
  1168
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1169
  shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x = f y \<Longrightarrow> l = 0"
wenzelm@63558
  1170
  by (auto dest!: DERIV_local_max)
huffman@21164
  1171
huffman@29975
  1172
wenzelm@60758
  1173
subsection \<open>Rolle's Theorem\<close>
huffman@29975
  1174
wenzelm@63558
  1175
text \<open>Lemma about introducing open ball in open interval\<close>
wenzelm@63558
  1176
lemma lemma_interval_lt: "a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a < y \<and> y < b)"
wenzelm@63558
  1177
  for a b x :: real
wenzelm@63558
  1178
  apply (simp add: abs_less_iff)
wenzelm@63558
  1179
  apply (insert linorder_linear [of "x - a" "b - x"])
wenzelm@63558
  1180
  apply safe
wenzelm@63558
  1181
   apply (rule_tac x = "x - a" in exI)
wenzelm@63558
  1182
   apply (rule_tac [2] x = "b - x" in exI)
wenzelm@63558
  1183
   apply auto
wenzelm@63558
  1184
  done
chaieb@27668
  1185
wenzelm@63558
  1186
lemma lemma_interval: "a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b)"
wenzelm@63558
  1187
  for a b x :: real
wenzelm@63558
  1188
  apply (drule lemma_interval_lt)
wenzelm@63558
  1189
   apply auto
wenzelm@63558
  1190
  apply force
wenzelm@63558
  1191
  done
huffman@21164
  1192
wenzelm@63558
  1193
text \<open>Rolle's Theorem.
huffman@21164
  1194
   If @{term f} is defined and continuous on the closed interval
wenzelm@61799
  1195
   \<open>[a,b]\<close> and differentiable on the open interval \<open>(a,b)\<close>,
wenzelm@63558
  1196
   and @{term "f a = f b"},
wenzelm@63558
  1197
   then there exists \<open>x0 \<in> (a,b)\<close> such that @{term "f' x0 = 0"}\<close>
huffman@21164
  1198
theorem Rolle:
wenzelm@63558
  1199
  fixes a b :: real
huffman@21164
  1200
  assumes lt: "a < b"
wenzelm@63558
  1201
    and eq: "f a = f b"
wenzelm@63558
  1202
    and con: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
wenzelm@63558
  1203
    and dif [rule_format]: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@63558
  1204
  shows "\<exists>z. a < z \<and> z < b \<and> DERIV f z :> 0"
huffman@21164
  1205
proof -
wenzelm@63558
  1206
  have le: "a \<le> b"
wenzelm@63558
  1207
    using lt by simp
huffman@21164
  1208
  from isCont_eq_Ub [OF le con]
wenzelm@63558
  1209
  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" and "a \<le> x" "x \<le> b"
huffman@21164
  1210
    by blast
huffman@21164
  1211
  from isCont_eq_Lb [OF le con]
wenzelm@63558
  1212
  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" and "a \<le> x'" "x' \<le> b"
huffman@21164
  1213
    by blast
wenzelm@63558
  1214
  consider "a < x" "x < b" | "x = a \<or> x = b"
wenzelm@63558
  1215
    using \<open>a \<le> x\<close> \<open>x \<le> b\<close> by arith
wenzelm@63558
  1216
  then show ?thesis
huffman@21164
  1217
  proof cases
wenzelm@63558
  1218
    case 1
wenzelm@63558
  1219
    \<comment>\<open>@{term f} attains its maximum within the interval\<close>
wenzelm@63558
  1220
    obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
wenzelm@63558
  1221
      using lemma_interval [OF 1] by blast
wenzelm@63558
  1222
    then have bound': "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
wenzelm@63558
  1223
      using x_max by blast
wenzelm@63558
  1224
    obtain l where der: "DERIV f x :> l"
wenzelm@63558
  1225
      using differentiableD [OF dif [OF conjI [OF 1]]] ..
wenzelm@63558
  1226
    \<comment>\<open>the derivative at a local maximum is zero\<close>
wenzelm@63558
  1227
    have "l = 0"
wenzelm@63558
  1228
      by (rule DERIV_local_max [OF der d bound'])
wenzelm@63558
  1229
    with 1 der show ?thesis by auto
huffman@21164
  1230
  next
wenzelm@63558
  1231
    case 2
wenzelm@63558
  1232
    then have fx: "f b = f x" by (auto simp add: eq)
wenzelm@63558
  1233
    consider "a < x'" "x' < b" | "x' = a \<or> x' = b"
wenzelm@63558
  1234
      using \<open>a \<le> x'\<close> \<open>x' \<le> b\<close> by arith
wenzelm@63558
  1235
    then show ?thesis
huffman@21164
  1236
    proof cases
wenzelm@63558
  1237
      case 1
wenzelm@63558
  1238
        \<comment> \<open>@{term f} attains its minimum within the interval\<close>
wenzelm@63558
  1239
      from lemma_interval [OF 1]
huffman@21164
  1240
      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
wenzelm@63558
  1241
        by blast
wenzelm@63558
  1242
      then have bound': "\<forall>y. \<bar>x' - y\<bar> < d \<longrightarrow> f x' \<le> f y"
wenzelm@63558
  1243
        using x'_min by blast
wenzelm@63558
  1244
      from differentiableD [OF dif [OF conjI [OF 1]]]
huffman@21164
  1245
      obtain l where der: "DERIV f x' :> l" ..
wenzelm@63558
  1246
      have "l = 0" by (rule DERIV_local_min [OF der d bound'])
wenzelm@63558
  1247
        \<comment> \<open>the derivative at a local minimum is zero\<close>
wenzelm@63558
  1248
      then show ?thesis using 1 der by auto
huffman@21164
  1249
    next
wenzelm@63558
  1250
      case 2
wenzelm@63558
  1251
        \<comment> \<open>@{term f} is constant throughout the interval\<close>
wenzelm@63558
  1252
      then have fx': "f b = f x'" by (auto simp: eq)
wenzelm@63558
  1253
      from dense [OF lt] obtain r where r: "a < r" "r < b" by blast
wenzelm@63558
  1254
      obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
wenzelm@63558
  1255
        using lemma_interval [OF r] by blast
wenzelm@63558
  1256
      have eq_fb: "f z = f b" if "a \<le> z" and "z \<le> b" for z
wenzelm@63558
  1257
      proof (rule order_antisym)
wenzelm@63558
  1258
        show "f z \<le> f b" by (simp add: fx x_max that)
wenzelm@63558
  1259
        show "f b \<le> f z" by (simp add: fx' x'_min that)
huffman@21164
  1260
      qed
wenzelm@63558
  1261
      have bound': "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> f r = f y"
huffman@21164
  1262
      proof (intro strip)
wenzelm@63558
  1263
        fix y :: real
wenzelm@63558
  1264
        assume lt: "\<bar>r - y\<bar> < d"
wenzelm@63558
  1265
        then have "f y = f b" by (simp add: eq_fb bound)
wenzelm@63558
  1266
        then show "f r = f y" by (simp add: eq_fb r order_less_imp_le)
huffman@21164
  1267
      qed
wenzelm@63558
  1268
      obtain l where der: "DERIV f r :> l"
wenzelm@63558
  1269
        using differentiableD [OF dif [OF conjI [OF r]]] ..
wenzelm@63558
  1270
      have "l = 0"
wenzelm@63558
  1271
        by (rule DERIV_local_const [OF der d bound'])
wenzelm@63558
  1272
        \<comment> \<open>the derivative of a constant function is zero\<close>
wenzelm@63558
  1273
      with r der show ?thesis by auto
huffman@21164
  1274
    qed
huffman@21164
  1275
  qed
huffman@21164
  1276
qed
huffman@21164
  1277
huffman@21164
  1278
wenzelm@63558
  1279
subsection \<open>Mean Value Theorem\<close>
huffman@21164
  1280
wenzelm@63558
  1281
lemma lemma_MVT: "f a - (f b - f a) / (b - a) * a = f b - (f b - f a) / (b - a) * b"
wenzelm@63558
  1282
  for a b :: real
hoelzl@51481
  1283
  by (cases "a = b") (simp_all add: field_simps)
huffman@21164
  1284
huffman@21164
  1285
theorem MVT:
wenzelm@63558
  1286
  fixes a b :: real
wenzelm@63558
  1287
  assumes lt: "a < b"
wenzelm@63558
  1288
    and con: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
wenzelm@63558
  1289
    and dif [rule_format]: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@63558
  1290
  shows "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
huffman@21164
  1291
proof -
wenzelm@63558
  1292
  let ?F = "\<lambda>x. f x - ((f b - f a) / (b - a)) * x"
wenzelm@63558
  1293
  have cont_f: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
hoelzl@56371
  1294
    using con by (fast intro: continuous_intros)
wenzelm@63558
  1295
  have dif_f: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
wenzelm@63558
  1296
  proof clarify
wenzelm@63558
  1297
    fix x :: real
wenzelm@63558
  1298
    assume x: "a < x" "x < b"
wenzelm@63558
  1299
    obtain l where der: "DERIV f x :> l"
wenzelm@63558
  1300
      using differentiableD [OF dif [OF conjI [OF x]]] ..
hoelzl@56181
  1301
    show "?F differentiable (at x)"
wenzelm@63558
  1302
      by (rule differentiableI [where D = "l - (f b - f a) / (b - a)"],
huffman@21164
  1303
          blast intro: DERIV_diff DERIV_cmult_Id der)
huffman@21164
  1304
  qed
wenzelm@63558
  1305
  from Rolle [where f = ?F, OF lt lemma_MVT cont_f dif_f]
wenzelm@63558
  1306
  obtain z where z: "a < z" "z < b" and der: "DERIV ?F z :> 0"
huffman@21164
  1307
    by blast
wenzelm@63558
  1308
  have "DERIV (\<lambda>x. ((f b - f a) / (b - a)) * x) z :> (f b - f a) / (b - a)"
huffman@21164
  1309
    by (rule DERIV_cmult_Id)
wenzelm@63558
  1310
  then have der_f: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z :> 0 + (f b - f a) / (b - a)"
huffman@21164
  1311
    by (rule DERIV_add [OF der])
huffman@21164
  1312
  show ?thesis
huffman@21164
  1313
  proof (intro exI conjI)
wenzelm@63558
  1314
    show "a < z" and "z < b" using z .
wenzelm@63558
  1315
    show "f b - f a = (b - a) * ((f b - f a) / (b - a))" by simp
wenzelm@63558
  1316
    show "DERIV f z :> ((f b - f a) / (b - a))" using der_f by simp
huffman@21164
  1317
  qed
huffman@21164
  1318
qed
huffman@21164
  1319
hoelzl@29803
  1320
lemma MVT2:
wenzelm@63558
  1321
  "a < b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> DERIV f x :> f' x \<Longrightarrow>
wenzelm@63558
  1322
    \<exists>z::real. a < z \<and> z < b \<and> (f b - f a = (b - a) * f' z)"
wenzelm@63558
  1323
  apply (drule MVT)
wenzelm@63558
  1324
    apply (blast intro: DERIV_isCont)
wenzelm@63558
  1325
   apply (force dest: order_less_imp_le simp add: real_differentiable_def)
wenzelm@63558
  1326
  apply (blast dest: DERIV_unique order_less_imp_le)
wenzelm@63558
  1327
  done
hoelzl@29803
  1328
huffman@21164
  1329
wenzelm@63558
  1330
text \<open>A function is constant if its derivative is 0 over an interval.\<close>
huffman@21164
  1331
huffman@21164
  1332
lemma DERIV_isconst_end:
wenzelm@63558
  1333
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1334
  shows "a < b \<Longrightarrow>
wenzelm@63558
  1335
    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
wenzelm@63558
  1336
    \<forall>x. a < x \<and> x < b \<longrightarrow> DERIV f x :> 0 \<Longrightarrow> f b = f a"
wenzelm@63558
  1337
  apply (drule (1) MVT)
wenzelm@63558
  1338
   apply (blast intro: differentiableI)
wenzelm@63558
  1339
  apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
wenzelm@63558
  1340
  done
huffman@21164
  1341
huffman@21164
  1342
lemma DERIV_isconst1:
wenzelm@63558
  1343
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1344
  shows "a < b \<Longrightarrow>
wenzelm@63558
  1345
    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
wenzelm@63558
  1346
    \<forall>x. a < x \<and> x < b \<longrightarrow> DERIV f x :> 0 \<Longrightarrow>
wenzelm@63558
  1347
    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x = f a"
wenzelm@63558
  1348
  apply safe
wenzelm@63558
  1349
  apply (drule_tac x = a in order_le_imp_less_or_eq)
wenzelm@63558
  1350
  apply safe
wenzelm@63558
  1351
  apply (drule_tac b = x in DERIV_isconst_end)
wenzelm@63558
  1352
    apply auto
wenzelm@63558
  1353
  done
huffman@21164
  1354
huffman@21164
  1355
lemma DERIV_isconst2:
wenzelm@63558
  1356
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1357
  shows "a < b \<Longrightarrow>
wenzelm@63558
  1358
    \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
wenzelm@63558
  1359
    \<forall>x. a < x \<and> x < b \<longrightarrow> DERIV f x :> 0 \<Longrightarrow>
wenzelm@63558
  1360
    a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> f x = f a"
wenzelm@63558
  1361
  by (blast dest: DERIV_isconst1)
huffman@21164
  1362
wenzelm@63558
  1363
lemma DERIV_isconst3:
wenzelm@63558
  1364
  fixes a b x y :: real
wenzelm@63558
  1365
  assumes "a < b"
wenzelm@63558
  1366
    and "x \<in> {a <..< b}"
wenzelm@63558
  1367
    and "y \<in> {a <..< b}"
wenzelm@63558
  1368
    and derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
hoelzl@29803
  1369
  shows "f x = f y"
hoelzl@29803
  1370
proof (cases "x = y")
hoelzl@29803
  1371
  case False
hoelzl@29803
  1372
  let ?a = "min x y"
hoelzl@29803
  1373
  let ?b = "max x y"
wenzelm@63558
  1374
hoelzl@29803
  1375
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
hoelzl@29803
  1376
  proof (rule allI, rule impI)
wenzelm@63558
  1377
    fix z :: real
wenzelm@63558
  1378
    assume "?a \<le> z \<and> z \<le> ?b"
wenzelm@63558
  1379
    then have "a < z" and "z < b"
wenzelm@63558
  1380
      using \<open>x \<in> {a <..< b}\<close> and \<open>y \<in> {a <..< b}\<close> by auto
wenzelm@63558
  1381
    then have "z \<in> {a<..<b}" by auto
wenzelm@63558
  1382
    then show "DERIV f z :> 0" by (rule derivable)
hoelzl@29803
  1383
  qed
wenzelm@63558
  1384
  then have isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
wenzelm@63558
  1385
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0"
wenzelm@63558
  1386
    using DERIV_isCont by auto
hoelzl@29803
  1387
wenzelm@60758
  1388
  have "?a < ?b" using \<open>x \<noteq> y\<close> by auto
hoelzl@29803
  1389
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
hoelzl@29803
  1390
  show ?thesis by auto
hoelzl@29803
  1391
qed auto
hoelzl@29803
  1392
huffman@21164
  1393
lemma DERIV_isconst_all:
wenzelm@63558
  1394
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1395
  shows "\<forall>x. DERIV f x :> 0 \<Longrightarrow> f x = f y"
wenzelm@63558
  1396
  apply (rule linorder_cases [of x y])
wenzelm@63558
  1397
    apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
wenzelm@63558
  1398
  done
huffman@21164
  1399
huffman@21164
  1400
lemma DERIV_const_ratio_const:
wenzelm@63558
  1401
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1402
  shows "a \<noteq> b \<Longrightarrow> \<forall>x. DERIV f x :> k \<Longrightarrow> f b - f a = (b - a) * k"
wenzelm@63558
  1403
  apply (rule linorder_cases [of a b])
wenzelm@63558
  1404
    apply auto
wenzelm@63558
  1405
   apply (drule_tac [!] f = f in MVT)
wenzelm@63558
  1406
       apply (auto dest: DERIV_isCont DERIV_unique simp: real_differentiable_def)
wenzelm@63558
  1407
  apply (auto dest: DERIV_unique simp: ring_distribs)
wenzelm@63558
  1408
  done
huffman@21164
  1409
huffman@21164
  1410
lemma DERIV_const_ratio_const2:
wenzelm@63558
  1411
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1412
  shows "a \<noteq> b \<Longrightarrow> \<forall>x. DERIV f x :> k \<Longrightarrow> (f b - f a) / (b - a) = k"
wenzelm@63558
  1413
  apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
wenzelm@63558
  1414
   apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
wenzelm@63558
  1415
  done
huffman@21164
  1416
wenzelm@63558
  1417
lemma real_average_minus_first [simp]: "(a + b) / 2 - a = (b - a) / 2"
wenzelm@63558
  1418
  for a b :: real
wenzelm@63558
  1419
  by simp
huffman@21164
  1420
wenzelm@63558
  1421
lemma real_average_minus_second [simp]: "(b + a) / 2 - a = (b - a) / 2"
wenzelm@63558
  1422
  for a b :: real
wenzelm@63558
  1423
  by simp
huffman@21164
  1424
wenzelm@63558
  1425
text \<open>Gallileo's "trick": average velocity = av. of end velocities.\<close>
huffman@21164
  1426
huffman@21164
  1427
lemma DERIV_const_average:
wenzelm@63558
  1428
  fixes v :: "real \<Rightarrow> real"
wenzelm@63558
  1429
    and a b :: real
wenzelm@63558
  1430
  assumes neq: "a \<noteq> b"
wenzelm@63558
  1431
    and der: "\<forall>x. DERIV v x :> k"
wenzelm@63558
  1432
  shows "v ((a + b) / 2) = (v a + v b) / 2"
huffman@21164
  1433
proof (cases rule: linorder_cases [of a b])
wenzelm@63558
  1434
  case equal
wenzelm@63558
  1435
  with neq show ?thesis by simp
huffman@21164
  1436
next
huffman@21164
  1437
  case less
huffman@21164
  1438
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1439
    by (rule DERIV_const_ratio_const2 [OF neq der])
wenzelm@63558
  1440
  then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k"
wenzelm@63558
  1441
    by simp
huffman@21164
  1442
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
wenzelm@63558
  1443
    by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq)
wenzelm@63558
  1444
  ultimately show ?thesis
wenzelm@63558
  1445
    using neq by force
huffman@21164
  1446
next
huffman@21164
  1447
  case greater
huffman@21164
  1448
  have "(v b - v a) / (b - a) = k"
huffman@21164
  1449
    by (rule DERIV_const_ratio_const2 [OF neq der])
wenzelm@63558
  1450
  then have "(b - a) * ((v b - v a) / (b - a)) = (b - a) * k"
wenzelm@63558
  1451
    by simp
huffman@21164
  1452
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
wenzelm@63558
  1453
    by (rule DERIV_const_ratio_const2 [OF _ der]) (simp add: neq)
wenzelm@63558
  1454
  ultimately show ?thesis
wenzelm@63558
  1455
    using neq by (force simp add: add.commute)
huffman@21164
  1456
qed
huffman@21164
  1457
wenzelm@63558
  1458
text \<open>
wenzelm@63558
  1459
  A function with positive derivative is increasing.
wenzelm@63558
  1460
  A simple proof using the MVT, by Jeremy Avigad. And variants.
wenzelm@63558
  1461
\<close>
lp15@56261
  1462
lemma DERIV_pos_imp_increasing_open:
wenzelm@63558
  1463
  fixes a b :: real
wenzelm@63558
  1464
    and f :: "real \<Rightarrow> real"
wenzelm@63558
  1465
  assumes "a < b"
wenzelm@63558
  1466
    and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
wenzelm@63558
  1467
    and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
paulson@33654
  1468
  shows "f a < f b"
paulson@33654
  1469
proof (rule ccontr)
wenzelm@63558
  1470
  assume f: "\<not> ?thesis"
wenzelm@63558
  1471
  have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
wenzelm@63558
  1472
    by (rule MVT) (use assms Deriv.differentiableI in \<open>force+\<close>)
wenzelm@63558
  1473
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" and "f b - f a = (b - a) * l"
paulson@33654
  1474
    by auto
wenzelm@63558
  1475
  with assms f have "\<not> l > 0"
huffman@36777
  1476
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
wenzelm@41550
  1477
  with assms z show False
lp15@56261
  1478
    by (metis DERIV_unique)
paulson@33654
  1479
qed
paulson@33654
  1480
lp15@56261
  1481
lemma DERIV_pos_imp_increasing:
wenzelm@63558
  1482
  fixes a b :: real
wenzelm@63558
  1483
    and f :: "real \<Rightarrow> real"
wenzelm@63558
  1484
  assumes "a < b"
wenzelm@63558
  1485
    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
lp15@56261
  1486
  shows "f a < f b"
wenzelm@63558
  1487
  by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)
lp15@56261
  1488
noschinl@45791
  1489
lemma DERIV_nonneg_imp_nondecreasing:
wenzelm@63558
  1490
  fixes a b :: real
wenzelm@63558
  1491
    and f :: "real \<Rightarrow> real"
wenzelm@63558
  1492
  assumes "a \<le> b"
wenzelm@63558
  1493
    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y \<ge> 0)"
paulson@33654
  1494
  shows "f a \<le> f b"
paulson@33654
  1495
proof (rule ccontr, cases "a = b")
wenzelm@63558
  1496
  assume "\<not> ?thesis" and "a = b"
wenzelm@41550
  1497
  then show False by auto
haftmann@37891
  1498
next
wenzelm@63558
  1499
  assume *: "\<not> ?thesis"
wenzelm@63558
  1500
  assume "a \<noteq> b"
wenzelm@63558
  1501
  with assms have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
wenzelm@33690
  1502
    apply -
wenzelm@33690
  1503
    apply (rule MVT)
wenzelm@33690
  1504
      apply auto
wenzelm@63558
  1505
     apply (metis DERIV_isCont)
wenzelm@63558
  1506
    apply (metis differentiableI less_le)
paulson@33654
  1507
    done
wenzelm@63558
  1508
  then obtain l z where lz: "a < z" "z < b" "DERIV f z :> l" and **: "f b - f a = (b - a) * l"
paulson@33654
  1509
    by auto
wenzelm@63558
  1510
  with * have "a < b" "f b < f a" by auto
wenzelm@63558
  1511
  with ** have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
wenzelm@63558
  1512
    (metis * add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
wenzelm@63558
  1513
  with assms lz show False
paulson@33654
  1514
    by (metis DERIV_unique order_less_imp_le)
paulson@33654
  1515
qed
paulson@33654
  1516
lp15@56261
  1517
lemma DERIV_neg_imp_decreasing_open:
wenzelm@63558
  1518
  fixes a b :: real
wenzelm@63558
  1519
    and f :: "real \<Rightarrow> real"
wenzelm@63558
  1520
  assumes "a < b"
wenzelm@63558
  1521
    and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y < 0)"
wenzelm@63558
  1522
    and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
lp15@56261
  1523
  shows "f a > f b"
lp15@56261
  1524
proof -
wenzelm@63558
  1525
  have "(\<lambda>x. -f x) a < (\<lambda>x. -f x) b"
wenzelm@63558
  1526
    apply (rule DERIV_pos_imp_increasing_open [of a b "\<lambda>x. -f x"])
lp15@56261
  1527
    using assms
wenzelm@63558
  1528
      apply auto
lp15@56261
  1529
    apply (metis field_differentiable_minus neg_0_less_iff_less)
lp15@56261
  1530
    done
wenzelm@63558
  1531
  then show ?thesis
lp15@56261
  1532
    by simp
lp15@56261
  1533
qed
lp15@56261
  1534
paulson@33654
  1535
lemma DERIV_neg_imp_decreasing:
wenzelm@63558
  1536
  fixes a b :: real
wenzelm@63558
  1537
    and f :: "real \<Rightarrow> real"
wenzelm@63558
  1538
  assumes "a < b"
wenzelm@63558
  1539
    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y < 0)"
paulson@33654
  1540
  shows "f a > f b"
wenzelm@63558
  1541
  by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)
paulson@33654
  1542
paulson@33654
  1543
lemma DERIV_nonpos_imp_nonincreasing:
wenzelm@63558
  1544
  fixes a b :: real
wenzelm@63558
  1545
    and f :: "real \<Rightarrow> real"
wenzelm@63558
  1546
  assumes "a \<le> b"
wenzelm@63558
  1547
    and "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> (\<exists>y. DERIV f x :> y \<and> y \<le> 0)"
paulson@33654
  1548
  shows "f a \<ge> f b"
paulson@33654
  1549
proof -
wenzelm@63558
  1550
  have "(\<lambda>x. -f x) a \<le> (\<lambda>x. -f x) b"
wenzelm@63558
  1551
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "\<lambda>x. -f x"])
wenzelm@33690
  1552
    using assms
wenzelm@63558
  1553
     apply auto
paulson@33654
  1554
    apply (metis DERIV_minus neg_0_le_iff_le)
paulson@33654
  1555
    done
wenzelm@63558
  1556
  then show ?thesis
paulson@33654
  1557
    by simp
paulson@33654
  1558
qed
huffman@21164
  1559
lp15@56289
  1560
lemma DERIV_pos_imp_increasing_at_bot:
wenzelm@63558
  1561
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1562
  assumes "\<And>x. x \<le> b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
wenzelm@63558
  1563
    and lim: "(f \<longlongrightarrow> flim) at_bot"
lp15@56289
  1564
  shows "flim < f b"
lp15@56289
  1565
proof -
lp15@63952
  1566
  have "\<exists>N. \<forall>n\<le>N. f n \<le> f (b - 1)"
lp15@56289
  1567
    apply (rule_tac x="b - 2" in exI)
lp15@56289
  1568
    apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289
  1569
    done
lp15@63952
  1570
  then have "flim \<le> f (b - 1)"
lp15@63952
  1571
     by (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder tendsto_upperbound [OF lim])
wenzelm@63558
  1572
  also have "\<dots> < f b"
lp15@56289
  1573
    by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
lp15@56289
  1574
  finally show ?thesis .
lp15@56289
  1575
qed
lp15@56289
  1576
lp15@56289
  1577
lemma DERIV_neg_imp_decreasing_at_top:
wenzelm@63558
  1578
  fixes f :: "real \<Rightarrow> real"
wenzelm@63558
  1579
  assumes der: "\<And>x. x \<ge> b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y < 0)"
wenzelm@63558
  1580
    and lim: "(f \<longlongrightarrow> flim) at_top"
lp15@56289
  1581
  shows "flim < f b"
lp15@56289
  1582
  apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
wenzelm@63558
  1583
   apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
lp15@56289
  1584
  apply (metis filterlim_at_top_mirror lim)
lp15@56289
  1585
  done
lp15@56289
  1586
wenzelm@60758
  1587
text \<open>Derivative of inverse function\<close>
huffman@23041
  1588
huffman@23041
  1589
lemma DERIV_inverse_function:
huffman@23041
  1590
  fixes f g :: "real \<Rightarrow> real"
huffman@23041
  1591
  assumes der: "DERIV f (g x) :> D"
wenzelm@63558
  1592
    and neq: "D \<noteq> 0"
wenzelm@63558
  1593
    and x: "a < x" "x < b"
wenzelm@63558
  1594
    and inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
wenzelm@63558
  1595
    and cont: "isCont g x"
huffman@23041
  1596
  shows "DERIV g x :> inverse D"
huffman@23041
  1597
unfolding DERIV_iff2
huffman@23044
  1598
proof (rule LIM_equal2)
huffman@23044
  1599
  show "0 < min (x - a) (b - x)"
wenzelm@63558
  1600
    using x by arith
huffman@23044
  1601
next
huffman@23041
  1602
  fix y
huffman@23044
  1603
  assume "norm (y - x) < min (x - a) (b - x)"
wenzelm@63558
  1604
  then have "a < y" and "y < b"
huffman@23044
  1605
    by (simp_all add: abs_less_iff)
wenzelm@63558
  1606
  then show "(g y - g x) / (y - x) = inverse ((f (g y) - x) / (g y - g x))"
huffman@23041
  1607
    by (simp add: inj)
huffman@23041
  1608
next
wenzelm@61976
  1609
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) \<midarrow>g x\<rightarrow> D"
huffman@23041
  1610
    by (rule der [unfolded DERIV_iff2])
wenzelm@63558
  1611
  then have 1: "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
wenzelm@63558
  1612
    using inj x by simp
huffman@23041
  1613
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
wenzelm@56219
  1614
  proof (rule exI, safe)
huffman@23044
  1615
    show "0 < min (x - a) (b - x)"
wenzelm@63558
  1616
      using x by simp
huffman@23041
  1617
  next
huffman@23041
  1618
    fix y
huffman@23044
  1619
    assume "norm (y - x) < min (x - a) (b - x)"
wenzelm@63558
  1620
    then have y: "a < y" "y < b"
huffman@23044
  1621
      by (simp_all add: abs_less_iff)
huffman@23041
  1622
    assume "g y = g x"
wenzelm@63558
  1623
    then have "f (g y) = f (g x)" by simp
wenzelm@63558
  1624
    then have "y = x" using inj y x by simp
huffman@23041
  1625
    also assume "y \<noteq> x"
huffman@23041
  1626
    finally show False by simp
huffman@23041
  1627
  qed
wenzelm@61976
  1628
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) \<midarrow>x\<rightarrow> D"
huffman@23041
  1629
    using cont 1 2 by (rule isCont_LIM_compose2)
wenzelm@63558
  1630
  then show "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x))) \<midarrow>x\<rightarrow> inverse D"
huffman@44568
  1631
    using neq by (rule tendsto_inverse)
huffman@23041
  1632
qed
huffman@23041
  1633
wenzelm@60758
  1634
subsection \<open>Generalized Mean Value Theorem\<close>
huffman@29975
  1635
huffman@21164
  1636
theorem GMVT:
huffman@21784
  1637
  fixes a b :: real
huffman@21164
  1638
  assumes alb: "a < b"
wenzelm@41550
  1639
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
hoelzl@56181
  1640
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
wenzelm@41550
  1641
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
hoelzl@56181
  1642
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
wenzelm@53381
  1643
  shows "\<exists>g'c f'c c.
wenzelm@63558
  1644
    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
  1645
proof -
wenzelm@63558
  1646
  let ?h = "\<lambda>x. (f b - f a) * g x - (g b - g a) * f x"
wenzelm@63558
  1647
  have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l"
wenzelm@63558
  1648
  proof (rule MVT)
wenzelm@63558
  1649
    from assms show "a < b" by simp
wenzelm@63558
  1650
    show "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
wenzelm@63558
  1651
      using fc gc by simp
wenzelm@63558
  1652
    show "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
wenzelm@63558
  1653
      using fd gd by simp
wenzelm@63558
  1654
  qed
wenzelm@63558
  1655
  then obtain l where l: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
wenzelm@63558
  1656
  then obtain c where c: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
huffman@21164
  1657
wenzelm@63558
  1658
  from c have cint: "a < c \<and> c < b" by auto
hoelzl@56181
  1659
  with gd have "g differentiable (at c)" by simp
wenzelm@63558
  1660
  then have "\<exists>D. DERIV g c :> D" by (rule differentiableD)
wenzelm@63558
  1661
  then obtain g'c where g'c: "DERIV g c :> g'c" ..
huffman@21164
  1662
wenzelm@63558
  1663
  from c have "a < c \<and> c < b" by auto
hoelzl@56181
  1664
  with fd have "f differentiable (at c)" by simp
wenzelm@63558
  1665
  then have "\<exists>D. DERIV f c :> D" by (rule differentiableD)
wenzelm@63558
  1666
  then obtain f'c where f'c: "DERIV f c :> f'c" ..
huffman@21164
  1667
wenzelm@63558
  1668
  from c have "DERIV ?h c :> l" by auto
hoelzl@41368
  1669
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
wenzelm@63558
  1670
    using g'c f'c by (auto intro!: derivative_eq_intros)
huffman@21164
  1671
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)
huffman@21164
  1672
wenzelm@63558
  1673
  have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))"
wenzelm@63558
  1674
  proof -
wenzelm@63558
  1675
    from c have "?h b - ?h a = (b - a) * l" by auto
wenzelm@53374
  1676
    also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
wenzelm@63558
  1677
    finally show ?thesis by simp
wenzelm@63558
  1678
  qed
wenzelm@63558
  1679
  moreover have "?h b - ?h a = 0"
wenzelm@63558
  1680
  proof -
huffman@21164
  1681
    have "?h b - ?h a =
wenzelm@63558
  1682
      ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
wenzelm@63558
  1683
      ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
nipkow@29667
  1684
      by (simp add: algebra_simps)
wenzelm@63558
  1685
    then show ?thesis  by auto
wenzelm@63558
  1686
  qed
huffman@21164
  1687
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
huffman@21164
  1688
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
wenzelm@63558
  1689
  then have "g'c * (f b - f a) = f'c * (g b - g a)" by simp
wenzelm@63558
  1690
  then have "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
wenzelm@63558
  1691
  with g'c f'c cint show ?thesis by auto
huffman@21164
  1692
qed
huffman@21164
  1693
hoelzl@50327
  1694
lemma GMVT':
hoelzl@50327
  1695
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1696
  assumes "a < b"
wenzelm@63558
  1697
    and isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
wenzelm@63558
  1698
    and isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
wenzelm@63558
  1699
    and DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
wenzelm@63558
  1700
    and DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
hoelzl@50327
  1701
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1702
proof -
hoelzl@50327
  1703
  have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
wenzelm@63558
  1704
      a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
hoelzl@56181
  1705
    using assms by (intro GMVT) (force simp: real_differentiable_def)+
hoelzl@50327
  1706
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
hoelzl@50327
  1707
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
hoelzl@50327
  1708
  then show ?thesis
hoelzl@50327
  1709
    by auto
hoelzl@50327
  1710
qed
hoelzl@50327
  1711
hoelzl@51529
  1712
wenzelm@60758
  1713
subsection \<open>L'Hopitals rule\<close>
hoelzl@51529
  1714
hoelzl@51641
  1715
lemma isCont_If_ge:
hoelzl@51641
  1716
  fixes a :: "'a :: linorder_topology"
wenzelm@63558
  1717
  shows "continuous (at_left a) g \<Longrightarrow> (f \<longlongrightarrow> g a) (at_right a) \<Longrightarrow>
wenzelm@63558
  1718
    isCont (\<lambda>x. if x \<le> a then g x else f x) a"
hoelzl@51641
  1719
  unfolding isCont_def continuous_within
hoelzl@51641
  1720
  apply (intro filterlim_split_at)
wenzelm@63558
  1721
   apply (subst filterlim_cong[OF refl refl, where g=g])
wenzelm@63558
  1722
    apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1723
  apply (subst filterlim_cong[OF refl refl, where g=f])
wenzelm@63558
  1724
   apply (simp_all add: eventually_at_filter less_le)
hoelzl@51641
  1725
  done
hoelzl@51641
  1726
hoelzl@50327
  1727
lemma lhopital_right_0:
hoelzl@50329
  1728
  fixes f0 g0 :: "real \<Rightarrow> real"
wenzelm@61973
  1729
  assumes f_0: "(f0 \<longlongrightarrow> 0) (at_right 0)"
wenzelm@63558
  1730
    and g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
wenzelm@63558
  1731
    and ev:
wenzelm@63558
  1732
      "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
wenzelm@63558
  1733
      "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
wenzelm@63558
  1734
      "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
wenzelm@63558
  1735
      "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
eberlm@63713
  1736
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) F (at_right 0)"
eberlm@63713
  1737
  shows "filterlim (\<lambda> x. f0 x / g0 x) F (at_right 0)"
hoelzl@50327
  1738
proof -
wenzelm@63040
  1739
  define f where [abs_def]: "f x = (if x \<le> 0 then 0 else f0 x)" for x
hoelzl@50329
  1740
  then have "f 0 = 0" by simp
hoelzl@50329
  1741
wenzelm@63040
  1742
  define g where [abs_def]: "g x = (if x \<le> 0 then 0 else g0 x)" for x
hoelzl@50329
  1743
  then have "g 0 = 0" by simp
hoelzl@50329
  1744
hoelzl@50329
  1745
  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
hoelzl@50329
  1746
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
hoelzl@50329
  1747
    using ev by eventually_elim auto
hoelzl@50329
  1748
  then obtain a where [arith]: "0 < a"
hoelzl@50329
  1749
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
hoelzl@50327
  1750
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50329
  1751
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
hoelzl@50329
  1752
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
wenzelm@56219
  1753
    unfolding eventually_at by (auto simp: dist_real_def)
hoelzl@50327
  1754
hoelzl@50329
  1755
  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
hoelzl@50329
  1756
    using g0_neq_0 by (simp add: g_def)
hoelzl@50329
  1757
wenzelm@63558
  1758
  have f: "DERIV f x :> (f' x)" if x: "0 < x" "x < a" for x
wenzelm@63558
  1759
    using that
wenzelm@63558
  1760
    by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
wenzelm@63558
  1761
      (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])
hoelzl@50329
  1762
wenzelm@63558
  1763
  have g: "DERIV g x :> (g' x)" if x: "0 < x" "x < a" for x
wenzelm@63558
  1764
    using that
wenzelm@63558
  1765
    by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
wenzelm@63558
  1766
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])
hoelzl@50329
  1767
hoelzl@50329
  1768
  have "isCont f 0"
hoelzl@51641
  1769
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
hoelzl@51641
  1770
hoelzl@50329
  1771
  have "isCont g 0"
hoelzl@51641
  1772
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
hoelzl@50329
  1773
hoelzl@50327
  1774
  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)"
wenzelm@63558
  1775
  proof (rule bchoice, rule ballI)
wenzelm@63558
  1776
    fix x
wenzelm@63558
  1777
    assume "x \<in> {0 <..< a}"
hoelzl@50327
  1778
    then have x[arith]: "0 < x" "x < a" by auto
wenzelm@60758
  1779
    with g'_neq_0 g_neq_0 \<open>g 0 = 0\<close> have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
hoelzl@50327
  1780
      by auto
hoelzl@50328
  1781
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
wenzelm@60758
  1782
      using \<open>isCont f 0\<close> f by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1783
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
wenzelm@60758
  1784
      using \<open>isCont g 0\<close> g by (auto intro: DERIV_isCont simp: le_less)
hoelzl@50328
  1785
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@60758
  1786
      using f g \<open>x < a\<close> by (intro GMVT') auto
wenzelm@53374
  1787
    then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
wenzelm@53374
  1788
      by blast
hoelzl@50327
  1789
    moreover
wenzelm@53374
  1790
    from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
hoelzl@50327
  1791
      by (simp add: field_simps)
hoelzl@50327
  1792
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
wenzelm@60758
  1793
      using \<open>f 0 = 0\<close> \<open>g 0 = 0\<close> by (auto intro!: exI[of _ c])
hoelzl@50327
  1794
  qed
wenzelm@53381
  1795
  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
  1796
  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
  1797
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
hoelzl@50327
  1798
  moreover
hoelzl@50327
  1799
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
hoelzl@50327
  1800
    by eventually_elim auto
wenzelm@61973
  1801
  then have "((\<lambda>x. norm (\<zeta> x)) \<longlongrightarrow> 0) (at_right 0)"
hoelzl@58729
  1802
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"]) auto
wenzelm@61973
  1803
  then have "(\<zeta> \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1804
    by (rule tendsto_norm_zero_cancel)
hoelzl@50327
  1805
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
lp15@61810
  1806
    by (auto elim!: eventually_mono simp: filterlim_at)
eberlm@63713
  1807
  from this lim have "filterlim (\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) F (at_right 0)"
hoelzl@50327
  1808
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
eberlm@63713
  1809
  ultimately have "filterlim (\<lambda>t. f t / g t) F (at_right 0)" (is ?P)
hoelzl@50328
  1810
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
lp15@61810
  1811
       (auto elim: eventually_mono)
hoelzl@50329
  1812
  also have "?P \<longleftrightarrow> ?thesis"
hoelzl@51641
  1813
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
hoelzl@50329
  1814
  finally show ?thesis .
hoelzl@50327
  1815
qed
hoelzl@50327
  1816
hoelzl@50330
  1817
lemma lhopital_right:
wenzelm@63558
  1818
  "(f \<longlongrightarrow> 0) (at_right x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1819
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1820
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1821
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1822
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
eberlm@63713
  1823
    filterlim (\<lambda> x. (f' x / g' x)) F (at_right x) \<Longrightarrow>
eberlm@63713
  1824
  filterlim (\<lambda> x. f x / g x) F (at_right x)"
wenzelm@63558
  1825
  for x :: real
hoelzl@50330
  1826
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1827
  by (rule lhopital_right_0)
hoelzl@50330
  1828
hoelzl@50330
  1829
lemma lhopital_left:
wenzelm@63558
  1830
  "(f \<longlongrightarrow> 0) (at_left x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1831
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1832
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1833
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1834
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
eberlm@63713
  1835
    filterlim (\<lambda> x. (f' x / g' x)) F (at_left x) \<Longrightarrow>
eberlm@63713
  1836
  filterlim (\<lambda> x. f x / g x) F (at_left x)"
wenzelm@63558
  1837
  for x :: real
hoelzl@50330
  1838
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@56479
  1839
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1840
hoelzl@50330
  1841
lemma lhopital:
wenzelm@63558
  1842
  "(f \<longlongrightarrow> 0) (at x) \<Longrightarrow> (g \<longlongrightarrow> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1843
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1844
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1845
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1846
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
eberlm@63713
  1847
    filterlim (\<lambda> x. (f' x / g' x)) F (at x) \<Longrightarrow>
eberlm@63713
  1848
  filterlim (\<lambda> x. f x / g x) F (at x)"
wenzelm@63558
  1849
  for x :: real
hoelzl@50330
  1850
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1851
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
hoelzl@50330
  1852
eberlm@63713
  1853
hoelzl@50327
  1854
lemma lhopital_right_0_at_top:
hoelzl@50327
  1855
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50327
  1856
  assumes g_0: "LIM x at_right 0. g x :> at_top"
wenzelm@63558
  1857
    and ev:
wenzelm@63558
  1858
      "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
wenzelm@63558
  1859
      "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
wenzelm@63558
  1860
      "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
wenzelm@63558
  1861
    and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
wenzelm@61973
  1862
  shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
hoelzl@50327
  1863
  unfolding tendsto_iff
hoelzl@50327
  1864
proof safe
wenzelm@63558
  1865
  fix e :: real
wenzelm@63558
  1866
  assume "0 < e"
hoelzl@50327
  1867
  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
wenzelm@63558
  1868
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)"
wenzelm@63558
  1869
    by simp
hoelzl@50327
  1870
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
hoelzl@50327
  1871
  obtain a where [arith]: "0 < a"
hoelzl@50327
  1872
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
hoelzl@50327
  1873
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
hoelzl@50327
  1874
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
hoelzl@50327
  1875
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
hoelzl@51641
  1876
    unfolding eventually_at_le by (auto simp: dist_real_def)
hoelzl@50327
  1877
wenzelm@63558
  1878
  from Df have "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
hoelzl@51641
  1879
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
hoelzl@50327
  1880
hoelzl@50327
  1881
  moreover
hoelzl@50328
  1882
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
lp15@61810
  1883
    using g_0 by (auto elim: eventually_mono simp: filterlim_at_top_dense)
hoelzl@50327
  1884
hoelzl@50327
  1885
  moreover
wenzelm@61973
  1886
  have inv_g: "((\<lambda>x. inverse (g x)) \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1887
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
hoelzl@50327
  1888
    by (rule filterlim_compose)
wenzelm@61973
  1889
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) \<longlongrightarrow> norm (1 - g a * 0)) (at_right 0)"
hoelzl@50327
  1890
    by (intro tendsto_intros)
wenzelm@61973
  1891
  then have "((\<lambda>x. norm (1 - g a / g x)) \<longlongrightarrow> 1) (at_right 0)"
hoelzl@50327
  1892
    by (simp add: inverse_eq_divide)
hoelzl@50327
  1893
  from this[unfolded tendsto_iff, rule_format, of 1]
hoelzl@50327
  1894
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
lp15@61810
  1895
    by (auto elim!: eventually_mono simp: dist_real_def)
hoelzl@50327
  1896
hoelzl@50327
  1897
  moreover
wenzelm@63558
  1898
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) \<longlongrightarrow> norm ((f a - x * g a) * 0))
wenzelm@63558
  1899
      (at_right 0)"
hoelzl@50327
  1900
    by (intro tendsto_intros)
wenzelm@61973
  1901
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) \<longlongrightarrow> 0) (at_right 0)"
hoelzl@50327
  1902
    by (simp add: inverse_eq_divide)
wenzelm@60758
  1903
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] \<open>0 < e\<close>
hoelzl@50327
  1904
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
hoelzl@50327
  1905
    by (auto simp: dist_real_def)
hoelzl@50327
  1906
hoelzl@50327
  1907
  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
hoelzl@50327
  1908
  proof eventually_elim
hoelzl@50327
  1909
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
hoelzl@50327
  1910
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
hoelzl@50327
  1911
hoelzl@50327
  1912
    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
hoelzl@50327
  1913
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
wenzelm@53381
  1914
    then obtain y where [arith]: "t < y" "y < a"
wenzelm@53381
  1915
      and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
wenzelm@53381
  1916
      by blast
wenzelm@53381
  1917
    from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
wenzelm@60758
  1918
      using \<open>g a < g t\<close> g'_neq_0[of y] by (auto simp add: field_simps)
hoelzl@50327
  1919
hoelzl@50327
  1920
    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
  1921
      by (simp add: field_simps)
hoelzl@50327
  1922
    have "norm (f t / g t - x) \<le>
hoelzl@50327
  1923
        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
  1924
      unfolding * by (rule norm_triangle_ineq)
hoelzl@50327
  1925
    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
  1926
      by (simp add: abs_mult D_eq dist_real_def)
hoelzl@50327
  1927
    also have "\<dots> < (e / 4) * 2 + e / 2"
wenzelm@60758
  1928
      using ineq Df[of y] \<open>0 < e\<close> by (intro add_le_less_mono mult_mono) auto
hoelzl@50327
  1929
    finally show "dist (f t / g t) x < e"
hoelzl@50327
  1930
      by (simp add: dist_real_def)
hoelzl@50327
  1931
  qed
hoelzl@50327
  1932
qed
hoelzl@50327
  1933
hoelzl@50330
  1934
lemma lhopital_right_at_top:
hoelzl@50330
  1935
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1936
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
hoelzl@50330
  1937
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
hoelzl@50330
  1938
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
wenzelm@61973
  1939
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_right x) \<Longrightarrow>
wenzelm@61973
  1940
    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_right x)"
hoelzl@50330
  1941
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
hoelzl@50330
  1942
  by (rule lhopital_right_0_at_top)
hoelzl@50330
  1943
hoelzl@50330
  1944
lemma lhopital_left_at_top:
wenzelm@63558
  1945
  "LIM x at_left x. g x :> at_top \<Longrightarrow>
hoelzl@50330
  1946
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
hoelzl@50330
  1947
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
hoelzl@50330
  1948
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
wenzelm@61973
  1949
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at_left x) \<Longrightarrow>
wenzelm@61973
  1950
    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at_left x)"
wenzelm@63558
  1951
  for x :: real
hoelzl@50330
  1952
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
hoelzl@56479
  1953
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
hoelzl@50330
  1954
hoelzl@50330
  1955
lemma lhopital_at_top:
hoelzl@50330
  1956
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
hoelzl@50330
  1957
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
hoelzl@50330
  1958
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
hoelzl@50330
  1959
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
wenzelm@61973
  1960
    ((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> y) (at x) \<Longrightarrow>
wenzelm@61973
  1961
    ((\<lambda> x. f x / g x) \<longlongrightarrow> y) (at x)"
hoelzl@50330
  1962
  unfolding eventually_at_split filterlim_at_split
hoelzl@50330
  1963
  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
  1964
hoelzl@50347
  1965
lemma lhospital_at_top_at_top:
hoelzl@50347
  1966
  fixes f g :: "real \<Rightarrow> real"
hoelzl@50347
  1967
  assumes g_0: "LIM x at_top. g x :> at_top"
wenzelm@63558
  1968
    and g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
wenzelm@63558
  1969
    and Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
wenzelm@63558
  1970
    and Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
wenzelm@63558
  1971
    and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
wenzelm@61973
  1972
  shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) at_top"
hoelzl@50347
  1973
  unfolding filterlim_at_top_to_right
hoelzl@50347
  1974
proof (rule lhopital_right_0_at_top)
hoelzl@50347
  1975
  let ?F = "\<lambda>x. f (inverse x)"
hoelzl@50347
  1976
  let ?G = "\<lambda>x. g (inverse x)"
hoelzl@50347
  1977
  let ?R = "at_right (0::real)"
hoelzl@50347
  1978
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
hoelzl@50347
  1979
  show "LIM x ?R. ?G x :> at_top"
hoelzl@50347
  1980
    using g_0 unfolding filterlim_at_top_to_right .
hoelzl@50347
  1981
  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
hoelzl@50347
  1982
    unfolding eventually_at_right_to_top
wenzelm@63558
  1983
    using Dg eventually_ge_at_top[where c=1]
hoelzl@50347
  1984
    apply eventually_elim
hoelzl@50347
  1985
    apply (rule DERIV_cong)
wenzelm@63558
  1986
     apply (rule DERIV_chain'[where f=inverse])
wenzelm@63558
  1987
      apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1988
    done
hoelzl@50347
  1989
  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
hoelzl@50347
  1990
    unfolding eventually_at_right_to_top
wenzelm@63558
  1991
    using Df eventually_ge_at_top[where c=1]
hoelzl@50347
  1992
    apply eventually_elim
hoelzl@50347
  1993
    apply (rule DERIV_cong)
wenzelm@63558
  1994
     apply (rule DERIV_chain'[where f=inverse])
wenzelm@63558
  1995
      apply (auto intro!:  DERIV_inverse)
hoelzl@50347
  1996
    done
hoelzl@50347
  1997
  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
hoelzl@50347
  1998
    unfolding eventually_at_right_to_top
wenzelm@63558
  1999
    using g' eventually_ge_at_top[where c=1]
hoelzl@50347
  2000
    by eventually_elim auto
wenzelm@61973
  2001
  show "((\<lambda>x. ?D f' x / ?D g' x) \<longlongrightarrow> x) ?R"
hoelzl@50347
  2002
    unfolding filterlim_at_right_to_top
hoelzl@50347
  2003
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
wenzelm@63558
  2004
    using eventually_ge_at_top[where c=1]
hoelzl@56479
  2005
    by eventually_elim simp
hoelzl@50347
  2006
qed
hoelzl@50347
  2007
eberlm@63713
  2008
lemma lhopital_right_at_top_at_top:
eberlm@63713
  2009
  fixes f g :: "real \<Rightarrow> real"
eberlm@63713
  2010
  assumes f_0: "LIM x at_right a. f x :> at_top"
eberlm@63713
  2011
  assumes g_0: "LIM x at_right a. g x :> at_top"
eberlm@63713
  2012
    and ev:
eberlm@63713
  2013
      "eventually (\<lambda>x. DERIV f x :> f' x) (at_right a)"
eberlm@63713
  2014
      "eventually (\<lambda>x. DERIV g x :> g' x) (at_right a)"
eberlm@63713
  2015
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at_right a)"
eberlm@63713
  2016
  shows "filterlim (\<lambda> x. f x / g x) at_top (at_right a)"
eberlm@63713
  2017
proof -
eberlm@63713
  2018
  from lim have pos: "eventually (\<lambda>x. f' x / g' x > 0) (at_right a)"
eberlm@63713
  2019
    unfolding filterlim_at_top_dense by blast
eberlm@63713
  2020
  have "((\<lambda>x. g x / f x) \<longlongrightarrow> 0) (at_right a)"
eberlm@63713
  2021
  proof (rule lhopital_right_at_top)
eberlm@63713
  2022
    from pos show "eventually (\<lambda>x. f' x \<noteq> 0) (at_right a)" by eventually_elim auto
eberlm@63713
  2023
    from tendsto_inverse_0_at_top[OF lim]
eberlm@63713
  2024
      show "((\<lambda>x. g' x / f' x) \<longlongrightarrow> 0) (at_right a)" by simp
eberlm@63713
  2025
  qed fact+
eberlm@63713
  2026
  moreover from f_0 g_0 
eberlm@63713
  2027
    have "eventually (\<lambda>x. f x > 0) (at_right a)" "eventually (\<lambda>x. g x > 0) (at_right a)"
eberlm@63713
  2028
    unfolding filterlim_at_top_dense by blast+
eberlm@63713
  2029
  hence "eventually (\<lambda>x. g x / f x > 0) (at_right a)" by eventually_elim simp
eberlm@63713
  2030
  ultimately have "filterlim (\<lambda>x. inverse (g x / f x)) at_top (at_right a)"
eberlm@63713
  2031
    by (rule filterlim_inverse_at_top)
eberlm@63713
  2032
  thus ?thesis by simp
eberlm@63713
  2033
qed
hoelzl@63717
  2034
eberlm@63713
  2035
lemma lhopital_right_at_top_at_bot:
eberlm@63713
  2036
  fixes f g :: "real \<Rightarrow> real"
eberlm@63713
  2037
  assumes f_0: "LIM x at_right a. f x :> at_top"
eberlm@63713
  2038
  assumes g_0: "LIM x at_right a. g x :> at_bot"
eberlm@63713
  2039
    and ev:
eberlm@63713
  2040
      "eventually (\<lambda>x. DERIV f x :> f' x) (at_right a)"
eberlm@63713
  2041
      "eventually (\<lambda>x. DERIV g x :> g' x) (at_right a)"
eberlm@63713
  2042
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at_right a)"
eberlm@63713
  2043
  shows "filterlim (\<lambda> x. f x / g x) at_bot (at_right a)"
eberlm@63713
  2044
proof -
eberlm@63713
  2045
  from ev(2) have ev': "eventually (\<lambda>x. DERIV (\<lambda>x. -g x) x :> -g' x) (at_right a)"
eberlm@63713
  2046
    by eventually_elim (auto intro: derivative_intros)
eberlm@63713
  2047
  have "filterlim (\<lambda>x. f x / (-g x)) at_top (at_right a)"
eberlm@63713
  2048
    by (rule lhopital_right_at_top_at_top[where f' = f' and g' = "\<lambda>x. -g' x"])
eberlm@63713
  2049
       (insert assms ev', auto simp: filterlim_uminus_at_bot)
eberlm@63713
  2050
  hence "filterlim (\<lambda>x. -(f x / g x)) at_top (at_right a)" by simp
eberlm@63713
  2051
  thus ?thesis by (simp add: filterlim_uminus_at_bot)
eberlm@63713
  2052
qed
eberlm@63713
  2053
eberlm@63713
  2054
lemma lhopital_left_at_top_at_top:
eberlm@63713
  2055
  fixes f g :: "real \<Rightarrow> real"
eberlm@63713
  2056
  assumes f_0: "LIM x at_left a. f x :> at_top"
eberlm@63713
  2057
  assumes g_0: "LIM x at_left a. g x :> at_top"
eberlm@63713
  2058
    and ev:
eberlm@63713
  2059
      "eventually (\<lambda>x. DERIV f x :> f' x) (at_left a)"
eberlm@63713
  2060
      "eventually (\<lambda>x. DERIV g x :> g' x) (at_left a)"
eberlm@63713
  2061
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at_left a)"
eberlm@63713
  2062
  shows "filterlim (\<lambda> x. f x / g x) at_top (at_left a)"
eberlm@63713
  2063
  by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror,
eberlm@63713
  2064
      rule lhopital_right_at_top_at_top[where f'="\<lambda>x. - f' (- x)"]) 
eberlm@63713
  2065
     (insert assms, auto simp: DERIV_mirror)
eberlm@63713
  2066
eberlm@63713
  2067
lemma lhopital_left_at_top_at_bot:
eberlm@63713
  2068
  fixes f g :: "real \<Rightarrow> real"
eberlm@63713
  2069
  assumes f_0: "LIM x at_left a. f x :> at_top"
eberlm@63713
  2070
  assumes g_0: "LIM x at_left a. g x :> at_bot"
eberlm@63713
  2071
    and ev:
eberlm@63713
  2072
      "eventually (\<lambda>x. DERIV f x :> f' x) (at_left a)"
eberlm@63713
  2073
      "eventually (\<lambda>x. DERIV g x :> g' x) (at_left a)"
eberlm@63713
  2074
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at_left a)"
eberlm@63713
  2075
  shows "filterlim (\<lambda> x. f x / g x) at_bot (at_left a)"
eberlm@63713
  2076
  by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror,
eberlm@63713
  2077
      rule lhopital_right_at_top_at_bot[where f'="\<lambda>x. - f' (- x)"]) 
eberlm@63713
  2078
     (insert assms, auto simp: DERIV_mirror)
eberlm@63713
  2079
eberlm@63713
  2080
lemma lhopital_at_top_at_top:
eberlm@63713
  2081
  fixes f g :: "real \<Rightarrow> real"
eberlm@63713
  2082
  assumes f_0: "LIM x at a. f x :> at_top"
eberlm@63713
  2083
  assumes g_0: "LIM x at a. g x :> at_top"
eberlm@63713
  2084
    and ev:
eberlm@63713
  2085
      "eventually (\<lambda>x. DERIV f x :> f' x) (at a)"
eberlm@63713
  2086
      "eventually (\<lambda>x. DERIV g x :> g' x) (at a)"
eberlm@63713
  2087
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at a)"
eberlm@63713
  2088
  shows "filterlim (\<lambda> x. f x / g x) at_top (at a)"
eberlm@63713
  2089
  using assms unfolding eventually_at_split filterlim_at_split
eberlm@63713
  2090
  by (auto intro!: lhopital_right_at_top_at_top[of f a g f' g'] 
eberlm@63713
  2091
                   lhopital_left_at_top_at_top[of f a g f' g'])
eberlm@63713
  2092
eberlm@63713
  2093
lemma lhopital_at_top_at_bot:
eberlm@63713
  2094
  fixes f g :: "real \<Rightarrow> real"
eberlm@63713
  2095
  assumes f_0: "LIM x at a. f x :> at_top"
eberlm@63713
  2096
  assumes g_0: "LIM x at a. g x :> at_bot"
eberlm@63713
  2097
    and ev:
eberlm@63713
  2098
      "eventually (\<lambda>x. DERIV f x :> f' x) (at a)"
eberlm@63713
  2099
      "eventually (\<lambda>x. DERIV g x :> g' x) (at a)"
eberlm@63713
  2100
    and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at a)"
eberlm@63713
  2101
  shows "filterlim (\<lambda> x. f x / g x) at_bot (at a)"
eberlm@63713
  2102
  using assms unfolding eventually_at_split filterlim_at_split
eberlm@63713
  2103
  by (auto intro!: lhopital_right_at_top_at_bot[of f a g f' g'] 
eberlm@63713
  2104
                   lhopital_left_at_top_at_bot[of f a g f' g'])
eberlm@63713
  2105
huffman@21164
  2106
end