(*  Title       : Deriv.thy

    Author      : Jacques D. Fleuriot

    Copyright   : 1998  University of Cambridge

    Author      : Brian Huffman

    Conversion to Isar and new proofs by Lawrence C Paulson, 2004

    GMVT by Benjamin Porter, 2005

*)

header{* Differentiation *}

theory Deriv

imports Limits

begin

definition

  -- {* Frechet derivative: D is derivative of function f at x within s *}

  has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow>  bool"

  (infixl "(has'_derivative)" 12)

where

  "(f has_derivative f') F \<longleftrightarrow>

    (bounded_linear f' \<and>

     ((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 0) F)"

lemma FDERIV_eq_rhs: "(f has_derivative f') F \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"

  by simp

ML {*

structure FDERIV_Intros = Named_Thms

(

  val name = @{binding FDERIV_intros}

  val description = "introduction rules for FDERIV"

)

*}

setup {*

  FDERIV_Intros.setup #>

  Global_Theory.add_thms_dynamic (@{binding FDERIV_eq_intros},

    map_filter (try (fn thm => @{thm FDERIV_eq_rhs} OF [thm])) o FDERIV_Intros.get o Context.proof_of);

*}

lemma FDERIV_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"

  by (simp add: has_derivative_def)

lemma FDERIV_ident[FDERIV_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"

  by (simp add: has_derivative_def tendsto_const)

lemma FDERIV_const[FDERIV_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"

  by (simp add: has_derivative_def tendsto_const)

lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..

lemma (in bounded_linear) FDERIV:

  "(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"

  using assms unfolding has_derivative_def

  apply safe

  apply (erule bounded_linear_compose [OF local.bounded_linear])

  apply (drule local.tendsto)

  apply (simp add: local.scaleR local.diff local.add local.zero)

  done

lemmas FDERIV_scaleR_right [FDERIV_intros] =

  bounded_linear.FDERIV [OF bounded_linear_scaleR_right]

lemmas FDERIV_scaleR_left [FDERIV_intros] =

  bounded_linear.FDERIV [OF bounded_linear_scaleR_left]

lemmas FDERIV_mult_right [FDERIV_intros] =

  bounded_linear.FDERIV [OF bounded_linear_mult_right]

lemmas FDERIV_mult_left [FDERIV_intros] =

  bounded_linear.FDERIV [OF bounded_linear_mult_left]

lemma FDERIV_add[simp, FDERIV_intros]:

  assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"

  shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"

  unfolding has_derivative_def

proof safe

  let ?x = "Lim F (\<lambda>x. x)"

  let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"

  have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"

    using f g by (intro tendsto_add) (auto simp: has_derivative_def)

  then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"

    by (simp add: field_simps scaleR_add_right scaleR_diff_right)

qed (blast intro: bounded_linear_add f g FDERIV_bounded_linear)

lemma FDERIV_setsum[simp, FDERIV_intros]:

  assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"

  shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"

proof cases

  assume "finite I" from this f show ?thesis

    by induct (simp_all add: f)

qed simp

lemma FDERIV_minus[simp, FDERIV_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"

  using FDERIV_scaleR_right[of f f' F "-1"] by simp

lemma FDERIV_diff[simp, FDERIV_intros]:

  "(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"

  by (simp only: diff_conv_add_uminus FDERIV_add FDERIV_minus)

abbreviation

  -- {* Frechet derivative: D is derivative of function f at x within s *}

  FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"

  ("(FDERIV (_)/ (_)/ : (_)/ :> (_))" [1000, 1000, 1000, 60] 60)

where

  "FDERIV f x : s :> f' \<equiv> (f has_derivative f') (at x within s)"

abbreviation

  fderiv_at ::

    "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"

    ("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)

where

  "FDERIV f x :> D \<equiv> FDERIV f x : UNIV :> D"

lemma FDERIV_def:

  "FDERIV f x : s :> f' \<longleftrightarrow>

    (bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s))"

  by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)

lemma FDERIV_iff_norm:

  "FDERIV f x : s :> f' \<longleftrightarrow>

    (bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ---> 0) (at x within s))"

  using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]

  by (simp add: FDERIV_def divide_inverse ac_simps)

lemma fderiv_def:

  "FDERIV f x :> D = (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"

  unfolding FDERIV_iff_norm LIM_offset_zero_iff[of _ _ x] by simp

lemma field_fderiv_def:

  fixes x :: "'a::real_normed_field"

  shows "FDERIV f x :> (\<lambda>h. h * D) = (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"

  apply (unfold fderiv_def)

  apply (simp add: bounded_linear_mult_left)

  apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])

  apply (subst diff_divide_distrib)

  apply (subst times_divide_eq_left [symmetric])

  apply (simp cong: LIM_cong)

  apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)

  done

lemma FDERIV_I:

  "bounded_linear f' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>

  FDERIV f x : s :> f'"

  by (simp add: FDERIV_def)

lemma FDERIV_I_sandwich:

  assumes e: "0 < e" and bounded: "bounded_linear f'"

    and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"

    and "(H ---> 0) (at x within s)"

  shows "FDERIV f x : s :> f'"

  unfolding FDERIV_iff_norm

proof safe

  show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"

  proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])

    show "(H ---> 0) (at x within s)" by fact

    show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"

      unfolding eventually_at using e sandwich by auto

  qed (auto simp: le_divide_eq tendsto_const)

qed fact

lemma FDERIV_subset: "FDERIV f x : s :> f' \<Longrightarrow> t \<subseteq> s \<Longrightarrow> FDERIV f x : t :> f'"

  by (auto simp add: FDERIV_iff_norm intro: tendsto_within_subset)

subsection {* Continuity *}

lemma FDERIV_continuous:

  assumes f: "FDERIV f x : s :> f'"

  shows "continuous (at x within s) f"

proof -

  from f interpret F: bounded_linear f' by (rule FDERIV_bounded_linear)

  note F.tendsto[tendsto_intros]

  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"

  have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"

    using f unfolding FDERIV_iff_norm by blast

  then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)

    by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)

  also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"

    by (intro filterlim_cong) (simp_all add: eventually_at_filter)

  finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"

    by (rule tendsto_norm_zero_cancel)

  then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"

    by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)

  then have "?L (\<lambda>y. f y - f x)"

    by simp

  from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis

    by (simp add: continuous_within)

qed

subsection {* Composition *}

lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (f ---> y) (inf (nhds x) (principal s))"

  unfolding tendsto_def eventually_inf_principal eventually_at_filter

  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)

lemma FDERIV_in_compose:

  assumes f: "FDERIV f x : s :> f'"

  assumes g: "FDERIV g (f x) : (f`s) :> g'"

  shows "FDERIV (\<lambda>x. g (f x)) x : s :> (\<lambda>x. g' (f' x))"

proof -

  from f interpret F: bounded_linear f' by (rule FDERIV_bounded_linear)

  from g interpret G: bounded_linear g' by (rule FDERIV_bounded_linear)

  from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast

  from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast

  note G.tendsto[tendsto_intros]

  let ?L = "\<lambda>f. (f ---> 0) (at x within s)"

  let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"

  let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"

  let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"

  def Nf \<equiv> "?N f f' x"

  def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"

  show ?thesis

  proof (rule FDERIV_I_sandwich[of 1])

    show "bounded_linear (\<lambda>x. g' (f' x))"

      using f g by (blast intro: bounded_linear_compose FDERIV_bounded_linear)

  next

    fix y::'a assume neq: "y \<noteq> x"

    have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"

      by (simp add: G.diff G.add field_simps)

    also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"

      by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)

    also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"

    proof (intro add_mono mult_left_mono)

      have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"

        by simp

      also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"

        by (rule norm_triangle_ineq)

      also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"

        using kF by (intro add_mono) simp

      finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"

        by (simp add: neq Nf_def field_simps)

    qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)

    finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .

  next

    have [tendsto_intros]: "?L Nf"

      using f unfolding FDERIV_iff_norm Nf_def ..

    from f have "(f ---> f x) (at x within s)"

      by (blast intro: FDERIV_continuous continuous_within[THEN iffD1])

    then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"

      unfolding filterlim_def

      by (simp add: eventually_filtermap eventually_at_filter le_principal)

    have "((?N g  g' (f x)) ---> 0) (at (f x) within f`s)"

      using g unfolding FDERIV_iff_norm ..

    then have g': "((?N g  g' (f x)) ---> 0) (inf (nhds (f x)) (principal (f`s)))"

      by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp

    have [tendsto_intros]: "?L Ng"

      unfolding Ng_def by (rule filterlim_compose[OF g' f'])

    show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) ---> 0) (at x within s)"

      by (intro tendsto_eq_intros) auto

  qed simp

qed

lemma FDERIV_compose:

  "FDERIV f x : s :> f' \<Longrightarrow> FDERIV g (f x) :> g' \<Longrightarrow> FDERIV (\<lambda>x. g (f x)) x : s :> (\<lambda>x. g' (f' x))"

  by (blast intro: FDERIV_in_compose FDERIV_subset)

lemma (in bounded_bilinear) FDERIV:

  assumes f: "FDERIV f x : s :> f'" and g: "FDERIV g x : s :> g'"

  shows "FDERIV (\<lambda>x. f x ** g x) x : s :> (\<lambda>h. f x ** g' h + f' h ** g x)"

proof -

  from bounded_linear.bounded [OF FDERIV_bounded_linear [OF f]]

  obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast

  from pos_bounded obtain K where K: "0 < K" and norm_prod:

    "\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast

  let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"

  let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"

  def Ng =="?N g g'" and Nf =="?N f f'"

  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)"

  let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"

  let ?F = "at x within s"

  show ?thesis

  proof (rule FDERIV_I_sandwich[of 1])

    show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"

      by (intro bounded_linear_add

        bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]

        FDERIV_bounded_linear [OF g] FDERIV_bounded_linear [OF f])

  next

    from g have "(g ---> g x) ?F"

      by (intro continuous_within[THEN iffD1] FDERIV_continuous)

    moreover from f g have "(Nf ---> 0) ?F" "(Ng ---> 0) ?F"

      by (simp_all add: FDERIV_iff_norm Ng_def Nf_def)

    ultimately have "(?fun2 ---> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"

      by (intro tendsto_intros) (simp_all add: LIM_zero_iff)

    then show "(?fun2 ---> 0) ?F"

      by simp

  next

    fix y::'d assume "y \<noteq> x"

    have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"

      by (simp add: diff_left diff_right add_left add_right field_simps)

    also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +

        norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"

      by (intro divide_right_mono mult_mono'

                order_trans [OF norm_triangle_ineq add_mono]

                order_trans [OF norm_prod mult_right_mono]

                mult_nonneg_nonneg order_refl norm_ge_zero norm_F

                K [THEN order_less_imp_le])

    also have "\<dots> = ?fun2 y"

      by (simp add: add_divide_distrib Ng_def Nf_def)

    finally show "?fun1 y \<le> ?fun2 y" .

  qed simp

qed

lemmas FDERIV_mult[simp, FDERIV_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]

lemmas FDERIV_scaleR[simp, FDERIV_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]

lemma FDERIV_setprod[simp, FDERIV_intros]:

  fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"

  assumes f: "\<And>i. i \<in> I \<Longrightarrow> FDERIV (f i) x : s :> f' i"

  shows "FDERIV (\<lambda>x. \<Prod>i\<in>I. f i x) x : s :> (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))"

proof cases

  assume "finite I" from this f show ?thesis

  proof induct

    case (insert i I)

    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)"

    have "FDERIV (\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) x : s :> ?P"

      using insert by (intro FDERIV_mult) auto

    also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"

      using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum_cong)

    finally show ?case

      using insert by simp

  qed simp  

qed simp

lemma FDERIV_power[simp, FDERIV_intros]:

  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"

  assumes f: "FDERIV f x : s :> f'"

  shows "FDERIV  (\<lambda>x. f x^n) x : s :> (\<lambda>y. of_nat n * f' y * f x^(n - 1))"

  using FDERIV_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)

lemma FDERIV_inverse':

  fixes x :: "'a::real_normed_div_algebra"

  assumes x: "x \<noteq> 0"

  shows "FDERIV inverse x : s :> (\<lambda>h. - (inverse x * h * inverse x))"

        (is "FDERIV ?inv x : s :> ?f")

proof (rule FDERIV_I_sandwich)

  show "bounded_linear (\<lambda>h. - (?inv x * h * ?inv x))"

    apply (rule bounded_linear_minus)

    apply (rule bounded_linear_mult_const)

    apply (rule bounded_linear_const_mult)

    apply (rule bounded_linear_ident)

    done

next

  show "0 < norm x" using x by simp

next

  show "((\<lambda>y. norm (?inv y - ?inv x) * norm (?inv x)) ---> 0) (at x within s)"

    apply (rule tendsto_mult_left_zero)

    apply (rule tendsto_norm_zero)

    apply (rule LIM_zero)

    apply (rule tendsto_inverse)

    apply (rule tendsto_ident_at)

    apply (rule x)

    done

next

  fix y::'a assume h: "y \<noteq> x" "dist y x < norm x"

  then have "y \<noteq> 0"

    by (auto simp: norm_conv_dist dist_commute)

  have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) = norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"

    apply (subst inverse_diff_inverse [OF `y \<noteq> 0` x])

    apply (subst minus_diff_minus)

    apply (subst norm_minus_cancel)

    apply (simp add: left_diff_distrib)

    done

  also have "\<dots> \<le> norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"

    apply (rule divide_right_mono [OF _ norm_ge_zero])

    apply (rule order_trans [OF norm_mult_ineq])

    apply (rule mult_right_mono [OF _ norm_ge_zero])

    apply (rule norm_mult_ineq)

    done

  also have "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"

    by simp

  finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>

      norm (?inv y - ?inv x) * norm (?inv x)" .

qed

lemma FDERIV_inverse[simp, FDERIV_intros]:

  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"

  assumes x:  "f x \<noteq> 0" and f: "FDERIV f x : s :> f'"

  shows "FDERIV (\<lambda>x. inverse (f x)) x : s :> (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))"

  using FDERIV_compose[OF f FDERIV_inverse', OF x] .

lemma FDERIV_divide[simp, FDERIV_intros]:

  fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"

  assumes g: "FDERIV g x : s :> g'"

  assumes x:  "f x \<noteq> 0" and f: "FDERIV f x : s :> f'"

  shows "FDERIV (\<lambda>x. g x / f x) x : s :> (\<lambda>h. - g x * (inverse (f x) * f' h * inverse (f x)) + g' h / f x)"

  using FDERIV_mult[OF g FDERIV_inverse[OF x f]]

  by (simp add: divide_inverse)

subsection {* Uniqueness *}

text {*

This can not generally shown for @{const FDERIV}, as we need to approach the point from

all directions. There is a proof in @{text Multivariate_Analysis} for @{text euclidean_space}.

*}

lemma FDERIV_zero_unique:

  assumes "FDERIV (\<lambda>x. 0) x :> F" shows "F = (\<lambda>h. 0)"

proof -

  interpret F: bounded_linear F

    using assms by (rule FDERIV_bounded_linear)

  let ?r = "\<lambda>h. norm (F h) / norm h"

  have *: "?r -- 0 --> 0"

    using assms unfolding fderiv_def by simp

  show "F = (\<lambda>h. 0)"

  proof

    fix h show "F h = 0"

    proof (rule ccontr)

      assume **: "F h \<noteq> 0"

      then have h: "h \<noteq> 0"

        by (clarsimp simp add: F.zero)

      with ** have "0 < ?r h"

        by (simp add: divide_pos_pos)

      from LIM_D [OF * this] obtain s where s: "0 < s"

        and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto

      from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..

      let ?x = "scaleR (t / norm h) h"

      have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all

      hence "?r ?x < ?r h" by (rule r)

      thus "False" using t h by (simp add: F.scaleR)

    qed

  qed

qed

lemma FDERIV_unique:

  assumes "FDERIV f x :> F" and "FDERIV f x :> F'" shows "F = F'"

proof -

  have "FDERIV (\<lambda>x. 0) x :> (\<lambda>h. F h - F' h)"

    using FDERIV_diff [OF assms] by simp

  hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"

    by (rule FDERIV_zero_unique)

  thus "F = F'"

    unfolding fun_eq_iff right_minus_eq .

qed

subsection {* Differentiability predicate *}

definition isDiff :: "'a filter \<Rightarrow> ('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool" where

  isDiff_def: "isDiff F f \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"

abbreviation differentiable_in :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> bool"

    ("(_) differentiable (_) in (_)"  [1000, 1000, 60] 60) where

  "f differentiable x in s \<equiv> isDiff (at x within s) f"

abbreviation differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool"

    (infixl "differentiable" 60) where

  "f differentiable x \<equiv> f differentiable x in UNIV"

lemma differentiable_subset: "f differentiable x in s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable x in t"

  unfolding isDiff_def by (blast intro: FDERIV_subset)

lemma differentiable_ident [simp]: "isDiff F (\<lambda>x. x)"

  unfolding isDiff_def by (blast intro: FDERIV_ident)

lemma differentiable_const [simp]: "isDiff F (\<lambda>z. a)"

  unfolding isDiff_def by (blast intro: FDERIV_const)

lemma differentiable_in_compose:

  "f differentiable (g x) in (g`s) \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f (g x)) differentiable x in s"

  unfolding isDiff_def by (blast intro: FDERIV_in_compose)

lemma differentiable_compose:

  "f differentiable (g x) \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f (g x)) differentiable x in s"

  by (blast intro: differentiable_in_compose differentiable_subset)

lemma differentiable_sum [simp]:

  "isDiff F f \<Longrightarrow> isDiff F g \<Longrightarrow> isDiff F (\<lambda>x. f x + g x)"

  unfolding isDiff_def by (blast intro: FDERIV_add)

lemma differentiable_minus [simp]:

  "isDiff F f \<Longrightarrow> isDiff F (\<lambda>x. - f x)"

  unfolding isDiff_def by (blast intro: FDERIV_minus)

lemma differentiable_diff [simp]:

  "isDiff F f \<Longrightarrow> isDiff F g \<Longrightarrow> isDiff F (\<lambda>x. f x - g x)"

  unfolding isDiff_def by (blast intro: FDERIV_diff)

lemma differentiable_mult [simp]:

  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"

  shows "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f x * g x) differentiable x in s"

  unfolding isDiff_def by (blast intro: FDERIV_mult)

lemma differentiable_inverse [simp]:

  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"

  shows "f differentiable x in s \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable x in s"

  unfolding isDiff_def by (blast intro: FDERIV_inverse)

lemma differentiable_divide [simp]:

  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"

  shows "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable x in s"

  unfolding divide_inverse using assms by simp

lemma differentiable_power [simp]:

  fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"

  shows "f differentiable x in s \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable x in s"

  unfolding isDiff_def by (blast intro: FDERIV_power)

lemma differentiable_scaleR [simp]:

  "f differentiable x in s \<Longrightarrow> g differentiable x in s \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable x in s"

  unfolding isDiff_def by (blast intro: FDERIV_scaleR)

definition 

  -- {*Differentiation: D is derivative of function f at x*}

  deriv ::

    "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a set \<Rightarrow> 'a \<Rightarrow> bool"

    ("(DERIV (_)/ (_)/ : (_)/ :> (_))" [1000, 1000, 1000, 60] 60)

where

  deriv_fderiv: "DERIV f x : s :> D = FDERIV f x : s :> (\<lambda>x. x * D)"

abbreviation

  -- {*Differentiation: D is derivative of function f at x*}

  deriv_at ::

    "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"

    ("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)

where

  "DERIV f x :> D \<equiv> DERIV f x : UNIV :> D"

lemma differentiable_def: "(f::real \<Rightarrow> real) differentiable x in s \<longleftrightarrow> (\<exists>D. DERIV f x : s :> D)"

proof safe

  assume "f differentiable x in s"

  then obtain f' where *: "FDERIV f x : s :> f'"

    unfolding isDiff_def by auto

  then obtain c where "f' = (\<lambda>x. x * c)"

    by (metis real_bounded_linear FDERIV_bounded_linear)

  with * show "\<exists>D. DERIV f x : s :> D"

    unfolding deriv_fderiv by auto

qed (auto simp: isDiff_def deriv_fderiv)

lemma differentiableE [elim?]:

  fixes f :: "real \<Rightarrow> real"

  assumes f: "f differentiable x in s" obtains df where "DERIV f x : s :> df"

  using assms by (auto simp: differentiable_def)

lemma differentiableD: "(f::real \<Rightarrow> real) differentiable x in s \<Longrightarrow> \<exists>D. DERIV f x : s :> D"

  by (auto elim: differentiableE)

lemma differentiableI: "DERIV f x : s :> D \<Longrightarrow> (f::real \<Rightarrow> real) differentiable x in s"

  by (force simp add: differentiable_def)

lemma DERIV_I_FDERIV: "FDERIV f x : s :> F \<Longrightarrow> (\<And>x. x * F' = F x) \<Longrightarrow> DERIV f x : s :> F'"

  by (simp add: deriv_fderiv)

lemma DERIV_D_FDERIV: "DERIV f x : s :> F \<Longrightarrow> FDERIV f x : s :> (\<lambda>x. x * F)"

  by (simp add: deriv_fderiv)

lemma deriv_def:

  "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"

  apply (simp add: deriv_fderiv fderiv_def bounded_linear_mult_left LIM_zero_iff[symmetric, of _ D])

  apply (subst (2) tendsto_norm_zero_iff[symmetric])

  apply (rule filterlim_cong)

  apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)

  done

subsection {* Derivatives *}

lemma DERIV_iff: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"

  by (simp add: deriv_def)

lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"

  by (simp add: deriv_def)

lemma DERIV_const [simp]: "DERIV (\<lambda>x. k) x : s :> 0"

  by (rule DERIV_I_FDERIV[OF FDERIV_const]) auto

lemma DERIV_ident [simp]: "DERIV (\<lambda>x. x) x : s :> 1"

  by (rule DERIV_I_FDERIV[OF FDERIV_ident]) auto

lemma DERIV_add: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x + g x) x : s :> D + E"

  by (rule DERIV_I_FDERIV[OF FDERIV_add]) (auto simp: field_simps dest: DERIV_D_FDERIV)

lemma DERIV_minus: "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. - f x) x : s :> - D"

  by (rule DERIV_I_FDERIV[OF FDERIV_minus]) (auto simp: field_simps dest: DERIV_D_FDERIV)

lemma DERIV_diff: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x - g x) x : s :> D - E"

  by (rule DERIV_I_FDERIV[OF FDERIV_diff]) (auto simp: field_simps dest: DERIV_D_FDERIV)

lemma DERIV_add_minus: "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x + - g x) x : s :> D + - E"

  by (simp only: DERIV_add DERIV_minus)

lemma DERIV_continuous: "DERIV f x : s :> D \<Longrightarrow> continuous (at x within s) f"

  by (drule FDERIV_continuous[OF DERIV_D_FDERIV]) simp

lemma DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"

  by (auto dest!: DERIV_continuous)

lemma DERIV_mult': "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x : s :> f x * E + D * g x"

  by (rule DERIV_I_FDERIV[OF FDERIV_mult]) (auto simp: field_simps dest: DERIV_D_FDERIV)

lemma DERIV_mult: "DERIV f x : s :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (\<lambda>x. f x * g x) x : s :> Da * g x + Db * f x"

  by (rule DERIV_I_FDERIV[OF FDERIV_mult]) (auto simp: field_simps dest: DERIV_D_FDERIV)

text {* Derivative of linear multiplication *}

lemma DERIV_cmult:

  "DERIV f x : s :> D ==> DERIV (%x. c * f x) x : s :> c*D"

  by (drule DERIV_mult' [OF DERIV_const], simp)

lemma DERIV_cmult_Id [simp]: "DERIV (op * c) x : s :> c"

  by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)

lemma DERIV_cdivide: "DERIV f x : s :> D ==> DERIV (%x. f x / c) x : s :> D / c"

  apply (subgoal_tac "DERIV (%x. (1 / c) * f x) x : s :> (1 / c) * D", force)

  apply (erule DERIV_cmult)

  done

lemma DERIV_unique:

  "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"

  unfolding deriv_def by (rule LIM_unique) 

lemma DERIV_setsum':

  "(\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x : s :> (f' x n)) \<Longrightarrow> DERIV (\<lambda>x. setsum (f x) S) x : s :> setsum (f' x) S"

  by (rule DERIV_I_FDERIV[OF FDERIV_setsum]) (auto simp: setsum_right_distrib dest: DERIV_D_FDERIV)

lemma DERIV_setsum:

  "finite S \<Longrightarrow> (\<And> n. n \<in> S \<Longrightarrow> DERIV (%x. f x n) x : s :> (f' x n)) \<Longrightarrow> DERIV (\<lambda>x. setsum (f x) S) x : s :> setsum (f' x) S"

  by (rule DERIV_setsum')

lemma DERIV_sumr [rule_format (no_asm)]: (* REMOVE *)

     "(\<forall>r. m \<le> r & r < (m + n) --> DERIV (%x. f r x) x : s :> (f' r x))

      --> DERIV (%x. \<Sum>n=m..<n::nat. f n x :: real) x : s :> (\<Sum>r=m..<n. f' r x)"

  by (auto intro: DERIV_setsum)

lemma DERIV_inverse':

  "DERIV f x : s :> D \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse (f x)) x : s :> - (inverse (f x) * D * inverse (f x))"

  by (rule DERIV_I_FDERIV[OF FDERIV_inverse]) (auto dest: DERIV_D_FDERIV)

text {* Power of @{text "-1"} *}

lemma DERIV_inverse:

  "x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse(x)) x : s :> - (inverse x ^ Suc (Suc 0))"

  by (drule DERIV_inverse' [OF DERIV_ident]) simp

text {* Derivative of inverse *}

lemma DERIV_inverse_fun:

  "DERIV f x : s :> d \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. inverse (f x)) x : s :> (- (d * inverse(f x ^ Suc (Suc 0))))"

  by (drule (1) DERIV_inverse') (simp add: mult_ac nonzero_inverse_mult_distrib)

text {* Derivative of quotient *}

lemma DERIV_divide:

  "DERIV f x : s :> D \<Longrightarrow> DERIV g x : s :> E \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>x. f x / g x) x : s :> (D * g x - f x * E) / (g x * g x)"

  by (rule DERIV_I_FDERIV[OF FDERIV_divide])

     (auto dest: DERIV_D_FDERIV simp: field_simps nonzero_inverse_mult_distrib divide_inverse)

lemma DERIV_quotient:

  "DERIV f x : s :> d \<Longrightarrow> DERIV g x : s :> e \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> DERIV (\<lambda>y. f y / g y) x : s :> (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))"

  by (drule (2) DERIV_divide) (simp add: mult_commute)

lemma DERIV_power_Suc:

  "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. f x ^ Suc n) x : s :> (1 + of_nat n) * (D * f x ^ n)"

  by (rule DERIV_I_FDERIV[OF FDERIV_power]) (auto simp: deriv_fderiv)

lemma DERIV_power:

  "DERIV f x : s :> D \<Longrightarrow> DERIV (\<lambda>x. f x ^ n) x : s :> of_nat n * (D * f x ^ (n - Suc 0))"

  by (rule DERIV_I_FDERIV[OF FDERIV_power]) (auto simp: deriv_fderiv)

lemma DERIV_pow: "DERIV (%x. x ^ n) x : s :> real n * (x ^ (n - Suc 0))"

  apply (cut_tac DERIV_power [OF DERIV_ident])

  apply (simp add: real_of_nat_def)

  done

lemma DERIV_chain': "DERIV f x : s :> D \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> DERIV (\<lambda>x. g (f x)) x : s :> E * D"

  using FDERIV_compose[of f "\<lambda>x. x * D" x s g "\<lambda>x. x * E"]

  by (auto simp: deriv_fderiv ac_simps dest: FDERIV_subset)

text {* Standard version *}

lemma DERIV_chain: "DERIV f (g x) :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (f o g) x : s :> Da * Db"

  by (drule (1) DERIV_chain', simp add: o_def mult_commute)

lemma DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> DERIV g x : s :> Db \<Longrightarrow> DERIV (%x. f (g x)) x : s :> Da * Db"

  by (auto dest: DERIV_chain simp add: o_def)

subsubsection {* @{text "DERIV_intros"} *}

ML {*

structure Deriv_Intros = Named_Thms

(

  val name = @{binding DERIV_intros}

  val description = "DERIV introduction rules"

)

*}

setup Deriv_Intros.setup

lemma DERIV_cong: "DERIV f x : s :> X \<Longrightarrow> X = Y \<Longrightarrow> DERIV f x : s :> Y"

  by simp

declare

  DERIV_const[THEN DERIV_cong, DERIV_intros]

  DERIV_ident[THEN DERIV_cong, DERIV_intros]

  DERIV_add[THEN DERIV_cong, DERIV_intros]

  DERIV_minus[THEN DERIV_cong, DERIV_intros]

  DERIV_mult[THEN DERIV_cong, DERIV_intros]

  DERIV_diff[THEN DERIV_cong, DERIV_intros]

  DERIV_inverse'[THEN DERIV_cong, DERIV_intros]

  DERIV_divide[THEN DERIV_cong, DERIV_intros]

  DERIV_power[where 'a=real, THEN DERIV_cong,

              unfolded real_of_nat_def[symmetric], DERIV_intros]

  DERIV_setsum[THEN DERIV_cong, DERIV_intros]

text{*Alternative definition for differentiability*}

lemma DERIV_LIM_iff:

  fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows

     "((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =

      ((%x. (f(x)-f(a)) / (x-a)) -- a --> D)"

apply (rule iffI)

apply (drule_tac k="- a" in LIM_offset)

apply simp

apply (drule_tac k="a" in LIM_offset)

apply (simp add: add_commute)

done

lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"

  by (simp add: deriv_def DERIV_LIM_iff)

lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>

    DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"

  unfolding DERIV_iff2

proof (rule filterlim_cong)

  assume *: "eventually (\<lambda>x. f x = g x) (nhds x)"

  moreover from * have "f x = g x" by (auto simp: eventually_nhds)

  moreover assume "x = y" "u = v"

  ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"

    by (auto simp: eventually_at_filter elim: eventually_elim1)

qed simp_all

lemma DERIV_shift:

  "(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"

  by (simp add: DERIV_iff field_simps)

lemma DERIV_mirror:

  "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"

  by (simp add: deriv_def filterlim_at_split filterlim_at_left_to_right

                tendsto_minus_cancel_left field_simps conj_commute)

text {* Caratheodory formulation of derivative at a point *}

lemma CARAT_DERIV:

  "(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)"

      (is "?lhs = ?rhs")

proof

  assume der: "DERIV f x :> l"

  show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"

  proof (intro exI conjI)

    let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"

    show "\<forall>z. f z - f x = ?g z * (z-x)" by simp

    show "isCont ?g x" using der

      by (simp add: isCont_iff DERIV_iff cong: LIM_equal [rule_format])

    show "?g x = l" by simp

  qed

next

  assume "?rhs"

  then obtain g where

    "(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast

  thus "(DERIV f x :> l)"

     by (auto simp add: isCont_iff DERIV_iff cong: LIM_cong)

qed

text {*

 Let's do the standard proof, though theorem

 @{text "LIM_mult2"} follows from a NS proof

*}

subsection {* Local extrema *}

text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}

lemma DERIV_pos_inc_right:

  fixes f :: "real => real"

  assumes der: "DERIV f x :> l"

      and l:   "0 < l"

  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"

proof -

  from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]

  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"

    by simp

  then obtain s

        where s:   "0 < s"

          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"

    by auto

  thus ?thesis

  proof (intro exI conjI strip)

    show "0<s" using s .

    fix h::real

    assume "0 < h" "h < s"

    with all [of h] show "f x < f (x+h)"

    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)

      assume "~ (f (x+h) - f x) / h < l" and h: "0 < h"

      with l

      have "0 < (f (x+h) - f x) / h" by arith

      thus "f x < f (x+h)"

  by (simp add: pos_less_divide_eq h)

    qed

  qed

qed

lemma DERIV_neg_dec_left:

  fixes f :: "real => real"

  assumes der: "DERIV f x :> l"

      and l:   "l < 0"

  shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"

proof -

  from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]

  have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"

    by simp

  then obtain s

        where s:   "0 < s"

          and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"

    by auto

  thus ?thesis

  proof (intro exI conjI strip)

    show "0<s" using s .

    fix h::real

    assume "0 < h" "h < s"

    with all [of "-h"] show "f x < f (x-h)"

    proof (simp add: abs_if pos_less_divide_eq split add: split_if_asm)

      assume " - ((f (x-h) - f x) / h) < l" and h: "0 < h"

      with l

      have "0 < (f (x-h) - f x) / h" by arith

      thus "f x < f (x-h)"

  by (simp add: pos_less_divide_eq h)

    qed

  qed

qed

lemma DERIV_pos_inc_left:

  fixes f :: "real => real"

  shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"

  apply (rule DERIV_neg_dec_left [of "%x. - f x" x "-l", simplified])

  apply (auto simp add: DERIV_minus)

  done

lemma DERIV_neg_dec_right:

  fixes f :: "real => real"

  shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"

  apply (rule DERIV_pos_inc_right [of "%x. - f x" x "-l", simplified])

  apply (auto simp add: DERIV_minus)

  done

lemma DERIV_local_max:

  fixes f :: "real => real"

  assumes der: "DERIV f x :> l"

      and d:   "0 < d"

      and le:  "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"

  shows "l = 0"

proof (cases rule: linorder_cases [of l 0])

  case equal thus ?thesis .

next

  case less

  from DERIV_neg_dec_left [OF der less]

  obtain d' where d': "0 < d'"

             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast

  from real_lbound_gt_zero [OF d d']

  obtain e where "0 < e \<and> e < d \<and> e < d'" ..

  with lt le [THEN spec [where x="x-e"]]

  show ?thesis by (auto simp add: abs_if)

next

  case greater

  from DERIV_pos_inc_right [OF der greater]

  obtain d' where d': "0 < d'"

             and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast

  from real_lbound_gt_zero [OF d d']

  obtain e where "0 < e \<and> e < d \<and> e < d'" ..

  with lt le [THEN spec [where x="x+e"]]

  show ?thesis by (auto simp add: abs_if)

qed

text{*Similar theorem for a local minimum*}

lemma DERIV_local_min:

  fixes f :: "real => real"

  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> f(y) |] ==> l = 0"

by (drule DERIV_minus [THEN DERIV_local_max], auto)

text{*In particular, if a function is locally flat*}

lemma DERIV_local_const:

  fixes f :: "real => real"

  shows "[| DERIV f x :> l; 0 < d; \<forall>y. \<bar>x-y\<bar> < d --> f(x) = f(y) |] ==> l = 0"

by (auto dest!: DERIV_local_max)

subsection {* Rolle's Theorem *}

text{*Lemma about introducing open ball in open interval*}

lemma lemma_interval_lt:

     "[| a < x;  x < b |]

      ==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"

apply (simp add: abs_less_iff)

apply (insert linorder_linear [of "x-a" "b-x"], safe)

apply (rule_tac x = "x-a" in exI)

apply (rule_tac [2] x = "b-x" in exI, auto)

done

lemma lemma_interval: "[| a < x;  x < b |] ==>

        \<exists>d::real. 0 < d &  (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"

apply (drule lemma_interval_lt, auto)

apply force

done

text{*Rolle's Theorem.

   If @{term f} is defined and continuous on the closed interval

   @{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},

   and @{term "f(a) = f(b)"},

   then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}

theorem Rolle:

  assumes lt: "a < b"

      and eq: "f(a) = f(b)"

      and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"

      and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable x"

  shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"

proof -

  have le: "a \<le> b" using lt by simp

  from isCont_eq_Ub [OF le con]

  obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"

             and alex: "a \<le> x" and xleb: "x \<le> b"

    by blast

  from isCont_eq_Lb [OF le con]

  obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"

              and alex': "a \<le> x'" and x'leb: "x' \<le> b"

    by blast

  show ?thesis

  proof cases

    assume axb: "a < x & x < b"

        --{*@{term f} attains its maximum within the interval*}

    hence ax: "a<x" and xb: "x<b" by arith + 

    from lemma_interval [OF ax xb]

    obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"

      by blast

    hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max

      by blast

    from differentiableD [OF dif [OF axb]]

    obtain l where der: "DERIV f x :> l" ..

    have "l=0" by (rule DERIV_local_max [OF der d bound'])

        --{*the derivative at a local maximum is zero*}

    thus ?thesis using ax xb der by auto

  next

    assume notaxb: "~ (a < x & x < b)"

    hence xeqab: "x=a | x=b" using alex xleb by arith

    hence fb_eq_fx: "f b = f x" by (auto simp add: eq)

    show ?thesis

    proof cases

      assume ax'b: "a < x' & x' < b"

        --{*@{term f} attains its minimum within the interval*}

      hence ax': "a<x'" and x'b: "x'<b" by arith+ 

      from lemma_interval [OF ax' x'b]

      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"

  by blast

      hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min

  by blast

      from differentiableD [OF dif [OF ax'b]]

      obtain l where der: "DERIV f x' :> l" ..

      have "l=0" by (rule DERIV_local_min [OF der d bound'])

        --{*the derivative at a local minimum is zero*}

      thus ?thesis using ax' x'b der by auto

    next

      assume notax'b: "~ (a < x' & x' < b)"

        --{*@{term f} is constant througout the interval*}

      hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith

      hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)

      from dense [OF lt]

      obtain r where ar: "a < r" and rb: "r < b" by blast

      from lemma_interval [OF ar rb]

      obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"

  by blast

      have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"

      proof (clarify)

        fix z::real

        assume az: "a \<le> z" and zb: "z \<le> b"

        show "f z = f b"

        proof (rule order_antisym)

          show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)

          show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)

        qed

      qed

      have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"

      proof (intro strip)

        fix y::real

        assume lt: "\<bar>r-y\<bar> < d"

        hence "f y = f b" by (simp add: eq_fb bound)