(*  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

subsection {* Frechet derivative *}

definition

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

  (infix "(has'_derivative)" 50)

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)"

text {*

  Usually the filter @{term F} is @{term "at x within s"}.  @{term "(f has_derivative D)

  (at x within s)"} means: @{term D} is the derivative of function @{term f} at point @{term x}

  within the set @{term s}. Where @{term s} is used to express left or right sided derivatives. In

  most cases @{term s} is either a variable or @{term UNIV}.

*}

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

  by simp

definition 

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

  (infix "(has'_field'_derivative)" 50)

where

  "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"

lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"

  by simp

definition

  has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"

  (infix "has'_vector'_derivative" 50)

where

  "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"

lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"

  by simp

ML {*

structure Derivative_Intros = Named_Thms

(

  val name = @{binding derivative_intros}

  val description = "structural introduction rules for derivatives"

)

*}

setup {*

  let

    val eq_thms = [@{thm has_derivative_eq_rhs}, @{thm DERIV_cong}, @{thm has_vector_derivative_eq_rhs}]

    fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms

  in

    Derivative_Intros.setup #>

    Global_Theory.add_thms_dynamic

      (@{binding derivative_eq_intros}, map_filter eq_rule o Derivative_Intros.get o Context.proof_of)

  end;

*}

text {*

  The following syntax is only used as a legacy syntax.

*}

abbreviation (input)

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

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

  by (simp add: has_derivative_def)

lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"

  using bounded_linear.linear[OF has_derivative_bounded_linear] .

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

  by (simp add: has_derivative_def tendsto_const)

lemma has_derivative_const[derivative_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) has_derivative:

  "(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 bounded_linear])

  apply (drule tendsto)

  apply (simp add: scaleR diff add zero)

  done

lemmas has_derivative_scaleR_right [derivative_intros] =

  bounded_linear.has_derivative [OF bounded_linear_scaleR_right]

lemmas has_derivative_scaleR_left [derivative_intros] =

  bounded_linear.has_derivative [OF bounded_linear_scaleR_left]

lemmas has_derivative_mult_right [derivative_intros] =

  bounded_linear.has_derivative [OF bounded_linear_mult_right]

lemmas has_derivative_mult_left [derivative_intros] =

  bounded_linear.has_derivative [OF bounded_linear_mult_left]

lemma has_derivative_add[simp, derivative_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 has_derivative_bounded_linear)

lemma has_derivative_setsum[simp, derivative_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 has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"

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

lemma has_derivative_diff[simp, derivative_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 has_derivative_add has_derivative_minus)

lemma has_derivative_at_within:

  "(f has_derivative f') (at x within s) \<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 has_derivative_iff_norm:

  "(f has_derivative f') (at x within s) \<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: has_derivative_at_within divide_inverse ac_simps)

lemma has_derivative_at:

  "(f has_derivative D) (at x) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. norm (f (x + h) - f x - D h) / norm h) -- 0 --> 0)"

  unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp

lemma field_has_derivative_at:

  fixes x :: "'a::real_normed_field"

  shows "(f has_derivative op * D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"

  apply (unfold has_derivative_at)

  apply (simp add: bounded_linear_mult_right)

  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 has_derivativeI:

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

  (f has_derivative f') (at x within s)"

  by (simp add: has_derivative_at_within)

lemma has_derivativeI_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 "(f has_derivative f') (at x within s)"

  unfolding has_derivative_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 has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"

  by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)

lemmas has_derivative_within_subset = has_derivative_subset 

subsection {* Continuity *}

lemma has_derivative_continuous:

  assumes f: "(f has_derivative f') (at x within s)"

  shows "continuous (at x within s) f"

proof -

  from f interpret F: bounded_linear f' by (rule has_derivative_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 has_derivative_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 has_derivative_in_compose:

  assumes f: "(f has_derivative f') (at x within s)"

  assumes g: "(g has_derivative g') (at (f x) within (f`s))"

  shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"

proof -

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

  from g interpret G: bounded_linear g' by (rule has_derivative_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 has_derivativeI_sandwich[of 1])

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

      using f g by (blast intro: bounded_linear_compose has_derivative_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 has_derivative_iff_norm Nf_def ..

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

      by (blast intro: has_derivative_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 has_derivative_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 has_derivative_compose:

  "(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>

  ((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"

  by (blast intro: has_derivative_in_compose has_derivative_subset)

lemma (in bounded_bilinear) FDERIV:

  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"

  shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"

proof -

  from bounded_linear.bounded [OF has_derivative_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 has_derivativeI_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]

        has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])

  next

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

      by (intro continuous_within[THEN iffD1] has_derivative_continuous)

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

      by (simp_all add: has_derivative_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 has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]

lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]

lemma has_derivative_setprod[simp, derivative_intros]:

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

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

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

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 "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"

      using insert by (intro has_derivative_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 has_derivative_power[simp, derivative_intros]:

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

  assumes f: "(f has_derivative f') (at x within s)"

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

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

lemma has_derivative_inverse':

  fixes x :: "'a::real_normed_div_algebra"

  assumes x: "x \<noteq> 0"

  shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"

        (is "(?inv has_derivative ?f) _")

proof (rule has_derivativeI_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 has_derivative_inverse[simp, derivative_intros]:

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

  assumes x:  "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"

  shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"

  using has_derivative_compose[OF f has_derivative_inverse', OF x] .

lemma has_derivative_divide[simp, derivative_intros]:

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

  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" 

  assumes x: "g x \<noteq> 0"

  shows "((\<lambda>x. f x / g x) has_derivative

                (\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"

  using has_derivative_mult[OF f has_derivative_inverse[OF x g]]

  by (simp add: field_simps)

text{*Conventional form requires mult-AC laws. Types real and complex only.*}

lemma has_derivative_divide'[derivative_intros]: 

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

  assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"

  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)"

proof -

  { fix h

    have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =

          (f' h * g x - f x * g' h) / (g x * g x)"

      by (simp add: field_simps x)

   }

  then show ?thesis

    using has_derivative_divide [OF f g] x

    by simp

qed

subsection {* Uniqueness *}

text {*

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

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

*}

lemma has_derivative_zero_unique:

  assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"

proof -

  interpret F: bounded_linear F

    using assms by (rule has_derivative_bounded_linear)

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

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

    using assms unfolding has_derivative_at by simp

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

  proof

    fix h show "F h = 0"

    proof (rule ccontr)

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

      hence h: "h \<noteq> 0" by (clarsimp simp add: F.zero)

      with ** have "0 < ?r h" by simp

      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 has_derivative_unique:

  assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"

proof -

  have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"

    using has_derivative_diff [OF assms] by simp

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

    by (rule has_derivative_zero_unique)

  thus "F = F'"

    unfolding fun_eq_iff right_minus_eq .

qed

subsection {* Differentiability predicate *}

definition

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

  (infix "differentiable" 50)

where

  "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"

lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"

  unfolding differentiable_def by (blast intro: has_derivative_subset)

lemmas differentiable_within_subset = differentiable_subset

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

  unfolding differentiable_def by (blast intro: has_derivative_ident)

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

  unfolding differentiable_def by (blast intro: has_derivative_const)

lemma differentiable_in_compose:

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

  unfolding differentiable_def by (blast intro: has_derivative_in_compose)

lemma differentiable_compose:

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

  by (blast intro: differentiable_in_compose differentiable_subset)

lemma differentiable_sum [simp, derivative_intros]:

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

  unfolding differentiable_def by (blast intro: has_derivative_add)

lemma differentiable_minus [simp, derivative_intros]:

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

  unfolding differentiable_def by (blast intro: has_derivative_minus)

lemma differentiable_diff [simp, derivative_intros]:

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

  unfolding differentiable_def by (blast intro: has_derivative_diff)

lemma differentiable_mult [simp, derivative_intros]:

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

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

  unfolding differentiable_def by (blast intro: has_derivative_mult)

lemma differentiable_inverse [simp, derivative_intros]:

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

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

  unfolding differentiable_def by (blast intro: has_derivative_inverse)

lemma differentiable_divide [simp, derivative_intros]:

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

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

  unfolding divide_inverse using assms by simp

lemma differentiable_power [simp, derivative_intros]:

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

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

  unfolding differentiable_def by (blast intro: has_derivative_power)

lemma differentiable_scaleR [simp, derivative_intros]:

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

  unfolding differentiable_def by (blast intro: has_derivative_scaleR)

lemma has_derivative_imp_has_field_derivative:

  "(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"

  unfolding has_field_derivative_def 

  by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult_commute)

lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"

  by (simp add: has_field_derivative_def)

lemma DERIV_subset: 

  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 

   \<Longrightarrow> (f has_field_derivative f') (at x within t)"

  by (simp add: has_field_derivative_def has_derivative_within_subset)

abbreviation (input)

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

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

where

  "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"

abbreviation 

  has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"

  (infix "(has'_real'_derivative)" 50)

where

  "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"

lemma real_differentiable_def:

  "f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"

proof safe

  assume "f differentiable at x within s"

  then obtain f' where *: "(f has_derivative f') (at x within s)"

    unfolding differentiable_def by auto

  then obtain c where "f' = (op * c)"

    by (metis real_bounded_linear has_derivative_bounded_linear mult_commute fun_eq_iff)

  with * show "\<exists>D. (f has_real_derivative D) (at x within s)"

    unfolding has_field_derivative_def by auto

qed (auto simp: differentiable_def has_field_derivative_def)

lemma real_differentiableE [elim?]:

  assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"

  using assms by (auto simp: real_differentiable_def)

lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>D. (f has_real_derivative D) (at x within s)"

  by (auto elim: real_differentiableE)

lemma differentiableI: "(f has_real_derivative D) (at x within s) \<Longrightarrow> f differentiable (at x within s)"

  by (force simp add: real_differentiable_def)

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

  apply (simp add: has_field_derivative_def has_derivative_at bounded_linear_mult_right 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

lemma mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"

  by (simp add: fun_eq_iff mult_commute)

subsection {* Derivatives *}

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, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto

lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto

lemma field_differentiable_add[derivative_intros]:

  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> 

    ((\<lambda>z. f z + g z) has_field_derivative f' + g') F"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])

     (auto simp: has_field_derivative_def field_simps mult_commute_abs)

corollary DERIV_add:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"

  by (rule field_differentiable_add)

lemma field_differentiable_minus[derivative_intros]:

  "(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])

     (auto simp: has_field_derivative_def field_simps mult_commute_abs)

corollary DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"

  by (rule field_differentiable_minus)

lemma field_differentiable_diff[derivative_intros]:

  "(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"

  by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)

corollary DERIV_diff:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"

  by (rule field_differentiable_diff)

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

  by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp

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

  by (rule DERIV_continuous)

lemma DERIV_continuous_on:

  "(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D) (at x)) \<Longrightarrow> continuous_on s f"

  by (metis DERIV_continuous continuous_at_imp_continuous_on)

lemma DERIV_mult':

  "(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])

     (auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)

lemma DERIV_mult[derivative_intros]:

  "(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])

     (auto simp: field_simps dest: has_field_derivative_imp_has_derivative)

text {* Derivative of linear multiplication *}

lemma DERIV_cmult:

  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"

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

lemma DERIV_cmult_right:

  "(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"

  using DERIV_cmult by (force simp add: mult_ac)

lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"

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

lemma DERIV_cdivide:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"

  using DERIV_cmult_right[of f D x s "1 / c"] by simp

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[derivative_intros]:

  "(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow> 

    ((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])

     (auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)

lemma DERIV_inverse'[derivative_intros]:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>

  ((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_inverse])

     (auto dest: has_field_derivative_imp_has_derivative)

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

lemma DERIV_inverse:

  "x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"

  by (drule DERIV_inverse' [OF DERIV_ident]) simp

text {* Derivative of inverse *}

lemma DERIV_inverse_fun:

  "(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>

  ((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"

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

text {* Derivative of quotient *}

lemma DERIV_divide[derivative_intros]:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow>

  (g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>

  ((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])

     (auto dest: has_field_derivative_imp_has_derivative simp: field_simps)

lemma DERIV_quotient:

  "(f has_field_derivative d) (at x within s) \<Longrightarrow>

  (g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> 

  ((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"

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

lemma DERIV_power_Suc:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])

     (auto simp: has_field_derivative_def)

lemma DERIV_power[derivative_intros]:

  "(f has_field_derivative D) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"

  by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])

     (auto simp: has_field_derivative_def)

lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"

  apply (cut_tac DERIV_power [OF DERIV_ident])

  apply (simp add: real_of_nat_def)

  done

lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow> 

  ((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"

  using has_derivative_compose[of f "op * D" x s g "op * E"]

  unfolding has_field_derivative_def mult_commute_abs ac_simps .

corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>

  ((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"

  by (rule DERIV_chain')

text {* Standard version *}

lemma DERIV_chain:

  "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow> 

  (f o g has_field_derivative Da * Db) (at x within s)"

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

lemma DERIV_image_chain: 

  "(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>

  (f o g has_field_derivative Da * Db) (at x within s)"

  using has_derivative_in_compose [of g "op * Db" x s f "op * Da "]

  by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)

(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)

lemma DERIV_chain_s:

  assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"

      and "DERIV f x :> f'" 

      and "f x \<in> s"

    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"

  by (metis (full_types) DERIV_chain' mult_commute assms)

lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)

  assumes "(\<And>x. DERIV g x :> g'(x))"

      and "DERIV f x :> f'" 

    shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"

  by (metis UNIV_I DERIV_chain_s [of UNIV] assms)

declare

  DERIV_power[where 'a=real, unfolded real_of_nat_def[symmetric], derivative_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_def 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: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)

  "(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_def 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_def 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" "-l" x, 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" "-l" x, 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