src/HOL/Deriv.thy

more on GTK;

(*  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_minus 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 add: diff_minus)
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 diff_minus [symmetric] 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 diff_minus
               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 add: diff_minus)
  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 diff_minus [symmetric]
    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 add: diff_minus)
  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 diff_minus [symmetric]
    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)
        thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
      qed
      from differentiableD [OF dif [OF conjI [OF ar rb]]]
      obtain l where der: "DERIV f r :> l" ..
      have "l=0" by (rule DERIV_local_const [OF der d bound'])
        --{*the derivative of a constant function is zero*}
      thus ?thesis using ar rb der by auto
    qed
  qed
qed


subsection{*Mean Value Theorem*}

lemma lemma_MVT:
     "f a - (f b - f a)/(b-a) * a = f b - (f b - f a)/(b-a) * (b::real)"
  by (cases "a = b") (simp_all add: field_simps)

theorem MVT:
  assumes lt:  "a < 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>l z::real. a < z & z < b & DERIV f z :> l &
                   (f(b) - f(a) = (b-a) * l)"
proof -
  let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
  have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
    using con by (fast intro: isCont_intros)
  have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable x"
  proof (clarify)
    fix x::real
    assume ax: "a < x" and xb: "x < b"
    from differentiableD [OF dif [OF conjI [OF ax xb]]]
    obtain l where der: "DERIV f x :> l" ..
    show "?F differentiable x"
      by (rule differentiableI [where D = "l - (f b - f a)/(b-a)"],
          blast intro: DERIV_diff DERIV_cmult_Id der)
  qed
  from Rolle [where f = ?F, OF lt lemma_MVT contF difF]
  obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
    by blast
  have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
    by (rule DERIV_cmult_Id)
  hence derF: "DERIV (\<lambda>x. ?F x + (f b - f a) / (b - a) * x) z
                   :> 0 + (f b - f a) / (b - a)"
    by (rule DERIV_add [OF der])
  show ?thesis
  proof (intro exI conjI)
    show "a < z" using az .
    show "z < b" using zb .
    show "f b - f a = (b - a) * ((f b - f a)/(b-a))" by (simp)
    show "DERIV f z :> ((f b - f a)/(b-a))"  using derF by simp
  qed
qed

lemma MVT2:
     "[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
      ==> \<exists>z::real. a < z & z < b & (f b - f a = (b - a) * f'(z))"
apply (drule MVT)
apply (blast intro: DERIV_isCont)
apply (force dest: order_less_imp_le simp add: differentiable_def)
apply (blast dest: DERIV_unique order_less_imp_le)
done


text{*A function is constant if its derivative is 0 over an interval.*}

lemma DERIV_isconst_end:
  fixes f :: "real => real"
  shows "[| a < b;
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
        ==> f b = f a"
apply (drule MVT, assumption)
apply (blast intro: differentiableI)
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
done

lemma DERIV_isconst1:
  fixes f :: "real => real"
  shows "[| a < b;
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
         \<forall>x. a < x & x < b --> DERIV f x :> 0 |]
        ==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
apply safe
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
apply (drule_tac b = x in DERIV_isconst_end, auto)
done

lemma DERIV_isconst2:
  fixes f :: "real => real"
  shows "[| a < b;
         \<forall>x. a \<le> x & x \<le> b --> isCont f x;
         \<forall>x. a < x & x < b --> DERIV f x :> 0;
         a \<le> x; x \<le> b |]
        ==> f x = f a"
apply (blast dest: DERIV_isconst1)
done

lemma DERIV_isconst3: fixes a b x y :: real
  assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
  assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
  shows "f x = f y"
proof (cases "x = y")
  case False
  let ?a = "min x y"
  let ?b = "max x y"
  
  have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
  proof (rule allI, rule impI)
    fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
    hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
    hence "z \<in> {a<..<b}" by auto
    thus "DERIV f z :> 0" by (rule derivable)
  qed
  hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
    and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto

  have "?a < ?b" using `x \<noteq> y` by auto
  from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
  show ?thesis by auto
qed auto

lemma DERIV_isconst_all:
  fixes f :: "real => real"
  shows "\<forall>x. DERIV f x :> 0 ==> f(x) = f(y)"
apply (rule linorder_cases [of x y])
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
done

lemma DERIV_const_ratio_const:
  fixes f :: "real => real"
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
apply (rule linorder_cases [of a b], auto)
apply (drule_tac [!] f = f in MVT)
apply (auto dest: DERIV_isCont DERIV_unique simp add: differentiable_def)
apply (auto dest: DERIV_unique simp add: ring_distribs diff_minus)
done

lemma DERIV_const_ratio_const2:
  fixes f :: "real => real"
  shows "[|a \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a))/(b-a) = k"
apply (rule_tac c1 = "b-a" in real_mult_right_cancel [THEN iffD1])
apply (auto dest!: DERIV_const_ratio_const simp add: mult_assoc)
done

lemma real_average_minus_first [simp]: "((a + b) /2 - a) = (b-a)/(2::real)"
by (simp)

lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
by (simp)

text{*Gallileo's "trick": average velocity = av. of end velocities*}

lemma DERIV_const_average:
  fixes v :: "real => real"
  assumes neq: "a \<noteq> (b::real)"
      and der: "\<forall>x. DERIV v x :> k"
  shows "v ((a + b)/2) = (v a + v b)/2"
proof (cases rule: linorder_cases [of a b])
  case equal with neq show ?thesis by simp
next
  case less
  have "(v b - v a) / (b - a) = k"
    by (rule DERIV_const_ratio_const2 [OF neq der])
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
  moreover have "(v ((a + b) / 2) - v a) / ((a + b) / 2 - a) = k"
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
  ultimately show ?thesis using neq by force
next
  case greater
  have "(v b - v a) / (b - a) = k"
    by (rule DERIV_const_ratio_const2 [OF neq der])
  hence "(b-a) * ((v b - v a) / (b-a)) = (b-a) * k" by simp
  moreover have " (v ((b + a) / 2) - v a) / ((b + a) / 2 - a) = k"
    by (rule DERIV_const_ratio_const2 [OF _ der], simp add: neq)
  ultimately show ?thesis using neq by (force simp add: add_commute)
qed

(* A function with positive derivative is increasing. 
   A simple proof using the MVT, by Jeremy Avigad. And variants.
*)
lemma DERIV_pos_imp_increasing:
  fixes a::real and b::real and f::"real => real"
  assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX y. DERIV f x :> y & y > 0)"
  shows "f a < f b"
proof (rule ccontr)
  assume f: "~ f a < f b"
  have "EX l z. a < z & z < b & DERIV f z :> l
      & f b - f a = (b - a) * l"
    apply (rule MVT)
      using assms
      apply auto
      apply (metis DERIV_isCont)
     apply (metis differentiableI less_le)
    done
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
      and "f b - f a = (b - a) * l"
    by auto
  with assms f have "~(l > 0)"
    by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
  with assms z show False
    by (metis DERIV_unique less_le)
qed

lemma DERIV_nonneg_imp_nondecreasing:
  fixes a::real and b::real and f::"real => real"
  assumes "a \<le> b" and
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
  shows "f a \<le> f b"
proof (rule ccontr, cases "a = b")
  assume "~ f a \<le> f b" and "a = b"
  then show False by auto
next
  assume A: "~ f a \<le> f b"
  assume B: "a ~= b"
  with assms have "EX l z. a < z & z < b & DERIV f z :> l
      & f b - f a = (b - a) * l"
    apply -
    apply (rule MVT)
      apply auto
      apply (metis DERIV_isCont)
     apply (metis differentiableI less_le)
    done
  then obtain l z where z: "a < z" "z < b" "DERIV f z :> l"
      and C: "f b - f a = (b - a) * l"
    by auto
  with A have "a < b" "f b < f a" by auto
  with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
    (metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
  with assms z show False
    by (metis DERIV_unique order_less_imp_le)
qed

lemma DERIV_neg_imp_decreasing:
  fixes a::real and b::real and f::"real => real"
  assumes "a < b" and
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y < 0)"
  shows "f a > f b"
proof -
  have "(%x. -f x) a < (%x. -f x) b"
    apply (rule DERIV_pos_imp_increasing [of a b "%x. -f x"])
    using assms
    apply auto
    apply (metis DERIV_minus neg_0_less_iff_less)
    done
  thus ?thesis
    by simp
qed

lemma DERIV_nonpos_imp_nonincreasing:
  fixes a::real and b::real and f::"real => real"
  assumes "a \<le> b" and
    "\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
  shows "f a \<ge> f b"
proof -
  have "(%x. -f x) a \<le> (%x. -f x) b"
    apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
    using assms
    apply auto
    apply (metis DERIV_minus neg_0_le_iff_le)
    done
  thus ?thesis
    by simp
qed

text {* Derivative of inverse function *}

lemma DERIV_inverse_function:
  fixes f g :: "real \<Rightarrow> real"
  assumes der: "DERIV f (g x) :> D"
  assumes neq: "D \<noteq> 0"
  assumes a: "a < x" and b: "x < b"
  assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
  assumes cont: "isCont g x"
  shows "DERIV g x :> inverse D"
unfolding DERIV_iff2
proof (rule LIM_equal2)
  show "0 < min (x - a) (b - x)"
    using a b by arith 
next
  fix y
  assume "norm (y - x) < min (x - a) (b - x)"
  hence "a < y" and "y < b" 
    by (simp_all add: abs_less_iff)
  thus "(g y - g x) / (y - x) =
        inverse ((f (g y) - x) / (g y - g x))"
    by (simp add: inj)
next
  have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
    by (rule der [unfolded DERIV_iff2])
  hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
    using inj a b by simp
  have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
  proof (safe intro!: exI)
    show "0 < min (x - a) (b - x)"
      using a b by simp
  next
    fix y
    assume "norm (y - x) < min (x - a) (b - x)"
    hence y: "a < y" "y < b"
      by (simp_all add: abs_less_iff)
    assume "g y = g x"
    hence "f (g y) = f (g x)" by simp
    hence "y = x" using inj y a b by simp
    also assume "y \<noteq> x"
    finally show False by simp
  qed
  have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
    using cont 1 2 by (rule isCont_LIM_compose2)
  thus "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x)))
        -- x --> inverse D"
    using neq by (rule tendsto_inverse)
qed

subsection {* Generalized Mean Value Theorem *}

theorem GMVT:
  fixes a b :: real
  assumes alb: "a < b"
    and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
    and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable x"
    and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
    and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable x"
  shows "\<exists>g'c f'c c.
    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)"
proof -
  let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
  from assms have "a < b" by simp
  moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
    using fc gc by simp
  moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable x"
    using fd gd by simp
  ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
  then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
  then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..

  from cdef have cint: "a < c \<and> c < b" by auto
  with gd have "g differentiable c" by simp
  hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
  then obtain g'c where g'cdef: "DERIV g c :> g'c" ..

  from cdef have "a < c \<and> c < b" by auto
  with fd have "f differentiable c" by simp
  hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
  then obtain f'c where f'cdef: "DERIV f c :> f'c" ..

  from cdef have "DERIV ?h c :> l" by auto
  moreover have "DERIV ?h c :>  g'c * (f b - f a) - f'c * (g b - g a)"
    using g'cdef f'cdef by (auto intro!: DERIV_intros)
  ultimately have leq: "l =  g'c * (f b - f a) - f'c * (g b - g a)" by (rule DERIV_unique)

  {
    from cdef have "?h b - ?h a = (b - a) * l" by auto
    also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
    finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
  }
  moreover
  {
    have "?h b - ?h a =
         ((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
          ((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
      by (simp add: algebra_simps)
    hence "?h b - ?h a = 0" by auto
  }
  ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
  with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
  hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
  hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: mult_ac)

  with g'cdef f'cdef cint show ?thesis by auto
qed

lemma GMVT':
  fixes f g :: "real \<Rightarrow> real"
  assumes "a < b"
  assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
  assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
  assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
  assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
  shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
proof -
  have "\<exists>g'c f'c c. 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"
    using assms by (intro GMVT) (force simp: differentiable_def)+
  then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
    using DERIV_f DERIV_g by (force dest: DERIV_unique)
  then show ?thesis
    by auto
qed


subsection {* L'Hopitals rule *}

lemma isCont_If_ge:
  fixes a :: "'a :: linorder_topology"
  shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> a then g x else f x) a"
  unfolding isCont_def continuous_within
  apply (intro filterlim_split_at)
  apply (subst filterlim_cong[OF refl refl, where g=g])
  apply (simp_all add: eventually_at_filter less_le)
  apply (subst filterlim_cong[OF refl refl, where g=f])
  apply (simp_all add: eventually_at_filter less_le)
  done

lemma lhopital_right_0:
  fixes f0 g0 :: "real \<Rightarrow> real"
  assumes f_0: "(f0 ---> 0) (at_right 0)"
  assumes g_0: "(g0 ---> 0) (at_right 0)"
  assumes ev:
    "eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
    "eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
    "eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
  shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
proof -
  def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
  then have "f 0 = 0" by simp

  def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
  then have "g 0 = 0" by simp

  have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
      DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
    using ev by eventually_elim auto
  then obtain a where [arith]: "0 < a"
    and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
    and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
    and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
    unfolding eventually_at eventually_at by (auto simp: dist_real_def)

  have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
    using g0_neq_0 by (simp add: g_def)

  { fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
         (auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
  note f = this

  { fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
      by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
         (auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x]) }
  note g = this

  have "isCont f 0"
    unfolding f_def by (intro isCont_If_ge f_0 continuous_const)

  have "isCont g 0"
    unfolding g_def by (intro isCont_If_ge g_0 continuous_const)

  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)"
  proof (rule bchoice, rule)
    fix x assume "x \<in> {0 <..< a}"
    then have x[arith]: "0 < x" "x < a" by auto
    with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a  \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
      by auto
    have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
      using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
    moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
      using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
    ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
      using f g `x < a` by (intro GMVT') auto
    then obtain c where *: "0 < c" "c < x" "(f x - f 0) * g' c = (g x - g 0) * f' c"
      by blast
    moreover
    from * g'(1)[of c] g'(2) have "(f x - f 0)  / (g x - g 0) = f' c / g' c"
      by (simp add: field_simps)
    ultimately show "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
      using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
  qed
  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)" ..
  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)"
    unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
  moreover
  from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
    by eventually_elim auto
  then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
    by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"])
       (auto intro: tendsto_const tendsto_ident_at)
  then have "(\<zeta> ---> 0) (at_right 0)"
    by (rule tendsto_norm_zero_cancel)
  with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
    by (auto elim!: eventually_elim1 simp: filterlim_at)
  from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
    by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
  ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
    by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
       (auto elim: eventually_elim1)
  also have "?P \<longleftrightarrow> ?thesis"
    by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
  finally show ?thesis .
qed

lemma lhopital_right:
  "((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
    eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
  ((\<lambda> x. f x / g x) ---> y) (at_right x)"
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
  by (rule lhopital_right_0)

lemma lhopital_left:
  "((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
    eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
  ((\<lambda> x. f x / g x) ---> y) (at_left x)"
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
  by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)

lemma lhopital:
  "((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
    eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
  ((\<lambda> x. f x / g x) ---> y) (at x)"
  unfolding eventually_at_split filterlim_at_split
  by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])

lemma lhopital_right_0_at_top:
  fixes f g :: "real \<Rightarrow> real"
  assumes g_0: "LIM x at_right 0. g x :> at_top"
  assumes ev:
    "eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
    "eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
    "eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
  shows "((\<lambda> x. f x / g x) ---> x) (at_right 0)"
  unfolding tendsto_iff
proof safe
  fix e :: real assume "0 < e"

  with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
  have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
  from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
  obtain a where [arith]: "0 < a"
    and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
    and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
    and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
    and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
    unfolding eventually_at_le by (auto simp: dist_real_def)
    

  from Df have
    "eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
    unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)

  moreover
  have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
    using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)

  moreover
  have inv_g: "((\<lambda>x. inverse (g x)) ---> 0) (at_right 0)"
    using tendsto_inverse_0 filterlim_mono[OF g_0 at_top_le_at_infinity order_refl]
    by (rule filterlim_compose)
  then have "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
    by (intro tendsto_intros)
  then have "((\<lambda>x. norm (1 - g a / g x)) ---> 1) (at_right 0)"
    by (simp add: inverse_eq_divide)
  from this[unfolded tendsto_iff, rule_format, of 1]
  have "eventually (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
    by (auto elim!: eventually_elim1 simp: dist_real_def)

  moreover
  from inv_g have "((\<lambda>t. norm ((f a - x * g a) * inverse (g t))) ---> norm ((f a - x * g a) * 0)) (at_right 0)"
    by (intro tendsto_intros)
  then have "((\<lambda>t. norm (f a - x * g a) / norm (g t)) ---> 0) (at_right 0)"
    by (simp add: inverse_eq_divide)
  from this[unfolded tendsto_iff, rule_format, of "e / 2"] `0 < e`
  have "eventually (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
    by (auto simp: dist_real_def)

  ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
  proof eventually_elim
    fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
    assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"

    have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
      using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
    then obtain y where [arith]: "t < y" "y < a"
      and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
      by blast
    from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
      using `g a < g t` g'_neq_0[of y] by (auto simp add: field_simps)

    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"
      by (simp add: field_simps)
    have "norm (f t / g t - x) \<le>
        norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
      unfolding * by (rule norm_triangle_ineq)
    also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
      by (simp add: abs_mult D_eq dist_real_def)
    also have "\<dots> < (e / 4) * 2 + e / 2"
      using ineq Df[of y] `0 < e` by (intro add_le_less_mono mult_mono) auto
    finally show "dist (f t / g t) x < e"
      by (simp add: dist_real_def)
  qed
qed

lemma lhopital_right_at_top:
  "LIM x at_right x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
    eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
    ((\<lambda> x. f x / g x) ---> y) (at_right x)"
  unfolding eventually_at_right_to_0[of _ x] filterlim_at_right_to_0[of _ _ x] DERIV_shift
  by (rule lhopital_right_0_at_top)

lemma lhopital_left_at_top:
  "LIM x at_left x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
    eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
    ((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
    ((\<lambda> x. f x / g x) ---> y) (at_left x)"
  unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
  by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)

lemma lhopital_at_top:
  "LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
    eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
    eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
    ((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
    ((\<lambda> x. f x / g x) ---> y) (at x)"
  unfolding eventually_at_split filterlim_at_split
  by (auto intro!: lhopital_right_at_top[of g x g' f f'] lhopital_left_at_top[of g x g' f f'])

lemma lhospital_at_top_at_top:
  fixes f g :: "real \<Rightarrow> real"
  assumes g_0: "LIM x at_top. g x :> at_top"
  assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
  assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
  assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
  assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
  shows "((\<lambda> x. f x / g x) ---> x) at_top"
  unfolding filterlim_at_top_to_right
proof (rule lhopital_right_0_at_top)
  let ?F = "\<lambda>x. f (inverse x)"
  let ?G = "\<lambda>x. g (inverse x)"
  let ?R = "at_right (0::real)"
  let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"

  show "LIM x ?R. ?G x :> at_top"
    using g_0 unfolding filterlim_at_top_to_right .

  show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
    unfolding eventually_at_right_to_top
    using Dg eventually_ge_at_top[where c="1::real"]
    apply eventually_elim
    apply (rule DERIV_cong)
    apply (rule DERIV_chain'[where f=inverse])
    apply (auto intro!:  DERIV_inverse)
    done

  show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
    unfolding eventually_at_right_to_top
    using Df eventually_ge_at_top[where c="1::real"]
    apply eventually_elim
    apply (rule DERIV_cong)
    apply (rule DERIV_chain'[where f=inverse])
    apply (auto intro!:  DERIV_inverse)
    done

  show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
    unfolding eventually_at_right_to_top
    using g' eventually_ge_at_top[where c="1::real"]
    by eventually_elim auto
    
  show "((\<lambda>x. ?D f' x / ?D g' x) ---> x) ?R"
    unfolding filterlim_at_right_to_top
    apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
    using eventually_ge_at_top[where c="1::real"]
    by eventually_elim simp
qed

end