src/HOL/Analysis/Derivative.thy

contributors

(*  Title:      HOL/Analysis/Derivative.thy
    Author:     John Harrison
    Author:     Robert Himmelmann, TU Muenchen (translation from HOL Light); tidied by LCP
*)

section \<open>Derivative\<close>

theory Derivative
  imports
    Bounded_Linear_Function
    Line_Segment
    Convex_Euclidean_Space
begin

declare bounded_linear_inner_left [intro]

declare has_derivative_bounded_linear[dest]

subsection \<open>Derivatives\<close>

lemma has_derivative_add_const:
  "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
  by (intro derivative_eq_intros) auto


subsection\<^marker>\<open>tag unimportant\<close> \<open>Derivative with composed bilinear function\<close>

text \<open>More explicit epsilon-delta forms.\<close>

proposition has_derivative_within':
  "(f has_derivative f')(at x within s) \<longleftrightarrow>
    bounded_linear f' \<and>
    (\<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
      norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
  unfolding has_derivative_within Lim_within dist_norm
  by (simp add: diff_diff_eq)

lemma has_derivative_at':
  "(f has_derivative f') (at x) 
   \<longleftrightarrow> bounded_linear f' \<and>
       (\<forall>e>0. \<exists>d>0. \<forall>x'. 0 < norm (x' - x) \<and> norm (x' - x) < d \<longrightarrow>
        norm (f x' - f x - f'(x' - x)) / norm (x' - x) < e)"
  using has_derivative_within' [of f f' x UNIV] by simp

lemma has_derivative_componentwise_within:
   "(f has_derivative f') (at a within S) \<longleftrightarrow>
    (\<forall>i \<in> Basis. ((\<lambda>x. f x \<bullet> i) has_derivative (\<lambda>x. f' x \<bullet> i)) (at a within S))"
  apply (simp add: has_derivative_within)
  apply (subst tendsto_componentwise_iff)
  apply (simp add: bounded_linear_componentwise_iff [symmetric] ball_conj_distrib)
  apply (simp add: algebra_simps)
  done

lemma has_derivative_at_withinI:
  "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f') (at x within s)"
  unfolding has_derivative_within' has_derivative_at'
  by blast

lemma has_derivative_right:
  fixes f :: "real \<Rightarrow> real"
    and y :: "real"
  shows "(f has_derivative ((*) y)) (at x within ({x <..} \<inter> I)) \<longleftrightarrow>
         ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x <..} \<inter> I))"
proof -
  have "((\<lambda>t. (f t - (f x + y * (t - x))) / \<bar>t - x\<bar>) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I)) \<longleftrightarrow>
    ((\<lambda>t. (f t - f x) / (t - x) - y) \<longlongrightarrow> 0) (at x within ({x<..} \<inter> I))"
    by (intro Lim_cong_within) (auto simp add: diff_divide_distrib add_divide_distrib)
  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f t - f x) / (t - x)) \<longlongrightarrow> y) (at x within ({x<..} \<inter> I))"
    by (simp add: Lim_null[symmetric])
  also have "\<dots> \<longleftrightarrow> ((\<lambda>t. (f x - f t) / (x - t)) \<longlongrightarrow> y) (at x within ({x<..} \<inter> I))"
    by (intro Lim_cong_within) (simp_all add: field_simps)
  finally show ?thesis
    by (simp add: bounded_linear_mult_right has_derivative_within)
qed

subsubsection \<open>Caratheodory characterization\<close>

lemma DERIV_caratheodory_within:
  "(f has_field_derivative l) (at x within S) \<longleftrightarrow>
   (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> continuous (at x within S) g \<and> g x = l)"
      (is "?lhs = ?rhs")
proof
  assume ?lhs
  show ?rhs
  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 "continuous (at x within S) ?g" using \<open>?lhs\<close>
      by (auto simp add: continuous_within has_field_derivative_iff cong: Lim_cong_within)
    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 "continuous (at x within S) g" and "g x = l" by blast
  thus ?lhs
    by (auto simp add: continuous_within has_field_derivative_iff cong: Lim_cong_within)
qed

subsection \<open>Differentiability\<close>

definition\<^marker>\<open>tag important\<close>
  differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool"
    (infix "differentiable'_on" 50)
  where "f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable (at x within s))"

lemma differentiableI: "(f has_derivative f') net \<Longrightarrow> f differentiable net"
  unfolding differentiable_def
  by auto

lemma differentiable_onD: "\<lbrakk>f differentiable_on S; x \<in> S\<rbrakk> \<Longrightarrow> f differentiable (at x within S)"
  using differentiable_on_def by blast

lemma differentiable_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
  unfolding differentiable_def
  using has_derivative_at_withinI
  by blast

lemma differentiable_at_imp_differentiable_on:
  "(\<And>x. x \<in> s \<Longrightarrow> f differentiable at x) \<Longrightarrow> f differentiable_on s"
  by (metis differentiable_at_withinI differentiable_on_def)

corollary\<^marker>\<open>tag unimportant\<close> differentiable_iff_scaleR:
  fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
  shows "f differentiable F \<longleftrightarrow> (\<exists>d. (f has_derivative (\<lambda>x. x *\<^sub>R d)) F)"
  by (auto simp: differentiable_def dest: has_derivative_linear linear_imp_scaleR)

lemma differentiable_on_eq_differentiable_at:
  "open s \<Longrightarrow> f differentiable_on s \<longleftrightarrow> (\<forall>x\<in>s. f differentiable at x)"
  unfolding differentiable_on_def
  by (metis at_within_interior interior_open)

lemma differentiable_transform_within:
  assumes "f differentiable (at x within s)"
    and "0 < d"
    and "x \<in> s"
    and "\<And>x'. \<lbrakk>x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
  shows "g differentiable (at x within s)"
   using assms has_derivative_transform_within unfolding differentiable_def
   by blast

lemma differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable_on S"
  by (simp add: differentiable_at_imp_differentiable_on)

lemma differentiable_on_id [simp, derivative_intros]: "id differentiable_on S"
  by (simp add: id_def)

lemma differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. c) differentiable_on S"
  by (simp add: differentiable_on_def)

lemma differentiable_on_mult [simp, derivative_intros]:
  fixes f :: "'M::real_normed_vector \<Rightarrow> 'a::real_normed_algebra"
  shows "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) differentiable_on S"
  unfolding differentiable_on_def differentiable_def
  using differentiable_def differentiable_mult by blast

lemma differentiable_on_compose:
   "\<lbrakk>g differentiable_on S; f differentiable_on (g ` S)\<rbrakk> \<Longrightarrow> (\<lambda>x. f (g x)) differentiable_on S"
by (simp add: differentiable_in_compose differentiable_on_def)

lemma bounded_linear_imp_differentiable_on: "bounded_linear f \<Longrightarrow> f differentiable_on S"
  by (simp add: differentiable_on_def bounded_linear_imp_differentiable)

lemma linear_imp_differentiable_on:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  shows "linear f \<Longrightarrow> f differentiable_on S"
by (simp add: differentiable_on_def linear_imp_differentiable)

lemma differentiable_on_minus [simp, derivative_intros]:
   "f differentiable_on S \<Longrightarrow> (\<lambda>z. -(f z)) differentiable_on S"
by (simp add: differentiable_on_def)

lemma differentiable_on_add [simp, derivative_intros]:
   "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) differentiable_on S"
by (simp add: differentiable_on_def)

lemma differentiable_on_diff [simp, derivative_intros]:
   "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) differentiable_on S"
by (simp add: differentiable_on_def)

lemma differentiable_on_inverse [simp, derivative_intros]:
  fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
  shows "f differentiable_on S \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> f x \<noteq> 0) \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable_on S"
by (simp add: differentiable_on_def)

lemma differentiable_on_scaleR [derivative_intros, simp]:
   "\<lbrakk>f differentiable_on S; g differentiable_on S\<rbrakk> \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable_on S"
  unfolding differentiable_on_def
  by (blast intro: differentiable_scaleR)

lemma has_derivative_sqnorm_at [derivative_intros, simp]:
  "((\<lambda>x. (norm x)\<^sup>2) has_derivative (\<lambda>x. 2 *\<^sub>R (a \<bullet> x))) (at a)"
  using bounded_bilinear.FDERIV  [of "(\<bullet>)" id id a _ id id]
  by (auto simp: inner_commute dot_square_norm bounded_bilinear_inner)

lemma differentiable_sqnorm_at [derivative_intros, simp]:
  fixes a :: "'a :: {real_normed_vector,real_inner}"
  shows "(\<lambda>x. (norm x)\<^sup>2) differentiable (at a)"
by (force simp add: differentiable_def intro: has_derivative_sqnorm_at)

lemma differentiable_on_sqnorm [derivative_intros, simp]:
  fixes S :: "'a :: {real_normed_vector,real_inner} set"
  shows "(\<lambda>x. (norm x)\<^sup>2) differentiable_on S"
by (simp add: differentiable_at_imp_differentiable_on)

lemma differentiable_norm_at [derivative_intros, simp]:
  fixes a :: "'a :: {real_normed_vector,real_inner}"
  shows "a \<noteq> 0 \<Longrightarrow> norm differentiable (at a)"
using differentiableI has_derivative_norm by blast

lemma differentiable_on_norm [derivative_intros, simp]:
  fixes S :: "'a :: {real_normed_vector,real_inner} set"
  shows "0 \<notin> S \<Longrightarrow> norm differentiable_on S"
by (metis differentiable_at_imp_differentiable_on differentiable_norm_at)


subsection \<open>Frechet derivative and Jacobian matrix\<close>

definition "frechet_derivative f net = (SOME f'. (f has_derivative f') net)"

proposition frechet_derivative_works:
  "f differentiable net \<longleftrightarrow> (f has_derivative (frechet_derivative f net)) net"
  unfolding frechet_derivative_def differentiable_def
  unfolding some_eq_ex[of "\<lambda> f' . (f has_derivative f') net"] ..

lemma linear_frechet_derivative: "f differentiable net \<Longrightarrow> linear (frechet_derivative f net)"
  unfolding frechet_derivative_works has_derivative_def
  by (auto intro: bounded_linear.linear)

lemma frechet_derivative_const [simp]: "frechet_derivative (\<lambda>x. c) (at a) = (\<lambda>x. 0)"
  using differentiable_const frechet_derivative_works has_derivative_const has_derivative_unique by blast

lemma frechet_derivative_id [simp]: "frechet_derivative id (at a) = id"
  using differentiable_def frechet_derivative_works has_derivative_id has_derivative_unique by blast

lemma frechet_derivative_ident [simp]: "frechet_derivative (\<lambda>x. x) (at a) = (\<lambda>x. x)"
  by (metis eq_id_iff frechet_derivative_id)


subsection \<open>Differentiability implies continuity\<close>

proposition differentiable_imp_continuous_within:
  "f differentiable (at x within s) \<Longrightarrow> continuous (at x within s) f"
  by (auto simp: differentiable_def intro: has_derivative_continuous)

lemma differentiable_imp_continuous_on:
  "f differentiable_on s \<Longrightarrow> continuous_on s f"
  unfolding differentiable_on_def continuous_on_eq_continuous_within
  using differentiable_imp_continuous_within by blast

lemma differentiable_on_subset:
  "f differentiable_on t \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable_on s"
  unfolding differentiable_on_def
  using differentiable_within_subset
  by blast

lemma differentiable_on_empty: "f differentiable_on {}"
  unfolding differentiable_on_def
  by auto

lemma has_derivative_continuous_on:
  "(\<And>x. x \<in> s \<Longrightarrow> (f has_derivative f' x) (at x within s)) \<Longrightarrow> continuous_on s f"
  by (auto intro!: differentiable_imp_continuous_on differentiableI simp: differentiable_on_def)

text \<open>Results about neighborhoods filter.\<close>

lemma eventually_nhds_metric_le:
  "eventually P (nhds a) = (\<exists>d>0. \<forall>x. dist x a \<le> d \<longrightarrow> P x)"
  unfolding eventually_nhds_metric by (safe, rule_tac x="d / 2" in exI, auto)

lemma le_nhds: "F \<le> nhds a \<longleftrightarrow> (\<forall>S. open S \<and> a \<in> S \<longrightarrow> eventually (\<lambda>x. x \<in> S) F)"
  unfolding le_filter_def eventually_nhds by (fast elim: eventually_mono)

lemma le_nhds_metric: "F \<le> nhds a \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist x a < e) F)"
  unfolding le_filter_def eventually_nhds_metric by (fast elim: eventually_mono)

lemma le_nhds_metric_le: "F \<le> nhds a \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist x a \<le> e) F)"
  unfolding le_filter_def eventually_nhds_metric_le by (fast elim: eventually_mono)

text \<open>Several results are easier using a "multiplied-out" variant.
(I got this idea from Dieudonne's proof of the chain rule).\<close>

lemma has_derivative_within_alt:
  "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
    (\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm(y - x) < d \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> e * norm (y - x))"
  unfolding has_derivative_within filterlim_def le_nhds_metric_le eventually_filtermap
    eventually_at dist_norm diff_diff_eq
  by (force simp add: linear_0 bounded_linear.linear pos_divide_le_eq)

lemma has_derivative_within_alt2:
  "(f has_derivative f') (at x within s) \<longleftrightarrow> bounded_linear f' \<and>
    (\<forall>e>0. eventually (\<lambda>y. norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)) (at x within s))"
  unfolding has_derivative_within filterlim_def le_nhds_metric_le eventually_filtermap
    eventually_at dist_norm diff_diff_eq
  by (force simp add: linear_0 bounded_linear.linear pos_divide_le_eq)

lemma has_derivative_at_alt:
  "(f has_derivative f') (at x) \<longleftrightarrow>
    bounded_linear f' \<and>
    (\<forall>e>0. \<exists>d>0. \<forall>y. norm(y - x) < d \<longrightarrow> norm (f y - f x - f'(y - x)) \<le> e * norm (y - x))"
  using has_derivative_within_alt[where s=UNIV]
  by simp


subsection \<open>The chain rule\<close>

proposition diff_chain_within[derivative_intros]:
  assumes "(f has_derivative f') (at x within s)"
    and "(g has_derivative g') (at (f x) within (f ` s))"
  shows "((g \<circ> f) has_derivative (g' \<circ> f'))(at x within s)"
  using has_derivative_in_compose[OF assms]
  by (simp add: comp_def)

lemma diff_chain_at[derivative_intros]:
  "(f has_derivative f') (at x) \<Longrightarrow>
    (g has_derivative g') (at (f x)) \<Longrightarrow> ((g \<circ> f) has_derivative (g' \<circ> f')) (at x)"
  using has_derivative_compose[of f f' x UNIV g g']
  by (simp add: comp_def)

lemma has_vector_derivative_within_open:
  "a \<in> S \<Longrightarrow> open S \<Longrightarrow>
    (f has_vector_derivative f') (at a within S) \<longleftrightarrow> (f has_vector_derivative f') (at a)"
  by (simp only: at_within_interior interior_open)

lemma field_vector_diff_chain_within:
 assumes Df: "(f has_vector_derivative f') (at x within S)"
     and Dg: "(g has_field_derivative g') (at (f x) within f ` S)"
 shows "((g \<circ> f) has_vector_derivative (f' * g')) (at x within S)"
using diff_chain_within[OF Df[unfolded has_vector_derivative_def]
                       Dg [unfolded has_field_derivative_def]]
 by (auto simp: o_def mult.commute has_vector_derivative_def)

lemma vector_derivative_diff_chain_within:
  assumes Df: "(f has_vector_derivative f') (at x within S)"
     and Dg: "(g has_derivative g') (at (f x) within f`S)"
  shows "((g \<circ> f) has_vector_derivative (g' f')) (at x within S)"
using diff_chain_within[OF Df[unfolded has_vector_derivative_def] Dg]
  linear.scaleR[OF has_derivative_linear[OF Dg]]
  unfolding has_vector_derivative_def o_def
  by (auto simp: o_def mult.commute has_vector_derivative_def)


subsection\<^marker>\<open>tag unimportant\<close> \<open>Composition rules stated just for differentiability\<close>

lemma differentiable_chain_at:
  "f differentiable (at x) \<Longrightarrow>
    g differentiable (at (f x)) \<Longrightarrow> (g \<circ> f) differentiable (at x)"
  unfolding differentiable_def
  by (meson diff_chain_at)

lemma differentiable_chain_within:
  "f differentiable (at x within S) \<Longrightarrow>
    g differentiable (at(f x) within (f ` S)) \<Longrightarrow> (g \<circ> f) differentiable (at x within S)"
  unfolding differentiable_def
  by (meson diff_chain_within)


subsection \<open>Uniqueness of derivative\<close>


text\<^marker>\<open>tag important\<close> \<open>
 The general result is a bit messy because we need approachability of the
 limit point from any direction. But OK for nontrivial intervals etc.
\<close>

proposition frechet_derivative_unique_within:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes 1: "(f has_derivative f') (at x within S)"
    and 2: "(f has_derivative f'') (at x within S)"
    and S: "\<And>i e. \<lbrakk>i\<in>Basis; e>0\<rbrakk> \<Longrightarrow> \<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> (x + d *\<^sub>R i) \<in> S"
  shows "f' = f''"
proof -
  note as = assms(1,2)[unfolded has_derivative_def]
  then interpret f': bounded_linear f' by auto
  from as interpret f'': bounded_linear f'' by auto
  have "x islimpt S" unfolding islimpt_approachable
  proof (intro allI impI)
    fix e :: real
    assume "e > 0"
    obtain d where "0 < \<bar>d\<bar>" and "\<bar>d\<bar> < e" and "x + d *\<^sub>R (SOME i. i \<in> Basis) \<in> S"
      using assms(3) SOME_Basis \<open>e>0\<close> by blast
    then show "\<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
      by (rule_tac x="x + d *\<^sub>R (SOME i. i \<in> Basis)" in bexI) (auto simp: dist_norm SOME_Basis nonzero_Basis)  qed
  then have *: "netlimit (at x within S) = x"
    by (simp add: Lim_ident_at trivial_limit_within)
  show ?thesis
  proof (rule linear_eq_stdbasis)
    show "linear f'" "linear f''"
      unfolding linear_conv_bounded_linear using as by auto
  next
    fix i :: 'a
    assume i: "i \<in> Basis"
    define e where "e = norm (f' i - f'' i)"
    show "f' i = f'' i"
    proof (rule ccontr)
      assume "f' i \<noteq> f'' i"
      then have "e > 0"
        unfolding e_def by auto
      obtain d where d:
        "0 < d"
        "(\<And>y. y\<in>S \<longrightarrow> 0 < dist y x \<and> dist y x < d \<longrightarrow>
          dist ((f y - f x - f' (y - x)) /\<^sub>R norm (y - x) -
              (f y - f x - f'' (y - x)) /\<^sub>R norm (y - x)) (0 - 0) < e)"
        using tendsto_diff [OF as(1,2)[THEN conjunct2]]
        unfolding * Lim_within
        using \<open>e>0\<close> by blast
      obtain c where c: "0 < \<bar>c\<bar>" "\<bar>c\<bar> < d \<and> x + c *\<^sub>R i \<in> S"
        using assms(3) i d(1) by blast
      have *: "norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R i)) =
        norm ((1 / \<bar>c\<bar>) *\<^sub>R (- (f' (c *\<^sub>R i)) + f'' (c *\<^sub>R i)))"
        unfolding scaleR_right_distrib by auto
      also have "\<dots> = norm ((1 / \<bar>c\<bar>) *\<^sub>R (c *\<^sub>R (- (f' i) + f'' i)))"
        unfolding f'.scaleR f''.scaleR
        unfolding scaleR_right_distrib scaleR_minus_right
        by auto
      also have "\<dots> = e"
        unfolding e_def
        using c(1)
        using norm_minus_cancel[of "f' i - f'' i"]
        by auto
      finally show False
        using c
        using d(2)[of "x + c *\<^sub>R i"]
        unfolding dist_norm
        unfolding f'.scaleR f''.scaleR f'.add f''.add f'.diff f''.diff
          scaleR_scaleR scaleR_right_diff_distrib scaleR_right_distrib
        using i
        by (auto simp: inverse_eq_divide)
    qed
  qed
qed

proposition frechet_derivative_unique_within_closed_interval:
  fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes ab: "\<And>i. i\<in>Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
    and x: "x \<in> cbox a b"
    and "(f has_derivative f' ) (at x within cbox a b)"
    and "(f has_derivative f'') (at x within cbox a b)"
  shows "f' = f''"
proof (rule frechet_derivative_unique_within)
  fix e :: real
  fix i :: 'a
  assume "e > 0" and i: "i \<in> Basis"
  then show "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R i \<in> cbox a b"
  proof (cases "x\<bullet>i = a\<bullet>i")
    case True
    with ab[of i] \<open>e>0\<close> x i show ?thesis
      by (rule_tac x="(min (b\<bullet>i - a\<bullet>i) e) / 2" in exI)
         (auto simp add: mem_box field_simps inner_simps inner_Basis)
  next
    case False
    moreover have "a \<bullet> i < x \<bullet> i"
      using False i mem_box(2) x by force
    moreover {
      have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> a\<bullet>i *2 + x\<bullet>i - a\<bullet>i"
        by auto
      also have "\<dots> = a\<bullet>i + x\<bullet>i"
        by auto
      also have "\<dots> \<le> 2 * (x\<bullet>i)"
        using \<open>a \<bullet> i < x \<bullet> i\<close> by auto
      finally have "a \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e \<le> x \<bullet> i * 2"
        by auto
    }
    moreover have "min (x \<bullet> i - a \<bullet> i) e \<ge> 0"
      by (simp add: \<open>0 < e\<close> \<open>a \<bullet> i < x \<bullet> i\<close> less_eq_real_def)
    then have "x \<bullet> i * 2 \<le> b \<bullet> i * 2 + min (x \<bullet> i - a \<bullet> i) e"
      using i mem_box(2) x by force
    ultimately show ?thesis
    using ab[of i] \<open>e>0\<close> x i 
      by (rule_tac x="- (min (x\<bullet>i - a\<bullet>i) e) / 2" in exI)
         (auto simp add: mem_box field_simps inner_simps inner_Basis)
  qed
qed (use assms in auto)

lemma frechet_derivative_unique_within_open_interval:
  fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes x: "x \<in> box a b"
    and f: "(f has_derivative f' ) (at x within box a b)" "(f has_derivative f'') (at x within box a b)"
  shows "f' = f''"
proof -
  have "at x within box a b = at x"
    by (metis x at_within_interior interior_open open_box)
  with f show "f' = f''"
    by (simp add: has_derivative_unique)
qed

lemma frechet_derivative_at:
  "(f has_derivative f') (at x) \<Longrightarrow> f' = frechet_derivative f (at x)"
  using differentiable_def frechet_derivative_works has_derivative_unique by blast

lemma frechet_derivative_compose:
  "frechet_derivative (f o g) (at x) = frechet_derivative (f) (at (g x)) o frechet_derivative g (at x)"
  if "g differentiable at x" "f differentiable at (g x)"
  by (metis diff_chain_at frechet_derivative_at frechet_derivative_works that)

lemma frechet_derivative_within_cbox:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes "\<And>i. i\<in>Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
    and "x \<in> cbox a b"
    and "(f has_derivative f') (at x within cbox a b)"
  shows "frechet_derivative f (at x within cbox a b) = f'"
  using assms
  by (metis Derivative.differentiableI frechet_derivative_unique_within_closed_interval frechet_derivative_works)

lemma frechet_derivative_transform_within_open:
  "frechet_derivative f (at x) = frechet_derivative g (at x)"
  if "f differentiable at x" "open X" "x \<in> X" "\<And>x. x \<in> X \<Longrightarrow> f x = g x"
  by (meson frechet_derivative_at frechet_derivative_works has_derivative_transform_within_open that)


subsection \<open>Derivatives of local minima and maxima are zero\<close>

lemma has_derivative_local_min:
  fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  assumes deriv: "(f has_derivative f') (at x)"
  assumes min: "eventually (\<lambda>y. f x \<le> f y) (at x)"
  shows "f' = (\<lambda>h. 0)"
proof
  fix h :: 'a
  interpret f': bounded_linear f'
    using deriv by (rule has_derivative_bounded_linear)
  show "f' h = 0"
  proof (cases "h = 0")
    case False
    from min obtain d where d1: "0 < d" and d2: "\<forall>y\<in>ball x d. f x \<le> f y"
      unfolding eventually_at by (force simp: dist_commute)
    have "FDERIV (\<lambda>r. x + r *\<^sub>R h) 0 :> (\<lambda>r. r *\<^sub>R h)"
      by (intro derivative_eq_intros) auto
    then have "FDERIV (\<lambda>r. f (x + r *\<^sub>R h)) 0 :> (\<lambda>k. f' (k *\<^sub>R h))"
      by (rule has_derivative_compose, simp add: deriv)
    then have "DERIV (\<lambda>r. f (x + r *\<^sub>R h)) 0 :> f' h"
      unfolding has_field_derivative_def by (simp add: f'.scaleR mult_commute_abs)
    moreover have "0 < d / norm h" using d1 and \<open>h \<noteq> 0\<close> by simp
    moreover have "\<forall>y. \<bar>0 - y\<bar> < d / norm h \<longrightarrow> f (x + 0 *\<^sub>R h) \<le> f (x + y *\<^sub>R h)"
      using \<open>h \<noteq> 0\<close> by (auto simp add: d2 dist_norm pos_less_divide_eq)
    ultimately show "f' h = 0"
      by (rule DERIV_local_min)
  qed simp
qed

lemma has_derivative_local_max:
  fixes f :: "'a::real_normed_vector \<Rightarrow> real"
  assumes "(f has_derivative f') (at x)"
  assumes "eventually (\<lambda>y. f y \<le> f x) (at x)"
  shows "f' = (\<lambda>h. 0)"
  using has_derivative_local_min [of "\<lambda>x. - f x" "\<lambda>h. - f' h" "x"]
  using assms unfolding fun_eq_iff by simp

lemma differential_zero_maxmin:
  fixes f::"'a::real_normed_vector \<Rightarrow> real"
  assumes "x \<in> S"
    and "open S"
    and deriv: "(f has_derivative f') (at x)"
    and mono: "(\<forall>y\<in>S. f y \<le> f x) \<or> (\<forall>y\<in>S. f x \<le> f y)"
  shows "f' = (\<lambda>v. 0)"
  using mono
proof
  assume "\<forall>y\<in>S. f y \<le> f x"
  with \<open>x \<in> S\<close> and \<open>open S\<close> have "eventually (\<lambda>y. f y \<le> f x) (at x)"
    unfolding eventually_at_topological by auto
  with deriv show ?thesis
    by (rule has_derivative_local_max)
next
  assume "\<forall>y\<in>S. f x \<le> f y"
  with \<open>x \<in> S\<close> and \<open>open S\<close> have "eventually (\<lambda>y. f x \<le> f y) (at x)"
    unfolding eventually_at_topological by auto
  with deriv show ?thesis
    by (rule has_derivative_local_min)
qed

lemma differential_zero_maxmin_component:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes k: "k \<in> Basis"
    and ball: "0 < e" "(\<forall>y \<in> ball x e. (f y)\<bullet>k \<le> (f x)\<bullet>k) \<or> (\<forall>y\<in>ball x e. (f x)\<bullet>k \<le> (f y)\<bullet>k)"
    and diff: "f differentiable (at x)"
  shows "(\<Sum>j\<in>Basis. (frechet_derivative f (at x) j \<bullet> k) *\<^sub>R j) = (0::'a)" (is "?D k = 0")
proof -
  let ?f' = "frechet_derivative f (at x)"
  have "x \<in> ball x e" using \<open>0 < e\<close> by simp
  moreover have "open (ball x e)" by simp
  moreover have "((\<lambda>x. f x \<bullet> k) has_derivative (\<lambda>h. ?f' h \<bullet> k)) (at x)"
    using bounded_linear_inner_left diff[unfolded frechet_derivative_works]
    by (rule bounded_linear.has_derivative)
  ultimately have "(\<lambda>h. frechet_derivative f (at x) h \<bullet> k) = (\<lambda>v. 0)"
    using ball(2) by (rule differential_zero_maxmin)
  then show ?thesis
    unfolding fun_eq_iff by simp
qed

subsection \<open>One-dimensional mean value theorem\<close>

lemma mvt_simple:
  fixes f :: "real \<Rightarrow> real"
  assumes "a < b"
    and derf: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x within {a..b})"
  shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
proof (rule mvt)
  have "f differentiable_on {a..b}"
    using derf unfolding differentiable_on_def differentiable_def by force
  then show "continuous_on {a..b} f"
    by (rule differentiable_imp_continuous_on)
  show "(f has_derivative f' x) (at x)" if "a < x" "x < b" for x
    by (metis at_within_Icc_at derf leI order.asym that)
qed (use assms in auto)

lemma mvt_very_simple:
  fixes f :: "real \<Rightarrow> real"
  assumes "a \<le> b"
    and derf: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x within {a..b})"
  shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
proof (cases "a = b")
  interpret bounded_linear "f' b"
    using assms(2) assms(1) by auto
  case True
  then show ?thesis
    by force
next
  case False
  then show ?thesis
    using mvt_simple[OF _ derf]
    by (metis \<open>a \<le> b\<close> atLeastAtMost_iff dual_order.order_iff_strict greaterThanLessThan_iff)
qed

text \<open>A nice generalization (see Havin's proof of 5.19 from Rudin's book).\<close>

lemma mvt_general:
  fixes f :: "real \<Rightarrow> 'a::real_inner"
  assumes "a < b"
    and contf: "continuous_on {a..b} f"
    and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
  shows "\<exists>x\<in>{a<..<b}. norm (f b - f a) \<le> norm (f' x (b - a))"
proof -
  have "\<exists>x\<in>{a<..<b}. (f b - f a) \<bullet> f b - (f b - f a) \<bullet> f a = (f b - f a) \<bullet> f' x (b - a)"
    apply (rule mvt [OF \<open>a < b\<close>, where f = "\<lambda>x. (f b - f a) \<bullet> f x"])
    apply (intro continuous_intros contf)
    using derf apply (auto intro: has_derivative_inner_right)
    done
  then obtain x where x: "x \<in> {a<..<b}"
    "(f b - f a) \<bullet> f b - (f b - f a) \<bullet> f a = (f b - f a) \<bullet> f' x (b - a)" ..
  show ?thesis
  proof (cases "f a = f b")
    case False
    have "norm (f b - f a) * norm (f b - f a) = (norm (f b - f a))\<^sup>2"
      by (simp add: power2_eq_square)
    also have "\<dots> = (f b - f a) \<bullet> (f b - f a)"
      unfolding power2_norm_eq_inner ..
    also have "\<dots> = (f b - f a) \<bullet> f' x (b - a)"
      using x(2) by (simp only: inner_diff_right)
    also have "\<dots> \<le> norm (f b - f a) * norm (f' x (b - a))"
      by (rule norm_cauchy_schwarz)
    finally show ?thesis
      using False x(1)
      by (auto simp add: mult_left_cancel)
  next
    case True
    then show ?thesis
      using \<open>a < b\<close> by (rule_tac x="(a + b) /2" in bexI) auto
  qed
qed


subsection \<open>More general bound theorems\<close>

proposition differentiable_bound_general:
  fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
  assumes "a < b"
    and f_cont: "continuous_on {a..b} f"
    and phi_cont: "continuous_on {a..b} \<phi>"
    and f': "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
    and phi': "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<phi> has_vector_derivative \<phi>' x) (at x)"
    and bnd: "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> norm (f' x) \<le> \<phi>' x"
  shows "norm (f b - f a) \<le> \<phi> b - \<phi> a"
proof -
  {
    fix x assume x: "a < x" "x < b"
    have "0 \<le> norm (f' x)" by simp
    also have "\<dots> \<le> \<phi>' x" using x by (auto intro!: bnd)
    finally have "0 \<le> \<phi>' x" .
  } note phi'_nonneg = this
  note f_tendsto = assms(2)[simplified continuous_on_def, rule_format]
  note phi_tendsto = assms(3)[simplified continuous_on_def, rule_format]
  {
    fix e::real assume "e > 0"
    define e2 where "e2 = e / 2"
    with \<open>e > 0\<close> have "e2 > 0" by simp
    let ?le = "\<lambda>x1. norm (f x1 - f a) \<le> \<phi> x1 - \<phi> a + e * (x1 - a) + e"
    define A where "A = {x2. a \<le> x2 \<and> x2 \<le> b \<and> (\<forall>x1\<in>{a ..< x2}. ?le x1)}"
    have A_subset: "A \<subseteq> {a..b}" by (auto simp: A_def)
    {
      fix x2
      assume a: "a \<le> x2" "x2 \<le> b" and le: "\<forall>x1\<in>{a..<x2}. ?le x1"
      have "?le x2" using \<open>e > 0\<close>
      proof cases
        assume "x2 \<noteq> a" with a have "a < x2" by simp
        have "at x2 within {a <..<x2}\<noteq> bot"
          using \<open>a < x2\<close>
          by (auto simp: trivial_limit_within islimpt_in_closure)
        moreover
        have "((\<lambda>x1. (\<phi> x1 - \<phi> a) + e * (x1 - a) + e) \<longlongrightarrow> (\<phi> x2 - \<phi> a) + e * (x2 - a) + e) (at x2 within {a <..<x2})"
          "((\<lambda>x1. norm (f x1 - f a)) \<longlongrightarrow> norm (f x2 - f a)) (at x2 within {a <..<x2})"
          using a
          by (auto intro!: tendsto_eq_intros f_tendsto phi_tendsto
            intro: tendsto_within_subset[where S="{a..b}"])
        moreover
        have "eventually (\<lambda>x. x > a) (at x2 within {a <..<x2})"
          by (auto simp: eventually_at_filter)
        hence "eventually ?le (at x2 within {a <..<x2})"
          unfolding eventually_at_filter
          by eventually_elim (insert le, auto)
        ultimately
        show ?thesis
          by (rule tendsto_le)
      qed simp
    } note le_cont = this
    have "a \<in> A"
      using assms by (auto simp: A_def)
    hence [simp]: "A \<noteq> {}" by auto
    have A_ivl: "\<And>x1 x2. x2 \<in> A \<Longrightarrow> x1 \<in> {a ..x2} \<Longrightarrow> x1 \<in> A"
      by (simp add: A_def)
    have [simp]: "bdd_above A" by (auto simp: A_def)
    define y where "y = Sup A"
    have "y \<le> b"
      unfolding y_def
      by (simp add: cSup_le_iff) (simp add: A_def)
     have leI: "\<And>x x1. a \<le> x1 \<Longrightarrow> x \<in> A \<Longrightarrow> x1 < x \<Longrightarrow> ?le x1"
       by (auto simp: A_def intro!: le_cont)
    have y_all_le: "\<forall>x1\<in>{a..<y}. ?le x1"
      by (auto simp: y_def less_cSup_iff leI)
    have "a \<le> y"
      by (metis \<open>a \<in> A\<close> \<open>bdd_above A\<close> cSup_upper y_def)
    have "y \<in> A"
      using y_all_le \<open>a \<le> y\<close> \<open>y \<le> b\<close>
      by (auto simp: A_def)
    hence "A = {a .. y}"
      using A_subset by (auto simp: subset_iff y_def cSup_upper intro: A_ivl)
    from le_cont[OF \<open>a \<le> y\<close> \<open>y \<le> b\<close> y_all_le] have le_y: "?le y" .
    have "y = b"
    proof (cases "a = y")
      case True
      with \<open>a < b\<close> have "y < b" by simp
      with \<open>a = y\<close> f_cont phi_cont \<open>e2 > 0\<close>
      have 1: "\<forall>\<^sub>F x in at y within {y..b}. dist (f x) (f y) < e2"
       and 2: "\<forall>\<^sub>F x in at y within {y..b}. dist (\<phi> x) (\<phi> y) < e2"
        by (auto simp: continuous_on_def tendsto_iff)
      have 3: "eventually (\<lambda>x. y < x) (at y within {y..b})"
        by (auto simp: eventually_at_filter)
      have 4: "eventually (\<lambda>x::real. x < b) (at y within {y..b})"
        using _ \<open>y < b\<close>
        by (rule order_tendstoD) (auto intro!: tendsto_eq_intros)
      from 1 2 3 4
      have eventually_le: "eventually (\<lambda>x. ?le x) (at y within {y .. b})"
      proof eventually_elim
        case (elim x1)
        have "norm (f x1 - f a) = norm (f x1 - f y)"
          by (simp add: \<open>a = y\<close>)
        also have "norm (f x1 - f y) \<le> e2"
          using elim \<open>a = y\<close> by (auto simp : dist_norm intro!:  less_imp_le)
        also have "\<dots> \<le> e2 + (\<phi> x1 - \<phi> a + e2 + e * (x1 - a))"
          using \<open>0 < e\<close> elim
          by (intro add_increasing2[OF add_nonneg_nonneg order.refl])
            (auto simp: \<open>a = y\<close> dist_norm intro!: mult_nonneg_nonneg)
        also have "\<dots> = \<phi> x1 - \<phi> a + e * (x1 - a) + e"
          by (simp add: e2_def)
        finally show "?le x1" .
      qed
      from this[unfolded eventually_at_topological] \<open>?le y\<close>
      obtain S where S: "open S" "y \<in> S" "\<And>x. x\<in>S \<Longrightarrow> x \<in> {y..b} \<Longrightarrow> ?le x"
        by metis
      from \<open>open S\<close> obtain d where d: "\<And>x. dist x y < d \<Longrightarrow> x \<in> S" "d > 0"
        by (force simp: dist_commute open_dist ball_def dest!: bspec[OF _ \<open>y \<in> S\<close>])
      define d' where "d' = min b (y + (d/2))"
      have "d' \<in> A"
        unfolding A_def
      proof safe
        show "a \<le> d'" using \<open>a = y\<close> \<open>0 < d\<close> \<open>y < b\<close> by (simp add: d'_def)
        show "d' \<le> b" by (simp add: d'_def)
        fix x1
        assume "x1 \<in> {a..<d'}"
        hence "x1 \<in> S" "x1 \<in> {y..b}"
          by (auto simp: \<open>a = y\<close> d'_def dist_real_def intro!: d )
        thus "?le x1"
          by (rule S)
      qed
      hence "d' \<le> y"
        unfolding y_def
        by (rule cSup_upper) simp
      then show "y = b" using \<open>d > 0\<close> \<open>y < b\<close>
        by (simp add: d'_def)
    next
      case False
      with \<open>a \<le> y\<close> have "a < y" by simp
      show "y = b"
      proof (rule ccontr)
        assume "y \<noteq> b"
        hence "y < b" using \<open>y \<le> b\<close> by simp
        let ?F = "at y within {y..<b}"
        from f' phi'
        have "(f has_vector_derivative f' y) ?F"
          and "(\<phi> has_vector_derivative \<phi>' y) ?F"
          using \<open>a < y\<close> \<open>y < b\<close>
          by (auto simp add: at_within_open[of _ "{a<..<b}"] has_vector_derivative_def
            intro!: has_derivative_subset[where s="{a<..<b}" and t="{y..<b}"])
        hence "\<forall>\<^sub>F x1 in ?F. norm (f x1 - f y - (x1 - y) *\<^sub>R f' y) \<le> e2 * \<bar>x1 - y\<bar>"
            "\<forall>\<^sub>F x1 in ?F. norm (\<phi> x1 - \<phi> y - (x1 - y) *\<^sub>R \<phi>' y) \<le> e2 * \<bar>x1 - y\<bar>"
          using \<open>e2 > 0\<close>
          by (auto simp: has_derivative_within_alt2 has_vector_derivative_def)
        moreover
        have "\<forall>\<^sub>F x1 in ?F. y \<le> x1" "\<forall>\<^sub>F x1 in ?F. x1 < b"
          by (auto simp: eventually_at_filter)
        ultimately
        have "\<forall>\<^sub>F x1 in ?F. norm (f x1 - f y) \<le> (\<phi> x1 - \<phi> y) + e * \<bar>x1 - y\<bar>"
          (is "\<forall>\<^sub>F x1 in ?F. ?le' x1")
        proof eventually_elim
          case (elim x1)
          from norm_triangle_ineq2[THEN order_trans, OF elim(1)]
          have "norm (f x1 - f y) \<le> norm (f' y) * \<bar>x1 - y\<bar> + e2 * \<bar>x1 - y\<bar>"
            by (simp add: ac_simps)
          also have "norm (f' y) \<le> \<phi>' y" using bnd \<open>a < y\<close> \<open>y < b\<close> by simp
          also have "\<phi>' y * \<bar>x1 - y\<bar> \<le> \<phi> x1 - \<phi> y + e2 * \<bar>x1 - y\<bar>"
            using elim by (simp add: ac_simps)
          finally
          have "norm (f x1 - f y) \<le> \<phi> x1 - \<phi> y + e2 * \<bar>x1 - y\<bar> + e2 * \<bar>x1 - y\<bar>"
            by (auto simp: mult_right_mono)
          thus ?case by (simp add: e2_def)
        qed
        moreover have "?le' y" by simp
        ultimately obtain S
        where S: "open S" "y \<in> S" "\<And>x. x\<in>S \<Longrightarrow> x \<in> {y..<b} \<Longrightarrow> ?le' x"
          unfolding eventually_at_topological
          by metis
        from \<open>open S\<close> obtain d where d: "\<And>x. dist x y < d \<Longrightarrow> x \<in> S" "d > 0"
          by (force simp: dist_commute open_dist ball_def dest!: bspec[OF _ \<open>y \<in> S\<close>])
        define d' where "d' = min ((y + b)/2) (y + (d/2))"
        have "d' \<in> A"
          unfolding A_def
        proof safe
          show "a \<le> d'" using \<open>a < y\<close> \<open>0 < d\<close> \<open>y < b\<close> by (simp add: d'_def)
          show "d' \<le> b" using \<open>y < b\<close> by (simp add: d'_def min_def)
          fix x1
          assume x1: "x1 \<in> {a..<d'}"
          show "?le x1"
          proof (cases "x1 < y")
            case True
            then show ?thesis
              using \<open>y \<in> A\<close> local.leI x1 by auto
          next
            case False
            hence x1': "x1 \<in> S" "x1 \<in> {y..<b}" using x1
              by (auto simp: d'_def dist_real_def intro!: d)
            have "norm (f x1 - f a) \<le> norm (f x1 - f y) + norm (f y - f a)"
              by (rule order_trans[OF _ norm_triangle_ineq]) simp
            also note S(3)[OF x1']
            also note le_y
            finally show "?le x1"
              using False by (auto simp: algebra_simps)
          qed
        qed
        hence "d' \<le> y"
          unfolding y_def by (rule cSup_upper) simp
        thus False using \<open>d > 0\<close> \<open>y < b\<close>
          by (simp add: d'_def min_def split: if_split_asm)
      qed
    qed
    with le_y have "norm (f b - f a) \<le> \<phi> b - \<phi> a + e * (b - a + 1)"
      by (simp add: algebra_simps)
  } note * = this
  show ?thesis
  proof (rule field_le_epsilon)
    fix e::real assume "e > 0"
    then show "norm (f b - f a) \<le> \<phi> b - \<phi> a + e"
      using *[of "e / (b - a + 1)"] \<open>a < b\<close> by simp
  qed
qed

lemma differentiable_bound:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "convex S"
    and derf: "\<And>x. x\<in>S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
    and B: "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x) \<le> B"
    and x: "x \<in> S"
    and y: "y \<in> S"
  shows "norm (f x - f y) \<le> B * norm (x - y)"
proof -
  let ?p = "\<lambda>u. x + u *\<^sub>R (y - x)"
  let ?\<phi> = "\<lambda>h. h * B * norm (x - y)"
  have *: "x + u *\<^sub>R (y - x) \<in> S" if "u \<in> {0..1}" for u
  proof -
    have "u *\<^sub>R y = u *\<^sub>R (y - x) + u *\<^sub>R x"
      by (simp add: scale_right_diff_distrib)
    then show "x + u *\<^sub>R (y - x) \<in> S"
      using that \<open>convex S\<close> x y by (simp add: convex_alt)
        (metis pth_b(2) pth_c(1) scaleR_collapse)
  qed
  have "\<And>z. z \<in> (\<lambda>u. x + u *\<^sub>R (y - x)) ` {0..1} \<Longrightarrow>
          (f has_derivative f' z) (at z within (\<lambda>u. x + u *\<^sub>R (y - x)) ` {0..1})"
    by (auto intro: * has_derivative_subset [OF derf])
  then have "continuous_on (?p ` {0..1}) f"
    unfolding continuous_on_eq_continuous_within
    by (meson has_derivative_continuous)
  with * have 1: "continuous_on {0 .. 1} (f \<circ> ?p)"
    by (intro continuous_intros)+
  {
    fix u::real assume u: "u \<in>{0 <..< 1}"
    let ?u = "?p u"
    interpret linear "(f' ?u)"
      using u by (auto intro!: has_derivative_linear derf *)
    have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within box 0 1)"
      by (intro derivative_intros has_derivative_subset [OF derf]) (use u * in auto)
    hence "((f \<circ> ?p) has_vector_derivative f' ?u (y - x)) (at u)"
      by (simp add: at_within_open[OF u open_greaterThanLessThan] scaleR has_vector_derivative_def o_def)
  } note 2 = this
  have 3: "continuous_on {0..1} ?\<phi>"
    by (rule continuous_intros)+
  have 4: "(?\<phi> has_vector_derivative B * norm (x - y)) (at u)" for u
    by (auto simp: has_vector_derivative_def intro!: derivative_eq_intros)
  {
    fix u::real assume u: "u \<in>{0 <..< 1}"
    let ?u = "?p u"
    interpret bounded_linear "(f' ?u)"
      using u by (auto intro!: has_derivative_bounded_linear derf *)
    have "norm (f' ?u (y - x)) \<le> onorm (f' ?u) * norm (y - x)"
      by (rule onorm) (rule bounded_linear)
    also have "onorm (f' ?u) \<le> B"
      using u by (auto intro!: assms(3)[rule_format] *)
    finally have "norm ((f' ?u) (y - x)) \<le> B * norm (x - y)"
      by (simp add: mult_right_mono norm_minus_commute)
  } note 5 = this
  have "norm (f x - f y) = norm ((f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 1 - (f \<circ> (\<lambda>u. x + u *\<^sub>R (y - x))) 0)"
    by (auto simp add: norm_minus_commute)
  also
  from differentiable_bound_general[OF zero_less_one 1, OF 3 2 4 5]
  have "norm ((f \<circ> ?p) 1 - (f \<circ> ?p) 0) \<le> B * norm (x - y)"
    by simp
  finally show ?thesis .
qed

lemma field_differentiable_bound:
  fixes S :: "'a::real_normed_field set"
  assumes cvs: "convex S"
      and df:  "\<And>z. z \<in> S \<Longrightarrow> (f has_field_derivative f' z) (at z within S)"
      and dn:  "\<And>z. z \<in> S \<Longrightarrow> norm (f' z) \<le> B"
      and "x \<in> S"  "y \<in> S"
    shows "norm(f x - f y) \<le> B * norm(x - y)"
  apply (rule differentiable_bound [OF cvs])
  apply (erule df [unfolded has_field_derivative_def])
  apply (rule onorm_le, simp_all add: norm_mult mult_right_mono assms)
  done

lemma
  differentiable_bound_segment:
  fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "\<And>t. t \<in> {0..1} \<Longrightarrow> x0 + t *\<^sub>R a \<in> G"
  assumes f': "\<And>x. x \<in> G \<Longrightarrow> (f has_derivative f' x) (at x within G)"
  assumes B: "\<And>x. x \<in> {0..1} \<Longrightarrow> onorm (f' (x0 + x *\<^sub>R a)) \<le> B"
  shows "norm (f (x0 + a) - f x0) \<le> norm a * B"
proof -
  let ?G = "(\<lambda>x. x0 + x *\<^sub>R a) ` {0..1}"
  have "?G = (+) x0 ` (\<lambda>x. x *\<^sub>R a) ` {0..1}" by auto
  also have "convex \<dots>"
    by (intro convex_translation convex_scaled convex_real_interval)
  finally have "convex ?G" .
  moreover have "?G \<subseteq> G" "x0 \<in> ?G" "x0 + a \<in> ?G" using assms by (auto intro: image_eqI[where x=1])
  ultimately show ?thesis
    using has_derivative_subset[OF f' \<open>?G \<subseteq> G\<close>] B
      differentiable_bound[of "(\<lambda>x. x0 + x *\<^sub>R a) ` {0..1}" f f' B "x0 + a" x0]
    by (force simp: ac_simps)
qed

lemma differentiable_bound_linearization:
  fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes S: "\<And>t. t \<in> {0..1} \<Longrightarrow> a + t *\<^sub>R (b - a) \<in> S"
  assumes f'[derivative_intros]: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
  assumes B: "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x - f' x0) \<le> B"
  assumes "x0 \<in> S"
  shows "norm (f b - f a - f' x0 (b - a)) \<le> norm (b - a) * B"
proof -
  define g where [abs_def]: "g x = f x - f' x0 x" for x
  have g: "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative (\<lambda>i. f' x i - f' x0 i)) (at x within S)"
    unfolding g_def using assms
    by (auto intro!: derivative_eq_intros
      bounded_linear.has_derivative[OF has_derivative_bounded_linear, OF f'])
  from B have "\<forall>x\<in>{0..1}. onorm (\<lambda>i. f' (a + x *\<^sub>R (b - a)) i - f' x0 i) \<le> B"
    using assms by (auto simp: fun_diff_def)
  with differentiable_bound_segment[OF S g] \<open>x0 \<in> S\<close>
  show ?thesis
    by (simp add: g_def field_simps linear_diff[OF has_derivative_linear[OF f']])
qed

lemma vector_differentiable_bound_linearization:
  fixes f::"real \<Rightarrow> 'b::real_normed_vector"
  assumes f': "\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)"
  assumes "closed_segment a b \<subseteq> S"
  assumes B: "\<And>x. x \<in> S \<Longrightarrow> norm (f' x - f' x0) \<le> B"
  assumes "x0 \<in> S"
  shows "norm (f b - f a - (b - a) *\<^sub>R f' x0) \<le> norm (b - a) * B"
  using assms
  by (intro differentiable_bound_linearization[of a b S f "\<lambda>x h. h *\<^sub>R f' x" x0 B])
    (force simp: closed_segment_real_eq has_vector_derivative_def
      scaleR_diff_right[symmetric] mult.commute[of B]
      intro!: onorm_le mult_left_mono)+


text \<open>In particular.\<close>

lemma has_derivative_zero_constant:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "convex s"
    and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
  shows "\<exists>c. \<forall>x\<in>s. f x = c"
proof -
  { fix x y assume "x \<in> s" "y \<in> s"
    then have "norm (f x - f y) \<le> 0 * norm (x - y)"
      using assms by (intro differentiable_bound[of s]) (auto simp: onorm_zero)
    then have "f x = f y"
      by simp }
  then show ?thesis
    by metis
qed

lemma has_field_derivative_zero_constant:
  assumes "convex s" "\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
  shows   "\<exists>c. \<forall>x\<in>s. f (x) = (c :: 'a :: real_normed_field)"
proof (rule has_derivative_zero_constant)
  have A: "(*) 0 = (\<lambda>_. 0 :: 'a)" by (intro ext) simp
  fix x assume "x \<in> s" thus "(f has_derivative (\<lambda>h. 0)) (at x within s)"
    using assms(2)[of x] by (simp add: has_field_derivative_def A)
qed fact

lemma
  has_vector_derivative_zero_constant:
  assumes "convex s"
  assumes "\<And>x. x \<in> s \<Longrightarrow> (f has_vector_derivative 0) (at x within s)"
  obtains c where "\<And>x. x \<in> s \<Longrightarrow> f x = c"
  using has_derivative_zero_constant[of s f] assms
  by (auto simp: has_vector_derivative_def)

lemma has_derivative_zero_unique:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "convex s"
    and "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
    and "x \<in> s" "y \<in> s"
  shows "f x = f y"
  using has_derivative_zero_constant[OF assms(1,2)] assms(3-) by force

lemma has_derivative_zero_unique_connected:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "open s" "connected s"
  assumes f: "\<And>x. x \<in> s \<Longrightarrow> (f has_derivative (\<lambda>x. 0)) (at x)"
  assumes "x \<in> s" "y \<in> s"
  shows "f x = f y"
proof (rule connected_local_const[where f=f, OF \<open>connected s\<close> \<open>x\<in>s\<close> \<open>y\<in>s\<close>])
  show "\<forall>a\<in>s. eventually (\<lambda>b. f a = f b) (at a within s)"
  proof
    fix a assume "a \<in> s"
    with \<open>open s\<close> obtain e where "0 < e" "ball a e \<subseteq> s"
      by (rule openE)
    then have "\<exists>c. \<forall>x\<in>ball a e. f x = c"
      by (intro has_derivative_zero_constant)
         (auto simp: at_within_open[OF _ open_ball] f)
    with \<open>0<e\<close> have "\<forall>x\<in>ball a e. f a = f x"
      by auto
    then show "eventually (\<lambda>b. f a = f b) (at a within s)"
      using \<open>0<e\<close> unfolding eventually_at_topological
      by (intro exI[of _ "ball a e"]) auto
  qed
qed

subsection \<open>Differentiability of inverse function (most basic form)\<close>

lemma has_derivative_inverse_basic:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes derf: "(f has_derivative f') (at (g y))"
    and ling': "bounded_linear g'"
    and "g' \<circ> f' = id"
    and contg: "continuous (at y) g"
    and "open T"
    and "y \<in> T"
    and fg: "\<And>z. z \<in> T \<Longrightarrow> f (g z) = z"
  shows "(g has_derivative g') (at y)"
proof -
  interpret f': bounded_linear f'
    using assms unfolding has_derivative_def by auto
  interpret g': bounded_linear g'
    using assms by auto
  obtain C where C: "0 < C" "\<And>x. norm (g' x) \<le> norm x * C"
    using bounded_linear.pos_bounded[OF assms(2)] by blast
  have lem1: "\<forall>e>0. \<exists>d>0. \<forall>z.
    norm (z - y) < d \<longrightarrow> norm (g z - g y - g'(z - y)) \<le> e * norm (g z - g y)"
  proof (intro allI impI)
    fix e :: real
    assume "e > 0"
    with C(1) have *: "e / C > 0" by auto
    obtain d0 where  "0 < d0" and d0:
        "\<And>u. norm (u - g y) < d0 \<Longrightarrow> norm (f u - f (g y) - f' (u - g y)) \<le> e / C * norm (u - g y)"
      using derf * unfolding has_derivative_at_alt by blast
    obtain d1 where "0 < d1" and d1: "\<And>x. \<lbrakk>0 < dist x y; dist x y < d1\<rbrakk> \<Longrightarrow> dist (g x) (g y) < d0"
      using contg \<open>0 < d0\<close> unfolding continuous_at Lim_at by blast
    obtain d2 where "0 < d2" and d2: "\<And>u. dist u y < d2 \<Longrightarrow> u \<in> T"
      using \<open>open T\<close> \<open>y \<in> T\<close> unfolding open_dist by blast
    obtain d where d: "0 < d" "d < d1" "d < d2"
      using field_lbound_gt_zero[OF \<open>0 < d1\<close> \<open>0 < d2\<close>] by blast
    show "\<exists>d>0. \<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
    proof (intro exI allI impI conjI)
      fix z
      assume as: "norm (z - y) < d"
      then have "z \<in> T"
        using d2 d unfolding dist_norm by auto
      have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
        unfolding g'.diff f'.diff
        unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] fg[OF \<open>z\<in>T\<close>]
        by (simp add: norm_minus_commute)
      also have "\<dots> \<le> norm (f (g z) - y - f' (g z - g y)) * C"
        by (rule C(2))
      also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
      proof -
        have "norm (g z - g y) < d0"
          by (metis as cancel_comm_monoid_add_class.diff_cancel d(2) \<open>0 < d0\<close> d1 diff_gt_0_iff_gt diff_strict_mono dist_norm dist_self zero_less_dist_iff)
        then show ?thesis
          by (metis C(1) \<open>y \<in> T\<close> d0 fg mult_le_cancel_iff1)
      qed
      also have "\<dots> \<le> e * norm (g z - g y)"
        using C by (auto simp add: field_simps)
      finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
        by simp
    qed (use d in auto)
  qed
  have *: "(0::real) < 1 / 2"
    by auto
  obtain d where "0 < d" and d:
      "\<And>z. norm (z - y) < d \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> 1/2 * norm (g z - g y)"
    using lem1 * by blast
  define B where "B = C * 2"
  have "B > 0"
    unfolding B_def using C by auto
  have lem2: "norm (g z - g y) \<le> B * norm (z - y)" if z: "norm(z - y) < d" for z
  proof -
    have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
      by (rule norm_triangle_sub)
    also have "\<dots> \<le> norm (g' (z - y)) + 1 / 2 * norm (g z - g y)"
      by (rule add_left_mono) (use d z in auto)
    also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
      by (rule add_right_mono) (use C in auto)
    finally show "norm (g z - g y) \<le> B * norm (z - y)"
      unfolding B_def
      by (auto simp add: field_simps)
  qed
  show ?thesis
    unfolding has_derivative_at_alt
  proof (intro conjI assms allI impI)
    fix e :: real
    assume "e > 0"
    then have *: "e / B > 0" by (metis \<open>B > 0\<close> divide_pos_pos)
    obtain d' where "0 < d'" and d':
        "\<And>z. norm (z - y) < d' \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> e / B * norm (g z - g y)"
      using lem1 * by blast
    obtain k where k: "0 < k" "k < d" "k < d'"
      using field_lbound_gt_zero[OF \<open>0 < d\<close> \<open>0 < d'\<close>] by blast
    show "\<exists>d>0. \<forall>ya. norm (ya - y) < d \<longrightarrow> norm (g ya - g y - g' (ya - y)) \<le> e * norm (ya - y)"
    proof (intro exI allI impI conjI)
      fix z
      assume as: "norm (z - y) < k"
      then have "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
        using d' k by auto
      also have "\<dots> \<le> e * norm (z - y)"
        unfolding times_divide_eq_left pos_divide_le_eq[OF \<open>B>0\<close>]
        using lem2[of z] k as \<open>e > 0\<close>
        by (auto simp add: field_simps)
      finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
        by simp
    qed (use k in auto)
  qed
qed

text\<^marker>\<open>tag unimportant\<close>\<open>Inverse function theorem for complex derivatives\<close>
lemma has_field_derivative_inverse_basic:
  shows "DERIV f (g y) :> f' \<Longrightarrow>
        f' \<noteq> 0 \<Longrightarrow>
        continuous (at y) g \<Longrightarrow>
        open t \<Longrightarrow>
        y \<in> t \<Longrightarrow>
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
        \<Longrightarrow> DERIV g y :> inverse (f')"
  unfolding has_field_derivative_def
  apply (rule has_derivative_inverse_basic)
  apply (auto simp:  bounded_linear_mult_right)
  done

text \<open>Simply rewrite that based on the domain point x.\<close>

lemma has_derivative_inverse_basic_x:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "(f has_derivative f') (at x)"
    and "bounded_linear g'"
    and "g' \<circ> f' = id"
    and "continuous (at (f x)) g"
    and "g (f x) = x"
    and "open T"
    and "f x \<in> T"
    and "\<And>y. y \<in> T \<Longrightarrow> f (g y) = y"
  shows "(g has_derivative g') (at (f x))"
  by (rule has_derivative_inverse_basic) (use assms in auto)

text \<open>This is the version in Dieudonne', assuming continuity of f and g.\<close>

lemma has_derivative_inverse_dieudonne:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "open S"
    and "open (f ` S)"
    and "continuous_on S f"
    and "continuous_on (f ` S) g"
    and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
    and "x \<in> S"
    and "(f has_derivative f') (at x)"
    and "bounded_linear g'"
    and "g' \<circ> f' = id"
  shows "(g has_derivative g') (at (f x))"
  apply (rule has_derivative_inverse_basic_x[OF assms(7-9) _ _ assms(2)])
  using assms(3-6)
  unfolding continuous_on_eq_continuous_at[OF assms(1)] continuous_on_eq_continuous_at[OF assms(2)]
  apply auto
  done

text \<open>Here's the simplest way of not assuming much about g.\<close>

proposition has_derivative_inverse:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "compact S"
    and "x \<in> S"
    and fx: "f x \<in> interior (f ` S)"
    and "continuous_on S f"
    and gf: "\<And>y. y \<in> S \<Longrightarrow> g (f y) = y"
    and "(f has_derivative f') (at x)"
    and "bounded_linear g'"
    and "g' \<circ> f' = id"
  shows "(g has_derivative g') (at (f x))"
proof -
  have *: "\<And>y. y \<in> interior (f ` S) \<Longrightarrow> f (g y) = y"
    by (metis gf image_iff interior_subset subsetCE)
  show ?thesis
    apply (rule has_derivative_inverse_basic_x[OF assms(6-8), where T = "interior (f ` S)"])
    apply (rule continuous_on_interior[OF _ fx])
    apply (rule continuous_on_inv)
    apply (simp_all add: assms *)
    done
qed


text \<open>Invertible derivative continuous at a point implies local
injectivity. It's only for this we need continuity of the derivative,
except of course if we want the fact that the inverse derivative is
also continuous. So if we know for some other reason that the inverse
function exists, it's OK.\<close>

proposition has_derivative_locally_injective:
  fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
  assumes "a \<in> S"
      and "open S"
      and bling: "bounded_linear g'"
      and "g' \<circ> f' a = id"
      and derf: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x)"
      and "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < e"
  obtains r where "r > 0" "ball a r \<subseteq> S" "inj_on f (ball a r)"
proof -
  interpret bounded_linear g'
    using assms by auto
  note f'g' = assms(4)[unfolded id_def o_def,THEN cong]
  have "g' (f' a (\<Sum>Basis)) = (\<Sum>Basis)" "(\<Sum>Basis) \<noteq> (0::'n)"
    using f'g' by auto
  then have *: "0 < onorm g'"
    unfolding onorm_pos_lt[OF assms(3)]
    by fastforce
  define k where "k = 1 / onorm g' / 2"
  have *: "k > 0"
    unfolding k_def using * by auto
  obtain d1 where d1:
      "0 < d1"
      "\<And>x. dist a x < d1 \<Longrightarrow> onorm (\<lambda>v. f' x v - f' a v) < k"
    using assms(6) * by blast
  from \<open>open S\<close> obtain d2 where "d2 > 0" "ball a d2 \<subseteq> S"
    using \<open>a\<in>S\<close> ..
  obtain d2 where d2: "0 < d2" "ball a d2 \<subseteq> S"
    using \<open>0 < d2\<close> \<open>ball a d2 \<subseteq> S\<close> by blast
  obtain d where d: "0 < d" "d < d1" "d < d2"
    using field_lbound_gt_zero[OF d1(1) d2(1)] by blast
  show ?thesis
  proof
    show "0 < d" by (fact d)
    show "ball a d \<subseteq> S"
      using \<open>d < d2\<close> \<open>ball a d2 \<subseteq> S\<close> by auto
    show "inj_on f (ball a d)"
    unfolding inj_on_def
    proof (intro strip)
      fix x y
      assume as: "x \<in> ball a d" "y \<in> ball a d" "f x = f y"
      define ph where [abs_def]: "ph w = w - g' (f w - f x)" for w
      have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
        unfolding ph_def o_def  by (simp add: diff f'g')
      have "norm (ph x - ph y) \<le> (1 / 2) * norm (x - y)"
      proof (rule differentiable_bound[OF convex_ball _ _ as(1-2)])
        fix u
        assume u: "u \<in> ball a d"
        then have "u \<in> S"
          using d d2 by auto
        have *: "(\<lambda>v. v - g' (f' u v)) = g' \<circ> (\<lambda>w. f' a w - f' u w)"
          unfolding o_def and diff
          using f'g' by auto
        have blin: "bounded_linear (f' a)"
          using \<open>a \<in> S\<close> derf by blast
        show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
          unfolding ph' * comp_def
          by (rule \<open>u \<in> S\<close> derivative_eq_intros has_derivative_at_withinI [OF derf] bounded_linear.has_derivative [OF blin]  bounded_linear.has_derivative [OF bling] |simp)+
        have **: "bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)"
          using \<open>u \<in> S\<close> blin bounded_linear_sub derf by auto
        then have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)"
          by (simp add: "*" bounded_linear_axioms onorm_compose)
        also have "\<dots> \<le> onorm g' * k"
          apply (rule mult_left_mono)
          using d1(2)[of u]
          using onorm_neg[where f="\<lambda>x. f' u x - f' a x"] d u onorm_pos_le[OF bling] apply (auto simp: algebra_simps)
          done
        also have "\<dots> \<le> 1 / 2"
          unfolding k_def by auto
        finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" .
      qed
      moreover have "norm (ph y - ph x) = norm (y - x)"
        by (simp add: as(3) ph_def)
      ultimately show "x = y"
        unfolding norm_minus_commute by auto
    qed
  qed
qed


subsection \<open>Uniformly convergent sequence of derivatives\<close>

lemma has_derivative_sequence_lipschitz_lemma:
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "convex S"
    and derf: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
    and nle: "\<And>n x h. \<lbrakk>n\<ge>N; x \<in> S\<rbrakk> \<Longrightarrow> norm (f' n x h - g' x h) \<le> e * norm h"
    and "0 \<le> e"
  shows "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
proof clarify
  fix m n x y
  assume as: "N \<le> m" "N \<le> n" "x \<in> S" "y \<in> S"
  show "norm ((f m x - f n x) - (f m y - f n y)) \<le> 2 * e * norm (x - y)"
  proof (rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF \<open>convex S\<close> _ _ as(3-4)])
    fix x
    assume "x \<in> S"
    show "((\<lambda>a. f m a - f n a) has_derivative (\<lambda>h. f' m x h - f' n x h)) (at x within S)"
      by (rule derivative_intros derf \<open>x\<in>S\<close>)+
    show "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e"
    proof (rule onorm_bound)
      fix h
      have "norm (f' m x h - f' n x h) \<le> norm (f' m x h - g' x h) + norm (f' n x h - g' x h)"
        using norm_triangle_ineq[of "f' m x h - g' x h" "- f' n x h + g' x h"]
        by (auto simp add: algebra_simps norm_minus_commute)
      also have "\<dots> \<le> e * norm h + e * norm h"
        using nle[OF \<open>N \<le> m\<close> \<open>x \<in> S\<close>, of h] nle[OF \<open>N \<le> n\<close> \<open>x \<in> S\<close>, of h]
        by (auto simp add: field_simps)
      finally show "norm (f' m x h - f' n x h) \<le> 2 * e * norm h"
        by auto
    qed (simp add: \<open>0 \<le> e\<close>)
  qed
qed

lemma has_derivative_sequence_Lipschitz:
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  assumes "convex S"
    and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
    and nle: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
    and "e > 0"
  shows "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
    norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
proof -
  have *: "2 * (e/2) = e"
    using \<open>e > 0\<close> by auto
  obtain N where "\<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> (e/2) * norm h"
    using nle \<open>e > 0\<close>
    unfolding eventually_sequentially
    by (metis less_divide_eq_numeral1(1) mult_zero_left)
  then show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
    apply (rule_tac x=N in exI)
    apply (rule has_derivative_sequence_lipschitz_lemma[where e="e/2", unfolded *])
    using assms \<open>e > 0\<close>
    apply auto
    done
qed

proposition has_derivative_sequence:
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
  assumes "convex S"
    and derf: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
    and nle: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
    and "x0 \<in> S"
    and lim: "((\<lambda>n. f n x0) \<longlongrightarrow> l) sequentially"
  shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x \<and> (g has_derivative g'(x)) (at x within S)"
proof -
  have lem1: "\<And>e. e > 0 \<Longrightarrow> \<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S.
      norm ((f m x - f n x) - (f m y - f n y)) \<le> e * norm (x - y)"
    using assms(1,2,3) by (rule has_derivative_sequence_Lipschitz)
  have "\<exists>g. \<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially"
  proof (intro ballI bchoice)
    fix x
    assume "x \<in> S"
    show "\<exists>y. (\<lambda>n. f n x) \<longlonglongrightarrow> y"
    unfolding convergent_eq_Cauchy
    proof (cases "x = x0")
      case True
      then show "Cauchy (\<lambda>n. f n x)"
        using LIMSEQ_imp_Cauchy[OF lim] by auto
    next
      case False
      show "Cauchy (\<lambda>n. f n x)"
        unfolding Cauchy_def
      proof (intro allI impI)
        fix e :: real
        assume "e > 0"
        hence *: "e / 2 > 0" "e / 2 / norm (x - x0) > 0" using False by auto
        obtain M where M: "\<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x0) (f n x0) < e / 2"
          using LIMSEQ_imp_Cauchy[OF lim] * unfolding Cauchy_def by blast
        obtain N where N:
          "\<forall>m\<ge>N. \<forall>n\<ge>N.
            \<forall>u\<in>S. \<forall>y\<in>S. norm (f m u - f n u - (f m y - f n y)) \<le>
              e / 2 / norm (x - x0) * norm (u - y)"
        using lem1 *(2) by blast
        show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e"
        proof (intro exI allI impI)
          fix m n
          assume as: "max M N \<le>m" "max M N\<le>n"
          have "dist (f m x) (f n x) \<le> norm (f m x0 - f n x0) + norm (f m x - f n x - (f m x0 - f n x0))"
            unfolding dist_norm
            by (rule norm_triangle_sub)
          also have "\<dots> \<le> norm (f m x0 - f n x0) + e / 2"
            using N \<open>x\<in>S\<close> \<open>x0\<in>S\<close> as False by fastforce
          also have "\<dots> < e / 2 + e / 2"
            by (rule add_strict_right_mono) (use as M in \<open>auto simp: dist_norm\<close>)
          finally show "dist (f m x) (f n x) < e"
            by auto
        qed
      qed
    qed
  qed
  then obtain g where g: "\<forall>x\<in>S. (\<lambda>n. f n x) \<longlonglongrightarrow> g x" ..
  have lem2: "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm ((f n x - f n y) - (g x - g y)) \<le> e * norm (x - y)" if "e > 0" for e
  proof -
    obtain N where
      N: "\<forall>m\<ge>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f m x - f n x - (f m y - f n y)) \<le> e * norm (x - y)"
      using lem1 \<open>e > 0\<close> by blast
    show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
    proof (intro exI ballI allI impI)
      fix n x y
      assume as: "N \<le> n" "x \<in> S" "y \<in> S"
      have "((\<lambda>m. norm (f n x - f n y - (f m x - f m y))) \<longlongrightarrow> norm (f n x - f n y - (g x - g y))) sequentially"
        by (intro tendsto_intros g[rule_format] as)
      moreover have "eventually (\<lambda>m. norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)) sequentially"
        unfolding eventually_sequentially
      proof (intro exI allI impI)
        fix m
        assume "N \<le> m"
        then show "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
          using N as by (auto simp add: algebra_simps)
      qed
      ultimately show "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)"
        by (simp add: tendsto_upperbound)
    qed
  qed
  have "\<forall>x\<in>S. ((\<lambda>n. f n x) \<longlongrightarrow> g x) sequentially \<and> (g has_derivative g' x) (at x within S)"
    unfolding has_derivative_within_alt2
  proof (intro ballI conjI allI impI)
    fix x
    assume "x \<in> S"
    then show "(\<lambda>n. f n x) \<longlonglongrightarrow> g x"
      by (simp add: g)
    have tog': "(\<lambda>n. f' n x u) \<longlonglongrightarrow> g' x u" for u
      unfolding filterlim_def le_nhds_metric_le eventually_filtermap dist_norm
    proof (intro allI impI)
      fix e :: real
      assume "e > 0"
      show "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e) sequentially"
      proof (cases "u = 0")
        case True
        have "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e * norm u) sequentially"
          using nle \<open>0 < e\<close> \<open>x \<in> S\<close> by (fast elim: eventually_mono)
        then show ?thesis
          using \<open>u = 0\<close> \<open>0 < e\<close> by (auto elim: eventually_mono)
      next
        case False
        with \<open>0 < e\<close> have "0 < e / norm u" by simp
        then have "eventually (\<lambda>n. norm (f' n x u - g' x u) \<le> e / norm u * norm u) sequentially"
          using nle \<open>x \<in> S\<close> by (fast elim: eventually_mono)
        then show ?thesis
          using \<open>u \<noteq> 0\<close> by simp
      qed
    qed
    show "bounded_linear (g' x)"
    proof
      fix x' y z :: 'a
      fix c :: real
      note lin = assms(2)[rule_format,OF \<open>x\<in>S\<close>,THEN has_derivative_bounded_linear]
      show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'"
        apply (rule tendsto_unique[OF trivial_limit_sequentially tog'])
        unfolding lin[THEN bounded_linear.linear, THEN linear_cmul]
        apply (intro tendsto_intros tog')
        done
      show "g' x (y + z) = g' x y + g' x z"
        apply (rule tendsto_unique[OF trivial_limit_sequentially tog'])
        unfolding lin[THEN bounded_linear.linear, THEN linear_add]
        apply (rule tendsto_add)
        apply (rule tog')+
        done
      obtain N where N: "\<forall>h. norm (f' N x h - g' x h) \<le> 1 * norm h"
        using nle \<open>x \<in> S\<close> unfolding eventually_sequentially by (fast intro: zero_less_one)
      have "bounded_linear (f' N x)"
        using derf \<open>x \<in> S\<close> by fast
      from bounded_linear.bounded [OF this]
      obtain K where K: "\<forall>h. norm (f' N x h) \<le> norm h * K" ..
      {
        fix h
        have "norm (g' x h) = norm (f' N x h - (f' N x h - g' x h))"
          by simp
        also have "\<dots> \<le> norm (f' N x h) + norm (f' N x h - g' x h)"
          by (rule norm_triangle_ineq4)
        also have "\<dots> \<le> norm h * K + 1 * norm h"
          using N K by (fast intro: add_mono)
        finally have "norm (g' x h) \<le> norm h * (K + 1)"
          by (simp add: ring_distribs)
      }
      then show "\<exists>K. \<forall>h. norm (g' x h) \<le> norm h * K" by fast
    qed
    show "eventually (\<lambda>y. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)) (at x within S)"
      if "e > 0" for e
    proof -
      have *: "e / 3 > 0"
        using that by auto
      obtain N1 where N1: "\<forall>n\<ge>N1. \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e / 3 * norm h"
        using nle * unfolding eventually_sequentially by blast
      obtain N2 where
          N2[rule_format]: "\<forall>n\<ge>N2. \<forall>x\<in>S. \<forall>y\<in>S. norm (f n x - f n y - (g x - g y)) \<le> e / 3 * norm (x - y)"
        using lem2 * by blast
      let ?N = "max N1 N2"
      have "eventually (\<lambda>y. norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)) (at x within S)"
        using derf[unfolded has_derivative_within_alt2] and \<open>x \<in> S\<close> and * by fast
      moreover have "eventually (\<lambda>y. y \<in> S) (at x within S)"
        unfolding eventually_at by (fast intro: zero_less_one)
      ultimately show "\<forall>\<^sub>F y in at x within S. norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
      proof (rule eventually_elim2)
        fix y
        assume "y \<in> S"
        assume "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)"
        moreover have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e / 3 * norm (y - x)"
          using N2[OF _ \<open>y \<in> S\<close> \<open>x \<in> S\<close>]
          by (simp add: norm_minus_commute)
        ultimately have "norm (g y - g x - f' ?N x (y - x)) \<le> 2 * e / 3 * norm (y - x)"
          using norm_triangle_le[of "g y - g x - (f ?N y - f ?N x)" "f ?N y - f ?N x - f' ?N x (y - x)" "2 * e / 3 * norm (y - x)"]
          by (auto simp add: algebra_simps)
        moreover
        have " norm (f' ?N x (y - x) - g' x (y - x)) \<le> e / 3 * norm (y - x)"
          using N1 \<open>x \<in> S\<close> by auto
        ultimately show "norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)"
          using norm_triangle_le[of "g y - g x - f' (max N1 N2) x (y - x)" "f' (max N1 N2) x (y - x) - g' x (y - x)"]
          by (auto simp add: algebra_simps)
      qed
    qed
  qed
  then show ?thesis by fast
qed

text \<open>Can choose to line up antiderivatives if we want.\<close>

lemma has_antiderivative_sequence:
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
  assumes "convex S"
    and der: "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
    and no: "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially.
       \<forall>x\<in>S. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
  shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
proof (cases "S = {}")
  case False
  then obtain a where "a \<in> S"
    by auto
  have *: "\<And>P Q. \<exists>g. \<forall>x\<in>S. P g x \<and> Q g x \<Longrightarrow> \<exists>g. \<forall>x\<in>S. Q g x"
    by auto
  show ?thesis
    apply (rule *)
    apply (rule has_derivative_sequence [OF \<open>convex S\<close> _ no, of "\<lambda>n x. f n x + (f 0 a - f n a)"])
       apply (metis assms(2) has_derivative_add_const)
    using \<open>a \<in> S\<close> 
      apply auto
    done
qed auto

lemma has_antiderivative_limit:
  fixes g' :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'b::banach"
  assumes "convex S"
    and "\<And>e. e>0 \<Longrightarrow> \<exists>f f'. \<forall>x\<in>S.
           (f has_derivative (f' x)) (at x within S) \<and> (\<forall>h. norm (f' x h - g' x h) \<le> e * norm h)"
  shows "\<exists>g. \<forall>x\<in>S. (g has_derivative g' x) (at x within S)"
proof -
  have *: "\<forall>n. \<exists>f f'. \<forall>x\<in>S.
    (f has_derivative (f' x)) (at x within S) \<and>
    (\<forall>h. norm(f' x h - g' x h) \<le> inverse (real (Suc n)) * norm h)"
    by (simp add: assms(2))
  obtain f where
    *: "\<And>x. \<exists>f'. \<forall>xa\<in>S. (f x has_derivative f' xa) (at xa within S) \<and>
        (\<forall>h. norm (f' xa h - g' xa h) \<le> inverse (real (Suc x)) * norm h)"
    using * by metis
  obtain f' where
    f': "\<And>x. \<forall>z\<in>S. (f x has_derivative f' x z) (at z within S) \<and>
            (\<forall>h. norm (f' x z h - g' z h) \<le> inverse (real (Suc x)) * norm h)"
    using * by metis
  show ?thesis
  proof (rule has_antiderivative_sequence[OF \<open>convex S\<close>, of f f'])
    fix e :: real
    assume "e > 0"
    obtain N where N: "inverse (real (Suc N)) < e"
      using reals_Archimedean[OF \<open>e>0\<close>] ..
    show "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>S.  \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"
        unfolding eventually_sequentially
    proof (intro exI allI ballI impI)
      fix n x h
      assume n: "N \<le> n" and x: "x \<in> S"
      have *: "inverse (real (Suc n)) \<le> e"
        apply (rule order_trans[OF _ N[THEN less_imp_le]])
        using n apply (auto simp add: field_simps)
        done
      show "norm (f' n x h - g' x h) \<le> e * norm h"
        by (meson "*" mult_right_mono norm_ge_zero order.trans x f')
    qed
  qed (use f' in auto)
qed


subsection \<open>Differentiation of a series\<close>

proposition has_derivative_series:
  fixes f :: "nat \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::banach"
  assumes "convex S"
    and "\<And>n x. x \<in> S \<Longrightarrow> ((f n) has_derivative (f' n x)) (at x within S)"
    and "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. norm (sum (\<lambda>i. f' i x h) {..<n} - g' x h) \<le> e * norm h"
    and "x \<in> S"
    and "(\<lambda>n. f n x) sums l"
  shows "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums (g x) \<and> (g has_derivative g' x) (at x within S)"
  unfolding sums_def
  apply (rule has_derivative_sequence[OF assms(1) _ assms(3)])
  apply (metis assms(2) has_derivative_sum)
  using assms(4-5)
  unfolding sums_def
  apply auto
  done

lemma has_field_derivative_series:
  fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
  assumes "convex S"
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
  assumes "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)"
  shows   "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
unfolding has_field_derivative_def
proof (rule has_derivative_series)
  show "\<forall>\<^sub>F n in sequentially.
       \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" if "e > 0" for e
    unfolding eventually_sequentially
  proof -
    from that assms(3) obtain N where N: "\<And>n x. n \<ge> N \<Longrightarrow> x \<in> S \<Longrightarrow> norm ((\<Sum>i<n. f' i x) - g' x) < e"
      unfolding uniform_limit_iff eventually_at_top_linorder dist_norm by blast
    {
      fix n :: nat and x h :: 'a assume nx: "n \<ge> N" "x \<in> S"
      have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) = norm ((\<Sum>i<n. f' i x) - g' x) * norm h"
        by (simp add: norm_mult [symmetric] ring_distribs sum_distrib_right)
      also from N[OF nx] have "norm ((\<Sum>i<n. f' i x) - g' x) \<le> e" by simp
      hence "norm ((\<Sum>i<n. f' i x) - g' x) * norm h \<le> e * norm h"
        by (intro mult_right_mono) simp_all
      finally have "norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" .
    }
    thus "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>S. \<forall>h. norm ((\<Sum>i<n. f' i x * h) - g' x * h) \<le> e * norm h" by blast
  qed
qed (use assms in \<open>auto simp: has_field_derivative_def\<close>)

lemma has_field_derivative_series':
  fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
  assumes "convex S"
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x within S)"
  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" "x \<in> interior S"
  shows   "summable (\<lambda>n. f n x)" "((\<lambda>x. \<Sum>n. f n x) has_field_derivative (\<Sum>n. f' n x)) (at x)"
proof -
  from \<open>x \<in> interior S\<close> have "x \<in> S" using interior_subset by blast
  define g' where [abs_def]: "g' x = (\<Sum>i. f' i x)" for x
  from assms(3) have "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
    by (simp add: uniformly_convergent_uniform_limit_iff suminf_eq_lim g'_def)
  from has_field_derivative_series[OF assms(1,2) this assms(4,5)] obtain g where g:
    "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
  from g(1)[OF \<open>x \<in> S\<close>] show "summable (\<lambda>n. f n x)" by (simp add: sums_iff)
  from g(2)[OF \<open>x \<in> S\<close>] \<open>x \<in> interior S\<close> have "(g has_field_derivative g' x) (at x)"
    by (simp add: at_within_interior[of x S])
  also have "(g has_field_derivative g' x) (at x) \<longleftrightarrow>
                ((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' x) (at x)"
    using eventually_nhds_in_nhd[OF \<open>x \<in> interior S\<close>] interior_subset[of S] g(1)
    by (intro DERIV_cong_ev) (auto elim!: eventually_mono simp: sums_iff)
  finally show "((\<lambda>x. \<Sum>n. f n x) has_field_derivative g' x) (at x)" .
qed

lemma differentiable_series:
  fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
  assumes "convex S" "open S"
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
  shows   "summable (\<lambda>n. f n x)" and "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)"
proof -
  from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
    unfolding uniformly_convergent_on_def by blast
  from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
  have "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
    by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
  then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
  from g[OF x] show "summable (\<lambda>n. f n x)" by (auto simp: summable_def)
  from g(2)[OF x] have g': "(g has_derivative (*) (g' x)) (at x)"
    by (simp add: has_field_derivative_def S)
  have "((\<lambda>x. \<Sum>n. f n x) has_derivative (*) (g' x)) (at x)"
    by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
       (insert g, auto simp: sums_iff)
  thus "(\<lambda>x. \<Sum>n. f n x) differentiable (at x)" unfolding differentiable_def
    by (auto simp: summable_def differentiable_def has_field_derivative_def)
qed

lemma differentiable_series':
  fixes f :: "nat \<Rightarrow> ('a :: {real_normed_field,banach}) \<Rightarrow> 'a"
  assumes "convex S" "open S"
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)"
  shows   "(\<lambda>x. \<Sum>n. f n x) differentiable (at x0)"
  using differentiable_series[OF assms, of x0] \<open>x0 \<in> S\<close> by blast+

subsection \<open>Derivative as a vector\<close>

text \<open>Considering derivative \<^typ>\<open>real \<Rightarrow> 'b::real_normed_vector\<close> as a vector.\<close>

definition "vector_derivative f net = (SOME f'. (f has_vector_derivative f') net)"

lemma vector_derivative_unique_within:
  assumes not_bot: "at x within S \<noteq> bot"
    and f': "(f has_vector_derivative f') (at x within S)"
    and f'': "(f has_vector_derivative f'') (at x within S)"
  shows "f' = f''"
proof -
  have "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
  proof (rule frechet_derivative_unique_within, simp_all)
    show "\<exists>d. d \<noteq> 0 \<and> \<bar>d\<bar> < e \<and> x + d \<in> S" if "0 < e"  for e
    proof -
      from that
      obtain x' where "x' \<in> S" "x' \<noteq> x" "\<bar>x' - x\<bar> < e"
        using islimpt_approachable_real[of x S] not_bot
        by (auto simp add: trivial_limit_within)
      then show ?thesis
        using eq_iff_diff_eq_0 by fastforce
    qed
  qed (use f' f'' in \<open>auto simp: has_vector_derivative_def\<close>)
  then show ?thesis
    unfolding fun_eq_iff by (metis scaleR_one)
qed

lemma vector_derivative_unique_at:
  "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f'') (at x) \<Longrightarrow> f' = f''"
  by (rule vector_derivative_unique_within) auto

lemma differentiableI_vector: "(f has_vector_derivative y) F \<Longrightarrow> f differentiable F"
  by (auto simp: differentiable_def has_vector_derivative_def)

proposition vector_derivative_works:
  "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net"
    (is "?l = ?r")
proof
  assume ?l
  obtain f' where f': "(f has_derivative f') net"
    using \<open>?l\<close> unfolding differentiable_def ..
  then interpret bounded_linear f'
    by auto
  show ?r
    unfolding vector_derivative_def has_vector_derivative_def
    by (rule someI[of _ "f' 1"]) (simp add: scaleR[symmetric] f')
qed (auto simp: vector_derivative_def has_vector_derivative_def differentiable_def)

lemma vector_derivative_within:
  assumes not_bot: "at x within S \<noteq> bot" and y: "(f has_vector_derivative y) (at x within S)"
  shows "vector_derivative f (at x within S) = y"
  using y
  by (intro vector_derivative_unique_within[OF not_bot vector_derivative_works[THEN iffD1] y])
     (auto simp: differentiable_def has_vector_derivative_def)

lemma frechet_derivative_eq_vector_derivative:
  assumes "f differentiable (at x)"
    shows  "(frechet_derivative f (at x)) = (\<lambda>r. r *\<^sub>R vector_derivative f (at x))"
using assms
by (auto simp: differentiable_iff_scaleR vector_derivative_def has_vector_derivative_def
         intro: someI frechet_derivative_at [symmetric])

lemma has_real_derivative:
  fixes f :: "real \<Rightarrow> real"
  assumes "(f has_derivative f') F"
  obtains c where "(f has_real_derivative c) F"
proof -
  obtain c where "f' = (\<lambda>x. x * c)"
    by (metis assms has_derivative_bounded_linear real_bounded_linear)
  then show ?thesis
    by (metis assms that has_field_derivative_def mult_commute_abs)
qed

lemma has_real_derivative_iff:
  fixes f :: "real \<Rightarrow> real"
  shows "(\<exists>c. (f has_real_derivative c) F) = (\<exists>D. (f has_derivative D) F)"
  by (metis has_field_derivative_def has_real_derivative)

lemma has_vector_derivative_cong_ev:
  assumes *: "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x = g x) (nhds x)" "f x = g x"
  shows "(f has_vector_derivative f') (at x within S) = (g has_vector_derivative f') (at x within S)"
  unfolding has_vector_derivative_def has_derivative_def
  using *
  apply (cases "at x within S \<noteq> bot")
  apply (intro refl conj_cong filterlim_cong)
  apply (auto simp: Lim_ident_at eventually_at_filter elim: eventually_mono)
  done

lemma islimpt_closure_open:
  fixes s :: "'a::perfect_space set"
  assumes "open s" and t: "t = closure s" "x \<in> t"
  shows "x islimpt t"
proof cases
  assume "x \<in> s"
  { fix T assume "x \<in> T" "open T"
    then have "open (s \<inter> T)"
      using \<open>open s\<close> by auto
    then have "s \<inter> T \<noteq> {x}"
      using not_open_singleton[of x] by auto
    with \<open>x \<in> T\<close> \<open>x \<in> s\<close> have "\<exists>y\<in>t. y \<in> T \<and> y \<noteq> x"
      using closure_subset[of s] by (auto simp: t) }
  then show ?thesis
    by (auto intro!: islimptI)
next
  assume "x \<notin> s" with t show ?thesis
    unfolding t closure_def by (auto intro: islimpt_subset)
qed

lemma vector_derivative_unique_within_closed_interval:
  assumes ab: "a < b" "x \<in> cbox a b"
  assumes D: "(f has_vector_derivative f') (at x within cbox a b)" "(f has_vector_derivative f'') (at x within cbox a b)"
  shows "f' = f''"
  using ab
  by (intro vector_derivative_unique_within[OF _ D])
     (auto simp: trivial_limit_within intro!: islimpt_closure_open[where s="{a <..< b}"])

lemma vector_derivative_at:
  "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
  by (intro vector_derivative_within at_neq_bot)

lemma has_vector_derivative_id_at [simp]: "vector_derivative (\<lambda>x. x) (at a) = 1"
  by (simp add: vector_derivative_at)

lemma vector_derivative_minus_at [simp]:
  "f differentiable at a
   \<Longrightarrow> vector_derivative (\<lambda>x. - f x) (at a) = - vector_derivative f (at a)"
  by (simp add: vector_derivative_at has_vector_derivative_minus vector_derivative_works [symmetric])

lemma vector_derivative_add_at [simp]:
  "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
   \<Longrightarrow> vector_derivative (\<lambda>x. f x + g x) (at a) = vector_derivative f (at a) + vector_derivative g (at a)"
  by (simp add: vector_derivative_at has_vector_derivative_add vector_derivative_works [symmetric])

lemma vector_derivative_diff_at [simp]:
  "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
   \<Longrightarrow> vector_derivative (\<lambda>x. f x - g x) (at a) = vector_derivative f (at a) - vector_derivative g (at a)"
  by (simp add: vector_derivative_at has_vector_derivative_diff vector_derivative_works [symmetric])

lemma vector_derivative_mult_at [simp]:
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
  shows  "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
   \<Longrightarrow> vector_derivative (\<lambda>x. f x * g x) (at a) = f a * vector_derivative g (at a) + vector_derivative f (at a) * g a"
  by (simp add: vector_derivative_at has_vector_derivative_mult vector_derivative_works [symmetric])

lemma vector_derivative_scaleR_at [simp]:
    "\<lbrakk>f differentiable at a; g differentiable at a\<rbrakk>
   \<Longrightarrow> vector_derivative (\<lambda>x. f x *\<^sub>R g x) (at a) = f a *\<^sub>R vector_derivative g (at a) + vector_derivative f (at a) *\<^sub>R g a"
apply (rule vector_derivative_at)
apply (rule has_vector_derivative_scaleR)
apply (auto simp: vector_derivative_works has_vector_derivative_def has_field_derivative_def mult_commute_abs)
done

lemma vector_derivative_within_cbox:
  assumes ab: "a < b" "x \<in> cbox a b"
  assumes f: "(f has_vector_derivative f') (at x within cbox a b)"
  shows "vector_derivative f (at x within cbox a b) = f'"
  by (intro vector_derivative_unique_within_closed_interval[OF ab _ f]
            vector_derivative_works[THEN iffD1] differentiableI_vector)
     fact

lemma vector_derivative_within_closed_interval:
  fixes f::"real \<Rightarrow> 'a::euclidean_space"
  assumes "a < b" and "x \<in> {a..b}"
  assumes "(f has_vector_derivative f') (at x within {a..b})"
  shows "vector_derivative f (at x within {a..b}) = f'"
  using assms vector_derivative_within_cbox
  by fastforce

lemma has_vector_derivative_within_subset:
  "(f has_vector_derivative f') (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f has_vector_derivative f') (at x within T)"
  by (auto simp: has_vector_derivative_def intro: has_derivative_subset)

lemma has_vector_derivative_at_within:
  "(f has_vector_derivative f') (at x) \<Longrightarrow> (f has_vector_derivative f') (at x within S)"
  unfolding has_vector_derivative_def
  by (rule has_derivative_at_withinI)

lemma has_vector_derivative_weaken:
  fixes x D and f g S T
  assumes f: "(f has_vector_derivative D) (at x within T)"
    and "x \<in> S" "S \<subseteq> T"
    and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
  shows "(g has_vector_derivative D) (at x within S)"
proof -
  have "(f has_vector_derivative D) (at x within S) \<longleftrightarrow> (g has_vector_derivative D) (at x within S)"
    unfolding has_vector_derivative_def has_derivative_iff_norm
    using assms by (intro conj_cong Lim_cong_within refl) auto
  then show ?thesis
    using has_vector_derivative_within_subset[OF f \<open>S \<subseteq> T\<close>] by simp
qed

lemma has_vector_derivative_transform_within:
  assumes "(f has_vector_derivative f') (at x within S)"
    and "0 < d"
    and "x \<in> S"
    and "\<And>x'. \<lbrakk>x'\<in>S; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'"
    shows "(g has_vector_derivative f') (at x within S)"
  using assms
  unfolding has_vector_derivative_def
  by (rule has_derivative_transform_within)

lemma has_vector_derivative_transform_within_open:
  assumes "(f has_vector_derivative f') (at x)"
    and "open S"
    and "x \<in> S"
    and "\<And>y. y\<in>S \<Longrightarrow> f y = g y"
  shows "(g has_vector_derivative f') (at x)"
  using assms
  unfolding has_vector_derivative_def
  by (rule has_derivative_transform_within_open)

lemma has_vector_derivative_transform:
  assumes "x \<in> S" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
  assumes f': "(f has_vector_derivative f') (at x within S)"
  shows "(g has_vector_derivative f') (at x within S)"
  using assms
  unfolding has_vector_derivative_def
  by (rule has_derivative_transform)

lemma vector_diff_chain_at:
  assumes "(f has_vector_derivative f') (at x)"
    and "(g has_vector_derivative g') (at (f x))"
  shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
  using assms has_vector_derivative_at_within has_vector_derivative_def vector_derivative_diff_chain_within by blast

lemma vector_diff_chain_within:
  assumes "(f has_vector_derivative f') (at x within s)"
    and "(g has_vector_derivative g') (at (f x) within f ` s)"
  shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
  using assms has_vector_derivative_def vector_derivative_diff_chain_within by blast

lemma vector_derivative_const_at [simp]: "vector_derivative (\<lambda>x. c) (at a) = 0"
  by (simp add: vector_derivative_at)

lemma vector_derivative_at_within_ivl:
  "(f has_vector_derivative f') (at x) \<Longrightarrow>
    a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> a<b \<Longrightarrow> vector_derivative f (at x within {a..b}) = f'"
  using has_vector_derivative_at_within vector_derivative_within_cbox by fastforce

lemma vector_derivative_chain_at:
  assumes "f differentiable at x" "(g differentiable at (f x))"
  shows "vector_derivative (g \<circ> f) (at x) =
         vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"
by (metis vector_diff_chain_at vector_derivative_at vector_derivative_works assms)

lemma field_vector_diff_chain_at:  (*thanks to Wenda Li*)
 assumes Df: "(f has_vector_derivative f') (at x)"
     and Dg: "(g has_field_derivative g') (at (f x))"
 shows "((g \<circ> f) has_vector_derivative (f' * g')) (at x)"
using diff_chain_at[OF Df[unfolded has_vector_derivative_def]
                       Dg [unfolded has_field_derivative_def]]
 by (auto simp: o_def mult.commute has_vector_derivative_def)

lemma vector_derivative_chain_within: 
  assumes "at x within S \<noteq> bot" "f differentiable (at x within S)" 
    "(g has_derivative g') (at (f x) within f ` S)" 
  shows "vector_derivative (g \<circ> f) (at x within S) =
        g' (vector_derivative f (at x within S)) "
  apply (rule vector_derivative_within [OF \<open>at x within S \<noteq> bot\<close>])
  apply (rule vector_derivative_diff_chain_within)
  using assms(2-3) vector_derivative_works
  by auto

subsection \<open>Field differentiability\<close>

definition\<^marker>\<open>tag important\<close> field_differentiable :: "['a \<Rightarrow> 'a::real_normed_field, 'a filter] \<Rightarrow> bool"
           (infixr "(field'_differentiable)" 50)
  where "f field_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"

lemma field_differentiable_imp_differentiable:
  "f field_differentiable F \<Longrightarrow> f differentiable F"
  unfolding field_differentiable_def differentiable_def 
  using has_field_derivative_imp_has_derivative by auto

lemma field_differentiable_imp_continuous_at:
    "f field_differentiable (at x within S) \<Longrightarrow> continuous (at x within S) f"
  by (metis DERIV_continuous field_differentiable_def)

lemma field_differentiable_within_subset:
    "\<lbrakk>f field_differentiable (at x within S); T \<subseteq> S\<rbrakk> \<Longrightarrow> f field_differentiable (at x within T)"
  by (metis DERIV_subset field_differentiable_def)

lemma field_differentiable_at_within:
    "\<lbrakk>f field_differentiable (at x)\<rbrakk>
     \<Longrightarrow> f field_differentiable (at x within S)"
  unfolding field_differentiable_def
  by (metis DERIV_subset top_greatest)

lemma field_differentiable_linear [simp,derivative_intros]: "((*) c) field_differentiable F"
  unfolding field_differentiable_def has_field_derivative_def mult_commute_abs
  by (force intro: has_derivative_mult_right)

lemma field_differentiable_const [simp,derivative_intros]: "(\<lambda>z. c) field_differentiable F"
  unfolding field_differentiable_def has_field_derivative_def
  using DERIV_const has_field_derivative_imp_has_derivative by blast

lemma field_differentiable_ident [simp,derivative_intros]: "(\<lambda>z. z) field_differentiable F"
  unfolding field_differentiable_def has_field_derivative_def
  using DERIV_ident has_field_derivative_def by blast

lemma field_differentiable_id [simp,derivative_intros]: "id field_differentiable F"
  unfolding id_def by (rule field_differentiable_ident)

lemma field_differentiable_minus [derivative_intros]:
  "f field_differentiable F \<Longrightarrow> (\<lambda>z. - (f z)) field_differentiable F"
  unfolding field_differentiable_def
  by (metis field_differentiable_minus)

lemma field_differentiable_add [derivative_intros]:
  assumes "f field_differentiable F" "g field_differentiable F"
    shows "(\<lambda>z. f z + g z) field_differentiable F"
  using assms unfolding field_differentiable_def
  by (metis field_differentiable_add)

lemma field_differentiable_add_const [simp,derivative_intros]:
     "(+) c field_differentiable F"
  by (simp add: field_differentiable_add)

lemma field_differentiable_sum [derivative_intros]:
  "(\<And>i. i \<in> I \<Longrightarrow> (f i) field_differentiable F) \<Longrightarrow> (\<lambda>z. \<Sum>i\<in>I. f i z) field_differentiable F"
  by (induct I rule: infinite_finite_induct)
     (auto intro: field_differentiable_add field_differentiable_const)

lemma field_differentiable_diff [derivative_intros]:
  assumes "f field_differentiable F" "g field_differentiable F"
    shows "(\<lambda>z. f z - g z) field_differentiable F"
  using assms unfolding field_differentiable_def
  by (metis field_differentiable_diff)

lemma field_differentiable_inverse [derivative_intros]:
  assumes "f field_differentiable (at a within S)" "f a \<noteq> 0"
  shows "(\<lambda>z. inverse (f z)) field_differentiable (at a within S)"
  using assms unfolding field_differentiable_def
  by (metis DERIV_inverse_fun)

lemma field_differentiable_mult [derivative_intros]:
  assumes "f field_differentiable (at a within S)"
          "g field_differentiable (at a within S)"
    shows "(\<lambda>z. f z * g z) field_differentiable (at a within S)"
  using assms unfolding field_differentiable_def
  by (metis DERIV_mult [of f _ a S g])

lemma field_differentiable_divide [derivative_intros]:
  assumes "f field_differentiable (at a within S)"
          "g field_differentiable (at a within S)"
          "g a \<noteq> 0"
    shows "(\<lambda>z. f z / g z) field_differentiable (at a within S)"
  using assms unfolding field_differentiable_def
  by (metis DERIV_divide [of f _ a S g])

lemma field_differentiable_power [derivative_intros]:
  assumes "f field_differentiable (at a within S)"
    shows "(\<lambda>z. f z ^ n) field_differentiable (at a within S)"
  using assms unfolding field_differentiable_def
  by (metis DERIV_power)

lemma field_differentiable_transform_within:
  "0 < d \<Longrightarrow>
        x \<in> S \<Longrightarrow>
        (\<And>x'. x' \<in> S \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
        f field_differentiable (at x within S)
        \<Longrightarrow> g field_differentiable (at x within S)"
  unfolding field_differentiable_def has_field_derivative_def
  by (blast intro: has_derivative_transform_within)

lemma field_differentiable_compose_within:
  assumes "f field_differentiable (at a within S)"
          "g field_differentiable (at (f a) within f`S)"
    shows "(g o f) field_differentiable (at a within S)"
  using assms unfolding field_differentiable_def
  by (metis DERIV_image_chain)

lemma field_differentiable_compose:
  "f field_differentiable at z \<Longrightarrow> g field_differentiable at (f z)
          \<Longrightarrow> (g o f) field_differentiable at z"
by (metis field_differentiable_at_within field_differentiable_compose_within)

lemma field_differentiable_within_open:
     "\<lbrakk>a \<in> S; open S\<rbrakk> \<Longrightarrow> f field_differentiable at a within S \<longleftrightarrow>
                          f field_differentiable at a"
  unfolding field_differentiable_def
  by (metis at_within_open)

lemma exp_scaleR_has_vector_derivative_right:
  "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative exp (t *\<^sub>R A) * A) (at t within T)"
  unfolding has_vector_derivative_def
proof (rule has_derivativeI)
  let ?F = "at t within (T \<inter> {t - 1 <..< t + 1})"
  have *: "at t within T = ?F"
    by (rule at_within_nhd[where S="{t - 1 <..< t + 1}"]) auto
  let ?e = "\<lambda>i x. (inverse (1 + real i) * inverse (fact i) * (x - t) ^ i) *\<^sub>R (A * A ^ i)"
  have "\<forall>\<^sub>F n in sequentially.
      \<forall>x\<in>T \<inter> {t - 1<..<t + 1}. norm (?e n x) \<le> norm (A ^ (n + 1) /\<^sub>R fact (n + 1))"
    apply (auto simp: algebra_split_simps intro!: eventuallyI)
    apply (rule mult_left_mono)
     apply (auto simp add: field_simps power_abs intro!: divide_right_mono power_le_one)
    done
  then have "uniform_limit (T \<inter> {t - 1<..<t + 1}) (\<lambda>n x. \<Sum>i<n. ?e i x) (\<lambda>x. \<Sum>i. ?e i x) sequentially"
    by (rule Weierstrass_m_test_ev) (intro summable_ignore_initial_segment summable_norm_exp)
  moreover
  have "\<forall>\<^sub>F x in sequentially. x > 0"
    by (metis eventually_gt_at_top)
  then have
    "\<forall>\<^sub>F n in sequentially. ((\<lambda>x. \<Sum>i<n. ?e i x) \<longlongrightarrow> A) ?F"
    by eventually_elim
      (auto intro!: tendsto_eq_intros
        simp: power_0_left if_distrib if_distribR
        cong: if_cong)
  ultimately
  have [tendsto_intros]: "((\<lambda>x. \<Sum>i. ?e i x) \<longlongrightarrow> A) ?F"
    by (auto intro!: swap_uniform_limit[where f="\<lambda>n x. \<Sum>i < n. ?e i x" and F = sequentially])
  have [tendsto_intros]: "((\<lambda>x. if x = t then 0 else 1) \<longlongrightarrow> 1) ?F"
    by (rule tendsto_eventually) (simp add: eventually_at_filter)
  have "((\<lambda>y. ((y - t) / abs (y - t)) *\<^sub>R ((\<Sum>n. ?e n y) - A)) \<longlongrightarrow> 0) (at t within T)"
    unfolding *
    by (rule tendsto_norm_zero_cancel) (auto intro!: tendsto_eq_intros)

  moreover have "\<forall>\<^sub>F x in at t within T. x \<noteq> t"
    by (simp add: eventually_at_filter)
  then have "\<forall>\<^sub>F x in at t within T. ((x - t) / \<bar>x - t\<bar>) *\<^sub>R ((\<Sum>n. ?e n x) - A) =
    (exp ((x - t) *\<^sub>R A) - 1 - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t)"
  proof eventually_elim
    case (elim x)
    have "(exp ((x - t) *\<^sub>R A) - 1 - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t) =
      ((\<Sum>n. (x - t) *\<^sub>R ?e n x) - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t)"
      unfolding exp_first_term
      by (simp add: ac_simps)
    also
    have "summable (\<lambda>n. ?e n x)"
    proof -
      from elim have "?e n x = (((x - t) *\<^sub>R A) ^ (n + 1)) /\<^sub>R fact (n + 1) /\<^sub>R (x - t)" for n
        by simp
      then show ?thesis
        by (auto simp only:
          intro!: summable_scaleR_right summable_ignore_initial_segment summable_exp_generic)
    qed
    then have "(\<Sum>n. (x - t) *\<^sub>R ?e n x) = (x - t) *\<^sub>R (\<Sum>n. ?e n x)"
      by (rule suminf_scaleR_right[symmetric])
    also have "(\<dots> - (x - t) *\<^sub>R A) /\<^sub>R norm (x - t) = (x - t) *\<^sub>R ((\<Sum>n. ?e n x) - A) /\<^sub>R norm (x - t)"
      by (simp add: algebra_simps)
    finally show ?case
      by simp (simp add: field_simps)
  qed

  ultimately have "((\<lambda>y. (exp ((y - t) *\<^sub>R A) - 1 - (y - t) *\<^sub>R A) /\<^sub>R norm (y - t)) \<longlongrightarrow> 0) (at t within T)"
    by (rule Lim_transform_eventually)
  from tendsto_mult_right_zero[OF this, where c="exp (t *\<^sub>R A)"]
  show "((\<lambda>y. (exp (y *\<^sub>R A) - exp (t *\<^sub>R A) - (y - t) *\<^sub>R (exp (t *\<^sub>R A) * A)) /\<^sub>R norm (y - t)) \<longlongrightarrow> 0)
      (at t within T)"
    by (rule Lim_transform_eventually)
      (auto simp: field_split_simps exp_add_commuting[symmetric])
qed (rule bounded_linear_scaleR_left)

lemma exp_times_scaleR_commute: "exp (t *\<^sub>R A) * A = A * exp (t *\<^sub>R A)"
  using exp_times_arg_commute[symmetric, of "t *\<^sub>R A"]
  by (auto simp: algebra_simps)

lemma exp_scaleR_has_vector_derivative_left: "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative A * exp (t *\<^sub>R A)) (at t)"
  using exp_scaleR_has_vector_derivative_right[of A t]
  by (simp add: exp_times_scaleR_commute)

lemma field_differentiable_series:
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
  assumes "convex S" "open S"
  assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
  assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
  assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
  shows  "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
proof -
  from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
    unfolding uniformly_convergent_on_def by blast
  from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
  have "\<exists>g. \<forall>x\<in>S. (\<lambda>n. f n x) sums g x \<and> (g has_field_derivative g' x) (at x within S)"
    by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
  then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
    "\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
  from g(2)[OF x] have g': "(g has_derivative (*) (g' x)) (at x)"
    by (simp add: has_field_derivative_def S)
  have "((\<lambda>x. \<Sum>n. f n x) has_derivative (*) (g' x)) (at x)"
    by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
       (insert g, auto simp: sums_iff)
  thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
    by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
qed

subsubsection\<^marker>\<open>tag unimportant\<close>\<open>Caratheodory characterization\<close>

lemma field_differentiable_caratheodory_at:
  "f field_differentiable (at z) \<longleftrightarrow>
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
  using CARAT_DERIV [of f]
  by (simp add: field_differentiable_def has_field_derivative_def)

lemma field_differentiable_caratheodory_within:
  "f field_differentiable (at z within s) \<longleftrightarrow>
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
  using DERIV_caratheodory_within [of f]
  by (simp add: field_differentiable_def has_field_derivative_def)


subsection \<open>Field derivative\<close>

definition\<^marker>\<open>tag important\<close> deriv :: "('a \<Rightarrow> 'a::real_normed_field) \<Rightarrow> 'a \<Rightarrow> 'a" where
  "deriv f x \<equiv> SOME D. DERIV f x :> D"

lemma DERIV_imp_deriv: "DERIV f x :> f' \<Longrightarrow> deriv f x = f'"
  unfolding deriv_def by (metis some_equality DERIV_unique)

lemma DERIV_deriv_iff_has_field_derivative:
  "DERIV f x :> deriv f x \<longleftrightarrow> (\<exists>f'. (f has_field_derivative f') (at x))"
  by (auto simp: has_field_derivative_def DERIV_imp_deriv)

lemma DERIV_deriv_iff_real_differentiable:
  fixes x :: real
  shows "DERIV f x :> deriv f x \<longleftrightarrow> f differentiable at x"
  unfolding differentiable_def by (metis DERIV_imp_deriv has_real_derivative_iff)

lemma deriv_cong_ev:
  assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
  shows   "deriv f x = deriv g y"
proof -
  have "(\<lambda>D. (f has_field_derivative D) (at x)) = (\<lambda>D. (g has_field_derivative D) (at y))"
    by (intro ext DERIV_cong_ev refl assms)
  thus ?thesis by (simp add: deriv_def assms)
qed

lemma higher_deriv_cong_ev:
  assumes "eventually (\<lambda>x. f x = g x) (nhds x)" "x = y"
  shows   "(deriv ^^ n) f x = (deriv ^^ n) g y"
proof -
  from assms(1) have "eventually (\<lambda>x. (deriv ^^ n) f x = (deriv ^^ n) g x) (nhds x)"
  proof (induction n arbitrary: f g)
    case (Suc n)
    from Suc.prems have "eventually (\<lambda>y. eventually (\<lambda>z. f z = g z) (nhds y)) (nhds x)"
      by (simp add: eventually_eventually)
    hence "eventually (\<lambda>x. deriv f x = deriv g x) (nhds x)"
      by eventually_elim (rule deriv_cong_ev, simp_all)
    thus ?case by (auto intro!: deriv_cong_ev Suc simp: funpow_Suc_right simp del: funpow.simps)
  qed auto
  from eventually_nhds_x_imp_x[OF this] assms(2) show ?thesis by simp
qed

lemma real_derivative_chain:
  fixes x :: real
  shows "f differentiable at x \<Longrightarrow> g differentiable at (f x)
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
  by (metis DERIV_deriv_iff_real_differentiable DERIV_chain DERIV_imp_deriv)
lemma field_derivative_eq_vector_derivative:
   "(deriv f x) = vector_derivative f (at x)"
by (simp add: mult.commute deriv_def vector_derivative_def has_vector_derivative_def has_field_derivative_def)

proposition field_differentiable_derivI:
    "f field_differentiable (at x) \<Longrightarrow> (f has_field_derivative deriv f x) (at x)"
by (simp add: field_differentiable_def DERIV_deriv_iff_has_field_derivative)

lemma vector_derivative_chain_at_general:
  assumes "f differentiable at x" "g field_differentiable at (f x)"
  shows "vector_derivative (g \<circ> f) (at x) = vector_derivative f (at x) * deriv g (f x)"
  apply (rule vector_derivative_at [OF field_vector_diff_chain_at])
  using assms vector_derivative_works by (auto simp: field_differentiable_derivI)

lemma DERIV_deriv_iff_field_differentiable:
  "DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
  unfolding field_differentiable_def by (metis DERIV_imp_deriv)

lemma deriv_chain:
  "f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
    \<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
  by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)

lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
  by (metis DERIV_imp_deriv DERIV_cmult_Id)

lemma deriv_uminus [simp]: "deriv (\<lambda>w. -w) = (\<lambda>z. -1)"
  using deriv_linear[of "-1"] by (simp del: deriv_linear)

lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
  by (metis DERIV_imp_deriv DERIV_ident)

lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"
  by (simp add: id_def)

lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
  by (metis DERIV_imp_deriv DERIV_const)

lemma deriv_add [simp]:
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
   \<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
  by (auto intro!: DERIV_imp_deriv derivative_intros)

lemma deriv_diff [simp]:
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
   \<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
  by (auto intro!: DERIV_imp_deriv derivative_intros)

lemma deriv_mult [simp]:
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
   \<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
  by (auto intro!: DERIV_imp_deriv derivative_eq_intros)

lemma deriv_cmult:
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
  by simp

lemma deriv_cmult_right:
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
  by simp

lemma deriv_inverse [simp]:
  "\<lbrakk>f field_differentiable at z; f z \<noteq> 0\<rbrakk>
   \<Longrightarrow> deriv (\<lambda>w. inverse (f w)) z = - deriv f z / f z ^ 2"
  unfolding DERIV_deriv_iff_field_differentiable[symmetric]
  by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: field_split_simps power2_eq_square)

lemma deriv_divide [simp]:
  "\<lbrakk>f field_differentiable at z; g field_differentiable at z; g z \<noteq> 0\<rbrakk>
   \<Longrightarrow> deriv (\<lambda>w. f w / g w) z = (deriv f z * g z - f z * deriv g z) / g z ^ 2"
  by (simp add: field_class.field_divide_inverse field_differentiable_inverse)
     (simp add: field_split_simps power2_eq_square)

lemma deriv_cdivide_right:
  "f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"
  by (simp add: field_class.field_divide_inverse)

lemma deriv_compose_linear:
  "f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
apply (rule DERIV_imp_deriv)
  unfolding DERIV_deriv_iff_field_differentiable [symmetric]
  by (metis (full_types) DERIV_chain2 DERIV_cmult_Id mult.commute)


lemma nonzero_deriv_nonconstant:
  assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
    shows "\<not> f constant_on S"
unfolding constant_on_def
by (metis \<open>df \<noteq> 0\<close> has_field_derivative_transform_within_open [OF df S] DERIV_const DERIV_unique)


subsection \<open>Relation between convexity and derivative\<close>

(* TODO: Generalise to real vector spaces? *)
proposition convex_on_imp_above_tangent:
  assumes convex: "convex_on A f" and connected: "connected A"
  assumes c: "c \<in> interior A" and x : "x \<in> A"
  assumes deriv: "(f has_field_derivative f') (at c within A)"
  shows   "f x - f c \<ge> f' * (x - c)"
proof (cases x c rule: linorder_cases)
  assume xc: "x > c"
  let ?A' = "interior A \<inter> {c<..}"
  from c have "c \<in> interior A \<inter> closure {c<..}" by auto
  also have "\<dots> \<subseteq> closure (interior A \<inter> {c<..})" by (intro open_Int_closure_subset) auto
  finally have "at c within ?A' \<noteq> bot" by (subst at_within_eq_bot_iff) auto
  moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) \<longlongrightarrow> f') (at c within ?A')"
    unfolding has_field_derivative_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le)
  moreover from eventually_at_right_real[OF xc]
    have "eventually (\<lambda>y. (f y - f c) / (y - c) \<le> (f x - f c) / (x - c)) (at_right c)"
  proof eventually_elim
    fix y assume y: "y \<in> {c<..<x}"
    with convex connected x c have "f y \<le> (f x - f c) / (x - c) * (y - c) + f c"
      using interior_subset[of A]
      by (intro convex_onD_Icc' convex_on_subset[OF convex] connected_contains_Icc) auto
    hence "f y - f c \<le> (f x - f c) / (x - c) * (y - c)" by simp
    thus "(f y - f c) / (y - c) \<le> (f x - f c) / (x - c)" using y xc by (simp add: field_split_simps)
  qed
  hence "eventually (\<lambda>y. (f y - f c) / (y - c) \<le> (f x - f c) / (x - c)) (at c within ?A')"
    by (blast intro: filter_leD at_le)
  ultimately have "f' \<le> (f x - f c) / (x - c)" by (simp add: tendsto_upperbound)
  thus ?thesis using xc by (simp add: field_simps)
next
  assume xc: "x < c"
  let ?A' = "interior A \<inter> {..<c}"
  from c have "c \<in> interior A \<inter> closure {..<c}" by auto
  also have "\<dots> \<subseteq> closure (interior A \<inter> {..<c})" by (intro open_Int_closure_subset) auto
  finally have "at c within ?A' \<noteq> bot" by (subst at_within_eq_bot_iff) auto
  moreover from deriv have "((\<lambda>y. (f y - f c) / (y - c)) \<longlongrightarrow> f') (at c within ?A')"
    unfolding has_field_derivative_iff using interior_subset[of A] by (blast intro: tendsto_mono at_le)
  moreover from eventually_at_left_real[OF xc]
    have "eventually (\<lambda>y. (f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)) (at_left c)"
  proof eventually_elim
    fix y assume y: "y \<in> {x<..<c}"
    with convex connected x c have "f y \<le> (f x - f c) / (c - x) * (c - y) + f c"
      using interior_subset[of A]
      by (intro convex_onD_Icc'' convex_on_subset[OF convex] connected_contains_Icc) auto
    hence "f y - f c \<le> (f x - f c) * ((c - y) / (c - x))" by simp
    also have "(c - y) / (c - x) = (y - c) / (x - c)" using y xc by (simp add: field_simps)
    finally show "(f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)" using y xc
      by (simp add: field_split_simps)
  qed
  hence "eventually (\<lambda>y. (f y - f c) / (y - c) \<ge> (f x - f c) / (x - c)) (at c within ?A')"
    by (blast intro: filter_leD at_le)
  ultimately have "f' \<ge> (f x - f c) / (x - c)" by (simp add: tendsto_lowerbound)
  thus ?thesis using xc by (simp add: field_simps)
qed simp_all


subsection \<open>Partial derivatives\<close>

lemma eventually_at_Pair_within_TimesI1:
  fixes x::"'a::metric_space"
  assumes "\<forall>\<^sub>F x' in at x within X. P x'"
  assumes "P x"
  shows "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P x'"
proof -
  from assms[unfolded eventually_at_topological]
  obtain S where S: "open S" "x \<in> S" "\<And>x'. x' \<in> X \<Longrightarrow> x' \<in> S \<Longrightarrow> P x'"
    by metis
  show "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P x'"
    unfolding eventually_at_topological
    by (auto intro!: exI[where x="S \<times> UNIV"] S open_Times)
qed

lemma eventually_at_Pair_within_TimesI2:
  fixes x::"'a::metric_space"
  assumes "\<forall>\<^sub>F y' in at y within Y. P y'" "P y"
  shows "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P y'"
proof -
  from assms[unfolded eventually_at_topological]
  obtain S where S: "open S" "y \<in> S" "\<And>y'. y' \<in> Y \<Longrightarrow> y' \<in> S \<Longrightarrow> P y'"
    by metis
  show "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. P y'"
    unfolding eventually_at_topological
    by (auto intro!: exI[where x="UNIV \<times> S"] S open_Times)
qed

proposition has_derivative_partialsI:
  fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector \<Rightarrow> 'c::real_normed_vector"
  assumes fx: "((\<lambda>x. f x y) has_derivative fx) (at x within X)"
  assumes fy: "\<And>x y. x \<in> X \<Longrightarrow> y \<in> Y \<Longrightarrow> ((\<lambda>y. f x y) has_derivative blinfun_apply (fy x y)) (at y within Y)"
  assumes fy_cont[unfolded continuous_within]: "continuous (at (x, y) within X \<times> Y) (\<lambda>(x, y). fy x y)"
  assumes "y \<in> Y" "convex Y"
  shows "((\<lambda>(x, y). f x y) has_derivative (\<lambda>(tx, ty). fx tx + fy x y ty)) (at (x, y) within X \<times> Y)"
proof (safe intro!: has_derivativeI tendstoI, goal_cases)
  case (2 e')
  interpret fx: bounded_linear "fx" using fx by (rule has_derivative_bounded_linear)
  define e where "e = e' / 9"
  have "e > 0" using \<open>e' > 0\<close> by (simp add: e_def)

  from fy_cont[THEN tendstoD, OF \<open>e > 0\<close>]
  have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. dist (fy x' y') (fy x y) < e"
    by (auto simp: split_beta')
  from this[unfolded eventually_at] obtain d' where
    "d' > 0"
    "\<And>x' y'. x' \<in> X \<Longrightarrow> y' \<in> Y \<Longrightarrow> (x', y') \<noteq> (x, y) \<Longrightarrow> dist (x', y') (x, y) < d' \<Longrightarrow>
      dist (fy x' y') (fy x y) < e"
    by auto
  then
  have d': "x' \<in> X \<Longrightarrow> y' \<in> Y \<Longrightarrow> dist (x', y') (x, y) < d' \<Longrightarrow> dist (fy x' y') (fy x y) < e"
    for x' y'
    using \<open>0 < e\<close>
    by (cases "(x', y') = (x, y)") auto
  define d where "d = d' / sqrt 2"
  have "d > 0" using \<open>0 < d'\<close> by (simp add: d_def)
  have d: "x' \<in> X \<Longrightarrow> y' \<in> Y \<Longrightarrow> dist x' x < d \<Longrightarrow> dist y' y < d \<Longrightarrow> dist (fy x' y') (fy x y) < e"
    for x' y'
    by (auto simp: dist_prod_def d_def intro!: d' real_sqrt_sum_squares_less)

  let ?S = "ball y d \<inter> Y"
  have "convex ?S"
    by (auto intro!: convex_Int \<open>convex Y\<close>)
  {
    fix x'::'a and y'::'b
    assume x': "x' \<in> X" and y': "y' \<in> Y"
    assume dx': "dist x' x < d" and dy': "dist y' y < d"
    have "norm (fy x' y' - fy x' y) \<le> dist (fy x' y') (fy x y) + dist (fy x' y) (fy x y)"
      by norm
    also have "dist (fy x' y') (fy x y) < e"
      by (rule d; fact)
    also have "dist (fy x' y) (fy x y) < e"
      by (auto intro!: d simp: dist_prod_def x' \<open>d > 0\<close> \<open>y \<in> Y\<close> dx')
    finally
    have "norm (fy x' y' - fy x' y) < e + e"
      by arith
    then have "onorm (blinfun_apply (fy x' y') - blinfun_apply (fy x' y)) < e + e"
      by (auto simp: norm_blinfun.rep_eq blinfun.diff_left[abs_def] fun_diff_def)
  } note onorm = this

  have ev_mem: "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. (x', y') \<in> X \<times> Y"
    using \<open>y \<in> Y\<close>
    by (auto simp: eventually_at intro!: zero_less_one)
  moreover
  have ev_dist: "\<forall>\<^sub>F xy in at (x, y) within X \<times> Y. dist xy (x, y) < d" if "d > 0" for d
    using eventually_at_ball[OF that]
    by (rule eventually_elim2) (auto simp: dist_commute intro!: eventually_True)
  note ev_dist[OF \<open>0 < d\<close>]
  ultimately
  have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y.
    norm (f x' y' - f x' y - (fy x' y) (y' - y)) \<le> norm (y' - y) * (e + e)"
  proof (eventually_elim, safe)
    fix x' y'
    assume "x' \<in> X" and y': "y' \<in> Y"
    assume dist: "dist (x', y') (x, y) < d"
    then have dx: "dist x' x < d" and dy: "dist y' y < d"
      unfolding dist_prod_def fst_conv snd_conv atomize_conj
      by (metis le_less_trans real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2)
    {
      fix t::real
      assume "t \<in> {0 .. 1}"
      then have "y + t *\<^sub>R (y' - y) \<in> closed_segment y y'"
        by (auto simp: closed_segment_def algebra_simps intro!: exI[where x=t])
      also
      have "\<dots> \<subseteq> ball y d \<inter> Y"
        using \<open>y \<in> Y\<close> \<open>0 < d\<close> dy y'
        by (intro \<open>convex ?S\<close>[unfolded convex_contains_segment, rule_format, of y y'])
          (auto simp: dist_commute)
      finally have "y + t *\<^sub>R (y' - y) \<in> ?S" .
    } note seg = this

    have "\<And>x. x \<in> ball y d \<inter> Y \<Longrightarrow> onorm (blinfun_apply (fy x' x) - blinfun_apply (fy x' y)) \<le> e + e"
      by (safe intro!: onorm less_imp_le \<open>x' \<in> X\<close> dx) (auto simp: dist_commute \<open>0 < d\<close> \<open>y \<in> Y\<close>)
    with seg has_derivative_subset[OF assms(2)[OF \<open>x' \<in> X\<close>]]
    show "norm (f x' y' - f x' y - (fy x' y) (y' - y)) \<le> norm (y' - y) * (e + e)"
      by (rule differentiable_bound_linearization[where S="?S"])
        (auto intro!: \<open>0 < d\<close> \<open>y \<in> Y\<close>)
  qed
  moreover
  let ?le = "\<lambda>x'. norm (f x' y - f x y - (fx) (x' - x)) \<le> norm (x' - x) * e"
  from fx[unfolded has_derivative_within, THEN conjunct2, THEN tendstoD, OF \<open>0 < e\<close>]
  have "\<forall>\<^sub>F x' in at x within X. ?le x'"
    by eventually_elim (simp, 
      simp add: dist_norm field_split_simps split: if_split_asm)
  then have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. ?le x'"
    by (rule eventually_at_Pair_within_TimesI1)
       (simp add: blinfun.bilinear_simps)
  moreover have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. norm ((x', y') - (x, y)) \<noteq> 0"
    unfolding norm_eq_zero right_minus_eq
    by (auto simp: eventually_at intro!: zero_less_one)
  moreover
  from fy_cont[THEN tendstoD, OF \<open>0 < e\<close>]
  have "\<forall>\<^sub>F x' in at x within X. norm (fy x' y - fy x y) < e"
    unfolding eventually_at
    using \<open>y \<in> Y\<close>
    by (auto simp: dist_prod_def dist_norm)
  then have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y. norm (fy x' y - fy x y) < e"
    by (rule eventually_at_Pair_within_TimesI1)
       (simp add: blinfun.bilinear_simps \<open>0 < e\<close>)
  ultimately
  have "\<forall>\<^sub>F (x', y') in at (x, y) within X \<times> Y.
            norm ((f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) /\<^sub>R
              norm ((x', y') - (x, y)))
            < e'"
    apply eventually_elim
  proof safe
    fix x' y'
    have "norm (f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) \<le>
        norm (f x' y' - f x' y - fy x' y (y' - y)) +
        norm (fy x y (y' - y) - fy x' y (y' - y)) +
        norm (f x' y - f x y - fx (x' - x))"
      by norm
    also
    assume nz: "norm ((x', y') - (x, y)) \<noteq> 0"
      and nfy: "norm (fy x' y - fy x y) < e"
    assume "norm (f x' y' - f x' y - blinfun_apply (fy x' y) (y' - y)) \<le> norm (y' - y) * (e + e)"
    also assume "norm (f x' y - f x y - (fx) (x' - x)) \<le> norm (x' - x) * e"
    also
    have "norm ((fy x y) (y' - y) - (fy x' y) (y' - y)) \<le> norm ((fy x y) - (fy x' y)) * norm (y' - y)"
      by (auto simp: blinfun.bilinear_simps[symmetric] intro!: norm_blinfun)
    also have "\<dots> \<le> (e + e) * norm (y' - y)"
      using \<open>e > 0\<close> nfy
      by (auto simp: norm_minus_commute intro!: mult_right_mono)
    also have "norm (x' - x) * e \<le> norm (x' - x) * (e + e)"
      using \<open>0 < e\<close> by simp
    also have "norm (y' - y) * (e + e) + (e + e) * norm (y' - y) + norm (x' - x) * (e + e) \<le>
        (norm (y' - y) + norm (x' - x)) * (4 * e)"
      using \<open>e > 0\<close>
      by (simp add: algebra_simps)
    also have "\<dots> \<le> 2 * norm ((x', y') - (x, y)) * (4 * e)"
      using \<open>0 < e\<close> real_sqrt_sum_squares_ge1[of "norm (x' - x)" "norm (y' - y)"]
        real_sqrt_sum_squares_ge2[of "norm (y' - y)" "norm (x' - x)"]
      by (auto intro!: mult_right_mono simp: norm_prod_def
        simp del: real_sqrt_sum_squares_ge1 real_sqrt_sum_squares_ge2)
    also have "\<dots> \<le> norm ((x', y') - (x, y)) * (8 * e)"
      by simp
    also have "\<dots> < norm ((x', y') - (x, y)) * e'"
      using \<open>0 < e'\<close> nz
      by (auto simp: e_def)
    finally show "norm ((f x' y' - f x y - (fx (x' - x) + fy x y (y' - y))) /\<^sub>R norm ((x', y') - (x, y))) < e'"
      by (simp add: dist_norm) (auto simp add: field_split_simps)
  qed
  then show ?case
    by eventually_elim (auto simp: dist_norm field_simps)
next
  from has_derivative_bounded_linear[OF fx]
  obtain fxb where "fx = blinfun_apply fxb"
    by (metis bounded_linear_Blinfun_apply)
  then show "bounded_linear (\<lambda>(tx, ty). fx tx + blinfun_apply (fy x y) ty)"
    by (auto intro!: bounded_linear_intros simp: split_beta')
qed


subsection\<^marker>\<open>tag unimportant\<close> \<open>Differentiable case distinction\<close>

lemma has_derivative_within_If_eq:
  "((\<lambda>x. if P x then f x else g x) has_derivative f') (at x within S) =
    (bounded_linear f' \<and>
     ((\<lambda>y.(if P y then (f y - ((if P x then f x else g x) + f' (y - x)))/\<^sub>R norm (y - x)
           else (g y - ((if P x then f x else g x) + f' (y - x)))/\<^sub>R norm (y - x)))
      \<longlongrightarrow> 0) (at x within S))"
  (is "_ = (_ \<and> (?if \<longlongrightarrow> 0) _)")
proof -
  have "(\<lambda>y. (1 / norm (y - x)) *\<^sub>R
           ((if P y then f y else g y) -
            ((if P x then f x else g x) + f' (y - x)))) = ?if"
    by (auto simp: inverse_eq_divide)
  thus ?thesis by (auto simp: has_derivative_within)
qed

lemma has_derivative_If_within_closures:
  assumes f': "x \<in> S \<union> (closure S \<inter> closure T) \<Longrightarrow>
    (f has_derivative f' x) (at x within S \<union> (closure S \<inter> closure T))"
  assumes g': "x \<in> T \<union> (closure S \<inter> closure T) \<Longrightarrow>
    (g has_derivative g' x) (at x within T \<union> (closure S \<inter> closure T))"
  assumes connect: "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f x = g x"
  assumes connect': "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f' x = g' x"
  assumes x_in: "x \<in> S \<union> T"
  shows "((\<lambda>x. if x \<in> S then f x else g x) has_derivative
      (if x \<in> S then f' x else g' x)) (at x within (S \<union> T))"
proof -
  from f' x_in interpret f': bounded_linear "if x \<in> S then f' x else (\<lambda>x. 0)"
    by (auto simp add: has_derivative_within)
  from g' interpret g': bounded_linear "if x \<in> T then g' x else (\<lambda>x. 0)"
    by (auto simp add: has_derivative_within)
  have bl: "bounded_linear (if x \<in> S then f' x else g' x)"
    using f'.scaleR f'.bounded f'.add g'.scaleR g'.bounded g'.add x_in
    by (unfold_locales; force)
  show ?thesis
    using f' g' closure_subset[of T] closure_subset[of S]
    unfolding has_derivative_within_If_eq
    by (intro conjI bl tendsto_If_within_closures x_in)
      (auto simp: has_derivative_within inverse_eq_divide connect connect' subsetD)
qed

lemma has_vector_derivative_If_within_closures:
  assumes x_in: "x \<in> S \<union> T"
  assumes "u = S \<union> T"
  assumes f': "x \<in> S \<union> (closure S \<inter> closure T) \<Longrightarrow>
    (f has_vector_derivative f' x) (at x within S \<union> (closure S \<inter> closure T))"
  assumes g': "x \<in> T \<union> (closure S \<inter> closure T) \<Longrightarrow>
    (g has_vector_derivative g' x) (at x within T \<union> (closure S \<inter> closure T))"
  assumes connect: "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f x = g x"
  assumes connect': "x \<in> closure S \<Longrightarrow> x \<in> closure T \<Longrightarrow> f' x = g' x"
  shows "((\<lambda>x. if x \<in> S then f x else g x) has_vector_derivative
    (if x \<in> S then f' x else g' x)) (at x within u)"
  unfolding has_vector_derivative_def assms
  using x_in
  apply (intro has_derivative_If_within_closures[where ?f' = "\<lambda>x a. a *\<^sub>R f' x" and ?g' = "\<lambda>x a. a *\<^sub>R g' x",
        THEN has_derivative_eq_rhs])
  subgoal by (rule f'[unfolded has_vector_derivative_def]; assumption)
  subgoal by (rule g'[unfolded has_vector_derivative_def]; assumption)
  by (auto simp: assms)

subsection\<^marker>\<open>tag important\<close>\<open>The Inverse Function Theorem\<close>

lemma linear_injective_contraction:
  assumes "linear f" "c < 1" and le: "\<And>x. norm (f x - x) \<le> c * norm x"
  shows "inj f"
  unfolding linear_injective_0[OF \<open>linear f\<close>]
proof safe
  fix x
  assume "f x = 0"
  with le [of x] have "norm x \<le> c * norm x"
    by simp
  then show "x = 0"
    using \<open>c < 1\<close> by (simp add: mult_le_cancel_right1)
qed

text\<open>From an online proof by J. Michael Boardman, Department of Mathematics, Johns Hopkins University\<close>
lemma inverse_function_theorem_scaled:
  fixes f::"'a::euclidean_space \<Rightarrow> 'a"
    and f'::"'a \<Rightarrow> ('a \<Rightarrow>\<^sub>L 'a)"
  assumes "open U"
    and derf: "\<And>x. x \<in> U \<Longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)"
    and contf: "continuous_on U f'"
    and "0 \<in> U" and [simp]: "f 0 = 0"
    and id: "f' 0 = id_blinfun"
  obtains U' V g g' where "open U'" "U' \<subseteq> U" "0 \<in> U'" "open V" "0 \<in> V" "homeomorphism U' V f g"
                "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
                "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f'(g y)))"
                "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f'(g y)))"
proof -
  obtain d1 where "cball 0 d1 \<subseteq> U" "d1 > 0"
    using \<open>open U\<close> \<open>0 \<in> U\<close> open_contains_cball by blast
  obtain d2 where d2: "\<And>x. \<lbrakk>x \<in> U; dist x 0 \<le> d2\<rbrakk> \<Longrightarrow> dist (f' x) (f' 0) < 1/2" "0 < d2"
    using continuous_onE [OF contf, of 0 "1/2"] by (metis \<open>0 \<in> U\<close> half_gt_zero_iff zero_less_one)
  obtain \<delta> where le: "\<And>x. norm x \<le> \<delta> \<Longrightarrow> dist (f' x) id_blinfun \<le> 1/2" and "0 < \<delta>"
    and subU: "cball 0 \<delta> \<subseteq> U"
  proof
    show "min d1 d2 > 0"
      by (simp add: \<open>0 < d1\<close> \<open>0 < d2\<close>)
    show "cball 0 (min d1 d2) \<subseteq> U"
      using \<open>cball 0 d1 \<subseteq> U\<close> by auto
    show "dist (f' x) id_blinfun \<le> 1/2" if "norm x \<le> min d1 d2" for x
      using \<open>cball 0 d1 \<subseteq> U\<close> d2 that id by fastforce
  qed
  let ?D = "cball 0 \<delta>"
  define V:: "'a set" where "V \<equiv> ball 0 (\<delta>/2)"
  have 4: "norm (f (x + h) - f x - h) \<le> 1/2 * norm h"
    if "x \<in> ?D" "x+h \<in> ?D" for x h
  proof -
    let ?w = "\<lambda>x. f x - x"
    have B: "\<And>x. x \<in> ?D \<Longrightarrow> onorm (blinfun_apply (f' x - id_blinfun)) \<le> 1/2"
      by (metis dist_norm le mem_cball_0 norm_blinfun.rep_eq)
    have "\<And>x. x \<in> ?D \<Longrightarrow> (?w has_derivative (blinfun_apply (f' x - id_blinfun))) (at x)"
      by (rule derivative_eq_intros derf subsetD [OF subU] | force simp: blinfun.diff_left)+
    then have Dw: "\<And>x. x \<in> ?D \<Longrightarrow> (?w has_derivative (blinfun_apply (f' x - id_blinfun))) (at x within ?D)"
      using has_derivative_at_withinI by blast
    have "norm (?w (x+h) - ?w x) \<le> (1/2) * norm h"
      using differentiable_bound [OF convex_cball Dw B] that by fastforce
    then show ?thesis
      by (auto simp: algebra_simps)
  qed
  have for_g: "\<exists>!x. norm x < \<delta> \<and> f x = y" if y: "norm y < \<delta>/2" for y
  proof -
    let ?u = "\<lambda>x. x + (y - f x)"
    have *: "norm (?u x) < \<delta>" if "x \<in> ?D" for x
    proof -
      have fxx: "norm (f x - x) \<le> \<delta>/2"
        using 4 [of 0 x] \<open>0 < \<delta>\<close> \<open>f 0 = 0\<close> that by auto
      have "norm (?u x) \<le> norm y + norm (f x - x)"
        by (metis add.commute add_diff_eq norm_minus_commute norm_triangle_ineq)
      also have "\<dots> < \<delta>/2 + \<delta>/2"
        using fxx y by auto
      finally show ?thesis
        by simp
    qed
    have "\<exists>!x \<in> ?D. ?u x = x"
    proof (rule banach_fix)
      show "cball 0 \<delta> \<noteq> {}"
        using \<open>0 < \<delta>\<close> by auto
      show "(\<lambda>x. x + (y - f x)) ` cball 0 \<delta> \<subseteq> cball 0 \<delta>"
        using * by force
      have "dist (x + (y - f x)) (xh + (y - f xh)) * 2 \<le> dist x xh"
        if "norm x \<le> \<delta>" and "norm xh \<le> \<delta>" for x xh
        using that 4 [of x "xh-x"] by (auto simp: dist_norm norm_minus_commute algebra_simps)
      then show "\<forall>x\<in>cball 0 \<delta>. \<forall>ya\<in>cball 0 \<delta>. dist (x + (y - f x)) (ya + (y - f ya)) \<le> (1/2) * dist x ya"
        by auto
    qed (auto simp: complete_eq_closed)
    then show ?thesis
      by (metis "*" add_cancel_right_right eq_iff_diff_eq_0 le_less mem_cball_0)
  qed
  define g where "g \<equiv> \<lambda>y. THE x. norm x < \<delta> \<and> f x = y"
  have g: "norm (g y) < \<delta> \<and> f (g y) = y" if "norm y < \<delta>/2" for y
    unfolding g_def using that theI' [OF for_g] by meson
  then have fg[simp]: "f (g y) = y" if "y \<in> V" for y
    using that by (auto simp: V_def)
  have 5: "norm (g y' - g y) \<le> 2 * norm (y' - y)" if "y \<in> V" "y' \<in> V" for y y'
  proof -
    have no: "norm (g y) \<le> \<delta>" "norm (g y') \<le> \<delta>" and [simp]: "f (g y) = y"
      using that g unfolding V_def by force+
    have "norm (g y' - g y) \<le> norm (g y' - g y - (y' - y)) + norm (y' - y)"
      by (simp add: add.commute norm_triangle_sub)
    also have "\<dots> \<le> (1/2) * norm (g y' - g y) + norm (y' - y)"
      using 4 [of "g y" "g y' - g y"] that no by (simp add: g norm_minus_commute V_def)
    finally show ?thesis
      by auto
  qed
  have contg: "continuous_on V g"
  proof
    fix y::'a and e::real
    assume "0 < e" and y: "y \<in> V"
    show "\<exists>d>0. \<forall>x'\<in>V. dist x' y < d \<longrightarrow> dist (g x') (g y) \<le> e"
    proof (intro exI conjI ballI impI)
      show "0 < e/2"
        by (simp add: \<open>0 < e\<close>)
    qed (use 5 y in \<open>force simp: dist_norm\<close>)
  qed
  show thesis
  proof
    define U' where "U' \<equiv> (f -` V) \<inter> ball 0 \<delta>"
    have contf: "continuous_on U f"
      using derf has_derivative_at_withinI by (fast intro: has_derivative_continuous_on)
    then have "continuous_on (ball 0 \<delta>) f"
      by (meson ball_subset_cball continuous_on_subset subU)
    then show "open U'"
      by (simp add: U'_def V_def Int_commute continuous_open_preimage)
    show "0 \<in> U'" "U' \<subseteq> U" "open V" "0 \<in> V"
      using \<open>0 < \<delta>\<close> subU by (auto simp: U'_def V_def)
    show hom: "homeomorphism U' V f g"
    proof
      show "continuous_on U' f"
        using \<open>U' \<subseteq> U\<close> contf continuous_on_subset by blast
      show "continuous_on V g"
        using contg by blast
      show "f ` U' \<subseteq> V"
        using U'_def by blast
      show "g ` V \<subseteq> U'"
        by (simp add: U'_def V_def g image_subset_iff)
      show "g (f x) = x" if "x \<in> U'" for x
        by (metis that fg Int_iff U'_def V_def for_g g mem_ball_0 vimage_eq)
      show "f (g y) = y" if "y \<in> V" for y
        using that by (simp add: g V_def)
    qed
    show bij: "bij (blinfun_apply (f'(g y)))" if "y \<in> V" for y
    proof -
      have inj: "inj (blinfun_apply (f' (g y)))"
      proof (rule linear_injective_contraction)
        show "linear (blinfun_apply (f' (g y)))"
          using blinfun.bounded_linear_right bounded_linear_def by blast
      next
        fix x
        have "norm (blinfun_apply (f' (g y)) x - x) = norm (blinfun_apply (f' (g y) - id_blinfun) x)"
          by (simp add: blinfun.diff_left)
        also have "\<dots> \<le> norm (f' (g y) - id_blinfun) * norm x"
          by (rule norm_blinfun)
        also have "\<dots> \<le> (1/2) * norm x"
        proof (rule mult_right_mono)
          show "norm (f' (g y) - id_blinfun) \<le> 1/2"
            using that g [of y] le by (auto simp: V_def dist_norm)
        qed auto
        finally show "norm (blinfun_apply (f' (g y)) x - x) \<le> (1/2) * norm x" .
      qed auto
      moreover
      have "surj (blinfun_apply (f' (g y)))"
        using blinfun.bounded_linear_right bounded_linear_def
        by (blast intro!: linear_inj_imp_surj [OF _ inj])
      ultimately show ?thesis
        using bijI by blast
    qed
    define g' where "g' \<equiv> \<lambda>y. inv (blinfun_apply (f'(g y)))"
    show "(g has_derivative g' y) (at y)" if "y \<in> V" for y
    proof -
      have gy: "g y \<in> U"
        using g subU that unfolding V_def by fastforce
      obtain e where e: "\<And>h. f (g y + h) = y + blinfun_apply (f' (g y)) h + e h"
        and e0: "(\<lambda>h. norm (e h) / norm h) \<midarrow>0\<rightarrow> 0"
        using iffD1 [OF has_derivative_iff_Ex derf [OF gy]] \<open>y \<in> V\<close> by auto
      have [simp]: "e 0 = 0"
        using e [of 0] that by simp
      let ?INV = "inv (blinfun_apply (f' (g y)))"
      have inj: "inj (blinfun_apply (f' (g y)))"
        using bij bij_betw_def that by blast
      have "(g has_derivative g' y) (at y within V)"
        unfolding has_derivative_at_within_iff_Ex [OF \<open>y \<in> V\<close> \<open>open V\<close>]
      proof
        show blinv: "bounded_linear (g' y)"
          unfolding g'_def using derf gy inj inj_linear_imp_inv_bounded_linear by blast
        define eg where "eg \<equiv> \<lambda>k. - ?INV (e (g (y+k) - g y))"
        have "g (y+k) = g y + g' y k + eg k" if "y + k \<in> V" for k
        proof -
          have "?INV k = ?INV (blinfun_apply (f' (g y)) (g (y+k) - g y) + e (g (y+k) - g y))"
            using e [of "g(y+k) - g y"] that by simp
          then have "g (y+k) = g y + ?INV k - ?INV (e (g (y+k) - g y))"
            using inj blinv by (simp add: linear_simps g'_def)
          then show ?thesis
            by (auto simp: eg_def g'_def)
        qed
        moreover have "(\<lambda>k. norm (eg k) / norm k) \<midarrow>0\<rightarrow> 0"
        proof (rule Lim_null_comparison)
          let ?g = "\<lambda>k. 2 * onorm ?INV * norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)"
          show "\<forall>\<^sub>F k in at 0. norm (norm (eg k) / norm k) \<le> ?g k"
            unfolding eventually_at_topological
          proof (intro exI conjI ballI impI)
            show "open ((+)(-y) ` V)"
              using \<open>open V\<close> open_translation by blast
            show "0 \<in> (+)(-y) ` V"
              by (simp add: that)
            show "norm (norm (eg k) / norm k) \<le> 2 * onorm (inv (blinfun_apply (f' (g y)))) * norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)"
              if "k \<in> (+)(-y) ` V" "k \<noteq> 0" for k
            proof -
              have "y+k \<in> V"
                using that by auto
              have "norm (norm (eg k) / norm k) \<le> onorm ?INV * norm (e (g (y+k) - g y)) / norm k"
                using blinv g'_def onorm by (force simp: eg_def divide_simps)
              also have "\<dots> = (norm (g (y+k) - g y) / norm k) * (onorm ?INV * (norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)))"
                by (simp add: divide_simps)
              also have "\<dots> \<le> 2 * (onorm ?INV * (norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)))"
                apply (rule mult_right_mono)
                using 5 [of y "y+k"] \<open>y \<in> V\<close> \<open>y + k \<in> V\<close>  onorm_pos_le [OF blinv]
                 apply (auto simp: divide_simps zero_le_mult_iff zero_le_divide_iff g'_def)
                done
              finally show "norm (norm (eg k) / norm k) \<le> 2 * onorm ?INV * norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)"
                by simp
            qed
          qed
          have 1: "(\<lambda>h. norm (e h) / norm h) \<midarrow>0\<rightarrow> (norm (e 0) / norm 0)"
            using e0 by auto
          have 2: "(\<lambda>k. g (y+k) - g y) \<midarrow>0\<rightarrow> 0"
            using contg \<open>open V\<close> \<open>y \<in> V\<close> LIM_offset_zero_iff LIM_zero_iff at_within_open continuous_on_def by fastforce
          from tendsto_compose [OF 1 2, simplified]
          have "(\<lambda>k. norm (e (g (y+k) - g y)) / norm (g (y+k) - g y)) \<midarrow>0\<rightarrow> 0" .
          from tendsto_mult_left [OF this] show "?g \<midarrow>0\<rightarrow> 0" by auto
        qed
        ultimately show "\<exists>e. (\<forall>k. y + k \<in> V \<longrightarrow> g (y+k) = g y + g' y k + e k) \<and> (\<lambda>k. norm (e k) / norm k) \<midarrow>0\<rightarrow> 0"
          by blast
      qed
      then show ?thesis
        by (metis \<open>open V\<close> at_within_open that)
    qed
    show "g' y = inv (blinfun_apply (f' (g y)))"
      if "y \<in> V" for y
      by (simp add: g'_def)
  qed
qed


text\<open>We need all this to justify the scaling and translations.\<close>
theorem inverse_function_theorem:
  fixes f::"'a::euclidean_space \<Rightarrow> 'a"
    and f'::"'a \<Rightarrow> ('a \<Rightarrow>\<^sub>L 'a)"
  assumes "open U"
    and derf: "\<And>x. x \<in> U \<Longrightarrow> (f has_derivative (blinfun_apply (f' x))) (at x)"
    and contf:  "continuous_on U f'"
    and "x0 \<in> U"
    and invf: "invf o\<^sub>L f' x0 = id_blinfun"
  obtains U' V g g' where "open U'" "U' \<subseteq> U" "x0 \<in> U'" "open V" "f x0 \<in> V" "homeomorphism U' V f g"
    "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
    "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (blinfun_apply (f'(g y)))"
    "\<And>y. y \<in> V \<Longrightarrow> bij (blinfun_apply (f'(g y)))"
proof -
  have apply1 [simp]: "\<And>i. blinfun_apply invf (blinfun_apply (f' x0) i) = i"
    by (metis blinfun_apply_blinfun_compose blinfun_apply_id_blinfun invf)
  have apply2 [simp]: "\<And>i. blinfun_apply (f' x0) (blinfun_apply invf i) = i"
    by (metis apply1 bij_inv_eq_iff blinfun_bij1 invf)
  have [simp]: "(range (blinfun_apply invf)) = UNIV"
    using apply1 surjI by blast
  let ?f = "invf \<circ> (\<lambda>x. (f \<circ> (+)x0)x - f x0)"
  let ?f' = "\<lambda>x. invf o\<^sub>L (f' (x + x0))"
  obtain U' V g g' where "open U'" and U': "U' \<subseteq> (+)(-x0) ` U" "0 \<in> U'"
    and "open V" "0 \<in> V" and hom: "homeomorphism U' V ?f g"
    and derg: "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
    and g': "\<And>y. y \<in> V \<Longrightarrow> g' y = inv (?f'(g y))"
    and bij: "\<And>y. y \<in> V \<Longrightarrow> bij (?f'(g y))"
  proof (rule inverse_function_theorem_scaled [of "(+)(-x0) ` U" ?f "?f'"])
    show ope: "open ((+) (- x0) ` U)"
      using \<open>open U\<close> open_translation by blast
    show "(?f has_derivative blinfun_apply (?f' x)) (at x)"
      if "x \<in> (+) (- x0) ` U" for x
      using that
      apply clarify
      apply (rule derf derivative_eq_intros | simp add: blinfun_compose.rep_eq)+
      done
    have YY: "(\<lambda>x. f' (x + x0)) \<midarrow>u-x0\<rightarrow> f' u"
      if "f' \<midarrow>u\<rightarrow> f' u" "u \<in> U" for u
      using that LIM_offset [where k = x0] by (auto simp: algebra_simps)
    then have "continuous_on ((+) (- x0) ` U) (\<lambda>x. f' (x + x0))"
      using contf \<open>open U\<close> Lim_at_imp_Lim_at_within
      by (fastforce simp: continuous_on_def at_within_open_NO_MATCH ope)
    then show "continuous_on ((+) (- x0) ` U) ?f'"
      by (intro continuous_intros) simp
  qed (auto simp: invf \<open>x0 \<in> U\<close>)
  show thesis
  proof
    let ?U' = "(+)x0 ` U'"
    let ?V = "((+)(f x0) \<circ> f' x0) ` V"
    let ?g = "(+)x0 \<circ> g \<circ> invf \<circ> (+)(- f x0)"
    let ?g' = "\<lambda>y. inv (blinfun_apply (f' (?g y)))"
    show oU': "open ?U'"
      by (simp add: \<open>open U'\<close> open_translation)
    show subU: "?U' \<subseteq> U"
      using ComplI \<open>U' \<subseteq> (+) (- x0) ` U\<close> by auto
    show "x0 \<in> ?U'"
      by (simp add: \<open>0 \<in> U'\<close>)
    show "open ?V"
      using blinfun_bij2 [OF invf]
      by (metis \<open>open V\<close> bij_is_surj blinfun.bounded_linear_right bounded_linear_def image_comp open_surjective_linear_image open_translation)
    show "f x0 \<in> ?V"
      using \<open>0 \<in> V\<close> image_iff by fastforce
    show "homeomorphism ?U' ?V f ?g"
    proof
      show "continuous_on ?U' f"
        by (meson subU continuous_on_eq_continuous_at derf has_derivative_continuous oU' subsetD)
      have "?f ` U' \<subseteq> V"
        using hom homeomorphism_image1 by blast
      then show "f ` ?U' \<subseteq> ?V"
        unfolding image_subset_iff
        by (clarsimp simp: image_def) (metis apply2 add.commute diff_add_cancel)
      show "?g ` ?V \<subseteq> ?U'"
        using hom invf by (auto simp: image_def homeomorphism_def)
      show "?g (f x) = x"
        if "x \<in> ?U'" for x
        using that hom homeomorphism_apply1 by fastforce
      have "continuous_on V g"
        using hom homeomorphism_def by blast
      then show "continuous_on ?V ?g"
        by (intro continuous_intros) (auto elim!: continuous_on_subset)
      have fg: "?f (g x) = x" if "x \<in> V" for x
        using hom homeomorphism_apply2 that by blast
      show "f (?g y) = y"
        if "y \<in> ?V" for y
        using that fg by (simp add: image_iff) (metis apply2 add.commute diff_add_cancel)
    qed
    show "(?g has_derivative ?g' y) (at y)" "bij (blinfun_apply (f' (?g y)))"
      if "y \<in> ?V" for y
    proof -
      have 1: "bij (blinfun_apply invf)"
        using blinfun_bij1 invf by blast
      then have 2: "bij (blinfun_apply (f' (x0 + g x)))" if "x \<in> V" for x
        by (metis add.commute bij bij_betw_comp_iff2 blinfun_compose.rep_eq that top_greatest)
      then show "bij (blinfun_apply (f' (?g y)))"
        using that by auto
      have "g' x \<circ> blinfun_apply invf = inv (blinfun_apply (f' (x0 + g x)))"
        if "x \<in> V" for x
        using that
        by (simp add: g' o_inv_distrib blinfun_compose.rep_eq 1 2 add.commute bij_is_inj flip: o_assoc)
      then show "(?g has_derivative ?g' y) (at y)"
        using that invf
        by clarsimp (rule derg derivative_eq_intros | simp flip: id_def)+
    qed
  qed auto
qed


subsection\<^marker>\<open>tag unimportant\<close> \<open>Piecewise differentiable functions\<close>

definition piecewise_differentiable_on
           (infixr "piecewise'_differentiable'_on" 50)
  where "f piecewise_differentiable_on i  \<equiv>
           continuous_on i f \<and>
           (\<exists>S. finite S \<and> (\<forall>x \<in> i - S. f differentiable (at x within i)))"

lemma piecewise_differentiable_on_imp_continuous_on:
    "f piecewise_differentiable_on S \<Longrightarrow> continuous_on S f"
by (simp add: piecewise_differentiable_on_def)

lemma piecewise_differentiable_on_subset:
    "f piecewise_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_differentiable_on T"
  using continuous_on_subset
  unfolding piecewise_differentiable_on_def
  apply safe
  apply (blast elim: continuous_on_subset)
  by (meson Diff_iff differentiable_within_subset subsetCE)

lemma differentiable_on_imp_piecewise_differentiable:
  fixes a:: "'a::{linorder_topology,real_normed_vector}"
  shows "f differentiable_on {a..b} \<Longrightarrow> f piecewise_differentiable_on {a..b}"
  apply (simp add: piecewise_differentiable_on_def differentiable_imp_continuous_on)
  apply (rule_tac x="{a,b}" in exI, simp add: differentiable_on_def)
  done

lemma differentiable_imp_piecewise_differentiable:
    "(\<And>x. x \<in> S \<Longrightarrow> f differentiable (at x within S))
         \<Longrightarrow> f piecewise_differentiable_on S"
by (auto simp: piecewise_differentiable_on_def differentiable_imp_continuous_on differentiable_on_def
         intro: differentiable_within_subset)

lemma piecewise_differentiable_const [iff]: "(\<lambda>x. z) piecewise_differentiable_on S"
  by (simp add: differentiable_imp_piecewise_differentiable)

lemma piecewise_differentiable_compose:
    "\<lbrakk>f piecewise_differentiable_on S; g piecewise_differentiable_on (f ` S);
      \<And>x. finite (S \<inter> f-`{x})\<rbrakk>
      \<Longrightarrow> (g \<circ> f) piecewise_differentiable_on S"
  apply (simp add: piecewise_differentiable_on_def, safe)
  apply (blast intro: continuous_on_compose2)
  apply (rename_tac A B)
  apply (rule_tac x="A \<union> (\<Union>x\<in>B. S \<inter> f-`{x})" in exI)
  apply (blast intro!: differentiable_chain_within)
  done

lemma piecewise_differentiable_affine:
  fixes m::real
  assumes "f piecewise_differentiable_on ((\<lambda>x. m *\<^sub>R x + c) ` S)"
  shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_differentiable_on S"
proof (cases "m = 0")
  case True
  then show ?thesis
    unfolding o_def
    by (force intro: differentiable_imp_piecewise_differentiable differentiable_const)
next
  case False
  show ?thesis
    apply (rule piecewise_differentiable_compose [OF differentiable_imp_piecewise_differentiable])
    apply (rule assms derivative_intros | simp add: False vimage_def real_vector_affinity_eq)+
    done
qed

lemma piecewise_differentiable_cases:
  fixes c::real
  assumes "f piecewise_differentiable_on {a..c}"
          "g piecewise_differentiable_on {c..b}"
           "a \<le> c" "c \<le> b" "f c = g c"
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_differentiable_on {a..b}"
proof -
  obtain S T where st: "finite S" "finite T"
               and fd: "\<And>x. x \<in> {a..c} - S \<Longrightarrow> f differentiable at x within {a..c}"
               and gd: "\<And>x. x \<in> {c..b} - T \<Longrightarrow> g differentiable at x within {c..b}"
    using assms
    by (auto simp: piecewise_differentiable_on_def)
  have finabc: "finite ({a,b,c} \<union> (S \<union> T))"
    by (metis \<open>finite S\<close> \<open>finite T\<close> finite_Un finite_insert finite.emptyI)
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
    using assms piecewise_differentiable_on_def by auto
  then have "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
                               OF closed_real_atLeastAtMost [of c b],
                               of f g "\<lambda>x. x\<le>c"]  assms
    by (force simp: ivl_disj_un_two_touch)
  moreover
  { fix x
    assume x: "x \<in> {a..b} - ({a,b,c} \<union> (S \<union> T))"
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b}" (is "?diff_fg")
    proof (cases x c rule: le_cases)
      case le show ?diff_fg
      proof (rule differentiable_transform_within [where d = "dist x c"])
        have "f differentiable at x"
          using x le fd [of x] at_within_interior [of x "{a..c}"] by simp
        then show "f differentiable at x within {a..b}"
          by (simp add: differentiable_at_withinI)
      qed (use x le st dist_real_def in auto)
    next
      case ge show ?diff_fg
      proof (rule differentiable_transform_within [where d = "dist x c"])
        have "g differentiable at x"
          using x ge gd [of x] at_within_interior [of x "{c..b}"] by simp
        then show "g differentiable at x within {a..b}"
          by (simp add: differentiable_at_withinI)
      qed (use x ge st dist_real_def in auto)
    qed
  }
  then have "\<exists>S. finite S \<and>
                 (\<forall>x\<in>{a..b} - S. (\<lambda>x. if x \<le> c then f x else g x) differentiable at x within {a..b})"
    by (meson finabc)
  ultimately show ?thesis
    by (simp add: piecewise_differentiable_on_def)
qed

lemma piecewise_differentiable_neg:
    "f piecewise_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_differentiable_on S"
  by (auto simp: piecewise_differentiable_on_def continuous_on_minus)

lemma piecewise_differentiable_add:
  assumes "f piecewise_differentiable_on i"
          "g piecewise_differentiable_on i"
    shows "(\<lambda>x. f x + g x) piecewise_differentiable_on i"
proof -
  obtain S T where st: "finite S" "finite T"
                       "\<forall>x\<in>i - S. f differentiable at x within i"
                       "\<forall>x\<in>i - T. g differentiable at x within i"
    using assms by (auto simp: piecewise_differentiable_on_def)
  then have "finite (S \<union> T) \<and> (\<forall>x\<in>i - (S \<union> T). (\<lambda>x. f x + g x) differentiable at x within i)"
    by auto
  moreover have "continuous_on i f" "continuous_on i g"
    using assms piecewise_differentiable_on_def by auto
  ultimately show ?thesis
    by (auto simp: piecewise_differentiable_on_def continuous_on_add)
qed

lemma piecewise_differentiable_diff:
    "\<lbrakk>f piecewise_differentiable_on S;  g piecewise_differentiable_on S\<rbrakk>
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_differentiable_on S"
  unfolding diff_conv_add_uminus
  by (metis piecewise_differentiable_add piecewise_differentiable_neg)


subsection\<open>The concept of continuously differentiable\<close>

text \<open>
John Harrison writes as follows:

``The usual assumption in complex analysis texts is that a path \<open>\<gamma>\<close> should be piecewise
continuously differentiable, which ensures that the path integral exists at least for any continuous
f, since all piecewise continuous functions are integrable. However, our notion of validity is
weaker, just piecewise differentiability\ldots{} [namely] continuity plus differentiability except on a
finite set\ldots{} [Our] underlying theory of integration is the Kurzweil-Henstock theory. In contrast to
the Riemann or Lebesgue theory (but in common with a simple notion based on antiderivatives), this
can integrate all derivatives.''

"Formalizing basic complex analysis." From Insight to Proof: Festschrift in Honour of Andrzej Trybulec.
Studies in Logic, Grammar and Rhetoric 10.23 (2007): 151-165.

And indeed he does not assume that his derivatives are continuous, but the penalty is unreasonably
difficult proofs concerning winding numbers. We need a self-contained and straightforward theorem
asserting that all derivatives can be integrated before we can adopt Harrison's choice.\<close>

definition\<^marker>\<open>tag important\<close> C1_differentiable_on :: "(real \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> real set \<Rightarrow> bool"
           (infix "C1'_differentiable'_on" 50)
  where
  "f C1_differentiable_on S \<longleftrightarrow>
   (\<exists>D. (\<forall>x \<in> S. (f has_vector_derivative (D x)) (at x)) \<and> continuous_on S D)"

lemma C1_differentiable_on_eq:
    "f C1_differentiable_on S \<longleftrightarrow>
     (\<forall>x \<in> S. f differentiable at x) \<and> continuous_on S (\<lambda>x. vector_derivative f (at x))"
     (is "?lhs = ?rhs")
proof
  assume ?lhs
  then show ?rhs
    unfolding C1_differentiable_on_def
    by (metis (no_types, lifting) continuous_on_eq  differentiableI_vector vector_derivative_at)
next
  assume ?rhs
  then show ?lhs
    using C1_differentiable_on_def vector_derivative_works by fastforce
qed

lemma C1_differentiable_on_subset:
  "f C1_differentiable_on T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> f C1_differentiable_on S"
  unfolding C1_differentiable_on_def  continuous_on_eq_continuous_within
  by (blast intro:  continuous_within_subset)

lemma C1_differentiable_compose:
  assumes fg: "f C1_differentiable_on S" "g C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
  shows "(g \<circ> f) C1_differentiable_on S"
proof -
  have "\<And>x. x \<in> S \<Longrightarrow> g \<circ> f differentiable at x"
    by (meson C1_differentiable_on_eq assms differentiable_chain_at imageI)
  moreover have "continuous_on S (\<lambda>x. vector_derivative (g \<circ> f) (at x))"
  proof (rule continuous_on_eq [of _ "\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x))"])
    show "continuous_on S (\<lambda>x. vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)))"
      using fg
      apply (clarsimp simp add: C1_differentiable_on_eq)
      apply (rule Limits.continuous_on_scaleR, assumption)
      by (metis (mono_tags, lifting) continuous_at_imp_continuous_on continuous_on_compose continuous_on_cong differentiable_imp_continuous_within o_def)
    show "\<And>x. x \<in> S \<Longrightarrow> vector_derivative f (at x) *\<^sub>R vector_derivative g (at (f x)) = vector_derivative (g \<circ> f) (at x)"
      by (metis (mono_tags, hide_lams) C1_differentiable_on_eq fg imageI vector_derivative_chain_at)
  qed
  ultimately show ?thesis
    by (simp add: C1_differentiable_on_eq)
qed

lemma C1_diff_imp_diff: "f C1_differentiable_on S \<Longrightarrow> f differentiable_on S"
  by (simp add: C1_differentiable_on_eq differentiable_at_imp_differentiable_on)

lemma C1_differentiable_on_ident [simp, derivative_intros]: "(\<lambda>x. x) C1_differentiable_on S"
  by (auto simp: C1_differentiable_on_eq)

lemma C1_differentiable_on_const [simp, derivative_intros]: "(\<lambda>z. a) C1_differentiable_on S"
  by (auto simp: C1_differentiable_on_eq)

lemma C1_differentiable_on_add [simp, derivative_intros]:
  "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x + g x) C1_differentiable_on S"
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)

lemma C1_differentiable_on_minus [simp, derivative_intros]:
  "f C1_differentiable_on S \<Longrightarrow> (\<lambda>x. - f x) C1_differentiable_on S"
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)

lemma C1_differentiable_on_diff [simp, derivative_intros]:
  "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x - g x) C1_differentiable_on S"
  unfolding C1_differentiable_on_eq  by (auto intro: continuous_intros)

lemma C1_differentiable_on_mult [simp, derivative_intros]:
  fixes f g :: "real \<Rightarrow> 'a :: real_normed_algebra"
  shows "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x * g x) C1_differentiable_on S"
  unfolding C1_differentiable_on_eq
  by (auto simp: continuous_on_add continuous_on_mult continuous_at_imp_continuous_on differentiable_imp_continuous_within)

lemma C1_differentiable_on_scaleR [simp, derivative_intros]:
  "f C1_differentiable_on S \<Longrightarrow> g C1_differentiable_on S \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) C1_differentiable_on S"
  unfolding C1_differentiable_on_eq
  by (rule continuous_intros | simp add: continuous_at_imp_continuous_on differentiable_imp_continuous_within)+


definition\<^marker>\<open>tag important\<close> piecewise_C1_differentiable_on
           (infixr "piecewise'_C1'_differentiable'_on" 50)
  where "f piecewise_C1_differentiable_on i  \<equiv>
           continuous_on i f \<and>
           (\<exists>S. finite S \<and> (f C1_differentiable_on (i - S)))"

lemma C1_differentiable_imp_piecewise:
    "f C1_differentiable_on S \<Longrightarrow> f piecewise_C1_differentiable_on S"
  by (auto simp: piecewise_C1_differentiable_on_def C1_differentiable_on_eq continuous_at_imp_continuous_on differentiable_imp_continuous_within)

lemma piecewise_C1_imp_differentiable:
    "f piecewise_C1_differentiable_on i \<Longrightarrow> f piecewise_differentiable_on i"
  by (auto simp: piecewise_C1_differentiable_on_def piecewise_differentiable_on_def
           C1_differentiable_on_def differentiable_def has_vector_derivative_def
           intro: has_derivative_at_withinI)

lemma piecewise_C1_differentiable_compose:
  assumes fg: "f piecewise_C1_differentiable_on S" "g piecewise_C1_differentiable_on (f ` S)" and fin: "\<And>x. finite (S \<inter> f-`{x})"
  shows "(g \<circ> f) piecewise_C1_differentiable_on S"
proof -
  have "continuous_on S (\<lambda>x. g (f x))"
    by (metis continuous_on_compose2 fg order_refl piecewise_C1_differentiable_on_def)
  moreover have "\<exists>T. finite T \<and> g \<circ> f C1_differentiable_on S - T"
  proof -
    obtain F where "finite F" and F: "f C1_differentiable_on S - F" and f: "f piecewise_C1_differentiable_on S"
      using fg by (auto simp: piecewise_C1_differentiable_on_def)
    obtain G where "finite G" and G: "g C1_differentiable_on f ` S - G" and g: "g piecewise_C1_differentiable_on f ` S"
      using fg by (auto simp: piecewise_C1_differentiable_on_def)
    show ?thesis
    proof (intro exI conjI)
      show "finite (F \<union> (\<Union>x\<in>G. S \<inter> f-`{x}))"
        using fin by (auto simp only: Int_Union \<open>finite F\<close> \<open>finite G\<close> finite_UN finite_imageI)
      show "g \<circ> f C1_differentiable_on S - (F \<union> (\<Union>x\<in>G. S \<inter> f -` {x}))"
        apply (rule C1_differentiable_compose)
          apply (blast intro: C1_differentiable_on_subset [OF F])
          apply (blast intro: C1_differentiable_on_subset [OF G])
        by (simp add:  C1_differentiable_on_subset G Diff_Int_distrib2 fin)
    qed
  qed
  ultimately show ?thesis
    by (simp add: piecewise_C1_differentiable_on_def)
qed

lemma piecewise_C1_differentiable_on_subset:
    "f piecewise_C1_differentiable_on S \<Longrightarrow> T \<le> S \<Longrightarrow> f piecewise_C1_differentiable_on T"
  by (auto simp: piecewise_C1_differentiable_on_def elim!: continuous_on_subset C1_differentiable_on_subset)

lemma C1_differentiable_imp_continuous_on:
  "f C1_differentiable_on S \<Longrightarrow> continuous_on S f"
  unfolding C1_differentiable_on_eq continuous_on_eq_continuous_within
  using differentiable_at_withinI differentiable_imp_continuous_within by blast

lemma C1_differentiable_on_empty [iff]: "f C1_differentiable_on {}"
  unfolding C1_differentiable_on_def
  by auto

lemma piecewise_C1_differentiable_affine:
  fixes m::real
  assumes "f piecewise_C1_differentiable_on ((\<lambda>x. m * x + c) ` S)"
  shows "(f \<circ> (\<lambda>x. m *\<^sub>R x + c)) piecewise_C1_differentiable_on S"
proof (cases "m = 0")
  case True
  then show ?thesis
    unfolding o_def by (auto simp: piecewise_C1_differentiable_on_def)
next
  case False
  have *: "\<And>x. finite (S \<inter> {y. m * y + c = x})"
    using False not_finite_existsD by fastforce
  show ?thesis
    apply (rule piecewise_C1_differentiable_compose [OF C1_differentiable_imp_piecewise])
    apply (rule * assms derivative_intros | simp add: False vimage_def)+
    done
qed

lemma piecewise_C1_differentiable_cases:
  fixes c::real
  assumes "f piecewise_C1_differentiable_on {a..c}"
          "g piecewise_C1_differentiable_on {c..b}"
           "a \<le> c" "c \<le> b" "f c = g c"
  shows "(\<lambda>x. if x \<le> c then f x else g x) piecewise_C1_differentiable_on {a..b}"
proof -
  obtain S T where st: "f C1_differentiable_on ({a..c} - S)"
                       "g C1_differentiable_on ({c..b} - T)"
                       "finite S" "finite T"
    using assms
    by (force simp: piecewise_C1_differentiable_on_def)
  then have f_diff: "f differentiable_on {a..<c} - S"
        and g_diff: "g differentiable_on {c<..b} - T"
    by (simp_all add: C1_differentiable_on_eq differentiable_at_withinI differentiable_on_def)
  have "continuous_on {a..c} f" "continuous_on {c..b} g"
    using assms piecewise_C1_differentiable_on_def by auto
  then have cab: "continuous_on {a..b} (\<lambda>x. if x \<le> c then f x else g x)"
    using continuous_on_cases [OF closed_real_atLeastAtMost [of a c],
                               OF closed_real_atLeastAtMost [of c b],
                               of f g "\<lambda>x. x\<le>c"]  assms
    by (force simp: ivl_disj_un_two_touch)
  { fix x
    assume x: "x \<in> {a..b} - insert c (S \<union> T)"
    have "(\<lambda>x. if x \<le> c then f x else g x) differentiable at x" (is "?diff_fg")
    proof (cases x c rule: le_cases)
      case le show ?diff_fg
        apply (rule differentiable_transform_within [where f=f and d = "dist x c"])
        using x dist_real_def le st by (auto simp: C1_differentiable_on_eq)
    next
      case ge show ?diff_fg
        apply (rule differentiable_transform_within [where f=g and d = "dist x c"])
        using dist_nz x dist_real_def ge st x by (auto simp: C1_differentiable_on_eq)
    qed
  }
  then have "(\<forall>x \<in> {a..b} - insert c (S \<union> T). (\<lambda>x. if x \<le> c then f x else g x) differentiable at x)"
    by auto
  moreover
  { assume fcon: "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative f (at x))"
       and gcon: "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative g (at x))"
    have "open ({a<..<c} - S)"  "open ({c<..<b} - T)"
      using st by (simp_all add: open_Diff finite_imp_closed)
    moreover have "continuous_on ({a<..<c} - S) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
    proof -
      have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative f (at x))            (at x)"
        if "a < x" "x < c" "x \<notin> S" for x
      proof -
        have f: "f differentiable at x"
          by (meson C1_differentiable_on_eq Diff_iff atLeastAtMost_iff less_eq_real_def st(1) that)
        show ?thesis
          using that
          apply (rule_tac f=f and d="dist x c" in has_vector_derivative_transform_within)
             apply (auto simp: dist_norm vector_derivative_works [symmetric] f)
          done
      qed
      then show ?thesis
        by (metis (no_types, lifting) continuous_on_eq [OF fcon] DiffE greaterThanLessThan_iff vector_derivative_at)
    qed
    moreover have "continuous_on ({c<..<b} - T) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
    proof -
      have "((\<lambda>x. if x \<le> c then f x else g x) has_vector_derivative vector_derivative g (at x))            (at x)"
        if "c < x" "x < b" "x \<notin> T" for x
      proof -
        have g: "g differentiable at x"
          by (metis C1_differentiable_on_eq DiffD1 DiffI atLeastAtMost_diff_ends greaterThanLessThan_iff st(2) that)
        show ?thesis
          using that
          apply (rule_tac f=g and d="dist x c" in has_vector_derivative_transform_within)
             apply (auto simp: dist_norm vector_derivative_works [symmetric] g)
          done
      qed
      then show ?thesis
        by (metis (no_types, lifting) continuous_on_eq [OF gcon] DiffE greaterThanLessThan_iff vector_derivative_at)
    qed
    ultimately have "continuous_on ({a<..<b} - insert c (S \<union> T))
        (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
      by (rule continuous_on_subset [OF continuous_on_open_Un], auto)
  } note * = this
  have "continuous_on ({a<..<b} - insert c (S \<union> T)) (\<lambda>x. vector_derivative (\<lambda>x. if x \<le> c then f x else g x) (at x))"
    using st
    by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset intro: *)
  ultimately have "\<exists>S. finite S \<and> ((\<lambda>x. if x \<le> c then f x else g x) C1_differentiable_on {a..b} - S)"
    apply (rule_tac x="{a,b,c} \<union> S \<union> T" in exI)
    using st  by (auto simp: C1_differentiable_on_eq elim!: continuous_on_subset)
  with cab show ?thesis
    by (simp add: piecewise_C1_differentiable_on_def)
qed

lemma piecewise_C1_differentiable_neg:
    "f piecewise_C1_differentiable_on S \<Longrightarrow> (\<lambda>x. -(f x)) piecewise_C1_differentiable_on S"
  unfolding piecewise_C1_differentiable_on_def
  by (auto intro!: continuous_on_minus C1_differentiable_on_minus)

lemma piecewise_C1_differentiable_add:
  assumes "f piecewise_C1_differentiable_on i"
          "g piecewise_C1_differentiable_on i"
    shows "(\<lambda>x. f x + g x) piecewise_C1_differentiable_on i"
proof -
  obtain S t where st: "finite S" "finite t"
                       "f C1_differentiable_on (i-S)"
                       "g C1_differentiable_on (i-t)"
    using assms by (auto simp: piecewise_C1_differentiable_on_def)
  then have "finite (S \<union> t) \<and> (\<lambda>x. f x + g x) C1_differentiable_on i - (S \<union> t)"
    by (auto intro: C1_differentiable_on_add elim!: C1_differentiable_on_subset)
  moreover have "continuous_on i f" "continuous_on i g"
    using assms piecewise_C1_differentiable_on_def by auto
  ultimately show ?thesis
    by (auto simp: piecewise_C1_differentiable_on_def continuous_on_add)
qed

lemma piecewise_C1_differentiable_diff:
    "\<lbrakk>f piecewise_C1_differentiable_on S;  g piecewise_C1_differentiable_on S\<rbrakk>
     \<Longrightarrow> (\<lambda>x. f x - g x) piecewise_C1_differentiable_on S"
  unfolding diff_conv_add_uminus
  by (metis piecewise_C1_differentiable_add piecewise_C1_differentiable_neg)

end