src/HOL/Multivariate_Analysis/Derivative.thy

generalize some lemmas about derivatives

(*  Title:                       HOL/Multivariate_Analysis/Derivative.thy
    Author:                      John Harrison
    Translation from HOL Light:  Robert Himmelmann, TU Muenchen
*)

header {* Multivariate calculus in Euclidean space. *}

theory Derivative
imports Brouwer_Fixpoint Operator_Norm
begin

(* Because I do not want to type this all the time *)
lemmas linear_linear = linear_conv_bounded_linear[THEN sym]

subsection {* Derivatives *}

text {* The definition is slightly tricky since we make it work over
  nets of a particular form. This lets us prove theorems generally and use 
  "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}

definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
(infixl "(has'_derivative)" 12) where
 "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
   (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"

lemma derivative_linear[dest]:"(f has_derivative f') net \<Longrightarrow> bounded_linear f'"
  unfolding has_derivative_def by auto

lemma DERIV_conv_has_derivative:"(DERIV f x :> f') = (f has_derivative op * f') (at (x::real))" (is "?l = ?r") proof 
  assume ?l note as = this[unfolded deriv_def LIM_def,rule_format]
  show ?r unfolding has_derivative_def Lim_at apply- apply(rule,rule mult.bounded_linear_right)
    apply safe apply(drule as,safe) apply(rule_tac x=s in exI) apply safe
    apply(erule_tac x="xa - x" in allE) unfolding dist_norm netlimit_at[of x] unfolding diff_0_right norm_scaleR
    by(auto simp add:field_simps) 
next assume ?r note this[unfolded has_derivative_def Lim_at] note as=conjunct2[OF this,rule_format]
  have *:"\<And>x xa f'. xa \<noteq> 0 \<Longrightarrow> \<bar>(f (xa + x) - f x) / xa - f'\<bar> = \<bar>(f (xa + x) - f x) - xa * f'\<bar> / \<bar>xa\<bar>" by(auto simp add:field_simps) 
  show ?l unfolding deriv_def LIM_def apply safe apply(drule as,safe)
    apply(rule_tac x=d in exI,safe) apply(erule_tac x="xa + x" in allE)
    unfolding dist_norm diff_0_right norm_scaleR
    unfolding dist_norm netlimit_at[of x] by(auto simp add:algebra_simps *) qed

lemma FDERIV_conv_has_derivative:"FDERIV f (x::'a::{real_normed_vector,perfect_space}) :> f' = (f has_derivative f') (at x)" (is "?l = ?r") proof 
  assume ?l note as = this[unfolded fderiv_def]
  show ?r unfolding has_derivative_def Lim_at apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
    fix e::real assume "e>0"
    guess d using as[THEN conjunct2,unfolded LIM_def,rule_format,OF`e>0`] ..
    thus "\<exists>d>0. \<forall>xa. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
      dist ((1 / norm (xa - netlimit (at x))) *\<^sub>R (f xa - (f (netlimit (at x)) + f' (xa - netlimit (at x))))) (0) < e"
      apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa - x" in allE)
      unfolding dist_norm netlimit_at[of x] by (auto simp add: diff_diff_eq) qed next
  assume ?r note as = this[unfolded has_derivative_def]
  show ?l unfolding fderiv_def LIM_def apply-apply(rule,rule as[THEN conjunct1]) proof(rule,rule)
    fix e::real assume "e>0"
    guess d using as[THEN conjunct2,unfolded Lim_at,rule_format,OF`e>0`] ..
    thus "\<exists>s>0. \<forall>xa. xa \<noteq> 0 \<and> dist xa 0 < s \<longrightarrow> dist (norm (f (x + xa) - f x - f' xa) / norm xa) 0 < e" apply-
      apply(rule_tac x=d in exI) apply(erule conjE,rule,assumption) apply rule apply(erule_tac x="xa + x" in allE)
      unfolding dist_norm netlimit_at[of x] by (auto simp add: diff_diff_eq add.commute) qed qed

subsection {* These are the only cases we'll care about, probably. *}

lemma has_derivative_within: "(f has_derivative f') (at x within s) \<longleftrightarrow>
         bounded_linear f' \<and> ((\<lambda>y. (1 / norm(y - x)) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x within s)"
  unfolding has_derivative_def and Lim by(auto simp add:netlimit_within)

lemma has_derivative_at: "(f has_derivative f') (at x) \<longleftrightarrow>
         bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) ---> 0) (at x)"
  apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within unfolding within_UNIV by auto

subsection {* More explicit epsilon-delta forms. *}

lemma 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
  unfolding diff_0_right 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)"
  apply(subst within_UNIV[THEN sym]) unfolding has_derivative_within' by auto

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

lemma has_derivative_within_open:
  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> ((f has_derivative f') (at a within s) \<longleftrightarrow> (f has_derivative f') (at a))"
  unfolding has_derivative_within has_derivative_at using Lim_within_open by auto

lemma bounded_linear_imp_linear: "bounded_linear f \<Longrightarrow> linear f" (* TODO: move elsewhere *)
proof -
  assume "bounded_linear f"
  then interpret f: bounded_linear f .
  show "linear f"
    by (simp add: f.add f.scaleR linear_def)
qed

lemma derivative_is_linear:
  "(f has_derivative f') net \<Longrightarrow> linear f'"
  by (rule derivative_linear [THEN bounded_linear_imp_linear])

subsection {* Combining theorems. *}

lemma (in bounded_linear) has_derivative: "(f has_derivative f) net"
  unfolding has_derivative_def apply(rule,rule bounded_linear_axioms)
  unfolding diff by(simp add: Lim_const)

lemma has_derivative_id: "((\<lambda>x. x) has_derivative (\<lambda>h. h)) net"
  apply(rule bounded_linear.has_derivative) using bounded_linear_ident[unfolded id_def] by simp

lemma has_derivative_const: "((\<lambda>x. c) has_derivative (\<lambda>h. 0)) net"
  unfolding has_derivative_def apply(rule,rule bounded_linear_zero) by(simp add: Lim_const)

lemma (in bounded_linear) cmul: shows "bounded_linear (\<lambda>x. (c::real) *\<^sub>R f x)"
proof -
  have "bounded_linear (\<lambda>x. c *\<^sub>R x)"
    by (rule scaleR.bounded_linear_right)
  moreover have "bounded_linear f" ..
  ultimately show ?thesis
    by (rule bounded_linear_compose)
qed

lemma has_derivative_cmul: assumes "(f has_derivative f') net" shows "((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net"
  unfolding has_derivative_def apply(rule,rule bounded_linear.cmul)
  using assms[unfolded has_derivative_def] using Lim_cmul[OF assms[unfolded has_derivative_def,THEN conjunct2]]
  unfolding scaleR_right_diff_distrib scaleR_right_distrib by auto 

lemma has_derivative_cmul_eq: assumes "c \<noteq> 0" 
  shows "(((\<lambda>x. c *\<^sub>R f(x)) has_derivative (\<lambda>h. c *\<^sub>R f'(h))) net \<longleftrightarrow> (f has_derivative f') net)"
  apply(rule) defer apply(rule has_derivative_cmul,assumption) 
  apply(drule has_derivative_cmul[where c="1/c"]) using assms by auto

lemma has_derivative_neg:
 "(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net"
  apply(drule has_derivative_cmul[where c="-1"]) by auto

lemma has_derivative_neg_eq: "((\<lambda>x. -(f x)) has_derivative (\<lambda>h. -(f' h))) net \<longleftrightarrow> (f has_derivative f') net"
  apply(rule, drule_tac[!] has_derivative_neg) by auto

lemma has_derivative_add: assumes "(f has_derivative f') net" "(g has_derivative g') net"
  shows "((\<lambda>x. f(x) + g(x)) has_derivative (\<lambda>h. f'(h) + g'(h))) net" proof-
  note as = assms[unfolded has_derivative_def]
  show ?thesis unfolding has_derivative_def apply(rule,rule bounded_linear_add)
    using Lim_add[OF as(1)[THEN conjunct2] as(2)[THEN conjunct2]] and as
    by (auto simp add:algebra_simps scaleR_right_diff_distrib scaleR_right_distrib) qed

lemma has_derivative_add_const:"(f has_derivative f') net \<Longrightarrow> ((\<lambda>x. f x + c) has_derivative f') net"
  apply(drule has_derivative_add) apply(rule has_derivative_const) by auto

lemma has_derivative_sub:
 "(f has_derivative f') net \<Longrightarrow> (g has_derivative g') net \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_derivative (\<lambda>h. f'(h) - g'(h))) net"
  apply(drule has_derivative_add) apply(drule has_derivative_neg,assumption) by(simp add:algebra_simps)

lemma has_derivative_setsum: assumes "finite s" "\<forall>a\<in>s. ((f a) has_derivative (f' a)) net"
  shows "((\<lambda>x. setsum (\<lambda>a. f a x) s) has_derivative (\<lambda>h. setsum (\<lambda>a. f' a h) s)) net"
  apply(induct_tac s rule:finite_subset_induct[where A=s]) apply(rule assms(1)) 
proof- fix x F assume as:"finite F" "x \<notin> F" "x\<in>s" "((\<lambda>x. \<Sum>a\<in>F. f a x) has_derivative (\<lambda>h. \<Sum>a\<in>F. f' a h)) net" 
  thus "((\<lambda>xa. \<Sum>a\<in>insert x F. f a xa) has_derivative (\<lambda>h. \<Sum>a\<in>insert x F. f' a h)) net"
    unfolding setsum_insert[OF as(1,2)] apply-apply(rule has_derivative_add) apply(rule assms(2)[rule_format]) by auto
qed(auto intro!: has_derivative_const)

lemma has_derivative_setsum_numseg:
  "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> ((f i) has_derivative (f' i)) net \<Longrightarrow>
  ((\<lambda>x. setsum (\<lambda>i. f i x) {m..n::nat}) has_derivative (\<lambda>h. setsum (\<lambda>i. f' i h) {m..n})) net"
  apply(rule has_derivative_setsum) by auto

subsection {* somewhat different results for derivative of scalar multiplier. *}

(** move **)
lemma linear_vmul_component:
  assumes lf: "linear f"
  shows "linear (\<lambda>x. f x $$ k *\<^sub>R v)"
  using lf
  by (auto simp add: linear_def algebra_simps)

lemma bounded_linear_euclidean_component: "bounded_linear (\<lambda>x. x $$ k)"
  unfolding euclidean_component_def
  by (rule inner.bounded_linear_right)

lemma has_derivative_vmul_component: fixes c::"'a::real_normed_vector \<Rightarrow> 'b::euclidean_space" and v::"'c::real_normed_vector"
  assumes "(c has_derivative c') net"
  shows "((\<lambda>x. c(x)$$k *\<^sub>R v) has_derivative (\<lambda>x. (c' x)$$k *\<^sub>R v)) net" proof-
  have *:"\<And>y. (c y $$ k *\<^sub>R v - (c (netlimit net) $$ k *\<^sub>R v + c' (y - netlimit net) $$ k *\<^sub>R v)) = 
        (c y $$ k - (c (netlimit net) $$ k + c' (y - netlimit net) $$ k)) *\<^sub>R v" 
    unfolding scaleR_left_diff_distrib scaleR_left_distrib by auto
  show ?thesis unfolding has_derivative_def and *
    apply (rule conjI)
    apply (rule bounded_linear_compose [OF scaleR.bounded_linear_left])
    apply (rule bounded_linear_compose [OF bounded_linear_euclidean_component])
    apply (rule derivative_linear [OF assms])
    apply(subst scaleR_zero_left[THEN sym, of v]) unfolding scaleR_scaleR apply(rule Lim_vmul)
    using assms[unfolded has_derivative_def] unfolding Lim o_def apply- apply(cases "trivial_limit net")
    apply(rule,assumption,rule disjI2,rule,rule) proof-
    have *:"\<And>x. x - 0 = (x::'a)" by auto 
    have **:"\<And>d x. d * (c x $$ k - (c (netlimit net) $$ k + c' (x - netlimit net) $$ k)) =
      (d *\<^sub>R (c x - (c (netlimit net) + c' (x - netlimit net) ))) $$k" by(auto simp add:field_simps)
    fix e assume "\<not> trivial_limit net" "0 < (e::real)"
    then have "eventually (\<lambda>x. dist ((1 / norm (x - netlimit net)) *\<^sub>R
      (c x - (c (netlimit net) + c' (x - netlimit net)))) 0 < e) net"
      using assms[unfolded has_derivative_def Lim] by auto
    thus "eventually (\<lambda>x. dist (1 / norm (x - netlimit net) *
      (c x $$ k - (c (netlimit net) $$ k + c' (x - netlimit net) $$ k))) 0 < e) net"
      proof (rule eventually_elim1)
      case goal1 thus ?case apply - unfolding dist_norm  apply(rule le_less_trans)
        prefer 2 apply assumption unfolding * **
        using component_le_norm[of "(1 / norm (x - netlimit net)) *\<^sub>R
          (c x - (c (netlimit net) + c' (x - netlimit net))) - 0" k] by auto
    qed
  qed
qed

lemma has_derivative_vmul_within: fixes c::"real \<Rightarrow> real"
  assumes "(c has_derivative c') (at x within s)"
  shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x within s)"
  using has_derivative_vmul_component[OF assms, of 0 v] by auto

lemma has_derivative_vmul_at: fixes c::"real \<Rightarrow> real"
  assumes "(c has_derivative c') (at x)"
  shows "((\<lambda>x. (c x) *\<^sub>R v) has_derivative (\<lambda>x. (c' x) *\<^sub>R v)) (at x)"
  using has_derivative_vmul_within[where s=UNIV] and assms by(auto simp add: within_UNIV)

lemma has_derivative_lift_dot:
  assumes "(f has_derivative f') net"
  shows "((\<lambda>x. inner v (f x)) has_derivative (\<lambda>t. inner v (f' t))) net" proof-
  show ?thesis using assms unfolding has_derivative_def apply- apply(erule conjE,rule)
    apply(rule bounded_linear_compose[of _ f']) apply(rule inner.bounded_linear_right,assumption)
    apply(drule Lim_inner[where a=v]) unfolding o_def
    by(auto simp add:inner.scaleR_right inner.add_right inner.diff_right) qed

lemmas has_derivative_intros = has_derivative_sub has_derivative_add has_derivative_cmul has_derivative_id has_derivative_const
   has_derivative_neg has_derivative_vmul_component has_derivative_vmul_at has_derivative_vmul_within has_derivative_cmul 
   bounded_linear.has_derivative has_derivative_lift_dot

subsection {* limit transformation for derivatives. *}

lemma has_derivative_transform_within:
  assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x within s)"
  shows "(g has_derivative f') (at x within s)"
  using assms(4) unfolding has_derivative_within apply- apply(erule conjE,rule,assumption)
  apply(rule Lim_transform_within[OF assms(1)]) defer apply assumption
  apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto

lemma has_derivative_transform_at:
  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_derivative f') (at x)"
  shows "(g has_derivative f') (at x)"
  apply(subst within_UNIV[THEN sym]) apply(rule has_derivative_transform_within[OF assms(1)])
  using assms(2-3) unfolding within_UNIV by auto

lemma has_derivative_transform_within_open:
  assumes "open s" "x \<in> s" "\<forall>y\<in>s. f y = g y" "(f has_derivative f') (at x)"
  shows "(g has_derivative f') (at x)"
  using assms(4) unfolding has_derivative_at apply- apply(erule conjE,rule,assumption)
  apply(rule Lim_transform_within_open[OF assms(1,2)]) defer apply assumption
  apply(rule,rule) apply(drule assms(3)[rule_format]) using assms(3)[rule_format, OF assms(2)] by auto

subsection {* differentiability. *}

no_notation Deriv.differentiable (infixl "differentiable" 60)

definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
  "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"

definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
  "f differentiable_on s \<equiv> (\<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_at_withinI: "f differentiable (at x) \<Longrightarrow> f differentiable (at x within s)"
  unfolding differentiable_def using has_derivative_at_within by blast

lemma differentiable_within_open: assumes "a \<in> s" "open s" shows 
  "f differentiable (at a within s) \<longleftrightarrow> (f differentiable (at a))"
  unfolding differentiable_def has_derivative_within_open[OF assms] by auto

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(auto simp add: differentiable_within_open)

lemma differentiable_transform_within:
  assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable (at x within s)"
  shows "g differentiable (at x within s)"
  using assms(4) unfolding differentiable_def by(auto intro!: has_derivative_transform_within[OF assms(1-3)])

lemma differentiable_transform_at:
  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "f differentiable at x"
  shows "g differentiable at x"
  using assms(3) unfolding differentiable_def using has_derivative_transform_at[OF assms(1-2)] by auto

subsection {* Frechet derivative and Jacobian matrix. *}

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

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

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

subsection {* Differentiability implies continuity. *}

lemma Lim_mul_norm_within: fixes f::"'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
  shows "(f ---> 0) (at a within s) \<Longrightarrow> ((\<lambda>x. norm(x - a) *\<^sub>R f(x)) ---> 0) (at a within s)"
  unfolding Lim_within apply(rule,rule) apply(erule_tac x=e in allE,erule impE,assumption,erule exE,erule conjE)
  apply(rule_tac x="min d 1" in exI) apply rule defer apply(rule,erule_tac x=x in ballE) unfolding dist_norm diff_0_right
  by(auto intro!: mult_strict_mono[of _ "1::real", unfolded mult_1_left])

lemma differentiable_imp_continuous_within: assumes "f differentiable (at x within s)" 
  shows "continuous (at x within s) f" proof-
  from assms guess f' unfolding differentiable_def has_derivative_within .. note f'=this
  then interpret bounded_linear f' by auto
  have *:"\<And>xa. x\<noteq>xa \<Longrightarrow> (f' \<circ> (\<lambda>y. y - x)) xa + norm (xa - x) *\<^sub>R ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) - ((f' \<circ> (\<lambda>y. y - x)) x + 0) = f xa - f x"
    using zero by auto
  have **:"continuous (at x within s) (f' \<circ> (\<lambda>y. y - x))"
    apply(rule continuous_within_compose) apply(rule continuous_intros)+
    by(rule linear_continuous_within[OF f'[THEN conjunct1]])
  show ?thesis unfolding continuous_within using f'[THEN conjunct2, THEN Lim_mul_norm_within]
    apply-apply(drule Lim_add) apply(rule **[unfolded continuous_within]) unfolding Lim_within and dist_norm
    apply(rule,rule) apply(erule_tac x=e in allE) apply(erule impE|assumption)+ apply(erule exE,rule_tac x=d in exI)
    by(auto simp add:zero * elim!:allE) qed

lemma differentiable_imp_continuous_at: "f differentiable at x \<Longrightarrow> continuous (at x) f"
 by(rule differentiable_imp_continuous_within[of _ x UNIV, unfolded within_UNIV])

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 has_derivative_within_subset:
 "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
  unfolding has_derivative_within using Lim_within_subset by blast

lemma differentiable_within_subset:
  "f differentiable (at x within t) \<Longrightarrow> s \<subseteq> t \<Longrightarrow> f differentiable (at x within s)"
  unfolding differentiable_def using has_derivative_within_subset 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

subsection {* Several results are easier using a "multiplied-out" variant.              *)
(* (I got this idea from Dieudonne's proof of the chain rule). *}

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))" (is "?lhs \<longleftrightarrow> ?rhs")
proof assume ?lhs thus ?rhs unfolding has_derivative_within apply-apply(erule conjE,rule,assumption)
    unfolding Lim_within apply(rule,erule_tac x=e in allE,rule,erule impE,assumption)
    apply(erule exE,rule_tac x=d in exI) apply(erule conjE,rule,assumption,rule,rule) proof-
    fix x y e d assume as:"0 < e" "0 < d" "norm (y - x) < d" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow>
      dist ((1 / norm (xa - x)) *\<^sub>R (f xa - (f x + f' (xa - x)))) 0 < e" "y \<in> s" "bounded_linear f'"
    then interpret bounded_linear f' by auto
    show "norm (f y - f x - f' (y - x)) \<le> e * norm (y - x)" proof(cases "y=x")
      case True thus ?thesis using `bounded_linear f'` by(auto simp add: zero) next
      case False hence "norm (f y - (f x + f' (y - x))) < e * norm (y - x)" using as(4)[rule_format, OF `y\<in>s`]
	unfolding dist_norm diff_0_right using as(3)
	using pos_divide_less_eq[OF False[unfolded dist_nz], unfolded dist_norm]
	by (auto simp add: linear_0 linear_sub)
      thus ?thesis by(auto simp add:algebra_simps) qed qed next
  assume ?rhs thus ?lhs unfolding has_derivative_within Lim_within apply-apply(erule conjE,rule,assumption)
    apply(rule,erule_tac x="e/2" in allE,rule,erule impE) defer apply(erule exE,rule_tac x=d in exI)
    apply(erule conjE,rule,assumption,rule,rule) unfolding dist_norm diff_0_right norm_scaleR
    apply(erule_tac x=xa in ballE,erule impE) proof-
    fix e d y assume "bounded_linear f'" "0 < e" "0 < d" "y \<in> s" "0 < norm (y - x) \<and> norm (y - x) < d"
        "norm (f y - f x - f' (y - x)) \<le> e / 2 * norm (y - x)"
    thus "\<bar>1 / norm (y - x)\<bar> * norm (f y - (f x + f' (y - x))) < e"
      apply(rule_tac le_less_trans[of _ "e/2"]) by(auto intro!:mult_imp_div_pos_le simp add:algebra_simps) qed auto qed

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] unfolding within_UNIV by auto

subsection {* The chain rule. *}

lemma diff_chain_within:
  assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at (f x) within (f ` s))"
  shows "((g o f) has_derivative (g' o f'))(at x within s)"
  unfolding has_derivative_within_alt apply(rule,rule bounded_linear_compose[unfolded o_def[THEN sym]])
  apply(rule assms(2)[unfolded has_derivative_def,THEN conjE],assumption)
  apply(rule assms(1)[unfolded has_derivative_def,THEN conjE],assumption) proof(rule,rule)
  note assms = assms[unfolded has_derivative_within_alt]
  fix e::real assume "0<e"
  guess B1 using bounded_linear.pos_bounded[OF assms(1)[THEN conjunct1, rule_format]] .. note B1 = this
  guess B2 using bounded_linear.pos_bounded[OF assms(2)[THEN conjunct1, rule_format]] .. note B2 = this
  have *:"e / 2 / B2 > 0" using `e>0` B2 apply-apply(rule divide_pos_pos) by auto
  guess d1 using assms(1)[THEN conjunct2, rule_format, OF *] .. note d1 = this
  have *:"e / 2 / (1 + B1) > 0" using `e>0` B1 apply-apply(rule divide_pos_pos) by auto
  guess de using assms(2)[THEN conjunct2, rule_format, OF *] .. note de = this
  guess d2 using assms(1)[THEN conjunct2, rule_format, OF zero_less_one] .. note d2 = this

  def d0 \<equiv> "(min d1 d2)/2" have d0:"d0>0" "d0 < d1" "d0 < d2" unfolding d0_def using d1 d2 by auto
  def d \<equiv> "(min d0 (de / (B1 + 1))) / 2" have "de * 2 / (B1 + 1) > de / (B1 + 1)" apply(rule divide_strict_right_mono) using B1 de by auto
  hence d:"d>0" "d < d1" "d < d2" "d < (de / (B1 + 1))" unfolding d_def using d0 d1 d2 de B1 by(auto intro!: divide_pos_pos simp add:min_less_iff_disj not_less)

  show "\<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" apply(rule_tac x=d in exI)
    proof(rule,rule `d>0`,rule,rule) 
    fix y assume as:"y \<in> s" "norm (y - x) < d"
    hence 1:"norm (f y - f x - f' (y - x)) \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x))" using d1 d2 d by auto

    have "norm (f y - f x) \<le> norm (f y - f x - f' (y - x)) + norm (f' (y - x))"
      using norm_triangle_sub[of "f y - f x" "f' (y - x)"] by(auto simp add:algebra_simps)
    also have "\<dots> \<le> norm (f y - f x - f' (y - x)) + B1 * norm (y - x)" apply(rule add_left_mono) using B1 by(auto simp add:algebra_simps)
    also have "\<dots> \<le> min (norm (y - x)) (e / 2 / B2 * norm (y - x)) + B1 * norm (y - x)" apply(rule add_right_mono) using d1 d2 d as by auto
    also have "\<dots> \<le> norm (y - x) + B1 * norm (y - x)" by auto
    also have "\<dots> = norm (y - x) * (1 + B1)" by(auto simp add:field_simps)
    finally have 3:"norm (f y - f x) \<le> norm (y - x) * (1 + B1)" by auto 

    hence "norm (f y - f x) \<le> d * (1 + B1)" apply- apply(rule order_trans,assumption,rule mult_right_mono) using as B1 by auto 
    also have "\<dots> < de" using d B1 by(auto simp add:field_simps) 
    finally have "norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 / (1 + B1) * norm (f y - f x)"
      apply-apply(rule de[THEN conjunct2,rule_format]) using `y\<in>s` using d as by auto 
    also have "\<dots> = (e / 2) * (1 / (1 + B1) * norm (f y - f x))" by auto 
    also have "\<dots> \<le> e / 2 * norm (y - x)" apply(rule mult_left_mono) using `e>0` and 3 using B1 and `e>0` by(auto simp add:divide_le_eq)
    finally have 4:"norm (g (f y) - g (f x) - g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
    
    interpret g': bounded_linear g' using assms(2) by auto
    interpret f': bounded_linear f' using assms(1) by auto
    have "norm (- g' (f' (y - x)) + g' (f y - f x)) = norm (g' (f y - f x - f' (y - x)))"
      by(auto simp add:algebra_simps f'.diff g'.diff g'.add)
    also have "\<dots> \<le> B2 * norm (f y - f x - f' (y - x))" using B2 by(auto simp add:algebra_simps)
    also have "\<dots> \<le> B2 * (e / 2 / B2 * norm (y - x))" apply(rule mult_left_mono) using as d1 d2 d B2 by auto 
    also have "\<dots> \<le> e / 2 * norm (y - x)" using B2 by auto
    finally have 5:"norm (- g' (f' (y - x)) + g' (f y - f x)) \<le> e / 2 * norm (y - x)" by auto
    
    have "norm (g (f y) - g (f x) - g' (f y - f x)) + norm (g (f y) - g (f x) - g' (f' (y - x)) - (g (f y) - g (f x) - g' (f y - f x))) \<le> e * norm (y - x)" using 5 4 by auto
    thus "norm ((g \<circ> f) y - (g \<circ> f) x - (g' \<circ> f') (y - x)) \<le> e * norm (y - x)" unfolding o_def apply- apply(rule order_trans, rule norm_triangle_sub) by assumption qed qed

lemma diff_chain_at:
  "(f has_derivative f') (at x) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow> ((g o f) has_derivative (g' o f')) (at x)"
  using diff_chain_within[of f f' x UNIV g g'] using has_derivative_within_subset[of g g' "f x" UNIV "range f"] unfolding within_UNIV by auto

subsection {* Composition rules stated just for differentiability. *}

lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
  unfolding differentiable_def using has_derivative_const by auto

lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
    unfolding differentiable_def using has_derivative_id by auto

lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
  unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto

lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
  unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto

lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
   \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
    unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
    apply(rule has_derivative_add) by auto

lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
  \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
  unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
    apply(rule has_derivative_sub) by auto 

lemma differentiable_setsum:
  assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
  shows "(\<lambda>x. setsum (\<lambda>a. f a x) s) differentiable net" proof-
  guess f' using bchoice[OF assms(2)[unfolded differentiable_def]] ..
  thus ?thesis unfolding differentiable_def apply- apply(rule,rule has_derivative_setsum[where f'=f'],rule assms(1)) by auto qed

lemma differentiable_setsum_numseg:
  shows "\<forall>i. m \<le> i \<and> i \<le> n \<longrightarrow> (f i) differentiable net \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) {m::nat..n}) differentiable net"
  apply(rule differentiable_setsum) using finite_atLeastAtMost[of n m] by auto

lemma differentiable_chain_at:
  "f differentiable (at x) \<Longrightarrow> g differentiable (at(f x)) \<Longrightarrow> (g o 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 o f) differentiable (at x within s)"
  unfolding differentiable_def by(meson diff_chain_within)

subsection {* Uniqueness of derivative.                                                 *)
(*                                                                           *)
(* The general result is a bit messy because we need approachability of the  *)
(* limit point from any direction. But OK for nontrivial intervals etc. *}
    
lemma frechet_derivative_unique_within: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes "(f has_derivative f') (at x within s)" "(f has_derivative f'') (at x within s)"
  "(\<forall>i<DIM('a). \<forall>e>0. \<exists>d. 0 < abs(d) \<and> abs(d) < e \<and> (x + d *\<^sub>R basis 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(rule,rule)
    fix e::real assume "0<e" guess d using assms(3)[rule_format,OF DIM_positive `e>0`] ..
    thus "\<exists>x'\<in>s. x' \<noteq> x \<and> dist x' x < e" apply(rule_tac x="x + d *\<^sub>R basis 0" in bexI)
      unfolding dist_norm by auto qed
  hence *:"netlimit (at x within s) = x" apply-apply(rule netlimit_within) unfolding trivial_limit_within by simp
  show ?thesis  apply(rule linear_eq_stdbasis) unfolding linear_conv_bounded_linear
    apply(rule as(1,2)[THEN conjunct1])+ proof(rule,rule,rule ccontr)
    fix i assume i:"i<DIM('a)" def e \<equiv> "norm (f' (basis i) - f'' (basis i))"
    assume "f' (basis i) \<noteq> f'' (basis i)" hence "e>0" unfolding e_def by auto
    guess d using Lim_sub[OF as(1,2)[THEN conjunct2], unfolded * Lim_within,rule_format,OF `e>0`] .. note d=this
    guess c using assms(3)[rule_format,OF i d[THEN conjunct1]] .. note c=this
    have *:"norm (- ((1 / \<bar>c\<bar>) *\<^sub>R f' (c *\<^sub>R basis i)) + (1 / \<bar>c\<bar>) *\<^sub>R f'' (c *\<^sub>R basis i)) = norm ((1 / abs c) *\<^sub>R (- (f' (c *\<^sub>R basis i)) + f'' (c *\<^sub>R basis i)))"
      unfolding scaleR_right_distrib by auto
    also have "\<dots> = norm ((1 / abs c) *\<^sub>R (c *\<^sub>R (- (f' (basis i)) + f'' (basis i))))"  
      unfolding f'.scaleR f''.scaleR unfolding scaleR_right_distrib scaleR_minus_right by auto
    also have "\<dots> = e" unfolding e_def using c[THEN conjunct1] using norm_minus_cancel[of "f' (basis i) - f'' (basis i)"] by (auto simp add: add.commute ab_diff_minus)
    finally show False using c using d[THEN conjunct2,rule_format,of "x + c *\<^sub>R basis 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 qed qed

lemma frechet_derivative_unique_at: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  shows "(f has_derivative f') (at x) \<Longrightarrow> (f has_derivative f'') (at x) \<Longrightarrow> f' = f''"
  apply(rule frechet_derivative_unique_within[of f f' x UNIV f'']) unfolding within_UNIV apply(assumption)+
  apply(rule,rule,rule,rule) apply(rule_tac x="e/2" in exI) by auto
 
lemma "isCont f x = continuous (at x) f" unfolding isCont_def LIM_def
  unfolding continuous_at Lim_at unfolding dist_nz by auto

lemma frechet_derivative_unique_within_closed_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" (is "x\<in>?I") and
  "(f has_derivative f' ) (at x within {a..b})" and
  "(f has_derivative f'') (at x within {a..b})"
  shows "f' = f''" apply(rule frechet_derivative_unique_within) apply(rule assms(3,4))+ proof(rule,rule,rule,rule)
  fix e::real and i assume "e>0" and i:"i<DIM('a)"
  thus "\<exists>d. 0 < \<bar>d\<bar> \<and> \<bar>d\<bar> < e \<and> x + d *\<^sub>R basis i \<in> {a..b}" proof(cases "x$$i=a$$i")
    case True thus ?thesis apply(rule_tac x="(min (b$$i - a$$i)  e) / 2" in exI)
      using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
      unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps)
  next note * = assms(2)[unfolded mem_interval,THEN spec[where x=i]]
    case False moreover have "a $$ i < x $$ i" using False * by auto
    moreover { have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> a$$i *2 + x$$i - a$$i" by auto
    also have "\<dots> = a$$i + x$$i" by auto also have "\<dots> \<le> 2 * x$$i" using * by auto 
    finally have "a $$ i * 2 + min (x $$ i - a $$ i) e \<le> x $$ i * 2" by auto }
    moreover have "min (x $$ i - a $$ i) e \<ge> 0" using * and `e>0` by auto
    hence "x $$ i * 2 \<le> b $$ i * 2 + min (x $$ i - a $$ i) e" using * by auto
    ultimately show ?thesis apply(rule_tac x="- (min (x$$i - a$$i) e) / 2" in exI)
      using assms(1)[THEN spec[where x=i]] and `e>0` and assms(2)
      unfolding mem_interval euclidean_simps basis_component using i by(auto simp add:field_simps) qed qed

lemma frechet_derivative_unique_within_open_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes "x \<in> {a<..<b}" "(f has_derivative f' ) (at x within {a<..<b})"
                         "(f has_derivative f'') (at x within {a<..<b})"
  shows "f' = f''"
proof -
  from assms(1) have *: "at x within {a<..<b} = at x"
    by (simp add: at_within_interior interior_open open_interval)
  from assms(2,3) [unfolded *] show "f' = f''"
    by (rule frechet_derivative_unique_at)
qed

lemma frechet_derivative_at: fixes f::"'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
  shows "(f has_derivative f') (at x) \<Longrightarrow> (f' = frechet_derivative f (at x))"
  apply(rule frechet_derivative_unique_at[of f],assumption)
  unfolding frechet_derivative_works[THEN sym] using differentiable_def by auto

lemma frechet_derivative_within_closed_interval: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector"
  assumes "\<forall>i<DIM('a). a$$i < b$$i" "x \<in> {a..b}" "(f has_derivative f') (at x within {a.. b})"
  shows "frechet_derivative f (at x within {a.. b}) = f'"
  apply(rule frechet_derivative_unique_within_closed_interval[where f=f]) 
  apply(rule assms(1,2))+ unfolding frechet_derivative_works[THEN sym]
  unfolding differentiable_def using assms(3) by auto 

subsection {* The traditional Rolle theorem in one dimension. *}

lemma linear_componentwise:
  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes lf: "linear f"
  shows "(f x) $$ j = (\<Sum>i<DIM('a). (x$$i) * (f (basis i)$$j))" (is "?lhs = ?rhs")
proof -
  have fA: "finite {..<DIM('a)}" by simp
  have "?rhs = (\<Sum>i<DIM('a). x$$i *\<^sub>R f (basis i))$$j"
    by (simp add: euclidean_simps)
  then show ?thesis
    unfolding linear_setsum_mul[OF lf fA, symmetric]
    unfolding euclidean_representation[symmetric] ..
qed

text {* We do not introduce @{text jacobian}, which is defined on matrices, instead we use
  the unfolding of it. *}

lemma jacobian_works:
  "(f::('a::euclidean_space) \<Rightarrow> ('b::euclidean_space)) differentiable net \<longleftrightarrow>
   (f has_derivative (\<lambda>h. \<chi>\<chi> i.
      \<Sum>j<DIM('a). frechet_derivative f net (basis j) $$ i * h $$ j)) net"
  (is "?differentiable \<longleftrightarrow> (f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net")
proof
  assume *: ?differentiable
  { fix h i
    have "?SUM h i = frechet_derivative f net h $$ i" using *
      by (auto intro!: setsum_cong
               simp: linear_componentwise[of _ h i] linear_frechet_derivative) }
  thus "(f has_derivative (\<lambda>h. \<chi>\<chi> i. ?SUM h i)) net"
    using * by (simp add: frechet_derivative_works)
qed (auto intro!: differentiableI)

lemma differential_zero_maxmin_component:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes k: "k < DIM('b)"
    and ball: "0 < e" "((\<forall>y \<in> ball x e. (f y)$$k \<le> (f x)$$k) \<or> (\<forall>y\<in>ball x e. (f x)$$k \<le> (f y)$$k))"
    and diff: "f differentiable (at x)"
  shows "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j) $$ k) = (0::'a)" (is "?D k = 0")
proof (rule ccontr)
  assume "?D k \<noteq> 0"
  then obtain j where j: "?D k $$ j \<noteq> 0" "j < DIM('a)"
    unfolding euclidean_lambda_beta euclidean_eq[of _ "0::'a"] by auto
  hence *: "\<bar>?D k $$ j\<bar> / 2 > 0" by auto
  note as = diff[unfolded jacobian_works has_derivative_at_alt]
  guess e' using as[THEN conjunct2, rule_format, OF *] .. note e' = this
  guess d using real_lbound_gt_zero[OF ball(1) e'[THEN conjunct1]] .. note d = this
  { fix c assume "abs c \<le> d"
    hence *:"norm (x + c *\<^sub>R basis j - x) < e'" using norm_basis[of j] d by auto
    let ?v = "(\<chi>\<chi> i. \<Sum>l<DIM('a). ?D i $$ l * (c *\<^sub>R basis j :: 'a) $$ l)"
    have if_dist: "\<And> P a b c. a * (if P then b else c) = (if P then a * b else a * c)" by auto
    have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le>
        norm (f (x + c *\<^sub>R basis j) - f x - ?v)" by (rule component_le_norm)
    also have "\<dots> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
      using e'[THEN conjunct2, rule_format, OF *] and norm_basis[of j] by fastsimp
    finally have "\<bar>(f (x + c *\<^sub>R basis j) - f x - ?v) $$ k\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>" by simp
    hence "\<bar>f (x + c *\<^sub>R basis j) $$ k - f x $$ k - c * ?D k $$ j\<bar> \<le> \<bar>?D k $$ j\<bar> / 2 * \<bar>c\<bar>"
      unfolding euclidean_simps euclidean_lambda_beta using j k
      by (simp add: if_dist setsum_cases field_simps) } note * = this
  have "x + d *\<^sub>R basis j \<in> ball x e" "x - d *\<^sub>R basis j \<in> ball x e"
    unfolding mem_ball dist_norm using norm_basis[of j] d by auto
  hence **:"((f (x - d *\<^sub>R basis j))$$k \<le> (f x)$$k \<and> (f (x + d *\<^sub>R basis j))$$k \<le> (f x)$$k) \<or>
         ((f (x - d *\<^sub>R basis j))$$k \<ge> (f x)$$k \<and> (f (x + d *\<^sub>R basis j))$$k \<ge> (f x)$$k)" using ball by auto
  have ***: "\<And>y y1 y2 d dx::real.
    (y1\<le>y\<and>y2\<le>y) \<or> (y\<le>y1\<and>y\<le>y2) \<Longrightarrow> d < abs dx \<Longrightarrow> abs(y1 - y - - dx) \<le> d \<Longrightarrow> (abs (y2 - y - dx) \<le> d) \<Longrightarrow> False" by arith
  show False apply(rule ***[OF **, where dx="d * ?D k $$ j" and d="\<bar>?D k $$ j\<bar> / 2 * \<bar>d\<bar>"])
    using *[of "-d"] and *[of d] and d[THEN conjunct1] and j unfolding mult_minus_left
    unfolding abs_mult diff_minus_eq_add scaleR.minus_left unfolding algebra_simps by (auto intro: mult_pos_pos)
qed

subsection {* In particular if we have a mapping into @{typ "real"}. *}

lemma differential_zero_maxmin:
  fixes f::"'a\<Colon>euclidean_space \<Rightarrow> real"
  assumes "x \<in> s" "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)"
proof -
  obtain e where e:"e>0" "ball x e \<subseteq> s" using `open s`[unfolded open_contains_ball] and `x \<in> s` by auto
  with differential_zero_maxmin_component[where 'b=real, of 0 e x f, simplified]
  have "(\<chi>\<chi> j. frechet_derivative f (at x) (basis j)) = (0::'a)"
    unfolding differentiable_def using mono deriv by auto
  with frechet_derivative_at[OF deriv, symmetric]
  have "\<forall>i<DIM('a). f' (basis i) = 0"
    by (simp add: euclidean_eq[of _ "0::'a"])
  with derivative_is_linear[OF deriv, THEN linear_componentwise, of _ 0]
  show ?thesis by (simp add: expand_fun_eq)
qed

lemma rolle: fixes f::"real\<Rightarrow>real"
  assumes "a < b" "f a = f b" "continuous_on {a..b} f"
  "\<forall>x\<in>{a<..<b}. (f has_derivative f'(x)) (at x)"
  shows "\<exists>x\<in>{a<..<b}. f' x = (\<lambda>v. 0)" proof-
  have "\<exists>x\<in>{a<..<b}. ((\<forall>y\<in>{a<..<b}. f x \<le> f y) \<or> (\<forall>y\<in>{a<..<b}. f y \<le> f x))" proof-
    have "(a + b) / 2 \<in> {a .. b}" using assms(1) by auto hence *:"{a .. b}\<noteq>{}" by auto
    guess d using continuous_attains_sup[OF compact_interval * assms(3)] .. note d=this
    guess c using continuous_attains_inf[OF compact_interval * assms(3)] .. note c=this
    show ?thesis proof(cases "d\<in>{a<..<b} \<or> c\<in>{a<..<b}")
      case True thus ?thesis apply(erule_tac disjE) apply(rule_tac x=d in bexI)
	apply(rule_tac[3] x=c in bexI) using d c by auto next def e \<equiv> "(a + b) /2"
      case False hence "f d = f c" using d c assms(2) by auto
      hence "\<And>x. x\<in>{a..b} \<Longrightarrow> f x = f d" using c d apply- apply(erule_tac x=x in ballE)+ by auto
      thus ?thesis apply(rule_tac x=e in bexI) unfolding e_def using assms(1) by auto qed qed
  then guess x .. note x=this
  hence "f' x = (\<lambda>v. 0)" apply(rule_tac differential_zero_maxmin[of x "{a<..<b}" f "f' x"])
    defer apply(rule open_interval)
    apply(rule assms(4)[unfolded has_derivative_at[THEN sym],THEN bspec[where x=x]],assumption)
    unfolding o_def apply(erule disjE,rule disjI2) by auto
  thus ?thesis apply(rule_tac x=x in bexI) unfolding o_def apply rule
    apply(drule_tac x=v in fun_cong) using x(1) by auto qed

subsection {* One-dimensional mean value theorem. *}

lemma mvt: fixes f::"real \<Rightarrow> real"
  assumes "a < b" "continuous_on {a .. b} f" "\<forall>x\<in>{a<..<b}. (f has_derivative (f' x)) (at x)"
  shows "\<exists>x\<in>{a<..<b}. (f b - f a = (f' x) (b - a))" proof-
  have "\<exists>x\<in>{a<..<b}. (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)"
    apply(rule rolle[OF assms(1), of "\<lambda>x. f x - (f b - f a) / (b - a) * x"]) defer
    apply(rule continuous_on_intros assms(2) continuous_on_cmul[where 'b=real, unfolded real_scaleR_def])+ proof
    fix x assume x:"x \<in> {a<..<b}"
    show "((\<lambda>x. f x - (f b - f a) / (b - a) * x) has_derivative (\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa)) (at x)"
      by(rule has_derivative_intros assms(3)[rule_format,OF x]
        has_derivative_cmul[where 'b=real, unfolded real_scaleR_def])+ 
  qed(insert assms(1), auto simp add:field_simps)
  then guess x .. thus ?thesis apply(rule_tac x=x in bexI) apply(drule fun_cong[of _ _ "b - a"]) by auto qed

lemma mvt_simple: fixes f::"real \<Rightarrow> real"
  assumes "a<b"  "\<forall>x\<in>{a..b}. (f has_derivative f' x) (at x within {a..b})"
  shows "\<exists>x\<in>{a<..<b}. f b - f a = f' x (b - a)"
  apply(rule mvt) apply(rule assms(1), rule differentiable_imp_continuous_on)
  unfolding differentiable_on_def differentiable_def defer proof 
  fix x assume x:"x \<in> {a<..<b}" show "(f has_derivative f' x) (at x)"
    unfolding has_derivative_within_open[OF x open_interval,THEN sym] 
    apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using x by auto qed(insert assms(2), auto)

lemma mvt_very_simple: fixes f::"real \<Rightarrow> real"
  assumes "a \<le> b" "\<forall>x\<in>{a..b}. (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 thus ?thesis apply(rule_tac x=a in bexI)
    using assms(2)[THEN bspec[where x=a]] unfolding has_derivative_def
    unfolding True using zero by auto next
  case False thus ?thesis using mvt_simple[OF _ assms(2)] using assms(1) by auto qed

subsection {* A nice generalization (see Havin's proof of 5.19 from Rudin's book). *}

lemma mvt_general: fixes f::"real\<Rightarrow>'a::euclidean_space"
  assumes "a<b" "continuous_on {a..b} f" "\<forall>x\<in>{a<..<b}. (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}. (op \<bullet> (f b - f a) \<circ> f) b - (op \<bullet> (f b - f a) \<circ> f) a = (f b - f a) \<bullet> f' x (b - a)"
    apply(rule mvt) apply(rule assms(1)) apply(rule continuous_on_inner continuous_on_intros assms(2))+ 
    unfolding o_def apply(rule,rule has_derivative_lift_dot) using assms(3) by auto
  then guess x .. note x=this
  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)^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 unfolding inner_simps by (auto simp add: inner_diff_left)
    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: real_mult_left_cancel) next
    case True thus ?thesis using assms(1) apply(rule_tac x="(a + b) /2" in bexI) by auto qed qed

subsection {* Still more general bound theorem. *}

lemma differentiable_bound: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes "convex s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x within s)" "\<forall>x\<in>s. 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)"
  have *:"\<And>u. u\<in>{0..1} \<Longrightarrow> x + u *\<^sub>R (y - x) \<in> s"
    using assms(1)[unfolded convex_alt,rule_format,OF x y] unfolding scaleR_left_diff_distrib scaleR_right_diff_distrib by(auto simp add:algebra_simps)
  hence 1:"continuous_on {0..1} (f \<circ> ?p)" apply- apply(rule continuous_on_intros continuous_on_vmul)+
    unfolding continuous_on_eq_continuous_within apply(rule,rule differentiable_imp_continuous_within)
    unfolding differentiable_def apply(rule_tac x="f' xa" in exI)
    apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) by auto
  have 2:"\<forall>u\<in>{0<..<1}. ((f \<circ> ?p) has_derivative f' (x + u *\<^sub>R (y - x)) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u)" proof rule case goal1
    let ?u = "x + u *\<^sub>R (y - x)"
    have "(f \<circ> ?p has_derivative (f' ?u) \<circ> (\<lambda>u. 0 + u *\<^sub>R (y - x))) (at u within {0<..<1})" 
      apply(rule diff_chain_within) apply(rule has_derivative_intros)+ 
      apply(rule has_derivative_within_subset) apply(rule assms(2)[rule_format]) using goal1 * by auto
    thus ?case unfolding has_derivative_within_open[OF goal1 open_interval] by auto qed
  guess u using mvt_general[OF zero_less_one 1 2] .. note u = this
  have **:"\<And>x y. x\<in>s \<Longrightarrow> norm (f' x y) \<le> B * norm y" proof- case goal1
    have "norm (f' x y) \<le> onorm (f' x) * norm y"
      using onorm(1)[OF derivative_is_linear[OF assms(2)[rule_format,OF goal1]]] by assumption
    also have "\<dots> \<le> B * norm y" apply(rule mult_right_mono)
      using assms(3)[rule_format,OF goal1] by(auto simp add:field_simps)
    finally show ?case by simp qed
  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 have "\<dots> \<le> norm (f' (x + u *\<^sub>R (y - x)) (y - x))" using u by auto
  also have "\<dots> \<le> B * norm(y - x)" apply(rule **) using * and u by auto
  finally show ?thesis by(auto simp add:norm_minus_commute) qed 

lemma differentiable_bound_real: fixes f::"real \<Rightarrow> real"
  assumes "convex s" "\<forall>x\<in>s. (f has_derivative f' x) (at x within s)" "\<forall>x\<in>s. 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)"
  using differentiable_bound[of s f f' B x y]
  unfolding Ball_def image_iff o_def using assms by auto

subsection {* In particular. *}

lemma has_derivative_zero_constant: fixes f::"real\<Rightarrow>real"
  assumes "convex s" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
  shows "\<exists>c. \<forall>x\<in>s. f x = c" proof(cases "s={}")
  case False then obtain x where "x\<in>s" by auto
  have "\<And>y. y\<in>s \<Longrightarrow> f x = f y" proof- case goal1
    thus ?case using differentiable_bound_real[OF assms(1-2), of 0 x y] and `x\<in>s`
    unfolding onorm_const by auto qed
  thus ?thesis apply(rule_tac x="f x" in exI) by auto qed auto

lemma has_derivative_zero_unique: fixes f::"real\<Rightarrow>real"
  assumes "convex s" "a \<in> s" "f a = c" "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)" "x\<in>s"
  shows "f x = c" using has_derivative_zero_constant[OF assms(1,4)] using assms(2-3,5) by auto

subsection {* Differentiability of inverse function (most basic form). *}

lemma has_derivative_inverse_basic: fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
  assumes "(f has_derivative f') (at (g y))" "bounded_linear g'" "g' \<circ> f' = id" "continuous (at y) g"
  "open t" "y \<in> t" "\<forall>z\<in>t. 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
  guess C using bounded_linear.pos_bounded[OF assms(2)] .. note C = this
(*  have fgid:"\<And>x. g' (f' x) = x" using assms(3) unfolding o_def id_def apply()*)
  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(rule,rule) case goal1
    have *:"e / C > 0" apply(rule divide_pos_pos) using `e>0` C by auto
    guess d0 using assms(1)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note d0=this
    guess d1 using assms(4)[unfolded continuous_at Lim_at,rule_format,OF d0[THEN conjunct1]] .. note d1=this
    guess d2 using assms(5)[unfolded open_dist,rule_format,OF assms(6)] .. note d2=this
    guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d=this
    thus ?case apply(rule_tac x=d in exI) apply rule defer proof(rule,rule)
      fix z assume as:"norm (z - y) < d" hence "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] 
	unfolding assms(7)[rule_format,OF `z\<in>t`] apply(subst norm_minus_cancel[THEN sym]) by auto
      also have "\<dots> \<le> norm(f (g z) - y - f' (g z - g y)) * C" by(rule C[THEN conjunct2,rule_format]) 
      also have "\<dots> \<le> (e / C) * norm (g z - g y) * C" apply(rule mult_right_mono)
	apply(rule d0[THEN conjunct2,rule_format,unfolded assms(7)[rule_format,OF `y\<in>t`]]) apply(cases "z=y") defer
	apply(rule d1[THEN conjunct2, unfolded dist_norm,rule_format]) using as d C d0 by auto
      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 auto qed
  have *:"(0::real) < 1 / 2" by auto guess d using lem1[rule_format,OF *] .. note d=this def B\<equiv>"C*2"
  have "B>0" unfolding B_def using C by auto
  have lem2:"\<forall>z. norm(z - y) < d \<longrightarrow> norm(g z - g y) \<le> B * norm(z - y)" proof(rule,rule) case goal1
    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)" apply(rule add_left_mono) using d and goal1 by auto
    also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)" apply(rule add_right_mono) using C by auto
    finally show ?case unfolding B_def by(auto simp add:field_simps) qed
  show ?thesis unfolding has_derivative_at_alt proof(rule,rule assms,rule,rule) case goal1
    hence *:"e/B >0" apply-apply(rule divide_pos_pos) using `B>0` by auto
    guess d' using lem1[rule_format,OF *] .. note d'=this
    guess k using real_lbound_gt_zero[OF d[THEN conjunct1] d'[THEN conjunct1]] .. note k=this
    show ?case apply(rule_tac x=k in exI,rule) defer proof(rule,rule) fix z assume as:"norm(z - y) < k"
      hence "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 `B>0`]
	using lem2[THEN spec[where x=z]] using k as using `e>0` by(auto simp add:field_simps)
      finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)" by simp qed(insert k, auto) qed qed

subsection {* Simply rewrite that based on the domain point x. *}

lemma has_derivative_inverse_basic_x: fixes f::"'b::euclidean_space \<Rightarrow> 'c::euclidean_space"
  assumes "(f has_derivative f') (at x)" "bounded_linear g'" "g' o f' = id"
  "continuous (at (f x)) g" "g(f x) = x" "open t" "f x \<in> t" "\<forall>y\<in>t. f(g y) = y"
  shows "(g has_derivative g') (at (f(x)))"
  apply(rule has_derivative_inverse_basic) using assms by auto

subsection {* This is the version in Dieudonne', assuming continuity of f and g. *}

lemma has_derivative_inverse_dieudonne: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes "open s" "open (f ` s)" "continuous_on s f" "continuous_on (f ` s) g" "\<forall>x\<in>s. g(f x) = x"
  (**) "x\<in>s" "(f has_derivative f') (at x)"  "bounded_linear g'" "g' o 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)] by auto

subsection {* Here's the simplest way of not assuming much about g. *}

lemma has_derivative_inverse: fixes f::"'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes "compact s" "x \<in> s" "f x \<in> interior(f ` s)" "continuous_on s f"
  "\<forall>y\<in>s. g(f y) = y" "(f has_derivative f') (at x)" "bounded_linear g'" "g' \<circ> f' = id"
  shows "(g has_derivative g') (at (f x))" proof-
  { fix y assume "y\<in>interior (f ` s)" 
    then obtain x where "x\<in>s" and *:"y = f x" unfolding image_iff using interior_subset by auto
    have "f (g y) = y" unfolding * and assms(5)[rule_format,OF `x\<in>s`] .. } note * = this
  show ?thesis apply(rule has_derivative_inverse_basic_x[OF assms(6-8)])
    apply(rule continuous_on_interior[OF _ assms(3)]) apply(rule continuous_on_inverse[OF assms(4,1)])
    apply(rule assms(2,5) assms(5)[rule_format] open_interior assms(3))+ by(rule, rule *, assumption)  qed

subsection {* Proving surjectivity via Brouwer fixpoint theorem. *}

lemma brouwer_surjective: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
  assumes "compact t" "convex t"  "t \<noteq> {}" "continuous_on t f"
  "\<forall>x\<in>s. \<forall>y\<in>t. x + (y - f y) \<in> t" "x\<in>s"
  shows "\<exists>y\<in>t. f y = x" proof-
  have *:"\<And>x y. f y = x \<longleftrightarrow> x + (y - f y) = y" by(auto simp add:algebra_simps)
  show ?thesis  unfolding * apply(rule brouwer[OF assms(1-3), of "\<lambda>y. x + (y - f y)"])
    apply(rule continuous_on_intros assms)+ using assms(4-6) by auto qed

lemma brouwer_surjective_cball: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
  assumes "0 < e" "continuous_on (cball a e) f"
  "\<forall>x\<in>s. \<forall>y\<in>cball a e. x + (y - f y) \<in> cball a e" "x\<in>s"
  shows "\<exists>y\<in>cball a e. f y = x" apply(rule brouwer_surjective) apply(rule compact_cball convex_cball)+
  unfolding cball_eq_empty using assms by auto 

text {* See Sussmann: "Multidifferential calculus", Theorem 2.1.1 *}

lemma sussmann_open_mapping: fixes f::"'a::euclidean_space \<Rightarrow> 'b::ordered_euclidean_space"
  assumes "open s" "continuous_on s f" "x \<in> s" 
  "(f has_derivative f') (at x)" "bounded_linear g'" "f' \<circ> g' = id"
  "t \<subseteq> s" "x \<in> interior t"
  shows "f x \<in> interior (f ` t)" proof- 
  interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
  interpret g':bounded_linear g' using assms by auto
  guess B using bounded_linear.pos_bounded[OF assms(5)] .. note B=this hence *:"1/(2*B)>0" by(auto intro!: divide_pos_pos)
  guess e0 using assms(4)[unfolded has_derivative_at_alt,THEN conjunct2,rule_format,OF *] .. note e0=this
  guess e1 using assms(8)[unfolded mem_interior_cball] .. note e1=this
  have *:"0<e0/B" "0<e1/B" apply(rule_tac[!] divide_pos_pos) using e0 e1 B by auto
  guess e using real_lbound_gt_zero[OF *] .. note e=this
  have "\<forall>z\<in>cball (f x) (e/2). \<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z"
    apply(rule,rule brouwer_surjective_cball[where s="cball (f x) (e/2)"])
    prefer 3 apply(rule,rule) proof- 
    show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))" unfolding g'.diff
      apply(rule continuous_on_compose[of _ _ f, unfolded o_def])
      apply(rule continuous_on_intros linear_continuous_on[OF assms(5)])+
      apply(rule continuous_on_subset[OF assms(2)]) apply(rule,unfold image_iff,erule bexE) proof-
      fix y z assume as:"y \<in>cball (f x) e"  "z = x + (g' y - g' (f x))"
      have "dist x z = norm (g' (f x) - g' y)" unfolding as(2) and dist_norm by auto
      also have "\<dots> \<le> norm (f x - y) * B" unfolding g'.diff[THEN sym] using B by auto
      also have "\<dots> \<le> e * B" using as(1)[unfolded mem_cball dist_norm] using B by auto
      also have "\<dots> \<le> e1" using e unfolding less_divide_eq using B by auto
      finally have "z\<in>cball x e1" unfolding mem_cball by force
      thus "z \<in> s" using e1 assms(7) by auto qed next
    fix y z assume as:"y \<in> cball (f x) (e / 2)" "z \<in> cball (f x) e"
    have "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by auto
    also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using as(2)[unfolded mem_cball dist_norm] and B unfolding norm_minus_commute by auto
    also have "\<dots> < e0" using e and B unfolding less_divide_eq by auto
    finally have *:"norm (x + g' (z - f x) - x) < e0" by auto
    have **:"f x + f' (x + g' (z - f x) - x) = z" using assms(6)[unfolded o_def id_def,THEN cong] by auto
    have "norm (f x - (y + (z - f (x + g' (z - f x))))) \<le> norm (f (x + g' (z - f x)) - z) + norm (f x - y)"
      using norm_triangle_ineq[of "f (x + g'(z - f x)) - z" "f x - y"] by(auto simp add:algebra_simps)
    also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + norm (f x - y)" using e0[THEN conjunct2,rule_format,OF *] unfolding algebra_simps ** by auto 
    also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2" using as(1)[unfolded mem_cball dist_norm] by auto
    also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2" using * and B by(auto simp add:field_simps)
    also have "\<dots> \<le> 1 / 2 * norm (z - f x) + e/2" by auto
    also have "\<dots> \<le> e/2 + e/2" apply(rule add_right_mono) using as(2)[unfolded mem_cball dist_norm] unfolding norm_minus_commute by auto
    finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e" unfolding mem_cball dist_norm by auto
  qed(insert e, auto) note lem = this
  show ?thesis unfolding mem_interior apply(rule_tac x="e/2" in exI)
    apply(rule,rule divide_pos_pos) prefer 3 proof 
    fix y assume "y \<in> ball (f x) (e/2)" hence *:"y\<in>cball (f x) (e/2)" by auto
    guess z using lem[rule_format,OF *] .. note z=this
    hence "norm (g' (z - f x)) \<le> norm (z - f x) * B" using B by(auto simp add:field_simps)
    also have "\<dots> \<le> e * B" apply(rule mult_right_mono) using z(1) unfolding mem_cball dist_norm norm_minus_commute using B by auto
    also have "\<dots> \<le> e1"  using e B unfolding less_divide_eq by auto
    finally have "x + g'(z - f x) \<in> t" apply- apply(rule e1[THEN conjunct2,unfolded subset_eq,rule_format]) 
      unfolding mem_cball dist_norm by auto
    thus "y \<in> f ` t" using z by auto qed(insert e, auto) qed

text {* Hence the following eccentric variant of the inverse function theorem.    *)
(* This has no continuity assumptions, but we do need the inverse function.  *)
(* We could put f' o g = I but this happens to fit with the minimal linear   *)
(* algebra theory I've set up so far. *}

(* move  before left_inverse_linear in Euclidean_Space*)

 lemma right_inverse_linear: fixes f::"'a::euclidean_space => 'a"
   assumes lf: "linear f" and gf: "f o g = id"
   shows "linear g"
 proof-
   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def expand_fun_eq)
     by metis 
   from linear_surjective_isomorphism[OF lf fi]
   obtain h:: "'a => 'a" where
     h: "linear h" "\<forall>x. h (f x) = x" "\<forall>x. f (h x) = x" by blast
   have "h = g" apply (rule ext) using gf h(2,3)
     apply (simp add: o_def id_def expand_fun_eq)
     by metis
   with h(1) show ?thesis by blast
 qed
 
lemma has_derivative_inverse_strong: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
  assumes "open s" "x \<in> s" "continuous_on s f"
  "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at x)" "f' o g' = id"
  shows "(g has_derivative g') (at (f x))" proof-
  have linf:"bounded_linear f'" using assms(5) unfolding has_derivative_def by auto
  hence ling:"bounded_linear g'" unfolding linear_conv_bounded_linear[THEN sym]
    apply- apply(rule right_inverse_linear) using assms(6) by auto 
  moreover have "g' \<circ> f' = id" using assms(6) linf ling unfolding linear_conv_bounded_linear[THEN sym]
    using linear_inverse_left by auto
  moreover have *:"\<forall>t\<subseteq>s. x\<in>interior t \<longrightarrow> f x \<in> interior (f ` t)" apply(rule,rule,rule,rule sussmann_open_mapping )
    apply(rule assms ling)+ by auto
  have "continuous (at (f x)) g" unfolding continuous_at Lim_at proof(rule,rule)
    fix e::real assume "e>0"
    hence "f x \<in> interior (f ` (ball x e \<inter> s))" using *[rule_format,of "ball x e \<inter> s"] `x\<in>s`
      by(auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
    then guess d unfolding mem_interior .. note d=this
    show "\<exists>d>0. \<forall>y. 0 < dist y (f x) \<and> dist y (f x) < d \<longrightarrow> dist (g y) (g (f x)) < e"
      apply(rule_tac x=d in exI) apply(rule,rule d[THEN conjunct1]) proof(rule,rule) case goal1
      hence "g y \<in> g ` f ` (ball x e \<inter> s)" using d[THEN conjunct2,unfolded subset_eq,THEN bspec[where x=y]]
	by(auto simp add:dist_commute)
      hence "g y \<in> ball x e \<inter> s" using assms(4) by auto
      thus "dist (g y) (g (f x)) < e" using assms(4)[rule_format,OF `x\<in>s`] by(auto simp add:dist_commute) qed qed
  moreover have "f x \<in> interior (f ` s)" apply(rule sussmann_open_mapping)
    apply(rule assms ling)+ using interior_open[OF assms(1)] and `x\<in>s` by auto
  moreover have "\<And>y. y \<in> interior (f ` s) \<Longrightarrow> f (g y) = y" proof- case goal1
    hence "y\<in>f ` s" using interior_subset by auto then guess z unfolding image_iff ..
    thus ?case using assms(4) by auto qed
  ultimately show ?thesis apply- apply(rule has_derivative_inverse_basic_x[OF assms(5)]) using assms by auto qed 

subsection {* A rewrite based on the other domain. *}

lemma has_derivative_inverse_strong_x: fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'a"
  assumes "open s" "g y \<in> s" "continuous_on s f"
  "\<forall>x\<in>s. g(f x) = x" "(f has_derivative f') (at (g y))" "f' o g' = id" "f(g y) = y"
  shows "(g has_derivative g') (at y)"
  using has_derivative_inverse_strong[OF assms(1-6)] unfolding assms(7) by simp

subsection {* On a region. *}

lemma has_derivative_inverse_on: fixes f::"'n::ordered_euclidean_space \<Rightarrow> 'n"
  assumes "open s" "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)" "\<forall>x\<in>s. g(f x) = x" "f'(x) o g'(x) = id" "x\<in>s"
  shows "(g has_derivative g'(x)) (at (f x))"
  apply(rule has_derivative_inverse_strong[where g'="g' x" and f=f]) apply(rule assms)+
  unfolding continuous_on_eq_continuous_at[OF assms(1)]
  apply(rule,rule differentiable_imp_continuous_at) unfolding differentiable_def using assms by auto

subsection {* Invertible derivative continous 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. *}

lemma bounded_linear_sub: "bounded_linear f \<Longrightarrow> bounded_linear g ==> bounded_linear (\<lambda>x. f x - g x)"
  using bounded_linear_add[of f "\<lambda>x. - g x"] bounded_linear_minus[of g] by(auto simp add:algebra_simps)

lemma has_derivative_locally_injective: fixes f::"'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
  assumes "a \<in> s" "open s" "bounded_linear g'" "g' o f'(a) = id"
  "\<forall>x\<in>s. (f has_derivative f'(x)) (at x)"
  "\<forall>e>0. \<exists>d>0. \<forall>x. dist a x < d \<longrightarrow> onorm(\<lambda>v. f' x v - f' a v) < e"
  obtains t where "a \<in> t" "open t" "\<forall>x\<in>t. \<forall>x'\<in>t. (f x' = f x) \<longrightarrow> (x' = x)" 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 (\<chi>\<chi> i.1)) = (\<chi>\<chi> i.1)" "(\<chi>\<chi> i.1) \<noteq> (0::'n)" defer 
    apply(subst euclidean_eq) using f'g' by auto
  hence *:"0 < onorm g'" unfolding onorm_pos_lt[OF assms(3)[unfolded linear_linear]] by fastsimp
  def k \<equiv> "1 / onorm g' / 2" have *:"k>0" unfolding k_def using * by auto
  guess d1 using assms(6)[rule_format,OF *] .. note d1=this
  from `open s` obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using `a\<in>s` ..
  obtain d2 where "d2>0" "ball a d2 \<subseteq> s" using assms(2,1) ..
  guess d2 using assms(2)[unfolded open_contains_ball,rule_format,OF `a\<in>s`] .. note d2=this
  guess d using real_lbound_gt_zero[OF d1[THEN conjunct1] d2[THEN conjunct1]] .. note d = this
  show ?thesis proof show "a\<in>ball a d" using d by auto
    show "\<forall>x\<in>ball a d. \<forall>x'\<in>ball a d. f x' = f x \<longrightarrow> x' = x" proof(intro strip)
      fix x y assume as:"x\<in>ball a d" "y\<in>ball a d" "f x = f y"
      def ph \<equiv> "\<lambda>w. w - g'(f w - f x)" have ph':"ph = g' \<circ> (\<lambda>w. f' a w - (f w - f x))"
	unfolding ph_def o_def unfolding diff using f'g' by(auto simp add:algebra_simps)
      have "norm (ph x - ph y) \<le> (1/2) * norm (x - y)"
	apply(rule differentiable_bound[OF convex_ball _ _ as(1-2), where f'="\<lambda>x v. v - g'(f' x v)"])
	apply(rule_tac[!] ballI) proof- fix u assume u:"u \<in> ball a d" hence "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
	show "(ph has_derivative (\<lambda>v. v - g' (f' u v))) (at u within ball a d)"
	  unfolding ph' * apply(rule diff_chain_within) defer apply(rule bounded_linear.has_derivative[OF assms(3)])
	  apply(rule has_derivative_intros) defer apply(rule has_derivative_sub[where g'="\<lambda>x.0",unfolded diff_0_right])
	  apply(rule has_derivative_at_within) using assms(5) and `u\<in>s` `a\<in>s`
          by(auto intro!: has_derivative_intros derivative_linear)
	have **:"bounded_linear (\<lambda>x. f' u x - f' a x)" "bounded_linear (\<lambda>x. f' a x - f' u x)" apply(rule_tac[!] bounded_linear_sub)
	  apply(rule_tac[!] derivative_linear) using assms(5) `u\<in>s` `a\<in>s` by auto
	have "onorm (\<lambda>v. v - g' (f' u v)) \<le> onorm g' * onorm (\<lambda>w. f' a w - f' u w)" unfolding * apply(rule onorm_compose)
	  unfolding linear_conv_bounded_linear by(rule assms(3) **)+ 
	also have "\<dots> \<le> onorm g' * k" apply(rule mult_left_mono) 
	  using d1[THEN conjunct2,rule_format,of u] using onorm_neg[OF **(1)[unfolded linear_linear]]
	  using d and u and onorm_pos_le[OF assms(3)[unfolded linear_linear]] by(auto simp add:algebra_simps) 
	also have "\<dots> \<le> 1/2" unfolding k_def by auto
	finally show "onorm (\<lambda>v. v - g' (f' u v)) \<le> 1 / 2" by assumption qed
      moreover have "norm (ph y - ph x) = norm (y - x)" apply(rule arg_cong[where f=norm])
	unfolding ph_def using diff unfolding as by auto
      ultimately show "x = y" unfolding norm_minus_commute by auto qed qed auto qed

subsection {* Uniformly convergent sequence of derivatives. *}

lemma has_derivative_sequence_lipschitz_lemma: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  "\<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
  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(default)+ 
  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)"
    apply(rule differentiable_bound[where f'="\<lambda>x h. f' m x h - f' n x h", OF assms(1) _ _ as(3-4)]) apply(rule_tac[!] ballI) proof-
    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 has_derivative_intros assms(2)[rule_format] `x\<in>s`)+
    { 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"] unfolding norm_minus_commute by(auto simp add:algebra_simps) 
      also have "\<dots> \<le> e * norm h+ e * norm h"  using assms(3)[rule_format,OF `N\<le>m` `x\<in>s`, of h] assms(3)[rule_format,OF `N\<le>n` `x\<in>s`, of h]
	by(auto simp add:field_simps)
      finally have "norm (f' m x h - f' n x h) \<le> 2 * e * norm h" by auto }
    thus "onorm (\<lambda>h. f' m x h - f' n x h) \<le> 2 * e" apply-apply(rule onorm(2)) apply(rule linear_compose_sub)
      unfolding linear_conv_bounded_linear using assms(2)[rule_format,OF `x\<in>s`, THEN derivative_linear] by auto qed qed

lemma has_derivative_sequence_lipschitz: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)" "0 < e"
  shows "\<forall>e>0. \<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(rule,rule)
  case goal1 have *:"2 * (1/2* e) = e" "1/2 * e >0" using `e>0` by auto
  guess N using assms(3)[rule_format,OF *(2)] ..
  thus ?case apply(rule_tac x=N in exI) apply(rule has_derivative_sequence_lipschitz_lemma[where e="1/2 *e", unfolded *]) using assms by auto qed

lemma has_derivative_sequence: fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(f' n x h - g' x h) \<le> e * norm(h)"
  "x0 \<in> s"  "((\<lambda>n. f n x0) ---> l) sequentially"
  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> (g has_derivative g'(x)) (at x within s)" proof-
  have lem1:"\<forall>e>0. \<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 has_derivative_sequence_lipschitz[where e="42::nat"]) apply(rule assms)+ by auto
  have "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) ---> g x) sequentially" apply(rule bchoice) unfolding convergent_eq_cauchy proof
    fix x assume "x\<in>s" show "Cauchy (\<lambda>n. f n x)" proof(cases "x=x0")
      case True thus ?thesis using convergent_imp_cauchy[OF assms(5)] by auto next
      case False show ?thesis unfolding Cauchy_def proof(rule,rule)
	fix e::real assume "e>0" hence *:"e/2>0" "e/2/norm(x-x0)>0" using False by(auto intro!:divide_pos_pos)
	guess M using convergent_imp_cauchy[OF assms(5), unfolded Cauchy_def, rule_format,OF *(1)] .. note M=this
	guess N using lem1[rule_format,OF *(2)] .. note N = this
	show " \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (f m x) (f n x) < e" apply(rule_tac x="max M N" in exI) proof(default+)
	  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[rule_format,OF _ _ `x\<in>s` `x0\<in>s`, of m n] and as and False by auto
	  also have "\<dots> < e / 2 + e / 2" apply(rule add_strict_right_mono) using as and M[rule_format] unfolding dist_norm by auto 
	  finally show "dist (f m x) (f n x) < e" by auto qed qed qed qed
  then guess g .. note g = this
  have lem2:"\<forall>e>0. \<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(rule,rule)
    fix e::real assume *:"e>0" guess N using lem1[rule_format,OF *] .. note N=this
    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)" apply(rule_tac x=N in exI) proof(default+)
      fix n x y assume as:"N \<le> n" "x \<in> s" "y \<in> s"
      have "eventually (\<lambda>xa. norm (f n x - f n y - (f xa x - f xa y)) \<le> e * norm (x - y)) sequentially" 
	unfolding eventually_sequentially apply(rule_tac x=N in exI) proof(rule,rule)
	fix m assume "N\<le>m" thus "norm (f n x - f n y - (f m x - f m y)) \<le> e * norm (x - y)"
	  using N[rule_format, of n m x y] and as by(auto simp add:algebra_simps) qed
      thus "norm (f n x - f n y - (g x - g y)) \<le> e * norm (x - y)" apply-
	apply(rule Lim_norm_ubound[OF trivial_limit_sequentially, where f="\<lambda>m. (f n x - f n y) - (f m x - f m y)"])
	apply(rule Lim_sub Lim_const g[rule_format] as)+ by assumption qed qed
  show ?thesis unfolding has_derivative_within_alt apply(rule_tac x=g in exI)
    apply(rule,rule,rule g[rule_format],assumption) proof fix x assume "x\<in>s"
    have lem3:"\<forall>u. ((\<lambda>n. f' n x u) ---> g' x u) sequentially" unfolding Lim_sequentially proof(rule,rule,rule)
      fix u and e::real assume "e>0" show "\<exists>N. \<forall>n\<ge>N. dist (f' n x u) (g' x u) < e" proof(cases "u=0")
	case True guess N using assms(3)[rule_format,OF `e>0`] .. note N=this
	show ?thesis apply(rule_tac x=N in exI) unfolding True 
	  using N[rule_format,OF _ `x\<in>s`,of _ 0] and `e>0` by auto next
	case False hence *:"e / 2 / norm u > 0" using `e>0` by(auto intro!: divide_pos_pos)
	guess N using assms(3)[rule_format,OF *] .. note N=this
	show ?thesis apply(rule_tac x=N in exI) proof(rule,rule) case goal1
	  show ?case unfolding dist_norm using N[rule_format,OF goal1 `x\<in>s`, of u] False `e>0`
	    by (auto simp add:field_simps) qed qed qed
    show "bounded_linear (g' x)" unfolding linear_linear linear_def apply(rule,rule,rule) defer proof(rule,rule)
      fix x' y z::"'m" and c::real
      note lin = assms(2)[rule_format,OF `x\<in>s`,THEN derivative_linear]
      show "g' x (c *\<^sub>R x') = c *\<^sub>R g' x x'" apply(rule Lim_unique[OF trivial_limit_sequentially])
	apply(rule lem3[rule_format])
        unfolding lin[unfolded bounded_linear_def bounded_linear_axioms_def,THEN conjunct2,THEN conjunct1,rule_format]
	apply(rule Lim_cmul) by(rule lem3[rule_format])
      show "g' x (y + z) = g' x y + g' x z" apply(rule Lim_unique[OF trivial_limit_sequentially])
	apply(rule lem3[rule_format]) unfolding lin[unfolded bounded_linear_def additive_def,THEN conjunct1,rule_format]
        apply(rule Lim_add) by(rule lem3[rule_format])+ qed 
    show "\<forall>e>0. \<exists>d>0. \<forall>y\<in>s. norm (y - x) < d \<longrightarrow> norm (g y - g x - g' x (y - x)) \<le> e * norm (y - x)" proof(rule,rule) case goal1
      have *:"e/3>0" using goal1 by auto guess N1 using assms(3)[rule_format,OF *] .. note N1=this
      guess N2 using lem2[rule_format,OF *] .. note N2=this
      guess d1 using assms(2)[unfolded has_derivative_within_alt, rule_format,OF `x\<in>s`, of "max N1 N2",THEN conjunct2,rule_format,OF *] .. note d1=this
      show ?case apply(rule_tac x=d1 in exI) apply(rule,rule d1[THEN conjunct1]) proof(rule,rule)
	fix y assume as:"y \<in> s" "norm (y - x) < d1" let ?N ="max N1 N2"
	have "norm (g y - g x - (f ?N y - f ?N x)) \<le> e /3 * norm (y - x)" apply(subst norm_minus_cancel[THEN sym])
	  using N2[rule_format, OF _ `y\<in>s` `x\<in>s`, of ?N] by auto moreover
	have "norm (f ?N y - f ?N x - f' ?N x (y - x)) \<le> e / 3 * norm (y - x)" using d1 and as by auto 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 `x\<in>s` 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 qed

subsection {* Can choose to line up antiderivatives if we want. *}

lemma has_antiderivative_sequence: fixes f::"nat\<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<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 assms(1) _ assms(3), of "\<lambda>n x. f n x + (f 0 a - f n a)"])
    apply(rule,rule) apply(rule has_derivative_add_const, rule assms(2)[rule_format], assumption)  
    apply(rule `a\<in>s`) by(auto intro!: Lim_const) qed auto

lemma has_antiderivative_limit: fixes g'::"'m::euclidean_space \<Rightarrow> 'm \<Rightarrow> 'n::euclidean_space"
  assumes "convex s" "\<forall>e>0. \<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))"
    apply(rule) using assms(2) apply(erule_tac x="inverse (real (Suc n))" in allE) by auto
  guess f using *[THEN choice] .. note * = this guess f' using *[THEN choice] .. note f=this
  show ?thesis apply(rule has_antiderivative_sequence[OF assms(1), of f f']) defer proof(rule,rule)
    fix e::real assume "0<e" guess  N using reals_Archimedean[OF `e>0`] .. note N=this 
    show "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm (f' n x h - g' x h) \<le> e * norm h"  apply(rule_tac x=N in exI) proof(default+) case goal1
      have *:"inverse (real (Suc n)) \<le> e" apply(rule order_trans[OF _ N[THEN less_imp_le]])
	using goal1(1) by(auto simp add:field_simps) 
      show ?case using f[rule_format,THEN conjunct2,OF goal1(2), of n, THEN spec[where x=h]] 
	apply(rule order_trans) using N * apply(cases "h=0") by auto qed qed(insert f,auto) qed

subsection {* Differentiation of a series. *}

definition sums_seq :: "(nat \<Rightarrow> 'a::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> (nat set) \<Rightarrow> bool"
(infixl "sums'_seq" 12) where "(f sums_seq l) s \<equiv> ((\<lambda>n. setsum f (s \<inter> {0..n})) ---> l) sequentially"

lemma has_derivative_series: fixes f::"nat \<Rightarrow> 'm::euclidean_space \<Rightarrow> 'n::euclidean_space"
  assumes "convex s" "\<forall>n. \<forall>x\<in>s. ((f n) has_derivative (f' n x)) (at x within s)"
  "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. norm(setsum (\<lambda>i. f' i x h) (k \<inter> {0..n}) - g' x h) \<le> e * norm(h)"
  "x\<in>s" "((\<lambda>n. f n x) sums_seq l) k"
  shows "\<exists>g. \<forall>x\<in>s. ((\<lambda>n. f n x) sums_seq (g x)) k \<and> (g has_derivative g'(x)) (at x within s)"
  unfolding sums_seq_def apply(rule has_derivative_sequence[OF assms(1) _ assms(3)]) apply(rule,rule)
  apply(rule has_derivative_setsum) defer apply(rule,rule assms(2)[rule_format],assumption)
  using assms(4-5) unfolding sums_seq_def by auto

subsection {* Derivative with composed bilinear function. *}

lemma has_derivative_bilinear_within:
  assumes "(f has_derivative f') (at x within s)" "(g has_derivative g') (at x within s)" "bounded_bilinear h"
  shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x within s)" proof-
  have "(g ---> g x) (at x within s)" apply(rule differentiable_imp_continuous_within[unfolded continuous_within])
    using assms(2) unfolding differentiable_def by auto moreover
  interpret f':bounded_linear f' using assms unfolding has_derivative_def by auto
  interpret g':bounded_linear g' using assms unfolding has_derivative_def by auto
  interpret h:bounded_bilinear h using assms by auto
  have "((\<lambda>y. f' (y - x)) ---> 0) (at x within s)" unfolding f'.zero[THEN sym]
    apply(rule Lim_linear[of "\<lambda>y. y - x" 0 "at x within s" f']) using Lim_sub[OF Lim_within_id Lim_const, of x x s]
    unfolding id_def using assms(1) unfolding has_derivative_def by auto
  hence "((\<lambda>y. f x + f' (y - x)) ---> f x) (at x within s)"
    using Lim_add[OF Lim_const, of "\<lambda>y. f' (y - x)" 0 "at x within s" "f x"] by auto ultimately
  have *:"((\<lambda>x'. h (f x + f' (x' - x)) ((1/(norm (x' - x))) *\<^sub>R (g x' - (g x + g' (x' - x))))
             + h ((1/ (norm (x' - x))) *\<^sub>R (f x' - (f x + f' (x' - x)))) (g x')) ---> h (f x) 0 + h 0 (g x)) (at x within s)"
    apply-apply(rule Lim_add) apply(rule_tac[!] Lim_bilinear[OF _ _ assms(3)]) using assms(1-2)  unfolding has_derivative_within by auto
  guess B using bounded_bilinear.pos_bounded[OF assms(3)] .. note B=this
  guess C using f'.pos_bounded .. note C=this
  guess D using g'.pos_bounded .. note D=this
  have bcd:"B * C * D > 0" using B C D by (auto intro!: mult_pos_pos)
  have **:"((\<lambda>y. (1/(norm(y - x))) *\<^sub>R (h (f'(y - x)) (g'(y - x)))) ---> 0) (at x within s)" unfolding Lim_within proof(rule,rule) case goal1
    hence "e/(B*C*D)>0" using B C D by(auto intro!:divide_pos_pos mult_pos_pos)
    thus ?case apply(rule_tac x="e/(B*C*D)" in exI,rule) proof(rule,rule,erule conjE)
      fix y assume as:"y \<in> s" "0 < dist y x" "dist y x < e / (B * C * D)"
      have "norm (h (f' (y - x)) (g' (y - x))) \<le> norm (f' (y - x)) * norm (g' (y - x)) * B" using B by auto
      also have "\<dots> \<le> (norm (y - x) * C) * (D * norm (y - x)) * B" apply(rule mult_right_mono)
	apply(rule mult_mono) using B C D by (auto simp add: field_simps intro!:mult_nonneg_nonneg)
      also have "\<dots> = (B * C * D * norm (y - x)) * norm (y - x)" by(auto simp add:field_simps)
      also have "\<dots> < e * norm (y - x)" apply(rule mult_strict_right_mono)
	using as(3)[unfolded dist_norm] and as(2) unfolding pos_less_divide_eq[OF bcd] by (auto simp add:field_simps)
      finally show "dist ((1 / norm (y - x)) *\<^sub>R h (f' (y - x)) (g' (y - x))) 0 < e"
	unfolding dist_norm apply-apply(cases "y = x") by(auto simp add:field_simps) qed qed
  have "bounded_linear (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))"
    apply (rule bounded_linear_add)
    apply (rule bounded_linear_compose [OF h.bounded_linear_right `bounded_linear g'`])
    apply (rule bounded_linear_compose [OF h.bounded_linear_left `bounded_linear f'`])
    done
  thus ?thesis using Lim_add[OF * **] unfolding has_derivative_within 
    unfolding g'.add f'.scaleR f'.add g'.scaleR f'.diff g'.diff
     h.add_right h.add_left scaleR_right_distrib h.scaleR_left h.scaleR_right h.diff_right h.diff_left
    scaleR_right_diff_distrib h.zero_right h.zero_left by(auto simp add:field_simps) qed

lemma has_derivative_bilinear_at:
  assumes "(f has_derivative f') (at x)" "(g has_derivative g') (at x)" "bounded_bilinear h"
  shows "((\<lambda>x. h (f x) (g x)) has_derivative (\<lambda>d. h (f x) (g' d) + h (f' d) (g x))) (at x)"
  using has_derivative_bilinear_within[of f f' x UNIV g g' h] unfolding within_UNIV using assms by auto

subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}

definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real net \<Rightarrow> bool)"
(infixl "has'_vector'_derivative" 12) where
 "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"

definition "vector_derivative f net \<equiv> (SOME f'. (f has_vector_derivative f') net)"

lemma vector_derivative_works: fixes f::"real \<Rightarrow> 'a::real_normed_vector"
  shows "f differentiable net \<longleftrightarrow> (f has_vector_derivative (vector_derivative f net)) net" (is "?l = ?r")
proof assume ?l guess f' using `?l`[unfolded differentiable_def] .. note f' = this
  then interpret bounded_linear f' by auto
  thus ?r unfolding vector_derivative_def has_vector_derivative_def
    apply-apply(rule someI_ex,rule_tac x="f' 1" in exI)
    using f' unfolding scaleR[THEN sym] by auto
next assume ?r thus ?l  unfolding vector_derivative_def has_vector_derivative_def differentiable_def by auto qed

lemma vector_derivative_unique_at: fixes f::"real\<Rightarrow> 'n::euclidean_space"
  assumes "(f has_vector_derivative f') (at x)" "(f has_vector_derivative f'') (at x)" shows "f' = f''" proof-
  have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')" apply(rule frechet_derivative_unique_at)
    using assms[unfolded has_vector_derivative_def] by auto
  show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
    hence "(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')" using * by auto
    ultimately show False unfolding expand_fun_eq by auto qed qed

lemma vector_derivative_unique_within_closed_interval: fixes f::"real \<Rightarrow> 'n::ordered_euclidean_space"
  assumes "a < b" "x \<in> {a..b}"
  "(f has_vector_derivative f') (at x within {a..b})"
  "(f has_vector_derivative f'') (at x within {a..b})" shows "f' = f''" proof-
  have *:"(\<lambda>x. x *\<^sub>R f') = (\<lambda>x. x *\<^sub>R f'')"
    apply(rule frechet_derivative_unique_within_closed_interval[of "a" "b"])
    using assms(3-)[unfolded has_vector_derivative_def] using assms(1-2) by auto
  show ?thesis proof(rule ccontr) assume "f' \<noteq> f''" moreover
    hence "(\<lambda>x. x *\<^sub>R f') 1 = (\<lambda>x. x *\<^sub>R f'') 1" using * by (auto simp: expand_fun_eq)
    ultimately show False unfolding o_def by auto qed qed

lemma vector_derivative_at: fixes f::"real \<Rightarrow> 'a::euclidean_space" shows
 "(f has_vector_derivative f') (at x) \<Longrightarrow> vector_derivative f (at x) = f'"
  apply(rule vector_derivative_unique_at) defer apply assumption
  unfolding vector_derivative_works[THEN sym] differentiable_def
  unfolding has_vector_derivative_def by auto

lemma vector_derivative_within_closed_interval: fixes f::"real \<Rightarrow> 'a::ordered_euclidean_space"
  assumes "a < b" "x \<in> {a..b}" "(f has_vector_derivative f') (at x within {a..b})"
  shows "vector_derivative f (at x within {a..b}) = f'"
  apply(rule vector_derivative_unique_within_closed_interval)
  using vector_derivative_works[unfolded differentiable_def]
  using assms by(auto simp add:has_vector_derivative_def)

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)"
  unfolding has_vector_derivative_def apply(rule has_derivative_within_subset) by auto

lemma has_vector_derivative_const: 
 "((\<lambda>x. c) has_vector_derivative 0) net"
  unfolding has_vector_derivative_def using has_derivative_const by auto

lemma has_vector_derivative_id: "((\<lambda>x::real. x) has_vector_derivative 1) net"
  unfolding has_vector_derivative_def using has_derivative_id by auto

lemma has_vector_derivative_cmul:  "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net"
  unfolding has_vector_derivative_def apply(drule has_derivative_cmul) by(auto simp add:algebra_simps)

lemma has_vector_derivative_cmul_eq: assumes "c \<noteq> 0"
  shows "(((\<lambda>x. c *\<^sub>R f x) has_vector_derivative (c *\<^sub>R f')) net \<longleftrightarrow> (f has_vector_derivative f') net)"
  apply rule apply(drule has_vector_derivative_cmul[where c="1/c"]) defer
  apply(rule has_vector_derivative_cmul) using assms by auto

lemma has_vector_derivative_neg:
 "(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. -(f x)) has_vector_derivative (- f')) net"
  unfolding has_vector_derivative_def apply(drule has_derivative_neg) by auto

lemma has_vector_derivative_add:
  assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
  shows "((\<lambda>x. f(x) + g(x)) has_vector_derivative (f' + g')) net"
  using has_derivative_add[OF assms[unfolded has_vector_derivative_def]]
  unfolding has_vector_derivative_def unfolding scaleR_right_distrib by auto

lemma has_vector_derivative_sub:
  assumes "(f has_vector_derivative f') net" "(g has_vector_derivative g') net"
  shows "((\<lambda>x. f(x) - g(x)) has_vector_derivative (f' - g')) net"
  using has_derivative_sub[OF assms[unfolded has_vector_derivative_def]]
  unfolding has_vector_derivative_def scaleR_right_diff_distrib by auto

lemma has_vector_derivative_bilinear_within:
  assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at x within s)" "bounded_bilinear h"
  shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x within s)" proof-
  interpret bounded_bilinear h using assms by auto 
  show ?thesis using has_derivative_bilinear_within[OF assms(1-2)[unfolded has_vector_derivative_def], of h]
    unfolding o_def has_vector_derivative_def
    using assms(3) unfolding scaleR_right scaleR_left scaleR_right_distrib by auto qed

lemma has_vector_derivative_bilinear_at:
  assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at x)" "bounded_bilinear h"
  shows "((\<lambda>x. h (f x) (g x)) has_vector_derivative (h (f x) g' + h f' (g x))) (at x)"
  apply(rule has_vector_derivative_bilinear_within[where s=UNIV, unfolded within_UNIV]) using assms by auto

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 apply(rule has_derivative_at_within) by auto

lemma has_vector_derivative_transform_within:
  assumes "0 < d" "x \<in> s" "\<forall>x'\<in>s. dist x' x < d \<longrightarrow> f x' = g x'" "(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_within)

lemma has_vector_derivative_transform_at:
  assumes "0 < d" "\<forall>x'. dist x' x < d \<longrightarrow> f x' = g x'" "(f has_vector_derivative f') (at x)"
  shows "(g has_vector_derivative f') (at x)"
  using assms unfolding has_vector_derivative_def by(rule has_derivative_transform_at)

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

lemma vector_diff_chain_at:
  assumes "(f has_vector_derivative f') (at x)" "(g has_vector_derivative g') (at (f x))"
  shows "((g \<circ> f) has_vector_derivative (f' *\<^sub>R g')) (at x)"
  using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_at[OF assms(1)[unfolded has_vector_derivative_def]])
  unfolding o_def scaleR.scaleR_left by auto

lemma vector_diff_chain_within:
  assumes "(f has_vector_derivative f') (at x within s)" "(g has_vector_derivative g') (at (f x) within f ` s)"
  shows "((g o f) has_vector_derivative (f' *\<^sub>R g')) (at x within s)"
  using assms(2) unfolding has_vector_derivative_def apply- apply(drule diff_chain_within[OF assms(1)[unfolded has_vector_derivative_def]])
  unfolding o_def scaleR.scaleR_left by auto

end