src/HOL/Analysis/Derivative.thy
 author paulson Sun, 20 May 2018 22:10:21 +0100 changeset 68241 39a311f50344 parent 68239 0764ee22a4d1 child 68527 2f4e2aab190a permissions -rw-r--r--
correcting the statements of the MVTs
```
(*  Title:      HOL/Analysis/Derivative.thy
Author:     John Harrison
Author:     Robert Himmelmann, TU Muenchen (translation from HOL Light); tidied by LCP
*)

section \<open>Multivariate calculus in Euclidean space\<close>

theory Derivative
imports Brouwer_Fixpoint Operator_Norm Uniform_Limit Bounded_Linear_Function
begin

declare bounded_linear_inner_left [intro]

declare has_derivative_bounded_linear[dest]

subsection \<open>Derivatives\<close>

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

subsection \<open>Derivative with composed bilinear function\<close>

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

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
by (simp add: diff_diff_eq)

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

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

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

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

subsubsection \<open>Caratheodory characterization\<close>

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

subsection \<open>Differentiability\<close>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

subsection \<open>Composition rules stated just for differentiability\<close>

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

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

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

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

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

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

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

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

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

subsection \<open>The traditional Rolle theorem in one dimension\<close>

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

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

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

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

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

theorem Rolle:
fixes f :: "real \<Rightarrow> real"
assumes "a < b"
and fab: "f a = f b"
and contf: "continuous_on {a..b} f"
and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
shows "\<exists>x\<in>{a <..< b}. f' x = (\<lambda>v. 0)"
proof -
have "\<exists>x\<in>box a b. (\<forall>y\<in>box a b. f x \<le> f y) \<or> (\<forall>y\<in>box a b. f y \<le> f x)"
proof -
have "(a + b) / 2 \<in> {a..b}"
using assms(1) by auto
then have *: "{a..b} \<noteq> {}"
by auto
obtain d where d: "d \<in>cbox a b" "\<forall>y\<in>cbox a b. f y \<le> f d"
using continuous_attains_sup[OF compact_Icc * contf] by auto
obtain c where c: "c \<in> cbox a b" "\<forall>y\<in>cbox a b. f c \<le> f y"
using continuous_attains_inf[OF compact_Icc * contf] by auto
show ?thesis
proof (cases "d \<in> box a b \<or> c \<in> box a b")
case True
then show ?thesis
by (metis c(2) d(2) box_subset_cbox subset_iff)
next
define e where "e = (a + b) /2"
case False
then have "f d = f c"
using d c fab by auto
with c d have "\<And>x. x \<in> {a..b} \<Longrightarrow> f x = f d"
by force
then show ?thesis
by (rule_tac x=e in bexI) (auto simp: e_def \<open>a < b\<close>)
qed
qed
then obtain x where x: "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)"
by auto
then have "f' x = (\<lambda>v. 0)"
apply (rule_tac differential_zero_maxmin[of x "box a b" f "f' x"])
using assms
apply auto
done
then show ?thesis
by (metis x(1))
qed

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

lemma mvt:
fixes f :: "real \<Rightarrow> real"
assumes "a < b"
and contf: "continuous_on {a..b} f"
and derf: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
shows "\<exists>x\<in>{a<..<b}. 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)"
proof (intro Rolle[OF \<open>a < b\<close>, of "\<lambda>x. f x - (f b - f a) / (b - a) * x"] ballI)
fix x
assume x: "a < x" "x < 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 (intro derivative_intros derf[OF x])
qed (use assms in \<open>auto intro!: continuous_intros simp: field_simps\<close>)
then obtain x where
"x \<in> {a <..< b}"
"(\<lambda>xa. f' x xa - (f b - f a) / (b - a) * xa) = (\<lambda>v. 0)" ..
then show ?thesis
by (metis (hide_lams) assms(1) diff_gt_0_iff_gt eq_iff_diff_eq_0
zero_less_mult_iff nonzero_mult_div_cancel_right not_real_square_gt_zero
times_divide_eq_left)
qed

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

lemma has_derivative_inverse_basic:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
assumes derf: "(f has_derivative f') (at (g y))"
and ling': "bounded_linear g'"
and "g' \<circ> f' = id"
and contg: "continuous (at y) g"
and "open T"
and "y \<in> T"
and fg: "\<And>z. z \<in> T \<Longrightarrow> f (g z) = z"
shows "(g has_derivative g') (at y)"
proof -
interpret f': bounded_linear f'
using assms unfolding has_derivative_def by auto
interpret g': bounded_linear g'
using assms by auto
obtain C where C: "0 < C" "\<And>x. norm (g' x) \<le> norm x * C"
using bounded_linear.pos_bounded[OF assms(2)] by blast
have lem1: "\<forall>e>0. \<exists>d>0. \<forall>z.
norm (z - y) < d \<longrightarrow> norm (g z - g y - g'(z - y)) \<le> e * norm (g z - g y)"
proof (intro allI impI)
fix e :: real
assume "e > 0"
with C(1) have *: "e / C > 0" by auto
obtain d0 where  "0 < d0" and d0:
"\<And>u. norm (u - g y) < d0 \<Longrightarrow> norm (f u - f (g y) - f' (u - g y)) \<le> e / C * norm (u - g y)"
using derf * unfolding has_derivative_at_alt by blast
obtain d1 where "0 < d1" and d1: "\<And>x. \<lbrakk>0 < dist x y; dist x y < d1\<rbrakk> \<Longrightarrow> dist (g x) (g y) < d0"
using contg \<open>0 < d0\<close> unfolding continuous_at Lim_at by blast
obtain d2 where "0 < d2" and d2: "\<And>u. dist u y < d2 \<Longrightarrow> u \<in> T"
using \<open>open T\<close> \<open>y \<in> T\<close> unfolding open_dist by blast
obtain d where d: "0 < d" "d < d1" "d < d2"
using real_lbound_gt_zero[OF \<open>0 < d1\<close> \<open>0 < d2\<close>] by blast
show "\<exists>d>0. \<forall>z. norm (z - y) < d \<longrightarrow> norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
proof (intro exI allI impI conjI)
fix z
assume as: "norm (z - y) < d"
then have "z \<in> T"
using d2 d unfolding dist_norm by auto
have "norm (g z - g y - g' (z - y)) \<le> norm (g' (f (g z) - y - f' (g z - g y)))"
unfolding g'.diff f'.diff
unfolding assms(3)[unfolded o_def id_def, THEN fun_cong] fg[OF \<open>z\<in>T\<close>]
by (simp add: norm_minus_commute)
also have "\<dots> \<le> norm (f (g z) - y - f' (g z - g y)) * C"
by (rule C(2))
also have "\<dots> \<le> (e / C) * norm (g z - g y) * C"
proof -
have "norm (g z - g y) < d0"
by (metis as cancel_comm_monoid_add_class.diff_cancel d(2) \<open>0 < d0\<close> d1 diff_gt_0_iff_gt diff_strict_mono dist_norm dist_self zero_less_dist_iff)
then show ?thesis
by (metis C(1) \<open>y \<in> T\<close> d0 fg real_mult_le_cancel_iff1)
qed
also have "\<dots> \<le> e * norm (g z - g y)"
using C by (auto simp add: field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (g z - g y)"
by simp
qed (use d in auto)
qed
have *: "(0::real) < 1 / 2"
by auto
obtain d where "0 < d" and d:
"\<And>z. norm (z - y) < d \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> 1/2 * norm (g z - g y)"
using lem1 * by blast
define B where "B = C * 2"
have "B > 0"
unfolding B_def using C by auto
have lem2: "norm (g z - g y) \<le> B * norm (z - y)" if z: "norm(z - y) < d" for z
proof -
have "norm (g z - g y) \<le> norm(g' (z - y)) + norm ((g z - g y) - g'(z - y))"
by (rule norm_triangle_sub)
also have "\<dots> \<le> norm (g' (z - y)) + 1 / 2 * norm (g z - g y)"
by (rule add_left_mono) (use d z in auto)
also have "\<dots> \<le> norm (z - y) * C + 1 / 2 * norm (g z - g y)"
by (rule add_right_mono) (use C in auto)
finally show "norm (g z - g y) \<le> B * norm (z - y)"
unfolding B_def
by (auto simp add: field_simps)
qed
show ?thesis
unfolding has_derivative_at_alt
proof (intro conjI assms allI impI)
fix e :: real
assume "e > 0"
then have *: "e / B > 0" by (metis \<open>B > 0\<close> divide_pos_pos)
obtain d' where "0 < d'" and d':
"\<And>z. norm (z - y) < d' \<Longrightarrow> norm (g z - g y - g' (z - y)) \<le> e / B * norm (g z - g y)"
using lem1 * by blast
obtain k where k: "0 < k" "k < d" "k < d'"
using real_lbound_gt_zero[OF \<open>0 < d\<close> \<open>0 < d'\<close>] by blast
show "\<exists>d>0. \<forall>ya. norm (ya - y) < d \<longrightarrow> norm (g ya - g y - g' (ya - y)) \<le> e * norm (ya - y)"
proof (intro exI allI impI conjI)
fix z
assume as: "norm (z - y) < k"
then have "norm (g z - g y - g' (z - y)) \<le> e / B * norm(g z - g y)"
using d' k by auto
also have "\<dots> \<le> e * norm (z - y)"
unfolding times_divide_eq_left pos_divide_le_eq[OF \<open>B>0\<close>]
using lem2[of z] k as \<open>e > 0\<close>
by (auto simp add: field_simps)
finally show "norm (g z - g y - g' (z - y)) \<le> e * norm (z - y)"
by simp
qed (use k in auto)
qed
qed

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

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

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

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

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

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

subsection \<open>Proving surjectivity via Brouwer fixpoint theorem\<close>

lemma brouwer_surjective:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "compact T"
and "convex T"
and "T \<noteq> {}"
and "continuous_on T f"
and "\<And>x y. \<lbrakk>x\<in>S; y\<in>T\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> T"
and "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 (intro continuous_intros)
using assms
apply auto
done
qed

lemma brouwer_surjective_cball:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "continuous_on (cball a e) f"
and "e > 0"
and "x \<in> S"
and "\<And>x y. \<lbrakk>x\<in>S; y\<in>cball a e\<rbrakk> \<Longrightarrow> x + (y - f y) \<in> cball a e"
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
apply auto
done

text \<open>See Sussmann: "Multidifferential calculus", Theorem 2.1.1\<close>

lemma sussmann_open_mapping:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
assumes "open S"
and contf: "continuous_on S f"
and "x \<in> S"
and derf: "(f has_derivative f') (at x)"
and "bounded_linear g'" "f' \<circ> g' = id"
and "T \<subseteq> S"
and x: "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
obtain B where B: "0 < B" "\<forall>x. norm (g' x) \<le> norm x * B"
using bounded_linear.pos_bounded[OF assms(5)] by blast
hence *: "1 / (2 * B) > 0" by auto
obtain e0 where e0:
"0 < e0"
"\<forall>y. norm (y - x) < e0 \<longrightarrow> norm (f y - f x - f' (y - x)) \<le> 1 / (2 * B) * norm (y - x)"
using derf unfolding has_derivative_at_alt
using * by blast
obtain e1 where e1: "0 < e1" "cball x e1 \<subseteq> T"
using mem_interior_cball x by blast
have *: "0 < e0 / B" "0 < e1 / B" using e0 e1 B by auto
obtain e where e: "0 < e" "e < e0 / B" "e < e1 / B"
using real_lbound_gt_zero[OF *] by blast
have lem: "\<exists>y\<in>cball (f x) e. f (x + g' (y - f x)) = z" if "z\<in>cball (f x) (e / 2)" for z
proof (rule brouwer_surjective_cball)
have z: "z \<in> S" if as: "y \<in>cball (f x) e" "z = x + (g' y - g' (f x))" for y z
proof-
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"
by (metis B(2) g'.diff)
also have "\<dots> \<le> e * B"
by (metis B(1) dist_norm mem_cball real_mult_le_cancel_iff1 that(1))
also have "\<dots> \<le> e1"
using B(1) e(3) pos_less_divide_eq by fastforce
finally have "z \<in> cball x e1"
by force
then show "z \<in> S"
using e1 assms(7) by auto
qed
show "continuous_on (cball (f x) e) (\<lambda>y. f (x + g' (y - f x)))"
unfolding g'.diff
proof (rule continuous_on_compose2 [OF _ _ order_refl, of _ _ f])
show "continuous_on ((\<lambda>y. x + (g' y - g' (f x))) ` cball (f x) e) f"
by (rule continuous_on_subset[OF contf]) (use z in blast)
show "continuous_on (cball (f x) e) (\<lambda>y. x + (g' y - g' (f x)))"
by (intro continuous_intros linear_continuous_on[OF \<open>bounded_linear g'\<close>])
qed
next
fix y z
assume y: "y \<in> cball (f x) (e / 2)" and z: "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"
by (metis B(1) z dist_norm mem_cball norm_minus_commute real_mult_le_cancel_iff1)
also have "\<dots> < e0"
using B(1) e(2) pos_less_divide_eq by blast
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(2)[rule_format, OF *]
by (simp only: algebra_simps **) auto
also have "\<dots> \<le> 1 / (B * 2) * norm (g' (z - f x)) + e/2"
using y by (auto simp: dist_norm)
also have "\<dots> \<le> 1 / (B * 2) * B * norm (z - f x) + e/2"
using * 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"
using B(1) \<open>norm (z - f x) * B \<le> e * B\<close> by auto
finally show "y + (z - f (x + g' (z - f x))) \<in> cball (f x) e"
by (auto simp: dist_norm)
qed (use e that in auto)
show ?thesis
unfolding mem_interior
proof (intro exI conjI subsetI)
fix y
assume "y \<in> ball (f x) (e / 2)"
then have *: "y \<in> cball (f x) (e / 2)"
by auto
obtain z where z: "z \<in> cball (f x) e" "f (x + g' (z - f x)) = y"
using lem * by blast
then have "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"
by (metis B(1) dist_norm mem_cball norm_minus_commute real_mult_le_cancel_iff1 z(1))
also have "\<dots> \<le> e1"
using e B unfolding less_divide_eq by auto
finally have "x + g'(z - f x) \<in> T"
by (metis add_diff_cancel diff_diff_add dist_norm e1(2) mem_cball norm_minus_commute subset_eq)
then show "y \<in> f ` T"
using z by auto
qed (use e in auto)
qed

text \<open>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 \<open>f' \<circ> g = I\<close> but this happens to fit with the minimal linear
algebra theory I've set up so far.\<close>

lemma has_derivative_inverse_strong:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "open S"
and "x \<in> S"
and contf: "continuous_on S f"
and gf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
and derf: "(f has_derivative f') (at x)"
and id: "f' \<circ> g' = id"
shows "(g has_derivative g') (at (f x))"
proof -
have linf: "bounded_linear f'"
using derf unfolding has_derivative_def by auto
then have ling: "bounded_linear g'"
unfolding linear_conv_bounded_linear[symmetric]
using id right_inverse_linear by blast
moreover have "g' \<circ> f' = id"
using id linf ling
unfolding linear_conv_bounded_linear[symmetric]
using linear_inverse_left
by auto
moreover have *: "\<And>T. \<lbrakk>T \<subseteq> S; x \<in> interior T\<rbrakk> \<Longrightarrow> f x \<in> interior (f ` T)"
apply (rule sussmann_open_mapping)
apply (rule assms ling)+
apply auto
done
have "continuous (at (f x)) g"
unfolding continuous_at Lim_at
proof (rule, rule)
fix e :: real
assume "e > 0"
then have "f x \<in> interior (f ` (ball x e \<inter> S))"
using *[rule_format,of "ball x e \<inter> S"] \<open>x \<in> S\<close>
by (auto simp add: interior_open[OF open_ball] interior_open[OF assms(1)])
then obtain d where d: "0 < d" "ball (f x) d \<subseteq> f ` (ball x e \<inter> S)"
unfolding mem_interior by blast
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"
proof (intro exI allI impI conjI)
fix y
assume "0 < dist y (f x) \<and> dist y (f x) < d"
then have "g y \<in> g ` f ` (ball x e \<inter> S)"
by (metis d(2) dist_commute mem_ball rev_image_eqI subset_iff)
then show "dist (g y) (g (f x)) < e"
using gf[OF \<open>x \<in> S\<close>]
by (simp add: assms(4) dist_commute image_iff)
qed (use d in auto)
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 \<open>x \<in> S\<close>
apply auto
done
moreover have "f (g y) = y" if "y \<in> interior (f ` S)" for y
by (metis gf imageE interiorE set_mp that)
ultimately show ?thesis using assms
by (metis has_derivative_inverse_basic_x open_interior)
qed

text \<open>A rewrite based on the other domain.\<close>

lemma has_derivative_inverse_strong_x:
fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
assumes "open S"
and "g y \<in> S"
and "continuous_on S f"
and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
and "(f has_derivative f') (at (g y))"
and "f' \<circ> g' = id"
and "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

text \<open>On a region.\<close>

lemma has_derivative_inverse_on:
fixes f :: "'n::euclidean_space \<Rightarrow> 'n"
assumes "open S"
and derf: "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f'(x)) (at x)"
and "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
and "f' x \<circ> g' x = id"
and "x \<in> S"
shows "(g has_derivative g'(x)) (at (f x))"
proof (rule has_derivative_inverse_strong[where g'="g' x" and f=f])
show "continuous_on S f"
unfolding continuous_on_eq_continuous_at[OF \<open>open S\<close>]
using derf has_derivative_continuous by blast
qed (use assms in auto)

text \<open>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.\<close>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

text \<open>Considering derivative @{typ "real \<Rightarrow> 'b::real_normed_vector"} as a vector.\<close>

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

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

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

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

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

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

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

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

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

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

definition deriv :: "('a \<Rightarrow> 'a::real_normed_field) \<Rightarrow> 'a \<Rightarrow> 'a" where
"deriv f x \<equiv> SOME D. DERIV f x :> D"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

subsection\<open>The notion of being field differentiable\<close>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

subsection \<open>Partial derivatives\<close>

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

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

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

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

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

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

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

subsection \<open>Differentiable case distinction\<close>

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

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

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

end
```