src/HOL/Deriv.thy
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NEWS;
```
(*  Title:      HOL/Deriv.thy
Author:     Jacques D. Fleuriot, University of Cambridge, 1998
Author:     Brian Huffman
Author:     Lawrence C Paulson, 2004
Author:     Benjamin Porter, 2005
*)

section \<open>Differentiation\<close>

theory Deriv
imports Limits
begin

subsection \<open>Frechet derivative\<close>

definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow>
('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool"  (infix "(has'_derivative)" 50)
where "(f has_derivative f') F \<longleftrightarrow>
bounded_linear f' \<and>
((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> 0) F"

text \<open>
Usually the filter \<^term>\<open>F\<close> is \<^term>\<open>at x within s\<close>.  \<^term>\<open>(f has_derivative D)
(at x within s)\<close> means: \<^term>\<open>D\<close> is the derivative of function \<^term>\<open>f\<close> at point \<^term>\<open>x\<close>
within the set \<^term>\<open>s\<close>. Where \<^term>\<open>s\<close> is used to express left or right sided derivatives. In
most cases \<^term>\<open>s\<close> is either a variable or \<^term>\<open>UNIV\<close>.
\<close>

text \<open>These are the only cases we'll care about, probably.\<close>

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

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

definition has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
(infix "(has'_field'_derivative)" 50)
where "(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative (*) D) F"

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

definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
(infix "has'_vector'_derivative" 50)
where "(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"

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

named_theorems derivative_intros "structural introduction rules for derivatives"
setup \<open>
let
val eq_thms = @{thms has_derivative_eq_rhs DERIV_cong has_vector_derivative_eq_rhs}
fun eq_rule thm = get_first (try (fn eq_thm => eq_thm OF [thm])) eq_thms
in
(\<^binding>\<open>derivative_eq_intros\<close>,
fn context =>
Named_Theorems.get (Context.proof_of context) \<^named_theorems>\<open>derivative_intros\<close>
|> map_filter eq_rule)
end
\<close>

text \<open>
The following syntax is only used as a legacy syntax.
\<close>
abbreviation (input)
FDERIV :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow>  ('a \<Rightarrow> 'b) \<Rightarrow> bool"
("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where "FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"

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

lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
using bounded_linear.linear[OF has_derivative_bounded_linear] .

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

lemma has_derivative_id [derivative_intros, simp]: "(id has_derivative id) (at a)"
by (metis eq_id_iff has_derivative_ident)

lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 0)) F"

lemma (in bounded_linear) bounded_linear: "bounded_linear f" ..

lemma (in bounded_linear) has_derivative:
"(g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
unfolding has_derivative_def
by (auto simp add: bounded_linear_compose [OF bounded_linear] scaleR diff dest: tendsto)

lemmas has_derivative_scaleR_right [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_scaleR_right]

lemmas has_derivative_scaleR_left [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_scaleR_left]

lemmas has_derivative_mult_right [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_mult_right]

lemmas has_derivative_mult_left [derivative_intros] =
bounded_linear.has_derivative [OF bounded_linear_mult_left]

lemmas has_derivative_of_real[derivative_intros, simp] =
bounded_linear.has_derivative[OF bounded_linear_of_real]

assumes f: "(f has_derivative f') F"
and g: "(g has_derivative g') F"
shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
unfolding has_derivative_def
proof safe
let ?x = "Lim F (\<lambda>x. x)"
let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
have "((\<lambda>x. ?D f f' x + ?D g g' x) \<longlongrightarrow> (0 + 0)) F"
using f g by (intro tendsto_add) (auto simp: has_derivative_def)
then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) \<longlongrightarrow> 0) F"
qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)

lemma has_derivative_sum[simp, derivative_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F) \<Longrightarrow>
((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
by (induct I rule: infinite_finite_induct) simp_all

lemma has_derivative_minus[simp, derivative_intros]:
"(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
using has_derivative_scaleR_right[of f f' F "-1"] by simp

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

lemma has_derivative_at_within:
"(f has_derivative f') (at x within s) \<longleftrightarrow>
(bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) \<longlongrightarrow> 0) (at x within s))"
proof (cases "at x within s = bot")
case True
then show ?thesis
by (metis (no_types, lifting) has_derivative_within tendsto_bot)
next
case False
then show ?thesis
qed

lemma has_derivative_iff_norm:
"(f has_derivative f') (at x within s) \<longleftrightarrow>
bounded_linear f' \<and> ((\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
by (simp add: has_derivative_at_within divide_inverse ac_simps)

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

lemma field_has_derivative_at:
fixes x :: "'a::real_normed_field"
shows "(f has_derivative (*) D) (at x) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D" (is "?lhs = ?rhs")
proof -
have "?lhs = (\<lambda>h. norm (f (x + h) - f x - D * h) / norm h) \<midarrow>0 \<rightarrow> 0"
also have "... = (\<lambda>y. norm ((f (x + y) - f x - D * y) / y)) \<midarrow>0\<rightarrow> 0"
by (simp cong: LIM_cong flip: nonzero_norm_divide)
also have "... = (\<lambda>y. norm ((f (x + y) - f x) / y - D / y * y)) \<midarrow>0\<rightarrow> 0"
by (simp only: diff_divide_distrib times_divide_eq_left [symmetric])
also have "... = ?rhs"
by (simp add: tendsto_norm_zero_iff LIM_zero_iff cong: LIM_cong)
finally show ?thesis .
qed

lemma has_derivative_iff_Ex:
"(f has_derivative f') (at x) \<longleftrightarrow>
bounded_linear f' \<and> (\<exists>e. (\<forall>h. f (x+h) = f x + f' h + e h) \<and> ((\<lambda>h. norm (e h) / norm h) \<longlongrightarrow> 0) (at 0))"
unfolding has_derivative_at by force

lemma has_derivative_at_within_iff_Ex:
assumes "x \<in> S" "open S"
shows "(f has_derivative f') (at x within S) \<longleftrightarrow>
bounded_linear f' \<and> (\<exists>e. (\<forall>h. x+h \<in> S \<longrightarrow> f (x+h) = f x + f' h + e h) \<and> ((\<lambda>h. norm (e h) / norm h) \<longlongrightarrow> 0) (at 0))"
(is "?lhs = ?rhs")
proof safe
show "bounded_linear f'"
if "(f has_derivative f') (at x within S)"
using has_derivative_bounded_linear that by blast
show "\<exists>e. (\<forall>h. x + h \<in> S \<longrightarrow> f (x + h) = f x + f' h + e h) \<and> (\<lambda>h. norm (e h) / norm h) \<midarrow>0\<rightarrow> 0"
if "(f has_derivative f') (at x within S)"
by (metis (full_types) assms that has_derivative_iff_Ex at_within_open)
show "(f has_derivative f') (at x within S)"
if "bounded_linear f'"
and eq [rule_format]: "\<forall>h. x + h \<in> S \<longrightarrow> f (x + h) = f x + f' h + e h"
and 0: "(\<lambda>h. norm (e (h::'a)::'b) / norm h) \<midarrow>0\<rightarrow> 0"
for e
proof -
have 1: "f y - f x = f' (y-x) + e (y-x)" if "y \<in> S" for y
using eq [of "y-x"] that by simp
have 2: "((\<lambda>y. norm (e (y-x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
by (simp add: "0" assms tendsto_offset_zero_iff)
have "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within S)"
by (simp add: Lim_cong_within 1 2)
then show ?thesis
by (simp add: has_derivative_iff_norm \<open>bounded_linear f'\<close>)
qed
qed

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

lemma has_derivativeI_sandwich:
assumes e: "0 < e"
and bounded: "bounded_linear f'"
and sandwich: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow>
norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
and "(H \<longlongrightarrow> 0) (at x within s)"
shows "(f has_derivative f') (at x within s)"
unfolding has_derivative_iff_norm
proof safe
show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within s)"
proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
show "(H \<longlongrightarrow> 0) (at x within s)" by fact
show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
unfolding eventually_at using e sandwich by auto
qed (auto simp: le_divide_eq)
qed fact

lemma has_derivative_subset:
"(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)

lemma has_derivative_within_singleton_iff:
"(f has_derivative g) (at x within {x}) \<longleftrightarrow> bounded_linear g"
by (auto intro!: has_derivativeI_sandwich[where e=1] has_derivative_bounded_linear)

subsubsection \<open>Limit transformation for derivatives\<close>

lemma has_derivative_transform_within:
assumes "(f has_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_derivative f') (at x within s)"
using assms
unfolding has_derivative_within
by (force simp add: intro: Lim_transform_within)

lemma has_derivative_transform_within_open:
assumes "(f has_derivative f') (at x within t)"
and "open s"
and "x \<in> s"
and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
shows "(g has_derivative f') (at x within t)"
using assms unfolding has_derivative_within
by (force simp add: intro: Lim_transform_within_open)

lemma has_derivative_transform:
assumes "x \<in> s" "\<And>x. x \<in> s \<Longrightarrow> g x = f x"
assumes "(f has_derivative f') (at x within s)"
shows "(g has_derivative f') (at x within s)"
using assms
by (intro has_derivative_transform_within[OF _ zero_less_one, where g=g]) auto

lemma has_derivative_transform_eventually:
assumes "(f has_derivative f') (at x within s)"
"(\<forall>\<^sub>F x' in at x within s. f x' = g x')"
assumes "f x = g x" "x \<in> s"
shows "(g has_derivative f') (at x within s)"
using assms
proof -
from assms(2,3) obtain d where "d > 0" "\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x'"
by (force simp: eventually_at)
from has_derivative_transform_within[OF assms(1) this(1) assms(4) this(2)]
show ?thesis .
qed

lemma has_field_derivative_transform_within:
assumes "(f has_field_derivative f') (at a within S)"
and "0 < d"
and "a \<in> S"
and "\<And>x. \<lbrakk>x \<in> S; dist x a < d\<rbrakk> \<Longrightarrow> f x = g x"
shows "(g has_field_derivative f') (at a within S)"
using assms unfolding has_field_derivative_def
by (metis has_derivative_transform_within)

lemma has_field_derivative_transform_within_open:
assumes "(f has_field_derivative f') (at a)"
and "open S" "a \<in> S"
and "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
shows "(g has_field_derivative f') (at a)"
using assms unfolding has_field_derivative_def
by (metis has_derivative_transform_within_open)

subsection \<open>Continuity\<close>

lemma has_derivative_continuous:
assumes f: "(f has_derivative f') (at x within s)"
shows "continuous (at x within s) f"
proof -
from f interpret F: bounded_linear f'
by (rule has_derivative_bounded_linear)
note F.tendsto[tendsto_intros]
let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
using f unfolding has_derivative_iff_norm by blast
then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
by (intro filterlim_cong) (simp_all add: eventually_at_filter)
finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
by (rule tendsto_norm_zero_cancel)
then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
then have "?L (\<lambda>y. f y - f x)"
by simp
from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
qed

subsection \<open>Composition\<close>

lemma tendsto_at_iff_tendsto_nhds_within:
"f x = y \<Longrightarrow> (f \<longlongrightarrow> y) (at x within s) \<longleftrightarrow> (f \<longlongrightarrow> y) (inf (nhds x) (principal s))"
unfolding tendsto_def eventually_inf_principal eventually_at_filter
by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)

lemma has_derivative_in_compose:
assumes f: "(f has_derivative f') (at x within s)"
and g: "(g has_derivative g') (at (f x) within (f`s))"
shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
proof -
from f interpret F: bounded_linear f'
by (rule has_derivative_bounded_linear)
from g interpret G: bounded_linear g'
by (rule has_derivative_bounded_linear)
from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF"
by fast
from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG"
by fast
note G.tendsto[tendsto_intros]

let ?L = "\<lambda>f. (f \<longlongrightarrow> 0) (at x within s)"
let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
define Nf where "Nf = ?N f f' x"
define Ng where [abs_def]: "Ng y = ?N g g' (f x) (f y)" for y

show ?thesis
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (\<lambda>x. g' (f' x))"
using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
next
fix y :: 'a
assume neq: "y \<noteq> x"
have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
also have "\<dots> \<le> norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
also have "\<dots> \<le> Nf y * kG + Ng y * (Nf y + kF)"
have "norm (f y - f x) = norm (?D f f' x y + f' (y - x))"
by simp
also have "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
by (rule norm_triangle_ineq)
also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
using kF by (intro add_mono) simp
finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
by (simp add: neq Nf_def field_simps)
qed (use kG in \<open>simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps\<close>)
finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
next
have [tendsto_intros]: "?L Nf"
using f unfolding has_derivative_iff_norm Nf_def ..
from f have "(f \<longlongrightarrow> f x) (at x within s)"
by (blast intro: has_derivative_continuous continuous_within[THEN iffD1])
then have f': "LIM x at x within s. f x :> inf (nhds (f x)) (principal (f`s))"
unfolding filterlim_def
by (simp add: eventually_filtermap eventually_at_filter le_principal)

have "((?N g  g' (f x)) \<longlongrightarrow> 0) (at (f x) within f`s)"
using g unfolding has_derivative_iff_norm ..
then have g': "((?N g  g' (f x)) \<longlongrightarrow> 0) (inf (nhds (f x)) (principal (f`s)))"
by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp

have [tendsto_intros]: "?L Ng"
unfolding Ng_def by (rule filterlim_compose[OF g' f'])
show "((\<lambda>y. Nf y * kG + Ng y * (Nf y + kF)) \<longlongrightarrow> 0) (at x within s)"
by (intro tendsto_eq_intros) auto
qed simp
qed

lemma has_derivative_compose:
"(f has_derivative f') (at x within s) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
by (blast intro: has_derivative_in_compose has_derivative_subset)

lemma has_derivative_in_compose2:
assumes "\<And>x. x \<in> t \<Longrightarrow> (g has_derivative g' x) (at x within t)"
assumes "f ` s \<subseteq> t" "x \<in> s"
assumes "(f has_derivative f') (at x within s)"
shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>y. g' (f x) (f' y))) (at x within s)"
using assms
by (auto intro: has_derivative_subset intro!: has_derivative_in_compose[of f f' x s g])

lemma (in bounded_bilinear) FDERIV:
assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
proof -
from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast

from pos_bounded obtain K
where K: "0 < K" and norm_prod: "\<And>a b. norm (a ** b) \<le> norm a * norm b * K"
by fast
let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
define Ng where "Ng = ?N g g'"
define Nf where "Nf = ?N f f'"

let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
let ?F = "at x within s"

show ?thesis
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
bounded_linear_compose [OF bounded_linear_right] bounded_linear_compose [OF bounded_linear_left]
has_derivative_bounded_linear [OF g] has_derivative_bounded_linear [OF f])
next
from g have "(g \<longlongrightarrow> g x) ?F"
by (intro continuous_within[THEN iffD1] has_derivative_continuous)
moreover from f g have "(Nf \<longlongrightarrow> 0) ?F" "(Ng \<longlongrightarrow> 0) ?F"
by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
ultimately have "(?fun2 \<longlongrightarrow> norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
then show "(?fun2 \<longlongrightarrow> 0) ?F"
by simp
next
fix y :: 'd
assume "y \<noteq> x"
have "?fun1 y =
norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
by (intro divide_right_mono mult_mono'
order_trans [OF norm_prod mult_right_mono]
mult_nonneg_nonneg order_refl norm_ge_zero norm_F
K [THEN order_less_imp_le])
also have "\<dots> = ?fun2 y"
finally show "?fun1 y \<le> ?fun2 y" .
qed simp
qed

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

lemma has_derivative_prod[simp, derivative_intros]:
fixes f :: "'i \<Rightarrow> 'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
shows "(\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within S)) \<Longrightarrow>
((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within S)"
proof (induct I rule: infinite_finite_induct)
case infinite
then show ?case by simp
next
case empty
then show ?case by simp
next
case (insert i I)
let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within S)"
using insert by (intro has_derivative_mult) auto
also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
using insert(1,2)
by (auto simp add: sum_distrib_left insert_Diff_if intro!: ext sum.cong)
finally show ?case
using insert by simp
qed

lemma has_derivative_power[simp, derivative_intros]:
fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
assumes f: "(f has_derivative f') (at x within S)"
shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within S)"
using has_derivative_prod[OF f, of "{..< n}"] by (simp add: prod_constant ac_simps)

lemma has_derivative_inverse':
fixes x :: "'a::real_normed_div_algebra"
assumes x: "x \<noteq> 0"
shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within S)"
(is "(_ has_derivative ?f) _")
proof (rule has_derivativeI_sandwich)
show "bounded_linear (\<lambda>h. - (inverse x * h * inverse x))"
by (simp add: bounded_linear_minus bounded_linear_mult_const bounded_linear_mult_right)
show "0 < norm x" using x by simp
have "(inverse \<longlongrightarrow> inverse x) (at x within S)"
using tendsto_inverse tendsto_ident_at x by auto
then show "((\<lambda>y. norm (inverse y - inverse x) * norm (inverse x)) \<longlongrightarrow> 0) (at x within S)"
by (simp add: LIM_zero_iff tendsto_mult_left_zero tendsto_norm_zero)
next
fix y :: 'a
assume h: "y \<noteq> x" "dist y x < norm x"
then have "y \<noteq> 0" by auto
have "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x)
= norm (- (inverse y * (y - x) * inverse x - inverse x * (y - x) * inverse x)) /
norm (y - x)"
by (simp add: \<open>y \<noteq> 0\<close> inverse_diff_inverse x)
also have "... = norm ((inverse y - inverse x) * (y - x) * inverse x) / norm (y - x)"
also have "\<dots> \<le> norm (inverse y - inverse x) * norm (y - x) * norm (inverse x) / norm (y - x)"
also have "\<dots> = norm (inverse y - inverse x) * norm (inverse x)"
by simp
finally show "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x) \<le>
norm (inverse y - inverse x) * norm (inverse x)" .
qed

lemma has_derivative_inverse[simp, derivative_intros]:
fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
assumes x:  "f x \<noteq> 0"
and f: "(f has_derivative f') (at x within S)"
shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x))))
(at x within S)"
using has_derivative_compose[OF f has_derivative_inverse', OF x] .

lemma has_derivative_divide[simp, derivative_intros]:
fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
assumes f: "(f has_derivative f') (at x within S)"
and g: "(g has_derivative g') (at x within S)"
assumes x: "g x \<noteq> 0"
shows "((\<lambda>x. f x / g x) has_derivative
(\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within S)"
using has_derivative_mult[OF f has_derivative_inverse[OF x g]]

lemma has_derivative_power_int':
fixes x :: "'a::real_normed_field"
assumes x: "x \<noteq> 0"
shows "((\<lambda>x. power_int x n) has_derivative (\<lambda>y. y * (of_int n * power_int x (n - 1)))) (at x within S)"
proof (cases n rule: int_cases4)
case (nonneg n)
thus ?thesis using x
by (cases "n = 0") (auto intro!: derivative_eq_intros simp: field_simps power_int_diff fun_eq_iff
simp flip: power_Suc)
next
case (neg n)
thus ?thesis using x
by (auto intro!: derivative_eq_intros simp: field_simps power_int_diff power_int_minus
qed

lemma has_derivative_power_int[simp, derivative_intros]:
fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
assumes x:  "f x \<noteq> 0"
and f: "(f has_derivative f') (at x within S)"
shows "((\<lambda>x. power_int (f x) n) has_derivative (\<lambda>h. f' h * (of_int n * power_int (f x) (n - 1))))
(at x within S)"
using has_derivative_compose[OF f has_derivative_power_int', OF x] .

text \<open>Conventional form requires mult-AC laws. Types real and complex only.\<close>

lemma has_derivative_divide'[derivative_intros]:
fixes f :: "_ \<Rightarrow> 'a::real_normed_field"
assumes f: "(f has_derivative f') (at x within S)"
and g: "(g has_derivative g') (at x within S)"
and x: "g x \<noteq> 0"
shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within S)"
proof -
have "f' h / g x - f x * (inverse (g x) * g' h * inverse (g x)) =
(f' h * g x - f x * g' h) / (g x * g x)" for h
then show ?thesis
using has_derivative_divide [OF f g] x
by simp
qed

subsection \<open>Uniqueness\<close>

text \<open>
This can not generally shown for \<^const>\<open>has_derivative\<close>, as we need to approach the point from
all directions. There is a proof in \<open>Analysis\<close> for \<open>euclidean_space\<close>.
\<close>

lemma has_derivative_at2: "(f has_derivative f') (at x) \<longleftrightarrow>
bounded_linear f' \<and> ((\<lambda>y. (1 / (norm(y - x))) *\<^sub>R (f y - (f x + f' (y - x)))) \<longlongrightarrow> 0) (at x)"
using has_derivative_within [of f f' x UNIV]
by simp

lemma has_derivative_zero_unique:
assumes "((\<lambda>x. 0) has_derivative F) (at x)"
shows "F = (\<lambda>h. 0)"
proof -
interpret F: bounded_linear F
using assms by (rule has_derivative_bounded_linear)
let ?r = "\<lambda>h. norm (F h) / norm h"
have *: "?r \<midarrow>0\<rightarrow> 0"
using assms unfolding has_derivative_at by simp
show "F = (\<lambda>h. 0)"
proof
show "F h = 0" for h
proof (rule ccontr)
assume **: "\<not> ?thesis"
then have h: "h \<noteq> 0"
with ** have "0 < ?r h"
by simp
from LIM_D [OF * this] obtain S
where S: "0 < S" and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < S \<Longrightarrow> ?r x < ?r h"
by auto
from dense [OF S] obtain t where t: "0 < t \<and> t < S" ..
let ?x = "scaleR (t / norm h) h"
have "?x \<noteq> 0" and "norm ?x < S"
using t h by simp_all
then have "?r ?x < ?r h"
by (rule r)
then show False
using t h by (simp add: F.scaleR)
qed
qed
qed

lemma has_derivative_unique:
assumes "(f has_derivative F) (at x)"
and "(f has_derivative F') (at x)"
shows "F = F'"
proof -
have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
using has_derivative_diff [OF assms] by simp
then have "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
by (rule has_derivative_zero_unique)
then show "F = F'"
unfolding fun_eq_iff right_minus_eq .
qed

lemma has_derivative_Uniq: "\<exists>\<^sub>\<le>\<^sub>1F. (f has_derivative F) (at x)"

subsection \<open>Differentiability predicate\<close>

definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
(infix "differentiable" 50)
where "f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"

lemma differentiable_subset:
"f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
unfolding differentiable_def by (blast intro: has_derivative_subset)

lemmas differentiable_within_subset = differentiable_subset

lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_ident)

lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_const)

lemma differentiable_in_compose:
"f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
(\<lambda>x. f (g x)) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_in_compose)

lemma differentiable_compose:
"f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
(\<lambda>x. f (g x)) differentiable (at x within s)"
by (blast intro: differentiable_in_compose differentiable_subset)

"f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_add)

lemma differentiable_sum[simp, derivative_intros]:
assumes "finite s" "\<forall>a\<in>s. (f a) differentiable net"
shows "(\<lambda>x. sum (\<lambda>a. f a x) s) differentiable net"
proof -
from bchoice[OF assms(2)[unfolded differentiable_def]]
show ?thesis
by (auto intro!: has_derivative_sum simp: differentiable_def)
qed

lemma differentiable_minus [simp, derivative_intros]:
"f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_minus)

lemma differentiable_diff [simp, derivative_intros]:
"f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_diff)

lemma differentiable_mult [simp, derivative_intros]:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
(\<lambda>x. f x * g x) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_mult)

lemma differentiable_inverse [simp, derivative_intros]:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
(\<lambda>x. inverse (f x)) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_inverse)

lemma differentiable_divide [simp, derivative_intros]:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
unfolding divide_inverse by simp

lemma differentiable_power [simp, derivative_intros]:
fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_power)

lemma differentiable_power_int [simp, derivative_intros]:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
(\<lambda>x. power_int (f x) n) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_power_int)

lemma differentiable_scaleR [simp, derivative_intros]:
"f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow>
(\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_scaleR)

lemma has_derivative_imp_has_field_derivative:
"(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
unfolding has_field_derivative_def
by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)

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

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

lemma has_field_derivative_at_within:
"(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
using DERIV_subset by blast

abbreviation (input)
DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where "DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"

abbreviation has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
(infix "(has'_real'_derivative)" 50)
where "(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"

lemma real_differentiable_def:
"f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
proof safe
assume "f differentiable at x within s"
then obtain f' where *: "(f has_derivative f') (at x within s)"
unfolding differentiable_def by auto
then obtain c where "f' = ((*) c)"
by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
unfolding has_field_derivative_def by auto
qed (auto simp: differentiable_def has_field_derivative_def)

lemma real_differentiableE [elim?]:
assumes f: "f differentiable (at x within s)"
obtains df where "(f has_real_derivative df) (at x within s)"
using assms by (auto simp: real_differentiable_def)

lemma has_field_derivative_iff:
"(f has_field_derivative D) (at x within S) \<longleftrightarrow>
((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"
proof -
have "((\<lambda>y. norm (f y - f x - D * (y - x)) / norm (y - x)) \<longlongrightarrow> 0) (at x within S)
= ((\<lambda>y. (f y - f x) / (y - x) - D) \<longlongrightarrow> 0) (at x within S)"
apply (subst tendsto_norm_zero_iff[symmetric], rule filterlim_cong)
apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
done
then show ?thesis
by (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right LIM_zero_iff)
qed

lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"
unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..

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

lemma DERIV_compose_FDERIV:
fixes f::"real\<Rightarrow>real"
assumes "DERIV f (g x) :> f'"
assumes "(g has_derivative g') (at x within s)"
shows "((\<lambda>x. f (g x)) has_derivative (\<lambda>x. g' x * f')) (at x within s)"
using assms has_derivative_compose[of g g' x s f "(*) f'"]
by (auto simp: has_field_derivative_def ac_simps)

subsection \<open>Vector derivative\<close>

lemma has_field_derivative_iff_has_vector_derivative:
"(f has_field_derivative y) F \<longleftrightarrow> (f has_vector_derivative y) F"
unfolding has_vector_derivative_def has_field_derivative_def real_scaleR_def mult_commute_abs ..

lemma has_field_derivative_subset:
"(f has_field_derivative y) (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow>
(f has_field_derivative y) (at x within t)"
unfolding has_field_derivative_def by (rule has_derivative_subset)

lemma has_vector_derivative_const[simp, derivative_intros]: "((\<lambda>x. c) has_vector_derivative 0) net"
by (auto simp: has_vector_derivative_def)

lemma has_vector_derivative_id[simp, derivative_intros]: "((\<lambda>x. x) has_vector_derivative 1) net"
by (auto simp: has_vector_derivative_def)

lemma has_vector_derivative_minus[derivative_intros]:
"(f has_vector_derivative f') net \<Longrightarrow> ((\<lambda>x. - f x) has_vector_derivative (- f')) net"
by (auto simp: has_vector_derivative_def)

"(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
((\<lambda>x. f x + g x) has_vector_derivative (f' + g')) net"
by (auto simp: has_vector_derivative_def scaleR_right_distrib)

lemma has_vector_derivative_sum[derivative_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i has_vector_derivative f' i) net) \<Longrightarrow>
((\<lambda>x. \<Sum>i\<in>I. f i x) has_vector_derivative (\<Sum>i\<in>I. f' i)) net"
by (auto simp: has_vector_derivative_def fun_eq_iff scaleR_sum_right intro!: derivative_eq_intros)

lemma has_vector_derivative_diff[derivative_intros]:
"(f has_vector_derivative f') net \<Longrightarrow> (g has_vector_derivative g') net \<Longrightarrow>
((\<lambda>x. f x - g x) has_vector_derivative (f' - g')) net"
by (auto simp: has_vector_derivative_def scaleR_diff_right)

"((\<lambda>t. g t + z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
apply (intro iffI)
apply (force dest: has_vector_derivative_diff [where g = "\<lambda>t. z", OF _ has_vector_derivative_const])
apply (force dest: has_vector_derivative_add [OF _ has_vector_derivative_const])
done

lemma has_vector_derivative_diff_const:
"((\<lambda>t. g t - z) has_vector_derivative f') net = ((\<lambda>t. g t) has_vector_derivative f') net"
using has_vector_derivative_add_const [where z = "-z"]
by simp

lemma (in bounded_linear) has_vector_derivative:
assumes "(g has_vector_derivative g') F"
shows "((\<lambda>x. f (g x)) has_vector_derivative f g') F"
using has_derivative[OF assms[unfolded has_vector_derivative_def]]

lemma (in bounded_bilinear) has_vector_derivative:
assumes "(f has_vector_derivative f') (at x within s)"
and "(g has_vector_derivative g') (at x within s)"
shows "((\<lambda>x. f x ** g x) has_vector_derivative (f x ** g' + f' ** g x)) (at x within s)"
using FDERIV[OF assms(1-2)[unfolded has_vector_derivative_def]]
by (simp add: has_vector_derivative_def scaleR_right scaleR_left scaleR_right_distrib)

lemma has_vector_derivative_scaleR[derivative_intros]:
"(f has_field_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
((\<lambda>x. f x *\<^sub>R g x) has_vector_derivative (f x *\<^sub>R g' + f' *\<^sub>R g x)) (at x within s)"
unfolding has_field_derivative_iff_has_vector_derivative
by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_scaleR])

lemma has_vector_derivative_mult[derivative_intros]:
"(f has_vector_derivative f') (at x within s) \<Longrightarrow> (g has_vector_derivative g') (at x within s) \<Longrightarrow>
((\<lambda>x. f x * g x) has_vector_derivative (f x * g' + f' * g x)) (at x within s)"
for f g :: "real \<Rightarrow> 'a::real_normed_algebra"
by (rule bounded_bilinear.has_vector_derivative[OF bounded_bilinear_mult])

lemma has_vector_derivative_of_real[derivative_intros]:
"(f has_field_derivative D) F \<Longrightarrow> ((\<lambda>x. of_real (f x)) has_vector_derivative (of_real D)) F"
by (rule bounded_linear.has_vector_derivative[OF bounded_linear_of_real])

lemma has_vector_derivative_real_field:
"(f has_field_derivative f') (at (of_real a)) \<Longrightarrow> ((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a within s)"
using has_derivative_compose[of of_real of_real a _ f "(*) f'"]
by (simp add: scaleR_conv_of_real ac_simps has_vector_derivative_def has_field_derivative_def)

lemma has_vector_derivative_continuous:
"(f has_vector_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
by (auto intro: has_derivative_continuous simp: has_vector_derivative_def)

lemma continuous_on_vector_derivative:
"(\<And>x. x \<in> S \<Longrightarrow> (f has_vector_derivative f' x) (at x within S)) \<Longrightarrow> continuous_on S f"
by (auto simp: continuous_on_eq_continuous_within intro!: has_vector_derivative_continuous)

lemma has_vector_derivative_mult_right[derivative_intros]:
fixes a :: "'a::real_normed_algebra"
shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. a * f x) has_vector_derivative (a * x)) F"
by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_right])

lemma has_vector_derivative_mult_left[derivative_intros]:
fixes a :: "'a::real_normed_algebra"
shows "(f has_vector_derivative x) F \<Longrightarrow> ((\<lambda>x. f x * a) has_vector_derivative (x * a)) F"
by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left])

subsection \<open>Derivatives\<close>

lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow> D"

lemma has_field_derivativeD:
"(f has_field_derivative D) (at x within S) \<Longrightarrow>
((\<lambda>y. (f y - f x) / (y - x)) \<longlongrightarrow> D) (at x within S)"

lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto

lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto

"(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow>
((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
(auto simp: has_field_derivative_def field_simps mult_commute_abs)

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

lemma field_differentiable_minus[derivative_intros]:
"(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
(auto simp: has_field_derivative_def field_simps mult_commute_abs)

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

lemma field_differentiable_diff[derivative_intros]:
"(f has_field_derivative f') F \<Longrightarrow>
(g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"

corollary DERIV_diff:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
(g has_field_derivative E) (at x within s) \<Longrightarrow>
((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
by (rule field_differentiable_diff)

lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp

corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
by (rule DERIV_continuous)

lemma DERIV_atLeastAtMost_imp_continuous_on:
assumes "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y"
shows "continuous_on {a..b} f"
by (meson DERIV_isCont assms atLeastAtMost_iff continuous_at_imp_continuous_at_within continuous_on_eq_continuous_within)

lemma DERIV_continuous_on:
"(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative (D x)) (at x within s)) \<Longrightarrow> continuous_on s f"
unfolding continuous_on_eq_continuous_within
by (intro continuous_at_imp_continuous_on ballI DERIV_continuous)

lemma DERIV_mult':
"(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
(auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)

lemma DERIV_mult[derivative_intros]:
"(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
(auto simp: field_simps dest: has_field_derivative_imp_has_derivative)

text \<open>Derivative of linear multiplication\<close>

lemma DERIV_cmult:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
by (drule DERIV_mult' [OF DERIV_const]) simp

lemma DERIV_cmult_right:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
using DERIV_cmult by (auto simp add: ac_simps)

lemma DERIV_cmult_Id [simp]: "((*) c has_field_derivative c) (at x within s)"
using DERIV_ident [THEN DERIV_cmult, where c = c and x = x] by simp

lemma DERIV_cdivide:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
using DERIV_cmult_right[of f D x s "1 / c"] by simp

lemma DERIV_unique: "DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
unfolding DERIV_def by (rule LIM_unique)

lemma DERIV_Uniq: "\<exists>\<^sub>\<le>\<^sub>1D. DERIV f x :> D"

lemma DERIV_sum[derivative_intros]:
"(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow>
((\<lambda>x. sum (f x) S) has_field_derivative sum (f' x) S) F"
by (rule has_derivative_imp_has_field_derivative [OF has_derivative_sum])
(auto simp: sum_distrib_left mult_commute_abs dest: has_field_derivative_imp_has_derivative)

lemma DERIV_inverse'[derivative_intros]:
assumes "(f has_field_derivative D) (at x within s)"
and "f x \<noteq> 0"
shows "((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))
(at x within s)"
proof -
have "(f has_derivative (\<lambda>x. x * D)) = (f has_derivative (*) D)"
by (rule arg_cong [of "\<lambda>x. x * D"]) (simp add: fun_eq_iff)
with assms have "(f has_derivative (\<lambda>x. x * D)) (at x within s)"
by (auto dest!: has_field_derivative_imp_has_derivative)
then show ?thesis using \<open>f x \<noteq> 0\<close>
by (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse)
qed

text \<open>Power of \<open>-1\<close>\<close>

lemma DERIV_inverse:
"x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
by (drule DERIV_inverse' [OF DERIV_ident]) simp

text \<open>Derivative of inverse\<close>

lemma DERIV_inverse_fun:
"(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)

text \<open>Derivative of quotient\<close>

lemma DERIV_divide[derivative_intros]:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
(g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
(auto dest: has_field_derivative_imp_has_derivative simp: field_simps)

lemma DERIV_quotient:
"(f has_field_derivative d) (at x within s) \<Longrightarrow>
(g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
by (drule (2) DERIV_divide) (simp add: mult.commute)

lemma DERIV_power_Suc:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
(auto simp: has_field_derivative_def)

lemma DERIV_power[derivative_intros]:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
(auto simp: has_field_derivative_def)

lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
using DERIV_power [OF DERIV_ident] by simp

lemma DERIV_power_int [derivative_intros]:
assumes [derivative_intros]: "(f has_field_derivative d) (at x within s)" and [simp]: "f x \<noteq> 0"
shows   "((\<lambda>x. power_int (f x) n) has_field_derivative
(of_int n * power_int (f x) (n - 1) * d)) (at x within s)"
proof (cases n rule: int_cases4)
case (nonneg n)
thus ?thesis
by (cases "n = 0")
(auto intro!: derivative_eq_intros simp: field_simps power_int_diff
next
case (neg n)
thus ?thesis
by (auto intro!: derivative_eq_intros simp: field_simps power_int_diff power_int_minus
qed

lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow>
((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
using has_derivative_compose[of f "(*) D" x s g "(*) E"]
by (simp only: has_field_derivative_def mult_commute_abs ac_simps)

corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
by (rule DERIV_chain')

text \<open>Standard version\<close>

lemma DERIV_chain:
"DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
(f \<circ> g has_field_derivative Da * Db) (at x within s)"
by (drule (1) DERIV_chain', simp add: o_def mult.commute)

lemma DERIV_image_chain:
"(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow>
(g has_field_derivative Db) (at x within s) \<Longrightarrow>
(f \<circ> g has_field_derivative Da * Db) (at x within s)"
using has_derivative_in_compose [of g "(*) Db" x s f "(*) Da "]
by (simp add: has_field_derivative_def o_def mult_commute_abs ac_simps)

(*These two are from HOL Light: HAS_COMPLEX_DERIVATIVE_CHAIN*)
lemma DERIV_chain_s:
assumes "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
and "DERIV f x :> f'"
and "f x \<in> s"
shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
by (metis (full_types) DERIV_chain' mult.commute assms)

lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
assumes "(\<And>x. DERIV g x :> g'(x))"
and "DERIV f x :> f'"
shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
by (metis UNIV_I DERIV_chain_s [of UNIV] assms)

text \<open>Alternative definition for differentiability\<close>

lemma DERIV_LIM_iff:
fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a"
shows "((\<lambda>h. (f (a + h) - f a) / h) \<midarrow>0\<rightarrow> D) = ((\<lambda>x. (f x - f a) / (x - a)) \<midarrow>a\<rightarrow> D)" (is "?lhs = ?rhs")
proof
assume ?lhs
then have "(\<lambda>x. (f (a + (x + - a)) - f a) / (x + - a)) \<midarrow>0 - - a\<rightarrow> D"
by (rule LIM_offset)
then show ?rhs
by simp
next
assume ?rhs
then have "(\<lambda>x. (f (x+a) - f a) / ((x+a) - a)) \<midarrow>a-a\<rightarrow> D"
by (rule LIM_offset)
then show ?lhs
qed

lemma has_field_derivative_cong_ev:
assumes "x = y"
and *: "eventually (\<lambda>x. x \<in> S \<longrightarrow> f x = g x) (nhds x)"
and "u = v" "S = t" "x \<in> S"
shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative v) (at y within t)"
unfolding has_field_derivative_iff
proof (rule filterlim_cong)
from assms have "f y = g y"
by (auto simp: eventually_nhds)
with * show "\<forall>\<^sub>F z in at x within S. (f z - f x) / (z - x) = (g z - g y) / (z - y)"
unfolding eventually_at_filter
by eventually_elim (auto simp: assms \<open>f y = g y\<close>)

lemma has_field_derivative_cong_eventually:
assumes "eventually (\<lambda>x. f x = g x) (at x within S)" "f x = g x"
shows "(f has_field_derivative u) (at x within S) = (g has_field_derivative u) (at x within S)"
unfolding has_field_derivative_iff
proof (rule tendsto_cong)
show "\<forall>\<^sub>F y in at x within S. (f y - f x) / (y - x) = (g y - g x) / (y - x)"
using assms by (auto elim: eventually_mono)
qed

lemma DERIV_cong_ev:
"x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
by (rule has_field_derivative_cong_ev) simp_all

lemma DERIV_shift:
"(f has_field_derivative y) (at (x + z)) = ((\<lambda>x. f (x + z)) has_field_derivative y) (at x)"

lemma DERIV_mirror: "(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x)) x :> - y)"
for f :: "real \<Rightarrow> real" and x y :: real
by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
tendsto_minus_cancel_left field_simps conj_commute)

lemma floor_has_real_derivative:
fixes f :: "real \<Rightarrow> 'a::{floor_ceiling,order_topology}"
assumes "isCont f x"
and "f x \<notin> \<int>"
shows "((\<lambda>x. floor (f x)) has_real_derivative 0) (at x)"
proof (subst DERIV_cong_ev[OF refl _ refl])
show "((\<lambda>_. floor (f x)) has_real_derivative 0) (at x)"
by simp
have "\<forall>\<^sub>F y in at x. \<lfloor>f y\<rfloor> = \<lfloor>f x\<rfloor>"
by (rule eventually_floor_eq[OF assms[unfolded continuous_at]])
then show "\<forall>\<^sub>F y in nhds x. real_of_int \<lfloor>f y\<rfloor> = real_of_int \<lfloor>f x\<rfloor>"
unfolding eventually_at_filter
by eventually_elim auto
qed

lemmas has_derivative_floor[derivative_intros] =
floor_has_real_derivative[THEN DERIV_compose_FDERIV]

lemma continuous_floor:
fixes x::real
shows "x \<notin> \<int> \<Longrightarrow> continuous (at x) (real_of_int \<circ> floor)"
using floor_has_real_derivative [where f=id]
by (auto simp: o_def has_field_derivative_def intro: has_derivative_continuous)

lemma continuous_frac:
fixes x::real
assumes "x \<notin> \<int>"
shows "continuous (at x) frac"
proof -
have "isCont (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x"
using continuous_floor [OF assms] by (simp add: o_def)
then have *: "continuous (at x) (\<lambda>x. x - real_of_int \<lfloor>x\<rfloor>)"
by (intro continuous_intros)
moreover have "\<forall>\<^sub>F x in nhds x. frac x = x - real_of_int \<lfloor>x\<rfloor>"
ultimately show ?thesis
by (simp add: LIM_imp_LIM frac_def isCont_def)
qed

text \<open>Caratheodory formulation of derivative at a point\<close>

lemma CARAT_DERIV:
"(DERIV f x :> l) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
(is "?lhs = ?rhs")
proof
assume ?lhs
show "\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l"
proof (intro exI conjI)
let ?g = "(\<lambda>z. if z = x then l else (f z - f x) / (z-x))"
show "\<forall>z. f z - f x = ?g z * (z - x)"
by simp
show "isCont ?g x"
using \<open>?lhs\<close> by (simp add: isCont_iff DERIV_def cong: LIM_equal [rule_format])
show "?g x = l"
by simp
qed
next
assume ?rhs
then show ?lhs
by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
qed

subsection \<open>Local extrema\<close>

text \<open>If \<^term>\<open>0 < f' x\<close> then \<^term>\<open>x\<close> is Locally Strictly Increasing At The Right.\<close>

lemma has_real_derivative_pos_inc_right:
fixes f :: "real \<Rightarrow> real"
assumes der: "(f has_real_derivative l) (at x within S)"
and l: "0 < l"
shows "\<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x + h)"
using assms
proof -
from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
obtain s where s: "0 < s"
and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < l"
by (auto simp: dist_real_def)
then show ?thesis
proof (intro exI conjI strip)
show "0 < s" by (rule s)
next
fix h :: real
assume "0 < h" "h < s" "x + h \<in> S"
with all [of "x + h"] show "f x < f (x+h)"
proof (simp add: abs_if dist_real_def pos_less_divide_eq split: if_split_asm)
assume "\<not> (f (x + h) - f x) / h < l" and h: "0 < h"
with l have "0 < (f (x + h) - f x) / h"
by arith
then show "f x < f (x + h)"
qed
qed
qed

lemma DERIV_pos_inc_right:
fixes f :: "real \<Rightarrow> real"
assumes der: "DERIV f x :> l"
and l: "0 < l"
shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x + h)"
using has_real_derivative_pos_inc_right[OF assms]
by auto

lemma has_real_derivative_neg_dec_left:
fixes f :: "real \<Rightarrow> real"
assumes der: "(f has_real_derivative l) (at x within S)"
and "l < 0"
shows "\<exists>d > 0. \<forall>h > 0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f x < f (x - h)"
proof -
from \<open>l < 0\<close> have l: "- l > 0"
by simp
from der [THEN has_field_derivativeD, THEN tendstoD, OF l, unfolded eventually_at]
obtain s where s: "0 < s"
and all: "\<And>xa. xa\<in>S \<Longrightarrow> xa \<noteq> x \<and> dist xa x < s \<longrightarrow> \<bar>(f xa - f x) / (xa - x) - l\<bar> < - l"
by (auto simp: dist_real_def)
then show ?thesis
proof (intro exI conjI strip)
show "0 < s" by (rule s)
next
fix h :: real
assume "0 < h" "h < s" "x - h \<in> S"
with all [of "x - h"] show "f x < f (x-h)"
proof (simp add: abs_if pos_less_divide_eq dist_real_def split: if_split_asm)
assume "- ((f (x-h) - f x) / h) < l" and h: "0 < h"
with l have "0 < (f (x-h) - f x) / h"
by arith
then show "f x < f (x - h)"
qed
qed
qed

lemma DERIV_neg_dec_left:
fixes f :: "real \<Rightarrow> real"
assumes der: "DERIV f x :> l"
and l: "l < 0"
shows "\<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x < f (x - h)"
using has_real_derivative_neg_dec_left[OF assms]
by auto

lemma has_real_derivative_pos_inc_left:
fixes f :: "real \<Rightarrow> real"
shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> 0 < l \<Longrightarrow>
\<exists>d>0. \<forall>h>0. x - h \<in> S \<longrightarrow> h < d \<longrightarrow> f (x - h) < f x"
by (rule has_real_derivative_neg_dec_left [of "\<lambda>x. - f x" "-l" x S, simplified])

lemma DERIV_pos_inc_left:
fixes f :: "real \<Rightarrow> real"
shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f (x - h) < f x"
using has_real_derivative_pos_inc_left
by blast

lemma has_real_derivative_neg_dec_right:
fixes f :: "real \<Rightarrow> real"
shows "(f has_real_derivative l) (at x within S) \<Longrightarrow> l < 0 \<Longrightarrow>
\<exists>d > 0. \<forall>h > 0. x + h \<in> S \<longrightarrow> h < d \<longrightarrow> f x > f (x + h)"
by (rule has_real_derivative_pos_inc_right [of "\<lambda>x. - f x" "-l" x S, simplified])

lemma DERIV_neg_dec_right:
fixes f :: "real \<Rightarrow> real"
shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d \<longrightarrow> f x > f (x + h)"
using has_real_derivative_neg_dec_right by blast

lemma DERIV_local_max:
fixes f :: "real \<Rightarrow> real"
assumes der: "DERIV f x :> l"
and d: "0 < d"
and le: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
shows "l = 0"
proof (cases rule: linorder_cases [of l 0])
case equal
then show ?thesis .
next
case less
from DERIV_neg_dec_left [OF der less]
obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x - h)"
by blast
obtain e where "0 < e \<and> e < d \<and> e < d'"
using field_lbound_gt_zero [OF d d']  ..
with lt le [THEN spec [where x="x - e"]] show ?thesis
next
case greater
from DERIV_pos_inc_right [OF der greater]
obtain d' where d': "0 < d'" and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)"
by blast
obtain e where "0 < e \<and> e < d \<and> e < d'"
using field_lbound_gt_zero [OF d d'] ..
with lt le [THEN spec [where x="x + e"]] show ?thesis
qed

text \<open>Similar theorem for a local minimum\<close>
lemma DERIV_local_min:
fixes f :: "real \<Rightarrow> real"
shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x \<le> f y \<Longrightarrow> l = 0"
by (drule DERIV_minus [THEN DERIV_local_max]) auto

text\<open>In particular, if a function is locally flat\<close>
lemma DERIV_local_const:
fixes f :: "real \<Rightarrow> real"
shows "DERIV f x :> l \<Longrightarrow> 0 < d \<Longrightarrow> \<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f x = f y \<Longrightarrow> l = 0"
by (auto dest!: DERIV_local_max)

subsection \<open>Rolle's Theorem\<close>

text \<open>Lemma about introducing open ball in open interval\<close>
lemma lemma_interval_lt:
fixes a b x :: real
assumes "a < x" "x < b"
shows "\<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a < y \<and> y < b)"
using linorder_linear [of "x - a" "b - x"]
proof
assume "x - a \<le> b - x"
with assms show ?thesis
by (rule_tac x = "x - a" in exI) auto
next
assume "b - x \<le> x - a"
with assms show ?thesis
by (rule_tac x = "b - x" in exI) auto
qed

lemma lemma_interval: "a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>d. 0 < d \<and> (\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b)"
for a b x :: real
by (force dest: lemma_interval_lt)

text \<open>Rolle's Theorem.
If \<^term>\<open>f\<close> is defined and continuous on the closed interval
\<open>[a,b]\<close> and differentiable on the open interval \<open>(a,b)\<close>,
and \<^term>\<open>f a = f b\<close>,
then there exists \<open>x0 \<in> (a,b)\<close> such that \<^term>\<open>f' x0 = 0\<close>\<close>
theorem Rolle_deriv:
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>z. a < z \<and> z < b \<and> f' z = (\<lambda>v. 0)"
proof -
have le: "a \<le> b"
using \<open>a < b\<close> by simp
have "(a + b) / 2 \<in> {a..b}"
using assms(1) by auto
then have *: "{a..b} \<noteq> {}"
by auto
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x" and "a \<le> x" "x \<le> b"
using continuous_attains_sup[OF compact_Icc * contf]
by (meson atLeastAtMost_iff)
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z" and "a \<le> x'" "x' \<le> b"
using continuous_attains_inf[OF compact_Icc * contf] by (meson atLeastAtMost_iff)
consider "a < x" "x < b" | "x = a \<or> x = b"
using \<open>a \<le> x\<close> \<open>x \<le> b\<close> by arith
then show ?thesis
proof cases
case 1
\<comment> \<open>\<^term>\<open>f\<close> attains its maximum within the interval\<close>
then obtain l where der: "DERIV f x :> l"
using derf differentiable_def real_differentiable_def by blast
obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
using lemma_interval [OF 1] by blast
then have bound': "\<forall>y. \<bar>x - y\<bar> < d \<longrightarrow> f y \<le> f x"
using x_max by blast
\<comment> \<open>the derivative at a local maximum is zero\<close>
have "l = 0"
by (rule DERIV_local_max [OF der d bound'])
with 1 der derf [of x] show ?thesis
by (metis has_derivative_unique has_field_derivative_def mult_zero_left)
next
case 2
then have fx: "f b = f x" by (auto simp add: fab)
consider "a < x'" "x' < b" | "x' = a \<or> x' = b"
using \<open>a \<le> x'\<close> \<open>x' \<le> b\<close> by arith
then show ?thesis
proof cases
case 1
\<comment> \<open>\<^term>\<open>f\<close> attains its minimum within the interval\<close>
then obtain l where der: "DERIV f x' :> l"
using derf differentiable_def real_differentiable_def by blast
from lemma_interval [OF 1]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
then have bound': "\<forall>y. \<bar>x' - y\<bar> < d \<longrightarrow> f x' \<le> f y"
using x'_min by blast
have "l = 0" by (rule DERIV_local_min [OF der d bound'])
\<comment> \<open>the derivative at a local minimum is zero\<close>
then show ?thesis using 1 der derf [of x']
by (metis has_derivative_unique has_field_derivative_def mult_zero_left)
next
case 2
\<comment> \<open>\<^term>\<open>f\<close> is constant throughout the interval\<close>
then have fx': "f b = f x'" by (auto simp: fab)
from dense [OF \<open>a < b\<close>] obtain r where r: "a < r" "r < b" by blast
obtain d where d: "0 < d" and bound: "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
using lemma_interval [OF r] by blast
have eq_fb: "f z = f b" if "a \<le> z" and "z \<le> b" for z
proof (rule order_antisym)
show "f z \<le> f b" by (simp add: fx x_max that)
show "f b \<le> f z" by (simp add: fx' x'_min that)
qed
have bound': "\<forall>y. \<bar>r - y\<bar> < d \<longrightarrow> f r = f y"
proof (intro strip)
fix y :: real
assume lt: "\<bar>r - y\<bar> < d"
then have "f y = f b" by (simp add: eq_fb bound)
then show "f r = f y" by (simp add: eq_fb r order_less_imp_le)
qed
obtain l where der: "DERIV f r :> l"
using derf differentiable_def r(1) r(2) real_differentiable_def by blast
have "l = 0"
by (rule DERIV_local_const [OF der d bound'])
\<comment> \<open>the derivative of a constant function is zero\<close>
with r der derf [of r] show ?thesis
by (metis has_derivative_unique has_field_derivative_def mult_zero_left)
qed
qed
qed

corollary Rolle:
fixes a b :: real
assumes ab: "a < b" "f a = f b" "continuous_on {a..b} f"
and dif [rule_format]: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable (at x)"
shows "\<exists>z. a < z \<and> z < b \<and> DERIV f z :> 0"
proof -
obtain f' where f': "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x)"
using dif unfolding differentiable_def by metis
then have "\<exists>z. a < z \<and> z < b \<and> f' z = (\<lambda>v. 0)"
by (metis Rolle_deriv [OF ab])
then show ?thesis
using f' has_derivative_imp_has_field_derivative by fastforce
qed

subsection \<open>Mean Value Theorem\<close>

theorem 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)"
obtains \<xi> where "a < \<xi>" "\<xi> < b" "f b - f a = (f' \<xi>) (b - a)"
proof -
have "\<exists>x. a < x \<and> x < b \<and> (\<lambda>y. f' x y - (f b - f a) / (b - a) * y) = (\<lambda>v. 0)"
proof (intro Rolle_deriv[OF \<open>a < b\<close>])
fix x
assume x: "a < x" "x < b"
show "((\<lambda>x. f x - (f b - f a) / (b - a) * x)
has_derivative (\<lambda>y. f' x y - (f b - f a) / (b - a) * y)) (at x)"
by (intro derivative_intros derf[OF x])
qed (use assms in \<open>auto intro!: continuous_intros simp: field_simps\<close>)
then obtain \<xi> where
"a < \<xi>" "\<xi> < b" "(\<lambda>y. f' \<xi> y - (f b - f a) / (b - a) * y) = (\<lambda>v. 0)"
by metis
then show ?thesis
less_irrefl nonzero_eq_divide_eq)
qed

theorem MVT:
fixes a b :: real
assumes lt: "a < b"
and contf: "continuous_on {a..b} f"
and dif: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable (at x)"
shows "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
proof -
obtain f' :: "real \<Rightarrow> real \<Rightarrow> real"
where derf: "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (f has_derivative f' x) (at x)"
using dif unfolding differentiable_def by metis
then obtain z where "a < z" "z < b" "f b - f a = (f' z) (b - a)"
using mvt [OF lt contf] by blast
then show ?thesis
(metis derf dif has_derivative_unique has_field_derivative_imp_has_derivative real_differentiable_def)
qed

corollary MVT2:
assumes "a < b" and der: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> DERIV f x :> f' x"
shows "\<exists>z::real. a < z \<and> z < b \<and> (f b - f a = (b - a) * f' z)"
proof -
have "\<exists>l z. a < z \<and>
z < b \<and>
(f has_real_derivative l) (at z) \<and>
f b - f a = (b - a) * l"
proof (rule MVT [OF \<open>a < b\<close>])
show "continuous_on {a..b} f"
by (meson DERIV_continuous atLeastAtMost_iff continuous_at_imp_continuous_on der)
show "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> f differentiable (at x)"
using assms by (force dest: order_less_imp_le simp add: real_differentiable_def)
qed
with assms show ?thesis
by (blast dest: DERIV_unique order_less_imp_le)
qed

lemma pos_deriv_imp_strict_mono:
assumes "\<And>x. (f has_real_derivative f' x) (at x)"
assumes "\<And>x. f' x > 0"
shows   "strict_mono f"
proof (rule strict_monoI)
fix x y :: real assume xy: "x < y"
from assms and xy have "\<exists>z>x. z < y \<and> f y - f x = (y - x) * f' z"
by (intro MVT2) (auto dest: connectedD_interval)
then obtain z where z: "z > x" "z < y" "f y - f x = (y - x) * f' z" by blast
note \<open>f y - f x = (y - x) * f' z\<close>
also have "(y - x) * f' z > 0" using xy assms by (intro mult_pos_pos) auto
finally show "f x < f y" by simp
qed

proposition  deriv_nonneg_imp_mono:
assumes deriv: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_real_derivative g' x) (at x)"
assumes nonneg: "\<And>x. x \<in> {a..b} \<Longrightarrow> g' x \<ge> 0"
assumes ab: "a \<le> b"
shows "g a \<le> g b"
proof (cases "a < b")
assume "a < b"
from deriv have "\<And>x. \<lbrakk>x \<ge> a; x \<le> b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x)" by simp
with MVT2[OF \<open>a < b\<close>] and deriv
obtain \<xi> where \<xi>_ab: "\<xi> > a" "\<xi> < b" and g_ab: "g b - g a = (b - a) * g' \<xi>" by blast
from \<xi>_ab ab nonneg have "(b - a) * g' \<xi> \<ge> 0" by simp
with g_ab show ?thesis by simp
qed (insert ab, simp)

subsubsection \<open>A function is constant if its derivative is 0 over an interval.\<close>

lemma DERIV_isconst_end:
fixes f :: "real \<Rightarrow> real"
assumes "a < b" and contf: "continuous_on {a..b} f"
and 0: "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> DERIV f x :> 0"
shows "f b = f a"
using MVT [OF \<open>a < b\<close>] "0" DERIV_unique contf real_differentiable_def
by (fastforce simp: algebra_simps)

lemma DERIV_isconst2:
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> DERIV f x :> 0"
and "a \<le> x" "x \<le> b"
shows "f x = f a"
proof (cases "a < x")
case True
have *: "continuous_on {a..x} f"
using \<open>x \<le> b\<close> contf continuous_on_subset by fastforce
show ?thesis
by (rule DERIV_isconst_end [OF True *]) (use \<open>x \<le> b\<close> derf in auto)
qed (use \<open>a \<le> x\<close> in auto)

lemma DERIV_isconst3:
fixes a b x y :: real
assumes "a < b"
and "x \<in> {a <..< b}"
and "y \<in> {a <..< b}"
and derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
shows "f x = f y"
proof (cases "x = y")
case False
let ?a = "min x y"
let ?b = "max x y"
have *: "DERIV f z :> 0" if "?a \<le> z" "z \<le> ?b" for z
proof -
have "a < z" and "z < b"
using that \<open>x \<in> {a <..< b}\<close> and \<open>y \<in> {a <..< b}\<close> by auto
then have "z \<in> {a<..<b}" by auto
then show "DERIV f z :> 0" by (rule derivable)
qed
have isCont: "continuous_on {?a..?b} f"
by (meson * DERIV_continuous_on atLeastAtMost_iff has_field_derivative_at_within)
have DERIV: "\<And>z. \<lbrakk>?a < z; z < ?b\<rbrakk> \<Longrightarrow> DERIV f z :> 0"
using * by auto
have "?a < ?b" using \<open>x \<noteq> y\<close> by auto
from DERIV_isconst2[OF this isCont DERIV, of x] and DERIV_isconst2[OF this isCont DERIV, of y]
show ?thesis by auto
qed auto

lemma DERIV_isconst_all:
fixes f :: "real \<Rightarrow> real"
shows "\<forall>x. DERIV f x :> 0 \<Longrightarrow> f x = f y"
apply (rule linorder_cases [of x y])
apply (metis DERIV_continuous DERIV_isconst_end continuous_at_imp_continuous_on)+
done

lemma DERIV_const_ratio_const:
fixes f :: "real \<Rightarrow> real"
assumes "a \<noteq> b" and df: "\<And>x. DERIV f x :> k"
shows "f b - f a = (b - a) * k"
proof (cases a b rule: linorder_cases)
case less
show ?thesis
using MVT [OF less] df
by (metis DERIV_continuous DERIV_unique continuous_at_imp_continuous_on real_differentiable_def)
next
case greater
have  "f a - f b = (a - b) * k"
using MVT [OF greater] df
by (metis DERIV_continuous DERIV_unique continuous_at_imp_continuous_on real_differentiable_def)
then show ?thesis
qed auto

lemma DERIV_const_ratio_const2:
fixes f :: "real \<Rightarrow> real"
assumes "a \<noteq> b" and df: "\<And>x. DERIV f x :> k"
shows "(f b - f a) / (b - a) = k"
using DERIV_const_ratio_const [OF assms] \<open>a \<noteq> b\<close> by auto

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

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

text \<open>Gallileo's "trick": average velocity = av. of end velocities.\<close>

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

subsubsection\<open>A function with positive derivative is increasing\<close>
text \<open>A simple proof using the MVT, by Jeremy Avigad. And variants.\<close>
lemma DERIV_pos_imp_increasing_open:
fixes a b :: real
and f :: "real \<Rightarrow> real"
assumes "a < b"
and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
and con: "continuous_on {a..b} f"
shows "f a < f b"
proof (rule ccontr)
assume f: "\<not> ?thesis"
have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
by (rule MVT) (use assms real_differentiable_def in \<open>force+\<close>)
then obtain l z where z: "a < z" "z < b" "DERIV f z :> l" and "f b - f a = (b - a) * l"
by auto
with assms f have "\<not> l > 0"
by (metis linorder_not_le mult_le_0_iff diff_le_0_iff_le)
with assms z show False
by (metis DERIV_unique)
qed

lemma DERIV_pos_imp_increasing:
fixes a b :: real and f :: "real \<Rightarrow> real"
assumes "a < b"
and der: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y > 0"
shows "f a < f b"
by (metis less_le_not_le DERIV_atLeastAtMost_imp_continuous_on DERIV_pos_imp_increasing_open [OF \<open>a < b\<close>] der)

lemma DERIV_nonneg_imp_nondecreasing:
fixes a b :: real
and f :: "real \<Rightarrow> real"
assumes "a \<le> b"
and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y \<ge> 0"
shows "f a \<le> f b"
proof (rule ccontr, cases "a = b")
assume "\<not> ?thesis" and "a = b"
then show False by auto
next
assume *: "\<not> ?thesis"
assume "a \<noteq> b"
with \<open>a \<le> b\<close> have "a < b"
by linarith
moreover have "continuous_on {a..b} f"
by (meson DERIV_isCont assms(2) atLeastAtMost_iff continuous_at_imp_continuous_on)
ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV f z :> l \<and> f b - f a = (b - a) * l"
using assms MVT [OF \<open>a < b\<close>, of f] real_differentiable_def less_eq_real_def by blast
then obtain l z where lz: "a < z" "z < b" "DERIV f z :> l" and **: "f b - f a = (b - a) * l"
by auto
with * have "a < b" "f b < f a" by auto
with ** have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
with assms lz show False
by (metis DERIV_unique order_less_imp_le)
qed

lemma DERIV_neg_imp_decreasing_open:
fixes a b :: real
and f :: "real \<Rightarrow> real"
assumes "a < b"
and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y < 0"
and con: "continuous_on {a..b} f"
shows "f a > f b"
proof -
have "(\<lambda>x. -f x) a < (\<lambda>x. -f x) b"
proof (rule DERIV_pos_imp_increasing_open [of a b])
show "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> \<exists>y. ((\<lambda>x. - f x) has_real_derivative y) (at x) \<and> 0 < y"
using assms
by simp (metis field_differentiable_minus neg_0_less_iff_less)
show "continuous_on {a..b} (\<lambda>x. - f x)"
using con continuous_on_minus by blast
qed (use assms in auto)
then show ?thesis
by simp
qed

lemma DERIV_neg_imp_decreasing:
fixes a b :: real and f :: "real \<Rightarrow> real"
assumes "a < b"
and der: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y < 0"
shows "f a > f b"
by (metis less_le_not_le DERIV_atLeastAtMost_imp_continuous_on DERIV_neg_imp_decreasing_open [OF \<open>a < b\<close>] der)

lemma DERIV_nonpos_imp_nonincreasing:
fixes a b :: real
and f :: "real \<Rightarrow> real"
assumes "a \<le> b"
and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y \<le> 0"
shows "f a \<ge> f b"
proof -
have "(\<lambda>x. -f x) a \<le> (\<lambda>x. -f x) b"
using DERIV_nonneg_imp_nondecreasing [of a b "\<lambda>x. -f x"] assms DERIV_minus by fastforce
then show ?thesis
by simp
qed

lemma DERIV_pos_imp_increasing_at_bot:
fixes f :: "real \<Rightarrow> real"
assumes "\<And>x. x \<le> b \<Longrightarrow> (\<exists>y. DERIV f x :> y \<and> y > 0)"
and lim: "(f \<longlongrightarrow> flim) at_bot"
shows "flim < f b"
proof -
have "\<exists>N. \<forall>n\<le>N. f n \<le> f (b - 1)"
by (rule_tac x="b - 2" in exI) (force intro: order.strict_implies_order DERIV_pos_imp_increasing assms)
then have "flim \<le> f (b - 1)"
by (auto simp: eventually_at_bot_linorder tendsto_upperbound [OF lim])
also have "\<dots> < f b"
by (force intro: DERIV_pos_imp_increasing [where f=f] assms)
finally show ?thesis .
qed

lemma DERIV_neg_imp_decreasing_at_top:
fixes f :: "real \<Rightarrow> real"
assumes der: "\<And>x. x \<ge> b \<Longrightarrow> \<exists>y. DERIV f x :> y \<and> y < 0"
and lim: "(f \<longlongrightarrow> flim) at_top"
shows "flim < f b"
apply (rule DERIV_pos_imp_increasing_at_bot [where f = "\<lambda>i. f (-i)" and b = "-b", simplified])
apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
apply (metis filterlim_at_top_mirror lim)
done

text \<open>Derivative of inverse function\<close>

lemma DERIV_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes der: "DERIV f (g x) :> D"
and neq: "D \<noteq> 0"
and x: "a < x" "x < b"
and inj: "\<And>y. \<lbrakk>a < y; y < b\<rbrakk> \<Longrightarrow> f (g y) = y"
and cont: "isCont g x"
shows "DERIV g x :> inverse D"
unfolding has_field_derivative_iff
proof (rule LIM_equal2)
show "0 < min (x - a) (b - x)"
using x by arith
next
fix y
assume "norm (y - x) < min (x - a) (b - x)"
then have "a < y" and "y < b"
then show "(g y - g x) / (y - x) = inverse ((f (g y) - x) / (g y - g x))"
next
have "(\<lambda>z. (f z - f (g x)) / (z - g x)) \<midarrow>g x\<rightarrow> D"
by (rule der [unfolded has_field_derivative_iff])
then have 1: "(\<lambda>z. (f z - x) / (z - g x)) \<midarrow>g x\<rightarrow> D"
using inj x by simp
have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
proof (rule exI, safe)
show "0 < min (x - a) (b - x)"
using x by simp
next
fix y
assume "norm (y - x) < min (x - a) (b - x)"
then have y: "a < y" "y < b"
assume "g y = g x"
then have "f (g y) = f (g x)" by simp
then have "y = x" using inj y x by simp
also assume "y \<noteq> x"
finally show False by simp
qed
have "(\<lambda>y. (f (g y) - x) / (g y - g x)) \<midarrow>x\<rightarrow> D"
using cont 1 2 by (rule isCont_LIM_compose2)
then show "(\<lambda>y. inverse ((f (g y) - x) / (g y - g x))) \<midarrow>x\<rightarrow> inverse D"
using neq by (rule tendsto_inverse)
qed

subsection \<open>Generalized Mean Value Theorem\<close>

theorem GMVT:
fixes a b :: real
assumes alb: "a < b"
and fc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
shows "\<exists>g'c f'c c.
DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
proof -
let ?h = "\<lambda>x. (f b - f a) * g x - (g b - g a) * f x"
have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l"
proof (rule MVT)
from assms show "a < b" by simp
show "continuous_on {a..b} ?h"
by (simp add: continuous_at_imp_continuous_on fc gc)
show "\<And>x. \<lbrakk>a < x; x < b\<rbrakk> \<Longrightarrow> ?h differentiable (at x)"
using fd gd by simp
qed
then obtain l where l: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
then obtain c where c: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..

from c have cint: "a < c \<and> c < b" by auto
then obtain g'c where g'c: "DERIV g c :> g'c"
using gd real_differentiable_def by blast
from c have "a < c \<and> c < b" by auto
then obtain f'c where f'c: "DERIV f c :> f'c"
using fd real_differentiable_def by blast

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

have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))"
proof -
from c have "?h b - ?h a = (b - a) * l" by auto
also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
finally show ?thesis by simp
qed
moreover have "?h b - ?h a = 0"
proof -
have "?h b - ?h a =
((f b)*(g b) - (f a)*(g b) - (g b)*(f b) + (g a)*(f b)) -
((f b)*(g a) - (f a)*(g a) - (g b)*(f a) + (g a)*(f a))"
then show ?thesis  by auto
qed
ultimately have "(b - a) * (g'c * (f b - f a) - f'c * (g b - g a)) = 0" by auto
with alb have "g'c * (f b - f a) - f'c * (g b - g a) = 0" by simp
then have "g'c * (f b - f a) = f'c * (g b - g a)" by simp
then have "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
with g'c f'c cint show ?thesis by auto
qed

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

subsection \<open>L'Hopitals rule\<close>

lemma isCont_If_ge:
fixes a :: "'a :: linorder_topology"
assumes "continuous (at_left a) g" and f: "(f \<longlongrightarrow> g a) (at_right a)"
shows "isCont (\<lambda>x. if x \<le> a then g x else f x) a" (is "isCont ?gf a")
proof -
have g: "(g \<longlongrightarrow> g a) (at_left a)"
using assms continuous_within by blast
show ?thesis
unfolding isCont_def continuous_within
proof (intro filterlim_split_at; simp)
show "(?gf \<longlongrightarrow> g a) (at_left a)"
by (subst filterlim_cong[OF refl refl, where g=g]) (simp_all add: eventually_at_filter less_le g)
show "(?gf \<longlongrightarrow> g a) (at_right a)"
by (subst filterlim_cong[OF refl refl, where g=f]) (simp_all add: eventually_at_filter less_le f)
qed
qed

lemma lhopital_right_0:
fixes f0 g0 :: "real \<Rightarrow> real"
assumes f_0: "(f0 \<longlongrightarrow> 0) (at_right 0)"
and g_0: "(g0 \<longlongrightarrow> 0) (at_right 0)"
and ev:
"eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
"eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
"eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
"eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) F (at_right 0)"
shows "filterlim (\<lambda> x. f0 x / g0 x) F (at_right 0)"
proof -
define f where [abs_def]: "f x = (if x \<le> 0 then 0 else f0 x)" for x
then have "f 0 = 0" by simp

define g where [abs_def]: "g x = (if x \<le> 0 then 0 else g0 x)" for x
then have "g 0 = 0" by simp

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

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

have f: "DERIV f x :> (f' x)" if x: "0 < x" "x < a" for x
using that
by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ f0[OF x]])
(auto simp: f_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])

have g: "DERIV g x :> (g' x)" if x: "0 < x" "x < a" for x
using that
by (intro DERIV_cong_ev[THEN iffD1, OF _ _ _ g0[OF x]])
(auto simp: g_def eventually_nhds_metric dist_real_def intro!: exI[of _ x])

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

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

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

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

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

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

lemma lhopital_right_0_at_top:
fixes f g :: "real \<Rightarrow> real"
assumes g_0: "LIM x at_right 0. g x :> at_top"
and ev:
"eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
"eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) (at_right 0)"
shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) (at_right 0)"
unfolding tendsto_iff
proof safe
fix e :: real
assume "0 < e"
with lim[unfolded tendsto_iff, rule_format, of "e / 4"]
have "eventually (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)"
by simp
from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
obtain a where [arith]: "0 < a"
and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
unfolding eventually_at_le by (auto simp: dist_real_def)

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

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

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

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

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

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

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

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

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

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

lemma lhospital_at_top_at_top:
fixes f g :: "real \<Rightarrow> real"
assumes g_0: "LIM x at_top. g x :> at_top"
and g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
and Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
and Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
and lim: "((\<lambda> x. (f' x / g' x)) \<longlongrightarrow> x) at_top"
shows "((\<lambda> x. f x / g x) \<longlongrightarrow> x) at_top"
unfolding filterlim_at_top_to_right
proof (rule lhopital_right_0_at_top)
let ?F = "\<lambda>x. f (inverse x)"
let ?G = "\<lambda>x. g (inverse x)"
let ?R = "at_right (0::real)"
let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
show "LIM x ?R. ?G x :> at_top"
using g_0 unfolding filterlim_at_top_to_right .
show "eventually (\<lambda>x. DERIV ?G x  :> ?D g' x) ?R"
unfolding eventually_at_right_to_top
using Dg eventually_ge_at_top[where c=1]
by eventually_elim (rule derivative_eq_intros DERIV_chain'[where f=inverse] | simp)+
show "eventually (\<lambda>x. DERIV ?F x  :> ?D f' x) ?R"
unfolding eventually_at_right_to_top
using Df eventually_ge_at_top[where c=1]
by eventually_elim (rule derivative_eq_intros DERIV_chain'[where f=inverse] | simp)+
show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
unfolding eventually_at_right_to_top
using g' eventually_ge_at_top[where c=1]
by eventually_elim auto
show "((\<lambda>x. ?D f' x / ?D g' x) \<longlongrightarrow> x) ?R"
unfolding filterlim_at_right_to_top
apply (intro filterlim_cong[THEN iffD2, OF refl refl _ lim])
using eventually_ge_at_top[where c=1]
by eventually_elim simp
qed

lemma lhopital_right_at_top_at_top:
fixes f g :: "real \<Rightarrow> real"
assumes f_0: "LIM x at_right a. f x :> at_top"
assumes g_0: "LIM x at_right a. g x :> at_top"
and ev:
"eventually (\<lambda>x. DERIV f x :> f' x) (at_right a)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at_right a)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at_right a)"
shows "filterlim (\<lambda> x. f x / g x) at_top (at_right a)"
proof -
from lim have pos: "eventually (\<lambda>x. f' x / g' x > 0) (at_right a)"
unfolding filterlim_at_top_dense by blast
have "((\<lambda>x. g x / f x) \<longlongrightarrow> 0) (at_right a)"
proof (rule lhopital_right_at_top)
from pos show "eventually (\<lambda>x. f' x \<noteq> 0) (at_right a)" by eventually_elim auto
from tendsto_inverse_0_at_top[OF lim]
show "((\<lambda>x. g' x / f' x) \<longlongrightarrow> 0) (at_right a)" by simp
qed fact+
moreover from f_0 g_0
have "eventually (\<lambda>x. f x > 0) (at_right a)" "eventually (\<lambda>x. g x > 0) (at_right a)"
unfolding filterlim_at_top_dense by blast+
hence "eventually (\<lambda>x. g x / f x > 0) (at_right a)" by eventually_elim simp
ultimately have "filterlim (\<lambda>x. inverse (g x / f x)) at_top (at_right a)"
by (rule filterlim_inverse_at_top)
thus ?thesis by simp
qed

lemma lhopital_right_at_top_at_bot:
fixes f g :: "real \<Rightarrow> real"
assumes f_0: "LIM x at_right a. f x :> at_top"
assumes g_0: "LIM x at_right a. g x :> at_bot"
and ev:
"eventually (\<lambda>x. DERIV f x :> f' x) (at_right a)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at_right a)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at_right a)"
shows "filterlim (\<lambda> x. f x / g x) at_bot (at_right a)"
proof -
from ev(2) have ev': "eventually (\<lambda>x. DERIV (\<lambda>x. -g x) x :> -g' x) (at_right a)"
by eventually_elim (auto intro: derivative_intros)
have "filterlim (\<lambda>x. f x / (-g x)) at_top (at_right a)"
by (rule lhopital_right_at_top_at_top[where f' = f' and g' = "\<lambda>x. -g' x"])
(insert assms ev', auto simp: filterlim_uminus_at_bot)
hence "filterlim (\<lambda>x. -(f x / g x)) at_top (at_right a)" by simp
thus ?thesis by (simp add: filterlim_uminus_at_bot)
qed

lemma lhopital_left_at_top_at_top:
fixes f g :: "real \<Rightarrow> real"
assumes f_0: "LIM x at_left a. f x :> at_top"
assumes g_0: "LIM x at_left a. g x :> at_top"
and ev:
"eventually (\<lambda>x. DERIV f x :> f' x) (at_left a)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at_left a)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at_left a)"
shows "filterlim (\<lambda> x. f x / g x) at_top (at_left a)"
by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror,
rule lhopital_right_at_top_at_top[where f'="\<lambda>x. - f' (- x)"])
(insert assms, auto simp: DERIV_mirror)

lemma lhopital_left_at_top_at_bot:
fixes f g :: "real \<Rightarrow> real"
assumes f_0: "LIM x at_left a. f x :> at_top"
assumes g_0: "LIM x at_left a. g x :> at_bot"
and ev:
"eventually (\<lambda>x. DERIV f x :> f' x) (at_left a)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at_left a)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at_left a)"
shows "filterlim (\<lambda> x. f x / g x) at_bot (at_left a)"
by (insert assms, unfold eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror,
rule lhopital_right_at_top_at_bot[where f'="\<lambda>x. - f' (- x)"])
(insert assms, auto simp: DERIV_mirror)

lemma lhopital_at_top_at_top:
fixes f g :: "real \<Rightarrow> real"
assumes f_0: "LIM x at a. f x :> at_top"
assumes g_0: "LIM x at a. g x :> at_top"
and ev:
"eventually (\<lambda>x. DERIV f x :> f' x) (at a)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at a)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_top (at a)"
shows "filterlim (\<lambda> x. f x / g x) at_top (at a)"
using assms unfolding eventually_at_split filterlim_at_split
by (auto intro!: lhopital_right_at_top_at_top[of f a g f' g']
lhopital_left_at_top_at_top[of f a g f' g'])

lemma lhopital_at_top_at_bot:
fixes f g :: "real \<Rightarrow> real"
assumes f_0: "LIM x at a. f x :> at_top"
assumes g_0: "LIM x at a. g x :> at_bot"
and ev:
"eventually (\<lambda>x. DERIV f x :> f' x) (at a)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at a)"
and lim: "filterlim (\<lambda> x. (f' x / g' x)) at_bot (at a)"
shows "filterlim (\<lambda> x. f x / g x) at_bot (at a)"
using assms unfolding eventually_at_split filterlim_at_split
by (auto intro!: lhopital_right_at_top_at_bot[of f a g f' g']
lhopital_left_at_top_at_bot[of f a g f' g'])

end
```