# Theory Deriv

```(*  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 ‹Differentiation›

theory Deriv
imports Limits
begin

subsection ‹Frechet derivative›

definition has_derivative :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒
('a ⇒ 'b) ⇒ 'a filter ⇒ bool"  (infix "(has'_derivative)" 50)
where "(f has_derivative f') F ⟷
bounded_linear f' ∧
((λy. ((f y - f (Lim F (λx. x))) - f' (y - Lim F (λx. x))) /⇩R norm (y - Lim F (λx. x))) ⤏ 0) F"

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

text ‹These are the only cases we'll care about, probably.›

lemma has_derivative_within: "(f has_derivative f') (at x within s) ⟷
bounded_linear f' ∧ ((λy. (1 / norm(y - x)) *⇩R (f y - (f x + f' (y - x)))) ⤏ 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 ⟹ f' = g' ⟹ (f has_derivative g') F"
by simp

definition has_field_derivative :: "('a::real_normed_field ⇒ 'a) ⇒ 'a ⇒ 'a filter ⇒ bool"
(infix "(has'_field'_derivative)" 50)
where "(f has_field_derivative D) F ⟷ (f has_derivative (*) D) F"

lemma DERIV_cong: "(f has_field_derivative X) F ⟹ X = Y ⟹ (f has_field_derivative Y) F"
by simp

definition has_vector_derivative :: "(real ⇒ 'b::real_normed_vector) ⇒ 'b ⇒ real filter ⇒ bool"
(infix "has'_vector'_derivative" 50)
where "(f has_vector_derivative f') net ⟷ (f has_derivative (λx. x *⇩R f')) net"

lemma has_vector_derivative_eq_rhs:
"(f has_vector_derivative X) F ⟹ X = Y ⟹ (f has_vector_derivative Y) F"
by simp

named_theorems derivative_intros "structural introduction rules for derivatives"
setup ‹
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>‹derivative_eq_intros›,
fn context =>
Named_Theorems.get (Context.proof_of context) \<^named_theorems>‹derivative_intros›
|> map_filter eq_rule)
end
›

text ‹
The following syntax is only used as a legacy syntax.
›
abbreviation (input)
FDERIV :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ 'a ⇒  ('a ⇒ 'b) ⇒ bool"
("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where "FDERIV f x :> f' ≡ (f has_derivative f') (at x)"

lemma has_derivative_bounded_linear: "(f has_derivative f') F ⟹ bounded_linear f'"

lemma has_derivative_linear: "(f has_derivative f') F ⟹ linear f'"
using bounded_linear.linear[OF has_derivative_bounded_linear] .

lemma has_derivative_ident[derivative_intros, simp]: "((λx. x) has_derivative (λx. x)) F"

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

lemma shift_has_derivative_id: "((+) d has_derivative (λx. x)) F"
using has_derivative_def by fastforce

lemma has_derivative_const[derivative_intros, simp]: "((λx. c) has_derivative (λx. 0)) F"

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

lemma (in bounded_linear) has_derivative:
"(g has_derivative g') F ⟹ ((λx. f (g x)) has_derivative (λx. 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 "((λx. f x + g x) has_derivative (λx. f' x + g' x)) F"
unfolding has_derivative_def
proof safe
let ?x = "Lim F (λx. x)"
let ?D = "λf f' y. ((f y - f ?x) - f' (y - ?x)) /⇩R norm (y - ?x)"
have "((λx. ?D f f' x + ?D g g' x) ⤏ (0 + 0)) F"
using f g by (intro tendsto_add) (auto simp: has_derivative_def)
then show "(?D (λx. f x + g x) (λx. f' x + g' x) ⤏ 0) F"
qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)

lemma has_derivative_sum[simp, derivative_intros]:
"(⋀i. i ∈ I ⟹ (f i has_derivative f' i) F) ⟹
((λx. ∑i∈I. f i x) has_derivative (λx. ∑i∈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 ⟹ ((λx. - f x) has_derivative (λx. - 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 ⟹ (g has_derivative g') F ⟹
((λx. f x - g x) has_derivative (λx. f' x - g' x)) F"

lemma has_derivative_at_within:
"(f has_derivative f') (at x within s) ⟷
(bounded_linear f' ∧ ((λy. ((f y - f x) - f' (y - x)) /⇩R norm (y - x)) ⤏ 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) ⟷
bounded_linear f' ∧ ((λy. norm ((f y - f x) - f' (y - x)) / norm (y - x)) ⤏ 0) (at x within s)"
using tendsto_norm_zero_iff[of _ "at x within s", where 'b="'b", symmetric]
by (simp add: has_derivative_at_within divide_inverse ac_simps)

lemma has_derivative_at:
"(f has_derivative D) (at x) ⟷
(bounded_linear D ∧ (λh. norm (f (x + h) - f x - D h) / norm h) ─0→ 0)"

lemma field_has_derivative_at:
fixes x :: "'a::real_normed_field"
shows "(f has_derivative (*) D) (at x) ⟷ (λh. (f (x + h) - f x) / h) ─0→ D" (is "?lhs = ?rhs")
proof -
have "?lhs = (λh. norm (f (x + h) - f x - D * h) / norm h) ─0 → 0"
also have "... = (λy. norm ((f (x + y) - f x - D * y) / y)) ─0→ 0"
by (simp cong: LIM_cong flip: nonzero_norm_divide)
also have "... = (λy. norm ((f (x + y) - f x) / y - D / y * y)) ─0→ 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) ⟷
bounded_linear f' ∧ (∃e. (∀h. f (x+h) = f x + f' h + e h) ∧ ((λh. norm (e h) / norm h) ⤏ 0) (at 0))"
unfolding has_derivative_at by force

lemma has_derivative_at_within_iff_Ex:
assumes "x ∈ S" "open S"
shows "(f has_derivative f') (at x within S) ⟷
bounded_linear f' ∧ (∃e. (∀h. x+h ∈ S ⟶ f (x+h) = f x + f' h + e h) ∧ ((λh. norm (e h) / norm h) ⤏ 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 "∃e. (∀h. x + h ∈ S ⟶ f (x + h) = f x + f' h + e h) ∧ (λh. norm (e h) / norm h) ─0→ 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]: "∀h. x + h ∈ S ⟶ f (x + h) = f x + f' h + e h"
and 0: "(λh. norm (e (h::'a)::'b) / norm h) ─0→ 0"
for e
proof -
have 1: "f y - f x = f' (y-x) + e (y-x)" if "y ∈ S" for y
using eq [of "y-x"] that by simp
have 2: "((λy. norm (e (y-x)) / norm (y - x)) ⤏ 0) (at x within S)"
by (simp add: "0" assms tendsto_offset_zero_iff)
have "((λy. norm (f y - f x - f' (y - x)) / norm (y - x)) ⤏ 0) (at x within S)"
by (simp add: Lim_cong_within 1 2)
then show ?thesis
by (simp add: has_derivative_iff_norm ‹bounded_linear f'›)
qed
qed

lemma has_derivativeI:
"bounded_linear f' ⟹
((λy. ((f y - f x) - f' (y - x)) /⇩R norm (y - x)) ⤏ 0) (at x within s) ⟹
(f has_derivative f') (at x within s)"

lemma has_derivativeI_sandwich:
assumes e: "0 < e"
and bounded: "bounded_linear f'"
and sandwich: "(⋀y. y ∈ s ⟹ y ≠ x ⟹ dist y x < e ⟹
norm ((f y - f x) - f' (y - x)) / norm (y - x) ≤ H y)"
and "(H ⤏ 0) (at x within s)"
shows "(f has_derivative f') (at x within s)"
unfolding has_derivative_iff_norm
proof safe
show "((λy. norm (f y - f x - f' (y - x)) / norm (y - x)) ⤏ 0) (at x within s)"
proof (rule tendsto_sandwich[where f="λx. 0"])
show "(H ⤏ 0) (at x within s)" by fact
show "eventually (λn. norm (f n - f x - f' (n - x)) / norm (n - x) ≤ 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) ⟹ t ⊆ s ⟹ (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}) ⟷ bounded_linear g"
by (auto intro!: has_derivativeI_sandwich[where e=1] has_derivative_bounded_linear)

subsubsection ‹Limit transformation for derivatives›

lemma has_derivative_transform_within:
assumes "(f has_derivative f') (at x within s)"
and "0 < d"
and "x ∈ s"
and "⋀x'. ⟦x' ∈ s; dist x' x < d⟧ ⟹ 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 ∈ s"
and "⋀x. x∈s ⟹ 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 ∈ s" "⋀x. x ∈ s ⟹ 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)"
"(∀⇩F x' in at x within s. f x' = g x')"
assumes "f x = g x" "x ∈ s"
shows "(g has_derivative f') (at x within s)"
using assms
proof -
from assms(2,3) obtain d where "d > 0" "⋀x'. x' ∈ s ⟹ dist x' x < d ⟹ 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 ∈ S"
and "⋀x. ⟦x ∈ S; dist x a < d⟧ ⟹ 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 ∈ S"
and "⋀x. x ∈ S ⟹ 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 ‹Continuity›

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 = "λf. (f ⤏ 0) (at x within s)"
have "?L (λy. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
using f unfolding has_derivative_iff_norm by blast
then have "?L (λy. 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 ⟷ ?L (λy. norm ((f y - f x) - f' (y - x)))"
by (intro filterlim_cong) (simp_all add: eventually_at_filter)
finally have "?L (λy. (f y - f x) - f' (y - x))"
by (rule tendsto_norm_zero_cancel)
then have "?L (λy. ((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 (λy. f y - f x)"
by simp
from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
qed

subsection ‹Composition›

lemma tendsto_at_iff_tendsto_nhds_within:
"f x = y ⟹ (f ⤏ y) (at x within s) ⟷ (f ⤏ y) (inf (nhds x) (principal s))"
unfolding tendsto_def eventually_inf_principal eventually_at_filter
by (intro ext all_cong imp_cong) (auto elim!: eventually_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 "((λx. g (f x)) has_derivative (λx. 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: "⋀x. norm (f' x) ≤ norm x * kF"
by fast
from G.bounded obtain kG where kG: "⋀x. norm (g' x) ≤ norm x * kG"
by fast
note G.tendsto[tendsto_intros]

let ?L = "λf. (f ⤏ 0) (at x within s)"
let ?D = "λf f' x y. (f y - f x) - f' (y - x)"
let ?N = "λf f' x y. norm (?D f f' x y) / norm (y - x)"
let ?gf = "λx. g (f x)" and ?gf' = "λx. 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 (λx. g' (f' x))"
using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
next
fix y :: 'a
assume neq: "y ≠ 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 "… ≤ norm (g' (?D f f' x y)) / norm (y - x) + Ng y * (norm (f y - f x) / norm (y - x))"
also have "… ≤ 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 "… ≤ norm (?D f f' x y) + norm (f' (y - x))"
by (rule norm_triangle_ineq)
also have "… ≤ 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) ≤ Nf y + kF"
by (simp add: neq Nf_def field_simps)
qed (use kG in ‹simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps›)
finally show "?N ?gf ?gf' x y ≤ 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 ⤏ 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)) ⤏ 0) (at (f x) within f`s)"
using g unfolding has_derivative_iff_norm ..
then have g': "((?N g  g' (f x)) ⤏ 0) (inf (nhds (f x)) (principal (f`s)))"
by (rule tendsto_at_iff_tendsto_nhds_within[THEN iffD1, rotated]) simp

have [tendsto_intros]: "?L Ng"
unfolding Ng_def by (rule filterlim_compose[OF g' f'])
show "((λy. Nf y * kG + Ng y * (Nf y + kF)) ⤏ 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) ⟹ (g has_derivative g') (at (f x)) ⟹
((λx. g (f x)) has_derivative (λx. g' (f' x))) (at x within s)"
by (blast intro: has_derivative_in_compose has_derivative_subset)

lemma has_derivative_in_compose2:
assumes "⋀x. x ∈ t ⟹ (g has_derivative g' x) (at x within t)"
assumes "f ` s ⊆ t" "x ∈ s"
assumes "(f has_derivative f') (at x within s)"
shows "((λx. g (f x)) has_derivative (λy. 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 "((λx. f x ** g x) has_derivative (λh. 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: "⋀x. norm (f' x) ≤ norm x * KF" by fast

from pos_bounded obtain K
where K: "0 < K" and norm_prod: "⋀a b. norm (a ** b) ≤ norm a * norm b * K"
by fast
let ?D = "λf f' y. f y - f x - f' (y - x)"
let ?N = "λf 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 = "λy. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
let ?fun2 = "λy. 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 (λh. 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 ⤏ g x) ?F"
by (intro continuous_within[THEN iffD1] has_derivative_continuous)
moreover from f g have "(Nf ⤏ 0) ?F" "(Ng ⤏ 0) ?F"
by (simp_all add: has_derivative_iff_norm Ng_def Nf_def)
ultimately have "(?fun2 ⤏ norm (f x) * 0 * K + 0 * norm (g x) * K + KF * norm (0::'b) * K) ?F"
by (intro tendsto_intros) (simp_all add: LIM_zero_iff)
then show "(?fun2 ⤏ 0) ?F"
by simp
next
fix y :: 'd
assume "y ≠ 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 "… ≤ (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 "… = ?fun2 y"
finally show "?fun1 y ≤ ?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 ⇒ 'a::real_normed_vector ⇒ 'b::real_normed_field"
shows "(⋀i. i ∈ I ⟹ (f i has_derivative f' i) (at x within S)) ⟹
((λx. ∏i∈I. f i x) has_derivative (λy. ∑i∈I. f' i y * (∏j∈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 = "λy. f i x * (∑i∈I. f' i y * (∏j∈I - {i}. f j x)) + (f' i y) * (∏i∈I. f i x)"
have "((λx. f i x * (∏i∈I. f i x)) has_derivative ?P) (at x within S)"
using insert by (intro has_derivative_mult) auto
also have "?P = (λy. ∑i'∈insert i I. f' i' y * (∏j∈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 ⇒ 'b :: real_normed_field"
assumes f: "(f has_derivative f') (at x within S)"
shows "((λx. f x^n) has_derivative (λy. 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 ≠ 0"
shows "(inverse has_derivative (λh. - (inverse x * h * inverse x))) (at x within S)"
(is "(_ has_derivative ?f) _")
proof (rule has_derivativeI_sandwich)
show "bounded_linear (λh. - (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 ⤏ inverse x) (at x within S)"
using tendsto_inverse tendsto_ident_at x by auto
then show "((λy. norm (inverse y - inverse x) * norm (inverse x)) ⤏ 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 ≠ x" "dist y x < norm x"
then have "y ≠ 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: ‹y ≠ 0› inverse_diff_inverse x)
also have "... = norm ((inverse y - inverse x) * (y - x) * inverse x) / norm (y - x)"
also have "… ≤ norm (inverse y - inverse x) * norm (y - x) * norm (inverse x) / norm (y - x)"
also have "… = norm (inverse y - inverse x) * norm (inverse x)"
by simp
finally show "norm (inverse y - inverse x - ?f (y -x)) / norm (y - x) ≤
norm (inverse y - inverse x) * norm (inverse x)" .
qed

lemma has_derivative_inverse[simp, derivative_intros]:
fixes f :: "_ ⇒ 'a::real_normed_div_algebra"
assumes x:  "f x ≠ 0"
and f: "(f has_derivative f') (at x within S)"
shows "((λx. inverse (f x)) has_derivative (λh. - (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 :: "_ ⇒ '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 ≠ 0"
shows "((λx. f x / g x) has_derivative
(λh. - 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 ≠ 0"
shows "((λx. power_int x n) has_derivative (λy. 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 :: "_ ⇒ 'a::real_normed_field"
assumes x:  "f x ≠ 0"
and f: "(f has_derivative f') (at x within S)"
shows "((λx. power_int (f x) n) has_derivative (λh. 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 ‹Conventional form requires mult-AC laws. Types real and complex only.›

lemma has_derivative_divide'[derivative_intros]:
fixes f :: "_ ⇒ '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 ≠ 0"
shows "((λx. f x / g x) has_derivative (λh. (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 ‹Uniqueness›

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

lemma has_derivative_at2: "(f has_derivative f') (at x) ⟷
bounded_linear f' ∧ ((λy. (1 / (norm(y - x))) *⇩R (f y - (f x + f' (y - x)))) ⤏ 0) (at x)"
using has_derivative_within [of f f' x UNIV]
by simp

lemma has_derivative_zero_unique:
assumes "((λx. 0) has_derivative F) (at x)"
shows "F = (λh. 0)"
proof -
interpret F: bounded_linear F
using assms by (rule has_derivative_bounded_linear)
let ?r = "λh. norm (F h) / norm h"
have *: "?r ─0→ 0"
using assms unfolding has_derivative_at by simp
show "F = (λh. 0)"
proof
show "F h = 0" for h
proof (rule ccontr)
assume **: "¬ ?thesis"
then have h: "h ≠ 0"
with ** have "0 < ?r h"
by simp
from LIM_D [OF * this] obtain S
where S: "0 < S" and r: "⋀x. x ≠ 0 ⟹ norm x < S ⟹ ?r x < ?r h"
by auto
from dense [OF S] obtain t where t: "0 < t ∧ t < S" ..
let ?x = "scaleR (t / norm h) h"
have "?x ≠ 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 "((λx. 0) has_derivative (λh. F h - F' h)) (at x)"
using has_derivative_diff [OF assms] by simp
then have "(λh. F h - F' h) = (λh. 0)"
by (rule has_derivative_zero_unique)
then show "F = F'"
unfolding fun_eq_iff right_minus_eq .
qed

lemma has_derivative_Uniq: "∃⇩≤⇩1F. (f has_derivative F) (at x)"

subsection ‹Differentiability predicate›

definition differentiable :: "('a::real_normed_vector ⇒ 'b::real_normed_vector) ⇒ 'a filter ⇒ bool"
(infix "differentiable" 50)
where "f differentiable F ⟷ (∃D. (f has_derivative D) F)"

lemma differentiable_subset:
"f differentiable (at x within s) ⟹ t ⊆ s ⟹ 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]: "(λx. x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_ident)

lemma differentiable_const [simp, derivative_intros]: "(λz. a) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_const)

lemma differentiable_in_compose:
"f differentiable (at (g x) within (g`s)) ⟹ g differentiable (at x within s) ⟹
(λx. 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)) ⟹ g differentiable (at x within s) ⟹
(λx. f (g x)) differentiable (at x within s)"
by (blast intro: differentiable_in_compose differentiable_subset)

"f differentiable F ⟹ g differentiable F ⟹ (λx. f x + g x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_add)

lemma differentiable_sum[simp, derivative_intros]:
assumes "finite s" "∀a∈s. (f a) differentiable net"
shows "(λx. sum (λa. 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 ⟹ (λx. - f x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_minus)

lemma differentiable_diff [simp, derivative_intros]:
"f differentiable F ⟹ g differentiable F ⟹ (λx. 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 ⇒ 'b::real_normed_algebra"
shows "f differentiable (at x within s) ⟹ g differentiable (at x within s) ⟹
(λx. f x * g x) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_mult)

lemma differentiable_cmult_left_iff [simp]:
fixes c::"'a::real_normed_field"
shows "(λt. c * q t) differentiable at t ⟷ c = 0 ∨ (λt. q t) differentiable at t" (is "?lhs = ?rhs")
proof
assume L: ?lhs
{assume "c ≠ 0"
then have "q differentiable at t"
using differentiable_mult [OF differentiable_const L, of concl: "1/c"] by auto
} then show ?rhs
by auto
qed auto

lemma differentiable_cmult_right_iff [simp]:
fixes c::"'a::real_normed_field"
shows "(λt. q t * c) differentiable at t ⟷ c = 0 ∨ (λt. q t) differentiable at t" (is "?lhs = ?rhs")
by (simp add: mult.commute flip: differentiable_cmult_left_iff)

lemma differentiable_inverse [simp, derivative_intros]:
fixes f :: "'a::real_normed_vector ⇒ 'b::real_normed_field"
shows "f differentiable (at x within s) ⟹ f x ≠ 0 ⟹
(λx. 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 ⇒ 'b::real_normed_field"
shows "f differentiable (at x within s) ⟹ g differentiable (at x within s) ⟹
g x ≠ 0 ⟹ (λx. 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 ⇒ 'b::real_normed_field"
shows "f differentiable (at x within s) ⟹ (λx. 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 ⇒ 'b::real_normed_field"
shows "f differentiable (at x within s) ⟹ f x ≠ 0 ⟹
(λx. 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) ⟹ g differentiable (at x within s) ⟹
(λx. f x *⇩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 ⟹ (⋀x. x * D' = D x) ⟹ (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 ⟹ (f has_derivative (*) D) F"

lemma DERIV_subset:
"(f has_field_derivative f') (at x within s) ⟹ t ⊆ s ⟹
(f has_field_derivative f') (at x within t)"

lemma has_field_derivative_at_within:
"(f has_field_derivative f') (at x) ⟹ (f has_field_derivative f') (at x within s)"
using DERIV_subset by blast

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

abbreviation has_real_derivative :: "(real ⇒ real) ⇒ real ⇒ real filter ⇒ bool"
(infix "(has'_real'_derivative)" 50)
where "(f has_real_derivative D) F ≡ (f has_field_derivative D) F"

lemma real_differentiable_def:
"f differentiable at x within s ⟷ (∃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 "∃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) ⟷
((λy. (f y - f x) / (y - x)) ⤏ D) (at x within S)"
proof -
have "((λy. norm (f y - f x - D * (y - x)) / norm (y - x)) ⤏ 0) (at x within S)
= ((λy. (f y - f x) / (y - x) - D) ⤏ 0) (at x within S)"
by (smt (verit, best) Lim_cong_within divide_diff_eq_iff norm_divide right_minus_eq tendsto_norm_zero_iff)
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 ⟷ (λh. (f (x + h) - f x) / h) ─0→ D"
unfolding field_has_derivative_at has_field_derivative_def has_field_derivative_iff ..

text ‹due to Christian Pardillo Laursen, replacing a proper epsilon-delta horror›
lemma field_derivative_lim_unique:
assumes f: "(f has_field_derivative df) (at z)"
and s: "s ⇢ 0"  "⋀n. s n ≠ 0"
and a: "(λn. (f (z + s n) - f z) / s n) ⇢ a"
shows "df = a"
proof -
have "((λk. (f (z + k) - f z) / k) ⤏ df) (at 0)"
using f by (simp add: DERIV_def)
with s have "((λn. (f (z + s n) - f z) / s n) ⇢ df)"
by (simp flip: LIMSEQ_SEQ_conv)
then show ?thesis
using a by (rule LIMSEQ_unique)
qed

lemma mult_commute_abs: "(λx. x * c) = (*) c"
for c :: "'a::ab_semigroup_mult"

lemma DERIV_compose_FDERIV:
fixes f::"real⇒real"
assumes "DERIV f (g x) :> f'"
assumes "(g has_derivative g') (at x within s)"
shows "((λx. f (g x)) has_derivative (λx. 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 ‹Vector derivative›

text ‹It's for real derivatives only, and not obviously generalisable to field derivatives›
lemma has_real_derivative_iff_has_vector_derivative:
"(f has_real_derivative y) F ⟷ (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) ⟹ t ⊆ s ⟹
(f has_field_derivative y) (at x within t)"
by (fact DERIV_subset)

lemma has_vector_derivative_const[simp, derivative_intros]: "((λx. c) has_vector_derivative 0) net"
by (auto simp: has_vector_derivative_def)

lemma has_vector_derivative_id[simp, derivative_intros]: "((λx. 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 ⟹ ((λx. - f x) has_vector_derivative (- f')) net"
by (auto simp: has_vector_derivative_def)

"(f has_vector_derivative f') net ⟹ (g has_vector_derivative g') net ⟹
((λx. 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]:
"(⋀i. i ∈ I ⟹ (f i has_vector_derivative f' i) net) ⟹
((λx. ∑i∈I. f i x) has_vector_derivative (∑i∈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 ⟹ (g has_vector_derivative g') net ⟹
((λx. f x - g x) has_vector_derivative (f' - g')) net"
by (auto simp: has_vector_derivative_def scaleR_diff_right)

"((λt. g t + z) has_vector_derivative f') net = ((λt. g t) has_vector_derivative f') net"
apply (intro iffI)
apply (force dest: has_vector_derivative_diff [where g = "λt. 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:
"((λt. g t - z) has_vector_derivative f') net = ((λt. 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 "((λx. 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 "((λx. 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) ⟹ (g has_vector_derivative g') (at x within s) ⟹
((λx. f x *⇩R g x) has_vector_derivative (f x *⇩R g' + f' *⇩R g x)) (at x within s)"
unfolding has_real_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) ⟹ (g has_vector_derivative g') (at x within s) ⟹
((λx. f x * g x) has_vector_derivative (f x * g' + f' * g x)) (at x within s)"
for f g :: "real ⇒ '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 ⟹ ((λx. 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)) ⟹ ((λx. 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) ⟹ continuous (at x within s) f"
by (auto intro: has_derivative_continuous simp: has_vector_derivative_def)

lemma continuous_on_vector_derivative:
"(⋀x. x ∈ S ⟹ (f has_vector_derivative f' x) (at x within S)) ⟹ 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 ⟹ ((λx. 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 ⟹ ((λx. f x * a) has_vector_derivative (x * a)) F"
by (rule bounded_linear.has_vector_derivative[OF bounded_linear_mult_left])

lemma has_vector_derivative_divide[derivative_intros]:
fixes a :: "'a::real_normed_field"
shows "(f has_vector_derivative x) F ⟹ ((λx. f x / a) has_vector_derivative (x / a)) F"
using has_vector_derivative_mult_left [of f x F "inverse a"]

subsection ‹Derivatives›

lemma DERIV_D: "DERIV f x :> D ⟹ (λh. (f (x + h) - f x) / h) ─0→ D"

lemma has_field_derivativeD:
"(f has_field_derivative D) (at x within S) ⟹
((λy. (f y - f x) / (y - x)) ⤏ D) (at x within S)"

lemma DERIV_const [simp, derivative_intros]: "((λx. 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]: "((λx. 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 ⟹ (g has_field_derivative g') F ⟹
((λz. 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) ⟹ (g has_field_derivative E) (at x within s) ⟹
((λx. 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 ⟹ ((λz. - (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) ⟹
((λx. - 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 ⟹
(g has_field_derivative g') F ⟹ ((λz. f z - g z) has_field_derivative f' - g') F"

corollary DERIV_diff:
"(f has_field_derivative D) (at x within s) ⟹
(g has_field_derivative E) (at x within s) ⟹
((λx. 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) ⟹ 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 ⟹ isCont f x"
by (rule DERIV_continuous)

lemma DERIV_atLeastAtMost_imp_continuous_on:
assumes "⋀x. ⟦a ≤ x; x ≤ b⟧ ⟹ ∃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:
"(⋀x. x ∈ s ⟹ (f has_field_derivative (D x)) (at x within s)) ⟹ 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) ⟹ (g has_field_derivative E) (at x within s) ⟹
((λx. 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) ⟹ (g has_field_derivative Db) (at x within s) ⟹
((λx. 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 ‹Derivative of linear multiplication›

lemma DERIV_cmult:
"(f has_field_derivative D) (at x within s) ⟹
((λx. 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) ⟹
((λx. 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) ⟹
((λx. 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 ⟹ DERIV f x :> E ⟹ D = E"
unfolding DERIV_def by (rule LIM_unique)

lemma DERIV_Uniq: "∃⇩≤⇩1D. DERIV f x :> D"

lemma DERIV_sum[derivative_intros]:
"(⋀ n. n ∈ S ⟹ ((λx. f x n) has_field_derivative (f' x n)) F) ⟹
((λx. 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 ≠ 0"
shows "((λx. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x)))
(at x within s)"
proof -
have "(f has_derivative (λx. x * D)) = (f has_derivative (*) D)"
by (rule arg_cong [of "λx. x * D"]) (simp add: fun_eq_iff)
with assms have "(f has_derivative (λx. x * D)) (at x within s)"
by (auto dest!: has_field_derivative_imp_has_derivative)
then show ?thesis using ‹f x ≠ 0›
by (auto intro: has_derivative_imp_has_field_derivative has_derivative_inverse)
qed

text ‹Power of ‹-1››

lemma DERIV_inverse:
"x ≠ 0 ⟹ ((λx. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
by (drule DERIV_inverse' [OF DERIV_ident]) simp

text ‹Derivative of inverse›

lemma DERIV_inverse_fun:
"(f has_field_derivative d) (at x within s) ⟹ f x ≠ 0 ⟹
((λx. 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 ‹Derivative of quotient›

lemma DERIV_divide[derivative_intros]:
"(f has_field_derivative D) (at x within s) ⟹
(g has_field_derivative E) (at x within s) ⟹ g x ≠ 0 ⟹
((λx. 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) ⟹
(g has_field_derivative e) (at x within s)⟹ g x ≠ 0 ⟹
((λy. 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) ⟹
((λx. 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) ⟹
((λx. 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: "((λx. 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 ≠ 0"
shows   "((λx. 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) ⟹ DERIV g (f x) :> E ⟹
((λx. 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 ⟹ (g has_field_derivative Db) (at x within s) ⟹
((λx. f (g x)) has_field_derivative Da * Db) (at x within s)"
by (rule DERIV_chain')

text ‹Standard version›

lemma DERIV_chain:
"DERIV f (g x) :> Da ⟹ (g has_field_derivative Db) (at x within s) ⟹
(f ∘ 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)) ⟹
(g has_field_derivative Db) (at x within s) ⟹
(f ∘ 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 "(⋀x. x ∈ s ⟹ DERIV g x :> g'(x))"
and "DERIV f x :> f'"
and "f x ∈ s"
shows "DERIV (λx. 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 "(⋀x. DERIV g x :> g'(x))"
and "DERIV f x :> f'"
shows "DERIV (λx. g(f x)) x :> f' * g'(f x)"
by (metis UNIV_I DERIV_chain_s [of UNIV] assms)

text ‹Alternative definition for differentiability›

lemma DERIV_LIM_iff:
fixes f :: "'a::{real_normed_vector,inverse} ⇒ 'a"
shows "((λh. (f (a + h) - f a) / h) ─0→ D) = ((λx. (f x - f a) / (x - a)) ─a→ D)" (is "?lhs = ?rhs")
proof
assume ?lhs
then have "(λx. (f (a + (x + - a)) - f a) / (x + - a)) ─0 - - a→ D"
by (rule LIM_offset)
then show ?rhs
by simp
next
assume ?rhs
then have "(λx. (f (x+a) - f a) / ((x+a) - a)) ─a-a→ D"
by (rule LIM_offset)
then show ?lhs
qed

lemma has_field_derivative_cong_ev:
assumes "x = y"
and *: "eventually (λx. x ∈ S ⟶ f x = g x) (nhds x)"
and "u = v" "S = t" "x ∈ 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 "∀⇩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 ‹f y = g y›)

lemma has_field_derivative_cong_eventually:
assumes "eventually (λx. 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 "∀⇩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 ⟹ eventually (λx. f x = g x) (nhds x) ⟹ u = v ⟹
DERIV f x :> u ⟷ DERIV g y :> v"
by (rule has_field_derivative_cong_ev) simp_all

lemma DERIV_mirror: "(DERIV f (- x) :> y) ⟷ (DERIV (λx. f (- x)) x :> - y)"
for f :: "real ⇒ 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 DERIV_shift:
"(f has_field_derivative y) (at (x + z)) = ((λx. f (x + z)) has_field_derivative y) (at x)"

lemma DERIV_at_within_shift_lemma:
assumes "(f has_field_derivative y) (at (z+x) within (+) z ` S)"
shows "(f ∘ (+)z has_field_derivative y) (at x within S)"
proof -
have "((+)z has_field_derivative 1) (at x within S)"
by (rule derivative_eq_intros | simp)+
with assms DERIV_image_chain show ?thesis
by (metis mult.right_neutral)
qed

lemma DERIV_at_within_shift:
"(f has_field_derivative y) (at (z+x) within (+) z ` S) ⟷
((λx. f (z+x)) has_field_derivative y) (at x within S)"   (is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
using DERIV_at_within_shift_lemma unfolding o_def by blast
next
have [simp]: "(λx. x - z) ` (+) z ` S = S"
by force
assume R: ?rhs
have "(f ∘ (+) z ∘ (+) (- z) has_field_derivative y) (at (z + x) within (+) z ` S)"
by (rule DERIV_at_within_shift_lemma) (use R in ‹simp add: o_def›)
then show ?lhs
qed

lemma floor_has_real_derivative:
fixes f :: "real ⇒ 'a::{floor_ceiling,order_topology}"
assumes "isCont f x"
and "f x ∉ ℤ"
shows "((λx. floor (f x)) has_real_derivative 0) (at x)"
proof (subst DERIV_cong_ev[OF refl _ refl])
show "((λ_. floor (f x)) has_real_derivative 0) (at x)"
by simp
have "∀⇩F y in at x. ⌊f y⌋ = ⌊f x⌋"
by (rule eventually_floor_eq[OF assms[unfolded continuous_at]])
then show "∀⇩F y in nhds x. real_of_int ⌊f y⌋ = real_of_int ⌊f x⌋"
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 ∉ ℤ ⟹ continuous (at x) (real_of_int ∘ 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 ∉ ℤ"
shows "continuous (at x) frac"
proof -
have "isCont (λx. real_of_int ⌊x⌋) x"
using continuous_floor [OF assms] by (simp add: o_def)
then have *: "continuous (at x) (λx. x - real_of_int ⌊x⌋)"
by (intro continuous_intros)
moreover have "∀⇩F x in nhds x. frac x = x - real_of_int ⌊x⌋"
ultimately show ?thesis
by (simp add: LIM_imp_LIM frac_def isCont_def)
qed

text ‹Caratheodory formulation of derivative at a point›

lemma CARAT_DERIV:
"(DERIV f x :> l) ⟷ (∃g. (∀z. f z - f x = g z * (z - x)) ∧ isCont g x ∧ g x = l)"
(is "?lhs = ?rhs")
proof
assume ?lhs
show "∃g. (∀z. f z - f x = g z * (z - x)) ∧ isCont g x ∧ g x = l"
proof (intro exI conjI)
let ?g = "(λz. if z = x then l else (f z - f x) / (z-x))"
show "∀z. f z - f x = ?g z * (z - x)"
by simp
show "isCont ?g x"
using ‹?lhs› 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 ‹Local extrema›

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

lemma has_real_derivative_pos_inc_right:
fixes f :: "real ⇒ real"
assumes der: "(f has_real_derivative l) (at x within S)"
and l: "0 < l"
shows "∃d > 0. ∀h > 0. x + h ∈ S ⟶ h < d ⟶ 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: "⋀xa. xa∈S ⟹ xa ≠ x ∧ dist xa x < s ⟶ ¦(f xa - f x) / (xa - x) - l¦ < 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 ∈ 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 "¬ (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 ⇒ real"
assumes der: "DERIV f x :> l"
and l: "0 < l"
shows "∃d > 0. ∀h > 0. h < d ⟶ 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 ⇒ real"
assumes der: "(f has_real_derivative l) (at x within S)"
and "l < 0"
shows "∃d > 0. ∀h > 0. x - h ∈ S ⟶ h < d ⟶ f x < f (x - h)"
proof -
from ‹l < 0› have l: "- l > 0"
by simp
from der [THEN ha```