# Theory Deriv

theory Deriv
imports Limits
```(*  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}.
›

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 op * 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'"
by (simp add: has_derivative_def)

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"
by (simp add: has_derivative_def)

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]: "((λx. c) has_derivative (λx. 0)) F"
by (simp add: has_derivative_def)

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
apply safe
apply (erule bounded_linear_compose [OF bounded_linear])
apply (drule tendsto)
apply (simp add: scaleR diff add zero)
done

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]

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))"
by (cases "at x within s = bot") (simp_all add: has_derivative_def Lim_ident_at)

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)"
unfolding has_derivative_iff_norm LIM_offset_zero_iff[of _ _ x] by simp

lemma field_has_derivative_at:
fixes x :: "'a::real_normed_field"
shows "(f has_derivative op * D) (at x) ⟷ (λh. (f (x + h) - f x) / h) ─0→ D"
apply (unfold has_derivative_at)
apply (simp add: bounded_linear_mult_right)
apply (simp cong: LIM_cong add: nonzero_norm_divide [symmetric])
apply (subst diff_divide_distrib)
apply (subst times_divide_eq_left [symmetric])
apply (simp cong: LIM_cong)
apply (simp add: tendsto_norm_zero_iff LIM_zero_iff)
done

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)"
by (simp add: has_derivative_at_within)

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)

lemmas has_derivative_within_subset = has_derivative_subset

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
by (simp add: continuous_within)
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))"
by (simp add: add_divide_distrib[symmetric] divide_right_mono norm_triangle_ineq G.zero Ng_def)
also have "… ≤ Nf y * kG + Ng y * (Nf y + kF)"
proof (intro add_mono mult_left_mono)
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 (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_triangle_ineq add_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 "(?inv has_derivative ?f) _")
proof (rule has_derivativeI_sandwich)
show "bounded_linear (λh. - (?inv x * h * ?inv x))"
apply (rule bounded_linear_minus)
apply (rule bounded_linear_mult_const)
apply (rule bounded_linear_const_mult)
apply (rule bounded_linear_ident)
done
show "0 < norm x" using x by simp
show "((λy. norm (?inv y - ?inv x) * norm (?inv x)) ⤏ 0) (at x within s)"
apply (rule tendsto_mult_left_zero)
apply (rule tendsto_norm_zero)
apply (rule LIM_zero)
apply (rule tendsto_inverse)
apply (rule tendsto_ident_at)
apply (rule x)
done
next
fix y :: 'a
assume h: "y ≠ x" "dist y x < norm x"
then have "y ≠ 0" by auto
have "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) =
norm ((?inv y - ?inv x) * (y - x) * ?inv x) / norm (y - x)"
apply (subst inverse_diff_inverse [OF ‹y ≠ 0› x])
apply (subst minus_diff_minus)
apply (subst norm_minus_cancel)
apply (simp add: left_diff_distrib)
done
also have "… ≤ norm (?inv y - ?inv x) * norm (y - x) * norm (?inv x) / norm (y - x)"
apply (rule divide_right_mono [OF _ norm_ge_zero])
apply (rule order_trans [OF norm_mult_ineq])
apply (rule mult_right_mono [OF _ norm_ge_zero])
apply (rule norm_mult_ineq)
done
also have "… = norm (?inv y - ?inv x) * norm (?inv x)"
by simp
finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) ≤
norm (?inv y - ?inv x) * norm (?inv 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]]
by (simp add: field_simps)

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
by (simp add: field_simps x)
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_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"
by (auto simp add: F.zero)
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

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)

lemma differentiable_sum [simp, derivative_intros]:
"f differentiable F ⟹ g differentiable F ⟹ (λx. f x + g x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_add)

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_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_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 op * D) F"
by (simp add: has_field_derivative_def)

lemma DERIV_subset:
"(f has_field_derivative f') (at x within s) ⟹ t ⊆ s ⟹
(f has_field_derivative f') (at x within t)"
by (simp add: has_field_derivative_def has_derivative_within_subset)

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' = (op * 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 differentiableD:
"f differentiable (at x within s) ⟹ ∃D. (f has_real_derivative D) (at x within s)"
by (auto elim: real_differentiableE)

lemma differentiableI:
"(f has_real_derivative D) (at x within s) ⟹ f differentiable (at x within s)"
by (force simp add: 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)"
apply (simp add: has_field_derivative_def has_derivative_iff_norm bounded_linear_mult_right
LIM_zero_iff[symmetric, of _ D])
apply (subst (2) tendsto_norm_zero_iff[symmetric])
apply (rule filterlim_cong)
apply (simp_all add: eventually_at_filter field_simps nonzero_norm_divide)
done

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 ..

lemma mult_commute_abs: "(λx. x * c) = op * c"
for c :: "'a::ab_semigroup_mult"
by (simp add: fun_eq_iff mult.commute)

subsection ‹Vector derivative›

lemma has_field_derivative_iff_has_vector_derivative:
"(f has_field_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)"
unfolding has_field_derivative_def by (rule has_derivative_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 (drule has_vector_derivative_diff [where g = "λt. z", OF _ has_vector_derivative_const])
apply simp
apply (drule has_vector_derivative_add [OF _ has_vector_derivative_const])
apply simp
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]]
by (simp add: has_vector_derivative_def scaleR)

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_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) ⟹ (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_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 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])

subsection ‹Derivatives›

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

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)"
by (simp add: has_field_derivative_iff)

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"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
(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_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]: "(op * 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_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 op * 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_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 "op * D" x s g "op * 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 "op * Db" x s f "op * 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)"
apply (rule iffI)
apply (drule_tac k="- a" in LIM_offset)
apply simp
apply (drule_tac k="a" in LIM_offset)
done

lemmas DERIV_iff2 = has_field_derivative_iff

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 DERIV_iff2
proof (rule filterlim_cong)
from assms have "f y = g y"
by (auto simp: eventually_nhds)
with * show "∀⇩F xa in at x within s. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)"
unfolding eventually_at_filter
by eventually_elim (auto simp: assms ‹f y = g y›)
qed (simp_all add: assms)

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_shift:
"(f has_field_derivative y) (at (x + z)) = ((λx. f (x + z)) has_field_derivative y) (at x)"
by (simp add: DERIV_def field_simps)

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 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

text ‹Caratheodory formulation of derivative at a point›

lemma CARAT_DERIV: (*FIXME: SUPERSEDED BY THE ONE IN Deriv.thy. But still used by NSA/HDeriv.thy*)
"(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 obtain g where "(∀z. f z - f x = g z * (z - x))" and "isCont g x" and "g x = l"
by blast
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)"
by (simp add: pos_less_divide_eq 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 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 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)"
by (simp add: pos_less_divide_eq h)
qed
qed
qed

lemma DERIV_neg_dec_left:
fixes f :: "real ⇒ real"
assumes der: "DERIV f x :> l"
and l: "l < 0"
shows "∃d > 0. ∀h > 0. h < d ⟶ 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 ⇒ real"
shows "(f has_real_derivative l) (at x within S) ⟹ 0 < l ⟹
∃d>0. ∀h>0. x - h ∈ S ⟶ h < d ⟶ f (x - h) < f x"
by (rule has_real_derivative_neg_dec_left [of "λx. - f x" "-l" x S, simplified])
(auto simp add: DERIV_minus)

lemma DERIV_pos_inc_left:
fixes f :: "real ⇒ real"
shows "DERIV f x :> l ⟹ 0 < l ⟹ ∃d > 0. ∀h > 0. h < d ⟶ f (x - h) < f x"
using has_real_derivative_pos_inc_left
by blast

lemma has_real_derivative_neg_dec_right:
fixes f :: "real ⇒ real"
shows "(f has_real_derivative l) (at x within S) ⟹ l < 0 ⟹
∃d > 0. ∀h > 0. x + h ∈ S ⟶ h < d ⟶ f x > f (x + h)"
by (rule has_real_derivative_pos_inc_right [of "λx. - f x" "-l" x S, simplified])
(auto simp add: DERIV_minus)

lemma DERIV_neg_dec_right:
fixes f :: "real ⇒ real"
shows "DERIV f x :> l ⟹ l < 0 ⟹ ∃d > 0. ∀h > 0. h < d ⟶ f x > f (x + h)"
using has_real_derivative_neg_dec_right by blast

lemma DERIV_local_max:
fixes f :: "real ⇒ real"
assumes der: "DERIV f x :> l"
and d: "0 < d"
and le: "∀y. ¦x - y¦ < d ⟶ f y ≤ 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: "∀h > 0. h < d' ⟶ f x < f (x - h)"
by blast
obtain e where "0 < e ∧ e < d ∧ e < d'"
using real_lbound_gt_zero [OF d d']  ..
with lt le [THEN spec [where x="x - e"]] show ?thesis
by (auto simp add: abs_if)
next
case greater
from DERIV_pos_inc_right [OF der greater]
obtain d' where d': "0 < d'" and lt: "∀h > 0. h < d' ⟶ f x < f (x + h)"
by blast
obtain e where "0 < e ∧ e < d ∧ e < d'"
using real_lbound_gt_zero [OF d d'] ..
with lt le [THEN spec [where x="x + e"]] show ?thesis
by (auto simp add: abs_if)
qed

text ‹Similar theorem for a local minimum›
lemma DERIV_local_min:
fixes f :: "real ⇒ real"
shows "DERIV f x :> l ⟹ 0 < d ⟹ ∀y. ¦x - y¦ < d ⟶ f x ≤ f y ⟹ l = 0"
by (drule DERIV_minus [THEN DERIV_local_max]) auto

text‹In particular, if a function is locally flat›
lemma DERIV_local_const:
fixes f :: "real ⇒ real"
shows "DERIV f x :> l ⟹ 0 < d ⟹ ∀y. ¦x - y¦ < d ⟶ f x = f y ⟹ l = 0"
by (auto dest!: DERIV_local_max)

subsection ‹Rolle's Theorem›

text ‹Lemma about introducing open ball in open interval›
lemma lemma_interval_lt: "a < x ⟹ x < b ⟹ ∃d. 0 < d ∧ (∀y. ¦x - y¦ < d ⟶ a < y ∧ y < b)"
for a b x :: real
apply (simp add: abs_less_iff)
apply (insert linorder_linear [of "x - a" "b - x"])
apply safe
apply (rule_tac x = "x - a" in exI)
apply (rule_tac [2] x = "b - x" in exI)
apply auto
done

lemma lemma_interval: "a < x ⟹ x < b ⟹ ∃d. 0 < d ∧ (∀y. ¦x - y¦ < d ⟶ a ≤ y ∧ y ≤ b)"
for a b x :: real
apply (drule lemma_interval_lt)
apply auto
apply force
done

text ‹Rolle's Theorem.
If @{term f} is defined and continuous on the closed interval
‹[a,b]› and differentiable on the open interval ‹(a,b)›,
and @{term "f a = f b"},
then there exists ‹x0 ∈ (a,b)› such that @{term "f' x0 = 0"}›
theorem Rolle:
fixes a b :: real
assumes lt: "a < b"
and eq: "f a = f b"
and con: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x"
and dif [rule_format]: "∀x. a < x ∧ x < b ⟶ f differentiable (at x)"
shows "∃z. a < z ∧ z < b ∧ DERIV f z :> 0"
proof -
have le: "a ≤ b"
using lt by simp
from isCont_eq_Ub [OF le con]
obtain x where x_max: "∀z. a ≤ z ∧ z ≤ b ⟶ f z ≤ f x" and "a ≤ x" "x ≤ b"
by blast
from isCont_eq_Lb [OF le con]
obtain x' where x'_min: "∀z. a ≤ z ∧ z ≤ b ⟶ f x' ≤ f z" and "a ≤ x'" "x' ≤ b"
by blast
consider "a < x" "x < b" | "x = a ∨ x = b"
using ‹a ≤ x› ‹x ≤ b› by arith
then show ?thesis
proof cases
case 1
―‹@{term f} attains its maximum within the interval›
obtain d where d: "0 < d" and bound: "∀y. ¦x - y¦ < d ⟶ a ≤ y ∧ y ≤ b"
using lemma_interval [OF 1] by blast
then have bound': "∀y. ¦x - y¦ < d ⟶ f y ≤ f x"
using x_max by blast
obtain l where der: "DERIV f x :> l"
using differentiableD [OF dif [OF conjI [OF 1]]] ..
―‹the derivative at a local maximum is zero›
have "l = 0"
by (rule DERIV_local_max [OF der d bound'])
with 1 der show ?thesis by auto
next
case 2
then have fx: "f b = f x" by (auto simp add: eq)
consider "a < x'" "x' < b" | "x' = a ∨ x' = b"
using ‹a ≤ x'› ‹x' ≤ b› by arith
then show ?thesis
proof cases
case 1
― ‹@{term f} attains its minimum within the interval›
from lemma_interval [OF 1]
obtain d where d: "0<d" and bound: "∀y. ¦x'-y¦ < d ⟶ a ≤ y ∧ y ≤ b"
by blast
then have bound': "∀y. ¦x' - y¦ < d ⟶ f x' ≤ f y"
using x'_min by blast
from differentiableD [OF dif [OF conjI [OF 1]]]
obtain l where der: "DERIV f x' :> l" ..
have "l = 0" by (rule DERIV_local_min [OF der d bound'])
― ‹the derivative at a local minimum is zero›
then show ?thesis using 1 der by auto
next
case 2
― ‹@{term f} is constant throughout the interval›
then have fx': "f b = f x'" by (auto simp: eq)
from dense [OF lt] obtain r where r: "a < r" "r < b" by blast
obtain d where d: "0 < d" and bound: "∀y. ¦r - y¦ < d ⟶ a ≤ y ∧ y ≤ b"
using lemma_interval [OF r] by blast
have eq_fb: "f z = f b" if "a ≤ z" and "z ≤ b" for z
proof (rule order_antisym)
show "f z ≤ f b" by (simp add: fx x_max that)
show "f b ≤ f z" by (simp add: fx' x'_min that)
qed
have bound': "∀y. ¦r - y¦ < d ⟶ f r = f y"
proof (intro strip)
fix y :: real
assume lt: "¦r - y¦ < 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 differentiableD [OF dif [OF conjI [OF r]]] ..
have "l = 0"
by (rule DERIV_local_const [OF der d bound'])
― ‹the derivative of a constant function is zero›
with r der show ?thesis by auto
qed
qed
qed

subsection ‹Mean Value Theorem›

lemma lemma_MVT: "f a - (f b - f a) / (b - a) * a = f b - (f b - f a) / (b - a) * b"
for a b :: real
by (cases "a = b") (simp_all add: field_simps)

theorem MVT:
fixes a b :: real
assumes lt: "a < b"
and con: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x"
and dif [rule_format]: "∀x. a < x ∧ x < b ⟶ f differentiable (at x)"
shows "∃l z. a < z ∧ z < b ∧ DERIV f z :> l ∧ f b - f a = (b - a) * l"
proof -
let ?F = "λx. f x - ((f b - f a) / (b - a)) * x"
have cont_f: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont ?F x"
using con by (fast intro: continuous_intros)
have dif_f: "∀x. a < x ∧ x < b ⟶ ?F differentiable (at x)"
proof clarify
fix x :: real
assume x: "a < x" "x < b"
obtain l where der: "DERIV f x :> l"
using differentiableD [OF dif [OF conjI [OF x]]] ..
show "?F differentiable (at x)"
by (rule differentiableI [where D = "l - (f b - f a) / (b - a)"],
blast intro: DERIV_diff DERIV_cmult_Id der)
qed
from Rolle [where f = ?F, OF lt lemma_MVT cont_f dif_f]
obtain z where z: "a < z" "z < b" and der: "DERIV ?F z :> 0"
by blast
have "DERIV (λx. ((f b - f a) / (b - a)) * x) z :> (f b - f a) / (b - a)"
by (rule DERIV_cmult_Id)
then have der_f: "DERIV (λx. ?F x + (f b - f a) / (b - a) * x) z :> 0 + (f b - f a) / (b - a)"
by (rule DERIV_add [OF der])
show ?thesis
proof (intro exI conjI)
show "a < z" and "z < b" using z .
show "f b - f a = (b - a) * ((f b - f a) / (b - a))" by simp
show "DERIV f z :> ((f b - f a) / (b - a))" using der_f by simp
qed
qed

lemma MVT2:
"a < b ⟹ ∀x. a ≤ x ∧ x ≤ b ⟶ DERIV f x :> f' x ⟹
∃z::real. a < z ∧ z < b ∧ (f b - f a = (b - a) * f' z)"
apply (drule MVT)
apply (blast intro: DERIV_isCont)
apply (force dest: order_less_imp_le simp add: real_differentiable_def)
apply (blast dest: DERIV_unique order_less_imp_le)
done

text ‹A function is constant if its derivative is 0 over an interval.›

lemma DERIV_isconst_end:
fixes f :: "real ⇒ real"
shows "a < b ⟹
∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹
∀x. a < x ∧ x < b ⟶ DERIV f x :> 0 ⟹ f b = f a"
apply (drule (1) MVT)
apply (blast intro: differentiableI)
apply (auto dest!: DERIV_unique simp add: diff_eq_eq)
done

lemma DERIV_isconst1:
fixes f :: "real ⇒ real"
shows "a < b ⟹
∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹
∀x. a < x ∧ x < b ⟶ DERIV f x :> 0 ⟹
∀x. a ≤ x ∧ x ≤ b ⟶ f x = f a"
apply safe
apply (drule_tac x = a in order_le_imp_less_or_eq)
apply safe
apply (drule_tac b = x in DERIV_isconst_end)
apply auto
done

lemma DERIV_isconst2:
fixes f :: "real ⇒ real"
shows "a < b ⟹
∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x ⟹
∀x. a < x ∧ x < b ⟶ DERIV f x :> 0 ⟹
a ≤ x ⟹ x ≤ b ⟹ f x = f a"
by (blast dest: DERIV_isconst1)

lemma DERIV_isconst3:
fixes a b x y :: real
assumes "a < b"
and "x ∈ {a <..< b}"
and "y ∈ {a <..< b}"
and derivable: "⋀x. x ∈ {a <..< b} ⟹ 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 "∀z. ?a ≤ z ∧ z ≤ ?b ⟶ DERIV f z :> 0"
proof (rule allI, rule impI)
fix z :: real
assume "?a ≤ z ∧ z ≤ ?b"
then have "a < z" and "z < b"
using ‹x ∈ {a <..< b}› and ‹y ∈ {a <..< b}› by auto
then have "z ∈ {a<..<b}" by auto
then show "DERIV f z :> 0" by (rule derivable)
qed
then have isCont: "∀z. ?a ≤ z ∧ z ≤ ?b ⟶ isCont f z"
and DERIV: "∀z. ?a < z ∧ z < ?b ⟶ DERIV f z :> 0"
using DERIV_isCont by auto

have "?a < ?b" using ‹x ≠ y› 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 ⇒ real"
shows "∀x. DERIV f x :> 0 ⟹ f x = f y"
apply (rule linorder_cases [of x y])
apply (blast intro: sym DERIV_isCont DERIV_isconst_end)+
done

lemma DERIV_const_ratio_const:
fixes f :: "real ⇒ real"
shows "a ≠ b ⟹ ∀x. DERIV f x :> k ⟹ f b - f a = (b - a) * k"
apply (rule linorder_cases [of a b])
apply auto
apply (drule_tac [!] f = f in MVT)
apply (auto dest: DERIV_isCont DERIV_unique simp: real_differentiable_def)
apply (auto dest: DERIV_unique simp: ring_distribs)
done

lemma DERIV_const_ratio_const2:
fixes f :: "real ⇒ real"
shows "a ≠ b ⟹ ∀x. DERIV f x :> k ⟹ (f b - f a) / (b - a) = k"
apply (rule_tac c1 = "b-a" in mult_right_cancel [THEN iffD1])
apply (auto dest!: DERIV_const_ratio_const simp add: mult.assoc)
done

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 ‹Gallileo's "trick": average velocity = av. of end velocities.›

lemma DERIV_const_average:
fixes v :: "real ⇒ real"
and a b :: real
assumes neq: "a ≠ b"
and der: "∀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
using neq by (force simp add: add.commute)
qed

text ‹
A function with positive derivative is increasing.
A simple proof using the MVT, by Jeremy Avigad. And variants.
›
lemma DERIV_pos_imp_increasing_open:
fixes a b :: real
and f :: "real ⇒ real"
assumes "a < b"
and "⋀x. a < x ⟹ x < b ⟹ (∃y. DERIV f x :> y ∧ y > 0)"
and con: "⋀x. a ≤ x ⟹ x ≤ b ⟹ isCont f x"
shows "f a < f b"
proof (rule ccontr)
assume f: "¬ ?thesis"
have "∃l z. a < z ∧ z < b ∧ DERIV f z :> l ∧ f b - f a = (b - a) * l"
by (rule MVT) (use assms Deriv.differentiableI in ‹force+›)
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 "¬ 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 ⇒ real"
assumes "a < b"
and "∀x. a ≤ x ∧ x ≤ b ⟶ (∃y. DERIV f x :> y ∧ y > 0)"
shows "f a < f b"
by (metis DERIV_pos_imp_increasing_open [of a b f] assms DERIV_continuous less_imp_le)

lemma DERIV_nonneg_imp_nondecreasing:
fixes a b :: real
and f :: "real ⇒ real"
assumes "a ≤ b"
and "∀x. a ≤ x ∧ x ≤ b ⟶ (∃y. DERIV f x :> y ∧ y ≥ 0)"
shows "f a ≤ f b"
proof (rule ccontr, cases "a = b")
assume "¬ ?thesis" and "a = b"
then show False by auto
next
assume *: "¬ ?thesis"
assume "a ≠ b"
with assms have "∃l z. a < z ∧ z < b ∧ DERIV f z :> l ∧ f b - f a = (b - a) * l"
apply -
apply (rule MVT)
apply auto
apply (metis DERIV_isCont)
apply (metis differentiableI less_le)
done
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 "¬ l ≥ 0" by (auto simp add: not_le algebra_simps)
(metis * add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
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 ⇒ real"
assumes "a < b"
and "⋀x. a < x ⟹ x < b ⟹ (∃y. DERIV f x :> y ∧ y < 0)"
and con: "⋀x. a ≤ x ⟹ x ≤ b ⟹ isCont f x"
shows "f a > f b"
proof -
have "(λx. -f x) a < (λx. -f x) b"
apply (rule DERIV_pos_imp_increasing_open [of a b "λx. -f x"])
using assms
apply auto
apply (metis field_differentiable_minus neg_0_less_iff_less)
done
then show ?thesis
by simp
qed

lemma DERIV_neg_imp_decreasing:
fixes a b :: real
and f :: "real ⇒ real"
assumes "a < b"
and "∀x. a ≤ x ∧ x ≤ b ⟶ (∃y. DERIV f x :> y ∧ y < 0)"
shows "f a > f b"
by (metis DERIV_neg_imp_decreasing_open [of a b f] assms DERIV_continuous less_imp_le)

lemma DERIV_nonpos_imp_nonincreasing:
fixes a b :: real
and f :: "real ⇒ real"
assumes "a ≤ b"
and "∀x. a ≤ x ∧ x ≤ b ⟶ (∃y. DERIV f x :> y ∧ y ≤ 0)"
shows "f a ≥ f b"
proof -
have "(λx. -f x) a ≤ (λx. -f x) b"
apply (rule DERIV_nonneg_imp_nondecreasing [of a b "λx. -f x"])
using assms
apply auto
apply (metis DERIV_minus neg_0_le_iff_le)
done
then show ?thesis
by simp
qed

lemma DERIV_pos_imp_increasing_at_bot:
fixes f :: "real ⇒ real"
assumes "⋀x. x ≤ b ⟹ (∃y. DERIV f x :> y ∧ y > 0)"
and lim: "(f ⤏ flim) at_bot"
shows "flim < f b"
proof -
have "∃N. ∀n≤N. f n ≤ f (b - 1)"
apply (rule_tac x="b - 2" in exI)
apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
done
then have "flim ≤ f (b - 1)"
by (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder tendsto_upperbound [OF lim])
also have "… < 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 ⇒ real"
assumes der: "⋀x. x ≥ b ⟹ (∃y. DERIV f x :> y ∧ y < 0)"
and lim: "(f ⤏ flim) at_top"
shows "flim < f b"
apply (rule DERIV_pos_imp_increasing_at_bot [where f = "λi. 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 ‹Derivative of inverse function›

lemma DERIV_inverse_function:
fixes f g :: "real ⇒ real"
assumes der: "DERIV f (g x) :> D"
and neq: "D ≠ 0"
and x: "a < x" "x < b"
and inj: "∀y. a < y ∧ y < b ⟶ f (g y) = y"
and cont: "isCont g x"
shows "DERIV g x :> inverse D"
unfolding DERIV_iff2
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"
by (simp_all add: abs_less_iff)
then show "(g y - g x) / (y - x) = inverse ((f (g y) - x) / (g y - g x))"
by (simp add: inj)
next
have "(λz. (f z - f (g x)) / (z - g x)) ─g x→ D"
by (rule der [unfolded DERIV_iff2])
then have 1: "(λz. (f z - x) / (z - g x)) ─g x→ D"
using inj x by simp
have 2: "∃d>0. ∀y. y ≠ x ∧ norm (y - x) < d ⟶ g y ≠ 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"
by (simp_all add: abs_less_iff)
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 ≠ x"
finally show False by simp
qed
have "(λy. (f (g y) - x) / (g y - g x)) ─x→ D"
using cont 1 2 by (rule isCont_LIM_compose2)
then show "(λy. inverse ((f (g y) - x) / (g y - g x))) ─x→ inverse D"
using neq by (rule tendsto_inverse)
qed

subsection ‹Generalized Mean Value Theorem›

theorem GMVT:
fixes a b :: real
assumes alb: "a < b"
and fc: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont f x"
and fd: "∀x. a < x ∧ x < b ⟶ f differentiable (at x)"
and gc: "∀x. a ≤ x ∧ x ≤ b ⟶ isCont g x"
and gd: "∀x. a < x ∧ x < b ⟶ g differentiable (at x)"
shows "∃g'c f'c c.
DERIV g c :> g'c ∧ DERIV f c :> f'c ∧ a < c ∧ c < b ∧ (f b - f a) * g'c = (g b - g a) * f'c"
proof -
let ?h = "λx. (f b - f a) * g x - (g b - g a) * f x"
have "∃l z. a < z ∧ z < b ∧ DERIV ?h z :> l ∧ ?h b - ?h a = (b - a) * l"
proof (rule MVT)
from assms show "a < b" by simp
show "∀x. a ≤ x ∧ x ≤ b ⟶ isCont ?h x"
using fc gc by simp
show "∀x. a < x ∧ x < b ⟶ ?h differentiable (at x)"
using fd gd by simp
qed
then obtain l where l: "∃z. a < z ∧ z < b ∧ DERIV ?h z :> l ∧ ?h b - ?h a = (b - a) * l" ..
then obtain c where c: "a < c ∧ c < b ∧ DERIV ?h c :> l ∧ ?h b - ?h a = (b - a) * l" ..

from c have cint: "a < c ∧ c < b" by auto
with gd have "g differentiable (at c)" by simp
then have "∃D. DERIV g c :> D" by (rule differentiableD)
then obtain g'c where g'c: "DERIV g c :> g'c" ..

from c have "a < c ∧ c < b" by auto
with fd have "f differentiable (at c)" by simp
then have "∃D. DERIV f c :> D" by (rule differentiableD)
then obtain f'c where f'c: "DERIV f c :> f'c" ..

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 "… = (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))"
by (simp add: algebra_simps)
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 ⇒ real"
assumes "a < b"
and isCont_f: "⋀z. a ≤ z ⟹ z ≤ b ⟹ isCont f z"
and isCont_g: "⋀z. a ≤ z ⟹ z ≤ b ⟹ isCont g z"
and DERIV_g: "⋀z. a < z ⟹ z < b ⟹ DERIV g z :> (g' z)"
and DERIV_f: "⋀z. a < z ⟹ z < b ⟹ DERIV f z :> (f' z)"
shows "∃c. a < c ∧ c < b ∧ (f b - f a) * g' c = (g b - g a) * f' c"
proof -
have "∃g'c f'c c. DERIV g c :> g'c ∧ DERIV f c :> f'c ∧
a < c ∧ c < b ∧ (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 ‹L'Hopitals rule›

lemma isCont_If_ge:
fixes a :: "'a :: linorder_topology"
shows "continuous (at_left a) g ⟹ (f ⤏ g a) (at_right a) ⟹
isCont (λx. if x ≤ a then g x else f x) a"
unfolding isCont_def continuous_within
apply (intro filterlim_split_at)
apply (subst filterlim_cong[OF refl refl, where g=g])
apply (simp_all add: eventually_at_filter less_le)
apply (subst filterlim_cong[OF refl refl, where g=f])
apply (simp_all add: eventually_at_filter less_le)
done

lemma lhopital_right_0:
fixes f0 g0 :: "real ⇒ real"
assumes f_0: "(f0 ⤏ 0) (at_right 0)"
and g_0: "(g0 ⤏ 0) (at_right 0)"
and ev:
"eventually (λx. g0 x ≠ 0) (at_right 0)"
"eventually (λx. g' x ≠ 0) (at_right 0)"
"eventually (λx. DERIV f0 x :> f' x) (at_right 0)"
"eventually (λx. DERIV g0 x :> g' x) (at_right 0)"
and lim: "filterlim (λ x. (f' x / g' x)) F (at_right 0)"
shows "filterlim (λ x. f0 x / g0 x) F (at_right 0)"
proof -
define f where [abs_def]: "f x = (if x ≤ 0 then 0 else f0 x)" for x
then have "f 0 = 0" by simp

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

have "eventually (λx. g0 x ≠ 0 ∧ g' x ≠ 0 ∧
DERIV f0 x :> (f' x) ∧ 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: "⋀x. 0 < x ⟹ x < a ⟹ g0 x ≠ 0"
and g'_neq_0: "⋀x. 0 < x ⟹ x < a ⟹ g' x ≠ 0"
and f0: "⋀x. 0 < x ⟹ x < a ⟹ DERIV f0 x :> (f' x)"
and g0: "⋀x. 0 < x ⟹ x < a ⟹ DERIV g0 x :> (g' x)"
unfolding eventually_at by (auto simp: dist_real_def)

have g_neq_0: "⋀x. 0 < x ⟹ x < a ⟹ g x ≠ 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 "∃ζ. ∀x∈{0 <..< a}. 0 < ζ x ∧ ζ x < x ∧ f x / g x = f' (ζ x) / g' (ζ x)"
proof (rule bchoice, rule ballI)
fix x
assume "x ∈ {0 <..< a}"
then have x[arith]: "0 < x" "x < a" by auto
with g'_neq_0 g_neq_0 ‹g 0 = 0› have g': "⋀x. 0 < x ⟹ x < a  ⟹ 0 ≠ g' x" "g 0 ≠ g x"
by auto
have "⋀x. 0 ≤ x ⟹ x < a ⟹ isCont f x"
using ‹isCont f 0› f by (auto intro: DERIV_isCont simp: le_less)
moreover have "⋀x. 0 ≤ x ⟹ x < a ⟹ isCont g x"
using ‹isCont g 0› g by (auto intro: DERIV_isCont simp: le_less)
ultimately have "∃c. 0 < c ∧ c < x ∧ (f x - f 0) * g' c = (g x - g 0) * f' c"
using f g ‹x < a› 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"
by (simp add: field_simps)
ultimately show "∃y. 0 < y ∧ y < x ∧ f x / g x = f' y / g' y"
using ‹f 0 = 0› ‹g 0 = 0› by (auto intro!: exI[of _ c])
qed
then obtain ζ where "∀x∈{0 <..< a}. 0 < ζ x ∧ ζ x < x ∧ f x / g x = f' (ζ x) / g' (ζ x)" ..
then have ζ: "eventually (λx. 0 < ζ x ∧ ζ x < x ∧ f x / g x = f' (ζ x) / g' (ζ x)) (at_right 0)"
unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
moreover
from ζ have "eventually (λx. norm (ζ x) ≤ x) (at_right 0)"
by eventually_elim auto
then have "((λx. norm (ζ x)) ⤏ 0) (at_right 0)"
by (rule_tac real_tendsto_sandwich[where f="λx. 0" and h="λx. x"]) auto
then have "(ζ ⤏ 0) (at_right 0)"
by (rule tendsto_norm_zero_cancel)
with ζ have "filterlim ζ (at_right 0) (at_right 0)"
by (auto elim!: eventually_mono simp: filterlim_at)
from this lim have "filterlim (λt. f' (ζ t) / g' (ζ t)) F (at_right 0)"
by (rule_tac filterlim_compose[of _ _ _ ζ])
ultimately have "filterlim (λt. 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 ⟷ ?thesis"
by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
finally show ?thesis .
qed

lemma lhopital_right:
"(f ⤏ 0) (at_right x) ⟹ (g ⤏ 0) (at_right x) ⟹
eventually (λx. g x ≠ 0) (at_right x) ⟹
eventually (λx. g' x ≠ 0) (at_right x) ⟹
eventually (λx. DERIV f x :> f' x) (at_right x) ⟹
eventually (λx. DERIV g x :> g' x) (at_right x) ⟹
filterlim (λ x. (f' x / g' x)) F (at_right x) ⟹
filterlim (λ 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 ⤏ 0) (at_left x) ⟹ (g ⤏ 0) (at_left x) ⟹
eventually (λx. g x ≠ 0) (at_left x) ⟹
eventually (λx. g' x ≠ 0) (at_left x) ⟹
eventually (λx. DERIV f x :> f' x) (at_left x) ⟹
eventually (λx. DERIV g x :> g' x) (at_left x) ⟹
filterlim (λ x. (f' x / g' x)) F (at_left x) ⟹
filterlim (λ 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'="λx. - f' (- x)"]) (auto simp: DERIV_mirror)

lemma lhopital:
"(f ⤏ 0) (at x) ⟹ (g ⤏ 0) (at x) ⟹
eventually (λx. g x ≠ 0) (at x) ⟹
eventually (λx. g' x ≠ 0) (at x) ⟹
eventually (λx. DERIV f x :> f' x) (at x) ⟹
eventually (λx. DERIV g x :> g' x) (at x) ⟹
filterlim (λ x. (f' x / g' x)) F (at x) ⟹
filterlim (λ 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 ⇒ real"
assumes g_0: "LIM x at_right 0. g x :> at_top"
and ev:
"eventually (λx. g' x ≠ 0) (at_right 0)"
"eventually (λx. DERIV f x :> f' x) (at_right 0)"
"eventually (λx. DERIV g x :> g' x) (at_right 0)"
and lim: "((λ x. (f' x / g' x)) ⤏ x) (at_right 0)"
shows "((λ x. f x / g x) ⤏ 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 (λt. 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: "⋀x. 0 < x ⟹ x < a ⟹ g' x ≠ 0"
and f0: "⋀x. 0 < x ⟹ x ≤ a ⟹ DERIV f x :> (f' x)"
and g0: "⋀x. 0 < x ⟹ x ≤ a ⟹ DERIV g x :> (g' x)"
and Df: "⋀t. 0 < t ⟹ t < a ⟹ dist (f' t / g' t) x < e / 4"
unfolding eventually_at_le by (auto simp: dist_real_def)

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

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

moreover
have inv_g: "((λx. inverse (g x)) ⤏ 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 "((λx. norm (1 - g a * inverse (g x))) ⤏ norm (1 - g a * 0)) (at_right 0)"
by (intro tendsto_intros)
then have "((λx. norm (1 - g a / g x)) ⤏ 1) (at_right 0)"
by (simp add: inverse_eq_divide)
from this[unfolded tendsto_iff, rule_format, of 1]
have "eventually (λx. norm (1 - g a / g x) < 2) (at_right 0)"
by (auto elim!: eventually_mono simp: dist_real_def)

moreover
from inv_g have "((λt. norm ((f a - x * g a) * inverse (g t))) ⤏ norm ((f a - x * g a) * 0))
(at_right 0)"
by (intro tendsto_intros)
then have "((λt. norm (f a - x * g a) / norm (g t)) ⤏ 0) (at_right 0)"
by (simp add: inverse_eq_divide)
from this[unfolded tendsto_iff, rule_format, of "e / 2"] ‹0 < e›
have "eventually (λt. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
by (auto simp: dist_real_def)

ultimately show "eventually (λt. 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 "∃y. t < y ∧ y < a ∧ (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 ‹g a < g t› 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"
by (simp add: field_simps)
have "norm (f t / g t - x) ≤
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 "… = 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 "… < (e / 4) * 2 + e / 2"
using ineq Df[of y] ‹0 < e› by (intro add_le_less_mono mult_mono) auto
finally show "dist (f t / g t) x < e"
by (simp add: dist_real_def)
qed
qed

lemma lhopital_right_at_top:
"LIM x at_right x. (g::real ⇒ real) x :> at_top ⟹
eventually (λx. g' x ≠ 0) (at_right x) ⟹
eventually (λx. DERIV f x :> f' x) (at_right x) ⟹
eventually (λx. DERIV g x :> g' x) (at_right x) ⟹
((λ x. (f' x / g' x)) ⤏ y) (at_right x) ⟹
((λ x. f x / g x) ⤏ 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 ⟹
eventually (λx. g' x ≠ 0) (at_left x) ⟹
eventually (λx. DERIV f x :> f' x) (at_left x) ⟹
eventually (λx. DERIV g x :> g' x) (at_left x) ⟹
((λ x. (f' x / g' x)) ⤏ y) (at_left x) ⟹
((λ x. f x / g x) ⤏ 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'="λx. - f' (- x)"]) (auto simp: DERIV_mirror)

lemma lhopital_at_top:
"LIM x at x. (g::real ⇒ real) x :> at_top ⟹
eventually (λx. g' x ≠ 0) (at x) ⟹
eventually (λx. DERIV f x :> f' x) (at x) ⟹
eventually (λx. DERIV g x :> g' x) (at x) ⟹
((λ x. (f' x / g' x)) ⤏ y) (at x) ⟹
((λ x. f x / g x) ⤏ 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 ⇒ real"
assumes g_0: "LIM x at_top. g x :> at_top"
and g': "eventually (λx. g' x ≠ 0) at_top"
and Df: "eventually (λx. DERIV f x :> f' x) at_top"
and Dg: "eventually (λx. DERIV g x :> g' x) at_top"
and lim: "((λ x. (f' x / g' x)) ⤏ x) at_top"
shows "((λ x. f x / g x) ⤏ x) at_top"
unfolding filterlim_at_top_to_right
proof (rule lhopital_right_0_at_top)
let ?F = "λx. f (inverse x)"
let ?G = "λx. g (inverse x)"
let ?R = "at_right (0::real)"
let ?D = "λf' 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 (λx. DERIV ?G x  :> ?D g' x) ?R"
unfolding eventually_at_right_to_top
using Dg eventually_ge_at_top[where c=1]
apply eventually_elim
apply (rule DERIV_cong)
apply (rule DERIV_chain'[where f=inverse])
apply (auto intro!:  DERIV_inverse)
done
show "eventually (λx. DERIV ?F x  :> ?D f' x) ?R"
unfolding eventually_at_right_to_top
using Df eventually_ge_at_top[where c=1]
apply eventually_elim
apply (rule DERIV_cong)
apply (rule DERIV_chain'[where f=inverse])
apply (auto intro!:  DERIV_inverse)
done
show "eventually (λx. ?D g' x ≠ 0) ?R"
unfolding eventually_at_right_to_top
using g' eventually_ge_at_top[where c=1]
by eventually_elim auto
show "((λx. ?D f' x / ?D g' x) ⤏ 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 ⇒ 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 (λx. DERIV f x :> f' x) (at_right a)"
"eventually (λx. DERIV g x :> g' x) (at_right a)"
and lim: "filterlim (λ x. (f' x / g' x)) at_top (at_right a)"
shows "filterlim (λ x. f x / g x) at_top (at_right a)"
proof -
from lim have pos: "eventually (λx. f' x / g' x > 0) (at_right a)"
unfolding filterlim_at_top_dense by blast
have "((λx. g x / f x) ⤏ 0) (at_right a)"
proof (rule lhopital_right_at_top)
from pos show "eventually (λx. f' x ≠ 0) (at_right a)" by eventually_elim auto
from tendsto_inverse_0_at_top[OF lim]
show "((λx. g' x / f' x) ⤏ 0) (at_right a)" by simp
qed fact+
moreover from f_0 g_0
have "eventually (λx. f x > 0) (at_right a)" "eventually (λx. g x > 0) (at_right a)"
unfolding filterlim_at_top_dense by blast+
hence "eventually (λx. g x / f x > 0) (at_right a)" by eventually_elim simp
ultimately have "filterlim (λx. 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 ⇒ 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 (λx. DERIV f x :> f' x) (at_right a)"
"eventually (λx. DERIV g x :> g' x) (at_right a)"
and lim: "filterlim (λ x. (f' x / g' x)) at_bot (at_right a)"
shows "filterlim (λ x. f x / g x) at_bot (at_right a)"
proof -
from ev(2) have ev': "eventually (λx. DERIV (λx. -g x) x :> -g' x) (at_right a)"
by eventually_elim (auto intro: derivative_intros)
have "filterlim (λx. f x / (-g x)) at_top (at_right a)"
by (rule lhopital_right_at_top_at_top[where f' = f' and g' = "λx. -g' x"])
(insert assms ev', auto simp: filterlim_uminus_at_bot)
hence "filterlim (λx. -(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 ⇒ 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 (λx. DERIV f x :> f' x) (at_left a)"
"eventually (λx. DERIV g x :> g' x) (at_left a)"
and lim: "filterlim (λ x. (f' x / g' x)) at_top (at_left a)"
shows "filterlim (λ 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'="λx. - f' (- x)"])
(insert assms, auto simp: DERIV_mirror)

lemma lhopital_left_at_top_at_bot:
fixes f g :: "real ⇒ 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 (λx. DERIV f x :> f' x) (at_left a)"
"eventually (λx. DERIV g x :> g' x) (at_left a)"
and lim: "filterlim (λ x. (f' x / g' x)) at_bot (at_left a)"
shows "filterlim (λ 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'="λx. - f' (- x)"])
(insert assms, auto simp: DERIV_mirror)

lemma lhopital_at_top_at_top:
fixes f g :: "real ⇒ 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 (λx. DERIV f x :> f' x) (at a)"
"eventually (λx. DERIV g x :> g' x) (at a)"
and lim: "filterlim (λ x. (f' x / g' x)) at_top (at a)"
shows "filterlim (λ 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 ⇒ 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 (λx. DERIV f x :> f' x) (at a)"
"eventually (λx. DERIV g x :> g' x) (at a)"
and lim: "filterlim (λ x. (f' x / g' x)) at_bot (at a)"
shows "filterlim (λ 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
```