(* Title : Deriv.thy
Author : Jacques D. Fleuriot
Copyright : 1998 University of Cambridge
Author : Brian Huffman
Conversion to Isar and new proofs by Lawrence C Paulson, 2004
GMVT by Benjamin Porter, 2005
*)
section{* Differentiation *}
theory Deriv
imports Limits
begin
subsection {* Frechet derivative *}
definition
has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> bool"
(infix "(has'_derivative)" 50)
where
"(f has_derivative f') F \<longleftrightarrow>
(bounded_linear f' \<and>
((\<lambda>y. ((f y - f (Lim F (\<lambda>x. x))) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) ---> 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 \<Longrightarrow> f' = g' \<Longrightarrow> (f has_derivative g') F"
by simp
definition
has_field_derivative :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a filter \<Rightarrow> bool"
(infix "(has'_field'_derivative)" 50)
where
"(f has_field_derivative D) F \<longleftrightarrow> (f has_derivative op * D) F"
lemma DERIV_cong: "(f has_field_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_field_derivative Y) F"
by simp
definition
has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> real filter \<Rightarrow> bool"
(infix "has'_vector'_derivative" 50)
where
"(f has_vector_derivative f') net \<longleftrightarrow> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
lemma has_vector_derivative_eq_rhs: "(f has_vector_derivative X) F \<Longrightarrow> X = Y \<Longrightarrow> (f has_vector_derivative Y) F"
by simp
named_theorems derivative_intros "structural introduction rules for derivatives"
setup {*
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
Global_Theory.add_thms_dynamic
(@{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 \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> bool"
("(FDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where
"FDERIV f x :> f' \<equiv> (f has_derivative f') (at x)"
lemma has_derivative_bounded_linear: "(f has_derivative f') F \<Longrightarrow> bounded_linear f'"
by (simp add: has_derivative_def)
lemma has_derivative_linear: "(f has_derivative f') F \<Longrightarrow> linear f'"
using bounded_linear.linear[OF has_derivative_bounded_linear] .
lemma has_derivative_ident[derivative_intros, simp]: "((\<lambda>x. x) has_derivative (\<lambda>x. x)) F"
by (simp add: has_derivative_def)
lemma has_derivative_const[derivative_intros, simp]: "((\<lambda>x. c) has_derivative (\<lambda>x. 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 \<Longrightarrow> ((\<lambda>x. f (g x)) has_derivative (\<lambda>x. f (g' x))) F"
using assms 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]
lemma has_derivative_add[simp, derivative_intros]:
assumes f: "(f has_derivative f') F" and g: "(g has_derivative g') F"
shows "((\<lambda>x. f x + g x) has_derivative (\<lambda>x. f' x + g' x)) F"
unfolding has_derivative_def
proof safe
let ?x = "Lim F (\<lambda>x. x)"
let ?D = "\<lambda>f f' y. ((f y - f ?x) - f' (y - ?x)) /\<^sub>R norm (y - ?x)"
have "((\<lambda>x. ?D f f' x + ?D g g' x) ---> (0 + 0)) F"
using f g by (intro tendsto_add) (auto simp: has_derivative_def)
then show "(?D (\<lambda>x. f x + g x) (\<lambda>x. f' x + g' x) ---> 0) F"
by (simp add: field_simps scaleR_add_right scaleR_diff_right)
qed (blast intro: bounded_linear_add f g has_derivative_bounded_linear)
lemma has_derivative_setsum[simp, derivative_intros]:
assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) F"
shows "((\<lambda>x. \<Sum>i\<in>I. f i x) has_derivative (\<lambda>x. \<Sum>i\<in>I. f' i x)) F"
proof cases
assume "finite I" from this f show ?thesis
by induct (simp_all add: f)
qed simp
lemma has_derivative_minus[simp, derivative_intros]: "(f has_derivative f') F \<Longrightarrow> ((\<lambda>x. - f x) has_derivative (\<lambda>x. - f' x)) F"
using has_derivative_scaleR_right[of f f' F "-1"] by simp
lemma has_derivative_diff[simp, derivative_intros]:
"(f has_derivative f') F \<Longrightarrow> (g has_derivative g') F \<Longrightarrow> ((\<lambda>x. f x - g x) has_derivative (\<lambda>x. f' x - g' x)) F"
by (simp only: diff_conv_add_uminus has_derivative_add has_derivative_minus)
lemma has_derivative_at_within:
"(f has_derivative f') (at x within s) \<longleftrightarrow>
(bounded_linear f' \<and> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 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) \<longleftrightarrow>
(bounded_linear f' \<and> ((\<lambda>y. 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) \<longleftrightarrow> (bounded_linear D \<and> (\<lambda>h. 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) \<longleftrightarrow> (\<lambda>h. (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' \<Longrightarrow> ((\<lambda>y. ((f y - f x) - f' (y - x)) /\<^sub>R norm (y - x)) ---> 0) (at x within s) \<Longrightarrow>
(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: "(\<And>y. y \<in> s \<Longrightarrow> y \<noteq> x \<Longrightarrow> dist y x < e \<Longrightarrow> norm ((f y - f x) - f' (y - x)) / norm (y - x) \<le> H y)"
and "(H ---> 0) (at x within s)"
shows "(f has_derivative f') (at x within s)"
unfolding has_derivative_iff_norm
proof safe
show "((\<lambda>y. norm (f y - f x - f' (y - x)) / norm (y - x)) ---> 0) (at x within s)"
proof (rule tendsto_sandwich[where f="\<lambda>x. 0"])
show "(H ---> 0) (at x within s)" by fact
show "eventually (\<lambda>n. norm (f n - f x - f' (n - x)) / norm (n - x) \<le> H n) (at x within s)"
unfolding eventually_at using e sandwich by auto
qed (auto simp: le_divide_eq)
qed fact
lemma has_derivative_subset: "(f has_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> (f has_derivative f') (at x within t)"
by (auto simp add: has_derivative_iff_norm intro: tendsto_within_subset)
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 = "\<lambda>f. (f ---> 0) (at x within s)"
have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x))"
using f unfolding has_derivative_iff_norm by blast
then have "?L (\<lambda>y. norm ((f y - f x) - f' (y - x)) / norm (y - x) * norm (y - x))" (is ?m)
by (rule tendsto_mult_zero) (auto intro!: tendsto_eq_intros)
also have "?m \<longleftrightarrow> ?L (\<lambda>y. norm ((f y - f x) - f' (y - x)))"
by (intro filterlim_cong) (simp_all add: eventually_at_filter)
finally have "?L (\<lambda>y. (f y - f x) - f' (y - x))"
by (rule tendsto_norm_zero_cancel)
then have "?L (\<lambda>y. ((f y - f x) - f' (y - x)) + f' (y - x))"
by (rule tendsto_eq_intros) (auto intro!: tendsto_eq_intros simp: F.zero)
then have "?L (\<lambda>y. f y - f x)"
by simp
from tendsto_add[OF this tendsto_const, of "f x"] show ?thesis
by (simp add: continuous_within)
qed
subsection {* Composition *}
lemma tendsto_at_iff_tendsto_nhds_within: "f x = y \<Longrightarrow> (f ---> y) (at x within s) \<longleftrightarrow> (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_elim1)
lemma has_derivative_in_compose:
assumes f: "(f has_derivative f') (at x within s)"
assumes g: "(g has_derivative g') (at (f x) within (f`s))"
shows "((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
proof -
from f interpret F: bounded_linear f' by (rule has_derivative_bounded_linear)
from g interpret G: bounded_linear g' by (rule has_derivative_bounded_linear)
from F.bounded obtain kF where kF: "\<And>x. norm (f' x) \<le> norm x * kF" by fast
from G.bounded obtain kG where kG: "\<And>x. norm (g' x) \<le> norm x * kG" by fast
note G.tendsto[tendsto_intros]
let ?L = "\<lambda>f. (f ---> 0) (at x within s)"
let ?D = "\<lambda>f f' x y. (f y - f x) - f' (y - x)"
let ?N = "\<lambda>f f' x y. norm (?D f f' x y) / norm (y - x)"
let ?gf = "\<lambda>x. g (f x)" and ?gf' = "\<lambda>x. g' (f' x)"
def Nf \<equiv> "?N f f' x"
def Ng \<equiv> "\<lambda>y. ?N g g' (f x) (f y)"
show ?thesis
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (\<lambda>x. g' (f' x))"
using f g by (blast intro: bounded_linear_compose has_derivative_bounded_linear)
next
fix y::'a assume neq: "y \<noteq> x"
have "?N ?gf ?gf' x y = norm (g' (?D f f' x y) + ?D g g' (f x) (f y)) / norm (y - x)"
by (simp add: G.diff G.add field_simps)
also have "\<dots> \<le> 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 "\<dots> \<le> 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 "\<dots> \<le> norm (?D f f' x y) + norm (f' (y - x))"
by (rule norm_triangle_ineq)
also have "\<dots> \<le> norm (?D f f' x y) + norm (y - x) * kF"
using kF by (intro add_mono) simp
finally show "norm (f y - f x) / norm (y - x) \<le> Nf y + kF"
by (simp add: neq Nf_def field_simps)
qed (insert kG, simp_all add: Ng_def Nf_def neq zero_le_divide_iff field_simps)
finally show "?N ?gf ?gf' x y \<le> Nf y * kG + Ng y * (Nf y + kF)" .
next
have [tendsto_intros]: "?L Nf"
using f unfolding has_derivative_iff_norm Nf_def ..
from f have "(f ---> 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 "((\<lambda>y. 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) \<Longrightarrow> (g has_derivative g') (at (f x)) \<Longrightarrow>
((\<lambda>x. g (f x)) has_derivative (\<lambda>x. g' (f' x))) (at x within s)"
by (blast intro: has_derivative_in_compose has_derivative_subset)
lemma (in bounded_bilinear) FDERIV:
assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
shows "((\<lambda>x. f x ** g x) has_derivative (\<lambda>h. f x ** g' h + f' h ** g x)) (at x within s)"
proof -
from bounded_linear.bounded [OF has_derivative_bounded_linear [OF f]]
obtain KF where norm_F: "\<And>x. norm (f' x) \<le> norm x * KF" by fast
from pos_bounded obtain K where K: "0 < K" and norm_prod:
"\<And>a b. norm (a ** b) \<le> norm a * norm b * K" by fast
let ?D = "\<lambda>f f' y. f y - f x - f' (y - x)"
let ?N = "\<lambda>f f' y. norm (?D f f' y) / norm (y - x)"
def Ng =="?N g g'" and Nf =="?N f f'"
let ?fun1 = "\<lambda>y. norm (f y ** g y - f x ** g x - (f x ** g' (y - x) + f' (y - x) ** g x)) / norm (y - x)"
let ?fun2 = "\<lambda>y. norm (f x) * Ng y * K + Nf y * norm (g y) * K + KF * norm (g y - g x) * K"
let ?F = "at x within s"
show ?thesis
proof (rule has_derivativeI_sandwich[of 1])
show "bounded_linear (\<lambda>h. f x ** g' h + f' h ** g x)"
by (intro bounded_linear_add
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 \<noteq> x"
have "?fun1 y = norm (f x ** ?D g g' y + ?D f f' y ** g y + f' (y - x) ** (g y - g x)) / norm (y - x)"
by (simp add: diff_left diff_right add_left add_right field_simps)
also have "\<dots> \<le> (norm (f x) * norm (?D g g' y) * K + norm (?D f f' y) * norm (g y) * K +
norm (y - x) * KF * norm (g y - g x) * K) / norm (y - x)"
by (intro divide_right_mono mult_mono'
order_trans [OF norm_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 "\<dots> = ?fun2 y"
by (simp add: add_divide_distrib Ng_def Nf_def)
finally show "?fun1 y \<le> ?fun2 y" .
qed simp
qed
lemmas has_derivative_mult[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_mult]
lemmas has_derivative_scaleR[simp, derivative_intros] = bounded_bilinear.FDERIV[OF bounded_bilinear_scaleR]
lemma has_derivative_setprod[simp, derivative_intros]:
fixes f :: "'i \<Rightarrow> 'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
assumes f: "\<And>i. i \<in> I \<Longrightarrow> (f i has_derivative f' i) (at x within s)"
shows "((\<lambda>x. \<Prod>i\<in>I. f i x) has_derivative (\<lambda>y. \<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x))) (at x within s)"
proof cases
assume "finite I" from this f show ?thesis
proof induct
case (insert i I)
let ?P = "\<lambda>y. f i x * (\<Sum>i\<in>I. f' i y * (\<Prod>j\<in>I - {i}. f j x)) + (f' i y) * (\<Prod>i\<in>I. f i x)"
have "((\<lambda>x. f i x * (\<Prod>i\<in>I. f i x)) has_derivative ?P) (at x within s)"
using insert by (intro has_derivative_mult) auto
also have "?P = (\<lambda>y. \<Sum>i'\<in>insert i I. f' i' y * (\<Prod>j\<in>insert i I - {i'}. f j x))"
using insert(1,2) by (auto simp add: setsum_right_distrib insert_Diff_if intro!: ext setsum.cong)
finally show ?case
using insert by simp
qed simp
qed simp
lemma has_derivative_power[simp, derivative_intros]:
fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
assumes f: "(f has_derivative f') (at x within s)"
shows "((\<lambda>x. f x^n) has_derivative (\<lambda>y. of_nat n * f' y * f x^(n - 1))) (at x within s)"
using has_derivative_setprod[OF f, of "{..< n}"] by (simp add: setprod_constant ac_simps)
lemma has_derivative_inverse':
fixes x :: "'a::real_normed_div_algebra"
assumes x: "x \<noteq> 0"
shows "(inverse has_derivative (\<lambda>h. - (inverse x * h * inverse x))) (at x within s)"
(is "(?inv has_derivative ?f) _")
proof (rule has_derivativeI_sandwich)
show "bounded_linear (\<lambda>h. - (?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
next
show "0 < norm x" using x by simp
next
show "((\<lambda>y. 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 \<noteq> x" "dist y x < norm x"
then have "y \<noteq> 0"
by (auto simp: norm_conv_dist dist_commute)
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 \<noteq> 0` x])
apply (subst minus_diff_minus)
apply (subst norm_minus_cancel)
apply (simp add: left_diff_distrib)
done
also have "\<dots> \<le> 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 "\<dots> = norm (?inv y - ?inv x) * norm (?inv x)"
by simp
finally show "norm (?inv y - ?inv x - ?f (y -x)) / norm (y - x) \<le>
norm (?inv y - ?inv x) * norm (?inv x)" .
qed
lemma has_derivative_inverse[simp, derivative_intros]:
fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
assumes x: "f x \<noteq> 0" and f: "(f has_derivative f') (at x within s)"
shows "((\<lambda>x. inverse (f x)) has_derivative (\<lambda>h. - (inverse (f x) * f' h * inverse (f x)))) (at x within s)"
using has_derivative_compose[OF f has_derivative_inverse', OF x] .
lemma has_derivative_divide[simp, derivative_intros]:
fixes f :: "_ \<Rightarrow> 'a::real_normed_div_algebra"
assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)"
assumes x: "g x \<noteq> 0"
shows "((\<lambda>x. f x / g x) has_derivative
(\<lambda>h. - f x * (inverse (g x) * g' h * inverse (g x)) + f' h / g x)) (at x within s)"
using has_derivative_mult[OF f has_derivative_inverse[OF x g]]
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 :: "_ \<Rightarrow> 'a::real_normed_field"
assumes f: "(f has_derivative f') (at x within s)" and g: "(g has_derivative g') (at x within s)" and x: "g x \<noteq> 0"
shows "((\<lambda>x. f x / g x) has_derivative (\<lambda>h. (f' h * g x - f x * g' h) / (g x * g x))) (at x within s)"
proof -
{ fix h
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)"
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 @{text Multivariate_Analysis} for @{text euclidean_space}.
*}
lemma has_derivative_zero_unique:
assumes "((\<lambda>x. 0) has_derivative F) (at x)" shows "F = (\<lambda>h. 0)"
proof -
interpret F: bounded_linear F
using assms by (rule has_derivative_bounded_linear)
let ?r = "\<lambda>h. norm (F h) / norm h"
have *: "?r -- 0 --> 0"
using assms unfolding has_derivative_at by simp
show "F = (\<lambda>h. 0)"
proof
fix h show "F h = 0"
proof (rule ccontr)
assume **: "F h \<noteq> 0"
hence h: "h \<noteq> 0" by (clarsimp simp add: F.zero)
with ** have "0 < ?r h" by simp
from LIM_D [OF * this] obtain s where s: "0 < s"
and r: "\<And>x. x \<noteq> 0 \<Longrightarrow> norm x < s \<Longrightarrow> ?r x < ?r h" by auto
from dense [OF s] obtain t where t: "0 < t \<and> t < s" ..
let ?x = "scaleR (t / norm h) h"
have "?x \<noteq> 0" and "norm ?x < s" using t h by simp_all
hence "?r ?x < ?r h" by (rule r)
thus "False" using t h by (simp add: F.scaleR)
qed
qed
qed
lemma has_derivative_unique:
assumes "(f has_derivative F) (at x)" and "(f has_derivative F') (at x)" shows "F = F'"
proof -
have "((\<lambda>x. 0) has_derivative (\<lambda>h. F h - F' h)) (at x)"
using has_derivative_diff [OF assms] by simp
hence "(\<lambda>h. F h - F' h) = (\<lambda>h. 0)"
by (rule has_derivative_zero_unique)
thus "F = F'"
unfolding fun_eq_iff right_minus_eq .
qed
subsection {* Differentiability predicate *}
definition
differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
(infix "differentiable" 50)
where
"f differentiable F \<longleftrightarrow> (\<exists>D. (f has_derivative D) F)"
lemma differentiable_subset: "f differentiable (at x within s) \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f differentiable (at x within t)"
unfolding differentiable_def by (blast intro: has_derivative_subset)
lemmas differentiable_within_subset = differentiable_subset
lemma differentiable_ident [simp, derivative_intros]: "(\<lambda>x. x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_ident)
lemma differentiable_const [simp, derivative_intros]: "(\<lambda>z. a) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_const)
lemma differentiable_in_compose:
"f differentiable (at (g x) within (g`s)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_in_compose)
lemma differentiable_compose:
"f differentiable (at (g x)) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f (g x)) differentiable (at x within s)"
by (blast intro: differentiable_in_compose differentiable_subset)
lemma differentiable_sum [simp, derivative_intros]:
"f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x + g x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_add)
lemma differentiable_minus [simp, derivative_intros]:
"f differentiable F \<Longrightarrow> (\<lambda>x. - f x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_minus)
lemma differentiable_diff [simp, derivative_intros]:
"f differentiable F \<Longrightarrow> g differentiable F \<Longrightarrow> (\<lambda>x. f x - g x) differentiable F"
unfolding differentiable_def by (blast intro: has_derivative_diff)
lemma differentiable_mult [simp, derivative_intros]:
fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_algebra"
shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x * g x) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_mult)
lemma differentiable_inverse [simp, derivative_intros]:
fixes f :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_inverse)
lemma differentiable_divide [simp, derivative_intros]:
fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow> (\<lambda>x. f x / g x) differentiable (at x within s)"
unfolding divide_inverse using assms by simp
lemma differentiable_power [simp, derivative_intros]:
fixes f g :: "'a :: real_normed_vector \<Rightarrow> 'b :: real_normed_field"
shows "f differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x ^ n) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_power)
lemma differentiable_scaleR [simp, derivative_intros]:
"f differentiable (at x within s) \<Longrightarrow> g differentiable (at x within s) \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) differentiable (at x within s)"
unfolding differentiable_def by (blast intro: has_derivative_scaleR)
lemma has_derivative_imp_has_field_derivative:
"(f has_derivative D) F \<Longrightarrow> (\<And>x. x * D' = D x) \<Longrightarrow> (f has_field_derivative D') F"
unfolding has_field_derivative_def
by (rule has_derivative_eq_rhs[of f D]) (simp_all add: fun_eq_iff mult.commute)
lemma has_field_derivative_imp_has_derivative: "(f has_field_derivative D) F \<Longrightarrow> (f has_derivative op * D) F"
by (simp add: has_field_derivative_def)
lemma DERIV_subset:
"(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s
\<Longrightarrow> (f has_field_derivative f') (at x within t)"
by (simp add: has_field_derivative_def has_derivative_within_subset)
abbreviation (input)
DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
where
"DERIV f x :> D \<equiv> (f has_field_derivative D) (at x)"
abbreviation
has_real_derivative :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> real filter \<Rightarrow> bool"
(infix "(has'_real'_derivative)" 50)
where
"(f has_real_derivative D) F \<equiv> (f has_field_derivative D) F"
lemma real_differentiable_def:
"f differentiable at x within s \<longleftrightarrow> (\<exists>D. (f has_real_derivative D) (at x within s))"
proof safe
assume "f differentiable at x within s"
then obtain f' where *: "(f has_derivative f') (at x within s)"
unfolding differentiable_def by auto
then obtain c where "f' = (op * c)"
by (metis real_bounded_linear has_derivative_bounded_linear mult.commute fun_eq_iff)
with * show "\<exists>D. (f has_real_derivative D) (at x within s)"
unfolding has_field_derivative_def by auto
qed (auto simp: differentiable_def has_field_derivative_def)
lemma real_differentiableE [elim?]:
assumes f: "f differentiable (at x within s)" obtains df where "(f has_real_derivative df) (at x within s)"
using assms by (auto simp: real_differentiable_def)
lemma differentiableD: "f differentiable (at x within s) \<Longrightarrow> \<exists>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) \<Longrightarrow> f differentiable (at x within s)"
by (force simp add: real_differentiable_def)
lemma DERIV_def: "DERIV f x :> D \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
apply (simp add: has_field_derivative_def has_derivative_at 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 mult_commute_abs: "(\<lambda>x. x * c) = op * (c::'a::ab_semigroup_mult)"
by (simp add: fun_eq_iff mult.commute)
subsection {* Derivatives *}
lemma DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) -- 0 --> D"
by (simp add: DERIV_def)
lemma DERIV_const [simp, derivative_intros]: "((\<lambda>x. k) has_field_derivative 0) F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_const]) auto
lemma DERIV_ident [simp, derivative_intros]: "((\<lambda>x. x) has_field_derivative 1) F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_ident]) auto
lemma field_differentiable_add[derivative_intros]:
"(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow>
((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
(auto simp: has_field_derivative_def field_simps mult_commute_abs)
corollary DERIV_add:
"(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
((\<lambda>x. f x + g x) has_field_derivative D + E) (at x within s)"
by (rule field_differentiable_add)
lemma field_differentiable_minus[derivative_intros]:
"(f has_field_derivative f') F \<Longrightarrow> ((\<lambda>z. - (f z)) has_field_derivative -f') F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
(auto simp: has_field_derivative_def field_simps mult_commute_abs)
corollary DERIV_minus: "(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. - f x) has_field_derivative -D) (at x within s)"
by (rule field_differentiable_minus)
lemma field_differentiable_diff[derivative_intros]:
"(f has_field_derivative f') F \<Longrightarrow> (g has_field_derivative g') F \<Longrightarrow> ((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)
corollary DERIV_diff:
"(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
((\<lambda>x. f x - g x) has_field_derivative D - E) (at x within s)"
by (rule field_differentiable_diff)
lemma DERIV_continuous: "(f has_field_derivative D) (at x within s) \<Longrightarrow> continuous (at x within s) f"
by (drule has_derivative_continuous[OF has_field_derivative_imp_has_derivative]) simp
corollary DERIV_isCont: "DERIV f x :> D \<Longrightarrow> isCont f x"
by (rule DERIV_continuous)
lemma DERIV_continuous_on:
"(\<And>x. x \<in> s \<Longrightarrow> (f has_field_derivative D) (at x)) \<Longrightarrow> continuous_on s f"
by (metis DERIV_continuous continuous_at_imp_continuous_on)
lemma DERIV_mult':
"(f has_field_derivative D) (at x within s) \<Longrightarrow> (g has_field_derivative E) (at x within s) \<Longrightarrow>
((\<lambda>x. f x * g x) has_field_derivative f x * E + D * g x) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
(auto simp: field_simps mult_commute_abs dest: has_field_derivative_imp_has_derivative)
lemma DERIV_mult[derivative_intros]:
"(f has_field_derivative Da) (at x within s) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
((\<lambda>x. f x * g x) has_field_derivative Da * g x + Db * f x) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_mult])
(auto simp: field_simps dest: has_field_derivative_imp_has_derivative)
text {* Derivative of linear multiplication *}
lemma DERIV_cmult:
"(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. c * f x) has_field_derivative c * D) (at x within s)"
by (drule DERIV_mult' [OF DERIV_const], simp)
lemma DERIV_cmult_right:
"(f has_field_derivative D) (at x within s) ==> ((\<lambda>x. f x * c) has_field_derivative D * c) (at x within s)"
using DERIV_cmult by (force simp add: ac_simps)
lemma DERIV_cmult_Id [simp]: "(op * c has_field_derivative c) (at x within s)"
by (cut_tac c = c and x = x in DERIV_ident [THEN DERIV_cmult], simp)
lemma DERIV_cdivide:
"(f has_field_derivative D) (at x within s) \<Longrightarrow> ((\<lambda>x. f x / c) has_field_derivative D / c) (at x within s)"
using DERIV_cmult_right[of f D x s "1 / c"] by simp
lemma DERIV_unique:
"DERIV f x :> D \<Longrightarrow> DERIV f x :> E \<Longrightarrow> D = E"
unfolding DERIV_def by (rule LIM_unique)
lemma DERIV_setsum[derivative_intros]:
"(\<And> n. n \<in> S \<Longrightarrow> ((\<lambda>x. f x n) has_field_derivative (f' x n)) F) \<Longrightarrow>
((\<lambda>x. setsum (f x) S) has_field_derivative setsum (f' x) S) F"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_setsum])
(auto simp: setsum_right_distrib mult_commute_abs dest: has_field_derivative_imp_has_derivative)
lemma DERIV_inverse'[derivative_intros]:
"(f has_field_derivative D) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
((\<lambda>x. inverse (f x)) has_field_derivative - (inverse (f x) * D * inverse (f x))) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_inverse])
(auto dest: has_field_derivative_imp_has_derivative)
text {* Power of @{text "-1"} *}
lemma DERIV_inverse:
"x \<noteq> 0 \<Longrightarrow> ((\<lambda>x. inverse(x)) has_field_derivative - (inverse x ^ Suc (Suc 0))) (at x within s)"
by (drule DERIV_inverse' [OF DERIV_ident]) simp
text {* Derivative of inverse *}
lemma DERIV_inverse_fun:
"(f has_field_derivative d) (at x within s) \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
((\<lambda>x. inverse (f x)) has_field_derivative (- (d * inverse(f x ^ Suc (Suc 0))))) (at x within s)"
by (drule (1) DERIV_inverse') (simp add: ac_simps nonzero_inverse_mult_distrib)
text {* Derivative of quotient *}
lemma DERIV_divide[derivative_intros]:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
(g has_field_derivative E) (at x within s) \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
((\<lambda>x. f x / g x) has_field_derivative (D * g x - f x * E) / (g x * g x)) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_divide])
(auto dest: has_field_derivative_imp_has_derivative simp: field_simps)
lemma DERIV_quotient:
"(f has_field_derivative d) (at x within s) \<Longrightarrow>
(g has_field_derivative e) (at x within s)\<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
((\<lambda>y. f y / g y) has_field_derivative (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))) (at x within s)"
by (drule (2) DERIV_divide) (simp add: mult.commute)
lemma DERIV_power_Suc:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. f x ^ Suc n) has_field_derivative (1 + of_nat n) * (D * f x ^ n)) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
(auto simp: has_field_derivative_def)
lemma DERIV_power[derivative_intros]:
"(f has_field_derivative D) (at x within s) \<Longrightarrow>
((\<lambda>x. f x ^ n) has_field_derivative of_nat n * (D * f x ^ (n - Suc 0))) (at x within s)"
by (rule has_derivative_imp_has_field_derivative[OF has_derivative_power])
(auto simp: has_field_derivative_def)
lemma DERIV_pow: "((\<lambda>x. x ^ n) has_field_derivative real n * (x ^ (n - Suc 0))) (at x within s)"
apply (cut_tac DERIV_power [OF DERIV_ident])
apply (simp add: real_of_nat_def)
done
lemma DERIV_chain': "(f has_field_derivative D) (at x within s) \<Longrightarrow> DERIV g (f x) :> E \<Longrightarrow>
((\<lambda>x. g (f x)) has_field_derivative E * D) (at x within s)"
using has_derivative_compose[of f "op * D" x s g "op * E"]
unfolding has_field_derivative_def mult_commute_abs ac_simps .
corollary DERIV_chain2: "DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
((\<lambda>x. f (g x)) has_field_derivative Da * Db) (at x within s)"
by (rule DERIV_chain')
text {* Standard version *}
lemma DERIV_chain:
"DERIV f (g x) :> Da \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
(f o g has_field_derivative Da * Db) (at x within s)"
by (drule (1) DERIV_chain', simp add: o_def mult.commute)
lemma DERIV_image_chain:
"(f has_field_derivative Da) (at (g x) within (g ` s)) \<Longrightarrow> (g has_field_derivative Db) (at x within s) \<Longrightarrow>
(f o 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 "(\<And>x. x \<in> s \<Longrightarrow> DERIV g x :> g'(x))"
and "DERIV f x :> f'"
and "f x \<in> s"
shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
by (metis (full_types) DERIV_chain' mult.commute assms)
lemma DERIV_chain3: (*HAS_COMPLEX_DERIVATIVE_CHAIN_UNIV*)
assumes "(\<And>x. DERIV g x :> g'(x))"
and "DERIV f x :> f'"
shows "DERIV (\<lambda>x. g(f x)) x :> f' * g'(f x)"
by (metis UNIV_I DERIV_chain_s [of UNIV] assms)
declare
DERIV_power[where 'a=real, unfolded real_of_nat_def[symmetric], derivative_intros]
text{*Alternative definition for differentiability*}
lemma DERIV_LIM_iff:
fixes f :: "'a::{real_normed_vector,inverse} \<Rightarrow> 'a" shows
"((%h. (f(a + h) - f(a)) / h) -- 0 --> D) =
((%x. (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)
apply (simp add: add.commute)
done
lemma DERIV_iff2: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) --x --> D"
by (simp add: DERIV_def DERIV_LIM_iff)
lemma DERIV_cong_ev: "x = y \<Longrightarrow> eventually (\<lambda>x. f x = g x) (nhds x) \<Longrightarrow> u = v \<Longrightarrow>
DERIV f x :> u \<longleftrightarrow> DERIV g y :> v"
unfolding DERIV_iff2
proof (rule filterlim_cong)
assume *: "eventually (\<lambda>x. f x = g x) (nhds x)"
moreover from * have "f x = g x" by (auto simp: eventually_nhds)
moreover assume "x = y" "u = v"
ultimately show "eventually (\<lambda>xa. (f xa - f x) / (xa - x) = (g xa - g y) / (xa - y)) (at x)"
by (auto simp: eventually_at_filter elim: eventually_elim1)
qed simp_all
lemma DERIV_shift:
"(DERIV f (x + z) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (x + z)) x :> y)"
by (simp add: DERIV_def field_simps)
lemma DERIV_mirror:
"(DERIV f (- x) :> y) \<longleftrightarrow> (DERIV (\<lambda>x. f (- x::real) :: real) x :> - y)"
by (simp add: DERIV_def filterlim_at_split filterlim_at_left_to_right
tendsto_minus_cancel_left field_simps conj_commute)
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) \<longleftrightarrow> (\<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isCont g x \<and> g x = l)"
(is "?lhs = ?rhs")
proof
assume der: "DERIV f x :> l"
show "\<exists>g. (\<forall>z. f z - f x = g z * (z-x)) \<and> isCont g x \<and> g x = l"
proof (intro exI conjI)
let ?g = "(%z. if z = x then l else (f z - f x) / (z-x))"
show "\<forall>z. f z - f x = ?g z * (z-x)" by simp
show "isCont ?g x" using der
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
"(\<forall>z. f z - f x = g z * (z-x))" and "isCont g x" and "g x = l" by blast
thus "(DERIV f x :> l)"
by (auto simp add: isCont_iff DERIV_def cong: LIM_cong)
qed
text {*
Let's do the standard proof, though theorem
@{text "LIM_mult2"} follows from a NS proof
*}
subsection {* Local extrema *}
text{*If @{term "0 < f'(x)"} then @{term x} is Locally Strictly Increasing At The Right*}
lemma DERIV_pos_inc_right:
fixes f :: "real => real"
assumes der: "DERIV f x :> l"
and l: "0 < l"
shows "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x + h)"
proof -
from l der [THEN DERIV_D, THEN LIM_D [where r = "l"]]
have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l)"
by simp
then obtain s
where s: "0 < s"
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < l"
by auto
thus ?thesis
proof (intro exI conjI strip)
show "0<s" using s .
fix h::real
assume "0 < h" "h < s"
with all [of h] show "f x < f (x+h)"
proof (simp add: abs_if pos_less_divide_eq split add: split_if_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
thus "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 "\<exists>d > 0. \<forall>h > 0. h < d --> f(x) < f(x-h)"
proof -
from l der [THEN DERIV_D, THEN LIM_D [where r = "-l"]]
have "\<exists>s > 0. (\<forall>z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l)"
by simp
then obtain s
where s: "0 < s"
and all: "!!z. z \<noteq> 0 \<and> \<bar>z\<bar> < s \<longrightarrow> \<bar>(f(x+z) - f x) / z - l\<bar> < -l"
by auto
thus ?thesis
proof (intro exI conjI strip)
show "0<s" using s .
fix h::real
assume "0 < h" "h < s"
with all [of "-h"] show "f x < f (x-h)"
proof (simp add: abs_if pos_less_divide_eq split add: split_if_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
thus "f x < f (x-h)"
by (simp add: pos_less_divide_eq h)
qed
qed
qed
lemma DERIV_pos_inc_left:
fixes f :: "real => real"
shows "DERIV f x :> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x - h) < f(x)"
apply (rule DERIV_neg_dec_left [of "%x. - f x" "-l" x, simplified])
apply (auto simp add: DERIV_minus)
done
lemma DERIV_neg_dec_right:
fixes f :: "real => real"
shows "DERIV f x :> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>d > 0. \<forall>h > 0. h < d --> f(x) > f(x + h)"
apply (rule DERIV_pos_inc_right [of "%x. - f x" "-l" x, simplified])
apply (auto simp add: DERIV_minus)
done
lemma DERIV_local_max:
fixes f :: "real => real"
assumes der: "DERIV f x :> l"
and d: "0 < d"
and le: "\<forall>y. \<bar>x-y\<bar> < d --> f(y) \<le> f(x)"
shows "l = 0"
proof (cases rule: linorder_cases [of l 0])
case equal thus ?thesis .
next
case less
from DERIV_neg_dec_left [OF der less]
obtain d' where d': "0 < d'"
and lt: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x-h)" by blast
from real_lbound_gt_zero [OF d d']
obtain e where "0 < e \<and> e < d \<and> e < 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: "\<forall>h > 0. h < d' \<longrightarrow> f x < f (x + h)" by blast
from real_lbound_gt_zero [OF d d']
obtain e where "0 < e \<and> e < d \<and> e < 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; \<forall>y. \<bar>x-y\<bar> < d --> f(x) \<le> 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; \<forall>y. \<bar>x-y\<bar> < 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 |]
==> \<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a < y & y < b)"
apply (simp add: abs_less_iff)
apply (insert linorder_linear [of "x-a" "b-x"], safe)
apply (rule_tac x = "x-a" in exI)
apply (rule_tac [2] x = "b-x" in exI, auto)
done
lemma lemma_interval: "[| a < x; x < b |] ==>
\<exists>d::real. 0 < d & (\<forall>y. \<bar>x-y\<bar> < d --> a \<le> y & y \<le> b)"
apply (drule lemma_interval_lt, auto)
apply force
done
text{*Rolle's Theorem.
If @{term f} is defined and continuous on the closed interval
@{text "[a,b]"} and differentiable on the open interval @{text "(a,b)"},
and @{term "f(a) = f(b)"},
then there exists @{text "x0 \<in> (a,b)"} such that @{term "f'(x0) = 0"}*}
theorem Rolle:
assumes lt: "a < b"
and eq: "f(a) = f(b)"
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
shows "\<exists>z::real. a < z & z < b & DERIV f z :> 0"
proof -
have le: "a \<le> b" using lt by simp
from isCont_eq_Ub [OF le con]
obtain x where x_max: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f z \<le> f x"
and alex: "a \<le> x" and xleb: "x \<le> b"
by blast
from isCont_eq_Lb [OF le con]
obtain x' where x'_min: "\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> f x' \<le> f z"
and alex': "a \<le> x'" and x'leb: "x' \<le> b"
by blast
show ?thesis
proof cases
assume axb: "a < x & x < b"
--{*@{term f} attains its maximum within the interval*}
hence ax: "a<x" and xb: "x<b" by arith +
from lemma_interval [OF ax xb]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
hence bound': "\<forall>y. \<bar>x-y\<bar> < d \<longrightarrow> f y \<le> f x" using x_max
by blast
from differentiableD [OF dif [OF axb]]
obtain l where der: "DERIV f x :> l" ..
have "l=0" by (rule DERIV_local_max [OF der d bound'])
--{*the derivative at a local maximum is zero*}
thus ?thesis using ax xb der by auto
next
assume notaxb: "~ (a < x & x < b)"
hence xeqab: "x=a | x=b" using alex xleb by arith
hence fb_eq_fx: "f b = f x" by (auto simp add: eq)
show ?thesis
proof cases
assume ax'b: "a < x' & x' < b"
--{*@{term f} attains its minimum within the interval*}
hence ax': "a<x'" and x'b: "x'<b" by arith+
from lemma_interval [OF ax' x'b]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
hence bound': "\<forall>y. \<bar>x'-y\<bar> < d \<longrightarrow> f x' \<le> f y" using x'_min
by blast
from differentiableD [OF dif [OF ax'b]]
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*}
thus ?thesis using ax' x'b der by auto
next
assume notax'b: "~ (a < x' & x' < b)"
--{*@{term f} is constant througout the interval*}
hence x'eqab: "x'=a | x'=b" using alex' x'leb by arith
hence fb_eq_fx': "f b = f x'" by (auto simp add: eq)
from dense [OF lt]
obtain r where ar: "a < r" and rb: "r < b" by blast
from lemma_interval [OF ar rb]
obtain d where d: "0<d" and bound: "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> a \<le> y \<and> y \<le> b"
by blast
have eq_fb: "\<forall>z. a \<le> z --> z \<le> b --> f z = f b"
proof (clarify)
fix z::real
assume az: "a \<le> z" and zb: "z \<le> b"
show "f z = f b"
proof (rule order_antisym)
show "f z \<le> f b" by (simp add: fb_eq_fx x_max az zb)
show "f b \<le> f z" by (simp add: fb_eq_fx' x'_min az zb)
qed
qed
have bound': "\<forall>y. \<bar>r-y\<bar> < d \<longrightarrow> f r = f y"
proof (intro strip)
fix y::real
assume lt: "\<bar>r-y\<bar> < d"
hence "f y = f b" by (simp add: eq_fb bound)
thus "f r = f y" by (simp add: eq_fb ar rb order_less_imp_le)
qed
from differentiableD [OF dif [OF conjI [OF ar rb]]]
obtain l where der: "DERIV f r :> l" ..
have "l=0" by (rule DERIV_local_const [OF der d bound'])
--{*the derivative of a constant function is zero*}
thus ?thesis using ar rb der 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::real)"
by (cases "a = b") (simp_all add: field_simps)
theorem MVT:
assumes lt: "a < b"
and con: "\<forall>x. a \<le> x & x \<le> b --> isCont f x"
and dif [rule_format]: "\<forall>x. a < x & x < b --> f differentiable (at x)"
shows "\<exists>l z::real. a < z & z < b & DERIV f z :> l &
(f(b) - f(a) = (b-a) * l)"
proof -
let ?F = "%x. f x - ((f b - f a) / (b-a)) * x"
have contF: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?F x"
using con by (fast intro: continuous_intros)
have difF: "\<forall>x. a < x \<and> x < b \<longrightarrow> ?F differentiable (at x)"
proof (clarify)
fix x::real
assume ax: "a < x" and xb: "x < b"
from differentiableD [OF dif [OF conjI [OF ax xb]]]
obtain l where der: "DERIV f x :> l" ..
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 contF difF]
obtain z where az: "a < z" and zb: "z < b" and der: "DERIV ?F z :> 0"
by blast
have "DERIV (%x. ((f b - f a)/(b-a)) * x) z :> (f b - f a)/(b-a)"
by (rule DERIV_cmult_Id)
hence derF: "DERIV (\<lambda>x. ?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" using az .
show "z < b" using zb .
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 derF by simp
qed
qed
lemma MVT2:
"[| a < b; \<forall>x. a \<le> x & x \<le> b --> DERIV f x :> f'(x) |]
==> \<exists>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;
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
\<forall>x. a < x & x < b --> DERIV f x :> 0 |]
==> f b = f a"
apply (drule MVT, assumption)
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;
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
\<forall>x. a < x & x < b --> DERIV f x :> 0 |]
==> \<forall>x. a \<le> x & x \<le> b --> f x = f a"
apply safe
apply (drule_tac x = a in order_le_imp_less_or_eq, safe)
apply (drule_tac b = x in DERIV_isconst_end, auto)
done
lemma DERIV_isconst2:
fixes f :: "real => real"
shows "[| a < b;
\<forall>x. a \<le> x & x \<le> b --> isCont f x;
\<forall>x. a < x & x < b --> DERIV f x :> 0;
a \<le> x; x \<le> b |]
==> f x = f a"
apply (blast dest: DERIV_isconst1)
done
lemma DERIV_isconst3: fixes a b x y :: real
assumes "a < b" and "x \<in> {a <..< b}" and "y \<in> {a <..< b}"
assumes derivable: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> DERIV f x :> 0"
shows "f x = f y"
proof (cases "x = y")
case False
let ?a = "min x y"
let ?b = "max x y"
have "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> DERIV f z :> 0"
proof (rule allI, rule impI)
fix z :: real assume "?a \<le> z \<and> z \<le> ?b"
hence "a < z" and "z < b" using `x \<in> {a <..< b}` and `y \<in> {a <..< b}` by auto
hence "z \<in> {a<..<b}" by auto
thus "DERIV f z :> 0" by (rule derivable)
qed
hence isCont: "\<forall>z. ?a \<le> z \<and> z \<le> ?b \<longrightarrow> isCont f z"
and DERIV: "\<forall>z. ?a < z \<and> z < ?b \<longrightarrow> DERIV f z :> 0" using DERIV_isCont by auto
have "?a < ?b" using `x \<noteq> 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 "\<forall>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 \<noteq> b; \<forall>x. DERIV f x :> k |] ==> (f(b) - f(a)) = (b-a) * k"
apply (rule linorder_cases [of a b], auto)
apply (drule_tac [!] f = f in MVT)
apply (auto dest: DERIV_isCont DERIV_unique simp add: real_differentiable_def)
apply (auto dest: DERIV_unique simp add: ring_distribs)
done
lemma DERIV_const_ratio_const2:
fixes f :: "real => real"
shows "[|a \<noteq> b; \<forall>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::real)"
by (simp)
lemma real_average_minus_second [simp]: "((b + a)/2 - a) = (b-a)/(2::real)"
by (simp)
text{*Gallileo's "trick": average velocity = av. of end velocities*}
lemma DERIV_const_average:
fixes v :: "real => real"
assumes neq: "a \<noteq> (b::real)"
and der: "\<forall>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])
hence "(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])
hence "(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
(* 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::real and b::real and f::"real => real"
assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
shows "f a < f b"
proof (rule ccontr)
assume f: "~ f a < f b"
have "EX l z. a < z & z < b & DERIV f z :> l
& f b - f a = (b - a) * l"
apply (rule MVT)
using assms Deriv.differentiableI
apply force+
done
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::real and b::real and f::"real => real"
assumes "a < b" and "\<forall>x. a \<le> x & x \<le> b --> (EX 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::real and b::real and f::"real => real"
assumes "a \<le> b" and
"\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<ge> 0)"
shows "f a \<le> f b"
proof (rule ccontr, cases "a = b")
assume "~ f a \<le> f b" and "a = b"
then show False by auto
next
assume A: "~ f a \<le> f b"
assume B: "a ~= b"
with assms have "EX 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 z: "a < z" "z < b" "DERIV f z :> l"
and C: "f b - f a = (b - a) * l"
by auto
with A have "a < b" "f b < f a" by auto
with C have "\<not> l \<ge> 0" by (auto simp add: not_le algebra_simps)
(metis A add_le_cancel_right assms(1) less_eq_real_def mult_right_mono add_left_mono linear order_refl)
with assms z show False
by (metis DERIV_unique order_less_imp_le)
qed
lemma DERIV_neg_imp_decreasing_open:
fixes a::real and b::real and f::"real => real"
assumes "a < b" and "\<And>x. a < x \<Longrightarrow> x < b \<Longrightarrow> (EX y. DERIV f x :> y & y < 0)"
and con: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> isCont f x"
shows "f a > f b"
proof -
have "(%x. -f x) a < (%x. -f x) b"
apply (rule DERIV_pos_imp_increasing_open [of a b "%x. -f x"])
using assms
apply auto
apply (metis field_differentiable_minus neg_0_less_iff_less)
done
thus ?thesis
by simp
qed
lemma DERIV_neg_imp_decreasing:
fixes a::real and b::real and f::"real => real"
assumes "a < b" and
"\<forall>x. a \<le> x & x \<le> b --> (\<exists>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::real and b::real and f::"real => real"
assumes "a \<le> b" and
"\<forall>x. a \<le> x & x \<le> b --> (\<exists>y. DERIV f x :> y & y \<le> 0)"
shows "f a \<ge> f b"
proof -
have "(%x. -f x) a \<le> (%x. -f x) b"
apply (rule DERIV_nonneg_imp_nondecreasing [of a b "%x. -f x"])
using assms
apply auto
apply (metis DERIV_minus neg_0_le_iff_le)
done
thus ?thesis
by simp
qed
lemma DERIV_pos_imp_increasing_at_bot:
fixes f :: "real => real"
assumes "\<And>x. x \<le> b \<Longrightarrow> (EX y. DERIV f x :> y & y > 0)"
and lim: "(f ---> flim) at_bot"
shows "flim < f b"
proof -
have "flim \<le> f (b - 1)"
apply (rule tendsto_ge_const [OF _ lim])
apply (auto simp: trivial_limit_at_bot_linorder eventually_at_bot_linorder)
apply (rule_tac x="b - 2" in exI)
apply (force intro: order.strict_implies_order DERIV_pos_imp_increasing [where f=f] assms)
done
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: "\<And>x. x \<ge> b \<Longrightarrow> (EX 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 = "\<lambda>i. f (-i)" and b = "-b", simplified])
apply (metis DERIV_mirror der le_minus_iff neg_0_less_iff_less)
apply (metis filterlim_at_top_mirror lim)
done
text {* Derivative of inverse function *}
lemma DERIV_inverse_function:
fixes f g :: "real \<Rightarrow> real"
assumes der: "DERIV f (g x) :> D"
assumes neq: "D \<noteq> 0"
assumes a: "a < x" and b: "x < b"
assumes inj: "\<forall>y. a < y \<and> y < b \<longrightarrow> f (g y) = y"
assumes 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 a b by arith
next
fix y
assume "norm (y - x) < min (x - a) (b - x)"
hence "a < y" and "y < b"
by (simp_all add: abs_less_iff)
thus "(g y - g x) / (y - x) =
inverse ((f (g y) - x) / (g y - g x))"
by (simp add: inj)
next
have "(\<lambda>z. (f z - f (g x)) / (z - g x)) -- g x --> D"
by (rule der [unfolded DERIV_iff2])
hence 1: "(\<lambda>z. (f z - x) / (z - g x)) -- g x --> D"
using inj a b by simp
have 2: "\<exists>d>0. \<forall>y. y \<noteq> x \<and> norm (y - x) < d \<longrightarrow> g y \<noteq> g x"
proof (rule exI, safe)
show "0 < min (x - a) (b - x)"
using a b by simp
next
fix y
assume "norm (y - x) < min (x - a) (b - x)"
hence y: "a < y" "y < b"
by (simp_all add: abs_less_iff)
assume "g y = g x"
hence "f (g y) = f (g x)" by simp
hence "y = x" using inj y a b by simp
also assume "y \<noteq> x"
finally show False by simp
qed
have "(\<lambda>y. (f (g y) - x) / (g y - g x)) -- x --> D"
using cont 1 2 by (rule isCont_LIM_compose2)
thus "(\<lambda>y. 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: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
and fd: "\<forall>x. a < x \<and> x < b \<longrightarrow> f differentiable (at x)"
and gc: "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont g x"
and gd: "\<forall>x. a < x \<and> x < b \<longrightarrow> g differentiable (at x)"
shows "\<exists>g'c f'c c.
DERIV g c :> g'c \<and> DERIV f c :> f'c \<and> a < c \<and> c < b \<and> ((f b - f a) * g'c) = ((g b - g a) * f'c)"
proof -
let ?h = "\<lambda>x. (f b - f a)*(g x) - (g b - g a)*(f x)"
from assms have "a < b" by simp
moreover have "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont ?h x"
using fc gc by simp
moreover have "\<forall>x. a < x \<and> x < b \<longrightarrow> ?h differentiable (at x)"
using fd gd by simp
ultimately have "\<exists>l z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" by (rule MVT)
then obtain l where ldef: "\<exists>z. a < z \<and> z < b \<and> DERIV ?h z :> l \<and> ?h b - ?h a = (b - a) * l" ..
then obtain c where cdef: "a < c \<and> c < b \<and> DERIV ?h c :> l \<and> ?h b - ?h a = (b - a) * l" ..
from cdef have cint: "a < c \<and> c < b" by auto
with gd have "g differentiable (at c)" by simp
hence "\<exists>D. DERIV g c :> D" by (rule differentiableD)
then obtain g'c where g'cdef: "DERIV g c :> g'c" ..
from cdef have "a < c \<and> c < b" by auto
with fd have "f differentiable (at c)" by simp
hence "\<exists>D. DERIV f c :> D" by (rule differentiableD)
then obtain f'c where f'cdef: "DERIV f c :> f'c" ..
from cdef 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'cdef f'cdef 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)
{
from cdef have "?h b - ?h a = (b - a) * l" by auto
also from leq have "\<dots> = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
finally have "?h b - ?h a = (b - a) * (g'c * (f b - f a) - f'c * (g b - g a))" by simp
}
moreover
{
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)
hence "?h b - ?h a = 0" by auto
}
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
hence "g'c * (f b - f a) = f'c * (g b - g a)" by simp
hence "(f b - f a) * g'c = (g b - g a) * f'c" by (simp add: ac_simps)
with g'cdef f'cdef cint show ?thesis by auto
qed
lemma GMVT':
fixes f g :: "real \<Rightarrow> real"
assumes "a < b"
assumes isCont_f: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont f z"
assumes isCont_g: "\<And>z. a \<le> z \<Longrightarrow> z \<le> b \<Longrightarrow> isCont g z"
assumes DERIV_g: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV g z :> (g' z)"
assumes DERIV_f: "\<And>z. a < z \<Longrightarrow> z < b \<Longrightarrow> DERIV f z :> (f' z)"
shows "\<exists>c. a < c \<and> c < b \<and> (f b - f a) * g' c = (g b - g a) * f' c"
proof -
have "\<exists>g'c f'c c. DERIV g c :> g'c \<and> DERIV f c :> f'c \<and>
a < c \<and> c < b \<and> (f b - f a) * g'c = (g b - g a) * f'c"
using assms by (intro GMVT) (force simp: real_differentiable_def)+
then obtain c where "a < c" "c < b" "(f b - f a) * g' c = (g b - g a) * f' c"
using DERIV_f DERIV_g by (force dest: DERIV_unique)
then show ?thesis
by auto
qed
subsection {* L'Hopitals rule *}
lemma isCont_If_ge:
fixes a :: "'a :: linorder_topology"
shows "continuous (at_left a) g \<Longrightarrow> (f ---> g a) (at_right a) \<Longrightarrow> isCont (\<lambda>x. if x \<le> 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 \<Rightarrow> real"
assumes f_0: "(f0 ---> 0) (at_right 0)"
assumes g_0: "(g0 ---> 0) (at_right 0)"
assumes ev:
"eventually (\<lambda>x. g0 x \<noteq> 0) (at_right 0)"
"eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
"eventually (\<lambda>x. DERIV f0 x :> f' x) (at_right 0)"
"eventually (\<lambda>x. DERIV g0 x :> g' x) (at_right 0)"
assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
shows "((\<lambda> x. f0 x / g0 x) ---> x) (at_right 0)"
proof -
def f \<equiv> "\<lambda>x. if x \<le> 0 then 0 else f0 x"
then have "f 0 = 0" by simp
def g \<equiv> "\<lambda>x. if x \<le> 0 then 0 else g0 x"
then have "g 0 = 0" by simp
have "eventually (\<lambda>x. g0 x \<noteq> 0 \<and> g' x \<noteq> 0 \<and>
DERIV f0 x :> (f' x) \<and> DERIV g0 x :> (g' x)) (at_right 0)"
using ev by eventually_elim auto
then obtain a where [arith]: "0 < a"
and g0_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g0 x \<noteq> 0"
and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
and f0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV f0 x :> (f' x)"
and g0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> DERIV g0 x :> (g' x)"
unfolding eventually_at by (auto simp: dist_real_def)
have g_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g x \<noteq> 0"
using g0_neq_0 by (simp add: g_def)
{ fix x assume x: "0 < x" "x < a" then have "DERIV f x :> (f' x)"
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]) }
note f = this
{ fix x assume x: "0 < x" "x < a" then have "DERIV g x :> (g' x)"
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]) }
note g = this
have "isCont f 0"
unfolding f_def by (intro isCont_If_ge f_0 continuous_const)
have "isCont g 0"
unfolding g_def by (intro isCont_If_ge g_0 continuous_const)
have "\<exists>\<zeta>. \<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)"
proof (rule bchoice, rule)
fix x assume "x \<in> {0 <..< a}"
then have x[arith]: "0 < x" "x < a" by auto
with g'_neq_0 g_neq_0 `g 0 = 0` have g': "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> 0 \<noteq> g' x" "g 0 \<noteq> g x"
by auto
have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont f x"
using `isCont f 0` f by (auto intro: DERIV_isCont simp: le_less)
moreover have "\<And>x. 0 \<le> x \<Longrightarrow> x < a \<Longrightarrow> isCont g x"
using `isCont g 0` g by (auto intro: DERIV_isCont simp: le_less)
ultimately have "\<exists>c. 0 < c \<and> c < x \<and> (f x - f 0) * g' c = (g x - g 0) * f' c"
using f g `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 "\<exists>y. 0 < y \<and> y < x \<and> f x / g x = f' y / g' y"
using `f 0 = 0` `g 0 = 0` by (auto intro!: exI[of _ c])
qed
then obtain \<zeta> where "\<forall>x\<in>{0 <..< a}. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)" ..
then have \<zeta>: "eventually (\<lambda>x. 0 < \<zeta> x \<and> \<zeta> x < x \<and> f x / g x = f' (\<zeta> x) / g' (\<zeta> x)) (at_right 0)"
unfolding eventually_at by (intro exI[of _ a]) (auto simp: dist_real_def)
moreover
from \<zeta> have "eventually (\<lambda>x. norm (\<zeta> x) \<le> x) (at_right 0)"
by eventually_elim auto
then have "((\<lambda>x. norm (\<zeta> x)) ---> 0) (at_right 0)"
by (rule_tac real_tendsto_sandwich[where f="\<lambda>x. 0" and h="\<lambda>x. x"]) auto
then have "(\<zeta> ---> 0) (at_right 0)"
by (rule tendsto_norm_zero_cancel)
with \<zeta> have "filterlim \<zeta> (at_right 0) (at_right 0)"
by (auto elim!: eventually_elim1 simp: filterlim_at)
from this lim have "((\<lambda>t. f' (\<zeta> t) / g' (\<zeta> t)) ---> x) (at_right 0)"
by (rule_tac filterlim_compose[of _ _ _ \<zeta>])
ultimately have "((\<lambda>t. f t / g t) ---> x) (at_right 0)" (is ?P)
by (rule_tac filterlim_cong[THEN iffD1, OF refl refl])
(auto elim: eventually_elim1)
also have "?P \<longleftrightarrow> ?thesis"
by (rule filterlim_cong) (auto simp: f_def g_def eventually_at_filter)
finally show ?thesis .
qed
lemma lhopital_right:
"((f::real \<Rightarrow> real) ---> 0) (at_right x) \<Longrightarrow> (g ---> 0) (at_right x) \<Longrightarrow>
eventually (\<lambda>x. g x \<noteq> 0) (at_right x) \<Longrightarrow>
eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
((\<lambda> 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)
lemma lhopital_left:
"((f::real \<Rightarrow> real) ---> 0) (at_left x) \<Longrightarrow> (g ---> 0) (at_left x) \<Longrightarrow>
eventually (\<lambda>x. g x \<noteq> 0) (at_left x) \<Longrightarrow>
eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
((\<lambda> x. f x / g x) ---> y) (at_left x)"
unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
by (rule lhopital_right[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
lemma lhopital:
"((f::real \<Rightarrow> real) ---> 0) (at x) \<Longrightarrow> (g ---> 0) (at x) \<Longrightarrow>
eventually (\<lambda>x. g x \<noteq> 0) (at x) \<Longrightarrow>
eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
((\<lambda> x. f x / g x) ---> y) (at x)"
unfolding eventually_at_split filterlim_at_split
by (auto intro!: lhopital_right[of f x g g' f'] lhopital_left[of f x g g' f'])
lemma lhopital_right_0_at_top:
fixes f g :: "real \<Rightarrow> real"
assumes g_0: "LIM x at_right 0. g x :> at_top"
assumes ev:
"eventually (\<lambda>x. g' x \<noteq> 0) (at_right 0)"
"eventually (\<lambda>x. DERIV f x :> f' x) (at_right 0)"
"eventually (\<lambda>x. DERIV g x :> g' x) (at_right 0)"
assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) (at_right 0)"
shows "((\<lambda> 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 (\<lambda>t. dist (f' t / g' t) x < e / 4) (at_right 0)" by simp
from eventually_conj[OF eventually_conj[OF ev(1) ev(2)] eventually_conj[OF ev(3) this]]
obtain a where [arith]: "0 < a"
and g'_neq_0: "\<And>x. 0 < x \<Longrightarrow> x < a \<Longrightarrow> g' x \<noteq> 0"
and f0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV f x :> (f' x)"
and g0: "\<And>x. 0 < x \<Longrightarrow> x \<le> a \<Longrightarrow> DERIV g x :> (g' x)"
and Df: "\<And>t. 0 < t \<Longrightarrow> t < a \<Longrightarrow> dist (f' t / g' t) x < e / 4"
unfolding eventually_at_le by (auto simp: dist_real_def)
from Df have
"eventually (\<lambda>t. t < a) (at_right 0)" "eventually (\<lambda>t::real. 0 < t) (at_right 0)"
unfolding eventually_at by (auto intro!: exI[of _ a] simp: dist_real_def)
moreover
have "eventually (\<lambda>t. 0 < g t) (at_right 0)" "eventually (\<lambda>t. g a < g t) (at_right 0)"
using g_0 by (auto elim: eventually_elim1 simp: filterlim_at_top_dense)
moreover
have inv_g: "((\<lambda>x. 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 "((\<lambda>x. norm (1 - g a * inverse (g x))) ---> norm (1 - g a * 0)) (at_right 0)"
by (intro tendsto_intros)
then have "((\<lambda>x. 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 (\<lambda>x. norm (1 - g a / g x) < 2) (at_right 0)"
by (auto elim!: eventually_elim1 simp: dist_real_def)
moreover
from inv_g have "((\<lambda>t. 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 "((\<lambda>t. 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 (\<lambda>t. norm (f a - x * g a) / norm (g t) < e / 2) (at_right 0)"
by (auto simp: dist_real_def)
ultimately show "eventually (\<lambda>t. dist (f t / g t) x < e) (at_right 0)"
proof eventually_elim
fix t assume t[arith]: "0 < t" "t < a" "g a < g t" "0 < g t"
assume ineq: "norm (1 - g a / g t) < 2" "norm (f a - x * g a) / norm (g t) < e / 2"
have "\<exists>y. t < y \<and> y < a \<and> (g a - g t) * f' y = (f a - f t) * g' y"
using f0 g0 t(1,2) by (intro GMVT') (force intro!: DERIV_isCont)+
then obtain y where [arith]: "t < y" "y < a"
and D_eq0: "(g a - g t) * f' y = (f a - f t) * g' y"
by blast
from D_eq0 have D_eq: "(f t - f a) / (g t - g a) = f' y / g' y"
using `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) \<le>
norm (((f t - f a) / (g t - g a) - x) * (1 - g a / g t)) + norm ((f a - x * g a) / g t)"
unfolding * by (rule norm_triangle_ineq)
also have "\<dots> = dist (f' y / g' y) x * norm (1 - g a / g t) + norm (f a - x * g a) / norm (g t)"
by (simp add: abs_mult D_eq dist_real_def)
also have "\<dots> < (e / 4) * 2 + e / 2"
using ineq Df[of y] `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 \<Rightarrow> real) x :> at_top \<Longrightarrow>
eventually (\<lambda>x. g' x \<noteq> 0) (at_right x) \<Longrightarrow>
eventually (\<lambda>x. DERIV f x :> f' x) (at_right x) \<Longrightarrow>
eventually (\<lambda>x. DERIV g x :> g' x) (at_right x) \<Longrightarrow>
((\<lambda> x. (f' x / g' x)) ---> y) (at_right x) \<Longrightarrow>
((\<lambda> 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::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
eventually (\<lambda>x. g' x \<noteq> 0) (at_left x) \<Longrightarrow>
eventually (\<lambda>x. DERIV f x :> f' x) (at_left x) \<Longrightarrow>
eventually (\<lambda>x. DERIV g x :> g' x) (at_left x) \<Longrightarrow>
((\<lambda> x. (f' x / g' x)) ---> y) (at_left x) \<Longrightarrow>
((\<lambda> x. f x / g x) ---> y) (at_left x)"
unfolding eventually_at_left_to_right filterlim_at_left_to_right DERIV_mirror
by (rule lhopital_right_at_top[where f'="\<lambda>x. - f' (- x)"]) (auto simp: DERIV_mirror)
lemma lhopital_at_top:
"LIM x at x. (g::real \<Rightarrow> real) x :> at_top \<Longrightarrow>
eventually (\<lambda>x. g' x \<noteq> 0) (at x) \<Longrightarrow>
eventually (\<lambda>x. DERIV f x :> f' x) (at x) \<Longrightarrow>
eventually (\<lambda>x. DERIV g x :> g' x) (at x) \<Longrightarrow>
((\<lambda> x. (f' x / g' x)) ---> y) (at x) \<Longrightarrow>
((\<lambda> 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 \<Rightarrow> real"
assumes g_0: "LIM x at_top. g x :> at_top"
assumes g': "eventually (\<lambda>x. g' x \<noteq> 0) at_top"
assumes Df: "eventually (\<lambda>x. DERIV f x :> f' x) at_top"
assumes Dg: "eventually (\<lambda>x. DERIV g x :> g' x) at_top"
assumes lim: "((\<lambda> x. (f' x / g' x)) ---> x) at_top"
shows "((\<lambda> x. f x / g x) ---> x) at_top"
unfolding filterlim_at_top_to_right
proof (rule lhopital_right_0_at_top)
let ?F = "\<lambda>x. f (inverse x)"
let ?G = "\<lambda>x. g (inverse x)"
let ?R = "at_right (0::real)"
let ?D = "\<lambda>f' x. f' (inverse x) * - (inverse x ^ Suc (Suc 0))"
show "LIM x ?R. ?G x :> at_top"
using g_0 unfolding filterlim_at_top_to_right .
show "eventually (\<lambda>x. DERIV ?G x :> ?D g' x) ?R"
unfolding eventually_at_right_to_top
using Dg eventually_ge_at_top[where c="1::real"]
apply eventually_elim
apply (rule DERIV_cong)
apply (rule DERIV_chain'[where f=inverse])
apply (auto intro!: DERIV_inverse)
done
show "eventually (\<lambda>x. DERIV ?F x :> ?D f' x) ?R"
unfolding eventually_at_right_to_top
using Df eventually_ge_at_top[where c="1::real"]
apply eventually_elim
apply (rule DERIV_cong)
apply (rule DERIV_chain'[where f=inverse])
apply (auto intro!: DERIV_inverse)
done
show "eventually (\<lambda>x. ?D g' x \<noteq> 0) ?R"
unfolding eventually_at_right_to_top
using g' eventually_ge_at_top[where c="1::real"]
by eventually_elim auto
show "((\<lambda>x. ?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::real"]
by eventually_elim simp
qed
end