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RIntExt.v
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RIntExt.v
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(*
Extra lemmas about Riemann integrals that are not found in the coquelicot library.
The most important for ProbCompCert is is_RInt_comp_noncont, which
is a lemma about change of variables in integrals. Coquelicot has a
change of variables lemmas (is_RInt_comp), but it requires that the
integrand be continuous. This continuity assumption seems hard to
satisfy in the context of ProbCompCert, where we would have to
argue that (1) the PDF of a program is continuous and (2) that the
iterated improprer integrals in the definition of the probability
distribution are also continuous.
In contrast, to avoid that, the is_RInt_comp_noncont lemma proved
here removes that continuity assumption.
*)
Require Export RelationClasses Morphisms Utf8.
From mathcomp Require Import ssreflect ssrbool eqtype.
From Coquelicot Require Import Hierarchy Markov Rcomplements Rbar Lub Lim_seq SF_seq Continuity Hierarchy RInt RInt_analysis Derive AutoDerive.
Require ClassicalEpsilon.
Require Import Reals.
Require Import Coqlib.
Require Import Psatz.
Require Import Program.Basics.
Require Import RealsExt Transforms.
Import Rbar.
(** Left/Right limits of integral boundary *)
Lemma is_RInt_upper_bound_left_lim a b f v :
Rlt a b ->
is_RInt f a b v ->
is_left_lim (RInt f a) b (RInt f a b).
Proof.
intros Hlt His.
unfold is_left_lim. split; first done.
unfold filterlim, filter_le, filtermap, Rbar_at_left, within, Rbar_locally, locally. intros P HP.
destruct HP as (eps&Heps).
cut (∃ eps' : posreal, ∀ y, ball b eps' y -> Rbar_lt y b -> ball (RInt f a b) eps (RInt f a y)).
{ intros (eps'&Heps'). exists eps'. intros y Hball Hbar. apply Heps, Heps'; auto. }
edestruct (ex_RInt_ub f a b) as (ub&Hub); first (econstructor; eauto).
assert (∀ t, a <= t <= b -> Rabs (RInt f t b) <= (b - t) * ub).
{ intros t Hle. apply abs_RInt_le_const; intuition.
{ apply: ex_RInt_Chasles_2.
{ split; eassumption. }
econstructor; eauto. }
apply Hub. split.
{ rewrite Rmin_left; nra. }
{ rewrite Rmax_right; nra. }
}
assert (0 <= ub).
{ specialize (Hub a). transitivity (norm (f a)).
{ apply norm_ge_0. }
{ apply Hub. rewrite Rmin_left ?Rmax_right; nra. }
}
assert (Heps': 0 < eps / (ub + 1)).
{ apply Rdiv_lt_0_compat.
{ destruct eps; auto. }
{ nra. }
}
set (eps' := (mkposreal _ Heps')).
edestruct (eps_squeeze_between a b eps') as (eps''&Hsmaller&Heps''); auto.
exists eps''.
rewrite /ball/=/AbsRing_ball/=/abs/=/minus/plus/opp/=.
intros y Hball Hlty.
assert (a <= y).
{ apply Heps''. auto. }
rewrite -(RInt_Chasles f a y b); swap 1 3.
{ eapply ex_RInt_Chasles_2; last first.
{ econstructor; eauto. }
split; nra. }
{ eapply ex_RInt_Chasles_1; last first.
{ econstructor; eauto. }
split; nra. }
eapply (Rle_lt_trans _ (Rabs (RInt f y b))).
{ right. rewrite -Rabs_Ropp. f_equal. rewrite /plus//=. nra. }
eapply (Rle_lt_trans); first (eapply H; nra).
rewrite Rabs_left in Hball; last first.
{ nra. }
assert (b - y < eps'' ) by nra.
apply (Rle_lt_trans _ (eps' * ub)).
{ apply Rmult_le_compat_r; nra. }
rewrite /eps' /=. rewrite /Rdiv. rewrite Rmult_assoc.
apply (Rlt_le_trans _ (eps * 1)); last nra.
apply Rmult_lt_compat_l; first by (destruct eps).
rewrite Rmult_comm. apply (Rdiv_lt_1 ub (ub + 1)); nra.
Qed.
Lemma is_RInt_lower_bound_right_lim a b f v :
Rlt a b ->
is_RInt f a b v ->
is_right_lim (λ x, RInt f x b) a (RInt f a b).
Proof.
intros Hlt His.
unfold is_right_lim. split; first done.
unfold filterlim, filter_le, filtermap, Rbar_at_right, within, Rbar_locally, locally. intros P HP.
destruct HP as (eps&Heps).
cut (∃ eps' : posreal, ∀ y, ball a eps' y -> Rbar_lt a y -> ball (RInt f a b) eps (RInt f y b)).
{ intros (eps'&Heps'). exists eps'. intros y Hball Hbar. apply Heps, Heps'; auto. }
edestruct (ex_RInt_ub f a b) as (ub&Hub); first (econstructor; eauto).
assert (∀ t, a <= t <= b -> Rabs (RInt f a t) <= (t - a) * ub).
{ intros t Hle. apply abs_RInt_le_const; intuition.
{ apply: ex_RInt_Chasles_1.
{ split; eassumption. }
econstructor; eauto. }
apply Hub. split.
{ rewrite Rmin_left; nra. }
{ rewrite Rmax_right; nra. }
}
assert (0 <= ub).
{ specialize (Hub a). transitivity (norm (f a)).
{ apply norm_ge_0. }
{ apply Hub. rewrite Rmin_left ?Rmax_right; nra. }
}
assert (Heps': 0 < eps / (ub + 1)).
{ apply Rdiv_lt_0_compat.
{ destruct eps; auto. }
{ nra. }
}
set (eps' := (mkposreal _ Heps')).
edestruct (eps_squeeze_between' a b eps') as (eps''&Hsmaller&Heps''); auto.
exists eps''.
rewrite /ball/=/AbsRing_ball/=/abs/=/minus/plus/opp/=.
intros y Hball Hlty.
assert (y <= b).
{ apply Heps''. auto. }
rewrite -(RInt_Chasles f a y b); swap 1 3.
{ eapply ex_RInt_Chasles_2; last first.
{ econstructor; eauto. }
split; nra. }
{ eapply ex_RInt_Chasles_1; last first.
{ econstructor; eauto. }
split; nra. }
eapply (Rle_lt_trans _ (Rabs (RInt f a y))).
{ right. rewrite -Rabs_Ropp. f_equal. rewrite /plus//=. nra. }
eapply (Rle_lt_trans); first (eapply H; nra).
rewrite Rabs_right in Hball; last first.
{ nra. }
assert (y - a < eps'' ) by nra.
apply (Rle_lt_trans _ (eps' * ub)).
{ apply Rmult_le_compat_r; nra. }
rewrite /eps' /=. rewrite /Rdiv. rewrite Rmult_assoc.
apply (Rlt_le_trans _ (eps * 1)); last nra.
apply Rmult_lt_compat_l; first by (destruct eps).
rewrite Rmult_comm. apply (Rdiv_lt_1 ub (ub + 1)); nra.
Qed.
Lemma is_RInt_comp' : ∀ (f : R → R) (g dg : R → R) (a b : R),
a <= b →
(∀ x : R, a <= x <= b → continuous f (g x)) →
(∀ x : R, a <= x <= b → is_derive g x (dg x) ∧ continuous dg x) →
is_RInt (λ y : R, scal (dg y) (f (g y))) a b (RInt f (g a) (g b)).
Proof.
intros f g dg a b Hle Hcont Hdiff. apply: is_RInt_comp.
- intros x. rewrite Rmin_left // Rmax_right //. apply Hcont.
- intros x. rewrite Rmin_left // Rmax_right //. apply Hdiff.
Qed.
Lemma ex_RInt_comp' : ∀ (f : R → R) (g dg : R → R) (a b : R),
a <= b →
(∀ x : R, a <= x <= b → continuous f (g x)) →
(∀ x : R, a <= x <= b → is_derive g x (dg x) ∧ continuous dg x) →
ex_RInt (λ y : R, scal (dg y) (f (g y))) a b.
Proof.
intros f g dg a b Hle Hcont Hdiff. eexists; apply: is_RInt_comp'; auto.
Qed.
Lemma RInt_comp' : ∀ (f : R → R) (g dg : R → R) (a b : R),
a <= b →
(∀ x : R, a <= x <= b → continuous f (g x)) →
(∀ x : R, a <= x <= b → is_derive g x (dg x) ∧ continuous dg x) →
RInt (λ y : R, scal (dg y) (f (g y))) a b = RInt f (g a) (g b).
Proof.
intros f g dg a b Hle Hcont Hdiff. apply: RInt_comp.
- intros x. rewrite Rmin_left // Rmax_right //. apply Hcont.
- intros x. rewrite Rmin_left // Rmax_right //. apply Hdiff.
Qed.
Lemma Rbar_at_left_interval a b (P: Rbar -> Prop) :
Rbar_lt a b ->
(∀ x, Rbar_lt a x -> Rbar_lt x b -> P x) ->
Rbar_at_left b P.
Proof.
intros Hlt HP. unfold Rbar_at_left, within.
apply open_Rbar_gt' in Hlt. move:Hlt. apply filter_imp.
intros. apply HP; auto.
Qed.
Lemma Rbar_at_right_interval a b (P: Rbar -> Prop) :
Rbar_lt a b ->
(∀ x, Rbar_lt a x -> Rbar_lt x b -> P x) ->
Rbar_at_right a P.
Proof.
intros Hlt HP. unfold Rbar_at_right, within.
apply open_Rbar_lt' in Hlt. move:Hlt. apply filter_imp.
intros. apply HP; auto.
Qed.
Lemma ball_interval_lb r eps x :
ball r eps x ->
r - eps < x.
Proof.
rewrite /ball/=/AbsRing_ball/abs/=/minus/plus/opp//=. apply Rabs_case; nra.
Qed.
Lemma ball_interval_ub r eps x :
ball r eps x ->
x < r + eps.
Proof.
rewrite /ball/=/AbsRing_ball/abs/=/minus/plus/opp//=. apply Rabs_case; nra.
Qed.
Lemma not_Rbar_at_left b P :
¬ Rbar_at_left b P →
match b with
| Finite r => ∀ eps : posreal, ∃ x, r - eps < x < r ∧ ¬ P x
| p_infty => ∀ M, ∃ x, M < x ∧ ¬ P x
| m_infty => True
end.
Proof.
intros Hneg.
destruct b; auto.
- intros eps. unfold Rbar_at_left, within, Rbar_locally, locally in Hneg.
specialize (Classical_Pred_Type.not_ex_all_not _ _ Hneg eps) => /= Heps.
apply Classical_Pred_Type.not_all_ex_not in Heps.
destruct Heps as (x&Hx).
assert (Hx': ¬ ((ball r eps x ∧ x < r) → P x)) by intuition.
eapply Classical_Prop.not_imply_elim in Hx'.
exists x.
split; last first.
{ eapply Classical_Prop.not_imply_elim2 in Hx; eauto. }
intuition.
apply ball_interval_lb; auto.
- intros M. unfold Rbar_at_left, within, Rbar_locally, locally in Hneg.
specialize (Classical_Pred_Type.not_ex_all_not _ _ Hneg M) => /= Heps.
apply Classical_Pred_Type.not_all_ex_not in Heps.
destruct Heps as (x&Hx).
assert (Hx': ¬ (M < x → P x)) by intuition.
eapply Classical_Prop.not_imply_elim in Hx'.
exists x.
split; last first.
{ eapply Classical_Prop.not_imply_elim2 in Hx; eauto. }
auto.
Qed.
Lemma not_Rbar_at_right b P :
¬ Rbar_at_right b P →
match b with
| Finite r => ∀ eps : posreal, ∃ x, r < x < r + eps ∧ ¬ P x
| p_infty => True
| m_infty => ∀ M, ∃ x, x < M ∧ ¬ P x
end.
Proof.
intros Hneg.
destruct b; auto.
- intros eps. unfold Rbar_at_right, within, Rbar_locally, locally in Hneg.
specialize (Classical_Pred_Type.not_ex_all_not _ _ Hneg eps) => /= Heps.
apply Classical_Pred_Type.not_all_ex_not in Heps.
destruct Heps as (x&Hx).
assert (Hx': ¬ ((ball r eps x ∧ r < x) → P x)) by intuition.
eapply Classical_Prop.not_imply_elim in Hx'.
exists x.
split; last first.
{ eapply Classical_Prop.not_imply_elim2 in Hx; eauto. }
intuition.
apply ball_interval_ub; auto.
- intros M. unfold Rbar_at_right, within, Rbar_locally, locally in Hneg.
specialize (Classical_Pred_Type.not_ex_all_not _ _ Hneg M) => /= Heps.
apply Classical_Pred_Type.not_all_ex_not in Heps.
destruct Heps as (x&Hx).
assert (Hx': ¬ (x < M → P x)) by intuition.
eapply Classical_Prop.not_imply_elim in Hx'.
exists x.
split; last first.
{ eapply Classical_Prop.not_imply_elim2 in Hx; eauto. }
auto.
Qed.
Lemma interval_inhabited (x y : R) : x < y -> ∃ z, x < z < y.
Proof.
intros.
edestruct (boule_of_interval x y) as (s&(r&Heq1&Heq2)); auto.
exists s. nra.
Qed.
Lemma Rbar_interval_inhabited (a b : Rbar) :
Rbar_lt a b -> ∃ (c : R), Rbar_lt a c /\ Rbar_lt c b.
Proof.
destruct a as [r | | ].
- destruct b as [ r' | |].
* simpl. apply interval_inhabited.
* intros _. exists (r + 1). simpl; split; auto; first nra.
* simpl; intuition.
- inversion 1.
- destruct b as [ r' | | ].
* intros _. exists (r' - 1). simpl; split; auto; last nra.
* intros _. exists 0; simpl; done.
* inversion 1.
Qed.
Lemma Rbar_at_left_witness (r: R) (eps: posreal) P:
Rbar_at_left r P -> ∃ x, r - eps < x < r ∧ P x.
Proof.
unfold Rbar_at_left, within, Rbar_locally, locally.
intros Hex. destruct Hex as (eps'&Heps').
set (lb := r - Rmin eps eps').
edestruct (interval_inhabited lb r) as (x&Hin).
{ rewrite /lb. apply Rmin_case; destruct eps, eps' => /=; nra. }
exists x. split.
{ move: Hin. rewrite /lb. apply Rmin_case_strong; destruct eps, eps' => /=; nra. }
apply Heps'; last by intuition.
rewrite /ball/=/AbsRing_ball/abs/=/minus/plus/opp//=.
{ move: Hin. rewrite /lb. apply Rabs_case; apply Rmin_case_strong; destruct eps, eps' => /=; nra. }
Qed.
Lemma Rbar_at_left_witness_above (r: R) y P:
Rbar_at_left r P -> y < r -> ∃ x, y < x < r ∧ P x.
Proof.
intros. assert (Hpos: 0 < r - y) by nra.
edestruct (Rbar_at_left_witness r (mkposreal _ Hpos) P) as (x&?&HP); auto.
exists x. split; auto.
simpl in H1. nra.
Qed.
Lemma Rbar_at_left_witness_above_p_infty y P:
Rbar_at_left p_infty P -> ∃ x, y < x ∧ P x.
Proof.
unfold Rbar_at_left, within, Rbar_locally.
intros HM. destruct HM as (M&HM). exists (Rmax (M + 1) (y + 1)).
split.
- apply (Rlt_le_trans _ (y + 1)); first nra.
apply Rmax_r.
- apply HM; simpl; auto.
- apply (Rlt_le_trans _ (M + 1)); first nra.
apply Rmax_l.
Qed.
Lemma Rbar_at_left_strict_monotone (t : R) (b : Rbar) g glim :
Rbar_lt t b →
(∀ (x y : R), Rbar_lt t x /\ x < y → Rbar_lt y b → g x < g y) →
is_left_lim g b glim ->
Rbar_at_left b (λ y : Rbar, Rbar_lt (g y) glim).
Proof.
unfold is_left_lim.
intros Hltb Ht (Hnm&Hlim).
apply Classical_Prop.NNPP. intros Hneg%not_Rbar_at_left.
destruct b; try congruence.
- unfold filterlim, filter_le, filtermap in Hlim.
assert (Hpos: 0 < r - t).
{ simpl in Hltb. nra. }
set (eps' := mkposreal _ Hpos).
specialize (Hneg eps').
assert (∃ x : R, (r - eps' < x ∧ x < r) ∧ Rbar_lt glim (g x)) as (x&Hrange&r0).
{
destruct Hneg as (x&Hrange&Hnlt).
apply Rbar_not_lt_le in Hnlt.
apply Rbar_le_lt_or_eq_dec in Hnlt.
destruct Hnlt as [Hlt|Heq].
{ exists x. split; eauto. }
destruct (interval_inhabited x r) as (x'&Hx'1&Hx'2); first nra.
exists x'.
split; first nra.
rewrite Heq. simpl. apply Ht; auto; split; try nra.
move: Hrange. rewrite /eps' /=. nra.
}
apply open_Rbar_lt' in r0. apply Hlim in r0.
eapply (Rbar_at_left_witness_above r x) in r0; try (intuition eauto; done).
destruct r0 as (y&Hrange'&Hlt).
simpl in Hlt. apply Rlt_not_le in Hlt.
apply Hlt. left. apply Ht; simpl; simpl in Hltb; try nra.
split; last by intuition.
move: Hrange. rewrite /eps' /=. nra.
- unfold filterlim, filter_le, filtermap in Hlim.
specialize (Hneg t).
assert (∃ x : R, t < x ∧ Rbar_lt glim (g x)) as (x&Hrange&r0).
{
destruct Hneg as (x&Hrange&Hnlt).
apply Rbar_not_lt_le in Hnlt.
apply Rbar_le_lt_or_eq_dec in Hnlt.
destruct Hnlt as [Hlt|Heq].
{ exists x. split; eauto. }
exists (x + 1).
split; first nra.
rewrite Heq. simpl. apply Ht; auto; split; simpl; try nra.
}
apply open_Rbar_lt' in r0. apply Hlim in r0.
eapply (Rbar_at_left_witness_above_p_infty x) in r0; try (intuition eauto; done).
destruct r0 as (y&Hrange'&Hlt).
simpl in Hlt. apply Rlt_not_le in Hlt.
apply Hlt. left. apply Ht; simpl; simpl in Hltb; try nra.
Qed.
Lemma Rbar_at_left_strict_monotone' (t : Rbar) (b : Rbar) g glim :
Rbar_lt t b →
(∀ (x y : R), Rbar_lt t x /\ x < y → Rbar_lt y b → g x < g y) →
is_left_lim g b glim ->
Rbar_at_left b (λ y : Rbar, Rbar_lt (g y) glim).
Proof.
intros Hlt Hmono Hlim.
destruct t.
- eapply Rbar_at_left_strict_monotone; eauto.
- inversion Hlt.
- destruct b as [r' | |].
{ eapply (Rbar_at_left_strict_monotone (Rmin 0 (r' - 1))); eauto.
{ simpl. apply Rmin_case_strong; nra. }
{ intros. apply Hmono; auto. split; intuition auto. }
}
{ eapply (Rbar_at_left_strict_monotone 0); eauto.
{ intros. apply Hmono; auto. split; intuition auto. }
}
{ inversion Hlt. }
Qed.
Lemma Rbar_at_right_witness (r: R) (eps: posreal) P:
Rbar_at_right r P -> ∃ x, r < x < r + eps ∧ P x.
Proof.
unfold Rbar_at_right, within, Rbar_locally, locally.
intros Hex. destruct Hex as (eps'&Heps').
set (lb := r + Rmin eps eps').
edestruct (interval_inhabited r lb) as (x&Hin).
{ rewrite /lb. apply Rmin_case; destruct eps, eps' => /=; nra. }
exists x. split.
{ move: Hin. rewrite /lb. apply Rmin_case_strong; destruct eps, eps' => /=; nra. }
apply Heps'; last by intuition.
rewrite /ball/=/AbsRing_ball/abs/=/minus/plus/opp//=.
{ move: Hin. rewrite /lb. apply Rabs_case; apply Rmin_case_strong; destruct eps, eps' => /=; nra. }
Qed.
Lemma Rbar_at_right_witness_above (r: R) y P:
Rbar_at_right r P -> r < y -> ∃ x, r < x < y ∧ P x.
Proof.
intros. assert (Hpos: 0 < y - r) by nra.
edestruct (Rbar_at_right_witness r (mkposreal _ Hpos) P) as (x&?&HP); auto.
exists x. split; auto.
simpl in H1. nra.
Qed.
Lemma Rbar_at_right_witness_above_m_infty y P:
Rbar_at_right m_infty P -> ∃ x, x < y ∧ P x.
Proof.
unfold Rbar_at_right, within, Rbar_locally.
intros HM. destruct HM as (M&HM). exists (Rmin (M - 1) (y - 1)).
split.
- apply (Rle_lt_trans _ (y - 1)); last nra.
apply Rmin_r.
- apply HM; simpl; auto.
- apply (Rle_lt_trans _ (M - 1)); last nra.
apply Rmin_l.
Qed.
Lemma Rbar_at_right_strict_monotone (t : R) (a : Rbar) g glim :
Rbar_lt a t →
(∀ x y, x < y /\ Rbar_lt y t → Rbar_lt a x → g x < g y) →
is_right_lim g a glim ->
Rbar_at_right a (λ y : Rbar, Rbar_lt glim (g y)).
Proof.
unfold is_left_lim.
intros Hltb Ht (Hnm&Hlim).
apply Classical_Prop.NNPP. intros Hneg%not_Rbar_at_right.
destruct a; try congruence.
- unfold filterlim, filter_le, filtermap in Hlim.
assert (Hpos: 0 < t - r).
{ simpl in Hltb. nra. }
set (eps' := mkposreal _ Hpos).
specialize (Hneg eps').
assert (∃ x : R, (r < x ∧ x < r + eps') ∧ Rbar_lt (g x) glim) as (x&Hrange&r0).
{
destruct Hneg as (x&Hrange&Hnlt).
apply Rbar_not_lt_le in Hnlt.
apply Rbar_le_lt_or_eq_dec in Hnlt.
destruct Hnlt as [Hlt|Heq].
{ exists x. split; eauto. }
destruct (interval_inhabited r x) as (x'&Hx'1&Hx'2); first nra.
exists x'.
split; first nra.
rewrite -Heq. simpl. apply Ht; auto; simpl; try nra. split; try nra.
move: Hrange. rewrite /eps' /=. nra.
}
apply open_Rbar_gt' in r0. apply Hlim in r0.
eapply (Rbar_at_right_witness_above r x) in r0; try (intuition eauto; done).
destruct r0 as (y&Hrange'&Hlt).
simpl in Hlt. apply Rlt_not_le in Hlt.
apply Hlt. left. apply Ht; simpl; simpl in Hltb; try nra.
split; first by intuition.
move: Hrange. rewrite /eps' /=. nra.
- unfold filterlim, filter_le, filtermap in Hlim.
specialize (Hneg t).
assert (∃ x : R, x < t ∧ Rbar_lt (g x) glim) as (x&Hrange&r0).
{
destruct Hneg as (x&Hrange&Hnlt).
apply Rbar_not_lt_le in Hnlt.
apply Rbar_le_lt_or_eq_dec in Hnlt.
destruct Hnlt as [Hlt|Heq].
{ exists x. split; eauto. }
exists (x - 1).
split; first nra.
rewrite -Heq. simpl. apply Ht; auto; split; simpl; try nra.
}
apply open_Rbar_gt' in r0. apply Hlim in r0.
eapply (Rbar_at_right_witness_above_m_infty x) in r0; try (intuition eauto; done).
destruct r0 as (y&Hrange'&Hlt).
simpl in Hlt. apply Rlt_not_le in Hlt.
apply Hlt. left. apply Ht; simpl; simpl in Hltb; try nra.
Qed.
Lemma Rbar_at_right_strict_monotone' (t : Rbar) (a : Rbar) g glim :
Rbar_lt a t →
(∀ x y, x < y /\ Rbar_lt y t → Rbar_lt a x → g x < g y) →
is_right_lim g a glim ->
Rbar_at_right a (λ y : Rbar, Rbar_lt glim (g y)).
Proof.
intros Hlt Hmono Hlim.
destruct t.
- eapply Rbar_at_right_strict_monotone; eauto.
- destruct a as [r' | |].
{ eapply (Rbar_at_right_strict_monotone (Rmax 0 (r' + 1))); eauto.
{ simpl. apply Rmax_case_strong; nra. }
{ intros. apply Hmono; auto. split; intuition auto. }
}
{ inversion Hlt. }
{ eapply (Rbar_at_right_strict_monotone 0); eauto.
{ intros. apply Hmono; auto. split; intuition auto. }
}
- destruct a; inversion Hlt.
Qed.
Section comp.
Lemma R_dist_plus_r1 x y z y':
R_dist x (y + z) <= R_dist x (y' + z) + R_dist y y'.
Proof.
rewrite /R_dist.
replace (y + z) with (y' + z + (y - y')) by nra.
replace (x - (y' + z + (y - y'))) with ((x - (y' + z)) - (y - y')) by nra.
rewrite -(Rabs_Ropp (y - y')).
etransitivity; last eapply Rabs_triang.
reflexivity.
Qed.
Lemma linear_interp_bound startv endv a b x :
a < b ->
a <= x <= b ->
Rmin startv endv <= startv + (endv - startv) * ((x -a)/(b-a)) <= Rmax startv endv.
Proof.
intros Hlt Hrange.
destruct (Rle_dec startv endv).
{
rewrite Rmin_left //. rewrite Rmax_right; last by nra.
split.
{ transitivity (startv + 0); first nra.
apply Rplus_le_compat_l.
apply Rmult_le_pos; first nra.
apply Rdiv_le_0_compat; nra.
}
{ transitivity (startv + (endv - startv) * 1); last nra.
apply Rplus_le_compat_l.
apply Rmult_le_compat_l; first nra.
apply Rle_div_l; nra.
}
}
{
rewrite Rmin_right; last nra. rewrite Rmax_left; last by nra.
split.
{ transitivity (startv + (endv - startv) * 1); first nra.
apply Rplus_le_compat_l.
apply Rmult_le_compat_neg_l; first nra.
apply Rle_div_l; nra.
}
{ transitivity (startv + 0); last nra.
apply Rplus_le_compat_l.
apply Rmult_le_0_r; first nra.
apply Rdiv_le_0_compat; nra.
}
}
Qed.
Lemma linear_interp_bound_abs startv endv a b x :
a < b ->
a <= x <= b ->
Rabs (startv + (endv - startv) * ((x -a)/(b-a))) <= Rmax (Rabs startv) (Rabs endv).
Proof.
intros Hlt Hrange.
specialize (linear_interp_bound startv endv a b x Hlt Hrange).
remember ((startv + (endv - startv) * ((x -a)/(b-a)))) as y.
clear Heqy.
apply Rmin_case_strong; repeat apply Rmax_case_strong; repeat (apply Rabs_case); try nra.
Qed.
Lemma continuity_linear_interp startv endv a b :
a < b ->
continuity (λ x0 : R, startv + (endv - startv) * ((x0 - a) * / (b - a))).
Proof.
intros Hlt.
apply continuity_plus.
{ apply continuity_const => ? //=. }
apply continuity_mult.
{ apply continuity_const => ? //=. }
apply continuity_mult.
{ rewrite /Rminus.
apply continuity_plus.
{ intros x. rewrite -continuous_continuity_pt. apply continuous_id. }
apply continuity_opp.
{ apply continuity_const => ? //=. }
}
{ apply continuity_const => ? //=. }
Qed.
Lemma constant_open_segment_ub_function_aux1 f a b v startv endv δ :
constant_D_eq f (open_interval a b) v ->
(f a <= startv) ->
(v <= startv) ->
(v <= endv) ->
(f b <= endv) ->
(0 < δ) ->
(a + δ < b - δ) ->
∃ g, (∀ x, a <= x <= b -> f x <= g x) /\
(∀ x, continuous g x) /\
(∀ x, a <= x <= b -> Rabs (g x) <= Rmax (Rmax (Rabs (f a)) (Rabs (f b)))
(Rmax (Rabs v) (Rmax (Rabs startv) (Rabs endv)))) /\
(∀ x, a + δ < x <= b - δ -> f x = g x) /\
g a = startv /\
g b = endv.
Proof.
intros Hconst Hle0 Hle0' Hle1 Hle2 Hdelt_nn Hdelt_range.
exists (λ x, match Rle_dec x (a + δ) with
| left _ => startv + (v - startv) * ((x - a) * / ((a + δ) - a))
| _ =>
match Rle_dec x (b - δ) with
| left _ => v
| _ =>
v + (endv - v) * ((x - (b - δ)) * / (b - (b - δ)))
end
end).
split.
{ intros x Hrange. destruct Rle_dec as [Hle|Hnle].
{
assert (x = a \/ a < x) as Hcases.
{ nra. }
destruct Hcases as [-> | Hstrict ].
{ field_simplify; last nra.
transitivity (startv); last by (right; field; nra).
auto.
}
assert (f x = v) as ->.
{ apply Hconst. split; nra. }
transitivity (startv + (v - startv) * 1).
{ right. field. }
apply Rplus_le_compat_l. apply Rmult_le_compat_neg_l.
{ nra. }
rewrite -Rdiv_le_1; first nra.
nra.
}
destruct Rle_dec.
{ right. apply Hconst. split; nra. }
assert (x = b \/ x < b) as [->|Hlt].
{ nra. }
{ field_simplify; nra. }
assert (f x = v) as ->.
{ apply Hconst. split; nra. }
transitivity (v + 0).
{ nra. }
apply Rplus_le_compat_l.
apply Rmult_le_pos; first nra.
apply Rdiv_le_0_compat; nra.
}
split.
{
intros x.
apply continuous_continuity_pt.
eapply piecewise_continuity'.
{
eapply continuity_linear_interp. nra.
}
{ eapply piecewise_continuity'.
{ apply continuity_const => ? //=. }
{ apply continuity_linear_interp. nra. }
nra.
}
destruct (Rle_dec _ _); try nra.
field. nra.
}
split.
{
intros x Hrange. destruct Rle_dec as [Hle|Hnle].
{ etransitivity; first eapply linear_interp_bound_abs; try nra.
{ repeat (apply Rmax_case_strong); nra. }
}
destruct Rle_dec as [Hle'|Hnle'].
{ repeat (apply Rmax_case_strong); nra. }
{ etransitivity; first eapply linear_interp_bound_abs; try nra.
repeat (apply Rmax_case_strong); nra. }
}
split.
{
intros x Hrange.
destruct Rle_dec; first nra.
destruct Rle_dec; last nra.
apply Hconst. rewrite /open_interval. split; nra.
}
split.
{
destruct Rle_dec; nra.
}
{
destruct Rle_dec; first nra.
destruct Rle_dec; first nra.
field. nra.
}
Qed.
Lemma constant_open_segment_ub_function f a b v (eps : posreal) startv endv :
a <= b ->
constant_D_eq f (open_interval a b) v ->
(f a <= startv) ->
(v <= startv) ->
(v <= endv) ->
(f b <= endv) ->
(a = b -> startv = endv) ->
∃ g, (∀ x, a <= x <= b -> f x <= g x) /\
(∀ x, continuous g x) /\
ex_RInt (λ x, g x - f x) a b /\
(RInt (λ x, g x - f x) a b < eps) /\
g a = startv /\
g b = endv.
Proof.
intros Hle0 Hconst Hlestartv1 Hlestartv2 Hle1 Hle2 Hcase.
assert (Hintegrablef: ∀ x y, a <= x /\ x <= y /\ y <= b -> ex_RInt f x y).
{
intros x y Hle.
apply: (ex_RInt_Chasles_2 _ a); first nra.
apply: (ex_RInt_Chasles_1 _ _ _ b); first nra.
eapply (ex_RInt_ext (λ x, v)).
{ rewrite Rmin_left // Rmax_right //. intros; symmetry; apply Hconst; split; nra. }
eapply ex_RInt_const.
}
destruct Hle0 as [Hlt|Heq]; last first.
{ subst. exists (λ x, endv).
split.
{ intros x ?. assert (x = b) by nra. subst. eauto. }
split.
{ intros x ?. apply continuous_const. }
split.
{ apply: ex_RInt_minus; last by (eapply Hintegrablef; nra). apply ex_RInt_const. }
{ rewrite RInt_point /zero/=. destruct eps => /=; nra. }
}
set (M := Rmax (Rmax (Rabs (f a)) (Rabs (f b)))
(Rmax (Rabs v) (Rmax (Rabs startv) (Rabs endv)))).
assert (Hfabs: ∀ x, a <= x <= b -> Rabs (f x) <= M).
{ intros ? Hrange. rewrite /M.
assert (f x = v \/ f x = f a \/ f x = f b) as [->|[-> | ->]].
{ destruct Hrange as ([?|?]&[?|?]); subst; try nra;
try (left; apply Hconst; split; nra).
}
{ repeat apply Rmax_case_strong; try nra. }
{ repeat apply Rmax_case_strong; try nra. }
{ repeat apply Rmax_case_strong; try nra. }
}
assert (M = 0 ∨ 0 < M) as [HM0|HMnz].
{ rewrite /M.
repeat (apply Rmin_case_strong || apply Rmax_case_strong || apply Rabs_case); try nra.
}
{ assert (f a = 0 /\ f b = 0 /\ v = 0 /\ startv = 0 /\ endv = 0) as (?&?&?&?&?).
{ move: HM0. rewrite /M.
repeat (apply Rmin_case_strong || apply Rmax_case_strong || apply Rabs_case); try nra. }
exists (λ x, 0).
assert (Hf0: ∀ x, a <= x <= b → f x = 0).
{ intros y (Hle1'&Hle2'). destruct Hle1'; last first.
{ subst. nra. }
destruct Hle2'; last first.
{ subst. nra. }
transitivity v; last done. apply Hconst. rewrite /open_interval; nra.
}
split.
{ intros. rewrite Hf0 //. nra. }
split.
{ apply continuous_const. }
split.
{ apply: ex_RInt_minus; last by (eapply Hintegrablef; nra). apply ex_RInt_const. }
split; last by (split; congruence).
eapply (Rle_lt_trans _ 0); last by (destruct eps => /=; nra).
right.
erewrite (RInt_ext _ (λ _, 0)).
{ rewrite RInt_const /scal/=/mult/=. nra. }
intros x. apply Rmin_case_strong; apply Rmax_case_strong; intros; rewrite Hf0 //; nra.
}
set (δ := Rmin ((b - a)/3) ((eps / (8 * M)))).
assert (0 < δ).
{ rewrite /δ.
apply Rmin_case_strong; first by nra.
intros _. apply Rdiv_lt_0_compat; destruct eps => /=; nra.
}
assert ((a + δ < b - δ)).
{ rewrite /δ. apply Rmin_case_strong; nra. }
edestruct (constant_open_segment_ub_function_aux1 f a b v startv endv δ) as (g&Hg1&Hg2&Hg3&Hg4&Hg5); eauto.
exists g.
assert (Hintegrableh: ∀ x y, a <= x /\ x <= y /\ y <= b -> ex_RInt (λ x, g x - f x) x y).
{
intros.
apply: (ex_RInt_Chasles_2 _ a); first nra.
apply: (ex_RInt_Chasles_1 _ _ _ b); first nra.
apply: ex_RInt_minus; last eapply Hintegrablef; try nra.
eapply ex_RInt_continuous; auto.
}
repeat (split; auto).
eapply Hintegrableh; eauto; nra.
eapply (Rle_lt_trans _ (RInt (λ x, g x - f x) a (a + δ) +
RInt (λ x, g x - f x) (a+δ) (b - δ) +
RInt (λ x, g x - f x) (b - δ) b)).
{
rewrite -(RInt_Chasles (λ x, g x - f x) a (a + δ)); try (eapply Hintegrableh; nra).
rewrite -(RInt_Chasles (λ x, g x - f x) (a + δ) (b - δ)); try (eapply Hintegrableh; nra).
rewrite /plus/=. right. field.
}
eapply (Rlt_le_trans _ (eps/2 + 0 + eps/2)); last first.
{ right. field. }
apply Rplus_lt_compat.
assert (RInt (λ x, g x - f x) (a + δ) (b - δ) = 0) as ->.
{ rewrite (RInt_ext _ (λ x, 0)); last first.
{ rewrite Rmin_left // ?Rmax_right //; try nra. intros. rewrite Hg4; try nra. }
rewrite RInt_const. rewrite /scal/=/mult/=. nra. }
apply Rplus_lt_compat_r.
{ eapply Rle_lt_trans; first eapply (RInt_le _ (λ x, 2 * M)); try nra.
{ apply Hintegrableh. nra. }
{ apply ex_RInt_const. }
{ intros.
transitivity (Rabs (g x) + Rabs (f x)).
{ repeat apply Rabs_case; nra. }
{ setoid_rewrite Hfabs; last nra. setoid_rewrite Hg3; last nra. right. rewrite /M. field. }
}
rewrite RInt_const. replace (a + δ -a) with δ by nra. rewrite /scal/=/mult/=.
eapply (Rle_lt_trans _ ((eps / (8 * M)) * (2 * M))).
{ apply Rmult_le_compat_r; first nra.
apply Rmin_r. }
field_simplify; last nra. assert (2 * eps / 8 = eps / 4) as -> by field.
rewrite /Rdiv. apply Rmult_lt_compat_l; first by (destruct eps => /=; nra).
nra.
}
{ eapply Rle_lt_trans; first eapply (RInt_le _ (λ x, 2 * M)); try nra.
{ apply Hintegrableh. nra. }
{ apply ex_RInt_const. }
{ intros.
transitivity (Rabs (g x) + Rabs (f x)).
{ repeat apply Rabs_case; nra. }
{ setoid_rewrite Hfabs; last nra. setoid_rewrite Hg3; last nra. right. rewrite /M. field. }
}
rewrite RInt_const. replace (b - (b - δ)) with δ by nra. rewrite /scal/=/mult/=.
eapply (Rle_lt_trans _ ((eps / (8 * M)) * (2 * M))).
{ apply Rmult_le_compat_r; first nra.
apply Rmin_r. }
field_simplify; last nra. assert (2 * eps / 8 = eps / 4) as -> by field.
rewrite /Rdiv. apply Rmult_lt_compat_l; first by (destruct eps => /=; nra).
nra.
}
Qed.
Lemma Riemann_integrable_SF_aux a b sf (eps: posreal) :
(∀ t : R, Rmin a b <= t <= Rmax a b → Rabs (sf t - sf t) <= {| fe := fct_cte 0; pre := StepFun_P4 a b 0 |} t)
∧ Rabs (RiemannInt_SF {| fe := fct_cte 0; pre := StepFun_P4 a b 0 |}) < eps.
Proof.
split.
- intros; replace (sf t - sf t) with 0 by nra; rewrite Rabs_right => //=; last nra; rewrite /fct_cte; nra.
- simpl. rewrite StepFun_P18 Rabs_right => //=; destruct eps => /=; nra.
Qed.
Lemma Riemann_integrable_SF a b (sf: StepFun a b) :
Riemann_integrable sf a b.
Proof.
rewrite /Riemann_integrable => eps.
exists sf.
exists (mkStepFun (StepFun_P4 a b 0)).
apply Riemann_integrable_SF_aux.
Defined.
Lemma StepFun_ub (f: R -> R) (a b : R) :
a <= b ->
IsStepFun f a b ->
∃ M, (∀ x, a <= x <= b -> f x <= M).
Proof.
rewrite /IsStepFun. intros Hle (l&Hsub).
revert a Hle Hsub.
induction l as [| x1 l IHl].
{ intros. destruct Hsub as (l'&Hadapted).
destruct Hadapted as (?&Hmin&?&Hfalse&_).
inversion Hfalse. }
{ intros a Hle Hsub.
destruct Hsub as (l'&Hadapted).
destruct l as [| x2 l].
{ destruct Hadapted as (?&Hmin&Hmax&Hlen&?).
destruct l' as [|]; last first.
{ simpl in Hlen. congruence. }
assert (a = b) as <-.
{ simpl in Hmin, Hmax. move: Hmin Hmax. apply Rmin_case_strong; apply Rmax_case_strong; nra. }
exists (f a). intros x ?. assert (x = a) by nra. subst; reflexivity. }
destruct l' as [|y l'].
{ destruct Hadapted as (?&Hmin&Hmax&Hlen&?).
simpl in Hlen. congruence. }
specialize (StepFun_P7 Hle Hadapted) => Hadapted_tl.
assert (Hlex2: x2 <= b).
{ destruct Hadapted as (Hordered&Hmin&Hmax&Hlen&?).
replace x2 with (RList.pos_Rl (x1 :: x2 :: l) 1) by auto.
rewrite Rmax_right // in Hmax. rewrite -Hmax.
apply RList.RList_P6; auto.
simpl. lia.
}
edestruct (IHl x2) as (M'&HM').
{ auto. }
{ eexists. eauto. }
destruct Hadapted as (HOrdered&Hmin&Hmax&Hlen&Hval).
assert (a = x1).
{ rewrite Rmin_left // in Hmin. }
exists (Rmax (f a) (Rmax y M')).
assert (Hconst: constant_D_eq f (open_interval x1 x2) y).
{ apply (Hval O). simpl. lia. }
intros z Hrange.
destruct (Rle_dec x2 z).
{ etransitivity; first eapply HM'; first by nra.
do 2 setoid_rewrite <-Rmax_r. reflexivity.
}
destruct (Req_appart_dec a z).
{ subst. apply Rmax_l. }
setoid_rewrite <-Rmax_r.
rewrite Hconst; first apply Rmax_l.
split; nra.
}
Qed.
Lemma StepFun_ub_fun_cont_aux (f: R -> R) (a b : R) (startv : R) (eps : posreal) :
a <= b ->
(∀ x, a <= x <= b -> f x <= startv) ->
IsStepFun f a b ->
∃ g : R -> R, (∀ x, continuous g x) /\ (∀ x, a <= x <= b -> f x <= g x) /\
ex_RInt (λ x, g x - f x) a b /\
RInt (λ x, g x - f x) a b < eps /\
g a = startv.
Proof.
rewrite /IsStepFun. intros Hle Hstartv (l&Hsub).
revert a startv eps Hle Hstartv Hsub.
induction l as [| x1 l IHl].
{ intros. destruct Hsub as (l'&Hadapted).
destruct Hadapted as (?&Hmin&?&Hfalse&_).
inversion Hfalse. }
{ intros a startv eps Hle Hstartv Hsub.
destruct Hsub as (l'&Hadapted).
destruct l as [| x2 l].
{ destruct Hadapted as (?&Hmin&Hmax&Hlen&?).
destruct l' as [|]; last first.
{ simpl in Hlen. congruence. }
assert (a = b) as <-.
{ simpl in Hmin, Hmax. move: Hmin Hmax. apply Rmin_case_strong; apply Rmax_case_strong; nra. }
subst.
exists (fct_cte (startv)).
split.
{ intros; apply continuous_continuity_pt. apply derivable_continuous, derivable_const. }
split.
{ intros x' ?. assert (x' = a) as -> by nra. rewrite /fct_cte//=. apply Hstartv; eauto. }
split.
{ apply ex_RInt_point. }
split.
{ rewrite RInt_point /zero/=. destruct eps; auto. }
rewrite /fct_cte//=. (* reflexivity. *)
}
destruct l' as [|y l'].
{ destruct Hadapted as (?&Hmin&Hmax&Hlen&?).
simpl in Hlen. congruence. }
specialize (StepFun_P7 Hle Hadapted) => Hadapted_tl.
assert (Hlex2: x2 <= b).
{ destruct Hadapted as (Hordered&Hmin&Hmax&Hlen&?).
replace x2 with (RList.pos_Rl (x1 :: x2 :: l) 1) by auto.
rewrite Rmax_right // in Hmax. rewrite -Hmax.
apply RList.RList_P6; auto.
simpl. lia.
}
assert (x1 = x2 \/ x1 ≠ x2) as [Heq|Hneq] by nra.
{
subst. eapply IHl; eauto. exists l'.
destruct Hadapted as (HOrdered&Hmin&Hmax&Hlen&Hval).
rewrite Rmin_left // in Hmin. simpl in Hmin. subst. eauto.
}
destruct Hadapted as (HOrdered&Hmin&Hmax&Hlen&Hval).
assert (a = x1).
{ rewrite Rmin_left // in Hmin. }
assert (a <= x2).
{ rewrite Rmin_left // in Hmin.
replace x2 with (RList.pos_Rl (x1 :: x2 :: l) 1) by auto.
rewrite -Hmin.
rewrite RList.RList_P6 in HOrdered * => HOrdered. eapply HOrdered; eauto.
simpl. lia.
}
edestruct (IHl x2 startv (mkposreal _ (is_pos_div_2 eps))) as (g&Hgcont&Hgub&Hex&Hint&?).
{ auto. }
{ intros. apply Hstartv; nra. }
{ eexists. eauto. }
assert (Hconst: constant_D_eq f (open_interval x1 x2) y).
{ apply (Hval O). simpl. lia. }
assert (y <= startv).
{ rewrite -(Hconst ((x1 + x2)/2)); first by (eapply Hstartv; nra).