Library Coq.Reals.PSeries_reg
Require Import Rbase.
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Ranalysis1.
Require Import MVT.
Require Import Max.
Require Import Even.
Require Import Lra.
Local Open Scope R_scope.
Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r.
Lemma Boule_convex : forall c d x y z,
Boule c d x -> Boule c d y -> x <= z <= y -> Boule c d z.
Proof.
intros c d x y z bx b_y intz.
unfold Boule in bx, b_y; apply Rabs_def2 in bx;
apply Rabs_def2 in b_y; apply Rabs_def1;
[apply Rle_lt_trans with (y - c);[apply Rplus_le_compat_r|]|
apply Rlt_le_trans with (x - c);[|apply Rplus_le_compat_r]];tauto.
Qed.
Definition boule_of_interval x y (h : x < y) :
{c :R & {r : posreal | c - r = x /\ c + r = y}}.
Proof.
exists ((x + y)/2).
assert (radius : 0 < (y - x)/2).
unfold Rdiv; apply Rmult_lt_0_compat.
apply Rlt_Rminus; assumption.
now apply Rinv_0_lt_compat, Rlt_0_2.
exists (mkposreal _ radius).
simpl; split; unfold Rdiv; field.
Qed.
Definition boule_in_interval x y z (h : x < z < y) :
{c : R & {r | Boule c r z /\ x < c - r /\ c + r < y}}.
Proof.
assert (cmp : x * /2 + z * /2 < z * /2 + y * /2).
destruct h as [h1 h2].
rewrite Rplus_comm; apply Rplus_lt_compat_l, Rmult_lt_compat_r.
apply Rinv_0_lt_compat, Rlt_0_2.
apply Rlt_trans with z; assumption.
destruct (boule_of_interval _ _ cmp) as [c [r [P1 P2]]].
assert (0 < /2) by (apply Rinv_0_lt_compat, Rlt_0_2).
exists c, r; split.
destruct h; unfold Boule; simpl; apply Rabs_def1.
apply Rplus_lt_reg_l with c; rewrite P2;
replace (c + (z - c)) with (z * / 2 + z * / 2) by field.
apply Rplus_lt_compat_l, Rmult_lt_compat_r;assumption.
apply Rplus_lt_reg_l with c; change (c + - r) with (c - r);
rewrite P1;
replace (c + (z - c)) with (z * / 2 + z * / 2) by field.
apply Rplus_lt_compat_r, Rmult_lt_compat_r;assumption.
destruct h; split.
replace x with (x * / 2 + x * / 2) by field; rewrite P1.
apply Rplus_lt_compat_l, Rmult_lt_compat_r;assumption.
replace y with (y * / 2 + y * /2) by field; rewrite P2.
apply Rplus_lt_compat_r, Rmult_lt_compat_r;assumption.
Qed.
Lemma Ball_in_inter : forall c1 c2 r1 r2 x,
Boule c1 r1 x -> Boule c2 r2 x ->
{r3 : posreal | forall y, Boule x r3 y -> Boule c1 r1 y /\ Boule c2 r2 y}.
Proof.
intros c1 c2 [r1 r1p] [r2 r2p] x; unfold Boule; simpl; intros in1 in2.
assert (Rmax (c1 - r1)(c2 - r2) < x).
apply Rmax_lub_lt;[revert in1 | revert in2]; intros h;
apply Rabs_def2 in h; destruct h as [_ u];
apply (fun h => Rplus_lt_reg_r _ _ _ (Rle_lt_trans _ _ _ h u)), Req_le; ring.
assert (x < Rmin (c1 + r1) (c2 + r2)).
apply Rmin_glb_lt;[revert in1 | revert in2]; intros h;
apply Rabs_def2 in h; destruct h as [u _];
apply (fun h => Rplus_lt_reg_r _ _ _ (Rlt_le_trans _ _ _ u h)), Req_le; ring.
assert (t: 0 < Rmin (x - Rmax (c1 - r1) (c2 - r2))
(Rmin (c1 + r1) (c2 + r2) - x)).
apply Rmin_glb_lt; apply Rlt_Rminus; assumption.
exists (mkposreal _ t).
apply Rabs_def2 in in1; destruct in1.
apply Rabs_def2 in in2; destruct in2.
assert (c1 - r1 <= Rmax (c1 - r1) (c2 - r2)) by apply Rmax_l.
assert (c2 - r2 <= Rmax (c1 - r1) (c2 - r2)) by apply Rmax_r.
assert (Rmin (c1 + r1) (c2 + r2) <= c1 + r1) by apply Rmin_l.
assert (Rmin (c1 + r1) (c2 + r2) <= c2 + r2) by apply Rmin_r.
assert (Rmin (x - Rmax (c1 - r1) (c2 - r2))
(Rmin (c1 + r1) (c2 + r2) - x) <= x - Rmax (c1 - r1) (c2 - r2))
by apply Rmin_l.
assert (Rmin (x - Rmax (c1 - r1) (c2 - r2))
(Rmin (c1 + r1) (c2 + r2) - x) <= Rmin (c1 + r1) (c2 + r2) - x)
by apply Rmin_r.
simpl.
intros y h; apply Rabs_def2 in h; destruct h as [h h'].
apply Rmin_Rgt in h; destruct h as [cmp1 cmp2].
apply Rplus_lt_reg_r in cmp2; apply Rmin_Rgt in cmp2.
rewrite Ropp_Rmin, Ropp_minus_distr in h'.
apply Rmax_Rlt in h'; destruct h' as [cmp3 cmp4];
apply Rplus_lt_reg_r in cmp3; apply Rmax_Rlt in cmp3;
split; apply Rabs_def1.
apply (fun h => Rplus_lt_reg_l _ _ _ (Rle_lt_trans _ _ _ h (proj1 cmp2))), Req_le;
ring.
apply (fun h => Rplus_lt_reg_l _ _ _ (Rlt_le_trans _ _ _ (proj1 cmp3) h)), Req_le;
ring.
apply (fun h => Rplus_lt_reg_l _ _ _ (Rle_lt_trans _ _ _ h (proj2 cmp2))), Req_le;
ring.
apply (fun h => Rplus_lt_reg_l _ _ _ (Rlt_le_trans _ _ _ (proj2 cmp3) h)), Req_le;
ring.
Qed.
Lemma Boule_center : forall x r, Boule x r x.
Proof.
intros x [r rpos]; unfold Boule, Rminus; simpl; rewrite Rplus_opp_r.
rewrite Rabs_pos_eq;[assumption | apply Rle_refl].
Qed.
Uniform convergence
Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R)
(r:posreal) : Prop :=
forall eps:R,
0 < eps ->
exists N : nat,
(forall (n:nat) (y:R),
(N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps).
(r:posreal) : Prop :=
forall eps:R,
0 < eps ->
exists N : nat,
(forall (n:nat) (y:R),
(N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps).
Normal convergence
Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type :=
{ An:nat -> R &
{ l:R |
Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\
(forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n) } }.
Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r.
Definition SFL (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
(y:R) : R := let (a,_) := cv y in a.
{ An:nat -> R &
{ l:R |
Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\
(forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n) } }.
Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r.
Definition SFL (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
(y:R) : R := let (a,_) := cv y in a.
In a complete space, normal convergence implies uniform convergence
Lemma CVN_CVU :
forall (fn:nat -> R -> R)
(cv:forall x:R, {l:R | Un_cv (fun N:nat => SP fn N x) l })
(r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r.
Proof.
intros; unfold CVU; intros.
unfold CVN_r in X.
elim X; intros An X0.
elim X0; intros s H0.
elim H0; intros.
cut (Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n - s) 0).
intro; unfold Un_cv in H3.
elim (H3 eps H); intros N0 H4.
exists N0; intros.
apply Rle_lt_trans with (Rabs (sum_f_R0 (fun k:nat => Rabs (An k)) n - s)).
rewrite <- (Rabs_Ropp (sum_f_R0 (fun k:nat => Rabs (An k)) n - s));
rewrite Ropp_minus_distr';
rewrite (Rabs_right (s - sum_f_R0 (fun k:nat => Rabs (An k)) n)).
eapply sum_maj1.
unfold SFL; case (cv y); intro.
trivial.
apply H1.
intro; elim H0; intros.
rewrite (Rabs_right (An n0)).
apply H8; apply H6.
apply Rle_ge; apply Rle_trans with (Rabs (fn n0 y)).
apply Rabs_pos.
apply H8; apply H6.
apply Rle_ge;
apply Rplus_le_reg_l with (sum_f_R0 (fun k:nat => Rabs (An k)) n).
rewrite Rplus_0_r; unfold Rminus; rewrite (Rplus_comm s);
rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l;
apply sum_incr.
apply H1.
intro; apply Rabs_pos.
unfold R_dist in H4; unfold Rminus in H4; rewrite Ropp_0 in H4.
assert (H7 := H4 n H5).
rewrite Rplus_0_r in H7; apply H7.
unfold Un_cv in H1; unfold Un_cv; intros.
elim (H1 _ H3); intros.
exists x; intros.
unfold R_dist; unfold R_dist in H4.
rewrite Rminus_0_r; apply H4; assumption.
Qed.
forall (fn:nat -> R -> R)
(cv:forall x:R, {l:R | Un_cv (fun N:nat => SP fn N x) l })
(r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r.
Proof.
intros; unfold CVU; intros.
unfold CVN_r in X.
elim X; intros An X0.
elim X0; intros s H0.
elim H0; intros.
cut (Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n - s) 0).
intro; unfold Un_cv in H3.
elim (H3 eps H); intros N0 H4.
exists N0; intros.
apply Rle_lt_trans with (Rabs (sum_f_R0 (fun k:nat => Rabs (An k)) n - s)).
rewrite <- (Rabs_Ropp (sum_f_R0 (fun k:nat => Rabs (An k)) n - s));
rewrite Ropp_minus_distr';
rewrite (Rabs_right (s - sum_f_R0 (fun k:nat => Rabs (An k)) n)).
eapply sum_maj1.
unfold SFL; case (cv y); intro.
trivial.
apply H1.
intro; elim H0; intros.
rewrite (Rabs_right (An n0)).
apply H8; apply H6.
apply Rle_ge; apply Rle_trans with (Rabs (fn n0 y)).
apply Rabs_pos.
apply H8; apply H6.
apply Rle_ge;
apply Rplus_le_reg_l with (sum_f_R0 (fun k:nat => Rabs (An k)) n).
rewrite Rplus_0_r; unfold Rminus; rewrite (Rplus_comm s);
rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l;
apply sum_incr.
apply H1.
intro; apply Rabs_pos.
unfold R_dist in H4; unfold Rminus in H4; rewrite Ropp_0 in H4.
assert (H7 := H4 n H5).
rewrite Rplus_0_r in H7; apply H7.
unfold Un_cv in H1; unfold Un_cv; intros.
elim (H1 _ H3); intros.
exists x; intros.
unfold R_dist; unfold R_dist in H4.
rewrite Rminus_0_r; apply H4; assumption.
Qed.
Each limit of a sequence of functions which converges uniformly is continue
Lemma CVU_continuity :
forall (fn:nat -> R -> R) (f:R -> R) (x:R) (r:posreal),
CVU fn f x r ->
(forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) ->
forall y:R, Boule x r y -> continuity_pt f y.
Proof.
intros; unfold continuity_pt; unfold continue_in;
unfold limit1_in; unfold limit_in;
simpl; unfold R_dist; intros.
unfold CVU in H.
cut (0 < eps / 3);
[ intro
| unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
elim (H _ H3); intros N0 H4.
assert (H5 := H0 N0 y H1).
cut (exists del : posreal, (forall h:R, Rabs h < del -> Boule x r (y + h))).
intro.
elim H6; intros del1 H7.
unfold continuity_pt in H5; unfold continue_in in H5; unfold limit1_in in H5;
unfold limit_in in H5; simpl in H5; unfold R_dist in H5.
elim (H5 _ H3); intros del2 H8.
set (del := Rmin del1 del2).
exists del; intros.
split.
unfold del; unfold Rmin; case (Rle_dec del1 del2); intro.
apply (cond_pos del1).
elim H8; intros; assumption.
intros;
apply Rle_lt_trans with (Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - f y)).
replace (f x0 - f y) with (f x0 - fn N0 x0 + (fn N0 x0 - f y));
[ apply Rabs_triang | ring ].
apply Rle_lt_trans with
(Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - fn N0 y) + Rabs (fn N0 y - f y)).
rewrite Rplus_assoc; apply Rplus_le_compat_l.
replace (fn N0 x0 - f y) with (fn N0 x0 - fn N0 y + (fn N0 y - f y));
[ apply Rabs_triang | ring ].
replace eps with (eps / 3 + eps / 3 + eps / 3).
repeat apply Rplus_lt_compat.
apply H4.
apply le_n.
replace x0 with (y + (x0 - y)); [ idtac | ring ]; apply H7.
elim H9; intros.
apply Rlt_le_trans with del.
assumption.
unfold del; apply Rmin_l.
elim H8; intros.
apply H11.
split.
elim H9; intros; assumption.
elim H9; intros; apply Rlt_le_trans with del.
assumption.
unfold del; apply Rmin_r.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H4.
apply le_n.
assumption.
apply Rmult_eq_reg_l with 3.
do 2 rewrite Rmult_plus_distr_l; unfold Rdiv; rewrite <- Rmult_assoc;
rewrite Rinv_r_simpl_m.
ring.
discrR.
discrR.
cut (0 < r - Rabs (x - y)).
intro; exists (mkposreal _ H6).
simpl; intros.
unfold Boule; replace (y + h - x) with (h + (y - x));
[ idtac | ring ]; apply Rle_lt_trans with (Rabs h + Rabs (y - x)).
apply Rabs_triang.
apply Rplus_lt_reg_l with (- Rabs (x - y)).
rewrite <- (Rabs_Ropp (y - x)); rewrite Ropp_minus_distr'.
replace (- Rabs (x - y) + r) with (r - Rabs (x - y)).
replace (- Rabs (x - y) + (Rabs h + Rabs (x - y))) with (Rabs h).
apply H7.
ring.
ring.
unfold Boule in H1; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr';
apply Rplus_lt_reg_l with (Rabs (y - x)).
rewrite Rplus_0_r; replace (Rabs (y - x) + (r - Rabs (y - x))) with (pos r);
[ apply H1 | ring ].
Qed.
Lemma continuity_pt_finite_SF :
forall (fn:nat -> R -> R) (N:nat) (x:R),
(forall n:nat, (n <= N)%nat -> continuity_pt (fn n) x) ->
continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x.
Proof.
intros; induction N as [| N HrecN].
simpl; apply (H 0%nat); apply le_n.
simpl;
replace (fun y:R => sum_f_R0 (fun k:nat => fn k y) N + fn (S N) y) with
((fun y:R => sum_f_R0 (fun k:nat => fn k y) N) + (fun y:R => fn (S N) y))%F;
[ idtac | reflexivity ].
apply continuity_pt_plus.
apply HrecN.
intros; apply H.
apply le_trans with N; [ assumption | apply le_n_Sn ].
apply (H (S N)); apply le_n.
Qed.
forall (fn:nat -> R -> R) (f:R -> R) (x:R) (r:posreal),
CVU fn f x r ->
(forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) ->
forall y:R, Boule x r y -> continuity_pt f y.
Proof.
intros; unfold continuity_pt; unfold continue_in;
unfold limit1_in; unfold limit_in;
simpl; unfold R_dist; intros.
unfold CVU in H.
cut (0 < eps / 3);
[ intro
| unfold Rdiv; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ].
elim (H _ H3); intros N0 H4.
assert (H5 := H0 N0 y H1).
cut (exists del : posreal, (forall h:R, Rabs h < del -> Boule x r (y + h))).
intro.
elim H6; intros del1 H7.
unfold continuity_pt in H5; unfold continue_in in H5; unfold limit1_in in H5;
unfold limit_in in H5; simpl in H5; unfold R_dist in H5.
elim (H5 _ H3); intros del2 H8.
set (del := Rmin del1 del2).
exists del; intros.
split.
unfold del; unfold Rmin; case (Rle_dec del1 del2); intro.
apply (cond_pos del1).
elim H8; intros; assumption.
intros;
apply Rle_lt_trans with (Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - f y)).
replace (f x0 - f y) with (f x0 - fn N0 x0 + (fn N0 x0 - f y));
[ apply Rabs_triang | ring ].
apply Rle_lt_trans with
(Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - fn N0 y) + Rabs (fn N0 y - f y)).
rewrite Rplus_assoc; apply Rplus_le_compat_l.
replace (fn N0 x0 - f y) with (fn N0 x0 - fn N0 y + (fn N0 y - f y));
[ apply Rabs_triang | ring ].
replace eps with (eps / 3 + eps / 3 + eps / 3).
repeat apply Rplus_lt_compat.
apply H4.
apply le_n.
replace x0 with (y + (x0 - y)); [ idtac | ring ]; apply H7.
elim H9; intros.
apply Rlt_le_trans with del.
assumption.
unfold del; apply Rmin_l.
elim H8; intros.
apply H11.
split.
elim H9; intros; assumption.
elim H9; intros; apply Rlt_le_trans with del.
assumption.
unfold del; apply Rmin_r.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H4.
apply le_n.
assumption.
apply Rmult_eq_reg_l with 3.
do 2 rewrite Rmult_plus_distr_l; unfold Rdiv; rewrite <- Rmult_assoc;
rewrite Rinv_r_simpl_m.
ring.
discrR.
discrR.
cut (0 < r - Rabs (x - y)).
intro; exists (mkposreal _ H6).
simpl; intros.
unfold Boule; replace (y + h - x) with (h + (y - x));
[ idtac | ring ]; apply Rle_lt_trans with (Rabs h + Rabs (y - x)).
apply Rabs_triang.
apply Rplus_lt_reg_l with (- Rabs (x - y)).
rewrite <- (Rabs_Ropp (y - x)); rewrite Ropp_minus_distr'.
replace (- Rabs (x - y) + r) with (r - Rabs (x - y)).
replace (- Rabs (x - y) + (Rabs h + Rabs (x - y))) with (Rabs h).
apply H7.
ring.
ring.
unfold Boule in H1; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr';
apply Rplus_lt_reg_l with (Rabs (y - x)).
rewrite Rplus_0_r; replace (Rabs (y - x) + (r - Rabs (y - x))) with (pos r);
[ apply H1 | ring ].
Qed.
Lemma continuity_pt_finite_SF :
forall (fn:nat -> R -> R) (N:nat) (x:R),
(forall n:nat, (n <= N)%nat -> continuity_pt (fn n) x) ->
continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x.
Proof.
intros; induction N as [| N HrecN].
simpl; apply (H 0%nat); apply le_n.
simpl;
replace (fun y:R => sum_f_R0 (fun k:nat => fn k y) N + fn (S N) y) with
((fun y:R => sum_f_R0 (fun k:nat => fn k y) N) + (fun y:R => fn (S N) y))%F;
[ idtac | reflexivity ].
apply continuity_pt_plus.
apply HrecN.
intros; apply H.
apply le_trans with N; [ assumption | apply le_n_Sn ].
apply (H (S N)); apply le_n.
Qed.
Continuity and normal convergence
Lemma SFL_continuity_pt :
forall (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
(r:posreal),
CVN_r fn r ->
(forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y) ->
forall y:R, Boule 0 r y -> continuity_pt (SFL fn cv) y.
Proof.
intros; eapply CVU_continuity.
apply CVN_CVU.
apply X.
intros; unfold SP; apply continuity_pt_finite_SF.
intros; apply H.
apply H1.
apply H0.
Qed.
Lemma SFL_continuity :
forall (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }),
CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv).
Proof.
intros; unfold continuity; intro.
cut (0 < Rabs x + 1);
[ intro | apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ] ].
cut (Boule 0 (mkposreal _ H0) x).
intro; eapply SFL_continuity_pt with (mkposreal _ H0).
apply X.
intros; apply (H n y).
apply H1.
unfold Boule; simpl; rewrite Rminus_0_r;
pattern (Rabs x) at 1; rewrite <- Rplus_0_r;
apply Rplus_lt_compat_l; apply Rlt_0_1.
Qed.
forall (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l })
(r:posreal),
CVN_r fn r ->
(forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y) ->
forall y:R, Boule 0 r y -> continuity_pt (SFL fn cv) y.
Proof.
intros; eapply CVU_continuity.
apply CVN_CVU.
apply X.
intros; unfold SP; apply continuity_pt_finite_SF.
intros; apply H.
apply H1.
apply H0.
Qed.
Lemma SFL_continuity :
forall (fn:nat -> R -> R)
(cv:forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }),
CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv).
Proof.
intros; unfold continuity; intro.
cut (0 < Rabs x + 1);
[ intro | apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ] ].
cut (Boule 0 (mkposreal _ H0) x).
intro; eapply SFL_continuity_pt with (mkposreal _ H0).
apply X.
intros; apply (H n y).
apply H1.
unfold Boule; simpl; rewrite Rminus_0_r;
pattern (Rabs x) at 1; rewrite <- Rplus_0_r;
apply Rplus_lt_compat_l; apply Rlt_0_1.
Qed.
As R is complete, normal convergence implies that (fn) is simply-uniformly convergent
Lemma CVN_R_CVS :
forall fn:nat -> R -> R,
CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }.
Proof.
intros; apply R_complete.
unfold SP; set (An := fun N:nat => fn N x).
change (Cauchy_crit_series An).
apply cauchy_abs.
unfold Cauchy_crit_series; apply CV_Cauchy.
unfold CVN_R in X; cut (0 < Rabs x + 1).
intro; assert (H0 := X (mkposreal _ H)).
unfold CVN_r in H0; elim H0; intros Bn H1.
elim H1; intros l H2.
elim H2; intros.
apply Rseries_CV_comp with Bn.
intro; split.
apply Rabs_pos.
unfold An; apply H4; unfold Boule; simpl;
rewrite Rminus_0_r.
pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
apply Rlt_0_1.
exists l.
cut (forall n:nat, 0 <= Bn n).
intro; unfold Un_cv in H3; unfold Un_cv; intros.
elim (H3 _ H6); intros.
exists x0; intros.
replace (sum_f_R0 Bn n) with (sum_f_R0 (fun k:nat => Rabs (Bn k)) n).
apply H7; assumption.
apply sum_eq; intros; apply Rabs_right; apply Rle_ge; apply H5.
intro; apply Rle_trans with (Rabs (An n)).
apply Rabs_pos.
unfold An; apply H4; unfold Boule; simpl;
rewrite Rminus_0_r; pattern (Rabs x) at 1;
rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1.
apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ].
Qed.
Lemma CVU_cv : forall f g c d, CVU f g c d ->
forall x, Boule c d x -> Un_cv (fun n => f n x) (g x).
Proof.
intros f g c d cvu x bx eps ep; destruct (cvu eps ep) as [N Pn].
exists N; intros n nN; rewrite R_dist_sym; apply Pn; assumption.
Qed.
Lemma CVU_ext_lim :
forall f g1 g2 c d, CVU f g1 c d -> (forall x, Boule c d x -> g1 x = g2 x) ->
CVU f g2 c d.
Proof.
intros f g1 g2 c d cvu q eps ep; destruct (cvu _ ep) as [N Pn].
exists N; intros; rewrite <- q; auto.
Qed.
Lemma CVU_derivable :
forall f f' g g' c d,
CVU f' g' c d ->
(forall x, Boule c d x -> Un_cv (fun n => f n x) (g x)) ->
(forall n x, Boule c d x -> derivable_pt_lim (f n) x (f' n x)) ->
forall x, Boule c d x -> derivable_pt_lim g x (g' x).
Proof.
intros f f' g g' c d cvu cvp dff' x bx.
set (rho_ :=
fun n y =>
if Req_EM_T y x then
f' n x
else ((f n y - f n x)/ (y - x))).
set (rho := fun y =>
if Req_EM_T y x then
g' x
else (g y - g x)/(y - x)).
assert (ctrho : forall n z, Boule c d z -> continuity_pt (rho_ n) z).
intros n z bz.
destruct (Req_EM_T x z) as [xz | xnz].
rewrite <- xz.
intros eps' ep'.
destruct (dff' n x bx eps' ep') as [alp Pa].
exists (pos alp);split;[apply cond_pos | ].
intros z'; unfold rho_, D_x, dist, R_met; simpl; intros [[_ xnz'] dxz'].
destruct (Req_EM_T z' x) as [abs | _].
case xnz'; symmetry; exact abs.
destruct (Req_EM_T x x) as [_ | abs];[ | case abs; reflexivity].
pattern z' at 1; replace z' with (x + (z' - x)) by ring.
apply Pa;[intros h; case xnz';
replace z' with (z' - x + x) by ring; rewrite h, Rplus_0_l;
reflexivity | exact dxz'].
destruct (Ball_in_inter c c d d z bz bz) as [delta Pd].
assert (dz : 0 < Rmin delta (Rabs (z - x))).
now apply Rmin_glb_lt;[apply cond_pos | apply Rabs_pos_lt; intros zx0; case xnz;
replace z with (z - x + x) by ring; rewrite zx0, Rplus_0_l].
assert (t' : forall y : R,
R_dist y z < Rmin delta (Rabs (z - x)) ->
(fun z : R => (f n z - f n x) / (z - x)) y = rho_ n y).
intros y dyz; unfold rho_; destruct (Req_EM_T y x) as [xy | xny].
rewrite xy in dyz.
destruct (Rle_dec delta (Rabs (z - x))).
rewrite Rmin_left, R_dist_sym in dyz; unfold R_dist in dyz; lra.
rewrite Rmin_right, R_dist_sym in dyz; unfold R_dist in dyz;
[case (Rlt_irrefl _ dyz) |apply Rlt_le, Rnot_le_gt; assumption].
reflexivity.
apply (continuity_pt_locally_ext (fun z => (f n z - f n x)/(z - x))
(rho_ n) _ z dz t'); clear t'.
apply continuity_pt_div.
apply continuity_pt_minus.
apply derivable_continuous_pt; eapply exist; apply dff'; assumption.
apply continuity_pt_const; intro; intro; reflexivity.
apply continuity_pt_minus;
[apply derivable_continuous_pt; exists 1; apply derivable_pt_lim_id
| apply continuity_pt_const; intro; reflexivity].
intros zx0; case xnz; replace z with (z - x + x) by ring.
rewrite zx0, Rplus_0_l; reflexivity.
assert (CVU rho_ rho c d ).
intros eps ep.
assert (ep8 : 0 < eps/8).
lra.
destruct (cvu _ ep8) as [N Pn1].
assert (cauchy1 : forall n p, (N <= n)%nat -> (N <= p)%nat ->
forall z, Boule c d z -> Rabs (f' n z - f' p z) < eps/4).
intros n p nN pN z bz; replace (eps/4) with (eps/8 + eps/8) by field.
rewrite <- Rabs_Ropp.
replace (-(f' n z - f' p z)) with (g' z - f' n z - (g' z - f' p z)) by ring.
apply Rle_lt_trans with (1 := Rabs_triang _ _); rewrite Rabs_Ropp.
apply Rplus_lt_compat; apply Pn1; assumption.
assert (step_2 : forall n p, (N <= n)%nat -> (N <= p)%nat ->
forall y, Boule c d y -> x <> y ->
Rabs ((f n y - f n x)/(y - x) - (f p y - f p x)/(y - x)) < eps/4).
intros n p nN pN y b_y xny.
assert (mm0 : (Rmin x y = x /\ Rmax x y = y) \/
(Rmin x y = y /\ Rmax x y = x)).
destruct (Rle_dec x y) as [H | H].
rewrite Rmin_left, Rmax_right.
left; split; reflexivity.
assumption.
assumption.
rewrite Rmin_right, Rmax_left.
right; split; reflexivity.
apply Rlt_le, Rnot_le_gt; assumption.
apply Rlt_le, Rnot_le_gt; assumption.
assert (mm : Rmin x y < Rmax x y).
destruct mm0 as [[q1 q2] | [q1 q2]]; generalize (Rminmax x y); rewrite q1, q2.
intros h; destruct h;[ assumption| contradiction].
intros h; destruct h as [h | h];[assumption | rewrite h in xny; case xny; reflexivity].
assert (dm : forall z, Rmin x y <= z <= Rmax x y ->
derivable_pt_lim (fun x => f n x - f p x) z (f' n z - f' p z)).
intros z intz; apply derivable_pt_lim_minus.
apply dff'; apply Boule_convex with (Rmin x y) (Rmax x y);
destruct mm0 as [[q1 q2] | [q1 q2]]; revert intz; rewrite ?q1, ?q2; intros;
try assumption.
apply dff'; apply Boule_convex with (Rmin x y) (Rmax x y);
destruct mm0 as [[q1 q2] | [q1 q2]]; revert intz; rewrite ?q1, ?q2; intros;
try assumption.
replace ((f n y - f n x) / (y - x) - (f p y - f p x) / (y - x))
with (((f n y - f p y) - (f n x - f p x))/(y - x)) by
(field; intros yx0; case xny; replace y with (y - x + x) by ring;
rewrite yx0, Rplus_0_l; reflexivity).
destruct (MVT_cor2 (fun x => f n x - f p x) (fun x => f' n x - f' p x)
(Rmin x y) (Rmax x y) mm dm) as [z [Pz inz]].
destruct mm0 as [[q1 q2] | [q1 q2]].
replace ((f n y - f p y - (f n x - f p x))/(y - x)) with
((f n (Rmax x y) - f p (Rmax x y) - (f n (Rmin x y) - f p (Rmin x y)))/
(Rmax x y - Rmin x y)) by (rewrite q1, q2; reflexivity).
unfold Rdiv; rewrite Pz, Rmult_assoc, Rinv_r, Rmult_1_r.
apply cauchy1; auto.
apply Boule_convex with (Rmin x y) (Rmax x y);
revert inz; rewrite ?q1, ?q2; intros;
try assumption.
split; apply Rlt_le; tauto.
rewrite q1, q2; apply Rminus_eq_contra, not_eq_sym; assumption.
replace ((f n y - f p y - (f n x - f p x))/(y - x)) with
((f n (Rmax x y) - f p (Rmax x y) - (f n (Rmin x y) - f p (Rmin x y)))/
(Rmax x y - Rmin x y)).
unfold Rdiv; rewrite Pz, Rmult_assoc, Rinv_r, Rmult_1_r.
apply cauchy1; auto.
apply Boule_convex with (Rmin x y) (Rmax x y);
revert inz; rewrite ?q1, ?q2; intros;
try assumption; split; apply Rlt_le; tauto.
rewrite q1, q2; apply Rminus_eq_contra; assumption.
rewrite q1, q2; field; split;
apply Rminus_eq_contra;[apply not_eq_sym |]; assumption.
assert (unif_ac :
forall n p, (N <= n)%nat -> (N <= p)%nat ->
forall y, Boule c d y ->
Rabs (rho_ n y - rho_ p y) <= eps/2).
intros n p nN pN y b_y.
destruct (Req_dec x y) as [xy | xny].
destruct (Ball_in_inter c c d d x bx bx) as [delta Pdelta].
destruct (ctrho n y b_y _ ep8) as [d' [dp Pd]].
destruct (ctrho p y b_y _ ep8) as [d2 [dp2 Pd2]].
assert (mmpos : 0 < (Rmin (Rmin d' d2) delta)/2).
apply Rmult_lt_0_compat; repeat apply Rmin_glb_lt; try assumption.
apply cond_pos.
apply Rinv_0_lt_compat, Rlt_0_2.
apply Rle_trans with (1 := R_dist_tri _ _ (rho_ n (y + Rmin (Rmin d' d2) delta/2))).
replace (eps/2) with (eps/8 + (eps/4 + eps/8)) by field.
apply Rplus_le_compat.
rewrite R_dist_sym; apply Rlt_le, Pd;split;[split;[exact I | ] | ].
apply Rminus_not_eq_right; rewrite Rplus_comm; unfold Rminus;
rewrite Rplus_assoc, Rplus_opp_r, Rplus_0_r; apply Rgt_not_eq; assumption.
simpl; unfold R_dist.
unfold Rminus; rewrite (Rplus_comm y), Rplus_assoc, Rplus_opp_r, Rplus_0_r.
rewrite Rabs_pos_eq;[ |apply Rlt_le; assumption ].
apply Rlt_le_trans with (Rmin (Rmin d' d2) delta);[lra | ].
apply Rle_trans with (Rmin d' d2); apply Rmin_l.
apply Rle_trans with (1 := R_dist_tri _ _ (rho_ p (y + Rmin (Rmin d' d2) delta/2))).
apply Rplus_le_compat.
apply Rlt_le.
replace (rho_ n (y + Rmin (Rmin d' d2) delta / 2)) with
((f n (y + Rmin (Rmin d' d2) delta / 2) - f n x)/
((y + Rmin (Rmin d' d2) delta / 2) - x)).
replace (rho_ p (y + Rmin (Rmin d' d2) delta / 2)) with
((f p (y + Rmin (Rmin d' d2) delta / 2) - f p x)/
((y + Rmin (Rmin d' d2) delta / 2) - x)).
apply step_2; auto; try lra.
assert (0 < pos delta) by (apply cond_pos).
apply Boule_convex with y (y + delta/2).
assumption.
destruct (Pdelta (y + delta/2)); auto.
rewrite xy; unfold Boule; rewrite Rabs_pos_eq; try lra; auto.
split; try lra.
apply Rplus_le_compat_l, Rmult_le_compat_r;[ | apply Rmin_r].
now apply Rlt_le, Rinv_0_lt_compat, Rlt_0_2.
unfold rho_.
destruct (Req_EM_T (y + Rmin (Rmin d' d2) delta/2) x) as [ymx | ymnx].
case (RIneq.Rle_not_lt _ _ (Req_le _ _ ymx)); lra.
reflexivity.
unfold rho_.
destruct (Req_EM_T (y + Rmin (Rmin d' d2) delta / 2) x) as [ymx | ymnx].
case (RIneq.Rle_not_lt _ _ (Req_le _ _ ymx)); lra.
reflexivity.
apply Rlt_le, Pd2; split;[split;[exact I | apply Rlt_not_eq; lra] | ].
simpl; unfold R_dist.
unfold Rminus; rewrite (Rplus_comm y), Rplus_assoc, Rplus_opp_r, Rplus_0_r.
rewrite Rabs_pos_eq;[ | lra].
apply Rlt_le_trans with (Rmin (Rmin d' d2) delta); [lra |].
apply Rle_trans with (Rmin d' d2).
solve[apply Rmin_l].
solve[apply Rmin_r].
apply Rlt_le, Rlt_le_trans with (eps/4);[ | lra].
unfold rho_; destruct (Req_EM_T y x); solve[auto].
assert (unif_ac' : forall p, (N <= p)%nat ->
forall y, Boule c d y -> Rabs (rho y - rho_ p y) < eps).
assert (cvrho : forall y, Boule c d y -> Un_cv (fun n => rho_ n y) (rho y)).
intros y b_y; unfold rho_, rho; destruct (Req_EM_T y x).
intros eps' ep'; destruct (cvu eps' ep') as [N2 Pn2].
exists N2; intros n nN2; rewrite R_dist_sym; apply Pn2; assumption.
apply CV_mult.
apply CV_minus.
apply cvp; assumption.
apply cvp; assumption.
intros eps' ep'; simpl; exists 0%nat; intros; rewrite R_dist_eq; assumption.
intros p pN y b_y.
replace eps with (eps/2 + eps/2) by field.
assert (ep2 : 0 < eps/2) by lra.
destruct (cvrho y b_y _ ep2) as [N2 Pn2].
apply Rle_lt_trans with (1 := R_dist_tri _ _ (rho_ (max N N2) y)).
apply Rplus_lt_le_compat.
solve[rewrite R_dist_sym; apply Pn2, Max.le_max_r].
apply unif_ac; auto; solve [apply Max.le_max_l].
exists N; intros; apply unif_ac'; solve[auto].
intros eps ep.
destruct (CVU_continuity _ _ _ _ H ctrho x bx eps ep) as [delta [dp Pd]].
exists (mkposreal _ dp); intros h hn0 dh.
replace ((g (x + h) - g x) / h) with (rho (x + h)).
replace (g' x) with (rho x).
apply Pd; unfold D_x, no_cond;split;[split;[solve[auto] | ] | ].
intros xxh; case hn0; replace h with (x + h - x) by ring; rewrite <- xxh; ring.
simpl; unfold R_dist; replace (x + h - x) with h by ring; exact dh.
unfold rho; destruct (Req_EM_T x x) as [ _ | abs];[ | case abs]; reflexivity.
unfold rho; destruct (Req_EM_T (x + h) x) as [abs | _];[ | ].
case hn0; replace h with (x + h - x) by ring; rewrite abs; ring.
replace (x + h - x) with h by ring; reflexivity.
Qed.
forall fn:nat -> R -> R,
CVN_R fn -> forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }.
Proof.
intros; apply R_complete.
unfold SP; set (An := fun N:nat => fn N x).
change (Cauchy_crit_series An).
apply cauchy_abs.
unfold Cauchy_crit_series; apply CV_Cauchy.
unfold CVN_R in X; cut (0 < Rabs x + 1).
intro; assert (H0 := X (mkposreal _ H)).
unfold CVN_r in H0; elim H0; intros Bn H1.
elim H1; intros l H2.
elim H2; intros.
apply Rseries_CV_comp with Bn.
intro; split.
apply Rabs_pos.
unfold An; apply H4; unfold Boule; simpl;
rewrite Rminus_0_r.
pattern (Rabs x) at 1; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
apply Rlt_0_1.
exists l.
cut (forall n:nat, 0 <= Bn n).
intro; unfold Un_cv in H3; unfold Un_cv; intros.
elim (H3 _ H6); intros.
exists x0; intros.
replace (sum_f_R0 Bn n) with (sum_f_R0 (fun k:nat => Rabs (Bn k)) n).
apply H7; assumption.
apply sum_eq; intros; apply Rabs_right; apply Rle_ge; apply H5.
intro; apply Rle_trans with (Rabs (An n)).
apply Rabs_pos.
unfold An; apply H4; unfold Boule; simpl;
rewrite Rminus_0_r; pattern (Rabs x) at 1;
rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1.
apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ].
Qed.
Lemma CVU_cv : forall f g c d, CVU f g c d ->
forall x, Boule c d x -> Un_cv (fun n => f n x) (g x).
Proof.
intros f g c d cvu x bx eps ep; destruct (cvu eps ep) as [N Pn].
exists N; intros n nN; rewrite R_dist_sym; apply Pn; assumption.
Qed.
Lemma CVU_ext_lim :
forall f g1 g2 c d, CVU f g1 c d -> (forall x, Boule c d x -> g1 x = g2 x) ->
CVU f g2 c d.
Proof.
intros f g1 g2 c d cvu q eps ep; destruct (cvu _ ep) as [N Pn].
exists N; intros; rewrite <- q; auto.
Qed.
Lemma CVU_derivable :
forall f f' g g' c d,
CVU f' g' c d ->
(forall x, Boule c d x -> Un_cv (fun n => f n x) (g x)) ->
(forall n x, Boule c d x -> derivable_pt_lim (f n) x (f' n x)) ->
forall x, Boule c d x -> derivable_pt_lim g x (g' x).
Proof.
intros f f' g g' c d cvu cvp dff' x bx.
set (rho_ :=
fun n y =>
if Req_EM_T y x then
f' n x
else ((f n y - f n x)/ (y - x))).
set (rho := fun y =>
if Req_EM_T y x then
g' x
else (g y - g x)/(y - x)).
assert (ctrho : forall n z, Boule c d z -> continuity_pt (rho_ n) z).
intros n z bz.
destruct (Req_EM_T x z) as [xz | xnz].
rewrite <- xz.
intros eps' ep'.
destruct (dff' n x bx eps' ep') as [alp Pa].
exists (pos alp);split;[apply cond_pos | ].
intros z'; unfold rho_, D_x, dist, R_met; simpl; intros [[_ xnz'] dxz'].
destruct (Req_EM_T z' x) as [abs | _].
case xnz'; symmetry; exact abs.
destruct (Req_EM_T x x) as [_ | abs];[ | case abs; reflexivity].
pattern z' at 1; replace z' with (x + (z' - x)) by ring.
apply Pa;[intros h; case xnz';
replace z' with (z' - x + x) by ring; rewrite h, Rplus_0_l;
reflexivity | exact dxz'].
destruct (Ball_in_inter c c d d z bz bz) as [delta Pd].
assert (dz : 0 < Rmin delta (Rabs (z - x))).
now apply Rmin_glb_lt;[apply cond_pos | apply Rabs_pos_lt; intros zx0; case xnz;
replace z with (z - x + x) by ring; rewrite zx0, Rplus_0_l].
assert (t' : forall y : R,
R_dist y z < Rmin delta (Rabs (z - x)) ->
(fun z : R => (f n z - f n x) / (z - x)) y = rho_ n y).
intros y dyz; unfold rho_; destruct (Req_EM_T y x) as [xy | xny].
rewrite xy in dyz.
destruct (Rle_dec delta (Rabs (z - x))).
rewrite Rmin_left, R_dist_sym in dyz; unfold R_dist in dyz; lra.
rewrite Rmin_right, R_dist_sym in dyz; unfold R_dist in dyz;
[case (Rlt_irrefl _ dyz) |apply Rlt_le, Rnot_le_gt; assumption].
reflexivity.
apply (continuity_pt_locally_ext (fun z => (f n z - f n x)/(z - x))
(rho_ n) _ z dz t'); clear t'.
apply continuity_pt_div.
apply continuity_pt_minus.
apply derivable_continuous_pt; eapply exist; apply dff'; assumption.
apply continuity_pt_const; intro; intro; reflexivity.
apply continuity_pt_minus;
[apply derivable_continuous_pt; exists 1; apply derivable_pt_lim_id
| apply continuity_pt_const; intro; reflexivity].
intros zx0; case xnz; replace z with (z - x + x) by ring.
rewrite zx0, Rplus_0_l; reflexivity.
assert (CVU rho_ rho c d ).
intros eps ep.
assert (ep8 : 0 < eps/8).
lra.
destruct (cvu _ ep8) as [N Pn1].
assert (cauchy1 : forall n p, (N <= n)%nat -> (N <= p)%nat ->
forall z, Boule c d z -> Rabs (f' n z - f' p z) < eps/4).
intros n p nN pN z bz; replace (eps/4) with (eps/8 + eps/8) by field.
rewrite <- Rabs_Ropp.
replace (-(f' n z - f' p z)) with (g' z - f' n z - (g' z - f' p z)) by ring.
apply Rle_lt_trans with (1 := Rabs_triang _ _); rewrite Rabs_Ropp.
apply Rplus_lt_compat; apply Pn1; assumption.
assert (step_2 : forall n p, (N <= n)%nat -> (N <= p)%nat ->
forall y, Boule c d y -> x <> y ->
Rabs ((f n y - f n x)/(y - x) - (f p y - f p x)/(y - x)) < eps/4).
intros n p nN pN y b_y xny.
assert (mm0 : (Rmin x y = x /\ Rmax x y = y) \/
(Rmin x y = y /\ Rmax x y = x)).
destruct (Rle_dec x y) as [H | H].
rewrite Rmin_left, Rmax_right.
left; split; reflexivity.
assumption.
assumption.
rewrite Rmin_right, Rmax_left.
right; split; reflexivity.
apply Rlt_le, Rnot_le_gt; assumption.
apply Rlt_le, Rnot_le_gt; assumption.
assert (mm : Rmin x y < Rmax x y).
destruct mm0 as [[q1 q2] | [q1 q2]]; generalize (Rminmax x y); rewrite q1, q2.
intros h; destruct h;[ assumption| contradiction].
intros h; destruct h as [h | h];[assumption | rewrite h in xny; case xny; reflexivity].
assert (dm : forall z, Rmin x y <= z <= Rmax x y ->
derivable_pt_lim (fun x => f n x - f p x) z (f' n z - f' p z)).
intros z intz; apply derivable_pt_lim_minus.
apply dff'; apply Boule_convex with (Rmin x y) (Rmax x y);
destruct mm0 as [[q1 q2] | [q1 q2]]; revert intz; rewrite ?q1, ?q2; intros;
try assumption.
apply dff'; apply Boule_convex with (Rmin x y) (Rmax x y);
destruct mm0 as [[q1 q2] | [q1 q2]]; revert intz; rewrite ?q1, ?q2; intros;
try assumption.
replace ((f n y - f n x) / (y - x) - (f p y - f p x) / (y - x))
with (((f n y - f p y) - (f n x - f p x))/(y - x)) by
(field; intros yx0; case xny; replace y with (y - x + x) by ring;
rewrite yx0, Rplus_0_l; reflexivity).
destruct (MVT_cor2 (fun x => f n x - f p x) (fun x => f' n x - f' p x)
(Rmin x y) (Rmax x y) mm dm) as [z [Pz inz]].
destruct mm0 as [[q1 q2] | [q1 q2]].
replace ((f n y - f p y - (f n x - f p x))/(y - x)) with
((f n (Rmax x y) - f p (Rmax x y) - (f n (Rmin x y) - f p (Rmin x y)))/
(Rmax x y - Rmin x y)) by (rewrite q1, q2; reflexivity).
unfold Rdiv; rewrite Pz, Rmult_assoc, Rinv_r, Rmult_1_r.
apply cauchy1; auto.
apply Boule_convex with (Rmin x y) (Rmax x y);
revert inz; rewrite ?q1, ?q2; intros;
try assumption.
split; apply Rlt_le; tauto.
rewrite q1, q2; apply Rminus_eq_contra, not_eq_sym; assumption.
replace ((f n y - f p y - (f n x - f p x))/(y - x)) with
((f n (Rmax x y) - f p (Rmax x y) - (f n (Rmin x y) - f p (Rmin x y)))/
(Rmax x y - Rmin x y)).
unfold Rdiv; rewrite Pz, Rmult_assoc, Rinv_r, Rmult_1_r.
apply cauchy1; auto.
apply Boule_convex with (Rmin x y) (Rmax x y);
revert inz; rewrite ?q1, ?q2; intros;
try assumption; split; apply Rlt_le; tauto.
rewrite q1, q2; apply Rminus_eq_contra; assumption.
rewrite q1, q2; field; split;
apply Rminus_eq_contra;[apply not_eq_sym |]; assumption.
assert (unif_ac :
forall n p, (N <= n)%nat -> (N <= p)%nat ->
forall y, Boule c d y ->
Rabs (rho_ n y - rho_ p y) <= eps/2).
intros n p nN pN y b_y.
destruct (Req_dec x y) as [xy | xny].
destruct (Ball_in_inter c c d d x bx bx) as [delta Pdelta].
destruct (ctrho n y b_y _ ep8) as [d' [dp Pd]].
destruct (ctrho p y b_y _ ep8) as [d2 [dp2 Pd2]].
assert (mmpos : 0 < (Rmin (Rmin d' d2) delta)/2).
apply Rmult_lt_0_compat; repeat apply Rmin_glb_lt; try assumption.
apply cond_pos.
apply Rinv_0_lt_compat, Rlt_0_2.
apply Rle_trans with (1 := R_dist_tri _ _ (rho_ n (y + Rmin (Rmin d' d2) delta/2))).
replace (eps/2) with (eps/8 + (eps/4 + eps/8)) by field.
apply Rplus_le_compat.
rewrite R_dist_sym; apply Rlt_le, Pd;split;[split;[exact I | ] | ].
apply Rminus_not_eq_right; rewrite Rplus_comm; unfold Rminus;
rewrite Rplus_assoc, Rplus_opp_r, Rplus_0_r; apply Rgt_not_eq; assumption.
simpl; unfold R_dist.
unfold Rminus; rewrite (Rplus_comm y), Rplus_assoc, Rplus_opp_r, Rplus_0_r.
rewrite Rabs_pos_eq;[ |apply Rlt_le; assumption ].
apply Rlt_le_trans with (Rmin (Rmin d' d2) delta);[lra | ].
apply Rle_trans with (Rmin d' d2); apply Rmin_l.
apply Rle_trans with (1 := R_dist_tri _ _ (rho_ p (y + Rmin (Rmin d' d2) delta/2))).
apply Rplus_le_compat.
apply Rlt_le.
replace (rho_ n (y + Rmin (Rmin d' d2) delta / 2)) with
((f n (y + Rmin (Rmin d' d2) delta / 2) - f n x)/
((y + Rmin (Rmin d' d2) delta / 2) - x)).
replace (rho_ p (y + Rmin (Rmin d' d2) delta / 2)) with
((f p (y + Rmin (Rmin d' d2) delta / 2) - f p x)/
((y + Rmin (Rmin d' d2) delta / 2) - x)).
apply step_2; auto; try lra.
assert (0 < pos delta) by (apply cond_pos).
apply Boule_convex with y (y + delta/2).
assumption.
destruct (Pdelta (y + delta/2)); auto.
rewrite xy; unfold Boule; rewrite Rabs_pos_eq; try lra; auto.
split; try lra.
apply Rplus_le_compat_l, Rmult_le_compat_r;[ | apply Rmin_r].
now apply Rlt_le, Rinv_0_lt_compat, Rlt_0_2.
unfold rho_.
destruct (Req_EM_T (y + Rmin (Rmin d' d2) delta/2) x) as [ymx | ymnx].
case (RIneq.Rle_not_lt _ _ (Req_le _ _ ymx)); lra.
reflexivity.
unfold rho_.
destruct (Req_EM_T (y + Rmin (Rmin d' d2) delta / 2) x) as [ymx | ymnx].
case (RIneq.Rle_not_lt _ _ (Req_le _ _ ymx)); lra.
reflexivity.
apply Rlt_le, Pd2; split;[split;[exact I | apply Rlt_not_eq; lra] | ].
simpl; unfold R_dist.
unfold Rminus; rewrite (Rplus_comm y), Rplus_assoc, Rplus_opp_r, Rplus_0_r.
rewrite Rabs_pos_eq;[ | lra].
apply Rlt_le_trans with (Rmin (Rmin d' d2) delta); [lra |].
apply Rle_trans with (Rmin d' d2).
solve[apply Rmin_l].
solve[apply Rmin_r].
apply Rlt_le, Rlt_le_trans with (eps/4);[ | lra].
unfold rho_; destruct (Req_EM_T y x); solve[auto].
assert (unif_ac' : forall p, (N <= p)%nat ->
forall y, Boule c d y -> Rabs (rho y - rho_ p y) < eps).
assert (cvrho : forall y, Boule c d y -> Un_cv (fun n => rho_ n y) (rho y)).
intros y b_y; unfold rho_, rho; destruct (Req_EM_T y x).
intros eps' ep'; destruct (cvu eps' ep') as [N2 Pn2].
exists N2; intros n nN2; rewrite R_dist_sym; apply Pn2; assumption.
apply CV_mult.
apply CV_minus.
apply cvp; assumption.
apply cvp; assumption.
intros eps' ep'; simpl; exists 0%nat; intros; rewrite R_dist_eq; assumption.
intros p pN y b_y.
replace eps with (eps/2 + eps/2) by field.
assert (ep2 : 0 < eps/2) by lra.
destruct (cvrho y b_y _ ep2) as [N2 Pn2].
apply Rle_lt_trans with (1 := R_dist_tri _ _ (rho_ (max N N2) y)).
apply Rplus_lt_le_compat.
solve[rewrite R_dist_sym; apply Pn2, Max.le_max_r].
apply unif_ac; auto; solve [apply Max.le_max_l].
exists N; intros; apply unif_ac'; solve[auto].
intros eps ep.
destruct (CVU_continuity _ _ _ _ H ctrho x bx eps ep) as [delta [dp Pd]].
exists (mkposreal _ dp); intros h hn0 dh.
replace ((g (x + h) - g x) / h) with (rho (x + h)).
replace (g' x) with (rho x).
apply Pd; unfold D_x, no_cond;split;[split;[solve[auto] | ] | ].
intros xxh; case hn0; replace h with (x + h - x) by ring; rewrite <- xxh; ring.
simpl; unfold R_dist; replace (x + h - x) with h by ring; exact dh.
unfold rho; destruct (Req_EM_T x x) as [ _ | abs];[ | case abs]; reflexivity.
unfold rho; destruct (Req_EM_T (x + h) x) as [abs | _];[ | ].
case hn0; replace h with (x + h - x) by ring; rewrite abs; ring.
replace (x + h - x) with h by ring; reflexivity.
Qed.