Library Coq.Reals.RiemannInt
Require Import Rfunctions.
Require Import SeqSeries.
Require Import Ranalysis.
Require Import Rbase.
Require Import RiemannInt_SF.
Require Import Classical_Prop.
Require Import Classical_Pred_Type.
Require Import Max. Open Local Scope R_scope.
Set Implicit Arguments.
Riemann's Integral
Definition Riemann_integrable (f:R -> R) (a b:R) : Type :=
forall eps:posreal,
sigT
(fun phi:StepFun a b =>
sigT
(fun psi:StepFun a b =>
(forall t:R,
Rmin a b <= t <= Rmax a b -> Rabs (f t - phi t) <= psi t) /\
Rabs (RiemannInt_SF psi) < eps)).
Definition phi_sequence (un:nat -> posreal) (f:R -> R)
(a b:R) (pr:Riemann_integrable f a b) (n:nat) :=
projT1 (pr (un n)).
Lemma phi_sequence_prop :
forall (un:nat -> posreal) (f:R -> R) (a b:R) (pr:Riemann_integrable f a b)
(N:nat),
sigT
(fun psi:StepFun a b =>
(forall t:R,
Rmin a b <= t <= Rmax a b ->
Rabs (f t - phi_sequence un pr N t) <= psi t) /\
Rabs (RiemannInt_SF psi) < un N).
Proof.
intros; apply (projT2 (pr (un N))).
Qed.
Lemma RiemannInt_P1 :
forall (f:R -> R) (a b:R),
Riemann_integrable f a b -> Riemann_integrable f b a.
Proof.
unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; intros;
elim p; clear p; intros; apply existT with (mkStepFun (StepFun_P6 (pre x)));
apply existT with (mkStepFun (StepFun_P6 (pre x0)));
elim p; clear p; intros; split.
intros; apply (H t); elim H1; clear H1; intros; split;
[ apply Rle_trans with (Rmin b a); try assumption; right;
unfold Rmin in |- *
| apply Rle_trans with (Rmax b a); try assumption; right;
unfold Rmax in |- * ];
(case (Rle_dec a b); case (Rle_dec b a); intros;
try reflexivity || apply Rle_antisym;
[ assumption | assumption | auto with real | auto with real ]).
generalize H0; unfold RiemannInt_SF in |- *; case (Rle_dec a b);
case (Rle_dec b a); intros;
(replace
(Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0))))
(subdivision (mkStepFun (StepFun_P6 (pre x0))))) with
(Int_SF (subdivision_val x0) (subdivision x0));
[ idtac
| apply StepFun_P17 with (fe x0) a b;
[ apply StepFun_P1
| apply StepFun_P2;
apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre x0)))) ] ]).
apply H1.
rewrite Rabs_Ropp; apply H1.
rewrite Rabs_Ropp in H1; apply H1.
apply H1.
Qed.
Lemma RiemannInt_P2 :
forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b),
Un_cv un 0 ->
a <= b ->
(forall n:nat,
(forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n) ->
sigT (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (vn N)) l).
Proof.
intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit in |- *;
intros; assert (H3 : 0 < eps / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist in |- *;
unfold R_dist in H4; elim (H1 n); elim (H1 m); intros;
replace (RiemannInt_SF (vn n) - RiemannInt_SF (vn m)) with
(RiemannInt_SF (vn n) + -1 * RiemannInt_SF (vn m));
[ idtac | ring ]; rewrite <- StepFun_P30;
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (vn n) (vn m)))))).
apply StepFun_P34; assumption.
apply Rle_lt_trans with
(RiemannInt_SF (mkStepFun (StepFun_P28 1 (wn n) (wn m)))).
apply StepFun_P37; try assumption.
intros; simpl in |- *;
apply Rle_trans with (Rabs (vn n x - f x) + Rabs (f x - vn m x)).
replace (vn n x + -1 * vn m x) with (vn n x - f x + (f x - vn m x));
[ apply Rabs_triang | ring ].
assert (H12 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
assert (H13 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
rewrite <- H12 in H11; pattern b at 2 in H11; rewrite <- H13 in H11;
rewrite Rmult_1_l; apply Rplus_le_compat.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9.
elim H11; intros; split; left; assumption.
apply H7.
elim H11; intros; split; left; assumption.
rewrite StepFun_P30; rewrite Rmult_1_l; apply Rlt_trans with (un n + un m).
apply Rle_lt_trans with
(Rabs (RiemannInt_SF (wn n)) + Rabs (RiemannInt_SF (wn m))).
apply Rplus_le_compat; apply RRle_abs.
apply Rplus_lt_compat; assumption.
apply Rle_lt_trans with (Rabs (un n) + Rabs (un m)).
apply Rplus_le_compat; apply RRle_abs.
replace (pos (un n)) with (un n - 0); [ idtac | ring ];
replace (pos (un m)) with (un m - 0); [ idtac | ring ];
rewrite (double_var eps); apply Rplus_lt_compat; apply H4;
assumption.
Qed.
Lemma RiemannInt_P3 :
forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b),
Un_cv un 0 ->
(forall n:nat,
(forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\
Rabs (RiemannInt_SF (wn n)) < un n) ->
sigT (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (vn N)) l).
Proof.
intros; case (Rle_dec a b); intro.
apply RiemannInt_P2 with f un wn; assumption.
assert (H1 : b <= a); auto with real.
set (vn' := fun n:nat => mkStepFun (StepFun_P6 (pre (vn n))));
set (wn' := fun n:nat => mkStepFun (StepFun_P6 (pre (wn n))));
assert
(H2 :
forall n:nat,
(forall t:R,
Rmin b a <= t <= Rmax b a -> Rabs (f t - vn' n t) <= wn' n t) /\
Rabs (RiemannInt_SF (wn' n)) < un n).
intro; elim (H0 n0); intros; split.
intros; apply (H2 t); elim H4; clear H4; intros; split;
[ apply Rle_trans with (Rmin b a); try assumption; right;
unfold Rmin in |- *
| apply Rle_trans with (Rmax b a); try assumption; right;
unfold Rmax in |- * ];
(case (Rle_dec a b); case (Rle_dec b a); intros;
try reflexivity || apply Rle_antisym;
[ assumption | assumption | auto with real | auto with real ]).
generalize H3; unfold RiemannInt_SF in |- *; case (Rle_dec a b);
case (Rle_dec b a); unfold wn' in |- *; intros;
(replace
(Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n0)))))
(subdivision (mkStepFun (StepFun_P6 (pre (wn n0)))))) with
(Int_SF (subdivision_val (wn n0)) (subdivision (wn n0)));
[ idtac
| apply StepFun_P17 with (fe (wn n0)) a b;
[ apply StepFun_P1
| apply StepFun_P2;
apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (wn n0))))) ] ]).
apply H4.
rewrite Rabs_Ropp; apply H4.
rewrite Rabs_Ropp in H4; apply H4.
apply H4.
assert (H3 := RiemannInt_P2 _ _ _ _ H H1 H2); elim H3; intros;
apply existT with (- x); unfold Un_cv in |- *; unfold Un_cv in p;
intros; elim (p _ H4); intros; exists x0; intros;
generalize (H5 _ H6); unfold R_dist, RiemannInt_SF in |- *;
case (Rle_dec b a); case (Rle_dec a b); intros.
elim n; assumption.
unfold vn' in H7;
replace (Int_SF (subdivision_val (vn n0)) (subdivision (vn n0))) with
(Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n0)))))
(subdivision (mkStepFun (StepFun_P6 (pre (vn n0))))));
[ unfold Rminus in |- *; rewrite Ropp_involutive; rewrite <- Rabs_Ropp;
rewrite Ropp_plus_distr; rewrite Ropp_involutive;
apply H7
| symmetry in |- *; apply StepFun_P17 with (fe (vn n0)) a b;
[ apply StepFun_P1
| apply StepFun_P2;
apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n0))))) ] ].
elim n1; assumption.
elim n2; assumption.
Qed.
Lemma RiemannInt_exists :
forall (f:R -> R) (a b:R) (pr:Riemann_integrable f a b)
(un:nat -> posreal),
Un_cv un 0 ->
sigT
(fun l:R => Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr N)) l).
Proof.
intros f; intros;
apply RiemannInt_P3 with
f un (fun n:nat => projT1 (phi_sequence_prop un pr n));
[ apply H | intro; apply (projT2 (phi_sequence_prop un pr n)) ].
Qed.
Lemma RiemannInt_P4 :
forall (f:R -> R) (a b l:R) (pr1 pr2:Riemann_integrable f a b)
(un vn:nat -> posreal),
Un_cv un 0 ->
Un_cv vn 0 ->
Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr1 N)) l ->
Un_cv (fun N:nat => RiemannInt_SF (phi_sequence vn pr2 N)) l.
Proof.
unfold Un_cv in |- *; unfold R_dist in |- *; intros f; intros;
assert (H3 : 0 < eps / 3).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H _ H3); clear H; intros N0 H; elim (H0 _ H3); clear H0; intros N1 H0;
elim (H1 _ H3); clear H1; intros N2 H1; set (N := max (max N0 N1) N2);
exists N; intros;
apply Rle_lt_trans with
(Rabs
(RiemannInt_SF (phi_sequence vn pr2 n) -
RiemannInt_SF (phi_sequence un pr1 n)) +
Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l)).
replace (RiemannInt_SF (phi_sequence vn pr2 n) - l) with
(RiemannInt_SF (phi_sequence vn pr2 n) -
RiemannInt_SF (phi_sequence un pr1 n) +
(RiemannInt_SF (phi_sequence un pr1 n) - l)); [ apply Rabs_triang | ring ].
replace eps with (2 * (eps / 3) + eps / 3).
apply Rplus_lt_compat.
elim (phi_sequence_prop vn pr2 n); intros psi_vn H5;
elim (phi_sequence_prop un pr1 n); intros psi_un H6;
replace
(RiemannInt_SF (phi_sequence vn pr2 n) -
RiemannInt_SF (phi_sequence un pr1 n)) with
(RiemannInt_SF (phi_sequence vn pr2 n) +
-1 * RiemannInt_SF (phi_sequence un pr1 n)); [ idtac | ring ];
rewrite <- StepFun_P30.
case (Rle_dec a b); intro.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32
(mkStepFun
(StepFun_P28 (-1) (phi_sequence vn pr2 n)
(phi_sequence un pr1 n)))))).
apply StepFun_P34; assumption.
apply Rle_lt_trans with
(RiemannInt_SF (mkStepFun (StepFun_P28 1 psi_un psi_vn))).
apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l;
apply Rle_trans with
(Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x)).
replace (phi_sequence vn pr2 n x + -1 * phi_sequence un pr1 n x) with
(phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x));
[ apply Rabs_triang | ring ].
assert (H10 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
assert (H11 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
rewrite (Rplus_comm (psi_un x)); apply Rplus_le_compat.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8.
rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
elim H6; intros; apply H8.
rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat.
apply Rlt_trans with (pos (un n)).
elim H6; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_un)).
apply RRle_abs.
assumption.
replace (pos (un n)) with (Rabs (un n - 0));
[ apply H; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_trans with (max N0 N1);
apply le_max_l
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
apply Rle_ge; left; apply (cond_pos (un n)) ].
apply Rlt_trans with (pos (vn n)).
elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)).
apply RRle_abs; assumption.
assumption.
replace (pos (vn n)) with (Rabs (vn n - 0));
[ apply H0; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_trans with (max N0 N1);
[ apply le_max_r | apply le_max_l ]
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
apply Rle_ge; left; apply (cond_pos (vn n)) ].
rewrite StepFun_P39; rewrite Rabs_Ropp;
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32
(mkStepFun
(StepFun_P6
(pre
(mkStepFun
(StepFun_P28 (-1) (phi_sequence vn pr2 n)
(phi_sequence un pr1 n))))))))).
apply StepFun_P34; try auto with real.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))).
apply StepFun_P37.
auto with real.
intros; simpl in |- *; rewrite Rmult_1_l;
apply Rle_trans with
(Rabs (phi_sequence vn pr2 n x - f x) +
Rabs (f x - phi_sequence un pr1 n x)).
replace (phi_sequence vn pr2 n x + -1 * phi_sequence un pr1 n x) with
(phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x));
[ apply Rabs_triang | ring ].
assert (H10 : Rmin a b = b).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ elim n0; assumption | reflexivity ].
assert (H11 : Rmax a b = a).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ elim n0; assumption | reflexivity ].
apply Rplus_le_compat.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8.
rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
elim H6; intros; apply H8.
rewrite H10; rewrite H11; elim H7; intros; split; left; assumption.
rewrite <-
(Ropp_involutive
(RiemannInt_SF
(mkStepFun
(StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))))
; rewrite <- StepFun_P39; rewrite StepFun_P30; rewrite Rmult_1_l;
rewrite double; rewrite Ropp_plus_distr; apply Rplus_lt_compat.
apply Rlt_trans with (pos (vn n)).
elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)).
rewrite <- Rabs_Ropp; apply RRle_abs.
assumption.
replace (pos (vn n)) with (Rabs (vn n - 0));
[ apply H0; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_trans with (max N0 N1);
[ apply le_max_r | apply le_max_l ]
| unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
left; apply (cond_pos (vn n)) ].
apply Rlt_trans with (pos (un n)).
elim H6; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_un)).
rewrite <- Rabs_Ropp; apply RRle_abs; assumption.
assumption.
replace (pos (un n)) with (Rabs (un n - 0));
[ apply H; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_trans with (max N0 N1);
apply le_max_l
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
apply Rle_ge; left; apply (cond_pos (un n)) ].
apply H1; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_max_r.
apply Rmult_eq_reg_l with 3;
[ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
Qed.
Lemma RinvN_pos : forall n:nat, 0 < / (INR n + 1).
Proof.
intro; apply Rinv_0_lt_compat; apply Rplus_le_lt_0_compat;
[ apply pos_INR | apply Rlt_0_1 ].
Qed.
Definition RinvN (N:nat) : posreal := mkposreal _ (RinvN_pos N).
Lemma RinvN_cv : Un_cv RinvN 0.
Proof.
unfold Un_cv in |- *; intros; assert (H0 := archimed (/ eps)); elim H0;
clear H0; intros; assert (H2 : (0 <= up (/ eps))%Z).
apply le_IZR; left; apply Rlt_trans with (/ eps);
[ apply Rinv_0_lt_compat; assumption | assumption ].
elim (IZN _ H2); intros; exists x; intros; unfold R_dist in |- *;
simpl in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; assert (H5 : 0 < INR n + 1).
apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
rewrite Rabs_right;
[ idtac
| left; change (0 < / (INR n + 1)) in |- *; apply Rinv_0_lt_compat;
assumption ]; apply Rle_lt_trans with (/ (INR x + 1)).
apply Rle_Rinv.
apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
assumption.
do 2 rewrite <- (Rplus_comm 1); apply Rplus_le_compat_l; apply le_INR;
apply H4.
rewrite <- (Rinv_involutive eps).
apply Rinv_lt_contravar.
apply Rmult_lt_0_compat.
apply Rinv_0_lt_compat; assumption.
apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ].
apply Rlt_trans with (INR x);
[ rewrite INR_IZR_INZ; rewrite <- H3; apply H0
| pattern (INR x) at 1 in |- *; rewrite <- Rplus_0_r;
apply Rplus_lt_compat_l; apply Rlt_0_1 ].
red in |- *; intro; rewrite H6 in H; elim (Rlt_irrefl _ H).
Qed.
Definition RiemannInt (f:R -> R) (a b:R) (pr:Riemann_integrable f a b) : R :=
match RiemannInt_exists pr RinvN RinvN_cv with
| existT a' b' => a'
end.
Lemma RiemannInt_P5 :
forall (f:R -> R) (a b:R) (pr1 pr2:Riemann_integrable f a b),
RiemannInt pr1 = RiemannInt pr2.
Proof.
intros; unfold RiemannInt in |- *;
case (RiemannInt_exists pr1 RinvN RinvN_cv);
case (RiemannInt_exists pr2 RinvN RinvN_cv); intros;
eapply UL_sequence;
[ apply u0
| apply RiemannInt_P4 with pr2 RinvN; apply RinvN_cv || assumption ].
Qed.
C°(
a,b) is included in L1(a,b)
Lemma maxN :
forall (a b:R) (del:posreal),
a < b ->
sigT (fun n:nat => a + INR n * del < b /\ b <= a + INR (S n) * del).
Proof.
intros; set (I := fun n:nat => a + INR n * del < b);
assert (H0 : exists n : nat, I n).
exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r;
assumption.
cut (Nbound I).
intro; assert (H2 := Nzorn H0 H1); elim H2; intros; exists x; elim p; intros;
split.
apply H3.
case (total_order_T (a + INR (S x) * del) b); intro.
elim s; intro.
assert (H5 := H4 (S x) a0); elim (le_Sn_n _ H5).
right; symmetry in |- *; assumption.
left; apply r.
assert (H1 : 0 <= (b - a) / del).
unfold Rdiv in |- *; apply Rmult_le_pos;
[ apply Rge_le; apply Rge_minus; apply Rle_ge; left; apply H
| left; apply Rinv_0_lt_compat; apply (cond_pos del) ].
elim (archimed ((b - a) / del)); intros;
assert (H4 : (0 <= up ((b - a) / del))%Z).
apply le_IZR; simpl in |- *; left; apply Rle_lt_trans with ((b - a) / del);
assumption.
assert (H5 := IZN _ H4); elim H5; clear H5; intros N H5;
unfold Nbound in |- *; exists N; intros; unfold I in H6;
apply INR_le; rewrite H5 in H2; rewrite <- INR_IZR_INZ in H2;
left; apply Rle_lt_trans with ((b - a) / del); try assumption;
apply Rmult_le_reg_l with (pos del);
[ apply (cond_pos del)
| unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ del));
rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; rewrite Rmult_comm; apply Rplus_le_reg_l with a;
replace (a + (b - a)) with b; [ left; assumption | ring ]
| assert (H7 := cond_pos del); red in |- *; intro; rewrite H8 in H7;
elim (Rlt_irrefl _ H7) ] ].
Qed.
Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) {struct N} : Rlist :=
match N with
| O => cons y nil
| S p => cons x (SubEquiN p (x + del) y del)
end.
Definition max_N (a b:R) (del:posreal) (h:a < b) : nat :=
match maxN del h with
| existT N H0 => N
end.
Definition SubEqui (a b:R) (del:posreal) (h:a < b) : Rlist :=
SubEquiN (S (max_N del h)) a b del.
Lemma Heine_cor1 :
forall (f:R -> R) (a b:R),
a < b ->
(forall x:R, a <= x <= b -> continuity_pt f x) ->
forall eps:posreal,
sigT
(fun delta:posreal =>
delta <= b - a /\
(forall x y:R,
a <= x <= b ->
a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps)).
Proof.
intro f; intros;
set
(E :=
fun l:R =>
0 < l <= b - a /\
(forall x y:R,
a <= x <= b ->
a <= y <= b -> Rabs (x - y) < l -> Rabs (f x - f y) < eps));
assert (H1 : bound E).
unfold bound in |- *; exists (b - a); unfold is_upper_bound in |- *; intros;
unfold E in H1; elim H1; clear H1; intros H1 _; elim H1;
intros; assumption.
assert (H2 : exists x : R, E x).
assert (H2 := Heine f (fun x:R => a <= x <= b) (compact_P3 a b) H0 eps);
elim H2; intros; exists (Rmin x (b - a)); unfold E in |- *;
split;
[ split;
[ unfold Rmin in |- *; case (Rle_dec x (b - a)); intro;
[ apply (cond_pos x) | apply Rlt_Rminus; assumption ]
| apply Rmin_r ]
| intros; apply H3; try assumption; apply Rlt_le_trans with (Rmin x (b - a));
[ assumption | apply Rmin_l ] ].
assert (H3 := completeness E H1 H2); elim H3; intros; cut (0 < x <= b - a).
intro; elim H4; clear H4; intros; apply existT with (mkposreal _ H4); split.
apply H5.
unfold is_lub in p; elim p; intros; unfold is_upper_bound in H6;
set (D := Rabs (x0 - y)); elim (classic (exists y : R, D < y /\ E y));
intro.
elim H11; intros; elim H12; clear H12; intros; unfold E in H13; elim H13;
intros; apply H15; assumption.
assert (H12 := not_ex_all_not _ (fun y:R => D < y /\ E y) H11);
assert (H13 : is_upper_bound E D).
unfold is_upper_bound in |- *; intros; assert (H14 := H12 x1);
elim (not_and_or (D < x1) (E x1) H14); intro.
case (Rle_dec x1 D); intro.
assumption.
elim H15; auto with real.
elim H15; assumption.
assert (H14 := H7 _ H13); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H10)).
unfold is_lub in p; unfold is_upper_bound in p; elim p; clear p; intros;
split.
elim H2; intros; assert (H7 := H4 _ H6); unfold E in H6; elim H6; clear H6;
intros H6 _; elim H6; intros; apply Rlt_le_trans with x0;
assumption.
apply H5; intros; unfold E in H6; elim H6; clear H6; intros H6 _; elim H6;
intros; assumption.
Qed.
Lemma Heine_cor2 :
forall (f:R -> R) (a b:R),
(forall x:R, a <= x <= b -> continuity_pt f x) ->
forall eps:posreal,
sigT
(fun delta:posreal =>
forall x y:R,
a <= x <= b ->
a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps).
Proof.
intro f; intros; case (total_order_T a b); intro.
elim s; intro.
assert (H0 := Heine_cor1 a0 H eps); elim H0; intros; apply existT with x;
elim p; intros; apply H2; assumption.
apply existT with (mkposreal _ Rlt_0_1); intros; assert (H3 : x = y);
[ elim H0; elim H1; intros; rewrite b0 in H3; rewrite b0 in H5;
apply Rle_antisym; apply Rle_trans with b; assumption
| rewrite H3; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
apply (cond_pos eps) ].
apply existT with (mkposreal _ Rlt_0_1); intros; elim H0; intros;
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H3 H4) r)).
Qed.
Lemma SubEqui_P1 :
forall (a b:R) (del:posreal) (h:a < b), pos_Rl (SubEqui del h) 0 = a.
Proof.
intros; unfold SubEqui in |- *; case (maxN del h); intros; reflexivity.
Qed.
Lemma SubEqui_P2 :
forall (a b:R) (del:posreal) (h:a < b),
pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))) = b.
Proof.
intros; unfold SubEqui in |- *; case (maxN del h); intros; clear a0;
cut
(forall (x:nat) (a:R) (del:posreal),
pos_Rl (SubEquiN (S x) a b del)
(pred (Rlength (SubEquiN (S x) a b del))) = b);
[ intro; apply H
| simple induction x0;
[ intros; reflexivity
| intros;
change
(pos_Rl (SubEquiN (S n) (a0 + del0) b del0)
(pred (Rlength (SubEquiN (S n) (a0 + del0) b del0))) = b)
in |- *; apply H ] ].
Qed.
Lemma SubEqui_P3 :
forall (N:nat) (a b:R) (del:posreal), Rlength (SubEquiN N a b del) = S N.
Proof.
simple induction N; intros;
[ reflexivity | simpl in |- *; rewrite H; reflexivity ].
Qed.
Lemma SubEqui_P4 :
forall (N:nat) (a b:R) (del:posreal) (i:nat),
(i < S N)%nat -> pos_Rl (SubEquiN (S N) a b del) i = a + INR i * del.
Proof.
simple induction N;
[ intros; inversion H; [ simpl in |- *; ring | elim (le_Sn_O _ H1) ]
| intros; induction i as [| i Hreci];
[ simpl in |- *; ring
| change
(pos_Rl (SubEquiN (S n) (a + del) b del) i = a + INR (S i) * del)
in |- *; rewrite H; [ rewrite S_INR; ring | apply lt_S_n; apply H0 ] ] ].
Qed.
Lemma SubEqui_P5 :
forall (a b:R) (del:posreal) (h:a < b),
Rlength (SubEqui del h) = S (S (max_N del h)).
Proof.
intros; unfold SubEqui in |- *; apply SubEqui_P3.
Qed.
Lemma SubEqui_P6 :
forall (a b:R) (del:posreal) (h:a < b) (i:nat),
(i < S (max_N del h))%nat -> pos_Rl (SubEqui del h) i = a + INR i * del.
Proof.
intros; unfold SubEqui in |- *; apply SubEqui_P4; assumption.
Qed.
Lemma SubEqui_P7 :
forall (a b:R) (del:posreal) (h:a < b), ordered_Rlist (SubEqui del h).
Proof.
intros; unfold ordered_Rlist in |- *; intros; rewrite SubEqui_P5 in H;
simpl in H; inversion H.
rewrite (SubEqui_P6 del h (i:=(max_N del h))).
replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))).
rewrite SubEqui_P2; unfold max_N in |- *; case (maxN del h); intros; left;
elim a0; intros; assumption.
rewrite SubEqui_P5; reflexivity.
apply lt_n_Sn.
repeat rewrite SubEqui_P6.
3: assumption.
2: apply le_lt_n_Sm; assumption.
apply Rplus_le_compat_l; rewrite S_INR; rewrite Rmult_plus_distr_r;
pattern (INR i * del) at 1 in |- *; rewrite <- Rplus_0_r;
apply Rplus_le_compat_l; rewrite Rmult_1_l; left;
apply (cond_pos del).
Qed.
Lemma SubEqui_P8 :
forall (a b:R) (del:posreal) (h:a < b) (i:nat),
(i < Rlength (SubEqui del h))%nat -> a <= pos_Rl (SubEqui del h) i <= b.
Proof.
intros; split.
pattern a at 1 in |- *; rewrite <- (SubEqui_P1 del h); apply RList_P5.
apply SubEqui_P7.
elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros; apply H1;
exists i; split; [ reflexivity | assumption ].
pattern b at 2 in |- *; rewrite <- (SubEqui_P2 del h); apply RList_P7;
[ apply SubEqui_P7
| elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros;
apply H1; exists i; split; [ reflexivity | assumption ] ].
Qed.
Lemma SubEqui_P9 :
forall (a b:R) (del:posreal) (f:R -> R) (h:a < b),
sigT
(fun g:StepFun a b =>
g b = f b /\
(forall i:nat,
(i < pred (Rlength (SubEqui del h)))%nat ->
constant_D_eq g
(co_interval (pos_Rl (SubEqui del h) i)
(pos_Rl (SubEqui del h) (S i)))
(f (pos_Rl (SubEqui del h) i)))).
Proof.
intros; apply StepFun_P38;
[ apply SubEqui_P7 | apply SubEqui_P1 | apply SubEqui_P2 ].
Qed.
Lemma RiemannInt_P6 :
forall (f:R -> R) (a b:R),
a < b ->
(forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b.
Proof.
intros; unfold Riemann_integrable in |- *; intro;
assert (H1 : 0 < eps / (2 * (b - a))).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos eps)
| apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
[ prove_sup0 | apply Rlt_Rminus; assumption ] ].
assert (H2 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; left; assumption ].
assert (H3 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; left; assumption ].
elim (Heine_cor2 H0 (mkposreal _ H1)); intros del H4;
elim (SubEqui_P9 del f H); intros phi [H5 H6]; split with phi;
split with (mkStepFun (StepFun_P4 a b (eps / (2 * (b - a)))));
split.
2: rewrite StepFun_P18; unfold Rdiv in |- *; rewrite Rinv_mult_distr.
2: do 2 rewrite Rmult_assoc; rewrite <- Rinv_l_sym.
2: rewrite Rmult_1_r; rewrite Rabs_right.
2: apply Rmult_lt_reg_l with 2.
2: prove_sup0.
2: rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym.
2: rewrite Rmult_1_l; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r;
rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps).
2: discrR.
2: apply Rle_ge; left; apply Rmult_lt_0_compat.
2: apply (cond_pos eps).
2: apply Rinv_0_lt_compat; prove_sup0.
2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H;
elim (Rlt_irrefl _ H).
2: discrR.
2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H;
elim (Rlt_irrefl _ H).
intros; rewrite H2 in H7; rewrite H3 in H7; simpl in |- *;
unfold fct_cte in |- *;
cut
(forall t:R,
a <= t <= b ->
t = b \/
(exists i : nat,
(i < pred (Rlength (SubEqui del H)))%nat /\
co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i))
t)).
intro; elim (H8 _ H7); intro.
rewrite H9; rewrite H5; unfold Rminus in |- *; rewrite Rplus_opp_r;
rewrite Rabs_R0; left; assumption.
elim H9; clear H9; intros I [H9 H10]; assert (H11 := H6 I H9 t H10);
rewrite H11; left; apply H4.
assumption.
apply SubEqui_P8; apply lt_trans with (pred (Rlength (SubEqui del H))).
assumption.
apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H9;
elim (lt_n_O _ H9).
unfold co_interval in H10; elim H10; clear H10; intros; rewrite Rabs_right.
rewrite SubEqui_P5 in H9; simpl in H9; inversion H9.
apply Rplus_lt_reg_r with (pos_Rl (SubEqui del H) (max_N del H)).
replace
(pos_Rl (SubEqui del H) (max_N del H) +
(t - pos_Rl (SubEqui del H) (max_N del H))) with t;
[ idtac | ring ]; apply Rlt_le_trans with b.
rewrite H14 in H12;
assert (H13 : S (max_N del H) = pred (Rlength (SubEqui del H))).
rewrite SubEqui_P5; reflexivity.
rewrite H13 in H12; rewrite SubEqui_P2 in H12; apply H12.
rewrite SubEqui_P6.
2: apply lt_n_Sn.
unfold max_N in |- *; case (maxN del H); intros; elim a0; clear a0;
intros _ H13; replace (a + INR x * del + del) with (a + INR (S x) * del);
[ assumption | rewrite S_INR; ring ].
apply Rplus_lt_reg_r with (pos_Rl (SubEqui del H) I);
replace (pos_Rl (SubEqui del H) I + (t - pos_Rl (SubEqui del H) I)) with t;
[ idtac | ring ];
replace (pos_Rl (SubEqui del H) I + del) with (pos_Rl (SubEqui del H) (S I)).
assumption.
repeat rewrite SubEqui_P6.
rewrite S_INR; ring.
assumption.
apply le_lt_n_Sm; assumption.
apply Rge_minus; apply Rle_ge; assumption.
intros; clear H0 H1 H4 phi H5 H6 t H7; case (Req_dec t0 b); intro.
left; assumption.
right; set (I := fun j:nat => a + INR j * del <= t0);
assert (H1 : exists n : nat, I n).
exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r; elim H8;
intros; assumption.
assert (H4 : Nbound I).
unfold Nbound in |- *; exists (S (max_N del H)); intros; unfold max_N in |- *;
case (maxN del H); intros; elim a0; clear a0; intros _ H5;
apply INR_le; apply Rmult_le_reg_l with (pos del).
apply (cond_pos del).
apply Rplus_le_reg_l with a; do 2 rewrite (Rmult_comm del);
apply Rle_trans with t0; unfold I in H4; try assumption;
apply Rle_trans with b; try assumption; elim H8; intros;
assumption.
elim (Nzorn H1 H4); intros N [H5 H6]; assert (H7 : (N < S (max_N del H))%nat).
unfold max_N in |- *; case (maxN del H); intros; apply INR_lt;
apply Rmult_lt_reg_l with (pos del).
apply (cond_pos del).
apply Rplus_lt_reg_r with a; do 2 rewrite (Rmult_comm del);
apply Rle_lt_trans with t0; unfold I in H5; try assumption;
elim a0; intros; apply Rlt_le_trans with b; try assumption;
elim H8; intros.
elim H11; intro.
assumption.
elim H0; assumption.
exists N; split.
rewrite SubEqui_P5; simpl in |- *; assumption.
unfold co_interval in |- *; split.
rewrite SubEqui_P6.
apply H5.
assumption.
inversion H7.
replace (S (max_N del H)) with (pred (Rlength (SubEqui del H))).
rewrite (SubEqui_P2 del H); elim H8; intros.
elim H11; intro.
assumption.
elim H0; assumption.
rewrite SubEqui_P5; reflexivity.
rewrite SubEqui_P6.
case (Rle_dec (a + INR (S N) * del) t0); intro.
assert (H11 := H6 (S N) r); elim (le_Sn_n _ H11).
auto with real.
apply le_lt_n_Sm; assumption.
Qed.
Lemma RiemannInt_P7 : forall (f:R -> R) (a:R), Riemann_integrable f a a.
Proof.
unfold Riemann_integrable in |- *; intro f; intros;
split with (mkStepFun (StepFun_P4 a a (f a)));
split with (mkStepFun (StepFun_P4 a a 0)); split.
intros; simpl in |- *; unfold fct_cte in |- *; replace t with a.
unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; right;
reflexivity.
generalize H; unfold Rmin, Rmax in |- *; case (Rle_dec a a); intros; elim H0;
intros; apply Rle_antisym; assumption.
rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; apply (cond_pos eps).
Qed.
Lemma continuity_implies_RiemannInt :
forall (f:R -> R) (a b:R),
a <= b ->
(forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b.
Proof.
intros; case (total_order_T a b); intro;
[ elim s; intro;
[ apply RiemannInt_P6; assumption | rewrite b0; apply RiemannInt_P7 ]
| elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)) ].
Qed.
Lemma RiemannInt_P8 :
forall (f:R -> R) (a b:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable f b a), RiemannInt pr1 = - RiemannInt pr2.
Proof.
intro f; intros; eapply UL_sequence.
unfold RiemannInt in |- *; case (RiemannInt_exists pr1 RinvN RinvN_cv);
intros; apply u.
unfold RiemannInt in |- *; case (RiemannInt_exists pr2 RinvN RinvN_cv);
intros;
cut
(exists psi1 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
cut
(exists psi2 : nat -> StepFun b a,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
intros; elim H; clear H; intros psi2 H; elim H0; clear H0; intros psi1 H0;
assert (H1 := RinvN_cv); unfold Un_cv in |- *; intros;
assert (H3 : 0 < eps / 3).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
unfold Un_cv in H1; elim (H1 _ H3); clear H1; intros N0 H1;
unfold R_dist in H1; simpl in H1;
assert (H4 : forall n:nat, (n >= N0)%nat -> RinvN n < eps / 3).
intros; assert (H5 := H1 _ H4);
replace (pos (RinvN n)) with (Rabs (/ (INR n + 1) - 0));
[ assumption
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
left; apply (cond_pos (RinvN n)) ].
clear H1; unfold Un_cv in u; elim (u _ H3); clear u; intros N1 H1;
exists (max N0 N1); intros; unfold R_dist in |- *;
apply Rle_lt_trans with
(Rabs
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n)) +
Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)).
rewrite <- (Rabs_Ropp (RiemannInt_SF (phi_sequence RinvN pr2 n) - x));
replace (RiemannInt_SF (phi_sequence RinvN pr1 n) - - x) with
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n) +
- (RiemannInt_SF (phi_sequence RinvN pr2 n) - x));
[ apply Rabs_triang | ring ].
replace eps with (2 * (eps / 3) + eps / 3).
apply Rplus_lt_compat.
rewrite (StepFun_P39 (phi_sequence RinvN pr2 n));
replace
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
- RiemannInt_SF (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n)))))
with
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
-1 *
RiemannInt_SF (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n)))));
[ idtac | ring ]; rewrite <- StepFun_P30.
case (Rle_dec a b); intro.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32
(mkStepFun
(StepFun_P28 (-1) (phi_sequence RinvN pr1 n)
(mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n))))))))).
apply StepFun_P34; assumption.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P28 1 (psi1 n) (mkStepFun (StepFun_P6 (pre (psi2 n))))))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; rewrite Rmult_1_l;
apply Rle_trans with
(Rabs (phi_sequence RinvN pr1 n x0 - f x0) +
Rabs (f x0 - phi_sequence RinvN pr2 n x0)).
replace (phi_sequence RinvN pr1 n x0 + -1 * phi_sequence RinvN pr2 n x0) with
(phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0));
[ apply Rabs_triang | ring ].
assert (H7 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
assert (H8 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
apply Rplus_le_compat.
elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9;
rewrite H7; rewrite H8.
elim H6; intros; split; left; assumption.
elim (H n); intros; apply H9; rewrite H7; rewrite H8.
elim H6; intros; split; left; assumption.
rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat.
elim (H0 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n)));
[ apply RRle_abs
| apply Rlt_trans with (pos (RinvN n));
[ assumption
| apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
[ apply le_max_l | assumption ] ] ].
elim (H n); intros;
rewrite <-
(Ropp_involutive (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi2 n))))))
; rewrite <- StepFun_P39;
apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n)));
[ rewrite <- Rabs_Ropp; apply RRle_abs
| apply Rlt_trans with (pos (RinvN n));
[ assumption
| apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
[ apply le_max_l | assumption ] ] ].
assert (Hyp : b <= a).
auto with real.
rewrite StepFun_P39; rewrite Rabs_Ropp;
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32
(mkStepFun
(StepFun_P6
(StepFun_P28 (-1) (phi_sequence RinvN pr1 n)
(mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n)))))))))).
apply StepFun_P34; assumption.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P28 1 (mkStepFun (StepFun_P6 (pre (psi1 n)))) (psi2 n)))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; rewrite Rmult_1_l;
apply Rle_trans with
(Rabs (phi_sequence RinvN pr1 n x0 - f x0) +
Rabs (f x0 - phi_sequence RinvN pr2 n x0)).
replace (phi_sequence RinvN pr1 n x0 + -1 * phi_sequence RinvN pr2 n x0) with
(phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0));
[ apply Rabs_triang | ring ].
assert (H7 : Rmin a b = b).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ elim n0; assumption | reflexivity ].
assert (H8 : Rmax a b = a).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ elim n0; assumption | reflexivity ].
apply Rplus_le_compat.
elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9;
rewrite H7; rewrite H8.
elim H6; intros; split; left; assumption.
elim (H n); intros; apply H9; rewrite H7; rewrite H8; elim H6; intros; split;
left; assumption.
rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat.
elim (H0 n); intros;
rewrite <-
(Ropp_involutive (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi1 n))))))
; rewrite <- StepFun_P39;
apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n)));
[ rewrite <- Rabs_Ropp; apply RRle_abs
| apply Rlt_trans with (pos (RinvN n));
[ assumption
| apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
[ apply le_max_l | assumption ] ] ].
elim (H n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n)));
[ apply RRle_abs
| apply Rlt_trans with (pos (RinvN n));
[ assumption
| apply H4; unfold ge in |- *; apply le_trans with (max N0 N1);
[ apply le_max_l | assumption ] ] ].
unfold R_dist in H1; apply H1; unfold ge in |- *;
apply le_trans with (max N0 N1); [ apply le_max_r | assumption ].
apply Rmult_eq_reg_l with 3;
[ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
rewrite Rmin_comm; rewrite RmaxSym;
apply (projT2 (phi_sequence_prop RinvN pr2 n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr1 n)).
Qed.
Lemma RiemannInt_P9 :
forall (f:R -> R) (a:R) (pr:Riemann_integrable f a a), RiemannInt pr = 0.
Proof.
intros; assert (H := RiemannInt_P8 pr pr); apply Rmult_eq_reg_l with 2;
[ rewrite Rmult_0_r; rewrite double; pattern (RiemannInt pr) at 2 in |- *;
rewrite H; apply Rplus_opp_r
| discrR ].
Qed.
Lemma Req_EM_T : forall r1 r2:R, {r1 = r2} + {r1 <> r2}.
Proof.
intros; elim (total_order_T r1 r2); intros;
[ elim a; intro;
[ right; red in |- *; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0)
| left; assumption ]
| right; red in |- *; intro; rewrite H in b; elim (Rlt_irrefl r2 b) ].
Qed.
Lemma RiemannInt_P10 :
forall (f g:R -> R) (a b l:R),
Riemann_integrable f a b ->
Riemann_integrable g a b ->
Riemann_integrable (fun x:R => f x + l * g x) a b.
Proof.
unfold Riemann_integrable in |- *; intros f g; intros; case (Req_EM_T l 0);
intro.
elim (X eps); intros; split with x; elim p; intros; split with x0; elim p0;
intros; split; try assumption; rewrite e; intros;
rewrite Rmult_0_l; rewrite Rplus_0_r; apply H; assumption.
assert (H : 0 < eps / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ].
assert (H0 : 0 < eps / (2 * Rabs l)).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos eps)
| apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
[ prove_sup0 | apply Rabs_pos_lt; assumption ] ].
elim (X (mkposreal _ H)); intros; elim (X0 (mkposreal _ H0)); intros;
split with (mkStepFun (StepFun_P28 l x x0)); elim p0;
elim p; intros; split with (mkStepFun (StepFun_P28 (Rabs l) x1 x2));
elim p1; elim p2; clear p1 p2 p0 p X X0; intros; split.
intros; simpl in |- *;
apply Rle_trans with (Rabs (f t - x t) + Rabs (l * (g t - x0 t))).
replace (f t + l * g t - (x t + l * x0 t)) with
(f t - x t + l * (g t - x0 t)); [ apply Rabs_triang | ring ].
apply Rplus_le_compat;
[ apply H3; assumption
| rewrite Rabs_mult; apply Rmult_le_compat_l;
[ apply Rabs_pos | apply H1; assumption ] ].
rewrite StepFun_P30;
apply Rle_lt_trans with
(Rabs (RiemannInt_SF x1) + Rabs (Rabs l * RiemannInt_SF x2)).
apply Rabs_triang.
rewrite (double_var eps); apply Rplus_lt_compat.
apply H4.
rewrite Rabs_mult; rewrite Rabs_Rabsolu; apply Rmult_lt_reg_l with (/ Rabs l).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym;
[ rewrite Rmult_1_l;
replace (/ Rabs l * (eps / 2)) with (eps / (2 * Rabs l));
[ apply H2
| unfold Rdiv in |- *; rewrite Rinv_mult_distr;
[ ring | discrR | apply Rabs_no_R0; assumption ] ]
| apply Rabs_no_R0; assumption ].
Qed.
Lemma RiemannInt_P11 :
forall (f:R -> R) (a b l:R) (un:nat -> posreal)
(phi1 phi2 psi1 psi2:nat -> StepFun a b),
Un_cv un 0 ->
(forall n:nat,
(forall t:R,
Rmin a b <= t <= Rmax a b -> Rabs (f t - phi1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < un n) ->
(forall n:nat,
(forall t:R,
Rmin a b <= t <= Rmax a b -> Rabs (f t - phi2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < un n) ->
Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) l ->
Un_cv (fun N:nat => RiemannInt_SF (phi2 N)) l.
Proof.
unfold Un_cv in |- *; intro f; intros; intros.
case (Rle_dec a b); intro Hyp.
assert (H4 : 0 < eps / 3).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H _ H4); clear H; intros N0 H.
elim (H2 _ H4); clear H2; intros N1 H2.
set (N := max N0 N1); exists N; intros; unfold R_dist in |- *.
apply Rle_lt_trans with
(Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) +
Rabs (RiemannInt_SF (phi1 n) - l)).
replace (RiemannInt_SF (phi2 n) - l) with
(RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n) +
(RiemannInt_SF (phi1 n) - l)); [ apply Rabs_triang | ring ].
replace eps with (2 * (eps / 3) + eps / 3).
apply Rplus_lt_compat.
replace (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) with
(RiemannInt_SF (phi2 n) + -1 * RiemannInt_SF (phi1 n));
[ idtac | ring ].
rewrite <- StepFun_P30.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (phi2 n) (phi1 n)))))).
apply StepFun_P34; assumption.
apply Rle_lt_trans with
(RiemannInt_SF (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n)))).
apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l.
apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)).
replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x));
[ apply Rabs_triang | ring ].
rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7.
assert (H10 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
assert (H11 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
assert (H11 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat.
apply Rlt_trans with (pos (un n)).
elim (H0 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))).
apply RRle_abs.
assumption.
replace (pos (un n)) with (R_dist (un n) 0).
apply H; unfold ge in |- *; apply le_trans with N; try assumption.
unfold N in |- *; apply le_max_l.
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply Rabs_right.
apply Rle_ge; left; apply (cond_pos (un n)).
apply Rlt_trans with (pos (un n)).
elim (H1 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))).
apply RRle_abs; assumption.
assumption.
replace (pos (un n)) with (R_dist (un n) 0).
apply H; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_max_l.
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
left; apply (cond_pos (un n)).
unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N;
try assumption; unfold N in |- *; apply le_max_r.
apply Rmult_eq_reg_l with 3;
[ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
assert (H4 : 0 < eps / 3).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H _ H4); clear H; intros N0 H.
elim (H2 _ H4); clear H2; intros N1 H2.
set (N := max N0 N1); exists N; intros; unfold R_dist in |- *.
apply Rle_lt_trans with
(Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) +
Rabs (RiemannInt_SF (phi1 n) - l)).
replace (RiemannInt_SF (phi2 n) - l) with
(RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n) +
(RiemannInt_SF (phi1 n) - l)); [ apply Rabs_triang | ring ].
assert (Hyp_b : b <= a).
auto with real.
replace eps with (2 * (eps / 3) + eps / 3).
apply Rplus_lt_compat.
replace (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) with
(RiemannInt_SF (phi2 n) + -1 * RiemannInt_SF (phi1 n));
[ idtac | ring ].
rewrite <- StepFun_P30.
rewrite StepFun_P39.
rewrite Rabs_Ropp.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32
(mkStepFun
(StepFun_P6
(pre (mkStepFun (StepFun_P28 (-1) (phi2 n) (phi1 n))))))))).
apply StepFun_P34; try assumption.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P6 (pre (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n))))))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; rewrite Rmult_1_l.
apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)).
replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x));
[ apply Rabs_triang | ring ].
rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7.
assert (H10 : Rmin a b = b).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ elim Hyp; assumption | reflexivity ].
assert (H11 : Rmax a b = a).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ elim Hyp; assumption | reflexivity ].
rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = b).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ elim Hyp; assumption | reflexivity ].
assert (H11 : Rmax a b = a).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ elim Hyp; assumption | reflexivity ].
rewrite H10; rewrite H11; elim H6; intros; split; left; assumption.
rewrite <-
(Ropp_involutive
(RiemannInt_SF
(mkStepFun
(StepFun_P6 (pre (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n))))))))
.
rewrite <- StepFun_P39.
rewrite StepFun_P30.
rewrite Rmult_1_l; rewrite double.
rewrite Ropp_plus_distr; apply Rplus_lt_compat.
apply Rlt_trans with (pos (un n)).
elim (H0 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))).
rewrite <- Rabs_Ropp; apply RRle_abs.
assumption.
replace (pos (un n)) with (R_dist (un n) 0).
apply H; unfold ge in |- *; apply le_trans with N; try assumption.
unfold N in |- *; apply le_max_l.
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply Rabs_right.
apply Rle_ge; left; apply (cond_pos (un n)).
apply Rlt_trans with (pos (un n)).
elim (H1 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))).
rewrite <- Rabs_Ropp; apply RRle_abs; assumption.
assumption.
replace (pos (un n)) with (R_dist (un n) 0).
apply H; unfold ge in |- *; apply le_trans with N; try assumption;
unfold N in |- *; apply le_max_l.
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
left; apply (cond_pos (un n)).
unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N;
try assumption; unfold N in |- *; apply le_max_r.
apply Rmult_eq_reg_l with 3;
[ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l;
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
Qed.
Lemma RiemannInt_P12 :
forall (f g:R -> R) (a b l:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable g a b)
(pr3:Riemann_integrable (fun x:R => f x + l * g x) a b),
a <= b -> RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2.
Proof.
intro f; intros; case (Req_dec l 0); intro.
pattern l at 2 in |- *; rewrite H0; rewrite Rmult_0_l; rewrite Rplus_0_r;
unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv);
case (RiemannInt_exists pr1 RinvN RinvN_cv); intros;
eapply UL_sequence;
[ apply u0
| set (psi1 := fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n));
set (psi2 := fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n));
apply RiemannInt_P11 with f RinvN (phi_sequence RinvN pr1) psi1 psi2;
[ apply RinvN_cv
| intro; apply (projT2 (phi_sequence_prop RinvN pr1 n))
| intro;
assert
(H1 :
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t + l * g t - phi_sequence RinvN pr3 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n);
[ apply (projT2 (phi_sequence_prop RinvN pr3 n))
| elim H1; intros; split; try assumption; intros;
replace (f t) with (f t + l * g t);
[ apply H2; assumption | rewrite H0; ring ] ]
| assumption ] ].
eapply UL_sequence.
unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv);
intros; apply u.
unfold Un_cv in |- *; intros; unfold RiemannInt in |- *;
case (RiemannInt_exists pr1 RinvN RinvN_cv);
case (RiemannInt_exists pr2 RinvN RinvN_cv); unfold Un_cv in |- *;
intros; assert (H2 : 0 < eps / 5).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (u0 _ H2); clear u0; intros N0 H3; assert (H4 := RinvN_cv);
unfold Un_cv in H4; elim (H4 _ H2); clear H4 H2; intros N1 H4;
assert (H5 : 0 < eps / (5 * Rabs l)).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption
| apply Rinv_0_lt_compat; apply Rmult_lt_0_compat;
[ prove_sup0 | apply Rabs_pos_lt; assumption ] ].
elim (u _ H5); clear u; intros N2 H6; assert (H7 := RinvN_cv);
unfold Un_cv in H7; elim (H7 _ H5); clear H7 H5; intros N3 H5;
unfold R_dist in H3, H4, H5, H6; set (N := max (max N0 N1) (max N2 N3)).
assert (H7 : forall n:nat, (n >= N1)%nat -> RinvN n < eps / 5).
intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0));
[ unfold RinvN in |- *; apply H4; assumption
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
left; apply (cond_pos (RinvN n)) ].
clear H4; assert (H4 := H7); clear H7;
assert (H7 : forall n:nat, (n >= N3)%nat -> RinvN n < eps / (5 * Rabs l)).
intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0));
[ unfold RinvN in |- *; apply H5; assumption
| unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right;
left; apply (cond_pos (RinvN n)) ].
clear H5; assert (H5 := H7); clear H7; exists N; intros;
unfold R_dist in |- *.
apply Rle_lt_trans with
(Rabs
(RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
l * RiemannInt_SF (phi_sequence RinvN pr2 n))) +
Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) - x0) +
Rabs l * Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)).
apply Rle_trans with
(Rabs
(RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
l * RiemannInt_SF (phi_sequence RinvN pr2 n))) +
Rabs
(RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 +
l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x))).
replace (RiemannInt_SF (phi_sequence RinvN pr3 n) - (x0 + l * x)) with
(RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
l * RiemannInt_SF (phi_sequence RinvN pr2 n)) +
(RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 +
l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)));
[ apply Rabs_triang | ring ].
rewrite Rplus_assoc; apply Rplus_le_compat_l; rewrite <- Rabs_mult;
replace
(RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 +
l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)) with
(RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 +
l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x));
[ apply Rabs_triang | ring ].
replace eps with (3 * (eps / 5) + eps / 5 + eps / 5).
repeat apply Rplus_lt_compat.
assert
(H7 :
exists psi1 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr1 n0)).
assert
(H8 :
exists psi2 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (g t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr2 n0)).
assert
(H9 :
exists psi3 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t + l * g t - phi_sequence RinvN pr3 n t) <= psi3 n t) /\
Rabs (RiemannInt_SF (psi3 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr3 n0)).
elim H7; clear H7; intros psi1 H7; elim H8; clear H8; intros psi2 H8; elim H9;
clear H9; intros psi3 H9;
replace
(RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
l * RiemannInt_SF (phi_sequence RinvN pr2 n))) with
(RiemannInt_SF (phi_sequence RinvN pr3 n) +
-1 *
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
l * RiemannInt_SF (phi_sequence RinvN pr2 n)));
[ idtac | ring ]; do 2 rewrite <- StepFun_P30; assert (H10 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
assert (H11 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
rewrite H10 in H7; rewrite H10 in H8; rewrite H10 in H9; rewrite H11 in H7;
rewrite H11 in H8; rewrite H11 in H9;
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32
(mkStepFun
(StepFun_P28 (-1) (phi_sequence RinvN pr3 n)
(mkStepFun
(StepFun_P28 l (phi_sequence RinvN pr1 n)
(phi_sequence RinvN pr2 n)))))))).
apply StepFun_P34; assumption.
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P28 1 (psi3 n)
(mkStepFun (StepFun_P28 (Rabs l) (psi1 n) (psi2 n)))))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; rewrite Rmult_1_l.
apply Rle_trans with
(Rabs (phi_sequence RinvN pr3 n x1 - (f x1 + l * g x1)) +
Rabs
(f x1 + l * g x1 +
-1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1))).
replace
(phi_sequence RinvN pr3 n x1 +
-1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1)) with
(phi_sequence RinvN pr3 n x1 - (f x1 + l * g x1) +
(f x1 + l * g x1 +
-1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1)));
[ apply Rabs_triang | ring ].
rewrite Rplus_assoc; apply Rplus_le_compat.
elim (H9 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr;
apply H13.
elim H12; intros; split; left; assumption.
apply Rle_trans with
(Rabs (f x1 - phi_sequence RinvN pr1 n x1) +
Rabs l * Rabs (g x1 - phi_sequence RinvN pr2 n x1)).
rewrite <- Rabs_mult;
replace
(f x1 +
(l * g x1 +
-1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1)))
with
(f x1 - phi_sequence RinvN pr1 n x1 +
l * (g x1 - phi_sequence RinvN pr2 n x1)); [ apply Rabs_triang | ring ].
apply Rplus_le_compat.
elim (H7 n); intros; apply H13.
elim H12; intros; split; left; assumption.
apply Rmult_le_compat_l;
[ apply Rabs_pos
| elim (H8 n); intros; apply H13; elim H12; intros; split; left; assumption ].
do 2 rewrite StepFun_P30; rewrite Rmult_1_l;
replace (3 * (eps / 5)) with (eps / 5 + (eps / 5 + eps / 5));
[ repeat apply Rplus_lt_compat | ring ].
apply Rlt_trans with (pos (RinvN n));
[ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi3 n)));
[ apply RRle_abs | elim (H9 n); intros; assumption ]
| apply H4; unfold ge in |- *; apply le_trans with N;
[ apply le_trans with (max N0 N1);
[ apply le_max_r | unfold N in |- *; apply le_max_l ]
| assumption ] ].
apply Rlt_trans with (pos (RinvN n));
[ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n)));
[ apply RRle_abs | elim (H7 n); intros; assumption ]
| apply H4; unfold ge in |- *; apply le_trans with N;
[ apply le_trans with (max N0 N1);
[ apply le_max_r | unfold N in |- *; apply le_max_l ]
| assumption ] ].
apply Rmult_lt_reg_l with (/ Rabs l).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_l; replace (/ Rabs l * (eps / 5)) with (eps / (5 * Rabs l)).
apply Rlt_trans with (pos (RinvN n));
[ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n)));
[ apply RRle_abs | elim (H8 n); intros; assumption ]
| apply H5; unfold ge in |- *; apply le_trans with N;
[ apply le_trans with (max N2 N3);
[ apply le_max_r | unfold N in |- *; apply le_max_r ]
| assumption ] ].
unfold Rdiv in |- *; rewrite Rinv_mult_distr;
[ ring | discrR | apply Rabs_no_R0; assumption ].
apply Rabs_no_R0; assumption.
apply H3; unfold ge in |- *; apply le_trans with (max N0 N1);
[ apply le_max_l
| apply le_trans with N; [ unfold N in |- *; apply le_max_l | assumption ] ].
apply Rmult_lt_reg_l with (/ Rabs l).
apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption.
rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym.
rewrite Rmult_1_l; replace (/ Rabs l * (eps / 5)) with (eps / (5 * Rabs l)).
apply H6; unfold ge in |- *; apply le_trans with (max N2 N3);
[ apply le_max_l
| apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ] ].
unfold Rdiv in |- *; rewrite Rinv_mult_distr;
[ ring | discrR | apply Rabs_no_R0; assumption ].
apply Rabs_no_R0; assumption.
apply Rmult_eq_reg_l with 5;
[ unfold Rdiv in |- *; do 2 rewrite Rmult_plus_distr_l;
do 3 rewrite (Rmult_comm 5); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
Qed.
Lemma RiemannInt_P13 :
forall (f g:R -> R) (a b l:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable g a b)
(pr3:Riemann_integrable (fun x:R => f x + l * g x) a b),
RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2.
Proof.
intros; case (Rle_dec a b); intro;
[ apply RiemannInt_P12; assumption
| assert (H : b <= a);
[ auto with real
| replace (RiemannInt pr3) with (- RiemannInt (RiemannInt_P1 pr3));
[ idtac | symmetry in |- *; apply RiemannInt_P8 ];
replace (RiemannInt pr2) with (- RiemannInt (RiemannInt_P1 pr2));
[ idtac | symmetry in |- *; apply RiemannInt_P8 ];
replace (RiemannInt pr1) with (- RiemannInt (RiemannInt_P1 pr1));
[ idtac | symmetry in |- *; apply RiemannInt_P8 ];
rewrite
(RiemannInt_P12 (RiemannInt_P1 pr1) (RiemannInt_P1 pr2)
(RiemannInt_P1 pr3) H); ring ] ].
Qed.
Lemma RiemannInt_P14 : forall a b c:R, Riemann_integrable (fct_cte c) a b.
Proof.
unfold Riemann_integrable in |- *; intros;
split with (mkStepFun (StepFun_P4 a b c));
split with (mkStepFun (StepFun_P4 a b 0)); split;
[ intros; simpl in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
rewrite Rabs_R0; unfold fct_cte in |- *; right;
reflexivity
| rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0;
apply (cond_pos eps) ].
Qed.
Lemma RiemannInt_P15 :
forall (a b c:R) (pr:Riemann_integrable (fct_cte c) a b),
RiemannInt pr = c * (b - a).
Proof.
intros; unfold RiemannInt in |- *; case (RiemannInt_exists pr RinvN RinvN_cv);
intros; eapply UL_sequence.
apply u.
set (phi1 := fun N:nat => phi_sequence RinvN pr N);
change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) (c * (b - a))) in |- *;
set (f := fct_cte c);
assert
(H1 :
exists psi1 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr n)).
elim H1; clear H1; intros psi1 H1;
set (phi2 := fun n:nat => mkStepFun (StepFun_P4 a b c));
set (psi2 := fun n:nat => mkStepFun (StepFun_P4 a b 0));
apply RiemannInt_P11 with f RinvN phi2 psi2 psi1;
try assumption.
apply RinvN_cv.
intro; split.
intros; unfold f in |- *; simpl in |- *; unfold Rminus in |- *;
rewrite Rplus_opp_r; rewrite Rabs_R0; unfold fct_cte in |- *;
right; reflexivity.
unfold psi2 in |- *; rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0;
apply (cond_pos (RinvN n)).
unfold Un_cv in |- *; intros; split with 0%nat; intros; unfold R_dist in |- *;
unfold phi2 in |- *; rewrite StepFun_P18; unfold Rminus in |- *;
rewrite Rplus_opp_r; rewrite Rabs_R0; apply H.
Qed.
Lemma RiemannInt_P16 :
forall (f:R -> R) (a b:R),
Riemann_integrable f a b -> Riemann_integrable (fun x:R => Rabs (f x)) a b.
Proof.
unfold Riemann_integrable in |- *; intro f; intros; elim (X eps); clear X;
intros phi [psi [H H0]]; split with (mkStepFun (StepFun_P32 phi));
split with psi; split; try assumption; intros; simpl in |- *;
apply Rle_trans with (Rabs (f t - phi t));
[ apply Rabs_triang_inv2 | apply H; assumption ].
Qed.
Lemma Rle_cv_lim :
forall (Un Vn:nat -> R) (l1 l2:R),
(forall n:nat, Un n <= Vn n) -> Un_cv Un l1 -> Un_cv Vn l2 -> l1 <= l2.
Proof.
intros; case (Rle_dec l1 l2); intro.
assumption.
assert (H2 : l2 < l1).
auto with real.
clear n; assert (H3 : 0 < (l1 - l2) / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply Rlt_Rminus; assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H1 _ H3); elim (H0 _ H3); clear H0 H1; unfold R_dist in |- *; intros;
set (N := max x x0); cut (Vn N < Un N).
intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (H N) H4)).
apply Rlt_trans with ((l1 + l2) / 2).
apply Rplus_lt_reg_r with (- l2);
replace (- l2 + (l1 + l2) / 2) with ((l1 - l2) / 2).
rewrite Rplus_comm; apply Rle_lt_trans with (Rabs (Vn N - l2)).
apply RRle_abs.
apply H1; unfold ge in |- *; unfold N in |- *; apply le_max_r.
apply Rmult_eq_reg_l with 2;
[ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2);
rewrite (Rmult_plus_distr_r (- l2) ((l1 + l2) * / 2) 2);
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
[ ring | discrR ]
| discrR ].
apply Ropp_lt_cancel; apply Rplus_lt_reg_r with l1;
replace (l1 + - ((l1 + l2) / 2)) with ((l1 - l2) / 2).
apply Rle_lt_trans with (Rabs (Un N - l1)).
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs.
apply H0; unfold ge in |- *; unfold N in |- *; apply le_max_l.
apply Rmult_eq_reg_l with 2;
[ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2);
rewrite (Rmult_plus_distr_r l1 (- ((l1 + l2) * / 2)) 2);
rewrite <- Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
Qed.
Lemma RiemannInt_P17 :
forall (f:R -> R) (a b:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable (fun x:R => Rabs (f x)) a b),
a <= b -> Rabs (RiemannInt pr1) <= RiemannInt pr2.
Proof.
intro f; intros; unfold RiemannInt in |- *;
case (RiemannInt_exists pr1 RinvN RinvN_cv);
case (RiemannInt_exists pr2 RinvN RinvN_cv); intros;
set (phi1 := phi_sequence RinvN pr1) in u0;
set (phi2 := fun N:nat => mkStepFun (StepFun_P32 (phi1 N)));
apply Rle_cv_lim with
(fun N:nat => Rabs (RiemannInt_SF (phi1 N)))
(fun N:nat => RiemannInt_SF (phi2 N)).
intro; unfold phi2 in |- *; apply StepFun_P34; assumption.
apply (continuity_seq Rabs (fun N:nat => RiemannInt_SF (phi1 N)) x0);
try assumption.
apply Rcontinuity_abs.
set (phi3 := phi_sequence RinvN pr2);
assert
(H0 :
exists psi3 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (Rabs (f t) - phi3 n t) <= psi3 n t) /\
Rabs (RiemannInt_SF (psi3 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr2 n)).
assert
(H1 :
exists psi2 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (Rabs (f t) - phi2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
assert
(H1 :
exists psi2 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b -> Rabs (f t - phi1 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr1 n)).
elim H1; clear H1; intros psi2 H1; split with psi2; intros; elim (H1 n);
clear H1; intros; split; try assumption.
intros; unfold phi2 in |- *; simpl in |- *;
apply Rle_trans with (Rabs (f t - phi1 n t)).
apply Rabs_triang_inv2.
apply H1; assumption.
elim H0; clear H0; intros psi3 H0; elim H1; clear H1; intros psi2 H1;
apply RiemannInt_P11 with (fun x:R => Rabs (f x)) RinvN phi3 psi3 psi2;
try assumption; apply RinvN_cv.
Qed.
Lemma RiemannInt_P18 :
forall (f g:R -> R) (a b:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable g a b),
a <= b ->
(forall x:R, a < x < b -> f x = g x) -> RiemannInt pr1 = RiemannInt pr2.
Proof.
intro f; intros; unfold RiemannInt in |- *;
case (RiemannInt_exists pr1 RinvN RinvN_cv);
case (RiemannInt_exists pr2 RinvN RinvN_cv); intros;
eapply UL_sequence.
apply u0.
set (phi1 := fun N:nat => phi_sequence RinvN pr1 N);
change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) x) in |- *;
assert
(H1 :
exists psi1 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr1 n)).
elim H1; clear H1; intros psi1 H1;
set (phi2 := fun N:nat => phi_sequence RinvN pr2 N).
set
(phi2_aux :=
fun (N:nat) (x:R) =>
match Req_EM_T x a with
| left _ => f a
| right _ =>
match Req_EM_T x b with
| left _ => f b
| right _ => phi2 N x
end
end).
cut (forall N:nat, IsStepFun (phi2_aux N) a b).
intro; set (phi2_m := fun N:nat => mkStepFun (X N)).
assert
(H2 :
exists psi2 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b -> Rabs (g t - phi2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr2 n)).
elim H2; clear H2; intros psi2 H2;
apply RiemannInt_P11 with f RinvN phi2_m psi2 psi1;
try assumption.
apply RinvN_cv.
intro; elim (H2 n); intros; split; try assumption.
intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *;
case (Req_EM_T t a); case (Req_EM_T t b); intros.
rewrite e0; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
apply Rle_trans with (Rabs (g t - phi2 n t)).
apply Rabs_pos.
pattern a at 3 in |- *; rewrite <- e0; apply H3; assumption.
rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
apply Rle_trans with (Rabs (g t - phi2 n t)).
apply Rabs_pos.
pattern a at 3 in |- *; rewrite <- e; apply H3; assumption.
rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
apply Rle_trans with (Rabs (g t - phi2 n t)).
apply Rabs_pos.
pattern b at 3 in |- *; rewrite <- e; apply H3; assumption.
replace (f t) with (g t).
apply H3; assumption.
symmetry in |- *; apply H0; elim H5; clear H5; intros.
assert (H7 : Rmin a b = a).
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n2; assumption ].
assert (H8 : Rmax a b = b).
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n2; assumption ].
rewrite H7 in H5; rewrite H8 in H6; split.
elim H5; intro; [ assumption | elim n1; symmetry in |- *; assumption ].
elim H6; intro; [ assumption | elim n0; assumption ].
cut (forall N:nat, RiemannInt_SF (phi2_m N) = RiemannInt_SF (phi2 N)).
intro; unfold Un_cv in |- *; intros; elim (u _ H4); intros; exists x1; intros;
rewrite (H3 n); apply H5; assumption.
intro; apply Rle_antisym.
apply StepFun_P37; try assumption.
intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *;
case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros.
elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4).
elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4).
elim H3; intros; rewrite e in H5; elim (Rlt_irrefl _ H5).
right; reflexivity.
apply StepFun_P37; try assumption.
intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *;
case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros.
elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4).
elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4).
elim H3; intros; rewrite e in H5; elim (Rlt_irrefl _ H5).
right; reflexivity.
intro; assert (H2 := pre (phi2 N)); unfold IsStepFun in H2;
unfold is_subdivision in H2; elim H2; clear H2; intros l [lf H2];
split with l; split with lf; unfold adapted_couple in H2;
decompose [and] H2; clear H2; unfold adapted_couple in |- *;
repeat split; try assumption.
intros; assert (H9 := H8 i H2); unfold constant_D_eq, open_interval in H9;
unfold constant_D_eq, open_interval in |- *; intros;
rewrite <- (H9 x1 H7); assert (H10 : a <= pos_Rl l i).
replace a with (Rmin a b).
rewrite <- H5; elim (RList_P6 l); intros; apply H10.
assumption.
apply le_O_n.
apply lt_trans with (pred (Rlength l)); [ assumption | apply lt_pred_n_n ].
apply neq_O_lt; intro; rewrite <- H12 in H6; discriminate.
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (H11 : pos_Rl l (S i) <= b).
replace b with (Rmax a b).
rewrite <- H4; elim (RList_P6 l); intros; apply H11.
assumption.
apply lt_le_S; assumption.
apply lt_pred_n_n; apply neq_O_lt; intro; rewrite <- H13 in H6; discriminate.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
elim H7; clear H7; intros; unfold phi2_aux in |- *; case (Req_EM_T x1 a);
case (Req_EM_T x1 b); intros.
rewrite e in H12; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H12)).
rewrite e in H7; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H10 H7)).
rewrite e in H12; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H12)).
reflexivity.
Qed.
Lemma RiemannInt_P19 :
forall (f g:R -> R) (a b:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable g a b),
a <= b ->
(forall x:R, a < x < b -> f x <= g x) -> RiemannInt pr1 <= RiemannInt pr2.
Proof.
intro f; intros; apply Rplus_le_reg_l with (- RiemannInt pr1);
rewrite Rplus_opp_l; rewrite Rplus_comm;
apply Rle_trans with (Rabs (RiemannInt (RiemannInt_P10 (-1) pr2 pr1))).
apply Rabs_pos.
replace (RiemannInt pr2 + - RiemannInt pr1) with
(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) pr2 pr1))).
apply
(RiemannInt_P17 (RiemannInt_P10 (-1) pr2 pr1)
(RiemannInt_P16 (RiemannInt_P10 (-1) pr2 pr1)));
assumption.
replace (RiemannInt pr2 + - RiemannInt pr1) with
(RiemannInt (RiemannInt_P10 (-1) pr2 pr1)).
apply RiemannInt_P18; try assumption.
intros; apply Rabs_right.
apply Rle_ge; apply Rplus_le_reg_l with (f x); rewrite Rplus_0_r;
replace (f x + (g x + -1 * f x)) with (g x); [ apply H0; assumption | ring ].
rewrite (RiemannInt_P12 pr2 pr1 (RiemannInt_P10 (-1) pr2 pr1));
[ ring | assumption ].
Qed.
Lemma FTC_P1 :
forall (f:R -> R) (a b:R),
a <= b ->
(forall x:R, a <= x <= b -> continuity_pt f x) ->
forall x:R, a <= x -> x <= b -> Riemann_integrable f a x.
Proof.
intros; apply continuity_implies_RiemannInt;
[ assumption
| intros; apply H0; elim H3; intros; split;
assumption || apply Rle_trans with x; assumption ].
Qed.
Definition primitive (f:R -> R) (a b:R) (h:a <= b)
(pr:forall x:R, a <= x -> x <= b -> Riemann_integrable f a x)
(x:R) : R :=
match Rle_dec a x with
| left r =>
match Rle_dec x b with
| left r0 => RiemannInt (pr x r r0)
| right _ => f b * (x - b) + RiemannInt (pr b h (Rle_refl b))
end
| right _ => f a * (x - a)
end.
Lemma RiemannInt_P20 :
forall (f:R -> R) (a b:R) (h:a <= b)
(pr:forall x:R, a <= x -> x <= b -> Riemann_integrable f a x)
(pr0:Riemann_integrable f a b),
RiemannInt pr0 = primitive h pr b - primitive h pr a.
Proof.
intros; replace (primitive h pr a) with 0.
replace (RiemannInt pr0) with (primitive h pr b).
ring.
unfold primitive in |- *; case (Rle_dec a b); case (Rle_dec b b); intros;
[ apply RiemannInt_P5
| elim n; right; reflexivity
| elim n; assumption
| elim n0; assumption ].
symmetry in |- *; unfold primitive in |- *; case (Rle_dec a a);
case (Rle_dec a b); intros;
[ apply RiemannInt_P9
| elim n; assumption
| elim n; right; reflexivity
| elim n0; right; reflexivity ].
Qed.
Lemma RiemannInt_P21 :
forall (f:R -> R) (a b c:R),
a <= b ->
b <= c ->
Riemann_integrable f a b ->
Riemann_integrable f b c -> Riemann_integrable f a c.
Proof.
unfold Riemann_integrable in |- *; intros f a b c Hyp1 Hyp2 X X0 eps.
assert (H : 0 < eps / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ].
elim (X (mkposreal _ H)); clear X; intros phi1 [psi1 H1];
elim (X0 (mkposreal _ H)); clear X0; intros phi2 [psi2 H2].
set
(phi3 :=
fun x:R =>
match Rle_dec a x with
| left _ =>
match Rle_dec x b with
| left _ => phi1 x
| right _ => phi2 x
end
| right _ => 0
end).
set
(psi3 :=
fun x:R =>
match Rle_dec a x with
| left _ =>
match Rle_dec x b with
| left _ => psi1 x
| right _ => psi2 x
end
| right _ => 0
end).
cut (IsStepFun phi3 a c).
intro; cut (IsStepFun psi3 a b).
intro; cut (IsStepFun psi3 b c).
intro; cut (IsStepFun psi3 a c).
intro; split with (mkStepFun X); split with (mkStepFun X2); simpl in |- *;
split.
intros; unfold phi3, psi3 in |- *; case (Rle_dec t b); case (Rle_dec a t);
intros.
elim H1; intros; apply H3.
replace (Rmin a b) with a.
replace (Rmax a b) with b.
split; assumption.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
elim n; replace a with (Rmin a c).
elim H0; intros; assumption.
unfold Rmin in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n0; apply Rle_trans with b; assumption ].
elim H2; intros; apply H3.
replace (Rmax b c) with (Rmax a c).
elim H0; intros; split; try assumption.
replace (Rmin b c) with b.
auto with real.
unfold Rmin in |- *; case (Rle_dec b c); intro;
[ reflexivity | elim n0; assumption ].
unfold Rmax in |- *; case (Rle_dec a c); case (Rle_dec b c); intros;
try (elim n0; assumption || elim n0; apply Rle_trans with b; assumption).
reflexivity.
elim n; replace a with (Rmin a c).
elim H0; intros; assumption.
unfold Rmin in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n1; apply Rle_trans with b; assumption ].
rewrite <- (StepFun_P43 X0 X1 X2).
apply Rle_lt_trans with
(Rabs (RiemannInt_SF (mkStepFun X0)) + Rabs (RiemannInt_SF (mkStepFun X1))).
apply Rabs_triang.
rewrite (double_var eps);
replace (RiemannInt_SF (mkStepFun X0)) with (RiemannInt_SF psi1).
replace (RiemannInt_SF (mkStepFun X1)) with (RiemannInt_SF psi2).
apply Rplus_lt_compat.
elim H1; intros; assumption.
elim H2; intros; assumption.
apply Rle_antisym.
apply StepFun_P37; try assumption.
simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
case (Rle_dec a x); case (Rle_dec x b); intros;
[ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0))
| right; reflexivity
| elim n; apply Rle_trans with b; [ assumption | left; assumption ]
| elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ].
apply StepFun_P37; try assumption.
simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
case (Rle_dec a x); case (Rle_dec x b); intros;
[ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0))
| right; reflexivity
| elim n; apply Rle_trans with b; [ assumption | left; assumption ]
| elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ].
apply Rle_antisym.
apply StepFun_P37; try assumption.
simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
case (Rle_dec a x); case (Rle_dec x b); intros;
[ right; reflexivity
| elim n; left; assumption
| elim n; left; assumption
| elim n0; left; assumption ].
apply StepFun_P37; try assumption.
simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros;
case (Rle_dec a x); case (Rle_dec x b); intros;
[ right; reflexivity
| elim n; left; assumption
| elim n; left; assumption
| elim n0; left; assumption ].
apply StepFun_P46 with b; assumption.
assert (H3 := pre psi2); unfold IsStepFun in H3; unfold is_subdivision in H3;
elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
split with lf1; unfold adapted_couple in H3; decompose [and] H3;
clear H3; unfold adapted_couple in |- *; repeat split;
try assumption.
intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
unfold constant_D_eq, open_interval in H9; intros;
rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x).
apply Rle_lt_trans with (pos_Rl l1 i).
replace b with (Rmin b c).
rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption.
apply le_O_n.
apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
discriminate.
unfold Rmin in |- *; case (Rle_dec b c); intro;
[ reflexivity | elim n; assumption ].
elim H7; intros; assumption.
case (Rle_dec a x); case (Rle_dec x b); intros;
[ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10))
| reflexivity
| elim n; apply Rle_trans with b; [ assumption | left; assumption ]
| elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ].
assert (H3 := pre psi1); unfold IsStepFun in H3; unfold is_subdivision in H3;
elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
split with lf1; unfold adapted_couple in H3; decompose [and] H3;
clear H3; unfold adapted_couple in |- *; repeat split;
try assumption.
intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
unfold constant_D_eq, open_interval in H9; intros;
rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b).
apply Rle_trans with (pos_Rl l1 (S i)).
elim H7; intros; left; assumption.
replace b with (Rmax a b).
rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption.
apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
discriminate.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (H11 : a <= x).
apply Rle_trans with (pos_Rl l1 i).
replace a with (Rmin a b).
rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption.
apply le_O_n.
apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6;
discriminate.
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
left; elim H7; intros; assumption.
case (Rle_dec a x); case (Rle_dec x b); intros; reflexivity || elim n;
assumption.
apply StepFun_P46 with b.
assert (H3 := pre phi1); unfold IsStepFun in H3; unfold is_subdivision in H3;
elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
split with lf1; unfold adapted_couple in H3; decompose [and] H3;
clear H3; unfold adapted_couple in |- *; repeat split;
try assumption.
intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
unfold constant_D_eq, open_interval in H9; intros;
rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b).
apply Rle_trans with (pos_Rl l1 (S i)).
elim H7; intros; left; assumption.
replace b with (Rmax a b).
rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption.
apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
discriminate.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
assert (H11 : a <= x).
apply Rle_trans with (pos_Rl l1 i).
replace a with (Rmin a b).
rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption.
apply le_O_n.
apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6;
discriminate.
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; assumption ].
left; elim H7; intros; assumption.
unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros;
reflexivity || elim n; assumption.
assert (H3 := pre phi2); unfold IsStepFun in H3; unfold is_subdivision in H3;
elim H3; clear H3; intros l1 [lf1 H3]; split with l1;
split with lf1; unfold adapted_couple in H3; decompose [and] H3;
clear H3; unfold adapted_couple in |- *; repeat split;
try assumption.
intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *;
unfold constant_D_eq, open_interval in H9; intros;
rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x).
apply Rle_lt_trans with (pos_Rl l1 i).
replace b with (Rmin b c).
rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption.
apply le_O_n.
apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n;
apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6;
discriminate.
unfold Rmin in |- *; case (Rle_dec b c); intro;
[ reflexivity | elim n; assumption ].
elim H7; intros; assumption.
unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros;
[ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10))
| reflexivity
| elim n; apply Rle_trans with b; [ assumption | left; assumption ]
| elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ].
Qed.
Lemma RiemannInt_P22 :
forall (f:R -> R) (a b c:R),
Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f a c.
Proof.
unfold Riemann_integrable in |- *; intros; elim (X eps); clear X;
intros phi [psi H0]; elim H; elim H0; clear H H0;
intros; assert (H3 : IsStepFun phi a c).
apply StepFun_P44 with b.
apply (pre phi).
split; assumption.
assert (H4 : IsStepFun psi a c).
apply StepFun_P44 with b.
apply (pre psi).
split; assumption.
split with (mkStepFun H3); split with (mkStepFun H4); split.
simpl in |- *; intros; apply H.
replace (Rmin a b) with (Rmin a c).
elim H5; intros; split; try assumption.
apply Rle_trans with (Rmax a c); try assumption.
replace (Rmax a b) with b.
replace (Rmax a c) with c.
assumption.
unfold Rmax in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n; assumption ].
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
unfold Rmin in |- *; case (Rle_dec a c); case (Rle_dec a b); intros;
[ reflexivity
| elim n; apply Rle_trans with c; assumption
| elim n; assumption
| elim n0; assumption ].
rewrite Rabs_right.
assert (H5 : IsStepFun psi c b).
apply StepFun_P46 with a.
apply StepFun_P6; assumption.
apply (pre psi).
replace (RiemannInt_SF (mkStepFun H4)) with
(RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)).
apply Rle_lt_trans with (RiemannInt_SF psi).
unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *;
rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0;
apply Ropp_ge_le_contravar; apply Rle_ge;
replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; unfold fct_cte in |- *;
apply Rle_trans with (Rabs (f x - phi x)).
apply Rabs_pos.
apply H.
replace (Rmin a b) with a.
replace (Rmax a b) with b.
elim H6; intros; split; left.
apply Rle_lt_trans with c; assumption.
assumption.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
rewrite StepFun_P18; ring.
apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)).
apply RRle_abs.
assumption.
assert (H6 : IsStepFun psi a b).
apply (pre psi).
replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)).
rewrite <- (StepFun_P43 H4 H5 H6); ring.
unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
eapply StepFun_P17.
apply StepFun_P1.
simpl in |- *; apply StepFun_P1.
apply Ropp_eq_compat; eapply StepFun_P17.
apply StepFun_P1.
simpl in |- *; apply StepFun_P1.
apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; unfold fct_cte in |- *;
apply Rle_trans with (Rabs (f x - phi x)).
apply Rabs_pos.
apply H.
replace (Rmin a b) with a.
replace (Rmax a b) with b.
elim H5; intros; split; left.
assumption.
apply Rlt_le_trans with c; assumption.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
rewrite StepFun_P18; ring.
Qed.
Lemma RiemannInt_P23 :
forall (f:R -> R) (a b c:R),
Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f c b.
Proof.
unfold Riemann_integrable in |- *; intros; elim (X eps); clear X;
intros phi [psi H0]; elim H; elim H0; clear H H0;
intros; assert (H3 : IsStepFun phi c b).
apply StepFun_P45 with a.
apply (pre phi).
split; assumption.
assert (H4 : IsStepFun psi c b).
apply StepFun_P45 with a.
apply (pre psi).
split; assumption.
split with (mkStepFun H3); split with (mkStepFun H4); split.
simpl in |- *; intros; apply H.
replace (Rmax a b) with (Rmax c b).
elim H5; intros; split; try assumption.
apply Rle_trans with (Rmin c b); try assumption.
replace (Rmin a b) with a.
replace (Rmin c b) with c.
assumption.
unfold Rmin in |- *; case (Rle_dec c b); intro;
[ reflexivity | elim n; assumption ].
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
unfold Rmax in |- *; case (Rle_dec c b); case (Rle_dec a b); intros;
[ reflexivity
| elim n; apply Rle_trans with c; assumption
| elim n; assumption
| elim n0; assumption ].
rewrite Rabs_right.
assert (H5 : IsStepFun psi a c).
apply StepFun_P46 with b.
apply (pre psi).
apply StepFun_P6; assumption.
replace (RiemannInt_SF (mkStepFun H4)) with
(RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)).
apply Rle_lt_trans with (RiemannInt_SF psi).
unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *;
rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0;
apply Ropp_ge_le_contravar; apply Rle_ge;
replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; unfold fct_cte in |- *;
apply Rle_trans with (Rabs (f x - phi x)).
apply Rabs_pos.
apply H.
replace (Rmin a b) with a.
replace (Rmax a b) with b.
elim H6; intros; split; left.
assumption.
apply Rlt_le_trans with c; assumption.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
rewrite StepFun_P18; ring.
apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)).
apply RRle_abs.
assumption.
assert (H6 : IsStepFun psi a b).
apply (pre psi).
replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)).
rewrite <- (StepFun_P43 H5 H4 H6); ring.
unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro.
eapply StepFun_P17.
apply StepFun_P1.
simpl in |- *; apply StepFun_P1.
apply Ropp_eq_compat; eapply StepFun_P17.
apply StepFun_P1.
simpl in |- *; apply StepFun_P1.
apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))).
apply StepFun_P37; try assumption.
intros; simpl in |- *; unfold fct_cte in |- *;
apply Rle_trans with (Rabs (f x - phi x)).
apply Rabs_pos.
apply H.
replace (Rmin a b) with a.
replace (Rmax a b) with b.
elim H5; intros; split; left.
apply Rle_lt_trans with c; assumption.
assumption.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n; apply Rle_trans with c; assumption ].
rewrite StepFun_P18; ring.
Qed.
Lemma RiemannInt_P24 :
forall (f:R -> R) (a b c:R),
Riemann_integrable f a b ->
Riemann_integrable f b c -> Riemann_integrable f a c.
Proof.
intros; case (Rle_dec a b); case (Rle_dec b c); intros.
apply RiemannInt_P21 with b; assumption.
case (Rle_dec a c); intro.
apply RiemannInt_P22 with b; try assumption.
split; [ assumption | auto with real ].
apply RiemannInt_P1; apply RiemannInt_P22 with b.
apply RiemannInt_P1; assumption.
split; auto with real.
case (Rle_dec a c); intro.
apply RiemannInt_P23 with b; try assumption.
split; auto with real.
apply RiemannInt_P1; apply RiemannInt_P23 with b.
apply RiemannInt_P1; assumption.
split; [ assumption | auto with real ].
apply RiemannInt_P1; apply RiemannInt_P21 with b;
auto with real || apply RiemannInt_P1; assumption.
Qed.
Lemma RiemannInt_P25 :
forall (f:R -> R) (a b c:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable f b c) (pr3:Riemann_integrable f a c),
a <= b -> b <= c -> RiemannInt pr1 + RiemannInt pr2 = RiemannInt pr3.
Proof.
intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; unfold RiemannInt in |- *;
case (RiemannInt_exists pr1 RinvN RinvN_cv);
case (RiemannInt_exists pr2 RinvN RinvN_cv);
case (RiemannInt_exists pr3 RinvN RinvN_cv); intros;
symmetry in |- *; eapply UL_sequence.
apply u.
unfold Un_cv in |- *; intros; assert (H0 : 0 < eps / 3).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (u1 _ H0); clear u1; intros N1 H1; elim (u0 _ H0); clear u0;
intros N2 H2;
cut
(Un_cv
(fun n:nat =>
RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n))) 0).
intro; elim (H3 _ H0); clear H3; intros N3 H3;
set (N0 := max (max N1 N2) N3); exists N0; intros;
unfold R_dist in |- *;
apply Rle_lt_trans with
(Rabs
(RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n))) +
Rabs
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n) - (x1 + x0))).
replace (RiemannInt_SF (phi_sequence RinvN pr3 n) - (x1 + x0)) with
(RiemannInt_SF (phi_sequence RinvN pr3 n) -
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n)) +
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n) - (x1 + x0)));
[ apply Rabs_triang | ring ].
replace eps with (eps / 3 + eps / 3 + eps / 3).
rewrite Rplus_assoc; apply Rplus_lt_compat.
unfold R_dist in H3; cut (n >= N3)%nat.
intro; assert (H6 := H3 _ H5); unfold Rminus in H6; rewrite Ropp_0 in H6;
rewrite Rplus_0_r in H6; apply H6.
unfold ge in |- *; apply le_trans with N0;
[ unfold N0 in |- *; apply le_max_r | assumption ].
apply Rle_lt_trans with
(Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) - x1) +
Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x0)).
replace
(RiemannInt_SF (phi_sequence RinvN pr1 n) +
RiemannInt_SF (phi_sequence RinvN pr2 n) - (x1 + x0)) with
(RiemannInt_SF (phi_sequence RinvN pr1 n) - x1 +
(RiemannInt_SF (phi_sequence RinvN pr2 n) - x0));
[ apply Rabs_triang | ring ].
apply Rplus_lt_compat.
unfold R_dist in H1; apply H1.
unfold ge in |- *; apply le_trans with N0;
[ apply le_trans with (max N1 N2);
[ apply le_max_l | unfold N0 in |- *; apply le_max_l ]
| assumption ].
unfold R_dist in H2; apply H2.
unfold ge in |- *; apply le_trans with N0;
[ apply le_trans with (max N1 N2);
[ apply le_max_r | unfold N0 in |- *; apply le_max_l ]
| assumption ].
apply Rmult_eq_reg_l with 3;
[ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l;
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
clear x u x0 x1 eps H H0 N1 H1 N2 H2;
assert
(H1 :
exists psi1 : nat -> StepFun a b,
(forall n:nat,
(forall t:R,
Rmin a b <= t /\ t <= Rmax a b ->
Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\
Rabs (RiemannInt_SF (psi1 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr1 n)).
assert
(H2 :
exists psi2 : nat -> StepFun b c,
(forall n:nat,
(forall t:R,
Rmin b c <= t /\ t <= Rmax b c ->
Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\
Rabs (RiemannInt_SF (psi2 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr2 n)).
assert
(H3 :
exists psi3 : nat -> StepFun a c,
(forall n:nat,
(forall t:R,
Rmin a c <= t /\ t <= Rmax a c ->
Rabs (f t - phi_sequence RinvN pr3 n t) <= psi3 n t) /\
Rabs (RiemannInt_SF (psi3 n)) < RinvN n)).
split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); intro;
apply (projT2 (phi_sequence_prop RinvN pr3 n)).
elim H1; clear H1; intros psi1 H1; elim H2; clear H2; intros psi2 H2; elim H3;
clear H3; intros psi3 H3; assert (H := RinvN_cv);
unfold Un_cv in |- *; intros; assert (H4 : 0 < eps / 3).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H _ H4); clear H; intros N0 H;
assert (H5 : forall n:nat, (n >= N0)%nat -> RinvN n < eps / 3).
intros;
replace (pos (RinvN n)) with
(R_dist (mkposreal (/ (INR n + 1)) (RinvN_pos n)) 0).
apply H; assumption.
unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge;
left; apply (cond_pos (RinvN n)).
exists N0; intros; elim (H1 n); elim (H2 n); elim (H3 n); clear H1 H2 H3;
intros; unfold R_dist in |- *; unfold Rminus in |- *;
rewrite Ropp_0; rewrite Rplus_0_r;
set (phi1 := phi_sequence RinvN pr1 n) in H8 |- *;
set (phi2 := phi_sequence RinvN pr2 n) in H3 |- *;
set (phi3 := phi_sequence RinvN pr3 n) in H1 |- *;
assert (H10 : IsStepFun phi3 a b).
apply StepFun_P44 with c.
apply (pre phi3).
split; assumption.
assert (H11 : IsStepFun (psi3 n) a b).
apply StepFun_P44 with c.
apply (pre (psi3 n)).
split; assumption.
assert (H12 : IsStepFun phi3 b c).
apply StepFun_P45 with a.
apply (pre phi3).
split; assumption.
assert (H13 : IsStepFun (psi3 n) b c).
apply StepFun_P45 with a.
apply (pre (psi3 n)).
split; assumption.
replace (RiemannInt_SF phi3) with
(RiemannInt_SF (mkStepFun H10) + RiemannInt_SF (mkStepFun H12)).
apply Rle_lt_trans with
(Rabs (RiemannInt_SF (mkStepFun H10) - RiemannInt_SF phi1) +
Rabs (RiemannInt_SF (mkStepFun H12) - RiemannInt_SF phi2)).
replace
(RiemannInt_SF (mkStepFun H10) + RiemannInt_SF (mkStepFun H12) +
- (RiemannInt_SF phi1 + RiemannInt_SF phi2)) with
(RiemannInt_SF (mkStepFun H10) - RiemannInt_SF phi1 +
(RiemannInt_SF (mkStepFun H12) - RiemannInt_SF phi2));
[ apply Rabs_triang | ring ].
replace (RiemannInt_SF (mkStepFun H10) - RiemannInt_SF phi1) with
(RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1))).
replace (RiemannInt_SF (mkStepFun H12) - RiemannInt_SF phi2) with
(RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))).
apply Rle_lt_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))) +
RiemannInt_SF
(mkStepFun
(StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))))).
apply Rle_trans with
(Rabs (RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1))) +
RiemannInt_SF
(mkStepFun
(StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))))).
apply Rplus_le_compat_l.
apply StepFun_P34; try assumption.
do 2
rewrite <-
(Rplus_comm
(RiemannInt_SF
(mkStepFun
(StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))))))
; apply Rplus_le_compat_l; apply StepFun_P34; try assumption.
apply Rle_lt_trans with
(RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H11) (psi1 n))) +
RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))).
apply Rle_trans with
(RiemannInt_SF
(mkStepFun
(StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))) +
RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))).
apply Rplus_le_compat_l; apply StepFun_P37; try assumption.
intros; simpl in |- *; rewrite Rmult_1_l;
apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi2 x)).
rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr;
replace (phi3 x + -1 * phi2 x) with (phi3 x - f x + (f x - phi2 x));
[ apply Rabs_triang | ring ].
apply Rplus_le_compat.
apply H1.
elim H14; intros; split.
replace (Rmin a c) with a.
apply Rle_trans with b; try assumption.
left; assumption.
unfold Rmin in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n0; apply Rle_trans with b; assumption ].
replace (Rmax a c) with c.
left; assumption.
unfold Rmax in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n0; apply Rle_trans with b; assumption ].
apply H3.
elim H14; intros; split.
replace (Rmin b c) with b.
left; assumption.
unfold Rmin in |- *; case (Rle_dec b c); intro;
[ reflexivity | elim n0; assumption ].
replace (Rmax b c) with c.
left; assumption.
unfold Rmax in |- *; case (Rle_dec b c); intro;
[ reflexivity | elim n0; assumption ].
do 2
rewrite <-
(Rplus_comm
(RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))))
; apply Rplus_le_compat_l; apply StepFun_P37; try assumption.
intros; simpl in |- *; rewrite Rmult_1_l;
apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi1 x)).
rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr;
replace (phi3 x + -1 * phi1 x) with (phi3 x - f x + (f x - phi1 x));
[ apply Rabs_triang | ring ].
apply Rplus_le_compat.
apply H1.
elim H14; intros; split.
replace (Rmin a c) with a.
left; assumption.
unfold Rmin in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n0; apply Rle_trans with b; assumption ].
replace (Rmax a c) with c.
apply Rle_trans with b.
left; assumption.
assumption.
unfold Rmax in |- *; case (Rle_dec a c); intro;
[ reflexivity | elim n0; apply Rle_trans with b; assumption ].
apply H8.
elim H14; intros; split.
replace (Rmin a b) with a.
left; assumption.
unfold Rmin in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
replace (Rmax a b) with b.
left; assumption.
unfold Rmax in |- *; case (Rle_dec a b); intro;
[ reflexivity | elim n0; assumption ].
do 2 rewrite StepFun_P30.
do 2 rewrite Rmult_1_l;
replace
(RiemannInt_SF (mkStepFun H11) + RiemannInt_SF (psi1 n) +
(RiemannInt_SF (mkStepFun H13) + RiemannInt_SF (psi2 n))) with
(RiemannInt_SF (psi3 n) + RiemannInt_SF (psi1 n) + RiemannInt_SF (psi2 n)).
replace eps with (eps / 3 + eps / 3 + eps / 3).
repeat rewrite Rplus_assoc; repeat apply Rplus_lt_compat.
apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi3 n))).
apply RRle_abs.
apply Rlt_trans with (pos (RinvN n)).
assumption.
apply H5; assumption.
apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))).
apply RRle_abs.
apply Rlt_trans with (pos (RinvN n)).
assumption.
apply H5; assumption.
apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))).
apply RRle_abs.
apply Rlt_trans with (pos (RinvN n)).
assumption.
apply H5; assumption.
apply Rmult_eq_reg_l with 3;
[ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l;
do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | discrR ]
| discrR ].
replace (RiemannInt_SF (psi3 n)) with
(RiemannInt_SF (mkStepFun (pre (psi3 n)))).
rewrite <- (StepFun_P43 H11 H13 (pre (psi3 n))); ring.
reflexivity.
rewrite StepFun_P30; ring.
rewrite StepFun_P30; ring.
apply (StepFun_P43 H10 H12 (pre phi3)).
Qed.
Lemma RiemannInt_P26 :
forall (f:R -> R) (a b c:R) (pr1:Riemann_integrable f a b)
(pr2:Riemann_integrable f b c) (pr3:Riemann_integrable f a c),
RiemannInt pr1 + RiemannInt pr2 = RiemannInt pr3.
Proof.
intros; case (Rle_dec a b); case (Rle_dec b c); intros.
apply RiemannInt_P25; assumption.
case (Rle_dec a c); intro.
assert (H : c <= b).
auto with real.
rewrite <- (RiemannInt_P25 pr3 (RiemannInt_P1 pr2) pr1 r0 H);
rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); ring.
assert (H : c <= a).
auto with real.
rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2));
rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr3) pr1 (RiemannInt_P1 pr2) H r);
rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); ring.
assert (H : b <= a).
auto with real.
case (Rle_dec a c); intro.
rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr1) pr3 pr2 H r0);
rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); ring.
assert (H0 : c <= a).
auto with real.
rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1));
rewrite <- (RiemannInt_P25 pr2 (RiemannInt_P1 pr3) (RiemannInt_P1 pr1) r H0);
rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); ring.
rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1));
rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2));
rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3));
rewrite <-
(RiemannInt_P25 (RiemannInt_P1 pr2) (RiemannInt_P1 pr1) (RiemannInt_P1 pr3))
; [ ring | auto with real | auto with real ].
Qed.
Lemma RiemannInt_P27 :
forall (f:R -> R) (a b x:R) (h:a <= b)
(C0:forall x:R, a <= x <= b -> continuity_pt f x),
a < x < b -> derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x).
Proof.
intro f; intros; elim H; clear H; intros; assert (H1 : continuity_pt f x).
apply C0; split; left; assumption.
unfold derivable_pt_lim in |- *; intros; assert (Hyp : 0 < eps / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H1 _ Hyp); unfold dist, D_x, no_cond in |- *; simpl in |- *;
unfold R_dist in |- *; intros; set (del := Rmin x0 (Rmin (b - x) (x - a)));
assert (H4 : 0 < del).
unfold del in |- *; unfold Rmin in |- *; case (Rle_dec (b - x) (x - a));
intro.
case (Rle_dec x0 (b - x)); intro;
[ elim H3; intros; assumption | apply Rlt_Rminus; assumption ].
case (Rle_dec x0 (x - a)); intro;
[ elim H3; intros; assumption | apply Rlt_Rminus; assumption ].
split with (mkposreal _ H4); intros;
assert (H7 : Riemann_integrable f x (x + h0)).
case (Rle_dec x (x + h0)); intro.
apply continuity_implies_RiemannInt; try assumption.
intros; apply C0; elim H7; intros; split.
apply Rle_trans with x; [ left; assumption | assumption ].
apply Rle_trans with (x + h0).
assumption.
left; apply Rlt_le_trans with (x + del).
apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h0);
[ apply RRle_abs | apply H6 ].
unfold del in |- *; apply Rle_trans with (x + Rmin (b - x) (x - a)).
apply Rplus_le_compat_l; apply Rmin_r.
pattern b at 2 in |- *; replace b with (x + (b - x));
[ apply Rplus_le_compat_l; apply Rmin_l | ring ].
apply RiemannInt_P1; apply continuity_implies_RiemannInt; auto with real.
intros; apply C0; elim H7; intros; split.
apply Rle_trans with (x + h0).
left; apply Rle_lt_trans with (x - del).
unfold del in |- *; apply Rle_trans with (x - Rmin (b - x) (x - a)).
pattern a at 1 in |- *; replace a with (x + (a - x)); [ idtac | ring ].
unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel.
rewrite Ropp_involutive; rewrite Ropp_plus_distr; rewrite Ropp_involutive;
rewrite (Rplus_comm x); apply Rmin_r.
unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel.
do 2 rewrite Ropp_involutive; apply Rmin_r.
unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel.
rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0);
[ rewrite <- Rabs_Ropp; apply RRle_abs | apply H6 ].
assumption.
apply Rle_trans with x; [ assumption | left; assumption ].
replace (primitive h (FTC_P1 h C0) (x + h0) - primitive h (FTC_P1 h C0) x)
with (RiemannInt H7).
replace (f x) with (RiemannInt (RiemannInt_P14 x (x + h0) (f x)) / h0).
replace
(RiemannInt H7 / h0 - RiemannInt (RiemannInt_P14 x (x + h0) (f x)) / h0)
with ((RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) / h0).
replace (RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) with
(RiemannInt (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))).
unfold Rdiv in |- *; rewrite Rabs_mult; case (Rle_dec x (x + h0)); intro.
apply Rle_lt_trans with
(RiemannInt
(RiemannInt_P16
(RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) *
Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply
(RiemannInt_P17 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))
(RiemannInt_P16
(RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))));
assumption.
apply Rle_lt_trans with
(RiemannInt (RiemannInt_P14 x (x + h0) (eps / 2)) * Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply RiemannInt_P19; try assumption.
intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x).
unfold fct_cte in |- *; case (Req_dec x x1); intro.
rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left;
assumption.
elim H3; intros; left; apply H11.
repeat split.
assumption.
rewrite Rabs_right.
apply Rplus_lt_reg_r with x; replace (x + (x1 - x)) with x1; [ idtac | ring ].
apply Rlt_le_trans with (x + h0).
elim H8; intros; assumption.
apply Rplus_le_compat_l; apply Rle_trans with del.
left; apply Rle_lt_trans with (Rabs h0); [ apply RRle_abs | assumption ].
unfold del in |- *; apply Rmin_l.
apply Rge_minus; apply Rle_ge; left; elim H8; intros; assumption.
unfold fct_cte in |- *; ring.
rewrite RiemannInt_P15.
rewrite Rmult_assoc; replace ((x + h0 - x) * Rabs (/ h0)) with 1.
rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
[ prove_sup0
| rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
rewrite Rabs_right.
replace (x + h0 - x) with h0; [ idtac | ring ].
apply Rinv_r_sym.
assumption.
apply Rle_ge; left; apply Rinv_0_lt_compat.
elim r; intro.
apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; assumption.
elim H5; symmetry in |- *; apply Rplus_eq_reg_l with x; rewrite Rplus_0_r;
assumption.
apply Rle_lt_trans with
(RiemannInt
(RiemannInt_P16
(RiemannInt_P1
(RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))) *
Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
replace
(RiemannInt (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) with
(-
RiemannInt
(RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))).
rewrite Rabs_Ropp;
apply
(RiemannInt_P17
(RiemannInt_P1
(RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))
(RiemannInt_P16
(RiemannInt_P1
(RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))));
auto with real.
symmetry in |- *; apply RiemannInt_P8.
apply Rle_lt_trans with
(RiemannInt (RiemannInt_P14 (x + h0) x (eps / 2)) * Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply RiemannInt_P19.
auto with real.
intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x).
unfold fct_cte in |- *; case (Req_dec x x1); intro.
rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left;
assumption.
elim H3; intros; left; apply H11.
repeat split.
assumption.
rewrite Rabs_left.
apply Rplus_lt_reg_r with (x1 - x0); replace (x1 - x0 + x0) with x1;
[ idtac | ring ].
replace (x1 - x0 + - (x1 - x)) with (x - x0); [ idtac | ring ].
apply Rle_lt_trans with (x + h0).
unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel.
rewrite Ropp_involutive; apply Rle_trans with (Rabs h0).
rewrite <- Rabs_Ropp; apply RRle_abs.
apply Rle_trans with del;
[ left; assumption | unfold del in |- *; apply Rmin_l ].
elim H8; intros; assumption.
apply Rplus_lt_reg_r with x; rewrite Rplus_0_r;
replace (x + (x1 - x)) with x1; [ elim H8; intros; assumption | ring ].
unfold fct_cte in |- *; ring.
rewrite RiemannInt_P15.
rewrite Rmult_assoc; replace ((x - (x + h0)) * Rabs (/ h0)) with 1.
rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
[ prove_sup0
| rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
rewrite Rabs_left.
replace (x - (x + h0)) with (- h0); [ idtac | ring ].
rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_mult_distr_r_reverse;
rewrite Ropp_involutive; apply Rinv_r_sym.
assumption.
apply Rinv_lt_0_compat.
assert (H8 : x + h0 < x).
auto with real.
apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; assumption.
rewrite
(RiemannInt_P13 H7 (RiemannInt_P14 x (x + h0) (f x))
(RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))
.
ring.
unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring.
rewrite RiemannInt_P15; apply Rmult_eq_reg_l with h0;
[ unfold Rdiv in |- *; rewrite (Rmult_comm h0); repeat rewrite Rmult_assoc;
rewrite <- Rinv_l_sym; [ ring | assumption ]
| assumption ].
cut (a <= x + h0).
cut (x + h0 <= b).
intros; unfold primitive in |- *.
case (Rle_dec a (x + h0)); case (Rle_dec (x + h0) b); case (Rle_dec a x);
case (Rle_dec x b); intros; try (elim n; assumption || left; assumption).
rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r0 r) H7 (FTC_P1 h C0 r2 r1)); ring.
apply Rplus_le_reg_l with (- x); replace (- x + (x + h0)) with h0;
[ idtac | ring ].
rewrite Rplus_comm; apply Rle_trans with (Rabs h0).
apply RRle_abs.
apply Rle_trans with del;
[ left; assumption
| unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a));
[ apply Rmin_r | apply Rmin_l ] ].
apply Ropp_le_cancel; apply Rplus_le_reg_l with x;
replace (x + - (x + h0)) with (- h0); [ idtac | ring ].
apply Rle_trans with (Rabs h0);
[ rewrite <- Rabs_Ropp; apply RRle_abs
| apply Rle_trans with del;
[ left; assumption
| unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a));
apply Rmin_r ] ].
Qed.
Lemma RiemannInt_P28 :
forall (f:R -> R) (a b x:R) (h:a <= b)
(C0:forall x:R, a <= x <= b -> continuity_pt f x),
a <= x <= b -> derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x).
Proof.
intro f; intros; elim h; intro.
elim H; clear H; intros; elim H; intro.
elim H1; intro.
apply RiemannInt_P27; split; assumption.
set
(f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b)));
rewrite H3.
assert (H4 : derivable_pt_lim f_b b (f b)).
unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0).
change
(derivable_pt_lim
((fct_cte (f b) * (id - fct_cte b))%F +
fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b (
f b + 0)) in |- *.
apply derivable_pt_lim_plus.
pattern (f b) at 2 in |- *;
replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_const.
replace 1 with (1 - 0); [ idtac | ring ].
apply derivable_pt_lim_minus.
apply derivable_pt_lim_id.
apply derivable_pt_lim_const.
unfold fct_cte in |- *; ring.
apply derivable_pt_lim_const.
ring.
unfold derivable_pt_lim in |- *; intros; elim (H4 _ H5); intros;
assert (H7 : continuity_pt f b).
apply C0; split; [ left; assumption | right; reflexivity ].
assert (H8 : 0 < eps / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H7 _ H8); unfold D_x, no_cond, dist in |- *; simpl in |- *;
unfold R_dist in |- *; intros; set (del := Rmin x0 (Rmin x1 (b - a)));
assert (H10 : 0 < del).
unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - a)); intros.
case (Rle_dec x0 x1); intro;
[ apply (cond_pos x0) | elim H9; intros; assumption ].
case (Rle_dec x0 (b - a)); intro;
[ apply (cond_pos x0) | apply Rlt_Rminus; assumption ].
split with (mkposreal _ H10); intros; case (Rcase_abs h0); intro.
assert (H14 : b + h0 < b).
pattern b at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
assumption.
assert (H13 : Riemann_integrable f (b + h0) b).
apply continuity_implies_RiemannInt.
left; assumption.
intros; apply C0; elim H13; intros; split; try assumption.
apply Rle_trans with (b + h0); try assumption.
apply Rplus_le_reg_l with (- a - h0).
replace (- a - h0 + a) with (- h0); [ idtac | ring ].
replace (- a - h0 + (b + h0)) with (b - a); [ idtac | ring ].
apply Rle_trans with del.
apply Rle_trans with (Rabs h0).
rewrite <- Rabs_Ropp; apply RRle_abs.
left; assumption.
unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
replace (primitive h (FTC_P1 h C0) (b + h0) - primitive h (FTC_P1 h C0) b)
with (- RiemannInt H13).
replace (f b) with (- RiemannInt (RiemannInt_P14 (b + h0) b (f b)) / h0).
rewrite <- Rabs_Ropp; unfold Rminus in |- *; unfold Rdiv in |- *;
rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_plus_distr;
repeat rewrite Ropp_involutive;
replace
(RiemannInt H13 * / h0 +
- RiemannInt (RiemannInt_P14 (b + h0) b (f b)) * / h0) with
((RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) / h0).
replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) with
(RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))).
unfold Rdiv in |- *; rewrite Rabs_mult;
apply Rle_lt_trans with
(RiemannInt
(RiemannInt_P16
(RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))) *
Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply
(RiemannInt_P17 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))
(RiemannInt_P16
(RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))));
left; assumption.
apply Rle_lt_trans with
(RiemannInt (RiemannInt_P14 (b + h0) b (eps / 2)) * Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply RiemannInt_P19.
left; assumption.
intros; replace (f x2 + -1 * fct_cte (f b) x2) with (f x2 - f b).
unfold fct_cte in |- *; case (Req_dec b x2); intro.
rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
left; assumption.
elim H9; intros; left; apply H18.
repeat split.
assumption.
rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; rewrite Rabs_right.
apply Rplus_lt_reg_r with (x2 - x1);
replace (x2 - x1 + (b - x2)) with (b - x1); [ idtac | ring ].
replace (x2 - x1 + x1) with x2; [ idtac | ring ].
apply Rlt_le_trans with (b + h0).
2: elim H15; intros; left; assumption.
unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel;
rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0).
rewrite <- Rabs_Ropp; apply RRle_abs.
apply Rlt_le_trans with del;
[ assumption
| unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a));
[ apply Rmin_r | apply Rmin_l ] ].
apply Rle_ge; left; apply Rlt_Rminus; elim H15; intros; assumption.
unfold fct_cte in |- *; ring.
rewrite RiemannInt_P15.
rewrite Rmult_assoc; replace ((b - (b + h0)) * Rabs (/ h0)) with 1.
rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
[ prove_sup0
| rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
rewrite Rabs_left.
apply Rmult_eq_reg_l with h0;
[ do 2 rewrite (Rmult_comm h0); rewrite Rmult_assoc;
rewrite Ropp_mult_distr_l_reverse; rewrite <- Rinv_l_sym;
[ ring | assumption ]
| assumption ].
apply Rinv_lt_0_compat; assumption.
rewrite
(RiemannInt_P13 H13 (RiemannInt_P14 (b + h0) b (f b))
(RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b))))
; ring.
unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring.
rewrite RiemannInt_P15.
rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_eq_reg_l with h0;
[ repeat rewrite (Rmult_comm h0); unfold Rdiv in |- *;
repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym;
[ ring | assumption ]
| assumption ].
cut (a <= b + h0).
cut (b + h0 <= b).
intros; unfold primitive in |- *; case (Rle_dec a (b + h0));
case (Rle_dec (b + h0) b); case (Rle_dec a b); case (Rle_dec b b);
intros; try (elim n; right; reflexivity) || (elim n; left; assumption).
rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r3 r2) H13 (FTC_P1 h C0 r1 r0)); ring.
elim n; assumption.
left; assumption.
apply Rplus_le_reg_l with (- a - h0).
replace (- a - h0 + a) with (- h0); [ idtac | ring ].
replace (- a - h0 + (b + h0)) with (b - a); [ idtac | ring ].
apply Rle_trans with del.
apply Rle_trans with (Rabs h0).
rewrite <- Rabs_Ropp; apply RRle_abs.
left; assumption.
unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
cut (primitive h (FTC_P1 h C0) b = f_b b).
intro; cut (primitive h (FTC_P1 h C0) (b + h0) = f_b (b + h0)).
intro; rewrite H13; rewrite H14; apply H6.
assumption.
apply Rlt_le_trans with del;
[ assumption | unfold del in |- *; apply Rmin_l ].
assert (H14 : b < b + h0).
pattern b at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
assert (H14 := Rge_le _ _ r); elim H14; intro.
assumption.
elim H11; symmetry in |- *; assumption.
unfold primitive in |- *; case (Rle_dec a (b + h0));
case (Rle_dec (b + h0) b); intros;
[ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H14))
| unfold f_b in |- *; reflexivity
| elim n; left; apply Rlt_trans with b; assumption
| elim n0; left; apply Rlt_trans with b; assumption ].
unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
rewrite Rmult_0_r; rewrite Rplus_0_l; unfold primitive in |- *;
case (Rle_dec a b); case (Rle_dec b b); intros;
[ apply RiemannInt_P5
| elim n; right; reflexivity
| elim n; left; assumption
| elim n; right; reflexivity ].
set (f_a := fun x:R => f a * (x - a)); rewrite <- H2;
assert (H3 : derivable_pt_lim f_a a (f a)).
unfold f_a in |- *;
change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a))
in |- *; pattern (f a) at 2 in |- *;
replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_const.
replace 1 with (1 - 0); [ idtac | ring ].
apply derivable_pt_lim_minus.
apply derivable_pt_lim_id.
apply derivable_pt_lim_const.
unfold fct_cte in |- *; ring.
unfold derivable_pt_lim in |- *; intros; elim (H3 _ H4); intros.
assert (H6 : continuity_pt f a).
apply C0; split; [ right; reflexivity | left; assumption ].
assert (H7 : 0 < eps / 2).
unfold Rdiv in |- *; apply Rmult_lt_0_compat;
[ assumption | apply Rinv_0_lt_compat; prove_sup0 ].
elim (H6 _ H7); unfold D_x, no_cond, dist in |- *; simpl in |- *;
unfold R_dist in |- *; intros.
set (del := Rmin x0 (Rmin x1 (b - a))).
assert (H9 : 0 < del).
unfold del in |- *; unfold Rmin in |- *.
case (Rle_dec x1 (b - a)); intros.
case (Rle_dec x0 x1); intro.
apply (cond_pos x0).
elim H8; intros; assumption.
case (Rle_dec x0 (b - a)); intro.
apply (cond_pos x0).
apply Rlt_Rminus; assumption.
split with (mkposreal _ H9).
intros; case (Rcase_abs h0); intro.
assert (H12 : a + h0 < a).
pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
assumption.
unfold primitive in |- *.
case (Rle_dec a (a + h0)); case (Rle_dec (a + h0) b); case (Rle_dec a a);
case (Rle_dec a b); intros;
try (elim n; left; assumption) || (elim n; right; reflexivity).
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H12)).
elim n; left; apply Rlt_trans with a; assumption.
rewrite RiemannInt_P9; replace 0 with (f_a a).
replace (f a * (a + h0 - a)) with (f_a (a + h0)).
apply H5; try assumption.
apply Rlt_le_trans with del;
[ assumption | unfold del in |- *; apply Rmin_l ].
unfold f_a in |- *; ring.
unfold f_a in |- *; ring.
elim n; left; apply Rlt_trans with a; assumption.
assert (H12 : a < a + h0).
pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
assert (H12 := Rge_le _ _ r); elim H12; intro.
assumption.
elim H10; symmetry in |- *; assumption.
assert (H13 : Riemann_integrable f a (a + h0)).
apply continuity_implies_RiemannInt.
left; assumption.
intros; apply C0; elim H13; intros; split; try assumption.
apply Rle_trans with (a + h0); try assumption.
apply Rplus_le_reg_l with (- b - h0).
replace (- b - h0 + b) with (- h0); [ idtac | ring ].
replace (- b - h0 + (a + h0)) with (a - b); [ idtac | ring ].
apply Ropp_le_cancel; rewrite Ropp_involutive; rewrite Ropp_minus_distr;
apply Rle_trans with del.
apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ].
unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r.
replace (primitive h (FTC_P1 h C0) (a + h0) - primitive h (FTC_P1 h C0) a)
with (RiemannInt H13).
replace (f a) with (RiemannInt (RiemannInt_P14 a (a + h0) (f a)) / h0).
replace
(RiemannInt H13 / h0 - RiemannInt (RiemannInt_P14 a (a + h0) (f a)) / h0)
with ((RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) / h0).
replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) with
(RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))).
unfold Rdiv in |- *; rewrite Rabs_mult;
apply Rle_lt_trans with
(RiemannInt
(RiemannInt_P16
(RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))) *
Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply
(RiemannInt_P17 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))
(RiemannInt_P16
(RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))));
left; assumption.
apply Rle_lt_trans with
(RiemannInt (RiemannInt_P14 a (a + h0) (eps / 2)) * Rabs (/ h0)).
do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l.
apply Rabs_pos.
apply RiemannInt_P19.
left; assumption.
intros; replace (f x2 + -1 * fct_cte (f a) x2) with (f x2 - f a).
unfold fct_cte in |- *; case (Req_dec a x2); intro.
rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0;
left; assumption.
elim H8; intros; left; apply H17; repeat split.
assumption.
rewrite Rabs_right.
apply Rplus_lt_reg_r with a; replace (a + (x2 - a)) with x2; [ idtac | ring ].
apply Rlt_le_trans with (a + h0).
elim H14; intros; assumption.
apply Rplus_le_compat_l; left; apply Rle_lt_trans with (Rabs h0).
apply RRle_abs.
apply Rlt_le_trans with del;
[ assumption
| unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a));
[ apply Rmin_r | apply Rmin_l ] ].
apply Rle_ge; left; apply Rlt_Rminus; elim H14; intros; assumption.
unfold fct_cte in |- *; ring.
rewrite RiemannInt_P15.
rewrite Rmult_assoc; replace ((a + h0 - a) * Rabs (/ h0)) with 1.
rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2;
[ prove_sup0
| rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc;
rewrite <- Rinv_r_sym;
[ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r;
rewrite double; apply Rplus_lt_compat_l; assumption
| discrR ] ].
rewrite Rabs_right.
rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym;
[ reflexivity | assumption ].
apply Rle_ge; left; apply Rinv_0_lt_compat; assert (H14 := Rge_le _ _ r);
elim H14; intro.
assumption.
elim H10; symmetry in |- *; assumption.
rewrite
(RiemannInt_P13 H13 (RiemannInt_P14 a (a + h0) (f a))
(RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a))))
; ring.
unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring.
rewrite RiemannInt_P15.
rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc;
rewrite Rplus_opp_r; rewrite Rplus_0_r; unfold Rdiv in |- *;
rewrite Rmult_assoc; rewrite <- Rinv_r_sym; [ ring | assumption ].
cut (a <= a + h0).
cut (a + h0 <= b).
intros; unfold primitive in |- *; case (Rle_dec a (a + h0));
case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b);
intros; try (elim n; right; reflexivity) || (elim n; left; assumption).
rewrite RiemannInt_P9; unfold Rminus in |- *; rewrite Ropp_0;
rewrite Rplus_0_r; apply RiemannInt_P5.
elim n; assumption.
elim n; assumption.
2: left; assumption.
apply Rplus_le_reg_l with (- a); replace (- a + (a + h0)) with h0;
[ idtac | ring ].
rewrite Rplus_comm; apply Rle_trans with del;
[ apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ]
| unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r ].
assert (H1 : x = a).
rewrite <- H0 in H; elim H; intros; apply Rle_antisym; assumption.
set (f_a := fun x:R => f a * (x - a)).
assert (H2 : derivable_pt_lim f_a a (f a)).
unfold f_a in |- *;
change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a))
in |- *; pattern (f a) at 2 in |- *;
replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_const.
replace 1 with (1 - 0); [ idtac | ring ].
apply derivable_pt_lim_minus.
apply derivable_pt_lim_id.
apply derivable_pt_lim_const.
unfold fct_cte in |- *; ring.
set
(f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b))).
assert (H3 : derivable_pt_lim f_b b (f b)).
unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0).
change
(derivable_pt_lim
((fct_cte (f b) * (id - fct_cte b))%F +
fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b (
f b + 0)) in |- *.
apply derivable_pt_lim_plus.
pattern (f b) at 2 in |- *;
replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1).
apply derivable_pt_lim_mult.
apply derivable_pt_lim_const.
replace 1 with (1 - 0); [ idtac | ring ].
apply derivable_pt_lim_minus.
apply derivable_pt_lim_id.
apply derivable_pt_lim_const.
unfold fct_cte in |- *; ring.
apply derivable_pt_lim_const.
ring.
unfold derivable_pt_lim in |- *; intros; elim (H2 _ H4); intros;
elim (H3 _ H4); intros; set (del := Rmin x0 x1).
assert (H7 : 0 < del).
unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x0 x1); intro.
apply (cond_pos x0).
apply (cond_pos x1).
split with (mkposreal _ H7); intros; case (Rcase_abs h0); intro.
assert (H10 : a + h0 < a).
pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l;
assumption.
rewrite H1; unfold primitive in |- *; case (Rle_dec a (a + h0));
case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b);
intros; try (elim n; right; assumption || reflexivity).
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H10)).
elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r2 H10)).
rewrite RiemannInt_P9; replace 0 with (f_a a).
replace (f a * (a + h0 - a)) with (f_a (a + h0)).
apply H5; try assumption.
apply Rlt_le_trans with del; try assumption.
unfold del in |- *; apply Rmin_l.
unfold f_a in |- *; ring.
unfold f_a in |- *; ring.
elim n; rewrite <- H0; left; assumption.
assert (H10 : a < a + h0).
pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l.
assert (H10 := Rge_le _ _ r); elim H10; intro.
assumption.
elim H8; symmetry in |- *; assumption.
rewrite H0 in H1; rewrite H1; unfold primitive in |- *;
case (Rle_dec a (b + h0)); case (Rle_dec (b + h0) b);
case (Rle_dec a b); case (Rle_dec b b); intros;
try (elim n; right; assumption || reflexivity).
rewrite H0 in H10; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r2 H10)).
repeat rewrite RiemannInt_P9.
replace (RiemannInt (FTC_P1 h C0 r1 r0)) with (f_b b).
fold (f_b (b + h0)) in |- *.
apply H6; try assumption.
apply Rlt_le_trans with del; try assumption.
unfold del in |- *; apply Rmin_r.
unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r;
rewrite Rmult_0_r; rewrite Rplus_0_l; apply RiemannInt_P5.
elim n; rewrite <- H0; left; assumption.
elim n0; rewrite <- H0; left; assumption.
Qed.
Lemma RiemannInt_P29 :
forall (f:R -> R) a b (h:a <= b)
(C0:forall x:R, a <= x <= b -> continuity_pt f x),
antiderivative f (primitive h (FTC_P1 h C0)) a b.
Proof.
intro f; intros; unfold antiderivative in |- *; split; try assumption; intros;
assert (H0 := RiemannInt_P28 h C0 H);
assert (H1 : derivable_pt (primitive h (FTC_P1 h C0)) x);
[ unfold derivable_pt in |- *; split with (f x); apply H0
| split with H1; symmetry in |- *; apply derive_pt_eq_0; apply H0 ].
Qed.
Lemma RiemannInt_P30 :
forall (f:R -> R) (a b:R),
a <= b ->
(forall x:R, a <= x <= b -> continuity_pt f x) ->
sigT (fun g:R -> R => antiderivative f g a b).
Proof.
intros; split with (primitive H (FTC_P1 H H0)); apply RiemannInt_P29.
Qed.
Record C1_fun : Type := mkC1
{c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}.
Lemma RiemannInt_P31 :
forall (f:C1_fun) (a b:R),
a <= b -> antiderivative (derive f (diff0 f)) f a b.
Proof.
intro f; intros; unfold antiderivative in |- *; split; try assumption; intros;
split with (diff0 f x); reflexivity.
Qed.
Lemma RiemannInt_P32 :
forall (f:C1_fun) (a b:R), Riemann_integrable (derive f (diff0 f)) a b.
Proof.
intro f; intros; case (Rle_dec a b); intro;
[ apply continuity_implies_RiemannInt; try assumption; intros;
apply (cont1 f)
| assert (H : b <= a);
[ auto with real
| apply RiemannInt_P1; apply continuity_implies_RiemannInt;
try assumption; intros; apply (cont1 f) ] ].
Qed.
Lemma RiemannInt_P33 :
forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b),
a <= b -> RiemannInt pr = f b - f a.
Proof.
intro f; intros;
assert
(H0 : forall x:R, a <= x <= b -> continuity_pt (derive f (diff0 f)) x).
intros; apply (cont1 f).
rewrite (RiemannInt_P20 H (FTC_P1 H H0) pr);
assert (H1 := RiemannInt_P29 H H0); assert (H2 := RiemannInt_P31 f H);
elim (antiderivative_Ucte (derive f (diff0 f)) _ _ _ _ H1 H2);
intros C H3; repeat rewrite H3;
[ ring
| split; [ right; reflexivity | assumption ]
| split; [ assumption | right; reflexivity ] ].
Qed.
Lemma FTC_Riemann :
forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b),
RiemannInt pr = f b - f a.
Proof.
intro f; intros; case (Rle_dec a b); intro;
[ apply RiemannInt_P33; assumption
| assert (H : b <= a);
[ auto with real
| assert (H0 := RiemannInt_P1 pr); rewrite (RiemannInt_P8 pr H0);
rewrite (RiemannInt_P33 _ H0 H); ring ] ].
Qed.