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Diffstat (limited to 'theories/Reals/RiemannInt.v')
| -rw-r--r-- | theories/Reals/RiemannInt.v | 3263 |
1 files changed, 3263 insertions, 0 deletions
diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v new file mode 100644 index 00000000..51323ac4 --- /dev/null +++ b/theories/Reals/RiemannInt.v @@ -0,0 +1,3263 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: RiemannInt.v,v 1.18.2.1 2004/07/16 19:31:13 herbelin Exp $ i*) + +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). +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. +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). +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). +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). +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. +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). +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. +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. +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). +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)). +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). +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. +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. +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. +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. +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)). +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. +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). +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. +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)))). +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. +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. +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. +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. +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. +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}. +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. + +(* L1([a,b]) is a vectorial space *) +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. +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. +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. +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. +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. +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). +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. +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. +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. +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); + 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. +fold phi1 in u0; + 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. +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. +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. +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. +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. +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. +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. +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. +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. +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); + fold phi1 in H8; set (phi2 := phi_sequence RinvN pr2 n); + fold phi2 in H3; set (phi3 := phi_sequence RinvN pr3 n); + fold phi2 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. +fold phi3 in H1; 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. +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). +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). +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. +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). +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. +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. +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. +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. +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. |