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Diffstat (limited to 'theories/Reals/PartSum.v')
| -rw-r--r-- | theories/Reals/PartSum.v | 603 |
1 files changed, 603 insertions, 0 deletions
diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v new file mode 100644 index 00000000..13070bde --- /dev/null +++ b/theories/Reals/PartSum.v @@ -0,0 +1,603 @@ +(************************************************************************) +(* 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: PartSum.v,v 1.11.2.1 2004/07/16 19:31:11 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import Rcomplete. +Require Import Max. +Open Local Scope R_scope. + +Lemma tech1 : + forall (An:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> 0 < An n) -> 0 < sum_f_R0 An N. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H; apply le_n. +simpl in |- *; apply Rplus_lt_0_compat. +apply HrecN; intros; apply H; apply le_S; assumption. +apply H; apply le_n. +Qed. + +(* Chasles' relation *) +Lemma tech2 : + forall (An:nat -> R) (m n:nat), + (m < n)%nat -> + sum_f_R0 An n = + sum_f_R0 An m + sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m). +intros; induction n as [| n Hrecn]. +elim (lt_n_O _ H). +cut ((m < n)%nat \/ m = n). +intro; elim H0; intro. +replace (sum_f_R0 An (S n)) with (sum_f_R0 An n + An (S n)); + [ idtac | reflexivity ]. +replace (S n - S m)%nat with (S (n - S m)). +replace (sum_f_R0 (fun i:nat => An (S m + i)%nat) (S (n - S m))) with + (sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m) + + An (S m + S (n - S m))%nat); [ idtac | reflexivity ]. +replace (S m + S (n - S m))%nat with (S n). +rewrite (Hrecn H1). +ring. +apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite S_INR; + rewrite minus_INR. +rewrite S_INR; ring. +apply lt_le_S; assumption. +apply INR_eq; rewrite S_INR; repeat rewrite minus_INR. +repeat rewrite S_INR; ring. +apply le_n_S; apply lt_le_weak; assumption. +apply lt_le_S; assumption. +rewrite H1; rewrite <- minus_n_n; simpl in |- *. +replace (n + 0)%nat with n; [ reflexivity | ring ]. +inversion H. +right; reflexivity. +left; apply lt_le_trans with (S m); [ apply lt_n_Sn | assumption ]. +Qed. + +(* Sum of geometric sequences *) +Lemma tech3 : + forall (k:R) (N:nat), + k <> 1 -> sum_f_R0 (fun i:nat => k ^ i) N = (1 - k ^ S N) / (1 - k). +intros; cut (1 - k <> 0). +intro; induction N as [| N HrecN]. +simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym. +reflexivity. +apply H0. +replace (sum_f_R0 (fun i:nat => k ^ i) (S N)) with + (sum_f_R0 (fun i:nat => k ^ i) N + k ^ S N); [ idtac | reflexivity ]; + rewrite HrecN; + replace ((1 - k ^ S N) / (1 - k) + k ^ S N) with + ((1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k)). +apply Rmult_eq_reg_l with (1 - k). +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ (1 - k))); + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ do 2 rewrite Rmult_1_l; simpl in |- *; ring | apply H0 ]. +apply H0. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k)); + repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply H0. +apply Rminus_eq_contra; red in |- *; intro; elim H; symmetry in |- *; + assumption. +Qed. + +Lemma tech4 : + forall (An:nat -> R) (k:R) (N:nat), + 0 <= k -> (forall i:nat, An (S i) < k * An i) -> An N <= An 0%nat * k ^ N. +intros; induction N as [| N HrecN]. +simpl in |- *; right; ring. +apply Rle_trans with (k * An N). +left; apply (H0 N). +replace (S N) with (N + 1)%nat; [ idtac | ring ]. +rewrite pow_add; simpl in |- *; rewrite Rmult_1_r; + replace (An 0%nat * (k ^ N * k)) with (k * (An 0%nat * k ^ N)); + [ idtac | ring ]; apply Rmult_le_compat_l. +assumption. +apply HrecN. +Qed. + +Lemma tech5 : + forall (An:nat -> R) (N:nat), sum_f_R0 An (S N) = sum_f_R0 An N + An (S N). +intros; reflexivity. +Qed. + +Lemma tech6 : + forall (An:nat -> R) (k:R) (N:nat), + 0 <= k -> + (forall i:nat, An (S i) < k * An i) -> + sum_f_R0 An N <= An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N. +intros; induction N as [| N HrecN]. +simpl in |- *; right; ring. +apply Rle_trans with (An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N + An (S N)). +rewrite tech5; do 2 rewrite <- (Rplus_comm (An (S N))); + apply Rplus_le_compat_l. +apply HrecN. +rewrite tech5; rewrite Rmult_plus_distr_l; apply Rplus_le_compat_l. +apply tech4; assumption. +Qed. + +Lemma tech7 : forall r1 r2:R, r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2. +intros; red in |- *; intro. +assert (H3 := Rmult_eq_compat_l r1 _ _ H2). +rewrite <- Rinv_r_sym in H3; [ idtac | assumption ]. +assert (H4 := Rmult_eq_compat_l r2 _ _ H3). +rewrite Rmult_1_r in H4; rewrite <- Rmult_assoc in H4. +rewrite Rinv_r_simpl_m in H4; [ idtac | assumption ]. +elim H1; symmetry in |- *; assumption. +Qed. + +Lemma tech11 : + forall (An Bn Cn:nat -> R) (N:nat), + (forall i:nat, An i = Bn i - Cn i) -> + sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H. +do 3 rewrite tech5; rewrite HrecN; rewrite (H (S N)); ring. +Qed. + +Lemma tech12 : + forall (An:nat -> R) (x l:R), + Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l -> + Pser An x l. +intros; unfold Pser in |- *; unfold infinit_sum in |- *; unfold Un_cv in H; + assumption. +Qed. + +Lemma scal_sum : + forall (An:nat -> R) (N:nat) (x:R), + x * sum_f_R0 An N = sum_f_R0 (fun i:nat => An i * x) N. +intros; induction N as [| N HrecN]. +simpl in |- *; ring. +do 2 rewrite tech5. +rewrite Rmult_plus_distr_l; rewrite <- HrecN; ring. +Qed. + +Lemma decomp_sum : + forall (An:nat -> R) (N:nat), + (0 < N)%nat -> + sum_f_R0 An N = An 0%nat + sum_f_R0 (fun i:nat => An (S i)) (pred N). +intros; induction N as [| N HrecN]. +elim (lt_irrefl _ H). +cut ((0 < N)%nat \/ N = 0%nat). +intro; elim H0; intro. +cut (S (pred N) = pred (S N)). +intro; rewrite <- H2. +do 2 rewrite tech5. +replace (S (S (pred N))) with (S N). +rewrite (HrecN H1); ring. +rewrite H2; simpl in |- *; reflexivity. +assert (H2 := O_or_S N). +elim H2; intros. +elim a; intros. +rewrite <- p. +simpl in |- *; reflexivity. +rewrite <- b in H1; elim (lt_irrefl _ H1). +rewrite H1; simpl in |- *; reflexivity. +inversion H. +right; reflexivity. +left; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. +Qed. + +Lemma plus_sum : + forall (An Bn:nat -> R) (N:nat), + sum_f_R0 (fun i:nat => An i + Bn i) N = sum_f_R0 An N + sum_f_R0 Bn N. +intros; induction N as [| N HrecN]. +simpl in |- *; ring. +do 3 rewrite tech5; rewrite HrecN; ring. +Qed. + +Lemma sum_eq : + forall (An Bn:nat -> R) (N:nat), + (forall i:nat, (i <= N)%nat -> An i = Bn i) -> + sum_f_R0 An N = sum_f_R0 Bn N. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H; apply le_n. +do 2 rewrite tech5; rewrite HrecN. +rewrite (H (S N)); [ reflexivity | apply le_n ]. +intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ]. +Qed. + +(* Unicity of the limit defined by convergent series *) +Lemma uniqueness_sum : + forall (An:nat -> R) (l1 l2:R), + infinit_sum An l1 -> infinit_sum An l2 -> l1 = l2. +unfold infinit_sum in |- *; intros. +case (Req_dec l1 l2); intro. +assumption. +cut (0 < Rabs ((l1 - l2) / 2)); [ intro | apply Rabs_pos_lt ]. +elim (H (Rabs ((l1 - l2) / 2)) H2); intros. +elim (H0 (Rabs ((l1 - l2) / 2)) H2); intros. +set (N := max x0 x); cut (N >= x0)%nat. +cut (N >= x)%nat. +intros; assert (H7 := H3 N H5); assert (H8 := H4 N H6). +cut (Rabs (l1 - l2) <= R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2). +intro; assert (H10 := Rplus_lt_compat _ _ _ _ H7 H8); + assert (H11 := Rle_lt_trans _ _ _ H9 H10); unfold Rdiv in H11; + rewrite Rabs_mult in H11. +cut (Rabs (/ 2) = / 2). +intro; rewrite H12 in H11; assert (H13 := double_var); unfold Rdiv in H13; + rewrite <- H13 in H11. +elim (Rlt_irrefl _ H11). +apply Rabs_right; left; change (0 < / 2) in |- *; apply Rinv_0_lt_compat; + cut (0%nat <> 2%nat); + [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR in |- *; + intro; assumption + | discriminate ]. +unfold R_dist in |- *; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1)); + rewrite Ropp_minus_distr'. +replace (l1 - l2) with (l1 - sum_f_R0 An N + (sum_f_R0 An N - l2)); + [ idtac | ring ]. +apply Rabs_triang. +unfold ge in |- *; unfold N in |- *; apply le_max_r. +unfold ge in |- *; unfold N in |- *; apply le_max_l. +unfold Rdiv in |- *; apply prod_neq_R0. +apply Rminus_eq_contra; assumption. +apply Rinv_neq_0_compat; discrR. +Qed. + +Lemma minus_sum : + forall (An Bn:nat -> R) (N:nat), + sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N. +intros; induction N as [| N HrecN]. +simpl in |- *; ring. +do 3 rewrite tech5; rewrite HrecN; ring. +Qed. + +Lemma sum_decomposition : + forall (An:nat -> R) (N:nat), + sum_f_R0 (fun l:nat => An (2 * l)%nat) (S N) + + sum_f_R0 (fun l:nat => An (S (2 * l))) N = sum_f_R0 An (2 * S N). +intros. +induction N as [| N HrecN]. +simpl in |- *; ring. +rewrite tech5. +rewrite (tech5 (fun l:nat => An (S (2 * l))) N). +replace (2 * S (S N))%nat with (S (S (2 * S N))). +rewrite (tech5 An (S (2 * S N))). +rewrite (tech5 An (2 * S N)). +rewrite <- HrecN. +ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR. +ring. +Qed. + +Lemma sum_Rle : + forall (An Bn:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> An n <= Bn n) -> + sum_f_R0 An N <= sum_f_R0 Bn N. +intros. +induction N as [| N HrecN]. +simpl in |- *; apply H. +apply le_n. +do 2 rewrite tech5. +apply Rle_trans with (sum_f_R0 An N + Bn (S N)). +apply Rplus_le_compat_l. +apply H. +apply le_n. +do 2 rewrite <- (Rplus_comm (Bn (S N))). +apply Rplus_le_compat_l. +apply HrecN. +intros; apply H. +apply le_trans with N; [ assumption | apply le_n_Sn ]. +Qed. + +Lemma Rsum_abs : + forall (An:nat -> R) (N:nat), + Rabs (sum_f_R0 An N) <= sum_f_R0 (fun l:nat => Rabs (An l)) N. +intros. +induction N as [| N HrecN]. +simpl in |- *. +right; reflexivity. +do 2 rewrite tech5. +apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))). +apply Rabs_triang. +do 2 rewrite <- (Rplus_comm (Rabs (An (S N)))). +apply Rplus_le_compat_l. +apply HrecN. +Qed. + +Lemma sum_cte : + forall (x:R) (N:nat), sum_f_R0 (fun _:nat => x) N = x * INR (S N). +intros. +induction N as [| N HrecN]. +simpl in |- *; ring. +rewrite tech5. +rewrite HrecN; repeat rewrite S_INR; ring. +Qed. + +(**********) +Lemma sum_growing : + forall (An Bn:nat -> R) (N:nat), + (forall n:nat, An n <= Bn n) -> sum_f_R0 An N <= sum_f_R0 Bn N. +intros. +induction N as [| N HrecN]. +simpl in |- *; apply H. +do 2 rewrite tech5. +apply Rle_trans with (sum_f_R0 An N + Bn (S N)). +apply Rplus_le_compat_l; apply H. +do 2 rewrite <- (Rplus_comm (Bn (S N))). +apply Rplus_le_compat_l; apply HrecN. +Qed. + +(**********) +Lemma Rabs_triang_gen : + forall (An:nat -> R) (N:nat), + Rabs (sum_f_R0 An N) <= sum_f_R0 (fun i:nat => Rabs (An i)) N. +intros. +induction N as [| N HrecN]. +simpl in |- *. +right; reflexivity. +do 2 rewrite tech5. +apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))). +apply Rabs_triang. +do 2 rewrite <- (Rplus_comm (Rabs (An (S N)))). +apply Rplus_le_compat_l; apply HrecN. +Qed. + +(**********) +Lemma cond_pos_sum : + forall (An:nat -> R) (N:nat), + (forall n:nat, 0 <= An n) -> 0 <= sum_f_R0 An N. +intros. +induction N as [| N HrecN]. +simpl in |- *; apply H. +rewrite tech5. +apply Rplus_le_le_0_compat. +apply HrecN. +apply H. +Qed. + +(* Cauchy's criterion for series *) +Definition Cauchy_crit_series (An:nat -> R) : Prop := + Cauchy_crit (fun N:nat => sum_f_R0 An N). + +(* If (|An|) satisfies the Cauchy's criterion for series, then (An) too *) +Lemma cauchy_abs : + forall An:nat -> R, + Cauchy_crit_series (fun i:nat => Rabs (An i)) -> Cauchy_crit_series An. +unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. +intros. +elim (H eps H0); intros. +exists x. +intros. +cut + (R_dist (sum_f_R0 An n) (sum_f_R0 An m) <= + R_dist (sum_f_R0 (fun i:nat => Rabs (An i)) n) + (sum_f_R0 (fun i:nat => Rabs (An i)) m)). +intro. +apply Rle_lt_trans with + (R_dist (sum_f_R0 (fun i:nat => Rabs (An i)) n) + (sum_f_R0 (fun i:nat => Rabs (An i)) m)). +assumption. +apply H1; assumption. +assert (H4 := lt_eq_lt_dec n m). +elim H4; intro. +elim a; intro. +rewrite (tech2 An n m); [ idtac | assumption ]. +rewrite (tech2 (fun i:nat => Rabs (An i)) n m); [ idtac | assumption ]. +unfold R_dist in |- *. +unfold Rminus in |- *. +do 2 rewrite Ropp_plus_distr. +do 2 rewrite <- Rplus_assoc. +do 2 rewrite Rplus_opp_r. +do 2 rewrite Rplus_0_l. +do 2 rewrite Rabs_Ropp. +rewrite + (Rabs_right (sum_f_R0 (fun i:nat => Rabs (An (S n + i)%nat)) (m - S n))) + . +set (Bn := fun i:nat => An (S n + i)%nat). +replace (fun i:nat => Rabs (An (S n + i)%nat)) with + (fun i:nat => Rabs (Bn i)). +apply Rabs_triang_gen. +unfold Bn in |- *; reflexivity. +apply Rle_ge. +apply cond_pos_sum. +intro; apply Rabs_pos. +rewrite b. +unfold R_dist in |- *. +unfold Rminus in |- *; do 2 rewrite Rplus_opp_r. +rewrite Rabs_R0; right; reflexivity. +rewrite (tech2 An m n); [ idtac | assumption ]. +rewrite (tech2 (fun i:nat => Rabs (An i)) m n); [ idtac | assumption ]. +unfold R_dist in |- *. +unfold Rminus in |- *. +do 2 rewrite Rplus_assoc. +rewrite (Rplus_comm (sum_f_R0 An m)). +rewrite (Rplus_comm (sum_f_R0 (fun i:nat => Rabs (An i)) m)). +do 2 rewrite Rplus_assoc. +do 2 rewrite Rplus_opp_l. +do 2 rewrite Rplus_0_r. +rewrite + (Rabs_right (sum_f_R0 (fun i:nat => Rabs (An (S m + i)%nat)) (n - S m))) + . +set (Bn := fun i:nat => An (S m + i)%nat). +replace (fun i:nat => Rabs (An (S m + i)%nat)) with + (fun i:nat => Rabs (Bn i)). +apply Rabs_triang_gen. +unfold Bn in |- *; reflexivity. +apply Rle_ge. +apply cond_pos_sum. +intro; apply Rabs_pos. +Qed. + +(**********) +Lemma cv_cauchy_1 : + forall An:nat -> R, + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l) -> + Cauchy_crit_series An. +intros. +elim X; intros. +unfold Un_cv in p. +unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. +intros. +cut (0 < eps / 2). +intro. +elim (p (eps / 2) H0); intros. +exists x0. +intros. +apply Rle_lt_trans with (R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x). +unfold R_dist in |- *. +replace (sum_f_R0 An n - sum_f_R0 An m) with + (sum_f_R0 An n - x + - (sum_f_R0 An m - x)); [ idtac | ring ]. +rewrite <- (Rabs_Ropp (sum_f_R0 An m - x)). +apply Rabs_triang. +apply Rlt_le_trans with (eps / 2 + eps / 2). +apply Rplus_lt_compat. +apply H1; assumption. +apply H1; assumption. +right; symmetry in |- *; apply double_var. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +Qed. + +Lemma cv_cauchy_2 : + forall An:nat -> R, + Cauchy_crit_series An -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros. +apply R_complete. +unfold Cauchy_crit_series in H. +exact H. +Qed. + +(**********) +Lemma sum_eq_R0 : + forall (An:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> An n = 0) -> sum_f_R0 An N = 0. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H; apply le_n. +rewrite tech5; rewrite HrecN; + [ rewrite Rplus_0_l; apply H; apply le_n + | intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ] ]. +Qed. + +Definition SP (fn:nat -> R -> R) (N:nat) (x:R) : R := + sum_f_R0 (fun k:nat => fn k x) N. + +(**********) +Lemma sum_incr : + forall (An:nat -> R) (N:nat) (l:R), + Un_cv (fun n:nat => sum_f_R0 An n) l -> + (forall n:nat, 0 <= An n) -> sum_f_R0 An N <= l. +intros; case (total_order_T (sum_f_R0 An N) l); intro. +elim s; intro. +left; apply a. +right; apply b. +cut (Un_growing (fun n:nat => sum_f_R0 An n)). +intro; set (l1 := sum_f_R0 An N). +fold l1 in r. +unfold Un_cv in H; cut (0 < l1 - l). +intro; elim (H _ H2); intros. +set (N0 := max x N); cut (N0 >= x)%nat. +intro; assert (H5 := H3 N0 H4). +cut (l1 <= sum_f_R0 An N0). +intro; unfold R_dist in H5; rewrite Rabs_right in H5. +cut (sum_f_R0 An N0 < l1). +intro; elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H7 H6)). +apply Rplus_lt_reg_r with (- l). +do 2 rewrite (Rplus_comm (- l)). +apply H5. +apply Rle_ge; apply Rplus_le_reg_l with l. +rewrite Rplus_0_r; replace (l + (sum_f_R0 An N0 - l)) with (sum_f_R0 An N0); + [ idtac | ring ]; apply Rle_trans with l1. +left; apply r. +apply H6. +unfold l1 in |- *; apply Rge_le; + apply (growing_prop (fun k:nat => sum_f_R0 An k)). +apply H1. +unfold ge, N0 in |- *; apply le_max_r. +unfold ge, N0 in |- *; apply le_max_l. +apply Rplus_lt_reg_r with l; rewrite Rplus_0_r; + replace (l + (l1 - l)) with l1; [ apply r | ring ]. +unfold Un_growing in |- *; intro; simpl in |- *; + pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; apply H0. +Qed. + +(**********) +Lemma sum_cv_maj : + forall (An:nat -> R) (fn:nat -> R -> R) (x l1 l2:R), + Un_cv (fun n:nat => SP fn n x) l1 -> + Un_cv (fun n:nat => sum_f_R0 An n) l2 -> + (forall n:nat, Rabs (fn n x) <= An n) -> Rabs l1 <= l2. +intros; case (total_order_T (Rabs l1) l2); intro. +elim s; intro. +left; apply a. +right; apply b. +cut (forall n0:nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0). +intro; cut (0 < (Rabs l1 - l2) / 2). +intro; unfold Un_cv in H, H0. +elim (H _ H3); intros Na H4. +elim (H0 _ H3); intros Nb H5. +set (N := max Na Nb). +unfold R_dist in H4, H5. +cut (Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2). +intro; cut (Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2). +intro; cut (sum_f_R0 An N < (Rabs l1 + l2) / 2). +intro; cut ((Rabs l1 + l2) / 2 < Rabs (SP fn N x)). +intro; cut (sum_f_R0 An N < Rabs (SP fn N x)). +intro; assert (H11 := H2 N). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H10)). +apply Rlt_trans with ((Rabs l1 + l2) / 2); assumption. +case (Rcase_abs (Rabs l1 - Rabs (SP fn N x))); intro. +apply Rlt_trans with (Rabs l1). +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite double; apply Rplus_lt_compat_l; apply r. +discrR. +apply (Rminus_lt _ _ r0). +rewrite (Rabs_right _ r0) in H7. +apply Rplus_lt_reg_r with ((Rabs l1 - l2) / 2 - Rabs (SP fn N x)). +replace ((Rabs l1 - l2) / 2 - Rabs (SP fn N x) + (Rabs l1 + l2) / 2) with + (Rabs l1 - Rabs (SP fn N x)). +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r; apply H7. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + rewrite <- (Rmult_comm (/ 2)); rewrite Rmult_minus_distr_l; + repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1 in |- *; + rewrite double_var; unfold Rdiv in |- *; ring. +case (Rcase_abs (sum_f_R0 An N - l2)); intro. +apply Rlt_trans with l2. +apply (Rminus_lt _ _ r0). +apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (double l2); unfold Rdiv in |- *; rewrite (Rmult_comm 2); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite (Rplus_comm (Rabs l1)); apply Rplus_lt_compat_l; + apply r. +discrR. +rewrite (Rabs_right _ r0) in H6; apply Rplus_lt_reg_r with (- l2). +replace (- l2 + (Rabs l1 + l2) / 2) with ((Rabs l1 - l2) / 2). +rewrite Rplus_comm; apply H6. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite Rmult_minus_distr_l; rewrite Rmult_plus_distr_r; + pattern l2 at 2 in |- *; rewrite double_var; + repeat rewrite (Rmult_comm (/ 2)); rewrite Ropp_plus_distr; + unfold Rdiv in |- *; ring. +apply Rle_lt_trans with (Rabs (SP fn N x - l1)). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply Rabs_triang_inv2. +apply H4; unfold ge, N in |- *; apply le_max_l. +apply H5; unfold ge, N in |- *; apply le_max_r. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with l2. +rewrite Rplus_0_r; replace (l2 + (Rabs l1 - l2)) with (Rabs l1); + [ apply r | ring ]. +apply Rinv_0_lt_compat; prove_sup0. +intros; induction n0 as [| n0 Hrecn0]. +unfold SP in |- *; simpl in |- *; apply H1. +unfold SP in |- *; simpl in |- *. +apply Rle_trans with + (Rabs (sum_f_R0 (fun k:nat => fn k x) n0) + Rabs (fn (S n0) x)). +apply Rabs_triang. +apply Rle_trans with (sum_f_R0 An n0 + Rabs (fn (S n0) x)). +do 2 rewrite <- (Rplus_comm (Rabs (fn (S n0) x))). +apply Rplus_le_compat_l; apply Hrecn0. +apply Rplus_le_compat_l; apply H1. +Qed.
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