diff options
Diffstat (limited to 'theories/Reals/Alembert.v')
| -rw-r--r-- | theories/Reals/Alembert.v | 47 |
1 files changed, 20 insertions, 27 deletions
diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v index a92b3584b..c4416e5d8 100644 --- a/theories/Reals/Alembert.v +++ b/theories/Reals/Alembert.v @@ -35,10 +35,8 @@ Proof. [ intro | apply Rinv_0_lt_compat; prove_sup0 ]. elim (H0 (/ 2) H1); intros. exists (sum_f_R0 An x + 2 * An (S x)). - unfold is_upper_bound; intros; unfold EUn in H3; elim H3; intros. - rewrite H4; assert (H5 := lt_eq_lt_dec x1 x). - elim H5; intros. - elim a; intro. + unfold is_upper_bound; intros; unfold EUn in H3; destruct H3 as (x1,->). + destruct (lt_eq_lt_dec x1 x) as [[| -> ]|]. replace (sum_f_R0 An x) with (sum_f_R0 An x1 + sum_f_R0 (fun i:nat => An (S x1 + i)%nat) (x - S x1)). pattern (sum_f_R0 An x1) at 1; rewrite <- Rplus_0_r; @@ -47,7 +45,7 @@ Proof. apply tech1; intros; apply H. apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. symmetry ; apply tech2; assumption. - rewrite b; pattern (sum_f_R0 An x) at 1; rewrite <- Rplus_0_r; + pattern (sum_f_R0 An x) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. left; apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. replace (sum_f_R0 An x1) with @@ -86,8 +84,8 @@ Proof. apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)). left; apply Rinv_0_lt_compat; prove_sup0. intro; cut (forall n:nat, (n >= x)%nat -> An (S n) < / 2 * An n). - intro; replace (S x + S i)%nat with (S (S x + i)). - apply H6; unfold ge; apply tech8. + intro H4; replace (S x + S i)%nat with (S (S x + i)). + apply H4; unfold ge; apply tech8. apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. intros; unfold R_dist in H2; apply Rmult_lt_reg_l with (/ An n). apply Rinv_0_lt_compat; apply H. @@ -101,17 +99,17 @@ Proof. unfold Rdiv; reflexivity. left; unfold Rdiv; change (0 < An (S n) * / An n); apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply H ]. - red; intro; assert (H8 := H n); rewrite H7 in H8; + intro H5; assert (H8 := H n); rewrite H5 in H8; elim (Rlt_irrefl _ H8). replace (S x + 0)%nat with (S x); [ reflexivity | ring ]. symmetry ; apply tech2; assumption. exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity. - intro X; elim X; intros. + intros (x,H1). exists x; apply Un_cv_crit_lub; [ unfold Un_growing; intro; rewrite tech5; pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply H - | apply p ]. + | apply H1 ]. Defined. Lemma Alembert_C2 : @@ -127,14 +125,12 @@ Proof. intro; cut (forall n:nat, 0 < Wn n). intro; cut (Un_cv (fun n:nat => Rabs (Vn (S n) / Vn n)) 0). intro; cut (Un_cv (fun n:nat => Rabs (Wn (S n) / Wn n)) 0). - intro; assert (H5 := Alembert_C1 Vn H1 H3). - assert (H6 := Alembert_C1 Wn H2 H4). - elim H5; intros. - elim H6; intros. + intro; pose proof (Alembert_C1 Vn H1 H3) as (x,p). + pose proof (Alembert_C1 Wn H2 H4) as (x0,p0). exists (x - x0); unfold Un_cv; unfold Un_cv in p; unfold Un_cv in p0; intros; cut (0 < eps / 2). - intro; elim (p (eps / 2) H8); clear p; intros. - elim (p0 (eps / 2) H8); clear p0; intros. + intro H6; destruct (p (eps / 2) H6) as (x1,H8). clear p. + destruct (p0 (eps / 2) H6) as (x2,H9). clear p0. set (N := max x1 x2). exists N; intros; replace (sum_f_R0 An n) with (sum_f_R0 Vn n - sum_f_R0 Wn n). @@ -146,9 +142,9 @@ Proof. apply Rabs_triang. rewrite Rabs_Ropp; apply Rlt_le_trans with (eps / 2 + eps / 2). apply Rplus_lt_compat. - unfold R_dist in H9; apply H9; unfold ge; apply le_trans with N; + unfold R_dist in H8; apply H8; unfold ge; apply le_trans with N; [ unfold N; apply le_max_l | assumption ]. - unfold R_dist in H10; apply H10; unfold ge; apply le_trans with N; + unfold R_dist in H9; apply H9; unfold ge; apply le_trans with N; [ unfold N; apply le_max_r | assumption ]. right; symmetry ; apply double_var. symmetry ; apply tech11; intro; unfold Vn, Wn; @@ -344,9 +340,8 @@ Proof. intros; set (Bn := fun i:nat => An i * x ^ i). cut (forall n:nat, Bn n <> 0). intro; cut (Un_cv (fun n:nat => Rabs (Bn (S n) / Bn n)) 0). - intro; assert (H4 := Alembert_C2 Bn H2 H3). - elim H4; intros. - exists x0; unfold Bn in p; apply tech12; assumption. + intro; destruct (Alembert_C2 Bn H2 H3) as (x0,H4). + exists x0; unfold Bn in H4; apply tech12; assumption. unfold Un_cv; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x). intro; elim (H1 (eps / Rabs x) H4); intros. exists x0; intros; unfold R_dist; unfold Rminus; @@ -431,9 +426,7 @@ Proof. unfold is_upper_bound; intros; unfold EUn in H6. elim H6; intros. rewrite H7. - assert (H8 := lt_eq_lt_dec x2 x0). - elim H8; intros. - elim a; intro. + destruct (lt_eq_lt_dec x2 x0) as [[| -> ]|]. replace (sum_f_R0 An x0) with (sum_f_R0 An x2 + sum_f_R0 (fun i:nat => An (S x2 + i)%nat) (x0 - S x2)). pattern (sum_f_R0 An x2) at 1; rewrite <- Rplus_0_r. @@ -446,7 +439,7 @@ Proof. replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. apply H. symmetry ; apply tech2; assumption. - rewrite b; pattern (sum_f_R0 An x0) at 1; rewrite <- Rplus_0_r; + pattern (sum_f_R0 An x0) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l. left; apply Rmult_lt_0_compat. apply Rinv_0_lt_compat; apply Rplus_lt_reg_l with x; rewrite Rplus_0_r; @@ -520,12 +513,12 @@ Proof. replace (S x0 + 0)%nat with (S x0); [ reflexivity | ring ]. symmetry ; apply tech2; assumption. exists (sum_f_R0 An 0); unfold EUn; exists 0%nat; reflexivity. - intro X; elim X; intros. + intros (x,H1). exists x; apply Un_cv_crit_lub; [ unfold Un_growing; intro; rewrite tech5; pattern (sum_f_R0 An n) at 1; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; apply H - | apply p ]. + | apply H1]. Qed. Lemma Alembert_C5 : |