diff options
Diffstat (limited to 'theories/Reals/Rtrigo_alt.v')
| -rw-r--r-- | theories/Reals/Rtrigo_alt.v | 63 |
1 files changed, 27 insertions, 36 deletions
diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v index 092bc30d0..55cb74e35 100644 --- a/theories/Reals/Rtrigo_alt.v +++ b/theories/Reals/Rtrigo_alt.v @@ -99,24 +99,22 @@ Proof. apply Rle_trans with 20. apply Rle_trans with 16. replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ]. - replace (a * a) with (Rsqr a); [ idtac | reflexivity ]. apply Rsqr_incr_1. assumption. assumption. - left; prove_sup0. - rewrite <- (Rplus_0_r 16); replace 20 with (16 + 4); - [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. - rewrite <- (Rplus_comm 20); pattern 20 at 1; rewrite <- Rplus_0_r; - apply Rplus_le_compat_l. + now apply IZR_le. + now apply IZR_le. + rewrite <- (Rplus_0_l 20) at 1; + apply Rplus_le_compat_r. apply Rplus_le_le_0_compat. - repeat apply Rmult_le_pos. - left; prove_sup0. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. apply Rmult_le_pos. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. + apply pos_INR. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. apply INR_fact_neq_0. apply INR_fact_neq_0. simpl; ring. @@ -182,16 +180,14 @@ Proof. replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ]. apply sum_eq; intros; unfold sin_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (- sum_f_R0 (tg_alt Un) (2 * n)) with (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ]. apply sum_eq; intros. unfold sin_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (2 * (n + 1))%nat with (S (S (2 * n))). reflexivity. ring. @@ -279,26 +275,23 @@ Proof. with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ]. apply Rle_trans with 12. apply Rle_trans with 4. - replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ]. - replace (a0 * a0) with (Rsqr a0); [ idtac | reflexivity ]. + change 4 with (Rsqr 2). apply Rsqr_incr_1. assumption. - discrR. assumption. - left; prove_sup0. - pattern 4 at 1; rewrite <- Rplus_0_r; replace 12 with (4 + 8); - [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. - rewrite <- (Rplus_comm 12); pattern 12 at 1; rewrite <- Rplus_0_r; - apply Rplus_le_compat_l. + now apply IZR_le. + now apply IZR_le. + rewrite <- (Rplus_0_l 12) at 1; + apply Rplus_le_compat_r. apply Rplus_le_le_0_compat. - repeat apply Rmult_le_pos. - left; prove_sup0. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. apply Rmult_le_pos. - left; prove_sup0. - replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. + apply pos_INR. + apply Rmult_le_pos. + now apply IZR_le. + apply pos_INR. apply INR_fact_neq_0. apply INR_fact_neq_0. simpl; ring. @@ -351,15 +344,13 @@ Proof. replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ]. apply sum_eq; intros; unfold cos_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ]; apply sum_eq; intros; unfold cos_term, Un, tg_alt; - replace ((-1) ^ S i) with (-1 * (-1) ^ i). + change ((-1) ^ S i) with (-1 * (-1) ^ i). unfold Rdiv; ring. - reflexivity. replace (2 * (n0 + 1))%nat with (S (S (2 * n0))). reflexivity. ring. |