diff options
Diffstat (limited to 'theories/Reals/Rtrigo_fun.v')
| -rw-r--r-- | theories/Reals/Rtrigo_fun.v | 18 |
1 files changed, 9 insertions, 9 deletions
diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index 6eec0329..cb53b534 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Rtrigo_fun.v 10710 2008-03-23 09:24:09Z herbelin $ i*) +(*i $Id$ i*) Require Import Rbase. Require Import Rfunctions. @@ -33,7 +33,7 @@ Proof. generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H2); replace (1 + (/ eps + -1)) with (/ eps); [ clear H2; intro | ring ]. rewrite (Rplus_comm 1 (INR n)) in H2; rewrite <- (S_INR n) in H2; - generalize (Rmult_gt_0_compat (/ INR (S n)) eps H1 H); + generalize (Rmult_gt_0_compat (/ INR (S n)) eps H1 H); intro; unfold Rgt in H3; generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H3 H2); intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H4; @@ -42,11 +42,11 @@ Proof. rewrite (Rmult_comm (/ INR (S n))) in H4; rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4; rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H4; - rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; + rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; assumption. apply Rlt_minus; unfold Rgt in a; rewrite <- Rinv_1; apply (Rinv_lt_contravar 1 eps); auto; - rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; + rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; assumption. unfold Rgt in H1; apply Rlt_le; assumption. unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. @@ -61,12 +61,12 @@ Proof. intro ; generalize (Rlt_le_trans (/ eps - 1) (INR x) (INR n) H4 - (le_INR x n H2)); + (le_INR x n H2)); clear H4; intro; unfold Rminus in H4; generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H4); replace (1 + (/ eps + -1)) with (/ eps); [ clear H4; intro | ring ]. rewrite (Rplus_comm 1 (INR n)) in H4; rewrite <- (S_INR n) in H4; - generalize (Rmult_gt_0_compat (/ INR (S n)) eps H3 H); + generalize (Rmult_gt_0_compat (/ INR (S n)) eps H3 H); intro; unfold Rgt in H5; generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H5 H4); intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H6; @@ -75,7 +75,7 @@ Proof. rewrite (Rmult_comm (/ INR (S n))) in H6; rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6; rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H6; - rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; + rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; assumption. cut (IZR (up (/ eps - 1)) = IZR (Z_of_nat x)); [ intro | rewrite H1; trivial ]. @@ -92,8 +92,8 @@ Proof. rewrite (Rinv_l eps (sym_not_eq (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) - ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); - intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; + ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); + intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; unfold Rgt in |- *; assumption. right; rewrite H0; rewrite Rinv_1; apply sym_eq; apply Rminus_diag_eq; auto. elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; |