diff options
Diffstat (limited to 'theories/Reals/Rlimit.v')
| -rw-r--r-- | theories/Reals/Rlimit.v | 48 |
1 files changed, 9 insertions, 39 deletions
diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v index e424a732..b14fcc4d 100644 --- a/theories/Reals/Rlimit.v +++ b/theories/Reals/Rlimit.v @@ -1,9 +1,11 @@ (************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *) +(* * The Coq Proof Assistant / The Coq Development Team *) +(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +(* <O___,, * (see CREDITS file for the list of authors) *) (* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(* * (see LICENSE file for the text of the license) *) (************************************************************************) (*********************************************************) @@ -29,59 +31,28 @@ Qed. Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps. Proof. intro esp. - assert (H := double_var esp). - unfold Rdiv in H. - symmetry ; exact H. + apply eq_sym, double_var. Qed. (*********) Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2. Proof. intro eps. - replace (2 + 2) with 4. - pattern eps at 3; rewrite double_var. - rewrite (Rmult_plus_distr_r (eps / 2) (eps / 2) (/ 2)). - unfold Rdiv. - repeat rewrite Rmult_assoc. - rewrite <- Rinv_mult_distr. - reflexivity. - discrR. - discrR. - ring. + field. Qed. (*********) Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps. Proof. intros. - pattern eps at 2; rewrite <- Rmult_1_r. - repeat rewrite (Rmult_comm eps). - apply Rmult_lt_compat_r. - exact H. - apply Rmult_lt_reg_l with 2. fourier. - rewrite Rmult_1_r; rewrite <- Rinv_r_sym. - fourier. - discrR. Qed. (*********) Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps. Proof. intros. - replace (2 + 2) with 4. - pattern eps at 2; rewrite <- Rmult_1_r. - repeat rewrite (Rmult_comm eps). - apply Rmult_lt_compat_r. - exact H. - apply Rmult_lt_reg_l with 4. - replace 4 with 4. - apply Rmult_lt_0_compat; fourier. - ring. - rewrite Rmult_1_r; rewrite <- Rinv_r_sym. fourier. - discrR. - ring. Qed. (*********) @@ -407,8 +378,7 @@ Proof. generalize (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0); unfold R_dist; intros; rewrite (Rabs_minus_sym (f x2) l) in H1; - rewrite (Rmult_comm 2 eps); rewrite (Rmult_plus_distr_l eps 1 1); - elim (Rmult_ne eps); intros a b; rewrite a; clear a b; + rewrite (Rmult_comm 2 eps); replace (eps *2) with (eps + eps) by ring; generalize (R_dist_tri l l' (f x2)); unfold R_dist; intros; apply |