diff options
Diffstat (limited to 'theories/Reals/Rtrigo_reg.v')
| -rw-r--r-- | theories/Reals/Rtrigo_reg.v | 32 |
1 files changed, 15 insertions, 17 deletions
diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v index b105ca69..139563bf 100644 --- a/theories/Reals/Rtrigo_reg.v +++ b/theories/Reals/Rtrigo_reg.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: Rtrigo_reg.v 9551 2007-01-29 15:13:35Z bgregoir $ i*) +(*i $Id: Rtrigo_reg.v 10710 2008-03-23 09:24:09Z herbelin $ i*) Require Import Rbase. Require Import Rfunctions. @@ -25,16 +25,15 @@ Proof. unfold CVN_R in |- *; intros. cut ((r:R) <> 0). intro hyp_r; unfold CVN_r in |- *. - apply existT with (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)). + exists (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)). cut - (sigT - (fun l:R => + { l:R | Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) - n) l)). + n) l }. intro X; elim X; intros. - apply existT with x. + exists x. split. apply p. intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult. @@ -124,7 +123,7 @@ Lemma continuity_cos : continuity cos. Proof. set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)). cut (CVN_R fn). - intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). + intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }). intro cv; cut (forall n:nat, continuity (fn n)). intro; cut (forall x:R, cos x = SFL fn cv x). intro; cut (continuity (SFL fn cv) -> continuity cos). @@ -144,7 +143,7 @@ Proof. case (cv x); case (exist_cos (Rsqr x)); intros. symmetry in |- *; eapply UL_sequence. apply u. - unfold cos_in in c; unfold infinit_sum in c; unfold Un_cv in |- *; intros. + unfold cos_in in c; unfold infinite_sum in c; unfold Un_cv in |- *; intros. elim (c _ H0); intros N0 H1. exists N0; intros. unfold R_dist in H1; unfold R_dist, SP in |- *. @@ -200,17 +199,16 @@ Lemma CVN_R_sin : CVN_R fn. Proof. unfold CVN_R in |- *; unfold CVN_r in |- *; intros fn H r. - apply existT with (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)). + exists (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)). cut - (sigT - (fun l:R => + { l:R | Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n) - l)). + l }. intro X; elim X; intros. - apply existT with x. + exists x. split. apply p. intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult; @@ -305,7 +303,7 @@ Proof. set (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)). cut (CVN_R fn). - intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). + intro; cut (forall x:R, { l:R | Un_cv (fun N:nat => SP fn N x) l }). intro cv. set (r := mkposreal _ Rlt_0_1). cut (CVN_r fn r). @@ -331,7 +329,7 @@ Proof. unfold Rdiv in |- *; rewrite (Rinv_r_simpl_m h x0 H6). eapply UL_sequence. apply u. - unfold sin_in in s; unfold sin_n, infinit_sum in s; + unfold sin_in in s; unfold sin_n, infinite_sum in s; unfold SP, fn, Un_cv in |- *; intros. elim (s _ H10); intros N0 H11. exists N0; intros. @@ -584,14 +582,14 @@ Qed. Lemma derivable_pt_sin : forall x:R, derivable_pt sin x. Proof. unfold derivable_pt in |- *; intro. - apply existT with (cos x). + exists (cos x). apply derivable_pt_lim_sin. Qed. Lemma derivable_pt_cos : forall x:R, derivable_pt cos x. Proof. unfold derivable_pt in |- *; intro. - apply existT with (- sin x). + exists (- sin x). apply derivable_pt_lim_cos. Qed. |