diff options
Diffstat (limited to 'theories/Reals/R_Ifp.v')
| -rw-r--r-- | theories/Reals/R_Ifp.v | 35 |
1 files changed, 14 insertions, 21 deletions
diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v index e9b1762af..46583d374 100644 --- a/theories/Reals/R_Ifp.v +++ b/theories/Reals/R_Ifp.v @@ -42,28 +42,23 @@ Qed. Lemma up_tech : forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r. Proof. - intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H; - rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1; - cut (1 = IZR 1); auto with zarith real. - intro; generalize H1; pattern 1 at 1; rewrite H; intro; clear H H1; - rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1)); - auto with zarith real. + intros. + apply tech_up with (1 := H0). + rewrite plus_IZR. + now apply Rplus_le_compat_r. Qed. (**********) Lemma fp_R0 : frac_part 0 = 0. Proof. - unfold frac_part; unfold Int_part; elim (archimed 0); intros; - unfold Rminus; elim (Rplus_ne (- IZR (up 0 - 1))); - intros a b; rewrite b; clear a b; rewrite <- Z_R_minus; - cut (up 0 = 1%Z). - intro; rewrite H1; - rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (eq_refl (IZR 1))); - apply Ropp_0. - elim (archimed 0); intros; clear H2; unfold Rgt in H1; - rewrite (Rminus_0_r (IZR (up 0))) in H0; generalize (lt_O_IZR (up 0) H1); - intro; clear H1; generalize (le_IZR_R1 (up 0) H0); - intro; clear H H0; omega. + unfold frac_part, Int_part. + replace (up 0) with 1%Z. + now rewrite <- minus_IZR. + destruct (archimed 0) as [H1 H2]. + apply lt_IZR in H1. + rewrite <- minus_IZR in H2. + apply le_IZR in H2. + omega. Qed. (**********) @@ -229,8 +224,7 @@ Proof. rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H; elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H; clear a b; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; - rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; - cut (1 = IZR 1); auto with zarith real. + rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H. rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H; generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H); intros; clear H H0; unfold Int_part at 1; @@ -497,8 +491,7 @@ Proof. in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0; elim (Rplus_ne (IZR (Int_part r1) + IZR (Int_part r2))); intros a b; rewrite a in H0; clear a b; elim (Rplus_ne (r1 + r2)); - intros a b; rewrite b in H0; clear a b; cut (1 = IZR 1); - auto with zarith real. + intros a b; rewrite b in H0; clear a b. rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0; rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1; rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1; |