diff options
Diffstat (limited to 'theories/Reals/RIneq.v')
| -rw-r--r-- | theories/Reals/RIneq.v | 14 |
1 files changed, 7 insertions, 7 deletions
diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index 7813c7b975..229e6018ca 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -19,7 +19,7 @@ Require Export Raxioms. Require Import Rpow_def. Require Import Zpower. Require Export ZArithRing. -Require Import Omega. +Require Import Lia. Require Export RealField. Local Open Scope Z_scope. @@ -1875,7 +1875,7 @@ Lemma eq_IZR : forall n m:Z, IZR n = IZR m -> n = m. Proof. intros z1 z2 H; generalize (Rminus_diag_eq (IZR z1) (IZR z2) H); rewrite (Z_R_minus z1 z2); intro; generalize (eq_IZR_R0 (z1 - z2) H0); - intro; omega. + intro; lia. Qed. (**********) @@ -1913,21 +1913,21 @@ Qed. Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m. Proof. intros m n H; apply Rnot_lt_ge; red; intro. - generalize (lt_IZR m n H0); intro; omega. + generalize (lt_IZR m n H0); intro; lia. Qed. Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m. Proof. intros m n H; apply Rnot_gt_le; red; intro. - unfold Rgt in H0; generalize (lt_IZR n m H0); intro; omega. + unfold Rgt in H0; generalize (lt_IZR n m H0); intro; lia. Qed. Lemma IZR_lt : forall n m:Z, (n < m)%Z -> IZR n < IZR m. Proof. intros m n H; cut (m <= n)%Z. intro H0; elim (IZR_le m n H0); intro; auto. - generalize (eq_IZR m n H1); intro; exfalso; omega. - omega. + generalize (eq_IZR m n H1); intro; exfalso; lia. + lia. Qed. Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2. @@ -1954,7 +1954,7 @@ Lemma one_IZR_r_R1 : forall r (n m:Z), r < IZR n <= r + 1 -> r < IZR m <= r + 1 -> n = m. Proof. intros r z x [H1 H2] [H3 H4]. - cut ((z - x)%Z = 0%Z); auto with zarith. + cut ((z - x)%Z = 0%Z). lia. apply one_IZR_lt1. rewrite <- Z_R_minus; split. replace (-1) with (r - (r + 1)). |