Commit b3e635ab authored by Guillaume Melquiond's avatar Guillaume Melquiond

Renamed Rabs_le theorems and added lt versions.

parent aa2517ea
......@@ -251,7 +251,7 @@ now apply Zrnd_monotone.
rewrite <- (Zrnd_Z2R rnd 0).
apply Zrnd_monotone.
apply Rle_trans with (Z2R (-1)). 2: now apply Z2R_le.
destruct (Rabs_le_r_inv _ _ Hx) as [Hx1|Hx1].
destruct (Rabs_ge_inv _ _ Hx) as [Hx1|Hx1].
exact Hx1.
elim Rle_not_lt with (1 := Hx1).
apply Rle_lt_trans with (2 := Hy).
......@@ -262,7 +262,7 @@ rewrite <- (Zrnd_Z2R rnd 0).
apply Zrnd_monotone.
apply Rle_trans with (Z2R 1).
now apply Z2R_le.
destruct (Rabs_le_r_inv _ _ Hy) as [Hy1|Hy1].
destruct (Rabs_ge_inv _ _ Hy) as [Hy1|Hy1].
elim Rle_not_lt with (1 := Hy1).
apply Rlt_le_trans with (2 := Hxy).
apply (Rabs_def2 _ _ Hx).
......
......@@ -177,7 +177,7 @@ apply Rle_refl.
apply Rsqrt_positivity.
Qed.
Theorem Rabs_le_l :
Theorem Rabs_le :
forall x y,
(-y <= x <= y)%R -> (Rabs x <= y)%R.
Proof.
......@@ -189,7 +189,7 @@ now rewrite Ropp_involutive.
exact Hxy.
Qed.
Theorem Rabs_le_l_inv :
Theorem Rabs_le_inv :
forall x y,
(Rabs x <= y)%R -> (-y <= x <= y)%R.
Proof.
......@@ -204,7 +204,7 @@ apply Rle_trans with (2 := Hxy).
apply RRle_abs.
Qed.
Theorem Rabs_le_r :
Theorem Rabs_ge :
forall x y,
(y <= -x \/ x <= y)%R -> (x <= Rabs y)%R.
Proof.
......@@ -218,12 +218,12 @@ apply Rle_trans with (1 := Hxy).
apply RRle_abs.
Qed.
Theorem Rabs_le_r_inv :
Theorem Rabs_ge_inv :
forall x y,
(x <= Rabs y)%R -> (y <= -x \/ x <= y)%R.
Proof.
unfold Rabs.
intros x y.
unfold Rabs.
case Rcase_abs ; intros Hy Hxy.
left.
apply Ropp_le_cancel.
......@@ -231,6 +231,49 @@ now rewrite Ropp_involutive.
now right.
Qed.
Theorem Rabs_lt :
forall x y,
(-y < x < y)%R -> (Rabs x < y)%R.
Proof.
intros x y (Hyx,Hxy).
now apply Rabs_def1.
Qed.
Theorem Rabs_lt_inv :
forall x y,
(Rabs x < y)%R -> (-y < x < y)%R.
Proof.
intros x y H.
now split ; eapply Rabs_def2.
Qed.
Theorem Rabs_gt :
forall x y,
(y < -x \/ x < y)%R -> (x < Rabs y)%R.
Proof.
intros x y [Hyx|Hxy].
rewrite <- Rabs_Ropp.
apply Rlt_le_trans with (Ropp y).
apply Ropp_lt_cancel.
now rewrite Ropp_involutive.
apply RRle_abs.
apply Rlt_le_trans with (1 := Hxy).
apply RRle_abs.
Qed.
Theorem Rabs_gt_inv :
forall x y,
(x < Rabs y)%R -> (y < -x \/ x < y)%R.
Proof.
intros x y.
unfold Rabs.
case Rcase_abs ; intros Hy Hxy.
left.
apply Ropp_lt_cancel.
now rewrite Ropp_involutive.
now right.
Qed.
End Rmissing.
Section Zmissing.
......
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