diff options
| author |  Guillaume Melquiond <guillaume.melquiond@inria.fr> | 2017-04-02 10:30:59 +0200 |
|---|---|---|
| committer | Guillaume Melquiond <guillaume.melquiond@inria.fr> | 2017-04-02 10:33:44 +0200 |
| commit | 58bc387700d1fe4856571e8fae5c1761f89adc38 (patch) | |
| tree | e0cf041a35ccbf5315d900e3bf05024bb38c8c96 /theories/QArith | |
| parent | 05421cef04206a18cb30f6d115d27e7cb25ba0bf (diff) | |
Simplify some proofs.
This commit does not modify the signature of the involved modules, only
the opaque proof terms.
One has to wonder how proofs can bitrot so much that several occurrences
of "replace 4 with 4" start appearing.
Diffstat (limited to 'theories/QArith')
| -rw-r--r-- | theories/QArith/Qreals.v | 30 |
1 files changed, 11 insertions, 19 deletions
diff --git a/theories/QArith/Qreals.v b/theories/QArith/Qreals.v index 048e409cd..5f04cf242 100644 --- a/theories/QArith/Qreals.v +++ b/theories/QArith/Qreals.v @@ -15,7 +15,8 @@ Definition Q2R (x : Q) : R := (IZR (Qnum x) * / IZR (QDen x))%R. Lemma IZR_nz : forall p : positive, IZR (Zpos p) <> 0%R. Proof. -intros; apply not_O_IZR; auto with qarith. +intros. +now apply not_O_IZR. Qed. Hint Resolve IZR_nz Rmult_integral_contrapositive. @@ -48,8 +49,7 @@ assert ((X1 * Y2)%R = (Y1 * X2)%R). apply IZR_eq; auto. clear H. field_simplify_eq; auto. -ring_simplify X1 Y2 (Y2 * X1)%R. -rewrite H0; ring. +rewrite H0; ring. Qed. Lemma Rle_Qle : forall x y : Q, (Q2R x <= Q2R y)%R -> x<=y. @@ -66,10 +66,8 @@ replace (X1 * Y2)%R with (X1 * / X2 * (X2 * Y2))%R; try (field; auto). replace (Y1 * X2)%R with (Y1 * / Y2 * (X2 * Y2))%R; try (field; auto). apply Rmult_le_compat_r; auto. apply Rmult_le_pos. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_le; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_le; - auto with zarith. +now apply IZR_le. +now apply IZR_le. Qed. Lemma Qle_Rle : forall x y : Q, x<=y -> (Q2R x <= Q2R y)%R. @@ -88,10 +86,8 @@ replace (X1 * / X2)%R with (X1 * Y2 * (/ X2 * / Y2))%R; try (field; auto). replace (Y1 * / Y2)%R with (Y1 * X2 * (/ X2 * / Y2))%R; try (field; auto). apply Rmult_le_compat_r; auto. apply Rmult_le_pos; apply Rlt_le; apply Rinv_0_lt_compat. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. +now apply IZR_lt. +now apply IZR_lt. Qed. Lemma Rlt_Qlt : forall x y : Q, (Q2R x < Q2R y)%R -> x<y. @@ -108,10 +104,8 @@ replace (X1 * Y2)%R with (X1 * / X2 * (X2 * Y2))%R; try (field; auto). replace (Y1 * X2)%R with (Y1 * / Y2 * (X2 * Y2))%R; try (field; auto). apply Rmult_lt_compat_r; auto. apply Rmult_lt_0_compat. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. +now apply IZR_lt. +now apply IZR_lt. Qed. Lemma Qlt_Rlt : forall x y : Q, x<y -> (Q2R x < Q2R y)%R. @@ -130,10 +124,8 @@ replace (X1 * / X2)%R with (X1 * Y2 * (/ X2 * / Y2))%R; try (field; auto). replace (Y1 * / Y2)%R with (Y1 * X2 * (/ X2 * / Y2))%R; try (field; auto). apply Rmult_lt_compat_r; auto. apply Rmult_lt_0_compat; apply Rinv_0_lt_compat. -unfold X2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. -unfold Y2; replace 0%R with (IZR 0); auto; apply IZR_lt; red; - auto with zarith. +now apply IZR_lt. +now apply IZR_lt. Qed. Lemma Q2R_plus : forall x y : Q, Q2R (x+y) = (Q2R x + Q2R y)%R. |