diff options
Diffstat (limited to 'theories/QArith')
| -rw-r--r-- | theories/QArith/QArith_base.v | 46 | ||||
| -rw-r--r-- | theories/QArith/Qabs.v | 8 | ||||
| -rw-r--r-- | theories/QArith/Qcanon.v | 4 | ||||
| -rw-r--r-- | theories/QArith/Qpower.v | 4 | ||||
| -rw-r--r-- | theories/QArith/Qreals.v | 4 | ||||
| -rw-r--r-- | theories/QArith/Qreduction.v | 14 | ||||
| -rw-r--r-- | theories/QArith/Qround.v | 12 |
7 files changed, 45 insertions, 47 deletions
diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v index 467f263be..35706e7fa 100644 --- a/theories/QArith/QArith_base.v +++ b/theories/QArith/QArith_base.v @@ -227,9 +227,7 @@ Infix "/" := Qdiv : Q_scope. (** A light notation for [Zpos] *) -Notation " ' x " := (Zpos x) (at level 20, no associativity) : Z_scope. - -Lemma Qmake_Qdiv a b : a#b==inject_Z a/inject_Z ('b). +Lemma Qmake_Qdiv a b : a#b==inject_Z a/inject_Z (Zpos b). Proof. unfold Qeq. simpl. ring. Qed. @@ -242,9 +240,9 @@ Proof. Open Scope Z_scope. intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; simpl in *. simpl_mult; ring_simplify. - replace (p1 * 'r2 * 'q2) with (p1 * 'q2 * 'r2) by ring. + replace (p1 * Zpos r2 * Zpos q2) with (p1 * Zpos q2 * Zpos r2) by ring. rewrite H. - replace (r1 * 'p2 * 'q2 * 's2) with (r1 * 's2 * 'p2 * 'q2) by ring. + replace (r1 * Zpos p2 * Zpos q2 * Zpos s2) with (r1 * Zpos s2 * Zpos p2 * Zpos q2) by ring. rewrite H0. ring. Close Scope Z_scope. @@ -255,7 +253,7 @@ Proof. unfold Qeq, Qopp; simpl. Open Scope Z_scope. intros x y H; simpl. - replace (- Qnum x * ' Qden y) with (- (Qnum x * ' Qden y)) by ring. + replace (- Qnum x * Zpos (Qden y)) with (- (Qnum x * Zpos (Qden y))) by ring. rewrite H; ring. Close Scope Z_scope. Qed. @@ -272,9 +270,9 @@ Proof. Open Scope Z_scope. intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; simpl in *. intros; simpl_mult; ring_simplify. - replace (q1 * s1 * 'p2) with (q1 * 'p2 * s1) by ring. + replace (q1 * s1 * Zpos p2) with (q1 * Zpos p2 * s1) by ring. rewrite <- H. - replace (p1 * r1 * 'q2 * 's2) with (r1 * 's2 * p1 * 'q2) by ring. + replace (p1 * r1 * Zpos q2 * Zpos s2) with (r1 * Zpos s2 * p1 * Zpos q2) by ring. rewrite H0. ring. Close Scope Z_scope. @@ -305,13 +303,13 @@ Proof. unfold Qeq, Qcompare. Open Scope Z_scope. intros (p1,p2) (q1,q2) H (r1,r2) (s1,s2) H'; simpl in *. - rewrite <- (Zcompare_mult_compat (q2*s2) (p1*'r2)). - rewrite <- (Zcompare_mult_compat (p2*r2) (q1*'s2)). - change ('(q2*s2)) with ('q2 * 's2). - change ('(p2*r2)) with ('p2 * 'r2). - replace ('q2 * 's2 * (p1*'r2)) with ((p1*'q2)*'r2*'s2) by ring. + rewrite <- (Zcompare_mult_compat (q2*s2) (p1*Zpos r2)). + rewrite <- (Zcompare_mult_compat (p2*r2) (q1*Zpos s2)). + change (Zpos (q2*s2)) with (Zpos q2 * Zpos s2). + change (Zpos (p2*r2)) with (Zpos p2 * Zpos r2). + replace (Zpos q2 * Zpos s2 * (p1*Zpos r2)) with ((p1*Zpos q2)*Zpos r2*Zpos s2) by ring. rewrite H. - replace ('q2 * 's2 * (r1*'p2)) with ((r1*'s2)*'q2*'p2) by ring. + replace (Zpos q2 * Zpos s2 * (r1*Zpos p2)) with ((r1*Zpos s2)*Zpos q2*Zpos p2) by ring. rewrite H'. f_equal; ring. Close Scope Z_scope. @@ -572,8 +570,8 @@ Lemma Qle_trans : forall x y z, x<=y -> y<=z -> x<=z. Proof. unfold Qle; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. Open Scope Z_scope. - apply Z.mul_le_mono_pos_r with ('y2); [easy|]. - apply Z.le_trans with (y1 * 'x2 * 'z2). + apply Z.mul_le_mono_pos_r with (Zpos y2); [easy|]. + apply Z.le_trans with (y1 * Zpos x2 * Zpos z2). - rewrite Z.mul_shuffle0. now apply Z.mul_le_mono_pos_r. - rewrite Z.mul_shuffle0, (Z.mul_shuffle0 z1). now apply Z.mul_le_mono_pos_r. @@ -620,8 +618,8 @@ Lemma Qle_lt_trans : forall x y z, x<=y -> y<z -> x<z. Proof. unfold Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. Open Scope Z_scope. - apply Z.mul_lt_mono_pos_r with ('y2); [easy|]. - apply Z.le_lt_trans with (y1 * 'x2 * 'z2). + apply Z.mul_lt_mono_pos_r with (Zpos y2); [easy|]. + apply Z.le_lt_trans with (y1 * Zpos x2 * Zpos z2). - rewrite Z.mul_shuffle0. now apply Z.mul_le_mono_pos_r. - rewrite Z.mul_shuffle0, (Z.mul_shuffle0 z1). now apply Z.mul_lt_mono_pos_r. @@ -632,8 +630,8 @@ Lemma Qlt_le_trans : forall x y z, x<y -> y<=z -> x<z. Proof. unfold Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. Open Scope Z_scope. - apply Z.mul_lt_mono_pos_r with ('y2); [easy|]. - apply Z.lt_le_trans with (y1 * 'x2 * 'z2). + apply Z.mul_lt_mono_pos_r with (Zpos y2); [easy|]. + apply Z.lt_le_trans with (y1 * Zpos x2 * Zpos z2). - rewrite Z.mul_shuffle0. now apply Z.mul_lt_mono_pos_r. - rewrite Z.mul_shuffle0, (Z.mul_shuffle0 z1). now apply Z.mul_le_mono_pos_r. @@ -723,9 +721,9 @@ Proof. match goal with |- ?a <= ?b => ring_simplify a b end. rewrite Z.add_comm. apply Z.add_le_mono. - match goal with |- ?a <= ?b => ring_simplify z1 t1 ('z2) ('t2) a b end. + match goal with |- ?a <= ?b => ring_simplify z1 t1 (Zpos z2) (Zpos t2) a b end. auto with zarith. - match goal with |- ?a <= ?b => ring_simplify x1 y1 ('x2) ('y2) a b end. + match goal with |- ?a <= ?b => ring_simplify x1 y1 (Zpos x2) (Zpos y2) a b end. auto with zarith. Close Scope Z_scope. Qed. @@ -740,9 +738,9 @@ Proof. match goal with |- ?a < ?b => ring_simplify a b end. rewrite Z.add_comm. apply Z.add_le_lt_mono. - match goal with |- ?a <= ?b => ring_simplify z1 t1 ('z2) ('t2) a b end. + match goal with |- ?a <= ?b => ring_simplify z1 t1 (Zpos z2) (Zpos t2) a b end. auto with zarith. - match goal with |- ?a < ?b => ring_simplify x1 y1 ('x2) ('y2) a b end. + match goal with |- ?a < ?b => ring_simplify x1 y1 (Zpos x2) (Zpos y2) a b end. do 2 (apply Z.mul_lt_mono_pos_r;try easy). Close Scope Z_scope. Qed. diff --git a/theories/QArith/Qabs.v b/theories/QArith/Qabs.v index 48be89417..31eb41bc9 100644 --- a/theories/QArith/Qabs.v +++ b/theories/QArith/Qabs.v @@ -28,8 +28,8 @@ intros [xn xd] [yn yd] H. simpl. unfold Qeq in *. simpl in *. -change (' yd)%Z with (Z.abs (' yd)). -change (' xd)%Z with (Z.abs (' xd)). +change (Zpos yd)%Z with (Z.abs (Zpos yd)). +change (Zpos xd)%Z with (Z.abs (Zpos xd)). repeat rewrite <- Z.abs_mul. congruence. Qed. @@ -88,8 +88,8 @@ unfold Qplus. unfold Qle. simpl. apply Z.mul_le_mono_nonneg_r;auto with *. -change (' yd)%Z with (Z.abs (' yd)). -change (' xd)%Z with (Z.abs (' xd)). +change (Zpos yd)%Z with (Z.abs (Zpos yd)). +change (Zpos xd)%Z with (Z.abs (Zpos xd)). repeat rewrite <- Z.abs_mul. apply Z.abs_triangle. Qed. diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v index e25f69c31..1510a7b82 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.v @@ -30,7 +30,7 @@ Lemma Qred_identity : Proof. intros (a,b) H; simpl in *. rewrite <- Z.ggcd_gcd in H. - generalize (Z.ggcd_correct_divisors a ('b)). + generalize (Z.ggcd_correct_divisors a (Zpos b)). destruct Z.ggcd as (g,(aa,bb)); simpl in *; subst. rewrite !Z.mul_1_l. now intros (<-,<-). Qed. @@ -39,7 +39,7 @@ Lemma Qred_identity2 : forall q:Q, Qred q = q -> Z.gcd (Qnum q) (QDen q) = 1%Z. Proof. intros (a,b) H; simpl in *. - generalize (Z.gcd_nonneg a ('b)) (Z.ggcd_correct_divisors a ('b)). + generalize (Z.gcd_nonneg a (Zpos b)) (Z.ggcd_correct_divisors a (Zpos b)). rewrite <- Z.ggcd_gcd. destruct Z.ggcd as (g,(aa,bb)); simpl in *. injection H as <- <-. intros H (_,H'). diff --git a/theories/QArith/Qpower.v b/theories/QArith/Qpower.v index 3fd78f092..010782209 100644 --- a/theories/QArith/Qpower.v +++ b/theories/QArith/Qpower.v @@ -90,7 +90,7 @@ rewrite Qinv_power. reflexivity. Qed. -Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z ('p))^n. +Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z (Zpos p))^n. Proof. intros n p. rewrite Qmake_Qdiv. @@ -190,7 +190,7 @@ unfold Z.succ. rewrite Zpower_exp; auto with *; try discriminate. rewrite Qpower_plus' by discriminate. rewrite <- IHn by discriminate. -replace (a^'n*a^1)%Z with (a^'n*a)%Z by ring. +replace (a^Zpos n*a^1)%Z with (a^Zpos n*a)%Z by ring. ring_simplify. reflexivity. Qed. diff --git a/theories/QArith/Qreals.v b/theories/QArith/Qreals.v index 14ab1700e..c83296259 100644 --- a/theories/QArith/Qreals.v +++ b/theories/QArith/Qreals.v @@ -167,8 +167,8 @@ unfold Qinv, Q2R, Qeq; intros (x1, x2). case x1; unfold Qnum, Qden. simpl; intros; elim H; trivial. intros; field; auto. intros; - change (IZR (Zneg x2)) with (- IZR (' x2))%R; - change (IZR (Zneg p)) with (- IZR (' p))%R; + change (IZR (Zneg x2)) with (- IZR (Zpos x2))%R; + change (IZR (Zneg p)) with (- IZR (Zpos p))%R; simpl; field; (*auto 8 with real.*) repeat split; auto; auto with real. Qed. diff --git a/theories/QArith/Qreduction.v b/theories/QArith/Qreduction.v index 7b08515d2..17307c827 100644 --- a/theories/QArith/Qreduction.v +++ b/theories/QArith/Qreduction.v @@ -21,14 +21,14 @@ Notation Z2P_correct := Z2Pos.id (only parsing). Definition Qred (q:Q) := let (q1,q2) := q in - let (r1,r2) := snd (Z.ggcd q1 ('q2)) + let (r1,r2) := snd (Z.ggcd q1 (Zpos q2)) in r1#(Z.to_pos r2). Lemma Qred_correct : forall q, (Qred q) == q. Proof. unfold Qred, Qeq; intros (n,d); simpl. - generalize (Z.ggcd_gcd n ('d)) (Z.gcd_nonneg n ('d)) - (Z.ggcd_correct_divisors n ('d)). + generalize (Z.ggcd_gcd n (Zpos d)) (Z.gcd_nonneg n (Zpos d)) + (Z.ggcd_correct_divisors n (Zpos d)). destruct (Z.ggcd n (Zpos d)) as (g,(nn,dd)); simpl. Open Scope Z_scope. intros Hg LE (Hn,Hd). rewrite Hd, Hn. @@ -45,13 +45,13 @@ Proof. unfold Qred, Qeq in *; simpl in *. Open Scope Z_scope. intros H. - generalize (Z.ggcd_gcd a ('b)) (Zgcd_is_gcd a ('b)) - (Z.gcd_nonneg a ('b)) (Z.ggcd_correct_divisors a ('b)). + generalize (Z.ggcd_gcd a (Zpos b)) (Zgcd_is_gcd a (Zpos b)) + (Z.gcd_nonneg a (Zpos b)) (Z.ggcd_correct_divisors a (Zpos b)). destruct (Z.ggcd a (Zpos b)) as (g,(aa,bb)). simpl. intros <- Hg1 Hg2 (Hg3,Hg4). assert (Hg0 : g <> 0) by (intro; now subst g). - generalize (Z.ggcd_gcd c ('d)) (Zgcd_is_gcd c ('d)) - (Z.gcd_nonneg c ('d)) (Z.ggcd_correct_divisors c ('d)). + generalize (Z.ggcd_gcd c (Zpos d)) (Zgcd_is_gcd c (Zpos d)) + (Z.gcd_nonneg c (Zpos d)) (Z.ggcd_correct_divisors c (Zpos d)). destruct (Z.ggcd c (Zpos d)) as (g',(cc,dd)). simpl. intros <- Hg'1 Hg'2 (Hg'3,Hg'4). assert (Hg'0 : g' <> 0) by (intro; now subst g'). diff --git a/theories/QArith/Qround.v b/theories/QArith/Qround.v index e4e974972..7c5ddbb6a 100644 --- a/theories/QArith/Qround.v +++ b/theories/QArith/Qround.v @@ -80,11 +80,11 @@ unfold Qlt. simpl. replace (n*1)%Z with n by ring. ring_simplify. -replace (n / ' d * ' d + ' d)%Z with - (('d * (n / 'd) + n mod 'd) + 'd - n mod 'd)%Z by ring. +replace (n / Zpos d * Zpos d + Zpos d)%Z with + ((Zpos d * (n / Zpos d) + n mod Zpos d) + Zpos d - n mod Zpos d)%Z by ring. rewrite <- Z_div_mod_eq; auto with*. rewrite <- Z.lt_add_lt_sub_r. -destruct (Z_mod_lt n ('d)); auto with *. +destruct (Z_mod_lt n (Zpos d)); auto with *. Qed. Hint Resolve Qlt_floor : qarith. @@ -107,9 +107,9 @@ Proof. intros [xn xd] [yn yd] Hxy. unfold Qle in *. simpl in *. -rewrite <- (Zdiv_mult_cancel_r xn ('xd) ('yd)); auto with *. -rewrite <- (Zdiv_mult_cancel_r yn ('yd) ('xd)); auto with *. -rewrite (Z.mul_comm ('yd) ('xd)). +rewrite <- (Zdiv_mult_cancel_r xn (Zpos xd) (Zpos yd)); auto with *. +rewrite <- (Zdiv_mult_cancel_r yn (Zpos yd) (Zpos xd)); auto with *. +rewrite (Z.mul_comm (Zpos yd) (Zpos xd)). apply Z_div_le; auto with *. Qed. |