diff options
Diffstat (limited to 'theories/QArith/QArith_base.v')
| -rw-r--r-- | theories/QArith/QArith_base.v | 535 |
1 files changed, 275 insertions, 260 deletions
diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v index 335466a6..66d16cfe 100644 --- a/theories/QArith/QArith_base.v +++ b/theories/QArith/QArith_base.v @@ -6,7 +6,7 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: QArith_base.v 8989 2006-06-25 22:17:49Z letouzey $ i*) +(*i $Id: QArith_base.v 9245 2006-10-17 12:53:34Z notin $ i*) Require Export ZArith. Require Export ZArithRing. @@ -87,7 +87,7 @@ Qed. Hint Unfold Qeq Qlt Qle: qarith. Hint Extern 5 (?X1 <> ?X2) => intro; discriminate: qarith. -(** Properties of equality. *) +(** * Properties of equality. *) Theorem Qeq_refl : forall x, x == x. Proof. @@ -104,8 +104,10 @@ Proof. unfold Qeq in |- *; intros. apply Zmult_reg_l with (QDen y). auto with qarith. -ring; rewrite H; ring. -rewrite Zmult_assoc; rewrite H0; ring. +transitivity (Qnum x * QDen y * QDen z)%Z; try ring. +rewrite H. +transitivity (Qnum y * QDen z * QDen x)%Z; try ring. +rewrite H0; ring. Qed. (** Furthermore, this equality is decidable: *) @@ -128,6 +130,9 @@ Hint Resolve (Seq_refl Q Qeq Q_Setoid): qarith. Hint Resolve (Seq_sym Q Qeq Q_Setoid): qarith. Hint Resolve (Seq_trans Q Qeq Q_Setoid): qarith. + +(** * Addition, multiplication and opposite *) + (** The addition, multiplication and opposite are defined in the straightforward way: *) @@ -160,133 +165,138 @@ Infix "/" := Qdiv : Q_scope. Notation " ' x " := (Zpos x) (at level 20, no associativity) : Z_scope. -(** Setoid compatibility results *) + +(** * Setoid compatibility results *) Add Morphism Qplus : Qplus_comp. Proof. -unfold Qeq, Qplus; simpl. -Open Scope Z_scope. -intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; simpl in *. -simpl_mult; ring. -replace (p1 * ('s2 * 'q2)) with (p1 * 'q2 * 's2) by ring. -rewrite H. -replace ('s2 * ('q2 * r1)) with (r1 * 's2 * 'q2) by ring. -rewrite H0. -ring. -Open Scope Q_scope. + unfold Qeq, Qplus; simpl. + 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. + rewrite H. + replace (r1 * 'p2 * 'q2 * 's2) with (r1 * 's2 * 'p2 * 'q2) by ring. + rewrite H0. + ring. + Close Scope Z_scope. Qed. Add Morphism Qopp : Qopp_comp. Proof. -unfold Qeq, Qopp; simpl. -intros; ring; rewrite H; ring. + unfold Qeq, Qopp; simpl. + Open Scope Z_scope. + intros. + replace (- Qnum x1 * ' Qden x2) with (- (Qnum x1 * ' Qden x2)) by ring. + rewrite H in |- *; ring. + Close Scope Z_scope. Qed. Add Morphism Qminus : Qminus_comp. Proof. -intros. -unfold Qminus. -rewrite H; rewrite H0; auto with qarith. + intros. + unfold Qminus. + rewrite H; rewrite H0; auto with qarith. Qed. Add Morphism Qmult : Qmult_comp. Proof. -unfold Qeq; simpl. -Open Scope Z_scope. -intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0; simpl in *. -intros; simpl_mult; ring. -replace ('p2 * (q1 * s1)) with (q1 * 'p2 * s1) by ring. -rewrite <- H. -replace ('s2 * ('q2 * r1)) with (r1 * 's2 * 'q2) by ring. -rewrite H0. -ring. -Open Scope Q_scope. + unfold Qeq; simpl. + 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. + rewrite <- H. + replace (p1 * r1 * 'q2 * 's2) with (r1 * 's2 * p1 * 'q2) by ring. + rewrite H0. + ring. + Close Scope Z_scope. Qed. Add Morphism Qinv : Qinv_comp. Proof. -unfold Qeq, Qinv; simpl. -Open Scope Z_scope. -intros (p1, p2) (q1, q2); simpl. -case p1; simpl. -intros. -assert (q1 = 0). - elim (Zmult_integral q1 ('p2)); auto with zarith. - intros; discriminate. -subst; auto. -case q1; simpl; intros; try discriminate. -rewrite (Pmult_comm p2 p); rewrite (Pmult_comm q2 p0); auto. -case q1; simpl; intros; try discriminate. -rewrite (Pmult_comm p2 p); rewrite (Pmult_comm q2 p0); auto. -Open Scope Q_scope. + unfold Qeq, Qinv; simpl. + Open Scope Z_scope. + intros (p1, p2) (q1, q2); simpl. + case p1; simpl. + intros. + assert (q1 = 0). + elim (Zmult_integral q1 ('p2)); auto with zarith. + intros; discriminate. + subst; auto. + case q1; simpl; intros; try discriminate. + rewrite (Pmult_comm p2 p); rewrite (Pmult_comm q2 p0); auto. + case q1; simpl; intros; try discriminate. + rewrite (Pmult_comm p2 p); rewrite (Pmult_comm q2 p0); auto. + Close Scope Z_scope. Qed. Add Morphism Qdiv : Qdiv_comp. Proof. -intros; unfold Qdiv. -rewrite H; rewrite H0; auto with qarith. + intros; unfold Qdiv. + rewrite H; rewrite H0; auto with qarith. Qed. Add Morphism Qle with signature Qeq ==> Qeq ==> iff as Qle_comp. Proof. -cut (forall x1 x2, x1==x2 -> forall x3 x4, x3==x4 -> x1<=x3 -> x2<=x4). -split; apply H; assumption || (apply Qeq_sym ; assumption). - -unfold Qeq, Qle; simpl. -Open Scope Z_scope. -intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0 H1; simpl in *. -apply Zmult_le_reg_r with ('p2). -unfold Zgt; auto. -replace (q1 * 's2 * 'p2) with (q1 * 'p2 * 's2) by ring. -rewrite <- H. -apply Zmult_le_reg_r with ('r2). -unfold Zgt; auto. -replace (s1 * 'q2 * 'p2 * 'r2) with (s1 * 'r2 * 'q2 * 'p2) by ring. -rewrite <- H0. -replace (p1 * 'q2 * 's2 * 'r2) with ('q2 * 's2 * (p1 * 'r2)) by ring. -replace (r1 * 's2 * 'q2 * 'p2) with ('q2 * 's2 * (r1 * 'p2)) by ring. -auto with zarith. -Open Scope Q_scope. + cut (forall x1 x2, x1==x2 -> forall x3 x4, x3==x4 -> x1<=x3 -> x2<=x4). + split; apply H; assumption || (apply Qeq_sym ; assumption). + + unfold Qeq, Qle; simpl. + Open Scope Z_scope. + intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0 H1; simpl in *. + apply Zmult_le_reg_r with ('p2). + unfold Zgt; auto. + replace (q1 * 's2 * 'p2) with (q1 * 'p2 * 's2) by ring. + rewrite <- H. + apply Zmult_le_reg_r with ('r2). + unfold Zgt; auto. + replace (s1 * 'q2 * 'p2 * 'r2) with (s1 * 'r2 * 'q2 * 'p2) by ring. + rewrite <- H0. + replace (p1 * 'q2 * 's2 * 'r2) with ('q2 * 's2 * (p1 * 'r2)) by ring. + replace (r1 * 's2 * 'q2 * 'p2) with ('q2 * 's2 * (r1 * 'p2)) by ring. + auto with zarith. + Close Scope Z_scope. Qed. Add Morphism Qlt with signature Qeq ==> Qeq ==> iff as Qlt_comp. Proof. -cut (forall x1 x2, x1==x2 -> forall x3 x4, x3==x4 -> x1<x3 -> x2<x4). -split; apply H; assumption || (apply Qeq_sym ; assumption). - -unfold Qeq, Qlt; simpl. -Open Scope Z_scope. -intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0 H1; simpl in *. -apply Zgt_lt. -generalize (Zlt_gt _ _ H1); clear H1; intro H1. -apply Zmult_gt_reg_r with ('p2); auto with zarith. -replace (q1 * 's2 * 'p2) with (q1 * 'p2 * 's2) by ring. -rewrite <- H. -apply Zmult_gt_reg_r with ('r2); auto with zarith. -replace (s1 * 'q2 * 'p2 * 'r2) with (s1 * 'r2 * 'q2 * 'p2) by ring. -rewrite <- H0. -replace (p1 * 'q2 * 's2 * 'r2) with ('q2 * 's2 * (p1 * 'r2)) by ring. -replace (r1 * 's2 * 'q2 * 'p2) with ('q2 * 's2 * (r1 * 'p2)) by ring. -apply Zlt_gt. -apply Zmult_gt_0_lt_compat_l; auto with zarith. -Open Scope Q_scope. + cut (forall x1 x2, x1==x2 -> forall x3 x4, x3==x4 -> x1<x3 -> x2<x4). + split; apply H; assumption || (apply Qeq_sym ; assumption). + + unfold Qeq, Qlt; simpl. + Open Scope Z_scope. + intros (p1, p2) (q1, q2) H (r1, r2) (s1, s2) H0 H1; simpl in *. + apply Zgt_lt. + generalize (Zlt_gt _ _ H1); clear H1; intro H1. + apply Zmult_gt_reg_r with ('p2); auto with zarith. + replace (q1 * 's2 * 'p2) with (q1 * 'p2 * 's2) by ring. + rewrite <- H. + apply Zmult_gt_reg_r with ('r2); auto with zarith. + replace (s1 * 'q2 * 'p2 * 'r2) with (s1 * 'r2 * 'q2 * 'p2) by ring. + rewrite <- H0. + replace (p1 * 'q2 * 's2 * 'r2) with ('q2 * 's2 * (p1 * 'r2)) by ring. + replace (r1 * 's2 * 'q2 * 'p2) with ('q2 * 's2 * (r1 * 'p2)) by ring. + apply Zlt_gt. + apply Zmult_gt_0_lt_compat_l; auto with zarith. + Close Scope Z_scope. Qed. Lemma Qcompare_egal_dec: forall n m p q : Q, - (n<m -> p<q) -> (n==m -> p==q) -> (n>m -> p>q) -> ((n?=m) = (p?=q)). + (n<m -> p<q) -> (n==m -> p==q) -> (n>m -> p>q) -> ((n?=m) = (p?=q)). Proof. -intros. -do 2 rewrite Qeq_alt in H0. -unfold Qeq, Qlt, Qcompare in *. -apply Zcompare_egal_dec; auto. -omega. + intros. + do 2 rewrite Qeq_alt in H0. + unfold Qeq, Qlt, Qcompare in *. + apply Zcompare_egal_dec; auto. + omega. Qed. Add Morphism Qcompare : Qcompare_comp. Proof. -intros; apply Qcompare_egal_dec; rewrite H; rewrite H0; auto. + intros; apply Qcompare_egal_dec; rewrite H; rewrite H0; auto. Qed. @@ -294,382 +304,387 @@ Qed. Lemma Q_apart_0_1 : ~ 1 == 0. Proof. - unfold Qeq; auto with qarith. + unfold Qeq; auto with qarith. Qed. +(** * Properties of [Qadd] *) + (** Addition is associative: *) Theorem Qplus_assoc : forall x y z, x+(y+z)==(x+y)+z. Proof. - intros (x1, x2) (y1, y2) (z1, z2). - unfold Qeq, Qplus; simpl; simpl_mult; ring. + intros (x1, x2) (y1, y2) (z1, z2). + unfold Qeq, Qplus; simpl; simpl_mult; ring. Qed. (** [0] is a neutral element for addition: *) Lemma Qplus_0_l : forall x, 0+x == x. Proof. - intros (x1, x2); unfold Qeq, Qplus; simpl; ring. + intros (x1, x2); unfold Qeq, Qplus; simpl; ring. Qed. Lemma Qplus_0_r : forall x, x+0 == x. Proof. - intros (x1, x2); unfold Qeq, Qplus; simpl. - rewrite Pmult_comm; simpl; ring. + intros (x1, x2); unfold Qeq, Qplus; simpl. + rewrite Pmult_comm; simpl; ring. Qed. (** Commutativity of addition: *) Theorem Qplus_comm : forall x y, x+y == y+x. Proof. - intros (x1, x2); unfold Qeq, Qplus; simpl. - intros; rewrite Pmult_comm; ring. + intros (x1, x2); unfold Qeq, Qplus; simpl. + intros; rewrite Pmult_comm; ring. Qed. -(** Properties of [Qopp] *) + +(** * Properties of [Qopp] *) Lemma Qopp_involutive : forall q, - -q == q. Proof. - red; simpl; intros; ring. + red; simpl; intros; ring. Qed. Theorem Qplus_opp_r : forall q, q+(-q) == 0. Proof. - red; simpl; intro; ring. + red; simpl; intro; ring. Qed. + +(** * Properties of [Qmult] *) + (** Multiplication is associative: *) Theorem Qmult_assoc : forall n m p, n*(m*p)==(n*m)*p. Proof. - intros; red; simpl; rewrite Pmult_assoc; ring. + intros; red; simpl; rewrite Pmult_assoc; ring. Qed. (** [1] is a neutral element for multiplication: *) Lemma Qmult_1_l : forall n, 1*n == n. Proof. - intro; red; simpl; destruct (Qnum n); auto. + intro; red; simpl; destruct (Qnum n); auto. Qed. Theorem Qmult_1_r : forall n, n*1==n. Proof. - intro; red; simpl. - rewrite Zmult_1_r with (n := Qnum n). - rewrite Pmult_comm; simpl; trivial. + intro; red; simpl. + rewrite Zmult_1_r with (n := Qnum n). + rewrite Pmult_comm; simpl; trivial. Qed. (** Commutativity of multiplication *) Theorem Qmult_comm : forall x y, x*y==y*x. Proof. - intros; red; simpl; rewrite Pmult_comm; ring. + intros; red; simpl; rewrite Pmult_comm; ring. Qed. -(** Distributivity *) +(** Distributivity over [Qadd] *) Theorem Qmult_plus_distr_r : forall x y z, x*(y+z)==(x*y)+(x*z). Proof. -intros (x1, x2) (y1, y2) (z1, z2). -unfold Qeq, Qmult, Qplus; simpl; simpl_mult; ring. + intros (x1, x2) (y1, y2) (z1, z2). + unfold Qeq, Qmult, Qplus; simpl; simpl_mult; ring. Qed. Theorem Qmult_plus_distr_l : forall x y z, (x+y)*z==(x*z)+(y*z). Proof. -intros (x1, x2) (y1, y2) (z1, z2). -unfold Qeq, Qmult, Qplus; simpl; simpl_mult; ring. + intros (x1, x2) (y1, y2) (z1, z2). + unfold Qeq, Qmult, Qplus; simpl; simpl_mult; ring. Qed. (** Integrality *) Theorem Qmult_integral : forall x y, x*y==0 -> x==0 \/ y==0. Proof. - intros (x1,x2) (y1,y2). - unfold Qeq, Qmult; simpl; intros. - destruct (Zmult_integral (x1*1)%Z (y1*1)%Z); auto. - rewrite <- H; ring. + intros (x1,x2) (y1,y2). + unfold Qeq, Qmult; simpl; intros. + destruct (Zmult_integral (x1*1)%Z (y1*1)%Z); auto. + rewrite <- H; ring. Qed. Theorem Qmult_integral_l : forall x y, ~ x == 0 -> x*y == 0 -> y == 0. Proof. - intros (x1, x2) (y1, y2). - unfold Qeq, Qmult; simpl; intros. - apply Zmult_integral_l with x1; auto with zarith. - rewrite <- H0; ring. + intros (x1, x2) (y1, y2). + unfold Qeq, Qmult; simpl; intros. + apply Zmult_integral_l with x1; auto with zarith. + rewrite <- H0; ring. Qed. -(** Inverse and division. *) +(** * Inverse and division. *) Theorem Qmult_inv_r : forall x, ~ x == 0 -> x*(/x) == 1. Proof. - intros (x1, x2); unfold Qeq, Qdiv, Qmult; case x1; simpl; - intros; simpl_mult; try ring. - elim H; auto. + intros (x1, x2); unfold Qeq, Qdiv, Qmult; case x1; simpl; + intros; simpl_mult; try ring. + elim H; auto. Qed. Lemma Qinv_mult_distr : forall p q, / (p * q) == /p * /q. Proof. -intros (x1,x2) (y1,y2); unfold Qeq, Qinv, Qmult; simpl. -destruct x1; simpl; auto; - destruct y1; simpl; auto. + intros (x1,x2) (y1,y2); unfold Qeq, Qinv, Qmult; simpl. + destruct x1; simpl; auto; + destruct y1; simpl; auto. Qed. Theorem Qdiv_mult_l : forall x y, ~ y == 0 -> (x*y)/y == x. Proof. - intros; unfold Qdiv. - rewrite <- (Qmult_assoc x y (Qinv y)). - rewrite (Qmult_inv_r y H). - apply Qmult_1_r. + intros; unfold Qdiv. + rewrite <- (Qmult_assoc x y (Qinv y)). + rewrite (Qmult_inv_r y H). + apply Qmult_1_r. Qed. Theorem Qmult_div_r : forall x y, ~ y == 0 -> y*(x/y) == x. Proof. - intros; unfold Qdiv. - rewrite (Qmult_assoc y x (Qinv y)). - rewrite (Qmult_comm y x). - fold (Qdiv (Qmult x y) y). - apply Qdiv_mult_l; auto. + intros; unfold Qdiv. + rewrite (Qmult_assoc y x (Qinv y)). + rewrite (Qmult_comm y x). + fold (Qdiv (Qmult x y) y). + apply Qdiv_mult_l; auto. Qed. -(** Properties of order upon Q. *) +(** * Properties of order upon Q. *) Lemma Qle_refl : forall x, x<=x. Proof. -unfold Qle; auto with zarith. + unfold Qle; auto with zarith. Qed. Lemma Qle_antisym : forall x y, x<=y -> y<=x -> x==y. Proof. -unfold Qle, Qeq; auto with zarith. + unfold Qle, Qeq; auto with zarith. Qed. 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 Zmult_le_reg_r with ('y2). -red; trivial. -apply Zle_trans with (y1 * 'x2 * 'z2). -replace (x1 * 'z2 * 'y2) with (x1 * 'y2 * 'z2) by ring. -apply Zmult_le_compat_r; auto with zarith. -replace (y1 * 'x2 * 'z2) with (y1 * 'z2 * 'x2) by ring. -replace (z1 * 'x2 * 'y2) with (z1 * 'y2 * 'x2) by ring. -apply Zmult_le_compat_r; auto with zarith. -Open Scope Q_scope. + unfold Qle; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. + Open Scope Z_scope. + apply Zmult_le_reg_r with ('y2). + red; trivial. + apply Zle_trans with (y1 * 'x2 * 'z2). + replace (x1 * 'z2 * 'y2) with (x1 * 'y2 * 'z2) by ring. + apply Zmult_le_compat_r; auto with zarith. + replace (y1 * 'x2 * 'z2) with (y1 * 'z2 * 'x2) by ring. + replace (z1 * 'x2 * 'y2) with (z1 * 'y2 * 'x2) by ring. + apply Zmult_le_compat_r; auto with zarith. + Close Scope Z_scope. Qed. Lemma Qlt_not_eq : forall x y, x<y -> ~ x==y. Proof. -unfold Qlt, Qeq; auto with zarith. + unfold Qlt, Qeq; auto with zarith. Qed. (** Large = strict or equal *) Lemma Qlt_le_weak : forall x y, x<y -> x<=y. Proof. -unfold Qle, Qlt; auto with zarith. + unfold Qle, Qlt; auto with zarith. Qed. 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 Zgt_lt. -apply Zmult_gt_reg_r with ('y2). -red; trivial. -apply Zgt_le_trans with (y1 * 'x2 * 'z2). -replace (y1 * 'x2 * 'z2) with (y1 * 'z2 * 'x2) by ring. -replace (z1 * 'x2 * 'y2) with (z1 * 'y2 * 'x2) by ring. -apply Zmult_gt_compat_r; auto with zarith. -replace (x1 * 'z2 * 'y2) with (x1 * 'y2 * 'z2) by ring. -apply Zmult_le_compat_r; auto with zarith. -Open Scope Q_scope. + unfold Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. + Open Scope Z_scope. + apply Zgt_lt. + apply Zmult_gt_reg_r with ('y2). + red; trivial. + apply Zgt_le_trans with (y1 * 'x2 * 'z2). + replace (y1 * 'x2 * 'z2) with (y1 * 'z2 * 'x2) by ring. + replace (z1 * 'x2 * 'y2) with (z1 * 'y2 * 'x2) by ring. + apply Zmult_gt_compat_r; auto with zarith. + replace (x1 * 'z2 * 'y2) with (x1 * 'y2 * 'z2) by ring. + apply Zmult_le_compat_r; auto with zarith. + Close Scope Z_scope. Qed. 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 Zgt_lt. -apply Zmult_gt_reg_r with ('y2). -red; trivial. -apply Zle_gt_trans with (y1 * 'x2 * 'z2). -replace (y1 * 'x2 * 'z2) with (y1 * 'z2 * 'x2) by ring. -replace (z1 * 'x2 * 'y2) with (z1 * 'y2 * 'x2) by ring. -apply Zmult_le_compat_r; auto with zarith. -replace (x1 * 'z2 * 'y2) with (x1 * 'y2 * 'z2) by ring. -apply Zmult_gt_compat_r; auto with zarith. -Open Scope Q_scope. + unfold Qle, Qlt; intros (x1, x2) (y1, y2) (z1, z2); simpl; intros. + Open Scope Z_scope. + apply Zgt_lt. + apply Zmult_gt_reg_r with ('y2). + red; trivial. + apply Zle_gt_trans with (y1 * 'x2 * 'z2). + replace (y1 * 'x2 * 'z2) with (y1 * 'z2 * 'x2) by ring. + replace (z1 * 'x2 * 'y2) with (z1 * 'y2 * 'x2) by ring. + apply Zmult_le_compat_r; auto with zarith. + replace (x1 * 'z2 * 'y2) with (x1 * 'y2 * 'z2) by ring. + apply Zmult_gt_compat_r; auto with zarith. + Close Scope Z_scope. Qed. Lemma Qlt_trans : forall x y z, x<y -> y<z -> x<z. Proof. -intros. -apply Qle_lt_trans with y; auto. -apply Qlt_le_weak; auto. + intros. + apply Qle_lt_trans with y; auto. + apply Qlt_le_weak; auto. Qed. (** [x<y] iff [~(y<=x)] *) Lemma Qnot_lt_le : forall x y, ~ x<y -> y<=x. Proof. -unfold Qle, Qlt; auto with zarith. + unfold Qle, Qlt; auto with zarith. Qed. Lemma Qnot_le_lt : forall x y, ~ x<=y -> y<x. Proof. -unfold Qle, Qlt; auto with zarith. + unfold Qle, Qlt; auto with zarith. Qed. Lemma Qlt_not_le : forall x y, x<y -> ~ y<=x. Proof. -unfold Qle, Qlt; auto with zarith. + unfold Qle, Qlt; auto with zarith. Qed. Lemma Qle_not_lt : forall x y, x<=y -> ~ y<x. Proof. -unfold Qle, Qlt; auto with zarith. + unfold Qle, Qlt; auto with zarith. Qed. Lemma Qle_lt_or_eq : forall x y, x<=y -> x<y \/ x==y. Proof. -unfold Qle, Qlt, Qeq; intros; apply Zle_lt_or_eq; auto. + unfold Qle, Qlt, Qeq; intros; apply Zle_lt_or_eq; auto. Qed. (** Some decidability results about orders. *) Lemma Q_dec : forall x y, {x<y} + {y<x} + {x==y}. Proof. -unfold Qlt, Qle, Qeq; intros. -exact (Z_dec' (Qnum x * QDen y) (Qnum y * QDen x)). + unfold Qlt, Qle, Qeq; intros. + exact (Z_dec' (Qnum x * QDen y) (Qnum y * QDen x)). Defined. Lemma Qlt_le_dec : forall x y, {x<y} + {y<=x}. Proof. -unfold Qlt, Qle; intros. -exact (Z_lt_le_dec (Qnum x * QDen y) (Qnum y * QDen x)). + unfold Qlt, Qle; intros. + exact (Z_lt_le_dec (Qnum x * QDen y) (Qnum y * QDen x)). Defined. (** Compatibility of operations with respect to order. *) Lemma Qopp_le_compat : forall p q, p<=q -> -q <= -p. Proof. -intros (a1,a2) (b1,b2); unfold Qle, Qlt; simpl. -do 2 rewrite <- Zopp_mult_distr_l; omega. + intros (a1,a2) (b1,b2); unfold Qle, Qlt; simpl. + do 2 rewrite <- Zopp_mult_distr_l; omega. Qed. Lemma Qle_minus_iff : forall p q, p <= q <-> 0 <= q+-p. Proof. -intros (x1,x2) (y1,y2); unfold Qle; simpl. -rewrite <- Zopp_mult_distr_l. -split; omega. + intros (x1,x2) (y1,y2); unfold Qle; simpl. + rewrite <- Zopp_mult_distr_l. + split; omega. Qed. Lemma Qlt_minus_iff : forall p q, p < q <-> 0 < q+-p. Proof. -intros (x1,x2) (y1,y2); unfold Qlt; simpl. -rewrite <- Zopp_mult_distr_l. -split; omega. + intros (x1,x2) (y1,y2); unfold Qlt; simpl. + rewrite <- Zopp_mult_distr_l. + split; omega. Qed. Lemma Qplus_le_compat : - forall x y z t, x<=y -> z<=t -> x+z <= y+t. -Proof. -unfold Qplus, Qle; intros (x1, x2) (y1, y2) (z1, z2) (t1, t2); - simpl; simpl_mult. -Open Scope Z_scope. -intros. -match goal with |- ?a <= ?b => ring a; ring b end. -apply Zplus_le_compat. -replace ('t2 * ('y2 * (z1 * 'x2))) with (z1 * 't2 * ('y2 * 'x2)) by ring. -replace ('z2 * ('x2 * (t1 * 'y2))) with (t1 * 'z2 * ('y2 * 'x2)) by ring. -apply Zmult_le_compat_r; auto with zarith. -replace ('t2 * ('y2 * ('z2 * x1))) with (x1 * 'y2 * ('z2 * 't2)) by ring. -replace ('z2 * ('x2 * ('t2 * y1))) with (y1 * 'x2 * ('z2 * 't2)) by ring. -apply Zmult_le_compat_r; auto with zarith. -Open Scope Q_scope. + forall x y z t, x<=y -> z<=t -> x+z <= y+t. +Proof. + unfold Qplus, Qle; intros (x1, x2) (y1, y2) (z1, z2) (t1, t2); + simpl; simpl_mult. + Open Scope Z_scope. + intros. + match goal with |- ?a <= ?b => ring_simplify a b end. + rewrite Zplus_comm. + apply Zplus_le_compat. + match goal with |- ?a <= ?b => ring_simplify z1 t1 ('z2) ('t2) a b end. + auto with zarith. + match goal with |- ?a <= ?b => ring_simplify x1 y1 ('x2) ('y2) a b end. + auto with zarith. + Close Scope Z_scope. Qed. Lemma Qmult_le_compat_r : forall x y z, x <= y -> 0 <= z -> x*z <= y*z. Proof. -intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl. -Open Scope Z_scope. -intros; simpl_mult. -replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring. -replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring. -apply Zmult_le_compat_r; auto with zarith. -Open Scope Q_scope. + intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl. + Open Scope Z_scope. + intros; simpl_mult. + replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring. + replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring. + apply Zmult_le_compat_r; auto with zarith. + Close Scope Z_scope. Qed. Lemma Qmult_lt_0_le_reg_r : forall x y z, 0 < z -> x*z <= y*z -> x <= y. Proof. -intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl. -Open Scope Z_scope. -simpl_mult. -replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring. -replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring. -intros; apply Zmult_le_reg_r with (c1*'c2); auto with zarith. -Open Scope Q_scope. + intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl. + Open Scope Z_scope. + simpl_mult. + replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring. + replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring. + intros; apply Zmult_le_reg_r with (c1*'c2); auto with zarith. + Close Scope Z_scope. Qed. Lemma Qmult_lt_compat_r : forall x y z, 0 < z -> x < y -> x*z < y*z. Proof. -intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl. -Open Scope Z_scope. -intros; simpl_mult. -replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring. -replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring. -apply Zmult_lt_compat_r; auto with zarith. -apply Zmult_lt_0_compat. -omega. -compute; auto. -Open Scope Q_scope. + intros (a1,a2) (b1,b2) (c1,c2); unfold Qle, Qlt; simpl. + Open Scope Z_scope. + intros; simpl_mult. + replace (a1*c1*('b2*'c2)) with ((a1*'b2)*(c1*'c2)) by ring. + replace (b1*c1*('a2*'c2)) with ((b1*'a2)*(c1*'c2)) by ring. + apply Zmult_lt_compat_r; auto with zarith. + apply Zmult_lt_0_compat. + omega. + compute; auto. + Close Scope Z_scope. Qed. -(** Rational to the n-th power *) +(** * Rational to the n-th power *) Fixpoint Qpower (q:Q)(n:nat) { struct n } : Q := - match n with - | O => 1 - | S n => q * (Qpower q n) - end. + match n with + | O => 1 + | S n => q * (Qpower q n) + end. Notation " q ^ n " := (Qpower q n) : Q_scope. Lemma Qpower_1 : forall n, 1^n == 1. Proof. -induction n; simpl; auto with qarith. -rewrite IHn; auto with qarith. + induction n; simpl; auto with qarith. + rewrite IHn; auto with qarith. Qed. Lemma Qpower_0 : forall n, n<>O -> 0^n == 0. Proof. -destruct n; simpl. -destruct 1; auto. -intros. -compute; auto. + destruct n; simpl. + destruct 1; auto. + intros. + compute; auto. Qed. Lemma Qpower_pos : forall p n, 0 <= p -> 0 <= p^n. Proof. -induction n; simpl; auto with qarith. -intros; compute; intro; discriminate. -intros. -apply Qle_trans with (0*(p^n)). -compute; intro; discriminate. -apply Qmult_le_compat_r; auto. + induction n; simpl; auto with qarith. + intros; compute; intro; discriminate. + intros. + apply Qle_trans with (0*(p^n)). + compute; intro; discriminate. + apply Qmult_le_compat_r; auto. Qed. Lemma Qinv_power_n : forall n p, (1#p)^n == /(inject_Z ('p))^n. Proof. -induction n. -compute; auto. -simpl. -intros; rewrite IHn; clear IHn. -unfold Qdiv; rewrite Qinv_mult_distr. -setoid_replace (1#p) with (/ inject_Z ('p)). -apply Qeq_refl. -compute; auto. + induction n. + compute; auto. + simpl. + intros; rewrite IHn; clear IHn. + unfold Qdiv; rewrite Qinv_mult_distr. + setoid_replace (1#p) with (/ inject_Z ('p)). + apply Qeq_refl. + compute; auto. Qed. |