diff options
Diffstat (limited to 'theories/QArith/QArith_base.v')
| -rw-r--r-- | theories/QArith/QArith_base.v | 145 |
1 files changed, 103 insertions, 42 deletions
diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v index 304fbf77..78cf2892 100644 --- a/theories/QArith/QArith_base.v +++ b/theories/QArith/QArith_base.v @@ -6,11 +6,11 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: QArith_base.v 10765 2008-04-08 16:15:23Z msozeau $ i*) +(*i $Id: QArith_base.v 11301 2008-08-04 19:41:18Z herbelin $ i*) Require Export ZArith. Require Export ZArithRing. -Require Export Setoid. +Require Export Setoid Bool. (** * Definition of [Q] and basic properties *) @@ -28,24 +28,24 @@ Ltac simpl_mult := repeat rewrite Zpos_mult_morphism. Notation "a # b" := (Qmake a b) (at level 55, no associativity) : Q_scope. -Definition inject_Z (x : Z) := Qmake x 1. +Definition inject_Z (x : Z) := Qmake x 1. Arguments Scope inject_Z [Z_scope]. -Notation " 'QDen' p " := (Zpos (Qden p)) (at level 20, no associativity) : Q_scope. +Notation QDen p := (Zpos (Qden p)). Notation " 0 " := (0#1) : Q_scope. Notation " 1 " := (1#1) : Q_scope. Definition Qeq (p q : Q) := (Qnum p * QDen q)%Z = (Qnum q * QDen p)%Z. Definition Qle (x y : Q) := (Qnum x * QDen y <= Qnum y * QDen x)%Z. Definition Qlt (x y : Q) := (Qnum x * QDen y < Qnum y * QDen x)%Z. -Notation Qgt := (fun a b : Q => Qlt b a). -Notation Qge := (fun a b : Q => Qle b a). +Notation Qgt a b := (Qlt b a) (only parsing). +Notation Qge a b := (Qle b a) (only parsing). -Infix "==" := Qeq (at level 70, no associativity) : Q_scope. +Infix "==" := Qeq (at level 70, no associativity) : Q_scope. Infix "<" := Qlt : Q_scope. -Infix ">" := Qgt : Q_scope. Infix "<=" := Qle : Q_scope. -Infix ">=" := Qge : Q_scope. +Notation "x > y" := (Qlt y x)(only parsing) : Q_scope. +Notation "x >= y" := (Qle y x)(only parsing) : Q_scope. Notation "x <= y <= z" := (x<=y/\y<=z) : Q_scope. (** Another approach : using Qcompare for defining order relations. *) @@ -84,7 +84,7 @@ rewrite Zcompare_Gt_Lt_antisym; auto. rewrite Zcompare_Gt_Lt_antisym in H; auto. Qed. -Hint Unfold Qeq Qlt Qle: qarith. +Hint Unfold Qeq Qlt Qle : qarith. Hint Extern 5 (?X1 <> ?X2) => intro; discriminate: qarith. (** * Properties of equality. *) @@ -94,7 +94,7 @@ Proof. auto with qarith. Qed. -Theorem Qeq_sym : forall x y, x == y -> y == x. +Theorem Qeq_sym : forall x y, x == y -> y == x. Proof. auto with qarith. Qed. @@ -102,7 +102,7 @@ Qed. Theorem Qeq_trans : forall x y z, x == y -> y == z -> x == z. Proof. unfold Qeq in |- *; intros. -apply Zmult_reg_l with (QDen y). +apply Zmult_reg_l with (QDen y). auto with qarith. transitivity (Qnum x * QDen y * QDen z)%Z; try ring. rewrite H. @@ -117,6 +117,44 @@ Proof. intros; case (Z_eq_dec (Qnum x * QDen y) (Qnum y * QDen x)); auto. Defined. +Definition Qeq_bool x y := + (Zeq_bool (Qnum x * QDen y) (Qnum y * QDen x))%Z. + +Definition Qle_bool x y := + (Zle_bool (Qnum x * QDen y) (Qnum y * QDen x))%Z. + +Lemma Qeq_bool_iff : forall x y, Qeq_bool x y = true <-> x == y. +Proof. + unfold Qeq_bool, Qeq; intros. + symmetry; apply Zeq_is_eq_bool. +Qed. + +Lemma Qeq_bool_eq : forall x y, Qeq_bool x y = true -> x == y. +Proof. + intros; rewrite <- Qeq_bool_iff; auto. +Qed. + +Lemma Qeq_eq_bool : forall x y, x == y -> Qeq_bool x y = true. +Proof. + intros; rewrite Qeq_bool_iff; auto. +Qed. + +Lemma Qeq_bool_neq : forall x y, Qeq_bool x y = false -> ~ x == y. +Proof. + intros x y H; rewrite <- Qeq_bool_iff, H; discriminate. +Qed. + +Lemma Qle_bool_iff : forall x y, Qle_bool x y = true <-> x <= y. +Proof. + unfold Qle_bool, Qle; intros. + symmetry; apply Zle_is_le_bool. +Qed. + +Lemma Qle_bool_imp_le : forall x y, Qle_bool x y = true -> x <= y. +Proof. + intros; rewrite <- Qle_bool_iff; auto. +Qed. + (** We now consider [Q] seen as a setoid. *) Definition Q_Setoid : Setoid_Theory Q Qeq. @@ -130,7 +168,7 @@ 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. -Theorem Qnot_eq_sym : forall x y, ~x == y -> ~y == x. +Theorem Qnot_eq_sym : forall x y, ~x == y -> ~y == x. Proof. auto with qarith. Qed. @@ -139,13 +177,13 @@ Hint Resolve Qnot_eq_sym : qarith. (** * Addition, multiplication and opposite *) -(** The addition, multiplication and opposite are defined +(** The addition, multiplication and opposite are defined in the straightforward way: *) Definition Qplus (x y : Q) := (Qnum x * QDen y + Qnum y * QDen x) # (Qden x * Qden y). -Definition Qmult (x y : Q) := (Qnum x * Qnum y) # (Qden x * Qden y). +Definition Qmult (x y : Q) := (Qnum x * Qnum y) # (Qden x * Qden y). Definition Qopp (x : Q) := (- Qnum x) # (Qden x). @@ -164,8 +202,8 @@ Infix "+" := Qplus : Q_scope. Notation "- x" := (Qopp x) : Q_scope. Infix "-" := Qminus : Q_scope. Infix "*" := Qmult : Q_scope. -Notation "/ x" := (Qinv x) : Q_scope. -Infix "/" := Qdiv : Q_scope. +Notation "/ x" := (Qinv x) : Q_scope. +Infix "/" := Qdiv : Q_scope. (** A light notation for [Zpos] *) @@ -181,7 +219,7 @@ Qed. (** * Setoid compatibility results *) -Add Morphism Qplus : Qplus_comp. +Add Morphism Qplus : Qplus_comp. Proof. unfold Qeq, Qplus; simpl. Open Scope Z_scope. @@ -208,7 +246,7 @@ Qed. Add Morphism Qminus : Qminus_comp. Proof. intros. - unfold Qminus. + unfold Qminus. rewrite H; rewrite H0; auto with qarith. Qed. @@ -232,11 +270,11 @@ Proof. Open Scope Z_scope. intros (p1, p2) (q1, q2); simpl. case p1; simpl. - intros. + intros. assert (q1 = 0). elim (Zmult_integral q1 ('p2)); auto with zarith. intros; discriminate. - subst; auto. + 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. @@ -254,7 +292,7 @@ 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 *. @@ -289,14 +327,26 @@ Proof. 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. + 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. +Add Morphism Qeq_bool with signature Qeq ==> Qeq ==> (@eq bool) as Qeqb_comp. +Proof. + intros; apply eq_true_iff_eq. + rewrite 2 Qeq_bool_iff, H, H0; split; auto with qarith. +Qed. + +Add Morphism Qle_bool with signature Qeq ==> Qeq ==> (@eq bool) as Qleb_comp. +Proof. + intros; apply eq_true_iff_eq. + rewrite 2 Qle_bool_iff, H, H0. + split; auto with qarith. +Qed. -Lemma Qcompare_egal_dec: forall n m p q : Q, +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)). Proof. intros. @@ -306,7 +356,6 @@ Proof. omega. Qed. - Add Morphism Qcompare : Qcompare_comp. Proof. intros; apply Qcompare_egal_dec; rewrite H; rewrite H0; auto. @@ -341,7 +390,7 @@ Lemma Qplus_0_r : forall x, x+0 == x. Proof. intros (x1, x2); unfold Qeq, Qplus; simpl. rewrite Pmult_comm; simpl; ring. -Qed. +Qed. (** Commutativity of addition: *) @@ -374,6 +423,18 @@ Proof. intros; red; simpl; rewrite Pmult_assoc; ring. Qed. +(** multiplication and zero *) + +Lemma Qmult_0_l : forall x , 0*x == 0. +Proof. + intros; compute; reflexivity. +Qed. + +Lemma Qmult_0_r : forall x , x*0 == 0. +Proof. + intros; red; simpl; ring. +Qed. + (** [1] is a neutral element for multiplication: *) Lemma Qmult_1_l : forall n, 1*n == n. @@ -385,7 +446,7 @@ 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. + rewrite Pmult_comm; simpl; trivial. Qed. (** Commutativity of multiplication *) @@ -427,7 +488,7 @@ Proof. rewrite <- H0; ring. Qed. -(** * Inverse and division. *) +(** * Inverse and division. *) Lemma Qinv_involutive : forall q, (/ / q) == q. Proof. @@ -438,13 +499,13 @@ 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. + 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 x1; simpl; auto; destruct y1; simpl; auto. Qed. @@ -485,10 +546,10 @@ Proof. 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. + 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. + apply Zmult_le_compat_r; auto with zarith. Close Scope Z_scope. Qed. @@ -516,9 +577,9 @@ Proof. 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. + 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. + apply Zmult_le_compat_r; auto with zarith. Close Scope Z_scope. Qed. @@ -532,9 +593,9 @@ Proof. 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. + 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. + apply Zmult_gt_compat_r; auto with zarith. Close Scope Z_scope. Qed. @@ -572,7 +633,7 @@ Proof. unfold Qle, Qlt, Qeq; intros; apply Zle_lt_or_eq; auto. Qed. -Hint Resolve Qle_not_lt Qlt_not_le Qnot_le_lt Qnot_lt_le +Hint Resolve Qle_not_lt Qlt_not_le Qnot_le_lt Qnot_lt_le Qlt_le_weak Qlt_not_eq Qle_antisym Qle_refl: qartih. (** Some decidability results about orders. *) @@ -679,7 +740,7 @@ Proof. intros [[|n|n] d] Ha; assumption. Qed. -Lemma Qle_shift_div_l : forall a b c, +Lemma Qle_shift_div_l : forall a b c, 0 < c -> a*c <= b -> a <= b/c. Proof. intros a b c Hc H. @@ -690,7 +751,7 @@ rewrite Qmult_div_r; try assumption. auto with *. Qed. -Lemma Qle_shift_inv_l : forall a c, +Lemma Qle_shift_inv_l : forall a c, 0 < c -> a*c <= 1 -> a <= /c. Proof. intros a c Hc H. @@ -699,7 +760,7 @@ change (a <= 1/c). apply Qle_shift_div_l; assumption. Qed. -Lemma Qle_shift_div_r : forall a b c, +Lemma Qle_shift_div_r : forall a b c, 0 < b -> a <= c*b -> a/b <= c. Proof. intros a b c Hc H. @@ -710,7 +771,7 @@ rewrite Qmult_div_r; try assumption. auto with *. Qed. -Lemma Qle_shift_inv_r : forall b c, +Lemma Qle_shift_inv_r : forall b c, 0 < b -> 1 <= c*b -> /b <= c. Proof. intros b c Hc H. @@ -772,7 +833,7 @@ Qed. (** * Rational to the n-th power *) -Definition Qpower_positive (q:Q)(p:positive) : Q := +Definition Qpower_positive (q:Q)(p:positive) : Q := pow_pos Qmult q p. Add Morphism Qpower_positive with signature Qeq ==> @eq _ ==> Qeq as Qpower_positive_comp. |