diff options
Diffstat (limited to 'theories/QArith/QArith_base.v')
| -rw-r--r-- | theories/QArith/QArith_base.v | 169 |
1 files changed, 163 insertions, 6 deletions
diff --git a/theories/QArith/QArith_base.v b/theories/QArith/QArith_base.v index 18b8823d..94ea4906 100644 --- a/theories/QArith/QArith_base.v +++ b/theories/QArith/QArith_base.v @@ -1,13 +1,11 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2011 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2010 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id: QArith_base.v 14641 2011-11-06 11:59:10Z herbelin $ i*) - Require Export ZArith. Require Export ZArithRing. Require Export Morphisms Setoid Bool. @@ -20,7 +18,7 @@ Record Q : Set := Qmake {Qnum : Z; Qden : positive}. Delimit Scope Q_scope with Q. Bind Scope Q_scope with Q. -Arguments Scope Qmake [Z_scope positive_scope]. +Arguments Qmake _%Z _%positive. Open Scope Q_scope. Ltac simpl_mult := repeat rewrite Zpos_mult_morphism. @@ -29,7 +27,7 @@ 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. -Arguments Scope inject_Z [Z_scope]. +Arguments inject_Z x%Z. Notation QDen p := (Zpos (Qden p)). Notation " 0 " := (0#1) : Q_scope. @@ -48,6 +46,13 @@ 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. +(** injection from Z is injective. *) + +Lemma inject_Z_injective (a b: Z): inject_Z a == inject_Z b <-> a = b. +Proof. + unfold Qeq. simpl. omega. +Qed. + (** Another approach : using Qcompare for defining order relations. *) Definition Qcompare (p q : Q) := (Qnum p * QDen q ?= Qnum q * QDen p)%Z. @@ -92,7 +97,7 @@ Proof. unfold "?=". intros. apply Zcompare_antisym. Qed. -Lemma Qcompare_spec : forall x y, CompSpec Qeq Qlt x y (x ?= y). +Lemma Qcompare_spec : forall x y, CompareSpec (x==y) (x<y) (y<x) (x ?= y). Proof. intros. destruct (x ?= y) as [ ]_eqn:H; constructor; auto. @@ -387,6 +392,26 @@ Proof. red; simpl; intro; ring. Qed. +(** Injectivity of addition (uses theory about Qopp above): *) + +Lemma Qplus_inj_r (x y z: Q): + x + z == y + z <-> x == y. +Proof. + split; intro E. + rewrite <- (Qplus_0_r x), <- (Qplus_0_r y). + rewrite <- (Qplus_opp_r z); auto. + do 2 rewrite Qplus_assoc. + rewrite E. reflexivity. + rewrite E. reflexivity. +Qed. + +Lemma Qplus_inj_l (x y z: Q): + z + x == z + y <-> x == y. +Proof. + rewrite (Qplus_comm z x), (Qplus_comm z y). + apply Qplus_inj_r. +Qed. + (** * Properties of [Qmult] *) @@ -462,6 +487,21 @@ Proof. rewrite <- H0; ring. Qed. + +(** * inject_Z is a ring homomorphism: *) + +Lemma inject_Z_plus (x y: Z): inject_Z (x + y) = inject_Z x + inject_Z y. +Proof. + unfold Qplus, inject_Z. simpl. f_equal. ring. +Qed. + +Lemma inject_Z_mult (x y: Z): inject_Z (x * y) = inject_Z x * inject_Z y. +Proof. reflexivity. Qed. + +Lemma inject_Z_opp (x: Z): inject_Z (- x) = - inject_Z x. +Proof. reflexivity. Qed. + + (** * Inverse and division. *) Lemma Qinv_involutive : forall q, (/ / q) == q. @@ -500,6 +540,25 @@ Proof. apply Qdiv_mult_l; auto. Qed. +(** Injectivity of Qmult (requires theory about Qinv above): *) + +Lemma Qmult_inj_r (x y z: Q): ~ z == 0 -> (x * z == y * z <-> x == y). +Proof. + intro z_ne_0. + split; intro E. + rewrite <- (Qmult_1_r x), <- (Qmult_1_r y). + rewrite <- (Qmult_inv_r z); auto. + do 2 rewrite Qmult_assoc. + rewrite E. reflexivity. + rewrite E. reflexivity. +Qed. + +Lemma Qmult_inj_l (x y z: Q): ~ z == 0 -> (z * x == z * y <-> x == y). +Proof. + rewrite (Qmult_comm z x), (Qmult_comm z y). + apply Qmult_inj_r. +Qed. + (** * Properties of order upon Q. *) Lemma Qle_refl : forall x, x<=x. @@ -539,6 +598,19 @@ Proof. unfold Qlt, Qeq; auto with zarith. Qed. +Lemma Zle_Qle (x y: Z): (x <= y)%Z = (inject_Z x <= inject_Z y). +Proof. + unfold Qle. intros. simpl. + do 2 rewrite Zmult_1_r. reflexivity. +Qed. + +Lemma Zlt_Qlt (x y: Z): (x < y)%Z = (inject_Z x < inject_Z y). +Proof. + unfold Qlt. intros. simpl. + do 2 rewrite Zmult_1_r. reflexivity. +Qed. + + (** Large = strict or equal *) Lemma Qle_lteq : forall x y, x<=y <-> x<y \/ x==y. @@ -677,6 +749,54 @@ Proof. Close Scope Z_scope. Qed. +Lemma Qplus_lt_le_compat : + forall x y z t, x<y -> z<=t -> x+z < y+t. +Proof. + unfold Qplus, Qle, Qlt; 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_lt_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. + assert (forall p, 0 < ' p) by reflexivity. + repeat (apply Zmult_lt_compat_r; auto). + Close Scope Z_scope. +Qed. + +Lemma Qplus_le_l (x y z: Q): x + z <= y + z <-> x <= y. +Proof. + split; intros. + rewrite <- (Qplus_0_r x), <- (Qplus_0_r y), <- (Qplus_opp_r z). + do 2 rewrite Qplus_assoc. + apply Qplus_le_compat; auto with *. + apply Qplus_le_compat; auto with *. +Qed. + +Lemma Qplus_le_r (x y z: Q): z + x <= z + y <-> x <= y. +Proof. + rewrite (Qplus_comm z x), (Qplus_comm z y). + apply Qplus_le_l. +Qed. + +Lemma Qplus_lt_l (x y z: Q): x + z < y + z <-> x < y. +Proof. + split; intros. + rewrite <- (Qplus_0_r x), <- (Qplus_0_r y), <- (Qplus_opp_r z). + do 2 rewrite Qplus_assoc. + apply Qplus_lt_le_compat; auto with *. + apply Qplus_lt_le_compat; auto with *. +Qed. + +Lemma Qplus_lt_r (x y z: Q): z + x < z + y <-> x < y. +Proof. + rewrite (Qplus_comm z x), (Qplus_comm z y). + apply Qplus_lt_l. +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. @@ -699,6 +819,19 @@ Proof. Close Scope Z_scope. Qed. +Lemma Qmult_le_r (x y z: Q): 0 < z -> (x*z <= y*z <-> x <= y). +Proof. + split; intro. + now apply Qmult_lt_0_le_reg_r with z. + apply Qmult_le_compat_r; auto with qarith. +Qed. + +Lemma Qmult_le_l (x y z: Q): 0 < z -> (z*x <= z*y <-> x <= y). +Proof. + rewrite (Qmult_comm z x), (Qmult_comm z y). + apply Qmult_le_r. +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. @@ -713,6 +846,30 @@ Proof. Close Scope Z_scope. Qed. +Lemma Qmult_lt_r: forall x y z, 0 < z -> (x*z < y*z <-> x < y). +Proof. + Open Scope Z_scope. + intros (a1,a2) (b1,b2) (c1,c2). + unfold Qle, Qlt; simpl. + 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. + assert (forall p, 0 < ' p) by reflexivity. + split; intros. + apply Zmult_lt_reg_r with (c1*'c2); auto with zarith. + apply Zmult_lt_0_compat; auto with zarith. + apply Zmult_lt_compat_r; auto with zarith. + apply Zmult_lt_0_compat. omega. + compute; auto. + Close Scope Z_scope. +Qed. + +Lemma Qmult_lt_l (x y z: Q): 0 < z -> (z*x < z*y <-> x < y). +Proof. + rewrite (Qmult_comm z x), (Qmult_comm z y). + apply Qmult_lt_r. +Qed. + Lemma Qmult_le_0_compat : forall a b, 0 <= a -> 0 <= b -> 0 <= a*b. Proof. intros a b Ha Hb. |