diff options
Diffstat (limited to 'theories/QArith/Qcanon.v')
| -rw-r--r-- | theories/QArith/Qcanon.v | 24 |
1 files changed, 12 insertions, 12 deletions
diff --git a/theories/QArith/Qcanon.v b/theories/QArith/Qcanon.v index a865a6cf..f7f83bf0 100644 --- a/theories/QArith/Qcanon.v +++ b/theories/QArith/Qcanon.v @@ -1,6 +1,6 @@ (************************************************************************) (* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2014 *) +(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2015 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -70,7 +70,6 @@ Qed. Definition Q2Qc (q:Q) : Qc := Qcmake (Qred q) (Qred_involutive q). Arguments Q2Qc q%Q. -Notation " !! " := Q2Qc : Qc_scope. Lemma Qc_is_canon : forall q q' : Qc, q == q' -> q = q'. Proof. @@ -87,8 +86,8 @@ Proof. Qed. Hint Resolve Qc_is_canon. -Notation " 0 " := (!!0) : Qc_scope. -Notation " 1 " := (!!1) : Qc_scope. +Notation " 0 " := (Q2Qc 0) : Qc_scope. +Notation " 1 " := (Q2Qc 1) : Qc_scope. Definition Qcle (x y : Qc) := (x <= y)%Q. Definition Qclt (x y : Qc) := (x < y)%Q. @@ -144,15 +143,15 @@ Defined. (** The addition, multiplication and opposite are defined in the straightforward way: *) -Definition Qcplus (x y : Qc) := !!(x+y). +Definition Qcplus (x y : Qc) := Q2Qc (x+y). Infix "+" := Qcplus : Qc_scope. -Definition Qcmult (x y : Qc) := !!(x*y). +Definition Qcmult (x y : Qc) := Q2Qc (x*y). Infix "*" := Qcmult : Qc_scope. -Definition Qcopp (x : Qc) := !!(-x). +Definition Qcopp (x : Qc) := Q2Qc (-x). Notation "- x" := (Qcopp x) : Qc_scope. Definition Qcminus (x y : Qc) := x+-y. Infix "-" := Qcminus : Qc_scope. -Definition Qcinv (x : Qc) := !!(/x). +Definition Qcinv (x : Qc) := Q2Qc (/x). Notation "/ x" := (Qcinv x) : Qc_scope. Definition Qcdiv (x y : Qc) := x*/y. Infix "/" := Qcdiv : Qc_scope. @@ -434,14 +433,14 @@ Qed. Lemma Qcmult_lt_0_le_reg_r : forall x y z, 0 < z -> x*z <= y*z -> x <= y. Proof. unfold Qcmult, Qcle, Qclt; intros; simpl in *. - repeat progress rewrite Qred_correct in * |-. + rewrite !Qred_correct in * |-. eapply Qmult_lt_0_le_reg_r; eauto. Qed. Lemma Qcmult_lt_compat_r : forall x y z, 0 < z -> x < y -> x*z < y*z. Proof. unfold Qcmult, Qclt; intros; simpl in *. - repeat progress rewrite Qred_correct in *. + rewrite !Qred_correct in *. eapply Qmult_lt_compat_r; eauto. Qed. @@ -460,13 +459,13 @@ Proof. induction n; simpl; auto with qarith. rewrite IHn; auto with qarith. Qed. - +Transparent Qred. Lemma Qcpower_0 : forall n, n<>O -> 0^n = 0. Proof. destruct n; simpl. destruct 1; auto. intros. - now apply Qc_is_canon. + now apply Qc_is_canon. Qed. Lemma Qcpower_pos : forall p n, 0 <= p -> 0 <= p^n. @@ -521,6 +520,7 @@ Add Field Qcfield : Qcft. (** A field tactic for rational numbers *) Example test_field : (forall x y : Qc, y<>0 -> (x/y)*y = x)%Qc. +Proof. intros. field. auto. |