diff options
Diffstat (limited to 'theories/Numbers/Rational/BigQ/BigQ.v')
| -rw-r--r-- | theories/Numbers/Rational/BigQ/BigQ.v | 78 |
1 files changed, 25 insertions, 53 deletions
diff --git a/theories/Numbers/Rational/BigQ/BigQ.v b/theories/Numbers/Rational/BigQ/BigQ.v index 82190f94..a2bc5e26 100644 --- a/theories/Numbers/Rational/BigQ/BigQ.v +++ b/theories/Numbers/Rational/BigQ/BigQ.v @@ -1,6 +1,6 @@ (************************************************************************) (* 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 1999-2012 *) (* \VV/ **************************************************************) (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) @@ -26,56 +26,31 @@ Module BigN_BigZ <: NType_ZType BigN.BigN BigZ. reflexivity. Qed. Definition Zabs_N := BigZ.to_N. - Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Zabs (BigZ.to_Z z). + Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Z.abs (BigZ.to_Z z). Proof. unfold Zabs_N; intros. - rewrite BigZ.spec_to_Z, Zmult_comm; apply Zsgn_Zabs. + rewrite BigZ.spec_to_Z, Z.mul_comm; apply Z.sgn_abs. Qed. End BigN_BigZ. (** This allows to build [BigQ] out of [BigN] and [BigQ] via [QMake] *) -Module BigQ <: QType <: OrderedTypeFull <: TotalOrder := - QMake.Make BigN BigZ BigN_BigZ <+ !QProperties <+ HasEqBool2Dec - <+ !MinMaxLogicalProperties <+ !MinMaxDecProperties. +Delimit Scope bigQ_scope with bigQ. -(** Notations about [BigQ] *) +Module BigQ <: QType <: OrderedTypeFull <: TotalOrder. + Include QMake.Make BigN BigZ BigN_BigZ [scope abstract_scope to bigQ_scope]. + Bind Scope bigQ_scope with t t_. + Include !QProperties <+ HasEqBool2Dec + <+ !MinMaxLogicalProperties <+ !MinMaxDecProperties. + Ltac order := Private_Tac.order. +End BigQ. -Notation bigQ := BigQ.t. +(** Notations about [BigQ] *) -Delimit Scope bigQ_scope with bigQ. -Bind Scope bigQ_scope with bigQ. -Bind Scope bigQ_scope with BigQ.t. -Bind Scope bigQ_scope with BigQ.t_. -(* Bind Scope has no retroactive effect, let's declare scopes by hand. *) -Arguments Scope BigQ.Qz [bigZ_scope]. -Arguments Scope BigQ.Qq [bigZ_scope bigN_scope]. -Arguments Scope BigQ.to_Q [bigQ_scope]. -Arguments Scope BigQ.red [bigQ_scope]. -Arguments Scope BigQ.opp [bigQ_scope]. -Arguments Scope BigQ.inv [bigQ_scope]. -Arguments Scope BigQ.square [bigQ_scope]. -Arguments Scope BigQ.add [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.sub [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.mul [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.div [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.eq [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.lt [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.le [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.eq [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.compare [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.min [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.max [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.eq_bool [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.power_pos [bigQ_scope positive_scope]. -Arguments Scope BigQ.power [bigQ_scope Z_scope]. -Arguments Scope BigQ.inv_norm [bigQ_scope]. -Arguments Scope BigQ.add_norm [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.sub_norm [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.mul_norm [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.div_norm [bigQ_scope bigQ_scope]. -Arguments Scope BigQ.power_norm [bigQ_scope bigQ_scope]. +Local Open Scope bigQ_scope. +Notation bigQ := BigQ.t. +Bind Scope bigQ_scope with bigQ BigQ.t BigQ.t_. (** As in QArith, we use [#] to denote fractions *) Notation "p # q" := (BigQ.Qq p q) (at level 55, no associativity) : bigQ_scope. Local Notation "0" := BigQ.zero : bigQ_scope. @@ -88,19 +63,17 @@ Infix "/" := BigQ.div : bigQ_scope. Infix "^" := BigQ.power : bigQ_scope. Infix "?=" := BigQ.compare : bigQ_scope. Infix "==" := BigQ.eq : bigQ_scope. -Notation "x != y" := (~x==y)%bigQ (at level 70, no associativity) : bigQ_scope. +Notation "x != y" := (~x==y) (at level 70, no associativity) : bigQ_scope. Infix "<" := BigQ.lt : bigQ_scope. Infix "<=" := BigQ.le : bigQ_scope. -Notation "x > y" := (BigQ.lt y x)(only parsing) : bigQ_scope. -Notation "x >= y" := (BigQ.le y x)(only parsing) : bigQ_scope. -Notation "x < y < z" := (x<y /\ y<z)%bigQ : bigQ_scope. -Notation "x < y <= z" := (x<y /\ y<=z)%bigQ : bigQ_scope. -Notation "x <= y < z" := (x<=y /\ y<z)%bigQ : bigQ_scope. -Notation "x <= y <= z" := (x<=y /\ y<=z)%bigQ : bigQ_scope. +Notation "x > y" := (BigQ.lt y x) (only parsing) : bigQ_scope. +Notation "x >= y" := (BigQ.le y x) (only parsing) : bigQ_scope. +Notation "x < y < z" := (x<y /\ y<z) : bigQ_scope. +Notation "x < y <= z" := (x<y /\ y<=z) : bigQ_scope. +Notation "x <= y < z" := (x<=y /\ y<z) : bigQ_scope. +Notation "x <= y <= z" := (x<=y /\ y<=z) : bigQ_scope. Notation "[ q ]" := (BigQ.to_Q q) : bigQ_scope. -Local Open Scope bigQ_scope. - (** [BigQ] is a field *) Lemma BigQfieldth : @@ -117,10 +90,10 @@ exact BigQ.div_mul_inv. exact BigQ.mul_inv_diag_l. Qed. Lemma BigQpowerth : - power_theory 1 BigQ.mul BigQ.eq Z_of_N BigQ.power. + power_theory 1 BigQ.mul BigQ.eq Z.of_N BigQ.power. Proof. constructor. intros. BigQ.qify. -replace ([r] ^ Z_of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n). +replace ([r] ^ Z.of_N n)%Q with (pow_N 1 Qmult [r] n)%Q by (now destruct n). destruct n. reflexivity. induction p; simpl; auto; rewrite ?BigQ.spec_mul, ?IHp; reflexivity. Qed. @@ -172,8 +145,7 @@ End TestField. (** [BigQ] can also benefit from an "order" tactic *) -Module BigQ_Order := !OrdersTac.MakeOrderTac BigQ. -Ltac bigQ_order := BigQ_Order.order. +Ltac bigQ_order := BigQ.order. Section TestOrder. Let test : forall x y : bigQ, x<=y -> y<=x -> x==y. |