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(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2016 *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(** * BigQ: an efficient implementation of rational numbers *)
(** Initial authors: Benjamin Gregoire, Laurent Thery, INRIA, 2007 *)
Require Export BigZ.
Require Import Field Qfield QSig QMake Orders GenericMinMax.
(** We choose for BigQ an implemention with
multiple representation of 0: 0, 1/0, 2/0 etc.
See [QMake.v] *)
(** First, we provide translations functions between [BigN] and [BigZ] *)
Module BigN_BigZ <: NType_ZType BigN.BigN BigZ.
Definition Z_of_N := BigZ.Pos.
Lemma spec_Z_of_N : forall n, BigZ.to_Z (Z_of_N n) = BigN.to_Z n.
Proof.
reflexivity.
Qed.
Definition Zabs_N := BigZ.to_N.
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, Z.mul_comm; apply Z.sgn_abs.
Qed.
End BigN_BigZ.
(** This allows building [BigQ] out of [BigN] and [BigQ] via [QMake] *)
Delimit Scope bigQ_scope with bigQ.
Module BigQ <: QType <: OrderedTypeFull <: TotalOrder.
Include QMake.Make BigN BigZ BigN_BigZ
<+ !QProperties <+ HasEqBool2Dec
<+ !MinMaxLogicalProperties <+ !MinMaxDecProperties.
Ltac order := Private_Tac.order.
End BigQ.
(** Notations about [BigQ] *)
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.
Local Notation "1" := BigQ.one : bigQ_scope.
Infix "+" := BigQ.add : bigQ_scope.
Infix "-" := BigQ.sub : bigQ_scope.
Notation "- x" := (BigQ.opp x) : bigQ_scope.
Infix "*" := BigQ.mul : bigQ_scope.
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) (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_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.
(** [BigQ] is a field *)
Lemma BigQfieldth :
field_theory 0 1 BigQ.add BigQ.mul BigQ.sub BigQ.opp
BigQ.div BigQ.inv BigQ.eq.
Proof.
constructor.
constructor.
exact BigQ.add_0_l. exact BigQ.add_comm. exact BigQ.add_assoc.
exact BigQ.mul_1_l. exact BigQ.mul_comm. exact BigQ.mul_assoc.
exact BigQ.mul_add_distr_r. exact BigQ.sub_add_opp.
exact BigQ.add_opp_diag_r. exact BigQ.neq_1_0.
exact BigQ.div_mul_inv. exact BigQ.mul_inv_diag_l.
Qed.
Declare Equivalent Keys pow_N pow_pos.
Lemma BigQpowerth :
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).
destruct n. reflexivity.
induction p; simpl; auto; rewrite ?BigQ.spec_mul, ?IHp; reflexivity.
Qed.
Ltac isBigQcst t :=
match t with
| BigQ.Qz ?t => isBigZcst t
| BigQ.Qq ?n ?d => match isBigZcst n with
| true => isBigNcst d
| false => constr:(false)
end
| BigQ.zero => constr:(true)
| BigQ.one => constr:(true)
| BigQ.minus_one => constr:(true)
| _ => constr:(false)
end.
Ltac BigQcst t :=
match isBigQcst t with
| true => constr:(t)
| false => constr:(NotConstant)
end.
Add Field BigQfield : BigQfieldth
(decidable BigQ.eqb_correct,
completeness BigQ.eqb_complete,
constants [BigQcst],
power_tac BigQpowerth [Qpow_tac]).
Section TestField.
Let ex1 : forall x y z, (x+y)*z == (x*z)+(y*z).
intros.
ring.
Qed.
Let ex8 : forall x, x ^ 2 == x*x.
intro.
ring.
Qed.
Let ex10 : forall x y, y!=0 -> (x/y)*y == x.
intros.
field.
auto.
Qed.
End TestField.
(** [BigQ] can also benefit from an "order" tactic *)
Ltac bigQ_order := BigQ.order.
Section TestOrder.
Let test : forall x y : bigQ, x<=y -> y<=x -> x==y.
Proof. bigQ_order. Qed.
End TestOrder.
(** We can also reason by switching to QArith thanks to tactic
BigQ.qify. *)
Section TestQify.
Let test : forall x : bigQ, 0+x == 1*x.
Proof. intro x. BigQ.qify. ring. Qed.
End TestQify.
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