The Coq Standard Library

# Library Coq.Numbers.Rational.BigQ.BigQ

# 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.
Definition Zabs_N := BigZ.to_N.
Lemma spec_Zabs_N : forall z, BigN.to_Z (Zabs_N z) = Z.abs (BigZ.to_Z z).
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.
As in QArith, we use # to denote fractions
Notation "p # q" := (BigQ.Qq p q) (at level 55, no associativity) : 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.
Lemma BigQpowerth :
power_theory 1 BigQ.mul BigQ.eq Z.of_N BigQ.power.
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).
Qed.
Let ex8 : forall x, x ^ 2 == x*x.
Qed.
Let ex10 : forall x y, y!=0 -> (x/y)*y == x.
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.
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.
End TestQify.