The Coq Standard Library

# Library Coq.QArith.Qreduction

Normalisation functions for rational numbers.
Require Export QArith_base.
Require Import Znumtheory.
Notation Z2P := Z.to_pos (compat "8.3").
Notation Z2P_correct := Z2Pos.id (compat "8.3").
Simplification of fractions using Z.gcd. This version can compute within Coq.
Definition Qred (q:Q) :=
let (q1,q2) := q in
let (r1,r2) := snd (Z.ggcd q1 ('q2))
in r1#(Z.to_pos r2).
Lemma Qred_correct : forall q, (Qred q) == q.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qred_complete : forall p q, p==q -> Qred p = Qred q.
Open Scope Z_scope.
Close Scope Z_scope.
Lemma Qred_eq_iff q q' : Qred q = Qred q' <-> q == q'.
Add Morphism Qred with signature (Qeq ==> Qeq) as Qred_comp.
Definition Qplus' (p q : Q) := Qred (Qplus p q).
Definition Qmult' (p q : Q) := Qred (Qmult p q).
Definition Qminus' x y := Qred (Qminus x y).
Lemma Qplus'_correct : forall p q : Q, (Qplus' p q)==(Qplus p q).
Lemma Qmult'_correct : forall p q : Q, (Qmult' p q)==(Qmult p q).
Lemma Qminus'_correct : forall p q : Q, (Qminus' p q)==(Qminus p q).
Add Morphism Qplus' with signature (Qeq ==> Qeq ==> Qeq) as Qplus'_comp.
Add Morphism Qmult' with signature (Qeq ==> Qeq ==> Qeq) as Qmult'_comp.
Add Morphism Qminus' with signature (Qeq ==> Qeq ==> Qeq) as Qminus'_comp.
Lemma Qred_opp: forall q, Qred (-q) = - (Qred q).
Theorem Qred_compare: forall x y,
Qcompare x y = Qcompare (Qred x) (Qred y).
Lemma Qred_le q q' : Qred q <= Qred q' <-> q <= q'.
Lemma Qred_lt q q' : Qred q < Qred q' <-> q < q'.