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 : 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' : Qplus'_comp.

Add Morphism Qmult' : Qmult'_comp.

Add Morphism Qminus' : 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'.