Library Coq.QArith.QOrderedType

Require Import QArith_base Equalities Orders OrdersTac.

Local Open Scope Q_scope.

DecidableType structure for rational numbers

Module Q_as_DT <: DecidableTypeFull.
 Definition t := Q.
 Definition eq := Qeq.
 Definition eq_equiv := Q_Setoid.
 Definition eqb := Qeq_bool.
 Definition eqb_eq := Qeq_bool_iff.

 Include BackportEq.
eq_refl, eq_sym, eq_trans
 Include HasEqBool2Dec.

End Q_as_DT.

Note that the last module fulfills by subtyping many other interfaces, such as DecidableType or EqualityType.

OrderedType structure for rational numbers

Module Q_as_OT <: OrderedTypeFull.
 Include Q_as_DT.
 Definition lt := Qlt.
 Definition le := Qle.
 Definition compare := Qcompare.

 Instance lt_strorder : StrictOrder Qlt.

 Instance lt_compat : Proper (Qeq==>Qeq==>iff) Qlt.

 Definition le_lteq := Qle_lteq.
 Definition compare_spec := Qcompare_spec.

End Q_as_OT.

An order tactic for Q numbers

Module QOrder := OTF_to_OrderTac Q_as_OT.
Ltac q_order := QOrder.order.

Note that q_order is domain-agnostic: it will not prove 1<=2 or x<=x+x, but rather things like x<=y -> y<=x -> x==y.