# 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.
eq_dec
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.