Library Coq.NArith.NOrderedType


Require Import BinNat Equalities Orders OrdersTac.

Local Open Scope N_scope.

DecidableType structure for N binary natural numbers


Module N_as_UBE <: UsualBoolEq.
 Definition t := N.
 Definition eq := @eq N.
 Definition eqb := Neqb.
 Definition eqb_eq := Neqb_eq.
End N_as_UBE.

Module N_as_DT <: UsualDecidableTypeFull := Make_UDTF N_as_UBE.

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

OrderedType structure for N numbers


Module N_as_OT <: OrderedTypeFull.
 Include N_as_DT.
 Definition lt := Nlt.
 Definition le := Nle.
 Definition compare := Ncompare.

 Instance lt_strorder : StrictOrder Nlt.

 Instance lt_compat : Proper (Logic.eq==>Logic.eq==>iff) Nlt.

 Definition le_lteq := Nle_lteq.
 Definition compare_spec := Ncompare_spec.

End N_as_OT.

Note that N_as_OT can also be seen as a UsualOrderedType and a OrderedType (and also as a DecidableType).

An order tactic for N numbers


Module NOrder := OTF_to_OrderTac N_as_OT.
Ltac n_order := NOrder.order.

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