# 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.