Library Coq.Arith.NatOrderedType

DecidableType structure for Peano numbers

Module Nat_as_UBE <: UsualBoolEq.
Definition t := nat.
Definition eq := @eq nat.
Definition eqb := beq_nat.
Definition eqb_eq := beq_nat_true_iff.
End Nat_as_UBE.

Module Nat_as_DT <: UsualDecidableTypeFull := Make_UDTF Nat_as_UBE.

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

Module Nat_as_OT <: OrderedTypeFull.
Include Nat_as_DT.
Definition lt := lt.
Definition le := le.
Definition compare := nat_compare.

Instance lt_strorder : StrictOrder lt.

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

Definition le_lteq := le_lt_or_eq_iff.
Definition compare_spec := nat_compare_spec.

End Nat_as_OT.

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

Module NatOrder := OTF_to_OrderTac Nat_as_OT.
Ltac nat_order := NatOrder.order.

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

Section Test.
Let test : forall x y : nat, x<=y -> y<=x -> x=y.
End Test.