Library Coq.PArith.POrderedType


Require Import BinPos Equalities Orders OrdersTac.

Local Open Scope positive_scope.

DecidableType structure for positive numbers


Module Positive_as_DT <: UsualDecidableTypeFull := Pos.

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

OrderedType structure for positive numbers


Module Positive_as_OT <: OrderedTypeFull := Pos.

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

An order tactic for positive numbers


Module PositiveOrder := OTF_to_OrderTac Positive_as_OT.
Ltac p_order := PositiveOrder.order.

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