Module SetOfPairsOfPositiveOrderedType <: OrderedType.
Definition t:= (positive * positive).
Definition eq (t1 t2 : t) :=
Pos.eq (fstpp(t1)) (fstpp(t2)) /\ Pos.eq (sndpp(t1)) (sndpp(t2)).
Include IsEq.
Definition eqb (t1 t2 : t) :=
Pos.eqb (fstpp(t1)) (fstpp(t2)) && Pos.eqb (sndpp(t1)) (sndpp(t2)).
Include HasEqBool2Dec.
Definition lt (t1 t2 : t) :=
let ft1 := fstpp(t1) in
let ft2 := fstpp(t2) in
let st1 := sndpp(t1) in
let st2 := sndpp(t2) in
if Pos.eqb ft1 ft2 then Pos.lt st1 st2
else Pos.lt ft1 ft2.

Lemma lt_antiref : forall x, ~ lt x x.