(* Unital *)
Instance sumAlg_unital:
@UnitalSeparationAlgebra _ nat_geR sum_Join.
(*Residual*)
Instance sumAlg_residual:
@ResidualSeparationAlgebra _ nat_geR sum_Join.
(** *Minimum Algebra on Positive integers*)
Record natPlus: Type:=
natp { nat_p:> nat;
is_pos: 0 < nat_p}.
Inductive sump_Join: natPlus -> natPlus -> natPlus -> Prop:=
|sump_j (x y z:natPlus): x + y = z -> sump_Join x y z.
Definition lep (n m:natPlus):= n <= m.

(* Specifications: *)
Lemma lep_preorder: RelationClasses.PreOrder lep.
Definition natPlus_R: Relation natPlus:= lep.