(* 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}.(** *Minimum Algebra on Positive integers*)
  
  Record natPlus: Type:=
   natp { nat_p:> nat;
      is_pos: 0 < nat_p}.(** *Minimum Algebra on Positive integers*)
  
  Record natPlus: Type:=
   natp { nat_p:> nat;
      is_pos: 0 < nat_p}.(** *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.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.

Lemma lep_preorder: RelationClasses.PreOrder lep.