Definition nat_leR: Relation nat := le.Definition nat_geR: Relation nat := ge.Instance po_nat_leR: PreOrder (@Krelation _ nat_leR) := Nat.le_preorder.Instance po_nat_geR: PreOrder (@Krelation _ nat_geR).Definition indexAlg: @SeparationAlgebra nat equiv_Join.(* upwards-closed*)
  Instance IndexAlg_uSA:
    @UpwardsClosedSeparationAlgebra _  nat_geR equiv_Join.(* Increasing*)
  Instance IndexAlg_increasing:
    @IncreasingSeparationAlgebra _ nat_geR equiv_Join.(* Residual*)
  Instance IndexAlg_residual:
    @ResidualSeparationAlgebra _ nat_geR equiv_Join.(* Unital *)
  Instance IndexAlg_unital:
    @UnitalSeparationAlgebra _ nat_geR equiv_Join.(* Minimum Algebra *)
  Inductive min_Join: nat -> nat -> nat -> Prop:=
  | min_j x y z: z <= x -> z <= y -> min_Join x y z.(* Minimum Algebra *)
  Inductive min_Join: nat -> nat -> nat -> Prop:=
  | min_j x y z: z <= x -> z <= y -> min_Join x y z.Definition minAlg: @SeparationAlgebra nat min_Join.

Lemma min_Join_is_downwards_closure: min_Join = @DownwardsClosure_J _ nat_geR equiv_Join.