Definition trivial_Join: Join worlds:=  (fun a b c => False).Definition trivial_SA: @SeparationAlgebra worlds trivial_Join.(* Trivial algebra is upwards closed *)
  (* Trivial is NOT downwards closed *)
  Definition trivial_uSA {R: Relation worlds}:
    @UpwardsClosedSeparationAlgebra worlds R (trivial_Join).(*Increasing*)
  Definition trivial_incrSA: @IncreasingSeparationAlgebra
                           worlds eq trivial_Join.Definition unit_Join: Join unit:=  (fun _ _ _ => True).Definition unit_SA: @SeparationAlgebra unit unit_Join.(* Unit algebra is upwards closed *)
  Definition unit_uSA:
    @UpwardsClosedSeparationAlgebra unit eq unit_Join.(* Unit algebra is downwards closed *)
  Definition unit_dSA:
    @DownwardsClosedSeparationAlgebra unit eq unit_Join.(*Increasing*)
  Instance unit_incrSA:
    @IncreasingSeparationAlgebra unit eq unit_Join.Instance unit_residual:
    @ResidualSeparationAlgebra unit eq unit_Join.Definition unit_unital:
    @UnitalSeparationAlgebra unit eq unit_Join.Definition equiv_Join: Join worlds:=  (fun a b c => a = c /\ b = c).Definition equiv_SA: @SeparationAlgebra worlds equiv_Join.(* Identity algebra is upwards closed *)
  (* Identity is NOT downwards closed *)
  Definition identity_uSA {R: Relation worlds}:
    @UpwardsClosedSeparationAlgebra worlds R equiv_Join.Definition equiv_incrSA: @IncreasingSeparationAlgebra
                           worlds eq equiv_Join.Definition ikiM_uSA {R: Relation worlds} {po_R: PreOrder Krelation} {ikiM: IdentityKripkeIntuitionisticModel worlds} {J: Join worlds}: UpwardsClosedSeparationAlgebra worlds.Definition ikiM_dSA {R: Relation worlds} {po_R: PreOrder Krelation} {ikiM: IdentityKripkeIntuitionisticModel worlds} {J: Join worlds}: DownwardsClosedSeparationAlgebra worlds.Inductive option_join {J: Join worlds}: option worlds -> option worlds -> option worlds -> Prop :=
  | None_None_join: option_join None None None
  | None_Some_join: forall a, option_join None (Some a) (Some a)
  | Some_None_join: forall a, option_join (Some a) None (Some a)
  | Some_Some_join: forall a b c, join a b c -> option_join (Some a) (Some b) (Some c).Inductive option_join {J: Join worlds}: option worlds -> option worlds -> option worlds -> Prop :=
  | None_None_join: option_join None None None
  | None_Some_join: forall a, option_join None (Some a) (Some a)
  | Some_None_join: forall a, option_join (Some a) None (Some a)
  | Some_Some_join: forall a b c, join a b c -> option_join (Some a) (Some b) (Some c).Inductive option_join {J: Join worlds}: option worlds -> option worlds -> option worlds -> Prop :=
  | None_None_join: option_join None None None
  | None_Some_join: forall a, option_join None (Some a) (Some a)
  | Some_None_join: forall a, option_join (Some a) None (Some a)
  | Some_Some_join: forall a b c, join a b c -> option_join (Some a) (Some b) (Some c).Inductive option_join {J: Join worlds}: option worlds -> option worlds -> option worlds -> Prop :=
  | None_None_join: option_join None None None
  | None_Some_join: forall a, option_join None (Some a) (Some a)
  | Some_None_join: forall a, option_join (Some a) None (Some a)
  | Some_Some_join: forall a b c, join a b c -> option_join (Some a) (Some b) (Some c).Inductive option_join {J: Join worlds}: option worlds -> option worlds -> option worlds -> Prop :=
  | None_None_join: option_join None None None
  | None_Some_join: forall a, option_join None (Some a) (Some a)
  | Some_None_join: forall a, option_join (Some a) None (Some a)
  | Some_Some_join: forall a b c, join a b c -> option_join (Some a) (Some b) (Some c).Definition option_Join {SA: Join worlds}: Join (option worlds):=
    (@option_join SA).Definition option_SA
             {J: Join worlds}
             {SA: SeparationAlgebra worlds}:
    @SeparationAlgebra (option worlds) (option_Join).

(* Specifications: *)
Lemma option_ord_uSA {R: Relation worlds} {J: Join worlds} (uSA: UpwardsClosedSeparationAlgebra worlds): @UpwardsClosedSeparationAlgebra (option worlds) (option01_relation Krelation) option_Join.
Lemma option_ord_dSA {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds} (dSA: DownwardsClosedSeparationAlgebra worlds) {incrSA: IncreasingSeparationAlgebra worlds}: @DownwardsClosedSeparationAlgebra (option worlds) (option01_relation Krelation) option_Join.
Lemma option_ord_incr_None {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: @increasing (option worlds) (option01_relation Krelation) option_Join None.
Lemma option_ord_res_None {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: forall n, @residue (option worlds) (option01_relation Krelation) option_Join n None.
Lemma option_ord_USA {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: @UnitalSeparationAlgebra (option worlds) (option01_relation Krelation) option_Join.
Lemma option_disj_uSA {R: Relation worlds} {J: Join worlds} (uSA: UpwardsClosedSeparationAlgebra worlds): @UpwardsClosedSeparationAlgebra (option worlds) (option00_relation Krelation) option_Join.
Lemma option_disj_dSA {R: Relation worlds} {J: Join worlds} {SA: SeparationAlgebra worlds} (dSA: DownwardsClosedSeparationAlgebra worlds): @DownwardsClosedSeparationAlgebra (option worlds) (option00_relation Krelation) option_Join.
Lemma option_disj_incr_None {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: @increasing (option worlds) (option00_relation Krelation) option_Join None.
Lemma option_disj_res_None {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: forall n, @residue (option worlds) (option00_relation Krelation) option_Join n None.
Lemma option_disj_USA {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: @UnitalSeparationAlgebra (option worlds) (option00_relation Krelation) option_Join.
Lemma option_disj_USA' {R: Relation worlds} {po_R: PreOrder Krelation} {J: Join worlds} {SA: SeparationAlgebra worlds}: @UnitalSeparationAlgebra' (option worlds) (option00_relation Krelation) option_Join.