Local Open Scope logic_base.
Local Open Scope syntax.
Local Open Scope kripke_model.
Class SepconSemantics
(L: Language)
{sepconL: SepconLanguage L}
(MD: Model)
{kMD: KripkeModel MD}
(M: Kmodel)
{R: Relation (Kworlds M)}
{J: Join (Kworlds M)}
(SM: Semantics L MD): Type :=
denote_sepcon: forall x y, Same_set _ (Kdenotation M (x * y)) (WeakSemantics.sepcon (Kdenotation M x) (Kdenotation M y)).
Class WandSemantics
(L: Language)
{wandL: WandLanguage L}
(MD: Model)
{kMD: KripkeModel MD}
(M: Kmodel)
{R: Relation (Kworlds M)}
{J: Join (Kworlds M)}
(SM: Semantics L MD): Type :=
denote_wand: forall x y, Same_set _ (Kdenotation M (x -* y)) (WeakSemantics.wand (Kdenotation M x) (Kdenotation M y)).
Class EmpSemantics
(L: Language)
{empL: EmpLanguage L}
(MD: Model)
{kMD: KripkeModel MD}
(M: Kmodel)
{R: Relation (Kworlds M)}
{J: Join (Kworlds M)}
(SM: Semantics L MD): Type :=
denote_emp: Same_set _ (Kdenotation M emp) WeakSemantics.emp.

Lemma sat_sepcon
{L: Language}
{sepconL: SepconLanguage L}
{MD: Model}
{kMD: KripkeModel MD}
{M: Kmodel}
{R: Relation (Kworlds M)}
{J: Join (Kworlds M)}
{SM: Semantics L MD}
{fsepconSM: SepconSemantics L MD M SM}:
forall m x y,
KRIPKE: M , m |= x * y <->
exists m1 m2, join m1 m2 m /\
KRIPKE: M , m1 |= x /\
KRIPKE: M, m2 |= y.