End Semantics.
Module SemanticsMono.
Program Definition impp {worlds: Type} {R: Relation worlds} {po_R: PreOrder Krelation} (X Y: MonoEnsemble worlds): MonoEnsemble worlds :=
Semantics.impp X Y.
Next Obligation.
apply (@Semantics.impp_closed worlds R po_R);
apply (proj2_sig _).
Defined.
End SemanticsMono.
Class KripkeMinimunSemantics (L: Language) {minL: MinimunLanguage L} (MD: Model) {kMD: KripkeModel MD} (M: Kmodel) {R: Relation (Kworlds M)} (SM: Semantics L MD) : Type := {
denote_impp: forall x y, Same_set _ (Kdenotation M (x --> y)) (Semantics.impp (Kdenotation M x) (Kdenotation M y))
}.

(* Specifications: *)
Lemma sat_impp {L: Language} {minL: MinimunLanguage L} {MD: Model} {kMD: KripkeModel MD} {M: Kmodel} {R: Relation (Kworlds M)} {SM: Semantics L MD} {kminSM: KripkeMinimunSemantics L MD M SM}: forall m x y, KRIPKE: M , m |= x --> y <-> (forall n, m <= n -> KRIPKE: M , n |= x -> KRIPKE: M , n |= y).