Local Open Scope logic_base.
Local Open Scope syntax.
Module Semantics.
Definition impp {model: Type} (X: Ensemble model) (Y: Ensemble model): Ensemble model :=
fun m => X m -> Y m.
End Semantics.
Class TrivialMinimunSemantics (L: Language) {minL: MinimunLanguage L} (MD: Model) (SM: Semantics L MD): Type := {
denote_impp: forall x y, Same_set _ (denotation (x --> y)) (Semantics.impp (denotation x) (denotation y))
}.

Lemma sat_impp {L: Language} {minL: MinimunLanguage L} {MD: Model} {SM: Semantics L MD} {tminSM: TrivialMinimunSemantics L MD SM}: forall m x y, m |= x --> y <-> (m |= x -> m |= y).