Local Open Scope logic_base.
Local Open Scope syntax.
Class PropositionalStable (L: Language) {minL: MinimunLanguage L} {pL: PropositionalLanguage L} (Gamma: ProofTheory L) (stable: expr -> Prop) := {
impp_stable: forall x y, stable x -> stable y -> stable (x --> y);
andp_stable: forall x y, stable x -> stable y -> stable (x && y);
orp_stable: forall x y, stable x -> stable y -> stable (x || y);
falsep_stable: stable FF;
stable_proper_iffp :> Proper ((fun x y => |-- x <--> y) ==> iff) stable
}.
Class ModalStable (L: Language) {minL: MinimunLanguage L} {mL: ModalLanguage L} (Gamma: ProofTheory L) (stable: expr -> Prop) := {
boxp_stable: forall x, stable x -> stable (boxp x)
}.
Class ModalAbsorbStable (L: Language) {minL: MinimunLanguage L} {mL: ModalLanguage L} (Gamma: ProofTheory L) (stable: expr -> Prop) := {
boxp_absorb_stable: forall x, stable x -> |-- x --> boxp x
}.
Class SeparationStable (L: Language) {minL: MinimunLanguage L} {sL: SeparationLanguage L} (Gamma: ProofTheory L) (stable: expr -> Prop) := {
sepcon_stable: forall x y, stable x -> stable y -> stable (x * y);
wand_stable: forall x y, stable x -> stable y -> stable (x -* y)
}.
Class SeparationAbsorbStable (L: Language) {minL: MinimunLanguage L} {pL: PropositionalLanguage L} {sL: SeparationLanguage L} (Gamma: ProofTheory L) (stable: expr -> Prop) := {
stable_andp_sepcon1: forall x y z, stable x -> |-- (x && y) * z <--> x && (y * z)
}.

(* Specifications: *)
Lemma iffp_stable {L: Language} {minL: MinimunLanguage L} {pL: PropositionalLanguage L} {Gamma: ProofTheory L} {stable: expr -> Prop} {pstable: PropositionalStable L Gamma stable}:
forall x y, stable x -> stable y -> stable (x <--> y).
Lemma truep_stable {L: Language} {minL: MinimunLanguage L} {pL: PropositionalLanguage L} {Gamma: ProofTheory L} {stable: expr -> Prop} {pstable: PropositionalStable L Gamma stable}:
stable TT.