class_instances.v

  rewrite big_opMS_commute. by apply big_sepMS_mono.
Qed.

(* FromLater *)
Global Instance from_laterN_later P : FromLaterN 1 (▷ P) P | 0.
Proof. by rewrite /FromLaterN. Qed.
Global Instance from_laterN_laterN n P : FromLaterN n (▷^n P) P | 0.
Proof. by rewrite /FromLaterN. Qed.

(* The instances below are used when stripping a specific number of laters, or
to balance laters in different branches of ∧, ∨ and ∗. *)
Global Instance from_laterN_0 P : FromLaterN 0 P P | 100. (* fallthrough *)
Proof. by rewrite /FromLaterN. Qed.
Global Instance from_laterN_later_S n P Q :
  FromLaterN n P Q → FromLaterN (S n) (▷ P) Q.
Proof. by rewrite /FromLaterN=><-. Qed.
Global Instance from_laterN_later_plus n m P Q :
  FromLaterN m P Q → FromLaterN (n + m) (▷^n P) Q.
Proof. rewrite /FromLaterN=><-. by rewrite laterN_plus. Qed.

Global Instance from_later_and n P1 P2 Q1 Q2 :
  FromLaterN n P1 Q1 → FromLaterN n P2 Q2 → FromLaterN n (P1 ∧ P2) (Q1 ∧ Q2).
Proof. intros ??; red. by rewrite laterN_and; apply and_mono. Qed.
Global Instance from_later_or n P1 P2 Q1 Q2 :
  FromLaterN n P1 Q1 → FromLaterN n P2 Q2 → FromLaterN n (P1 ∨ P2) (Q1 ∨ Q2).
Proof. intros ??; red. by rewrite laterN_or; apply or_mono. Qed.
Global Instance from_later_sep n P1 P2 Q1 Q2 :
  FromLaterN n P1 Q1 → FromLaterN n P2 Q2 → FromLaterN n (P1 ∗ P2) (Q1 ∗ Q2).
Proof. intros ??; red. by rewrite laterN_sep; apply sep_mono. Qed.

Global Instance from_later_persistently n P Q :
  FromLaterN n P Q → FromLaterN n (□ P) (□ Q).
Proof. by rewrite /FromLaterN laterN_persistently=> ->. Qed.

Global Instance from_later_forall {A} n (Φ Ψ : A → PROP) :
  (∀ x, FromLaterN n (Φ x) (Ψ x)) → FromLaterN n (∀ x, Φ x) (∀ x, Ψ x).
Proof. rewrite /FromLaterN laterN_forall=> ?. by apply forall_mono. Qed.
Global Instance from_later_exist {A} n (Φ Ψ : A → PROP) :
  Inhabited A → (∀ x, FromLaterN n (Φ x) (Ψ x)) →
  FromLaterN n (∃ x, Φ x) (∃ x, Ψ x).
Proof. intros ?. rewrite /FromLaterN laterN_exist=> ?. by apply exist_mono. Qed.
End sbi_instances.