Library iris.proofmode.class_instances_later

From stdpp Require Import nat_cancel.
From iris.bi Require Import bi.
From iris.proofmode Require Import modality_instances classes.
From iris.prelude Require Import options.
Import bi.

Section class_instances_later.
Context {PROP : bi}.
Implicit Types P Q R : PROP.

FromAssumption
FromPure
Global Instance from_pure_later a P φ : FromPure a P φ FromPure a ( P) φ.
Proof. rewrite /FromPure⇒ →. apply later_intro. Qed.
Global Instance from_pure_laterN a n P φ : FromPure a P φ FromPure a (▷^n P) φ.
Proof. rewrite /FromPure⇒ →. apply laterN_intro. Qed.
Global Instance from_pure_except_0 a P φ : FromPure a P φ FromPure a ( P) φ.
Proof. rewrite /FromPure⇒ →. apply except_0_intro. Qed.

IntoWand
Global Instance into_wand_later p q R P Q :
  IntoWand p q R P Q IntoWand p q ( R) ( P) ( Q).
Proof.
  rewrite /IntoWand /= ⇒ HR.
  by rewrite !later_intuitionistically_if_2 -later_wand HR.
Qed.
Global Instance into_wand_later_args p q R P Q :
  IntoWand p q R P Q IntoWand' p q R ( P) ( Q).
Proof.
  rewrite /IntoWand' /IntoWand /= ⇒ HR.
  by rewrite !later_intuitionistically_if_2
             (later_intro (□?p R)) -later_wand HR.
Qed.
Global Instance into_wand_laterN n p q R P Q :
  IntoWand p q R P Q IntoWand p q (▷^n R) (▷^n P) (▷^n Q).
Proof.
  rewrite /IntoWand /= ⇒ HR.
  by rewrite !laterN_intuitionistically_if_2 -laterN_wand HR.
Qed.
Global Instance into_wand_laterN_args n p q R P Q :
  IntoWand p q R P Q IntoWand' p q R (▷^n P) (▷^n Q).
Proof.
  rewrite /IntoWand' /IntoWand /= ⇒ HR.
  by rewrite !laterN_intuitionistically_if_2
             (laterN_intro _ (□?p R)) -laterN_wand HR.
Qed.

FromAnd
Global Instance from_and_later P Q1 Q2 :
  FromAnd P Q1 Q2 FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd⇒ <-. by rewrite later_and. Qed.
Global Instance from_and_laterN n P Q1 Q2 :
  FromAnd P Q1 Q2 FromAnd (▷^n P) (▷^n Q1) (▷^n Q2).
Proof. rewrite /FromAnd⇒ <-. by rewrite laterN_and. Qed.
Global Instance from_and_except_0 P Q1 Q2 :
  FromAnd P Q1 Q2 FromAnd ( P) ( Q1) ( Q2).
Proof. rewrite /FromAnd=><-. by rewrite except_0_and. Qed.

FromSep
Global Instance from_sep_later P Q1 Q2 :
  FromSep P Q1 Q2 FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep⇒ <-. by rewrite later_sep. Qed.
Global Instance from_sep_laterN n P Q1 Q2 :
  FromSep P Q1 Q2 FromSep (▷^n P) (▷^n Q1) (▷^n Q2).
Proof. rewrite /FromSep⇒ <-. by rewrite laterN_sep. Qed.
Global Instance from_sep_except_0 P Q1 Q2 :
  FromSep P Q1 Q2 FromSep ( P) ( Q1) ( Q2).
Proof. rewrite /FromSep=><-. by rewrite except_0_sep. Qed.

IntoAnd
Global Instance into_and_later p P Q1 Q2 :
  IntoAnd p P Q1 Q2 IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAndHP. apply intuitionistically_if_intro'.
  by rewrite later_intuitionistically_if_2 HP
             intuitionistically_if_elim later_and.
Qed.
Global Instance into_and_laterN n p P Q1 Q2 :
  IntoAnd p P Q1 Q2 IntoAnd p (▷^n P) (▷^n Q1) (▷^n Q2).
Proof.
  rewrite /IntoAndHP. apply intuitionistically_if_intro'.
  by rewrite laterN_intuitionistically_if_2 HP
             intuitionistically_if_elim laterN_and.
Qed.
Global Instance into_and_except_0 p P Q1 Q2 :
  IntoAnd p P Q1 Q2 IntoAnd p ( P) ( Q1) ( Q2).
Proof.
  rewrite /IntoAndHP. apply intuitionistically_if_intro'.
  by rewrite except_0_intuitionistically_if_2 HP
             intuitionistically_if_elim except_0_and.
Qed.

IntoSep
Global Instance into_sep_later P Q1 Q2 :
  IntoSep P Q1 Q2 IntoSep ( P) ( Q1) ( Q2).
Proof. rewrite /IntoSep⇒ →. by rewrite later_sep. Qed.
Global Instance into_sep_laterN n P Q1 Q2 :
  IntoSep P Q1 Q2 IntoSep (▷^n P) (▷^n Q1) (▷^n Q2).
Proof. rewrite /IntoSep⇒ →. by rewrite laterN_sep. Qed.
Global Instance into_sep_except_0 P Q1 Q2 :
  IntoSep P Q1 Q2 IntoSep ( P) ( Q1) ( Q2).
Proof. rewrite /IntoSep⇒ →. by rewrite except_0_sep. Qed.

Global Instance into_sep_affinely_later `{!Timeless (PROP:=PROP) emp} P Q1 Q2 :
  IntoSep P Q1 Q2 Affine Q1 Affine Q2
  IntoSep (<affine> P) (<affine> Q1) (<affine> Q2).
Proof.
  rewrite /IntoSep /= ⇒ → ??.
  rewrite -{1}(affine_affinely Q1) -{1}(affine_affinely Q2) later_sep !later_affinely_1.
  rewrite -except_0_sep /bi_except_0 affinely_or. apply or_elim, affinely_elim.
  rewrite -(idemp bi_and (<affine> False)%I) persistent_and_sep_1.
  by rewrite -(False_elim Q1) -(False_elim Q2).
Qed.

FromOr
Global Instance from_or_later P Q1 Q2 :
  FromOr P Q1 Q2 FromOr ( P) ( Q1) ( Q2).
Proof. rewrite /FromOr=><-. by rewrite later_or. Qed.
Global Instance from_or_laterN n P Q1 Q2 :
  FromOr P Q1 Q2 FromOr (▷^n P) (▷^n Q1) (▷^n Q2).
Proof. rewrite /FromOr=><-. by rewrite laterN_or. Qed.
Global Instance from_or_except_0 P Q1 Q2 :
  FromOr P Q1 Q2 FromOr ( P) ( Q1) ( Q2).
Proof. rewrite /FromOr=><-. by rewrite except_0_or. Qed.

IntoOr
Global Instance into_or_later P Q1 Q2 :
  IntoOr P Q1 Q2 IntoOr ( P) ( Q1) ( Q2).
Proof. rewrite /IntoOr=>->. by rewrite later_or. Qed.
Global Instance into_or_laterN n P Q1 Q2 :
  IntoOr P Q1 Q2 IntoOr (▷^n P) (▷^n Q1) (▷^n Q2).
Proof. rewrite /IntoOr=>->. by rewrite laterN_or. Qed.
Global Instance into_or_except_0 P Q1 Q2 :
  IntoOr P Q1 Q2 IntoOr ( P) ( Q1) ( Q2).
Proof. rewrite /IntoOr=>->. by rewrite except_0_or. Qed.

FromExist
Global Instance from_exist_later {A} P (Φ : A PROP) :
  FromExist P Φ FromExist ( P) (λ a, (Φ a))%I.
Proof.
  rewrite /FromExist⇒ <-. apply exist_elimx. apply later_mono, exist_intro.
Qed.
Global Instance from_exist_laterN {A} n P (Φ : A PROP) :
  FromExist P Φ FromExist (▷^n P) (λ a, ▷^n (Φ a))%I.
Proof.
  rewrite /FromExist⇒ <-. apply exist_elimx. apply laterN_mono, exist_intro.
Qed.
Global Instance from_exist_except_0 {A} P (Φ : A PROP) :
  FromExist P Φ FromExist ( P) (λ a, (Φ a))%I.
Proof. rewrite /FromExist⇒ <-. by rewrite except_0_exist_2. Qed.

IntoExist
Global Instance into_exist_later {A} P (Φ : A PROP) name :
  IntoExist P Φ name Inhabited A IntoExist ( P) (λ a, (Φ a))%I name.
Proof. rewrite /IntoExistHP ?. by rewrite HP later_exist. Qed.
Global Instance into_exist_laterN {A} n P (Φ : A PROP) name :
  IntoExist P Φ name Inhabited A IntoExist (▷^n P) (λ a, ▷^n (Φ a))%I name.
Proof. rewrite /IntoExistHP ?. by rewrite HP laterN_exist. Qed.
Global Instance into_exist_except_0 {A} P (Φ : A PROP) name :
  IntoExist P Φ name Inhabited A IntoExist ( P) (λ a, (Φ a))%I name.
Proof. rewrite /IntoExistHP ?. by rewrite HP except_0_exist. Qed.

IntoForall
Global Instance into_forall_later {A} P (Φ : A PROP) :
  IntoForall P Φ IntoForall ( P) (λ a, (Φ a))%I.
Proof. rewrite /IntoForallHP. by rewrite HP later_forall. Qed.

Global Instance into_forall_laterN {A} P (Φ : A PROP) n :
  IntoForall P Φ IntoForall (▷^n P) (λ a, ▷^n (Φ a))%I.
Proof. rewrite /IntoForallHP. by rewrite HP laterN_forall. Qed.

Global Instance into_forall_except_0 {A} P (Φ : A PROP) :
  IntoForall P Φ IntoForall ( P) (λ a, (Φ a))%I.
Proof. rewrite /IntoForallHP. by rewrite HP except_0_forall. Qed.

FromForall
Global Instance from_forall_later {A} P (Φ : A PROP) name :
  FromForall P Φ name FromForall ( P) (λ a, (Φ a))%I name.
Proof. rewrite /FromForall⇒ <-. by rewrite later_forall. Qed.

Global Instance from_forall_laterN {A} P (Φ : A PROP) n name :
  FromForall P Φ name FromForall (▷^n P) (λ a, ▷^n (Φ a))%I name.
Proof. rewrite /FromForall ⇒ <-. by rewrite laterN_forall. Qed.

Global Instance from_forall_except_0 {A} P (Φ : A PROP) name :
  FromForall P Φ name FromForall ( P) (λ a, (Φ a))%I name.
Proof. rewrite /FromForall⇒ <-. by rewrite except_0_forall. Qed.

IsExcept0
Global Instance is_except_0_except_0 P : IsExcept0 ( P).
Proof. by rewrite /IsExcept0 except_0_idemp. Qed.
Global Instance is_except_0_later P : IsExcept0 ( P).
Proof. by rewrite /IsExcept0 except_0_later. Qed.

FromModal
Global Instance from_modal_later P :
  FromModal True (modality_laterN 1) (▷^1 P) ( P) P.
Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_laterN n P :
  FromModal True (modality_laterN n) (▷^n P) (▷^n P) P.
Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_except_0 P :
  FromModal True modality_id ( P) ( P) P.
Proof. by rewrite /FromModal /= -except_0_intro. Qed.

IntoExcept0
Global Instance into_except_0_except_0 P : IntoExcept0 ( P) P.
Proof. by rewrite /IntoExcept0. Qed.
Global Instance into_except_0_later P : Timeless P IntoExcept0 ( P) P.
Proof. by rewrite /IntoExcept0. Qed.
Global Instance into_except_0_later_if p P : Timeless P IntoExcept0 (▷?p P) P.
Proof. rewrite /IntoExcept0. destruct p; auto using except_0_intro. Qed.

Global Instance into_except_0_affinely P Q :
  IntoExcept0 P Q IntoExcept0 (<affine> P) (<affine> Q).
Proof. rewrite /IntoExcept0⇒ →. by rewrite except_0_affinely_2. Qed.
Global Instance into_except_0_intuitionistically P Q :
  IntoExcept0 P Q IntoExcept0 ( P) ( Q).
Proof. rewrite /IntoExcept0⇒ →. by rewrite except_0_intuitionistically_2. Qed.
Global Instance into_except_0_absorbingly P Q :
  IntoExcept0 P Q IntoExcept0 (<absorb> P) (<absorb> Q).
Proof. rewrite /IntoExcept0⇒ →. by rewrite except_0_absorbingly. Qed.
Global Instance into_except_0_persistently P Q :
  IntoExcept0 P Q IntoExcept0 (<pers> P) (<pers> Q).
Proof. rewrite /IntoExcept0⇒ →. by rewrite except_0_persistently. Qed.

ElimModal
Global Instance elim_modal_timeless p P P' Q :
  IntoExcept0 P P' IsExcept0 Q ElimModal True p p P P' Q Q.
Proof.
  intros. rewrite /ElimModal (except_0_intro (_ -∗ _)) (into_except_0 P).
  by rewrite except_0_intuitionistically_if_2 -except_0_sep wand_elim_r.
Qed.

AddModal
Global Instance add_modal_later_except_0 P Q :
  Timeless P AddModal ( P) P ( Q) | 0.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ -∗ _)) (timeless P).
  by rewrite -except_0_sep wand_elim_r except_0_idemp.
Qed.
Global Instance add_modal_later P Q :
  Timeless P AddModal ( P) P ( Q) | 0.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ -∗ _)) (timeless P).
  by rewrite -except_0_sep wand_elim_r except_0_later.
Qed.
Global Instance add_modal_except_0 P Q : AddModal ( P) P ( Q) | 1.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ -∗ _)).
  by rewrite -except_0_sep wand_elim_r except_0_idemp.
Qed.
Global Instance add_modal_except_0_later P Q : AddModal ( P) P ( Q) | 1.
Proof.
  intros. rewrite /AddModal (except_0_intro (_ -∗ _)).
  by rewrite -except_0_sep wand_elim_r except_0_later.
Qed.

IntoAcc

IntoLater
Global Instance into_laterN_0 only_head P : IntoLaterN only_head 0 P P.
Proof. by rewrite /IntoLaterN /MaybeIntoLaterN. Qed.
Global Instance into_laterN_later only_head n n' m' P Q lQ :
  NatCancel n 1 n' m'
  
If canceling has failed (i.e. 1 = m'), we should make progress deeper into P, as such, we continue with the IntoLaterN class, which is required to make progress. If canceling has succeeded, we do not need to make further progress, but there may still be a left-over (i.e. n') to cancel more deeply into P, as such, we continue with MaybeIntoLaterN.
  TCIf (TCEq 1 m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q)
  MakeLaterN m' Q lQ
  IntoLaterN only_head n ( P) lQ | 2.
Proof.
  rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel.
  moveHn [_ ->|->] <-;
    by rewrite -later_laterN -laterN_plus -Hn Nat.add_comm.
Qed.
Global Instance into_laterN_laterN only_head n m n' m' P Q lQ :
  NatCancel n m n' m'
  TCIf (TCEq m m') (IntoLaterN only_head n' P Q) (MaybeIntoLaterN only_head n' P Q)
  MakeLaterN m' Q lQ
  IntoLaterN only_head n (▷^m P) lQ | 1.
Proof.
  rewrite /MakeLaterN /IntoLaterN /MaybeIntoLaterN /NatCancel.
  moveHn [_ ->|->] <-; by rewrite -!laterN_plus -Hn Nat.add_comm.
Qed.

Global Instance into_laterN_and_l n P1 P2 Q1 Q2 :
  IntoLaterN false n P1 Q1 MaybeIntoLaterN false n P2 Q2
  IntoLaterN false n (P1 P2) (Q1 Q2) | 10.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ → →. by rewrite laterN_and. Qed.
Global Instance into_laterN_and_r n P P2 Q2 :
  IntoLaterN false n P2 Q2 IntoLaterN false n (P P2) (P Q2) | 11.
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_and -(laterN_intro _ P).
Qed.

Global Instance into_laterN_forall {A} n (Φ Ψ : A PROP) :
  ( x, IntoLaterN false n (Φ x) (Ψ x))
  IntoLaterN false n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN laterN_forall⇒ ?. by apply forall_mono. Qed.
Global Instance into_laterN_exist {A} n (Φ Ψ : A PROP) :
  ( x, IntoLaterN false n (Φ x) (Ψ x))
  IntoLaterN false n ( x, Φ x) ( x, Ψ x).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN -laterN_exist_2⇒ ?. by apply exist_mono. Qed.

Global Instance into_laterN_or_l n P1 P2 Q1 Q2 :
  IntoLaterN false n P1 Q1 MaybeIntoLaterN false n P2 Q2
  IntoLaterN false n (P1 P2) (Q1 Q2) | 10.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ → →. by rewrite laterN_or. Qed.
Global Instance into_laterN_or_r n P P2 Q2 :
  IntoLaterN false n P2 Q2
  IntoLaterN false n (P P2) (P Q2) | 11.
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_or -(laterN_intro _ P).
Qed.

Global Instance into_later_affinely n P Q :
  IntoLaterN false n P Q IntoLaterN false n (<affine> P) (<affine> Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_affinely_2. Qed.
Global Instance into_later_intuitionistically n P Q :
  IntoLaterN false n P Q IntoLaterN false n ( P) ( Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_intuitionistically_2. Qed.
Global Instance into_later_absorbingly n P Q :
  IntoLaterN false n P Q IntoLaterN false n (<absorb> P) (<absorb> Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_absorbingly. Qed.
Global Instance into_later_persistently n P Q :
  IntoLaterN false n P Q IntoLaterN false n (<pers> P) (<pers> Q).
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_persistently. Qed.

Global Instance into_laterN_sep_l n P1 P2 Q1 Q2 :
  IntoLaterN false n P1 Q1 MaybeIntoLaterN false n P2 Q2
  IntoLaterN false n (P1 P2) (Q1 Q2) | 10.
Proof. rewrite /IntoLaterN /MaybeIntoLaterN⇒ → →. by rewrite laterN_sep. Qed.
Global Instance into_laterN_sep_r n P P2 Q2 :
  IntoLaterN false n P2 Q2
  IntoLaterN false n (P P2) (P Q2) | 11.
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ →. by rewrite laterN_sep -(laterN_intro _ P).
Qed.

Global Instance into_laterN_big_sepL n {A} (Φ Ψ : nat A PROP) (l: list A) :
  ( x k, IntoLaterN false n (Φ k x) (Ψ k x))
  IntoLaterN false n ([∗ list] k x l, Φ k x) ([∗ list] k x l, Ψ k x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ ?.
  rewrite big_opL_commute. by apply big_sepL_mono.
Qed.
Global Instance into_laterN_big_sepL2 n {A B} (Φ Ψ : nat A B PROP) l1 l2 :
  ( x1 x2 k, IntoLaterN false n (Φ k x1 x2) (Ψ k x1 x2))
  IntoLaterN false n ([∗ list] k y1;y2 l1;l2, Φ k y1 y2)
    ([∗ list] k y1;y2 l1;l2, Ψ k y1 y2).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ ?.
  rewrite -big_sepL2_laterN_2. by apply big_sepL2_mono.
Qed.
Global Instance into_laterN_big_sepM n `{Countable K} {A}
    (Φ Ψ : K A PROP) (m : gmap K A) :
  ( x k, IntoLaterN false n (Φ k x) (Ψ k x))
  IntoLaterN false n ([∗ map] k x m, Φ k x) ([∗ map] k x m, Ψ k x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ ?.
  rewrite big_opM_commute. by apply big_sepM_mono.
Qed.
Global Instance into_laterN_big_sepM2 n `{Countable K} {A B}
    (Φ Ψ : K A B PROP) (m1 : gmap K A) (m2 : gmap K B) :
  ( x1 x2 k, IntoLaterN false n (Φ k x1 x2) (Ψ k x1 x2))
  IntoLaterN false n ([∗ map] k x1;x2 m1;m2, Φ k x1 x2) ([∗ map] k x1;x2 m1;m2, Ψ k x1 x2).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterNHΦΨ.
  rewrite -big_sepM2_laterN_2. by apply big_sepM2_mono.
Qed.
Global Instance into_laterN_big_sepS n `{Countable A}
    (Φ Ψ : A PROP) (X : gset A) :
  ( x, IntoLaterN false n (Φ x) (Ψ x))
  IntoLaterN false n ([∗ set] x X, Φ x) ([∗ set] x X, Ψ x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ ?.
  rewrite big_opS_commute. by apply big_sepS_mono.
Qed.
Global Instance into_laterN_big_sepMS n `{Countable A}
    (Φ Ψ : A PROP) (X : gmultiset A) :
  ( x, IntoLaterN false n (Φ x) (Ψ x))
  IntoLaterN false n ([∗ mset] x X, Φ x) ([∗ mset] x X, Ψ x).
Proof.
  rewrite /IntoLaterN /MaybeIntoLaterN⇒ ?.
  rewrite big_opMS_commute. by apply big_sepMS_mono.
Qed.
End class_instances_later.