Library iris.proofmode.class_instances_frame

From iris.bi Require Import telescopes.
From iris.proofmode Require Import classes classes_make.
From iris.prelude Require Import options.
Import bi.

This file defines the instances that make up the framing machinery.

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

When framing R against itself, we leave True if possible (via frame_here_absorbing or frame_affinely_here_absorbing) since it is a weaker goal. Otherwise we leave emp via frame_here. Only if all those options fail, we start decomposing R, via instances like frame_exist. To ensure that, all other instances must have cost > 1.
Global Instance frame_here_absorbing p R : Absorbing R Frame p R R True | 0.
Proof. intros. by rewrite /Frame intuitionistically_if_elim sep_elim_l. Qed.
Global Instance frame_here p R : Frame p R R emp | 1.
Proof. intros. by rewrite /Frame intuitionistically_if_elim sep_elim_l. Qed.
Global Instance frame_affinely_here_absorbing p R :
  Absorbing R Frame p (<affine> R) R True | 0.
Proof.
  intros. rewrite /Frame intuitionistically_if_elim affinely_elim.
  apply sep_elim_l, _.
Qed.
Global Instance frame_affinely_here p R : Frame p (<affine> R) R emp | 1.
Proof.
  intros. rewrite /Frame intuitionistically_if_elim affinely_elim.
  apply sep_elim_l, _.
Qed.

Global Instance frame_here_pure_persistent a φ Q :
  FromPure a Q φ Frame true φ Q emp | 2.
Proof.
  rewrite /FromPure /Frame /= ⇒ <-. rewrite right_id.
  by rewrite -affinely_affinely_if intuitionistically_affinely.
Qed.
Global Instance frame_here_pure a φ Q :
  FromPure a Q φ
  TCOr (TCEq a false) (BiAffine PROP)
  Frame false φ Q emp | 2. Proof.
  rewrite /FromPure /Frame ⇒ <- [->|?] /=.
  - by rewrite right_id.
  - by rewrite right_id -affinely_affinely_if affine_affinely.
Qed.

Global Instance frame_embed `{BiEmbed PROP PROP'} p P Q (Q' : PROP') R :
  Frame p R P Q MakeEmbed Q Q'
  Frame p R P Q' | 2. Proof.
  rewrite /Frame /MakeEmbed ⇒ <- <-.
  rewrite embed_sep embed_intuitionistically_if_2 ⇒ //.
Qed.
Global Instance frame_pure_embed `{BiEmbed PROP PROP'} p P Q (Q' : PROP') φ :
  Frame p φ P Q MakeEmbed Q Q'
  Frame p φ P Q' | 2. Proof. rewrite /Frame /MakeEmbed -embed_pure. apply (frame_embed p P Q). Qed.

Global Instance frame_sep_persistent_l progress R P1 P2 Q1 Q2 Q' :
  Frame true R P1 Q1 MaybeFrame true R P2 Q2 progress MakeSep Q1 Q2 Q'
  Frame true R (P1 P2) Q' | 9.
Proof.
  rewrite /Frame /MaybeFrame /MakeSep /= ⇒ <- <- <-.
  rewrite {1}(intuitionistically_sep_dup R).
  by rewrite !assoc -(assoc _ _ _ Q1) -(comm _ Q1) assoc -(comm _ Q1).
Qed.
Global Instance frame_sep_l R P1 P2 Q Q' :
  Frame false R P1 Q MakeSep Q P2 Q' Frame false R (P1 P2) Q' | 9.
Proof. rewrite /Frame /MakeSep ⇒ <- <-. by rewrite assoc. Qed.
Global Instance frame_sep_r p R P1 P2 Q Q' :
  Frame p R P2 Q MakeSep P1 Q Q' Frame p R (P1 P2) Q' | 10.
Proof.
  rewrite /Frame /MakeSep ⇒ <- <-. by rewrite assoc -(comm _ P1) assoc.
Qed.

Global Instance frame_big_sepL_cons {A} p (Φ : nat A PROP) R Q l x l' :
  IsCons l x l'
  Frame p R (Φ 0 x [∗ list] k y l', Φ (S k) y) Q
  Frame p R ([∗ list] k y l, Φ k y) Q | 2. Proof. rewrite /IsCons=>->. by rewrite /Frame big_sepL_cons. Qed.
Global Instance frame_big_sepL_app {A} p (Φ : nat A PROP) R Q l l1 l2 :
  IsApp l l1 l2
  Frame p R (([∗ list] k y l1, Φ k y)
           [∗ list] k y l2, Φ (length l1 + k) y) Q
  Frame p R ([∗ list] k y l, Φ k y) Q | 2. Proof. rewrite /IsApp=>->. by rewrite /Frame big_sepL_app. Qed.

Global Instance frame_big_sepL2_cons {A B} p (Φ : nat A B PROP)
    R Q l1 x1 l1' l2 x2 l2' :
  IsCons l1 x1 l1' IsCons l2 x2 l2'
  Frame p R (Φ 0 x1 x2 [∗ list] k y1;y2 l1';l2', Φ (S k) y1 y2) Q
  Frame p R ([∗ list] k y1;y2 l1;l2, Φ k y1 y2) Q. Proof. rewrite /IsCons=>-> →. by rewrite /Frame big_sepL2_cons. Qed.
Global Instance frame_big_sepL2_app {A B} p (Φ : nat A B PROP)
    R Q l1 l1' l1'' l2 l2' l2'' :
  IsApp l1 l1' l1'' IsApp l2 l2' l2''
  Frame p R (([∗ list] k y1;y2 l1';l2', Φ k y1 y2)
           [∗ list] k y1;y2 l1'';l2'', Φ (length l1' + k) y1 y2) Q
  Frame p R ([∗ list] k y1;y2 l1;l2, Φ k y1 y2) Q.
Proof. rewrite /IsApp /Frame=>-> → →. apply wand_elim_l', big_sepL2_app. Qed.

Global Instance frame_big_sepMS_disj_union `{Countable A} p (Φ : A PROP) R Q X1 X2 :
  Frame p R (([∗ mset] y X1, Φ y) [∗ mset] y X2, Φ y) Q
  Frame p R ([∗ mset] y X1 X2, Φ y) Q | 2.
Proof. by rewrite /Frame big_sepMS_disj_union. Qed.

Global Instance frame_and p progress1 progress2 R P1 P2 Q1 Q2 Q' :
  MaybeFrame p R P1 Q1 progress1
  MaybeFrame p R P2 Q2 progress2
  TCEq (progress1 || progress2) true
  MakeAnd Q1 Q2 Q'
  Frame p R (P1 P2) Q' | 9.
Proof.
  rewrite /MaybeFrame /Frame /MakeAnd ⇒ <- <- _ <-.
  apply and_intro; [rewrite and_elim_l|rewrite and_elim_r]; done.
Qed.

We could in principle write the instance frame_or_spatial by a bunch of instances (omitting the parameter p = false):
Frame R P1 Q1 → Frame R P2 Q2 → Frame R (P1 ∨ P2) (Q1 ∨ Q2) Frame R P1 True → Frame R (P1 ∨ P2) P2 Frame R P2 True → Frame R (P1 ∨ P2) P1
The problem here is that Coq will try to infer Frame R P1 ? and Frame R P2 ? multiple times, whereas the current solution makes sure that said inference appears at most once.
If Coq would memorize the results of type class resolution, the solution with multiple instances would be preferred (and more Prolog-like).

Global Instance frame_or_spatial progress1 progress2 R P1 P2 Q1 Q2 Q :
  MaybeFrame false R P1 Q1 progress1 MaybeFrame false R P2 Q2 progress2
  TCOr (TCEq (progress1 && progress2) true) (TCOr
    (TCAnd (TCEq progress1 true) (TCEq Q1 True%I))
    (TCAnd (TCEq progress2 true) (TCEq Q2 True%I)))
  MakeOr Q1 Q2 Q
  Frame false R (P1 P2) Q | 9.
Proof. rewrite /Frame /MakeOr ⇒ <- <- _ <-. by rewrite -sep_or_l. Qed.

Global Instance frame_or_persistent progress1 progress2 R P1 P2 Q1 Q2 Q :
  MaybeFrame true R P1 Q1 progress1 MaybeFrame true R P2 Q2 progress2
  TCEq (progress1 || progress2) true
  MakeOr Q1 Q2 Q Frame true R (P1 P2) Q | 9.
Proof. rewrite /Frame /MakeOr ⇒ <- <- _ <-. by rewrite -sep_or_l. Qed.

Global Instance frame_wand p R P1 P2 Q2 :
  Frame p R P2 Q2 Frame p R (P1 -∗ P2) (P1 -∗ Q2) | 2.
Proof.
  rewrite /Frame⇒ ?. apply wand_intro_l.
  by rewrite assoc (comm _ P1) -assoc wand_elim_r.
Qed.

Global Instance frame_affinely p R P Q Q' :
  TCOr (TCEq p true) (Affine R)
  Frame p R P Q MakeAffinely Q Q'
  Frame p R (<affine> P) Q'. Proof.
  rewrite /Frame /MakeAffinely⇒ -[->|?] <- <- /=;
    by rewrite -{1}(affine_affinely (_ R)) affinely_sep_2.
Qed.

Global Instance frame_intuitionistically R P Q Q' :
  Frame true R P Q MakeIntuitionistically Q Q'
  Frame true R ( P) Q' | 2. Proof.
  rewrite /Frame /MakeIntuitionistically⇒ <- <- /=.
  rewrite -intuitionistically_sep_2 intuitionistically_idemp //.
Qed.

Global Instance frame_absorbingly p R P Q Q' :
  Frame p R P Q MakeAbsorbingly Q Q'
  Frame p R (<absorb> P) Q' | 2. Proof.
  rewrite /Frame /MakeAbsorbingly⇒ <- <- /=. by rewrite absorbingly_sep_r.
Qed.

Global Instance frame_persistently R P Q Q' :
  Frame true R P Q MakePersistently Q Q'
  Frame true R (<pers> P) Q' | 2. Proof.
  rewrite /Frame /MakePersistently⇒ <- <- /=.
  rewrite -persistently_and_intuitionistically_sep_l.
  by rewrite -persistently_sep_2 -persistently_and_sep_l_1
    persistently_affinely_elim persistently_idemp.
Qed.

Global Instance frame_exist {A} p R (Φ Ψ : A PROP) :
  ( a, Frame p R (Φ a) (Ψ a)) Frame p R ( x, Φ x) ( x, Ψ x) | 2.
Proof. rewrite /Frame⇒ ?. by rewrite sep_exist_l; apply exist_mono. Qed.
Global Instance frame_texist {TT : tele} p R (Φ Ψ : TT PROP) :
  ( x, Frame p R (Φ x) (Ψ x)) Frame p R (.. x, Φ x) (.. x, Ψ x) | 2.
Proof. rewrite /Frame !bi_texist_exist. apply frame_exist. Qed.
Global Instance frame_forall {A} p R (Φ Ψ : A PROP) :
  ( a, Frame p R (Φ a) (Ψ a)) Frame p R ( x, Φ x) ( x, Ψ x) | 2.
Proof. rewrite /Frame⇒ ?. by rewrite sep_forall_l; apply forall_mono. Qed.
Global Instance frame_tforall {TT : tele} p R (Φ Ψ : TT PROP) :
  ( x, Frame p R (Φ x) (Ψ x)) Frame p R (.. x, Φ x) (.. x, Ψ x) | 2.
Proof. rewrite /Frame !bi_tforall_forall. apply frame_forall. Qed.

Global Instance frame_impl_persistent R P1 P2 Q2 :
  Frame true R P2 Q2 Frame true R (P1 P2) (P1 Q2) | 2.
Proof.
  rewrite /Frame /= ⇒ ?. apply impl_intro_l.
  by rewrite -persistently_and_intuitionistically_sep_l assoc (comm _ P1) -assoc impl_elim_r
             persistently_and_intuitionistically_sep_l.
Qed.
Global Instance frame_impl R P1 P2 Q2 :
  Persistent P1 Absorbing P1
  Frame false R P2 Q2 Frame false R (P1 P2) (P1 Q2). Proof.
  rewrite /Frame /==> ???. apply impl_intro_l.
  rewrite {1}(persistent P1) persistently_and_intuitionistically_sep_l assoc.
  rewrite (comm _ ( P1)%I) -assoc -persistently_and_intuitionistically_sep_l.
  rewrite persistently_elim impl_elim_r //.
Qed.

Global Instance frame_eq_embed `{!BiEmbed PROP PROP', !BiInternalEq PROP,
    !BiInternalEq PROP', !BiEmbedInternalEq PROP PROP'}
    p P Q (Q' : PROP') {A : ofe} (a b : A) :
  Frame p (a b) P Q MakeEmbed Q Q'
  Frame p (a b) P Q'. Proof. rewrite /Frame /MakeEmbed -embed_internal_eq. apply (frame_embed p P Q). Qed.

Global Instance frame_later p R R' P Q Q' :
  TCNoBackTrack (MaybeIntoLaterN true 1 R' R)
  Frame p R P Q MakeLaterN 1 Q Q'
  Frame p R' ( P) Q'. Proof.
  rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-.
  by rewrite later_intuitionistically_if_2 later_sep.
Qed.
Global Instance frame_laterN p n R R' P Q Q' :
  TCNoBackTrack (MaybeIntoLaterN true n R' R)
  Frame p R P Q MakeLaterN n Q Q'
  Frame p R' (▷^n P) Q'. Proof.
  rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-.
  by rewrite laterN_intuitionistically_if_2 laterN_sep.
Qed.

Global Instance frame_bupd `{BiBUpd PROP} p R P Q :
  Frame p R P Q Frame p R (|==> P) (|==> Q) | 2.
Proof. rewrite /Frame=><-. by rewrite bupd_frame_l. Qed.
Global Instance frame_fupd `{BiFUpd PROP} p E1 E2 R P Q :
  Frame p R P Q Frame p R (|={E1,E2}=> P) (|={E1,E2}=> Q) | 2.
Proof. rewrite /Frame=><-. by rewrite fupd_frame_l. Qed.

Global Instance frame_except_0 p R P Q Q' :
  Frame p R P Q MakeExcept0 Q Q'
  Frame p R ( P) Q' | 2. Proof.
  rewrite /Frame /MakeExcept0=><- <-.
  by rewrite except_0_sep -(except_0_intro (□?p R)).
Qed.
End class_instances_frame.