Commit cd5f38bf authored by Robbert Krebbers's avatar Robbert Krebbers
Browse files

Fine-grained split of `class_instances` based on the structure of the `bi` folder.

parent 7a0ed0f8
......@@ -138,8 +138,12 @@ theories/proofmode/sel_patterns.v
theories/proofmode/tactics.v
theories/proofmode/notation.v
theories/proofmode/classes.v
theories/proofmode/class_instances_bi.v
theories/proofmode/class_instances_sbi.v
theories/proofmode/class_instances.v
theories/proofmode/class_instances_later.v
theories/proofmode/class_instances_updates.v
theories/proofmode/class_instances_embedding.v
theories/proofmode/class_instances_plainly.v
theories/proofmode/class_instances_internal_eq.v
theories/proofmode/frame_instances.v
theories/proofmode/monpred.v
theories/proofmode/modalities.v
......
From iris.bi Require Export bi.
From iris.proofmode Require Import classes class_instances_bi.
From iris.proofmode Require Import classes class_instances.
Set Default Proof Using "Type".
Class Fractional {PROP : bi} (Φ : Qp PROP) :=
......
From stdpp Require Import nat_cancel.
From iris.bi Require Import bi tactics telescopes.
From iris.proofmode Require Import base modality_instances classes ltac_tactics.
From iris.bi Require Import bi telescopes.
From iris.proofmode Require Import base modality_instances classes.
From iris.proofmode Require Import ltac_tactics.
Set Default Proof Using "Type".
Import bi.
......@@ -25,7 +26,7 @@ Proof. by rewrite /FromExist. Qed.
Hint Extern 0 (FromExist _ _) =>
notypeclasses refine (from_exist_exist _) : typeclass_instances.
Section bi_instances.
Section class_instances.
Context {PROP : bi}.
Implicit Types P Q R : PROP.
Implicit Types mP : option PROP.
......@@ -52,17 +53,6 @@ Proof.
apply as_emp_valid_forall.
Qed.
(* We add a useless hypothesis [BiEmbed PROP PROP'] in order to make
sure this instance is not used when there is no embedding between
PROP and PROP'.
The first [`{BiEmbed PROP PROP'}] is not considered as a premise by
Coq TC search mechanism because the rest of the hypothesis is dependent
on it. *)
Global Instance as_emp_valid_embed `{BiEmbed PROP PROP'} (φ : Prop) (P : PROP) :
BiEmbed PROP PROP'
AsEmpValid0 φ P AsEmpValid φ P.
Proof. rewrite /AsEmpValid0 /AsEmpValid=> _ ->. rewrite embed_emp_valid //. Qed.
(** FromAffinely *)
Global Instance from_affinely_affine P : Affine P FromAffinely P P.
Proof. intros. by rewrite /FromAffinely affinely_elim. Qed.
......@@ -155,10 +145,6 @@ Proof.
by rewrite bi_tforall_forall forall_elim.
Qed.
Global Instance from_assumption_bupd `{BiBUpd PROP} p P Q :
FromAssumption p P Q KnownRFromAssumption p P (|==> Q).
Proof. rewrite /KnownRFromAssumption /FromAssumption=>->. apply bupd_intro. Qed.
(** IntoPure *)
Global Instance into_pure_pure φ : @IntoPure PROP ⌜φ⌝ φ.
Proof. by rewrite /IntoPure. Qed.
......@@ -209,9 +195,6 @@ Proof. rewrite /IntoPure=> ->. by rewrite absorbingly_pure. Qed.
Global Instance into_pure_persistently P φ :
IntoPure P φ IntoPure (<pers> P) φ.
Proof. rewrite /IntoPure=> ->. apply: persistently_elim. Qed.
Global Instance into_pure_embed `{BiEmbed PROP PROP'} P φ :
IntoPure P φ IntoPure P φ.
Proof. rewrite /IntoPure=> ->. by rewrite embed_pure. Qed.
(** FromPure *)
Global Instance from_pure_emp : @FromPure PROP true emp True.
......@@ -303,13 +286,6 @@ Proof.
rewrite /FromPure=> <- /=. rewrite -affinely_affinely_if.
by rewrite -persistent_absorbingly_affinely_2.
Qed.
Global Instance from_pure_embed `{BiEmbed PROP PROP'} a P φ :
FromPure a P φ FromPure a P φ.
Proof. rewrite /FromPure=> <-. by rewrite -embed_pure embed_affinely_if_2. Qed.
Global Instance from_pure_bupd `{BiBUpd PROP} a P φ :
FromPure a P φ FromPure a (|==> P) φ.
Proof. rewrite /FromPure=> <-. apply bupd_intro. Qed.
(** IntoPersistent *)
Global Instance into_persistent_persistently p P Q :
......@@ -329,11 +305,6 @@ Proof.
eauto using persistently_mono, intuitionistically_elim,
intuitionistically_into_persistently_1.
Qed.
Global Instance into_persistent_embed `{BiEmbed PROP PROP'} p P Q :
IntoPersistent p P Q IntoPersistent p P Q | 0.
Proof.
rewrite /IntoPersistent -embed_persistently -embed_persistently_if=> -> //.
Qed.
Global Instance into_persistent_here P : IntoPersistent true P P | 1.
Proof. by rewrite /IntoPersistent. Qed.
Global Instance into_persistent_persistent P :
......@@ -361,34 +332,6 @@ Global Instance from_modal_absorbingly P :
FromModal modality_id (<absorb> P) (<absorb> P) P.
Proof. by rewrite /FromModal /= -absorbingly_intro. Qed.
(* When having a modality nested in an embedding, e.g. [ ⎡|==> P⎤ ], we prefer
the embedding over the modality. *)
Global Instance from_modal_embed `{BiEmbed PROP PROP'} (P : PROP) :
FromModal (@modality_embed PROP PROP' _) P P P.
Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_id_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_id sel P Q
FromModal modality_id sel P Q | 100.
Proof. by rewrite /FromModal /= =><-. Qed.
Global Instance from_modal_affinely_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_affinely sel P Q
FromModal modality_affinely sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_affinely_2. Qed.
Global Instance from_modal_persistently_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_persistently sel P Q
FromModal modality_persistently sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_persistently. Qed.
Global Instance from_modal_intuitionistically_embed `{BiEmbed PROP PROP'} `(sel : A) P Q :
FromModal modality_intuitionistically sel P Q
FromModal modality_intuitionistically sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_intuitionistically_2. Qed.
Global Instance from_modal_bupd `{BiBUpd PROP} P :
FromModal modality_id (|==> P) (|==> P) P.
Proof. by rewrite /FromModal /= -bupd_intro. Qed.
(** IntoWand *)
Global Instance into_wand_wand' p q (P Q P' Q' : PROP) :
IntoWand' p q (P - Q) P' Q' IntoWand p q (P - Q) P' Q' | 100.
......@@ -508,69 +451,16 @@ Global Instance into_wand_persistently_false q R P Q :
Absorbing R IntoWand false q R P Q IntoWand false q (<pers> R) P Q.
Proof. intros ?. by rewrite /IntoWand persistently_elim. Qed.
Global Instance into_wand_embed `{BiEmbed PROP PROP'} p q R P Q :
IntoWand p q R P Q IntoWand p q R P Q.
Proof. by rewrite /IntoWand !embed_intuitionistically_if_2 -embed_wand=> ->. Qed.
(* There are two versions for [IntoWand ⎡RR⎤ ...] with the argument being
[<affine> ⎡PP⎤]. When the wand [⎡RR⎤] resides in the intuitionistic context
the result of wand elimination will have the affine modality. Otherwise, it
won't. Note that when the wand [⎡RR⎤] is under an affine modality, the instance
[into_wand_affine] would already have been used. *)
Global Instance into_wand_affine_embed_true `{BiEmbed PROP PROP'} q (PP QQ RR : PROP) :
IntoWand true q RR PP QQ IntoWand true q RR (<affine> PP) (<affine> QQ) | 100.
Proof.
rewrite /IntoWand /=.
rewrite -(intuitionistically_idemp _ %I) embed_intuitionistically_2=> ->.
apply bi.wand_intro_l. destruct q; simpl.
- rewrite affinely_elim -(intuitionistically_idemp _ %I).
rewrite embed_intuitionistically_2 intuitionistically_sep_2 -embed_sep.
by rewrite wand_elim_r intuitionistically_affinely.
- by rewrite intuitionistically_affinely affinely_sep_2 -embed_sep wand_elim_r.
Qed.
Global Instance into_wand_affine_embed_false `{BiEmbed PROP PROP'} q (PP QQ RR : PROP) :
IntoWand false q RR (<affine> PP) QQ IntoWand false q RR (<affine> PP) QQ | 100.
Proof.
rewrite /IntoWand /= => ->.
by rewrite embed_affinely_2 embed_intuitionistically_if_2 embed_wand.
Qed.
Global Instance into_wand_bupd `{BiBUpd PROP} p q R P Q :
IntoWand false false R P Q IntoWand p q (|==> R) (|==> P) (|==> Q).
Proof.
rewrite /IntoWand /= => HR. rewrite !intuitionistically_if_elim HR.
apply wand_intro_l. by rewrite bupd_sep wand_elim_r.
Qed.
Global Instance into_wand_bupd_persistent `{BiBUpd PROP} p q R P Q :
IntoWand false q R P Q IntoWand p q (|==> R) P (|==> Q).
Proof.
rewrite /IntoWand /= => HR. rewrite intuitionistically_if_elim HR.
apply wand_intro_l. by rewrite bupd_frame_l wand_elim_r.
Qed.
Global Instance into_wand_bupd_args `{BiBUpd PROP} p q R P Q :
IntoWand p false R P Q IntoWand' p q R (|==> P) (|==> Q).
Proof.
rewrite /IntoWand' /IntoWand /= => ->.
apply wand_intro_l. by rewrite intuitionistically_if_elim bupd_wand_r.
Qed.
(** FromWand *)
Global Instance from_wand_wand P1 P2 : FromWand (P1 - P2) P1 P2.
Proof. by rewrite /FromWand. Qed.
Global Instance from_wand_wandM mP1 P2 :
FromWand (mP1 -? P2) (default emp mP1)%I P2.
Proof. by rewrite /FromWand wandM_sound. Qed.
Global Instance from_wand_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromWand P Q1 Q2 FromWand P Q1 Q2.
Proof. by rewrite /FromWand -embed_wand => <-. Qed.
(** FromImpl *)
Global Instance from_impl_impl P1 P2 : FromImpl (P1 P2) P1 P2.
Proof. by rewrite /FromImpl. Qed.
Global Instance from_impl_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromImpl P Q1 Q2 FromImpl P Q1 Q2.
Proof. by rewrite /FromImpl -embed_impl => <-. Qed.
(** FromAnd *)
Global Instance from_and_and P1 P2 : FromAnd (P1 P2) P1 P2 | 100.
......@@ -602,10 +492,6 @@ Global Instance from_and_persistently_sep P Q1 Q2 :
FromAnd (<pers> P) (<pers> Q1) (<pers> Q2) | 11.
Proof. rewrite /FromAnd=> <-. by rewrite -persistently_and persistently_and_sep. Qed.
Global Instance from_and_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromAnd P Q1 Q2 FromAnd P Q1 Q2.
Proof. by rewrite /FromAnd -embed_and => <-. Qed.
Global Instance from_and_big_sepL_cons_persistent {A} (Φ : nat A PROP) l x l' :
IsCons l x l'
Persistent (Φ 0 x)
......@@ -671,10 +557,6 @@ Global Instance from_sep_persistently P Q1 Q2 :
FromSep (<pers> P) (<pers> Q1) (<pers> Q2).
Proof. rewrite /FromSep=> <-. by rewrite persistently_sep_2. Qed.
Global Instance from_sep_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromSep P Q1 Q2 FromSep P Q1 Q2.
Proof. by rewrite /FromSep -embed_sep => <-. Qed.
Global Instance from_sep_big_sepL_cons {A} (Φ : nat A PROP) l x l' :
IsCons l x l'
FromSep ([ list] k y l, Φ k y) (Φ 0 x) ([ list] k y l', Φ (S k) y).
......@@ -703,10 +585,6 @@ Global Instance from_sep_big_sepMS_disj_union `{Countable A} (Φ : A → PROP) X
FromSep ([ mset] y X1 X2, Φ y) ([ mset] y X1, Φ y) ([ mset] y X2, Φ y).
Proof. by rewrite /FromSep big_sepMS_disj_union. Qed.
Global Instance from_sep_bupd `{BiBUpd PROP} P Q1 Q2 :
FromSep P Q1 Q2 FromSep (|==> P) (|==> Q1) (|==> Q2).
Proof. rewrite /FromSep=><-. apply bupd_sep. Qed.
(** IntoAnd *)
Global Instance into_and_and p P Q : IntoAnd p (P Q) P Q | 10.
Proof. by rewrite /IntoAnd intuitionistically_if_and. Qed.
......@@ -758,12 +636,6 @@ Proof.
- rewrite -persistently_and !intuitionistically_persistently_elim //.
- intros ->. by rewrite persistently_and.
Qed.
Global Instance into_and_embed `{BiEmbed PROP PROP'} p P Q1 Q2 :
IntoAnd p P Q1 Q2 IntoAnd p P Q1 Q2.
Proof.
rewrite /IntoAnd -embed_and=> HP. apply intuitionistically_if_intro'.
by rewrite embed_intuitionistically_if_2 HP intuitionistically_if_elim.
Qed.
(** IntoSep *)
Global Instance into_sep_sep P Q : IntoSep (P Q) P Q.
......@@ -796,10 +668,6 @@ Qed.
Global Instance into_sep_pure φ ψ : @IntoSep PROP ⌜φ ψ⌝ ⌜φ⌝ ⌜ψ⌝.
Proof. by rewrite /IntoSep pure_and persistent_and_sep_1. Qed.
Global Instance into_sep_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
IntoSep P Q1 Q2 IntoSep P Q1 Q2.
Proof. rewrite /IntoSep -embed_sep=> -> //. Qed.
Global Instance into_sep_affinely `{BiPositive PROP} P Q1 Q2 :
IntoSep P Q1 Q2 IntoSep (<affine> P) (<affine> Q1) (<affine> Q2) | 0.
Proof. rewrite /IntoSep /= => ->. by rewrite affinely_sep. Qed.
......@@ -875,16 +743,6 @@ Global Instance from_or_persistently P Q1 Q2 :
FromOr P Q1 Q2
FromOr (<pers> P) (<pers> Q1) (<pers> Q2).
Proof. rewrite /FromOr=> <-. by rewrite persistently_or. Qed.
Global Instance from_or_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
FromOr P Q1 Q2 FromOr P Q1 Q2.
Proof. by rewrite /FromOr -embed_or => <-. Qed.
Global Instance from_or_bupd `{BiBUpd PROP} P Q1 Q2 :
FromOr P Q1 Q2 FromOr (|==> P) (|==> Q1) (|==> Q2).
Proof.
rewrite /FromOr=><-.
apply or_elim; apply bupd_mono; auto using or_intro_l, or_intro_r.
Qed.
(** IntoOr *)
Global Instance into_or_or P Q : IntoOr (P Q) P Q.
......@@ -904,9 +762,6 @@ Global Instance into_or_persistently P Q1 Q2 :
IntoOr P Q1 Q2
IntoOr (<pers> P) (<pers> Q1) (<pers> Q2).
Proof. rewrite /IntoOr=>->. by rewrite persistently_or. Qed.
Global Instance into_or_embed `{BiEmbed PROP PROP'} P Q1 Q2 :
IntoOr P Q1 Q2 IntoOr P Q1 Q2.
Proof. by rewrite /IntoOr -embed_or => <-. Qed.
(** FromExist *)
Global Instance from_exist_texist {TT : tele} (Φ : TT PROP) :
......@@ -927,15 +782,6 @@ Proof. rewrite /FromExist=> <-. by rewrite absorbingly_exist. Qed.
Global Instance from_exist_persistently {A} P (Φ : A PROP) :
FromExist P Φ FromExist (<pers> P) (λ a, <pers> (Φ a))%I.
Proof. rewrite /FromExist=> <-. by rewrite persistently_exist. Qed.
Global Instance from_exist_embed `{BiEmbed PROP PROP'} {A} P (Φ : A PROP) :
FromExist P Φ FromExist P (λ a, ⎡Φ a%I).
Proof. by rewrite /FromExist -embed_exist => <-. Qed.
Global Instance from_exist_bupd `{BiBUpd PROP} {A} P (Φ : A PROP) :
FromExist P Φ FromExist (|==> P) (λ a, |==> Φ a)%I.
Proof.
rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
Qed.
(** IntoExist *)
Global Instance into_exist_exist {A} (Φ : A PROP) : IntoExist ( a, Φ a) Φ.
......@@ -971,9 +817,6 @@ Proof. rewrite /IntoExist=> HP. by rewrite HP absorbingly_exist. Qed.
Global Instance into_exist_persistently {A} P (Φ : A PROP) :
IntoExist P Φ IntoExist (<pers> P) (λ a, <pers> (Φ a))%I.
Proof. rewrite /IntoExist=> HP. by rewrite HP persistently_exist. Qed.
Global Instance into_exist_embed `{BiEmbed PROP PROP'} {A} P (Φ : A PROP) :
IntoExist P Φ IntoExist P (λ a, ⎡Φ a%I).
Proof. by rewrite /IntoExist -embed_exist => <-. Qed.
(** IntoForall *)
Global Instance into_forall_forall {A} (Φ : A PROP) : IntoForall ( a, Φ a) Φ.
......@@ -990,9 +833,6 @@ Proof. rewrite /IntoForall=> HP. by rewrite HP intuitionistically_forall. Qed.
Global Instance into_forall_persistently {A} P (Φ : A PROP) :
IntoForall P Φ IntoForall (<pers> P) (λ a, <pers> (Φ a))%I.
Proof. rewrite /IntoForall=> HP. by rewrite HP persistently_forall. Qed.
Global Instance into_forall_embed `{BiEmbed PROP PROP'} {A} P (Φ : A PROP) :
IntoForall P Φ IntoForall P (λ a, ⎡Φ a%I).
Proof. by rewrite /IntoForall -embed_forall => <-. Qed.
Global Instance into_forall_impl_pure a φ P Q :
FromPureT a P φ
......@@ -1062,13 +902,6 @@ Qed.
Global Instance from_forall_persistently {A} P (Φ : A PROP) :
FromForall P Φ FromForall (<pers> P) (λ a, <pers> (Φ a))%I.
Proof. rewrite /FromForall=> <-. by rewrite persistently_forall. Qed.
Global Instance from_forall_embed `{BiEmbed PROP PROP'} {A} P (Φ : A PROP) :
FromForall P Φ FromForall P (λ a, ⎡Φ a%I).
Proof. by rewrite /FromForall -embed_forall => <-. Qed.
(** IntoInv *)
Global Instance into_inv_embed {PROP' : bi} `{BiEmbed PROP PROP'} P N :
IntoInv P N IntoInv P N := {}.
(** ElimModal *)
Global Instance elim_modal_wand φ p p' P P' Q Q' R :
......@@ -1098,24 +931,6 @@ Proof.
absorbingly_sep_l wand_elim_r absorbing_absorbingly.
Qed.
Global Instance elim_modal_bupd `{BiBUpd PROP} p P Q :
ElimModal True p false (|==> P) P (|==> Q) (|==> Q).
Proof.
by rewrite /ElimModal
intuitionistically_if_elim bupd_frame_r wand_elim_r bupd_trans.
Qed.
Global Instance elim_modal_embed_bupd_goal `{BiEmbedBUpd PROP PROP'}
p p' φ (P P' : PROP') (Q Q' : PROP) :
ElimModal φ p p' P P' (|==> Q)%I (|==> Q')%I
ElimModal φ p p' P P' |==> Q |==> Q'.
Proof. by rewrite /ElimModal !embed_bupd. Qed.
Global Instance elim_modal_embed_bupd_hyp `{BiEmbedBUpd PROP PROP'}
p p' φ (P : PROP) (P' Q Q' : PROP') :
ElimModal φ p p' (|==> P)%I P' Q Q'
ElimModal φ p p' |==> P P' Q Q'.
Proof. by rewrite /ElimModal !embed_bupd. Qed.
(** AddModal *)
Global Instance add_modal_wand P P' Q R :
AddModal P P' Q AddModal P P' (R - Q).
......@@ -1134,18 +949,10 @@ Qed.
Global Instance add_modal_tforall {TT : tele} P P' (Φ : TT PROP) :
( x, AddModal P P' (Φ x)) AddModal P P' (.. x, Φ x).
Proof. rewrite /AddModal bi_tforall_forall. apply add_modal_forall. Qed.
Global Instance add_modal_embed_bupd_goal `{BiEmbedBUpd PROP PROP'}
(P P' : PROP') (Q : PROP) :
AddModal P P' (|==> Q)%I AddModal P P' |==> Q.
Proof. by rewrite /AddModal !embed_bupd. Qed.
Global Instance add_modal_bupd `{BiBUpd PROP} P Q : AddModal (|==> P) P (|==> Q).
Proof. by rewrite /AddModal bupd_frame_r wand_elim_r bupd_trans. Qed.
(** ElimInv *)
Global Instance elim_inv_acc_without_close {X : Type}
φ Pinv Pin
M1 M2 α β mγ Q (Q' : X PROP) :
φ Pinv Pin (M1 M2 : PROP PROP) α β mγ Q (Q' : X PROP) :
IntoAcc (X:=X) Pinv φ Pin M1 M2 α β mγ
ElimAcc (X:=X) M1 M2 α β mγ Q Q'
ElimInv φ Pinv Pin α None Q Q'.
......@@ -1161,8 +968,7 @@ Qed.
[None] or [Some _] there, so we want to reduce the combinator before showing the
goal to the user. *)
Global Instance elim_inv_acc_with_close {X : Type}
φ1 φ2 Pinv Pin
M1 M2 α β mγ Q Q' :
φ1 φ2 Pinv Pin (M1 M2 : PROP PROP) α β mγ Q Q' :
IntoAcc Pinv φ1 Pin M1 M2 α β mγ
( R, ElimModal φ2 false false (M1 R) R Q Q')
ElimInv (X:=X) (φ1 φ2) Pinv Pin
......@@ -1175,12 +981,4 @@ Proof.
iMod (Hacc with "Hinv Hin") as (x) "[Hα Hclose]"; first done.
iApply "Hcont". simpl. iSplitL "Hα"; done.
Qed.
(** IntoEmbed *)
Global Instance into_embed_embed {PROP' : bi} `{BiEmbed PROP PROP'} P :
IntoEmbed P P.
Proof. by rewrite /IntoEmbed. Qed.
Global Instance into_embed_affinely `{BiEmbedBUpd PROP PROP'} (P : PROP') (Q : PROP) :
IntoEmbed P Q IntoEmbed (<affine> P) (<affine> Q).
Proof. rewrite /IntoEmbed=> ->. by rewrite embed_affinely_2. Qed.
End bi_instances.
End class_instances.
From iris.bi Require Import bi.
From iris.proofmode Require Import modality_instances classes.
Set Default Proof Using "Type".
Import bi.
(** We add a useless hypothesis [BiEmbed PROP PROP'] in order to make sure this
instance is not used when there is no embedding between [PROP] and [PROP']. The
first [`{BiEmbed PROP PROP'}] is not considered as a premise by Coq TC search
mechanism because the rest of the hypothesis is dependent on it. *)
Global Instance as_emp_valid_embed `{!BiEmbed PROP PROP'} (φ : Prop) (P : PROP) :
BiEmbed PROP PROP'
AsEmpValid0 φ P AsEmpValid φ P.
Proof. rewrite /AsEmpValid0 /AsEmpValid=> _ ->. rewrite embed_emp_valid //. Qed.
Section class_instances_embedding.
Context `{!BiEmbed PROP PROP'}.
Implicit Types P Q R : PROP.
Global Instance into_pure_embed P φ :
IntoPure P φ IntoPure P φ.
Proof. rewrite /IntoPure=> ->. by rewrite embed_pure. Qed.
Global Instance from_pure_embed a P φ :
FromPure a P φ FromPure a P φ.
Proof. rewrite /FromPure=> <-. by rewrite -embed_pure embed_affinely_if_2. Qed.
Global Instance into_persistent_embed p P Q :
IntoPersistent p P Q IntoPersistent p P Q | 0.
Proof.
rewrite /IntoPersistent -embed_persistently -embed_persistently_if=> -> //.
Qed.
(* When having a modality nested in an embedding, e.g. [ ⎡|==> P⎤ ], we prefer
the embedding over the modality. *)
Global Instance from_modal_embed P :
FromModal (@modality_embed PROP PROP' _) P P P.
Proof. by rewrite /FromModal. Qed.
Global Instance from_modal_id_embed `(sel : A) P Q :
FromModal modality_id sel P Q
FromModal modality_id sel P Q | 100.
Proof. by rewrite /FromModal /= =><-. Qed.
Global Instance from_modal_affinely_embed `(sel : A) P Q :
FromModal modality_affinely sel P Q
FromModal modality_affinely sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_affinely_2. Qed.
Global Instance from_modal_persistently_embed `(sel : A) P Q :
FromModal modality_persistently sel P Q
FromModal modality_persistently sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_persistently. Qed.
Global Instance from_modal_intuitionistically_embed `(sel : A) P Q :
FromModal modality_intuitionistically sel P Q
FromModal modality_intuitionistically sel P Q | 100.
Proof. rewrite /FromModal /= =><-. by rewrite embed_intuitionistically_2. Qed.
Global Instance into_wand_embed p q R P Q :
IntoWand p q R P Q IntoWand p q R P Q.
Proof. by rewrite /IntoWand !embed_intuitionistically_if_2 -embed_wand=> ->. Qed.
(* There are two versions for [IntoWand ⎡R⎤ ...] with the argument being
[<affine> ⎡P⎤]. When the wand [⎡R⎤] resides in the intuitionistic context
the result of wand elimination will have the affine modality. Otherwise, it
won't. Note that when the wand [⎡R⎤] is under an affine modality, the instance
[into_wand_affine] would already have been used. *)
Global Instance into_wand_affine_embed_true q P Q R :
IntoWand true q R P Q IntoWand true q R (<affine> P) (<affine> Q) | 100.
Proof.
rewrite /IntoWand /=.
rewrite -(intuitionistically_idemp _ %I) embed_intuitionistically_2=> ->.
apply bi.wand_intro_l. destruct q; simpl.
- rewrite affinely_elim -(intuitionistically_idemp _ %I).
rewrite embed_intuitionistically_2 intuitionistically_sep_2 -embed_sep.
by rewrite wand_elim_r intuitionistically_affinely.
- by rewrite intuitionistically_affinely affinely_sep_2 -embed_sep wand_elim_r.
Qed.
Global Instance into_wand_affine_embed_false q P Q R :
IntoWand false q R (<affine> P) Q
IntoWand false q R (<affine> P) Q | 100.
Proof.
rewrite /IntoWand /= => ->.
by rewrite embed_affinely_2 embed_intuitionistically_if_2 embed_wand.
Qed.
Global Instance from_wand_embed P Q1 Q2 :
FromWand P Q1 Q2 FromWand P Q1 Q2.
Proof. by rewrite /FromWand -embed_wand => <-. Qed.
Global Instance from_impl_embed P Q1 Q2 :
FromImpl P Q1 Q2 FromImpl P Q1 Q2.
Proof. by rewrite /FromImpl -embed_impl => <-. Qed.
Global Instance from_and_embed P Q1 Q2 :
FromAnd P Q1 Q2 FromAnd P Q1 Q2.
Proof. by rewrite /FromAnd -embed_and => <-. Qed.
Global Instance from_sep_embed P Q1 Q2 :
FromSep P Q1 Q2 FromSep P Q1 Q2.
Proof. by rewrite /FromSep -embed_sep => <-. Qed.
Global Instance into_and_embed p P Q1 Q2 :
IntoAnd p P Q1 Q2 IntoAnd p P Q1 Q2.
Proof.
rewrite /IntoAnd -embed_and=> HP. apply intuitionistically_if_intro'.
by rewrite embed_intuitionistically_if_2 HP intuitionistically_if_elim.
Qed.
Global Instance into_sep_embed P Q1 Q2 :
IntoSep P Q1 Q2 IntoSep P Q1 Q2.
Proof. rewrite /IntoSep -embed_sep=> -> //. Qed.
Global Instance from_or_embed P Q1 Q2 :
FromOr P Q1 Q2 FromOr P Q1 Q2.
Proof. by rewrite /FromOr -embed_or => <-. Qed.
Global Instance into_or_embed P Q1 Q2 :
IntoOr P Q1 Q2 IntoOr P Q1 Q2.
Proof. by rewrite /IntoOr -embed_or => <-. Qed.
Global Instance from_exist_embed {A} P (Φ : A PROP) :
FromExist P Φ FromExist P (λ a, ⎡Φ a%I).
Proof. by rewrite /FromExist -embed_exist => <-. Qed.
Global Instance into_exist_embed {A} P (Φ : A PROP) :
IntoExist P Φ IntoExist P (λ a, ⎡Φ a%I).
Proof. by rewrite /IntoExist -embed_exist => <-. Qed.
Global Instance into_forall_embed {A} P (Φ : A PROP) :