Commit 2cfaecf6 authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan
Browse files

Get rid of KnownFrame and iFrame automatically instantiating evars.

parent 1ac24719
......@@ -378,8 +378,8 @@ Section proofmode_classes.
Global Instance frame_twp p s E e R Φ Ψ :
( v, Frame p R (Φ v) (Ψ v))
KnownFrame p R (WP e @ s; E [{ Φ }]) (WP e @ s; E [{ Ψ }]).
Proof. rewrite /KnownFrame /Frame=> HR. rewrite twp_frame_l. apply twp_mono, HR. Qed.
Frame p R (WP e @ s; E [{ Φ }]) (WP e @ s; E [{ Ψ }]).
Proof. rewrite /Frame=> HR. rewrite twp_frame_l. apply twp_mono, HR. Qed.
Global Instance is_except_0_wp s E e Φ : IsExcept0 (WP e @ s; E [{ Φ }]).
Proof. by rewrite /IsExcept0 -{2}fupd_twp -except_0_fupd -fupd_intro. Qed.
......
......@@ -372,8 +372,8 @@ Section proofmode_classes.
Global Instance frame_wp p s E e R Φ Ψ :
( v, Frame p R (Φ v) (Ψ v))
KnownFrame p R (WP e @ s; E {{ Φ }}) (WP e @ s; E {{ Ψ }}).
Proof. rewrite /KnownFrame /Frame=> HR. rewrite wp_frame_l. apply wp_mono, HR. Qed.
Frame p R (WP e @ s; E {{ Φ }}) (WP e @ s; E {{ Ψ }}).
Proof. rewrite /Frame=> HR. rewrite wp_frame_l. apply wp_mono, HR. Qed.
Global Instance is_except_0_wp s E e Φ : IsExcept0 (WP e @ s; E {{ Φ }}).
Proof. by rewrite /IsExcept0 -{2}fupd_wp -except_0_fupd -fupd_intro. Qed.
......
......@@ -716,25 +716,25 @@ Proof.
Qed.
(* Frame *)
Global Instance frame_here_absorbing p R : Absorbing R KnownFrame p R R True | 0.
Proof. intros. by rewrite /KnownFrame /Frame affinely_persistently_if_elim sep_elim_l. Qed.
Global Instance frame_here p R : KnownFrame p R R emp | 1.
Proof. intros. by rewrite /KnownFrame /Frame affinely_persistently_if_elim sep_elim_l. Qed.
Global Instance frame_here_absorbing p R : Absorbing R Frame p R R True | 0.
Proof. intros. by rewrite /Frame affinely_persistently_if_elim sep_elim_l. Qed.
Global Instance frame_here p R : Frame p R R emp | 1.
Proof. intros. by rewrite /Frame affinely_persistently_if_elim sep_elim_l. Qed.
Global Instance frame_affinely_here_absorbing p R :
Absorbing R KnownFrame p (bi_affinely R) R True | 0.
Absorbing R Frame p (bi_affinely R) R True | 0.
Proof.
intros. rewrite /KnownFrame /Frame affinely_persistently_if_elim affinely_elim.
intros. rewrite /Frame affinely_persistently_if_elim affinely_elim.
apply sep_elim_l, _.
Qed.
Global Instance frame_affinely_here p R : KnownFrame p (bi_affinely R) R emp | 1.
Global Instance frame_affinely_here p R : Frame p (bi_affinely R) R emp | 1.
Proof.
intros. rewrite /KnownFrame /Frame affinely_persistently_if_elim affinely_elim.
intros. rewrite /Frame affinely_persistently_if_elim affinely_elim.
apply sep_elim_l, _.
Qed.
Global Instance frame_here_pure p φ Q : FromPure false Q φ KnownFrame p ⌜φ⌝ Q True.
Global Instance frame_here_pure p φ Q : FromPure false Q φ Frame p ⌜φ⌝ Q True.
Proof.
rewrite /FromPure /KnownFrame /Frame=> <-.
rewrite /FromPure /Frame=> <-.
by rewrite affinely_persistently_if_elim sep_elim_l.
Qed.
......@@ -749,16 +749,14 @@ Global Instance make_embed_default `{BiEmbed PROP PROP'} P :
Proof. by rewrite /MakeEmbed. Qed.
Global Instance frame_embed `{BiEmbed PROP PROP'} p P Q (Q' : PROP') R :
Frame p R P Q MakeEmbed Q Q' KnownFrame p R P Q'.
Frame p R P Q MakeEmbed Q Q' Frame p R P Q'.
Proof.
rewrite /KnownFrame /Frame /MakeEmbed => <- <-.
rewrite /Frame /MakeEmbed => <- <-.
rewrite embed_sep embed_affinely_if embed_persistently_if => //.
Qed.
Global Instance frame_pure_embed `{BiEmbed PROP PROP'} p P Q (Q' : PROP') φ :
Frame p ⌜φ⌝ P Q MakeEmbed Q Q' KnownFrame p ⌜φ⌝ P Q'.
Proof.
rewrite /KnownFrame /Frame /MakeEmbed -embed_pure. apply (frame_embed p P Q).
Qed.
Frame p ⌜φ⌝ P Q MakeEmbed Q Q' Frame p ⌜φ⌝ P Q'.
Proof. rewrite /Frame /MakeEmbed -embed_pure. apply (frame_embed p P Q). Qed.
Global Instance make_sep_emp_l P : KnownLMakeSep emp P P.
Proof. apply left_id, _. Qed.
......@@ -777,31 +775,31 @@ Proof. by rewrite /MakeSep. 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'
KnownFrame true R (P1 P2) Q' | 9.
Frame true R (P1 P2) Q' | 9.
Proof.
rewrite /KnownFrame /Frame /MaybeFrame /MakeSep /= => <- <- <-.
rewrite /Frame /MaybeFrame /MakeSep /= => <- <- <-.
rewrite {1}(affinely_persistently_sep_dup R). solve_sep_entails.
Qed.
Global Instance frame_sep_l R P1 P2 Q Q' :
Frame false R P1 Q MakeSep Q P2 Q' KnownFrame false R (P1 P2) Q' | 9.
Proof. rewrite /KnownFrame /Frame /MakeSep => <- <-. by rewrite assoc. Qed.
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' KnownFrame p R (P1 P2) Q' | 10.
Frame p R P2 Q MakeSep P1 Q Q' Frame p R (P1 P2) Q' | 10.
Proof.
rewrite /KnownFrame /Frame /MakeSep => <- <-. by rewrite assoc -(comm _ P1) assoc.
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
KnownFrame p R ([ list] k y l, Φ k y) Q.
Proof. rewrite /IsCons=>->. by rewrite /KnownFrame /Frame big_sepL_cons. Qed.
Frame p R ([ list] k y l, Φ k y) Q.
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
KnownFrame p R ([ list] k y l, Φ k y) Q.
Proof. rewrite /IsApp=>->. by rewrite /KnownFrame /Frame big_opL_app. Qed.
Frame p R ([ list] k y l, Φ k y) Q.
Proof. rewrite /IsApp=>->. by rewrite /Frame big_opL_app. Qed.
Global Instance make_and_true_l P : KnownLMakeAnd True P P.
Proof. apply left_id, _. Qed.
......@@ -823,9 +821,9 @@ Global Instance frame_and p progress1 progress2 R P1 P2 Q1 Q2 Q' :
MaybeFrame p R P2 Q2 progress2
TCEq (progress1 || progress2) true
MakeAnd Q1 Q2 Q'
KnownFrame p R (P1 P2) Q' | 9.
Frame p R (P1 P2) Q' | 9.
Proof.
rewrite /MaybeFrame /KnownFrame /Frame /MakeAnd => <- <- _ <-. apply and_intro;
rewrite /MaybeFrame /Frame /MakeAnd => <- <- _ <-. apply and_intro;
[rewrite and_elim_l|rewrite and_elim_r]; done.
Qed.
......@@ -843,9 +841,9 @@ Proof. by rewrite /MakeOr. Qed.
(* We could in principle write the instance [frame_or_spatial] by a bunch of
instances, i.e. (omitting the parameter [p = false]):
Frame R P1 Q1 → Frame R P2 Q2 → KnownFrame R (P1 ∨ P2) (Q1 ∨ Q2)
Frame R P1 True → KnownFrame R (P1 ∨ P2) P2
Frame R P2 True → KnownFrame R (P1 ∨ P2) P1
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
......@@ -859,19 +857,19 @@ Global Instance frame_or_spatial progress1 progress2 R P1 P2 Q1 Q2 Q :
(TCAnd (TCEq progress1 true) (TCEq Q1 True%I))
(TCAnd (TCEq progress2 true) (TCEq Q2 True%I)))
MakeOr Q1 Q2 Q
KnownFrame false R (P1 P2) Q | 9.
Proof. rewrite /KnownFrame /Frame /MakeOr => <- <- _ <-. by rewrite -sep_or_l. Qed.
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 KnownFrame true R (P1 P2) Q | 9.
Proof. rewrite /KnownFrame /Frame /MakeOr => <- <- _ <-. by rewrite -sep_or_l. Qed.
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 KnownFrame p R (P1 - P2) (P1 - Q2).
Frame p R P2 Q2 Frame p R (P1 - P2) (P1 - Q2).
Proof.
rewrite /KnownFrame /Frame=> ?. apply wand_intro_l.
rewrite /Frame=> ?. apply wand_intro_l.
by rewrite assoc (comm _ P1) -assoc wand_elim_r.
Qed.
......@@ -883,9 +881,9 @@ Global Instance make_affinely_default P : MakeAffinely P (bi_affinely P) | 100.
Proof. by rewrite /MakeAffinely. Qed.
Global Instance frame_affinely R P Q Q' :
Frame true R P Q MakeAffinely Q Q' KnownFrame true R (bi_affinely P) Q'.
Frame true R P Q MakeAffinely Q Q' Frame true R (bi_affinely P) Q'.
Proof.
rewrite /KnownFrame /Frame /MakeAffinely=> <- <- /=.
rewrite /Frame /MakeAffinely=> <- <- /=.
by rewrite -{1}affinely_idemp affinely_sep_2.
Qed.
......@@ -901,9 +899,9 @@ Global Instance make_absorbingly_default P : MakeAbsorbingly P (bi_absorbingly P
Proof. by rewrite /MakeAbsorbingly. Qed.
Global Instance frame_absorbingly p R P Q Q' :
Frame p R P Q MakeAbsorbingly Q Q' KnownFrame p R (bi_absorbingly P) Q'.
Frame p R P Q MakeAbsorbingly Q Q' Frame p R (bi_absorbingly P) Q'.
Proof.
rewrite /KnownFrame /Frame /MakeAbsorbingly=> <- <- /=. by rewrite absorbingly_sep_r.
rewrite /Frame /MakeAbsorbingly=> <- <- /=. by rewrite absorbingly_sep_r.
Qed.
Global Instance make_persistently_true : @KnownMakePersistently PROP True True.
......@@ -918,45 +916,38 @@ Global Instance make_persistently_default P :
Proof. by rewrite /MakePersistently. Qed.
Global Instance frame_persistently R P Q Q' :
Frame true R P Q MakePersistently Q Q' KnownFrame true R (bi_persistently P) Q'.
Frame true R P Q MakePersistently Q Q' Frame true R (bi_persistently P) Q'.
Proof.
rewrite /KnownFrame /Frame /MakePersistently=> <- <- /=.
rewrite /Frame /MakePersistently=> <- <- /=.
rewrite -persistently_and_affinely_sep_l.
by rewrite -persistently_sep_2 -persistently_and_sep_l_1 persistently_affinely
persistently_idemp.
Qed.
Global Instance frame_exist {A} p R (Φ Ψ : A PROP) :
( a, Frame p R (Φ a) (Ψ a)) KnownFrame p R ( x, Φ x) ( x, Ψ x).
Proof. rewrite /KnownFrame /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
( a, Frame p R (Φ a) (Ψ a)) Frame p R ( x, Φ x) ( x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_exist_l; apply exist_mono. Qed.
Global Instance frame_forall {A} p R (Φ Ψ : A PROP) :
( a, Frame p R (Φ a) (Ψ a)) KnownFrame p R ( x, Φ x) ( x, Ψ x).
Proof. rewrite /KnownFrame /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.
( a, Frame p R (Φ a) (Ψ a)) Frame p R ( x, Φ x) ( x, Ψ x).
Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.
Global Instance frame_impl_persistent R P1 P2 Q2 :
Frame true R P2 Q2 KnownFrame true R (P1 P2) (P1 Q2).
Frame true R P2 Q2 Frame true R (P1 P2) (P1 Q2).
Proof.
rewrite /KnownFrame /Frame /= => ?. apply impl_intro_l.
rewrite /Frame /= => ?. apply impl_intro_l.
by rewrite -persistently_and_affinely_sep_l assoc (comm _ P1) -assoc impl_elim_r
persistently_and_affinely_sep_l.
Qed.
Global Instance frame_impl R P1 P2 Q2 :
Persistent P1 Absorbing P1
Frame false R P2 Q2 KnownFrame false R (P1 P2) (P1 Q2).
Frame false R P2 Q2 Frame false R (P1 P2) (P1 Q2).
Proof.
rewrite /KnownFrame /Frame /==> ???. apply impl_intro_l.
rewrite /Frame /==> ???. apply impl_intro_l.
rewrite {1}(persistent P1) persistently_and_affinely_sep_l assoc.
rewrite (comm _ ( P1)%I) -assoc -persistently_and_affinely_sep_l.
rewrite persistently_elim impl_elim_r //.
Qed.
(* In the case nothing else suceeded, then we try to instantiate the
evar [P] with [R ∗ Q]. This only does so in non-persistent mode:
The idea is that persistent assumptions will still be available
when [P] will be used, so there is no need to keep it in [P]. *)
Global Instance frame_sep_instantiate R Q : Frame false R (R Q) Q | 100.
Proof. by rewrite /Frame. Qed.
(* IntoEmbed *)
Global Instance into_embed_embed {PROP' : bi} `{BiEmbed PROP PROP'} P :
IntoEmbed P P.
......@@ -1054,7 +1045,7 @@ 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_affinely_persistently_if_2 -later_wand HR.
by rewrite !later_affinely_persistently_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).
......@@ -1067,7 +1058,7 @@ 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_affinely_persistently_if_2 -laterN_wand HR.
by rewrite !laterN_affinely_persistently_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).
......@@ -1506,8 +1497,8 @@ Proof. by rewrite /AddModal fupd_frame_r wand_elim_r fupd_trans. Qed.
(* Frame *)
Global Instance frame_eq_embed `{SbiEmbed PROP PROP'} p P Q (Q' : PROP')
{A : ofeT} (a b : A) :
Frame p (a b) P Q MakeEmbed Q Q' KnownFrame p (a b) P Q'.
Proof. rewrite /KnownFrame /Frame /MakeEmbed -embed_internal_eq. apply (frame_embed p P Q). Qed.
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 make_laterN_true n : @KnownMakeLaterN PROP n True True | 0.
Proof. by rewrite /KnownMakeLaterN /MakeLaterN laterN_True. Qed.
......@@ -1520,25 +1511,25 @@ Proof. by rewrite /MakeLaterN. Qed.
Global Instance frame_later p R R' P Q Q' :
NoBackTrack (MaybeIntoLaterN true 1 R' R)
Frame p R P Q MakeLaterN 1 Q Q' KnownFrame p R' ( P) Q'.
Frame p R P Q MakeLaterN 1 Q Q' Frame p R' ( P) Q'.
Proof.
rewrite /KnownFrame /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-.
rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-.
by rewrite later_affinely_persistently_if_2 later_sep.
Qed.
Global Instance frame_laterN p n R R' P Q Q' :
NoBackTrack (MaybeIntoLaterN true n R' R)
Frame p R P Q MakeLaterN n Q Q' KnownFrame p R' (^n P) Q'.
Frame p R P Q MakeLaterN n Q Q' Frame p R' (^n P) Q'.
Proof.
rewrite /KnownFrame /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-.
rewrite /Frame /MakeLaterN /MaybeIntoLaterN=>-[->] <- <-.
by rewrite laterN_affinely_persistently_if_2 laterN_sep.
Qed.
Global Instance frame_bupd `{BiBUpd PROP} p R P Q :
Frame p R P Q KnownFrame p R (|==> P) (|==> Q).
Proof. rewrite /KnownFrame /Frame=><-. by rewrite bupd_frame_l. Qed.
Frame p R P Q Frame p R (|==> P) (|==> Q).
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 KnownFrame p R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof. rewrite /KnownFrame /Frame=><-. by rewrite fupd_frame_l. Qed.
Frame p R P Q Frame p R (|={E1,E2}=> P) (|={E1,E2}=> Q).
Proof. rewrite /Frame=><-. by rewrite fupd_frame_l. Qed.
Global Instance make_except_0_True : @KnownMakeExcept0 PROP True True.
Proof. by rewrite /KnownMakeExcept0 /MakeExcept0 except_0_True. Qed.
......@@ -1546,9 +1537,9 @@ Global Instance make_except_0_default P : MakeExcept0 P (◇ P) | 100.
Proof. by rewrite /MakeExcept0. Qed.
Global Instance frame_except_0 p R P Q Q' :
Frame p R P Q MakeExcept0 Q Q' KnownFrame p R ( P) Q'.
Frame p R P Q MakeExcept0 Q Q' Frame p R ( P) Q'.
Proof.
rewrite /KnownFrame /Frame /MakeExcept0=><- <-.
rewrite /Frame /MakeExcept0=><- <-.
by rewrite except_0_sep -(except_0_intro (?p R)%I).
Qed.
......
......@@ -254,19 +254,7 @@ Proof. done. Qed.
Class Frame {PROP : bi} (p : bool) (R P Q : PROP) := frame : ?p R Q P.
Arguments Frame {_} _ _%I _%I _%I.
Arguments frame {_ _} _%I _%I _%I {_}.
(* [P] has - hint mode even though it is morally an input. In the case
P is an evar, it will be instantiated with [R * ?P'], where [?P']
is a new evar. This mechanism should be the only instance able to
instanciate [P] when it is an evar. *)
Hint Mode Frame + + ! - - : typeclass_instances.
(* [KnownFrame] is like [Frame], but with an [Hint Mode] that make
sure we do not instantiate [P]. *)
Class KnownFrame {PROP : bi} (p : bool) (R P Q : PROP) :=
known_frame :> Frame p R P Q.
Arguments KnownFrame {_} _ _%I _%I _%I.
Arguments known_frame {_ _} _%I _%I _%I {_}.
Hint Mode KnownFrame + + ! ! - : typeclass_instances.
Hint Mode Frame + + ! ! - : typeclass_instances.
(* The boolean [progress] indicates whether actual framing has been performed.
If it is [false], then the default instance [maybe_frame_default] below has been
......
......@@ -332,16 +332,16 @@ Qed.
search, and hence performance issues. *)
Global Instance frame_monPred_at p P Q 𝓠 R i j :
IsBiIndexRel i j Frame p R P Q MakeMonPredAt j Q 𝓠
KnownFrame p (R i) (P j) 𝓠.
Frame p (R i) (P j) 𝓠.
Proof.
rewrite /KnownFrame /Frame /MakeMonPredAt /bi_affinely_if /bi_persistently_if
rewrite /Frame /MakeMonPredAt /bi_affinely_if /bi_persistently_if
/IsBiIndexRel=> Hij <- <-.
destruct p; by rewrite Hij monPred_at_sep ?monPred_at_affinely ?monPred_at_persistently.
Qed.
Global Instance frame_monPred_at_embed i p P Q 𝓠 𝓡 :
Frame p ⎡𝓡⎤ P Q MakeMonPredAt i Q 𝓠 KnownFrame p 𝓡 (P i) 𝓠.
Frame p ⎡𝓡⎤ P Q MakeMonPredAt i Q 𝓠 Frame p 𝓡 (P i) 𝓠.
Proof.
rewrite /KnownFrame /Frame /MakeMonPredAt /bi_affinely_if /bi_persistently_if=> <- <-.
rewrite /Frame /MakeMonPredAt /bi_affinely_if /bi_persistently_if=> <- <-.
destruct p; by rewrite monPred_at_sep ?monPred_at_affinely
?monPred_at_persistently monPred_at_embed.
Qed.
......
......@@ -370,15 +370,15 @@ Proof.
Qed.
Lemma iFrame_with_evar_r P Q :
R, (P - Q - P R) R = (Q emp)%I.
R, (P - Q - P R) R = Q.
Proof.
eexists. split. iIntros "HP HQ". iFrame. iEmpIntro. done.
eexists. split. iIntros "HP HQ". iFrame. iApply "HQ". done.
Qed.
Lemma iFrame_with_evar_l P Q :
R, (P - Q - R P) R = (Q emp)%I.
R, (P - Q - R P) R = Q.
Proof.
eexists. split. iIntros "HP HQ". iFrame "HQ".
iSplitR. iEmpIntro. done. done.
eexists. split. iIntros "HP HQ". Fail iFrame "HQ".
by iSplitR "HP". done.
Qed.
Lemma iFrame_with_evar_persistent P Q :
R, (P - Q - P R Q) R = emp%I.
......
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