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Commit f763f227 authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan
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Reorganize bi.monpred. Add unfolding manual lemmas for monPred_at, and use... 

Reorganize bi.monpred. Add unfolding manual lemmas for monPred_at, and use them in proofmode.monpred. Add big op lemmas for monpred_at.
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theories/bi/embedding.v
+ 14
− 1
View file @ f763f227
From iris.algebra Require Import monoid.
From iris.bi Require Import interface derived_laws.
From iris.bi Require Import interface derived_laws big_op.
From stdpp Require Import hlist.
(* Embeddings are often used to *define* the connectives of the
......@@ -143,6 +143,19 @@ Section bi_embedding.
by split; [split|]; try apply _;
[setoid_rewrite bi_embed_sep|rewrite bi_embed_emp].
Qed.
Lemma bi_embed_big_sepL {A} (Φ : nat → A → PROP1) l :
⎡[∗ list] k↦x ∈ l, Φ k x⎤ ⊣⊢ [∗ list] k↦x ∈ l, ⎡Φ k x⎤.
Proof. apply (big_opL_commute _). Qed.
Lemma bi_embed_big_sepM `{Countable K} {A} (Φ : K → A → PROP1) (m : gmap K A) :
⎡[∗ map] k↦x ∈ m, Φ k x⎤ ⊣⊢ [∗ map] k↦x ∈ m, ⎡Φ k x⎤.
Proof. apply (big_opM_commute _). Qed.
Lemma bi_embed_big_sepS `{Countable A} (Φ : A → PROP1) (X : gset A) :
⎡[∗ set] y ∈ X, Φ y⎤ ⊣⊢ [∗ set] y ∈ X, ⎡Φ y⎤.
Proof. apply (big_opS_commute _). Qed.
Lemma bi_embed_big_sepMS `{Countable A} (Φ : A → PROP1) (X : gmultiset A) :
⎡[∗ mset] y ∈ X, Φ y⎤ ⊣⊢ [∗ mset] y ∈ X, ⎡Φ y⎤.
Proof. apply (big_opMS_commute _). Qed.
End bi_embedding.
Section sbi_embedding.
......
theories/bi/monpred.v
+ 135
− 65
View file @ f763f227
......@@ -123,8 +123,10 @@ Definition monPred_embed_aux : seal (@monPred_embed_def). by eexists. Qed.
Global Instance monPred_embed : BiEmbed PROP monPred := unseal monPred_embed_aux.
Definition monPred_embed_eq : bi_embed (A:=PROP) = _ := seal_eq _.
Definition monPred_pure (φ : Prop) : monPred := ⎡⌜φ⌝⎤%I.
Definition monPred_emp : monPred := ⎡emp⎤%I.
Definition monPred_pure (φ : Prop) : monPred := tc_opaque ⎡⌜φ⌝⎤%I.
Definition monPred_emp : monPred := tc_opaque ⎡emp⎤%I.
Definition monPred_internal_eq (A : ofeT) (a b : A) : monPred := tc_opaque ⎡a ≡ b⎤%I.
Definition monPred_plainly P : monPred := tc_opaque ⎡∀ i, bi_plainly (P i)⎤%I.
Program Definition monPred_and_def P Q : monPred :=
MonPred (λ i, P i ∧ Q i)%I _.
......@@ -160,12 +162,6 @@ Definition monPred_exist_aux : seal (@monPred_exist_def). by eexists. Qed.
Definition monPred_exist := unseal (@monPred_exist_aux).
Definition monPred_exist_eq : @monPred_exist = _ := seal_eq _.
Definition monPred_internal_eq_def (A : ofeT) (a b : A) : monPred :=
MonPred (λ _, bi_internal_eq a b) _.
Definition monPred_internal_eq_aux : seal (@monPred_internal_eq_def). by eexists. Qed.
Definition monPred_internal_eq := unseal (@monPred_internal_eq_aux).
Definition monPred_internal_eq_eq : @monPred_internal_eq = _ := seal_eq _.
Program Definition monPred_sep_def P Q : monPred :=
MonPred (λ i, P i ∗ Q i)%I _.
Next Obligation. solve_proper. Qed.
......@@ -186,8 +182,6 @@ Definition monPred_persistently_aux : seal (@monPred_persistently_def). by eexis
Definition monPred_persistently := unseal monPred_persistently_aux.
Definition monPred_persistently_eq : @monPred_persistently = _ := seal_eq _.
Definition monPred_plainly P : monPred := ⎡∀ i, bi_plainly (P i)⎤%I.
Program Definition monPred_in_def (i0 : I) : monPred :=
MonPred (λ i : I, ⌜i0 ⊑ i⌝%I) _.
Next Obligation. solve_proper. Qed.
......@@ -207,7 +201,6 @@ Definition monPred_ex_aux : seal (@monPred_ex_def). by eexists. Qed.
Definition monPred_ex := unseal (@monPred_ex_aux).
Definition monPred_ex_eq : @monPred_ex = _ := seal_eq _.
Definition monPred_bupd_def `{BUpd PROP} (P : monPred) : monPred :=
(* monPred_upclosed is not actually needed, since bupd is always
monotone. However, this depends on BUpdFacts, and we do not want
......@@ -247,7 +240,7 @@ End Sbi.
Module MonPred.
Definition unseal_eqs :=
(@monPred_and_eq, @monPred_or_eq, @monPred_impl_eq,
@monPred_forall_eq, @monPred_exist_eq, @monPred_internal_eq_eq,
@monPred_forall_eq, @monPred_exist_eq,
@monPred_sep_eq, @monPred_wand_eq,
@monPred_persistently_eq, @monPred_later_eq,
@monPred_in_eq, @monPred_all_eq, @monPred_ex_eq, @monPred_embed_eq,
......@@ -260,7 +253,7 @@ Ltac unseal :=
unfold sbi_emp, sbi_pure, sbi_and, sbi_or, sbi_impl, sbi_forall, sbi_exist,
sbi_internal_eq, sbi_sep, sbi_wand, sbi_persistently, sbi_plainly,
bi_plainly; simpl;
unfold monPred_pure, monPred_emp, monPred_plainly; simpl;
unfold monPred_pure, monPred_emp, monPred_internal_eq, monPred_plainly; simpl;
rewrite !unseal_eqs /=.
End MonPred.
Import MonPred.
......@@ -398,14 +391,18 @@ End canonical_sbi.
Section bi_facts.
Context {I : biIndex} {PROP : bi}.
Local Notation monPredI := (monPredI I PROP).
Local Notation monPred_at := (@monPred_at I PROP).
Implicit Types i : I.
Implicit Types P Q : monPred I PROP.
Implicit Types P Q : monPredI.
(** Instances *)
Global Instance monPred_at_mono :
Proper ((⊢) ==> (⊑) ==> (⊢)) (@monPred_at I PROP).
Proper ((⊢) ==> (⊑) ==> (⊢)) monPred_at.
Proof. by move=> ?? [?] ?? ->. Qed.
Global Instance monPred_at_mono_flip :
Proper (flip (⊢) ==> flip (⊑) ==> flip (⊢)) (@monPred_at I PROP).
Proper (flip (⊢) ==> flip (⊑) ==> flip (⊢)) monPred_at.
Proof. solve_proper. Qed.
Global Instance monPred_in_proper (R : relation I) :
......@@ -437,40 +434,22 @@ Global Instance monPred_ex_mono_flip :
Proper (flip (⊢) ==> flip (⊢)) (@monPred_ex I PROP).
Proof. solve_proper. Qed.
Global Instance monPred_positive : BiPositive PROP → BiPositive (monPredI I PROP).
Global Instance monPred_positive : BiPositive PROP → BiPositive monPredI.
Proof. split => ?. unseal. apply bi_positive. Qed.
Global Instance monPred_affine : BiAffine PROP → BiAffine (monPredI I PROP).
Global Instance monPred_affine : BiAffine PROP → BiAffine monPredI.
Proof. split => ?. unseal. by apply affine. Qed.
(* (∀ x, bottom ⊑ x) cannot be infered using typeclasses. So this is
not an instance. One should explicitely make an instance from this
lemma for each instantiation of monPred. *)
Lemma monPred_plainly_exist (bottom : I) :
(∀ x, bottom ⊑ x) → BiPlainlyExist PROP → BiPlainlyExist (monPredI I PROP).
(∀ x, bottom ⊑ x) → BiPlainlyExist PROP → BiPlainlyExist monPredI.
Proof.
intros ????. unseal. split=>? /=.
rewrite (bi.forall_elim bottom) plainly_exist_1. do 2 f_equiv.
apply bi.forall_intro=>?. by do 2 f_equiv.
Qed.
Lemma monPred_wand_force P Q i : (P -∗ Q) i -∗ (P i -∗ Q i).
Proof. unseal. rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim //. Qed.
Lemma monPred_impl_force P Q i : (P → Q) i -∗ (P i → Q i).
Proof. unseal. rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim //. Qed.
Lemma monPred_at_embed (P : PROP) (i : I) :
(bi_embed (B := monPred _ _) P) i ⊣⊢ P.
Proof. by unseal. Qed.
Lemma monPred_persistently_if_eq p P i:
(bi_persistently_if p P) i = bi_persistently_if p (P i).
Proof. rewrite /bi_persistently_if. unseal. by destruct p. Qed.
Lemma monPred_affinely_if_eq p P i:
(bi_affinely_if p P) i = bi_affinely_if p (P i).
Proof. rewrite /bi_affinely_if. destruct p => //. rewrite /bi_affinely. by unseal. Qed.
Global Instance monPred_at_persistent P i : Persistent P → Persistent (P i).
Proof. move => [] /(_ i). by unseal. Qed.
Global Instance monPred_at_plain P i : Plain P → Plain (P i).
......@@ -540,7 +519,7 @@ Proof.
by apply affine, bi.exist_affine.
Qed.
Global Instance monPred_bi_embedding : BiEmbedding PROP (monPredI I PROP).
Global Instance monPred_bi_embedding : BiEmbedding PROP monPredI.
Proof.
split; try apply _; unseal; try done.
- move =>?? /= [/(_ inhabitant) ?] //.
......@@ -552,6 +531,78 @@ Proof.
by apply bi.forall_intro. by rewrite bi.forall_elim.
Qed.
Global Instance monPred_bupd_facts `{BUpdFacts PROP} : BUpdFacts monPredI.
Proof.
split; unseal; unfold monPred_bupd_def, monPred_upclosed.
- intros n P Q HPQ. split=>/= i. by repeat f_equiv.
- intros P. split=>/= i. apply bi.forall_intro=>?. rewrite bi.pure_impl_forall.
apply bi.forall_intro=><-. apply bupd_intro.
- intros P Q HPQ. split=>/= i. by repeat f_equiv.
- intros P. split=>/= i. do 3 f_equiv. rewrite -(bupd_trans (P _))
bi.forall_elim bi.pure_impl_forall bi.forall_elim //.
- intros P Q. split=> /= i. apply bi.forall_intro=>?.
rewrite bi.pure_impl_forall. apply bi.forall_intro=><-.
rewrite -bupd_frame_r bi.forall_elim bi.pure_impl_forall bi.forall_elim //.
- intros P. split=> /= i. rewrite bi.forall_elim bi.pure_impl_forall
bi.forall_elim // -bi.plainly_forall bupd_plainly bi.forall_elim //.
Qed.
(** monPred_at unfolding laws *)
Lemma monPred_at_embed i (P : PROP) : monPred_at ⎡P⎤ i ⊣⊢ P.
Proof. by unseal. Qed.
Lemma monPred_at_pure i (φ : Prop) : monPred_at ⌜φ⌝ i ⊣⊢ ⌜φ⌝.
Proof. by apply monPred_at_embed. Qed.
Lemma monPred_at_emp i : monPred_at emp i ⊣⊢ emp.
Proof. by apply monPred_at_embed. Qed.
Lemma monPred_at_internal_eq {A : ofeT} i (a b : A) : monPred_at (a ≡ b) i ⊣⊢ a ≡ b.
Proof. by apply monPred_at_embed. Qed.
Lemma monPred_at_plainly i P : bi_plainly P i ⊣⊢ ∀ j, bi_plainly (P j).
Proof. by apply monPred_at_embed. Qed.
Lemma monPred_at_and i P Q : (P ∧ Q) i ⊣⊢ P i ∧ Q i.
Proof. by unseal. Qed.
Lemma monPred_at_or i P Q : (P ∨ Q) i ⊣⊢ P i ∨ Q i.
Proof. by unseal. Qed.
Lemma monPred_at_impl i P Q : (P → Q) i ⊣⊢ ∀ j, ⌜i ⊑ j⌝ → P j → Q j.
Proof. by unseal. Qed.
Lemma monPred_at_forall {A} i (Φ : A → monPredI) : (∀ x, Φ x) i ⊣⊢ ∀ x, Φ x i.
Proof. by unseal. Qed.
Lemma monPred_at_exist {A} i (Φ : A → monPredI) : (∃ x, Φ x) i ⊣⊢ ∃ x, Φ x i.
Proof. by unseal. Qed.
Lemma monPred_at_sep i P Q : (P ∗ Q) i ⊣⊢ P i ∗ Q i.
Proof. by unseal. Qed.
Lemma monPred_at_wand i P Q : (P -∗ Q) i ⊣⊢ ∀ j, ⌜i ⊑ j⌝ → P j -∗ Q j.
Proof. by unseal. Qed.
Lemma monPred_at_persistently i P : bi_persistently P i ⊣⊢ bi_persistently (P i).
Proof. by unseal. Qed.
Lemma monPred_at_in i j : monPred_at (monPred_in j) i ⊣⊢ ⌜j ⊑ i⌝.
Proof. by unseal. Qed.
Lemma monPred_at_all i P : monPred_all P i ⊣⊢ ∀ j, P j.
Proof. by unseal. Qed.
Lemma monPred_at_ex i P : monPred_ex P i ⊣⊢ ∃ j, P j.
Proof. by unseal. Qed.
Lemma monPred_at_bupd `{BUpdFacts PROP} i P : (|==> P) i ⊣⊢ |==> P i.
Proof.
unseal. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
- rewrite !bi.forall_elim //.
- do 2 apply bi.forall_intro=>?. by do 2 f_equiv.
Qed.
Lemma monPred_at_persistently_if i p P :
bi_persistently_if p P i ⊣⊢ bi_persistently_if p (P i).
Proof. destruct p=>//=. apply monPred_at_persistently. Qed.
Lemma monPred_at_affinely i P : bi_affinely P i ⊣⊢ bi_affinely (P i).
Proof. by rewrite /bi_affinely monPred_at_and monPred_at_emp. Qed.
Lemma monPred_at_affinely_if i p P :
bi_affinely_if p P i ⊣⊢ bi_affinely_if p (P i).
Proof. destruct p=>//=. apply monPred_at_affinely. Qed.
Lemma monPred_at_absorbingly i P : bi_absorbingly P i ⊣⊢ bi_absorbingly (P i).
Proof. by rewrite /bi_absorbingly monPred_at_sep monPred_at_pure. Qed.
Lemma monPred_wand_force i P Q : (P -∗ Q) i -∗ (P i -∗ Q i).
Proof. unseal. rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim //. Qed.
Lemma monPred_impl_force i P Q : (P → Q) i -∗ (P i → Q i).
Proof. unseal. rewrite bi.forall_elim bi.pure_impl_forall bi.forall_elim //. Qed.
(* Laws for monPred_all. *)
Lemma monPred_all_elim P : monPred_all P ⊢ P.
Proof. unseal. split=>?. apply bi.forall_elim. Qed.
Lemma monPred_all_idemp P : monPred_all (monPred_all P) ⊣⊢ monPred_all P.
......@@ -579,46 +630,57 @@ Proof.
eapply bi.pure_elim; [apply bi.and_elim_l|]=>?. rewrite bi.and_elim_r. by f_equiv.
Qed.
Global Instance monPred_bupd_facts `{BUpdFacts PROP} : BUpdFacts (monPredI I PROP).
Lemma monPred_equivI {PROP' : bi} P Q :
bi_internal_eq (PROP:=PROP') P Q ⊣⊢ ∀ i, P i ≡ Q i.
Proof.
split; unseal; unfold monPred_bupd_def, monPred_upclosed.
- intros n P Q HPQ. split=>/= i. by repeat f_equiv.
- intros P. split=>/= i. apply bi.forall_intro=>?. rewrite bi.pure_impl_forall.
apply bi.forall_intro=><-. apply bupd_intro.
- intros P Q HPQ. split=>/= i. by repeat f_equiv.
- intros P. split=>/= i. do 3 f_equiv. rewrite -(bupd_trans (P _))
bi.forall_elim bi.pure_impl_forall bi.forall_elim //.
- intros P Q. split=> /= i. apply bi.forall_intro=>?.
rewrite bi.pure_impl_forall. apply bi.forall_intro=><-.
rewrite -bupd_frame_r bi.forall_elim bi.pure_impl_forall bi.forall_elim //.
- intros P. split=> /= i. rewrite bi.forall_elim bi.pure_impl_forall
bi.forall_elim // -bi.plainly_forall bupd_plainly bi.forall_elim //.
Qed.
Lemma monPred_bupd_at `{BUpdFacts PROP} P i : (|==> P) i ⊣⊢ |==> P i.
Proof.
unseal. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
- rewrite !bi.forall_elim //.
- do 2 apply bi.forall_intro=>?. by do 2 f_equiv.
apply bi.equiv_spec. split.
- apply bi.forall_intro=>?. apply (bi.f_equiv (flip monPred_at _)).
- by rewrite -{2}(sig_monPred_sig P) -{2}(sig_monPred_sig Q)
-bi.f_equiv -bi.sig_equivI !bi.ofe_fun_equivI.
Qed.
Lemma monPred_bupd_embed `{BUpdFacts PROP} (P : PROP) :
⎡|==> P⎤ ⊣⊢ bupd (PROP:=monPred I PROP) ⎡P⎤.
⎡|==> P⎤ ⊣⊢ bupd (PROP:=monPredI) ⎡P⎤.
Proof.
unseal. split=>i /=. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
- by do 2 apply bi.forall_intro=>?.
- rewrite !bi.forall_elim //.
Qed.
(** Big op *)
Global Instance monPred_at_monoid_and_homomorphism i :
MonoidHomomorphism bi_and bi_and (≡) (flip monPred_at i).
Proof. split; [split|]; try apply _. apply monPred_at_and. apply monPred_at_pure. Qed.
Global Instance monPred_at_monoid_or_homomorphism :
MonoidHomomorphism bi_or bi_or (≡) (flip monPred_at i).
Proof. split; [split|]; try apply _. apply monPred_at_or. apply monPred_at_pure. Qed.
Global Instance monPred_at_monoid_sep_homomorphism :
MonoidHomomorphism bi_sep bi_sep (≡) (flip monPred_at i).
Proof. split; [split|]; try apply _. apply monPred_at_sep. apply monPred_at_emp. Qed.
Lemma monPred_at_big_sepL {A} i (Φ : nat → A → monPredI) l :
([∗ list] k↦x ∈ l, Φ k x) i ⊣⊢ [∗ list] k↦x ∈ l, Φ k x i.
Proof. apply (big_opL_commute (flip monPred_at i)). Qed.
Lemma monPred_at_big_sepM `{Countable K} {A} i (Φ : K → A → monPredI) (m : gmap K A) :
([∗ map] k↦x ∈ m, Φ k x) i ⊣⊢ [∗ map] k↦x ∈ m, Φ k x i.
Proof. apply (big_opM_commute (flip monPred_at i)). Qed.
Lemma monPred_at_big_sepS `{Countable A} i (Φ : A → monPredI) (X : gset A) :
([∗ set] y ∈ X, Φ y) i ⊣⊢ [∗ set] y ∈ X, Φ y i.
Proof. apply (big_opS_commute (flip monPred_at i)). Qed.
Lemma monPred_at_big_sepMS `{Countable A} i (Φ : A → monPredI) (X : gmultiset A) :
([∗ mset] y ∈ X, Φ y) i ⊣⊢ ([∗ mset] y ∈ X, Φ y i).
Proof. apply (big_opMS_commute (flip monPred_at i)). Qed.
End bi_facts.
Section sbi_facts.
Context {I : biIndex} {PROP : sbi}.
Local Notation monPredSI := (monPredSI I PROP).
Implicit Types i : I.
Implicit Types P Q : monPred I PROP.
Implicit Types P Q : monPredSI.
Global Instance monPred_at_timeless P i : Timeless P → Timeless (P i).
Proof. move => [] /(_ i). unfold Timeless. by unseal. Qed.
Global Instance monPred_embed_timeless (P : PROP) :
Timeless P → @Timeless (monPredSI I PROP) ⎡P⎤%I.
Proof. intros. split => ? /=. by unseal. Qed.
Global Instance monPred_in_timeless i0 : Timeless (@monPred_in I PROP i0).
Proof. split => ? /=. unseal. apply timeless, _. Qed.
Global Instance monPred_all_timeless P :
......@@ -634,10 +696,10 @@ Proof.
by apply timeless, bi.exist_timeless.
Qed.
Global Instance monPred_sbi_embedding : SbiEmbedding PROP (monPredSI I PROP).
Global Instance monPred_sbi_embedding : SbiEmbedding PROP monPredSI.
Proof. split; try apply _. by unseal. Qed.
Global Instance monPred_fupd_facts `{FUpdFacts PROP} : FUpdFacts (monPredSI I PROP).
Global Instance monPred_fupd_facts `{FUpdFacts PROP} : FUpdFacts monPredSI.
Proof.
split; first apply _; unseal; unfold monPred_fupd_def, monPred_upclosed.
- intros E1 E2 n P Q HPQ. split=>/= i. by repeat f_equiv.
......@@ -664,13 +726,21 @@ Proof.
- intros E P ?. split=>/= i. setoid_rewrite bi.pure_impl_forall.
do 2 setoid_rewrite bi.later_forall. do 4 f_equiv. apply later_fupd_plain, _.
Qed.
Lemma monPred_fupd_at `{FUpdFacts PROP} E1 E2 P i :
(** Unfolding lemmas *)
Lemma monPred_at_later i P : (▷ P) i ⊣⊢ ▷ P i.
Proof. by unseal. Qed.
Lemma monPred_at_fupd `{FUpdFacts PROP} i E1 E2 P :
(|={E1,E2}=> P) i ⊣⊢ |={E1,E2}=> P i.
Proof.
unseal. setoid_rewrite bi.pure_impl_forall. apply bi.equiv_spec; split.
- rewrite !bi.forall_elim //.
- do 2 apply bi.forall_intro=>?. by do 2 f_equiv.
Qed.
Lemma monPred_at_except_0 i P : (◇ P) i ⊣⊢ ◇ P i.
Proof. by unseal. Qed.
Lemma monPred_fupd_embed `{FUpdFacts PROP} E1 E2 (P : PROP) :
⎡|={E1,E2}=> P⎤ ⊣⊢ fupd E1 E2 (PROP:=monPred I PROP) ⎡P⎤.
Proof.
......
theories/proofmode/monpred.v
+ 86
− 74
View file @ f763f227
From iris.bi Require Export monpred.
From iris.proofmode Require Import tactics.
Import MonPred.
Class MakeMonPredAt {I : biIndex} {PROP : bi} (i : I)
(P : monPred I PROP) (𝓟 : PROP) :=
......@@ -25,39 +24,39 @@ Implicit Types φ : Prop.
Implicit Types i j : I.
Global Instance make_monPred_at_pure φ i : MakeMonPredAt i ⌜φ⌝ ⌜φ⌝.
Proof. rewrite /MakeMonPredAt. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_pure. Qed.
Global Instance make_monPred_at_internal_eq {A : ofeT} (x y : A) i :
MakeMonPredAt i (x ≡ y) (x ≡ y).
Proof. rewrite /MakeMonPredAt. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_internal_eq. Qed.
Global Instance make_monPred_at_emp i : MakeMonPredAt i emp emp.
Proof. rewrite /MakeMonPredAt. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_emp. Qed.
Global Instance make_monPred_at_sep i P 𝓟 Q 𝓠 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i Q 𝓠 →
MakeMonPredAt i (P ∗ Q) (𝓟 ∗ 𝓠).
Proof. rewrite /MakeMonPredAt=><-<-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_sep=><-<-. Qed.
Global Instance make_monPred_at_and i P 𝓟 Q 𝓠 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i Q 𝓠 →
MakeMonPredAt i (P ∧ Q) (𝓟 ∧ 𝓠).
Proof. rewrite /MakeMonPredAt=><-<-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_and=><-<-. Qed.
Global Instance make_monPred_at_or i P 𝓟 Q 𝓠 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i Q 𝓠 →
MakeMonPredAt i (P ∨ Q) (𝓟 ∨ 𝓠).
Proof. rewrite /MakeMonPredAt=><-<-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_or=><-<-. Qed.
Global Instance make_monPred_at_forall {A} i (Φ : A → monPred) (Ψ : A → PROP) :
(∀ a, MakeMonPredAt i (Φ a) (Ψ a)) → MakeMonPredAt i (∀ a, Φ a) (∀ a, Ψ a).
Proof. rewrite /MakeMonPredAt=>H. setoid_rewrite <- H. by unseal. Qed.
Proof. rewrite /MakeMonPredAt monPred_at_forall=>H. by setoid_rewrite <- H. Qed.
Global Instance make_monPred_at_exists {A} i (Φ : A → monPred) (Ψ : A → PROP) :
(∀ a, MakeMonPredAt i (Φ a) (Ψ a)) → MakeMonPredAt i (∃ a, Φ a) (∃ a, Ψ a).
Proof. rewrite /MakeMonPredAt=>H. setoid_rewrite <- H. by unseal. Qed.
Proof. rewrite /MakeMonPredAt monPred_at_exist=>H. by setoid_rewrite <- H. Qed.
Global Instance make_monPred_at_persistently i P 𝓟 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i (bi_persistently P) (bi_persistently 𝓟).
Proof. rewrite /MakeMonPredAt=><-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_persistently=><-. Qed.
Global Instance make_monPred_at_affinely i P 𝓟 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i (bi_affinely P) (bi_affinely 𝓟).
Proof. rewrite /MakeMonPredAt=><-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_affinely=><-. Qed.
Global Instance make_monPred_at_absorbingly i P 𝓟 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i (bi_absorbingly P) (bi_absorbingly 𝓟).
Proof. rewrite /MakeMonPredAt=><-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_absorbingly=><-. Qed.
Global Instance make_monPred_at_persistently_if i P 𝓟 p :
MakeMonPredAt i P 𝓟 →
MakeMonPredAt i (bi_persistently_if p P) (bi_persistently_if p 𝓟).
......@@ -66,15 +65,15 @@ Global Instance make_monPred_at_affinely_if i P 𝓟 p :
MakeMonPredAt i P 𝓟 →
MakeMonPredAt i (bi_affinely_if p P) (bi_affinely_if p 𝓟).
Proof. destruct p; simpl; apply _. Qed.
Global Instance make_monPred_at_embed i : MakeMonPredAt i ⎡P⎤ P.
Proof. rewrite /MakeMonPredAt. by unseal. Qed.
Global Instance make_monPred_at_embed i 𝓟 : MakeMonPredAt i ⎡𝓟⎤ 𝓟.
Proof. by rewrite /MakeMonPredAt monPred_at_embed. Qed.
Global Instance make_monPred_at_in i j : MakeMonPredAt j (monPred_in i) ⌜i ⊑ j⌝.
Proof. rewrite /MakeMonPredAt. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_in. Qed.
Global Instance make_monPred_at_default i P : MakeMonPredAt i P (P i) | 100.
Proof. by rewrite /MakeMonPredAt. Qed.
Global Instance make_monPred_at_bupd `{BUpdFacts PROP} i P 𝓟 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i (|==> P)%I (|==> 𝓟)%I.
Proof. by rewrite /MakeMonPredAt monPred_bupd_at=> <-. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_bupd=> <-. Qed.
Global Instance from_assumption_make_monPred_at_l p i j P 𝓟 :
MakeMonPredAt i P 𝓟 → IsBiIndexRel j i → FromAssumption p (P j) 𝓟.
......@@ -100,9 +99,9 @@ Global Instance as_valid_monPred_at φ P (Φ : I → PROP) :
AsValid φ P → (∀ i, MakeMonPredAt i P (Φ i)) → AsValid' φ (∀ i, Φ i) | 100.
Proof.
rewrite /MakeMonPredAt /AsValid' /AsValid /bi_valid=> -> EQ.
setoid_rewrite <-EQ. unseal; split.
- move=>[/= /bi.forall_intro //].
- move=>HP. split=>i. rewrite /= HP bi.forall_elim //.
setoid_rewrite <-EQ. split.
- move=>[H]. apply bi.forall_intro=>i. rewrite -H. by rewrite monPred_at_emp.
- move=>HP. split=>i. rewrite monPred_at_emp HP bi.forall_elim //.
Qed.
Global Instance as_valid_monPred_at_wand φ P Q (Φ Ψ : I → PROP) :
AsValid φ (P -∗ Q) →
......@@ -125,36 +124,32 @@ Proof.
- move=>HP. apply bi.equiv_wand_iff. split=>i. by iSplit; iIntros; iApply HP.
Qed.
Global Instance into_pure_monPred_at P φ i :
IntoPure P φ → IntoPure (P i) φ.
Proof. rewrite /IntoPure=>->. by unseal. Qed.
Global Instance from_pure_monPred_at P φ i :
FromPure P φ → FromPure (P i) φ.
Proof. rewrite /FromPure=><-. by unseal. Qed.
Global Instance into_pure_monPred_in i j :
@IntoPure PROP (monPred_in i j) (i ⊑ j).
Proof. rewrite /IntoPure. by unseal. Qed.
Global Instance from_pure_monPred_in i j :
@FromPure PROP (monPred_in i j) (i ⊑ j).
Proof. rewrite /FromPure. by unseal. Qed.
Global Instance into_pure_monPred_at P φ i : IntoPure P φ → IntoPure (P i) φ.
Proof. rewrite /IntoPure=>->. by rewrite monPred_at_pure. Qed.
Global Instance from_pure_monPred_at P φ i : FromPure P φ → FromPure (P i) φ.
Proof. rewrite /FromPure=><-. by rewrite monPred_at_pure. Qed.
Global Instance into_pure_monPred_in i j : @IntoPure PROP (monPred_in i j) (i ⊑ j).
Proof. by rewrite /IntoPure monPred_at_in. Qed.
Global Instance from_pure_monPred_in i j : @FromPure PROP (monPred_in i j) (i ⊑ j).
Proof. by rewrite /FromPure monPred_at_in. Qed.
Global Instance into_internal_eq_monPred_at {A : ofeT} (x y : A) P i :
IntoInternalEq P x y → IntoInternalEq (P i) x y.
Proof. rewrite /IntoInternalEq=> ->. by unseal. Qed.
Proof. rewrite /IntoInternalEq=> ->. by rewrite monPred_at_internal_eq. Qed.
Global Instance into_persistent_monPred_at p P Q 𝓠 i :
IntoPersistent p P Q → MakeMonPredAt i Q 𝓠 → IntoPersistent p (P i) 𝓠 | 0.
Proof.
rewrite /IntoPersistent /MakeMonPredAt /bi_persistently_if.
unseal=>-[/(_ i) ?] <-. by destruct p.
rewrite /IntoPersistent /MakeMonPredAt =>-[/(_ i) ?] <-.
by rewrite -monPred_at_persistently -monPred_at_persistently_if.
Qed.
Global Instance from_always_monPred_at a pe P Q 𝓠 i :
FromAlways a pe false P Q → MakeMonPredAt i Q 𝓠 →
FromAlways a pe false (P i) 𝓠 | 0.
Proof.
rewrite /FromAlways /MakeMonPredAt /bi_persistently_if /bi_affinely_if=><-.
by destruct a, pe=><-; try unseal.
rewrite /FromAlways /MakeMonPredAt /bi_persistently_if /bi_affinely_if=><- /=.
destruct a, pe=><- /=; by rewrite ?monPred_at_affinely ?monPred_at_persistently.
Qed.
Lemma into_wand_monPred_at_unknown_unknown p q R P 𝓟 Q 𝓠 i :
......@@ -163,7 +158,7 @@ Lemma into_wand_monPred_at_unknown_unknown p q R P 𝓟 Q 𝓠 i :
Proof.
rewrite /IntoWand /MakeMonPredAt /bi_affinely_if /bi_persistently_if.
destruct p, q=> /bi.wand_elim_l' [/(_ i) H] <- <-; apply bi.wand_intro_r;
revert H; unseal; done.
revert H; by rewrite monPred_at_sep ?monPred_at_affinely ?monPred_at_persistently.
Qed.
Lemma into_wand_monPred_at_unknown_known p q R P 𝓟 Q i j :
IsBiIndexRel i j → IntoWand p q R P Q →
......@@ -198,14 +193,14 @@ Global Instance from_forall_monPred_at_wand P Q (Φ Ψ : I → PROP) i :
(∀ j, MakeMonPredAt j P (Φ j)) → (∀ j, MakeMonPredAt j Q (Ψ j)) →
FromForall ((P -∗ Q) i)%I (λ j, ⌜i ⊑ j⌝ → Φ j -∗ Ψ j)%I.
Proof.
rewrite /FromForall /MakeMonPredAt. unseal=> H1 H2. do 2 f_equiv.
rewrite /FromForall /MakeMonPredAt monPred_at_wand=> H1 H2. do 2 f_equiv.
by rewrite H1 H2.
Qed.
Global Instance from_forall_monPred_at_impl P Q (Φ Ψ : I → PROP) i :
(∀ j, MakeMonPredAt j P (Φ j)) → (∀ j, MakeMonPredAt j Q (Ψ j)) →
FromForall ((P → Q) i)%I (λ j, ⌜i ⊑ j⌝ → Φ j → Ψ j)%I.
Proof.
rewrite /FromForall /MakeMonPredAt. unseal=> H1 H2. do 2 f_equiv.
rewrite /FromForall /MakeMonPredAt monPred_at_impl=> H1 H2. do 2 f_equiv.
by rewrite H1 H2 bi.pure_impl_forall.
Qed.
......@@ -219,84 +214,97 @@ Qed.
Global Instance from_and_monPred_at P Q1 𝓠1 Q2 𝓠2 i :
FromAnd P Q1 Q2 → MakeMonPredAt i Q1 𝓠1 → MakeMonPredAt i Q2 𝓠2 →
FromAnd (P i) 𝓠1 𝓠2.
Proof. rewrite /FromAnd /MakeMonPredAt /MakeMonPredAt=> <- <- <-. by unseal. Qed.
Proof.
rewrite /FromAnd /MakeMonPredAt /MakeMonPredAt=> <- <- <-.
by rewrite monPred_at_and.
Qed.
Global Instance into_and_monPred_at p P Q1 𝓠1 Q2 𝓠2 i :
IntoAnd p P Q1 Q2 → MakeMonPredAt i Q1 𝓠1 → MakeMonPredAt i Q2 𝓠2 →
IntoAnd p (P i) 𝓠1 𝓠2.
Proof.
rewrite /IntoAnd /MakeMonPredAt /bi_affinely_if /bi_persistently_if.
destruct p=>-[/(_ i) H] <- <-; revert H; unseal; done.
destruct p=>-[/(_ i) H] <- <-; revert H;
by rewrite ?monPred_at_affinely ?monPred_at_persistently monPred_at_and.
Qed.
Global Instance from_sep_monPred_at P Q1 𝓠1 Q2 𝓠2 i :
FromSep P Q1 Q2 → MakeMonPredAt i Q1 𝓠1 → MakeMonPredAt i Q2 𝓠2 →
FromSep (P i) 𝓠1 𝓠2.
Proof. rewrite /FromSep /MakeMonPredAt=> <- <- <-. by unseal. Qed.
Proof. rewrite /FromSep /MakeMonPredAt=> <- <- <-. by rewrite monPred_at_sep. Qed.
Global Instance into_sep_monPred_at P Q1 𝓠1 Q2 𝓠2 i :
IntoSep P Q1 Q2 → MakeMonPredAt i Q1 𝓠1 → MakeMonPredAt i Q2 𝓠2 →
IntoSep (P i) 𝓠1 𝓠2.
Proof. rewrite /IntoSep /MakeMonPredAt=> -> <- <-. by unseal. Qed.
Proof. rewrite /IntoSep /MakeMonPredAt=> -> <- <-. by rewrite monPred_at_sep. Qed.
Global Instance from_or_monPred_at P Q1 𝓠1 Q2 𝓠2 i :
FromOr P Q1 Q2 → MakeMonPredAt i Q1 𝓠1 → MakeMonPredAt i Q2 𝓠2 →
FromOr (P i) 𝓠1 𝓠2.
Proof. rewrite /FromOr /MakeMonPredAt=> <- <- <-. by unseal. Qed.
Proof. rewrite /FromOr /MakeMonPredAt=> <- <- <-. by rewrite monPred_at_or. Qed.
Global Instance into_or_monPred_at P Q1 𝓠1 Q2 𝓠2 i :
IntoOr P Q1 Q2 → MakeMonPredAt i Q1 𝓠1 → MakeMonPredAt i Q2 𝓠2 →
IntoOr (P i) 𝓠1 𝓠2.
Proof. rewrite /IntoOr /MakeMonPredAt=> -> <- <-. by unseal. Qed.
Proof. rewrite /IntoOr /MakeMonPredAt=> -> <- <-. by rewrite monPred_at_or. Qed.
Global Instance from_exist_monPred_at {A} P (Φ : A → monPred) (Ψ : A → PROP) i :
FromExist P Φ → (∀ a, MakeMonPredAt i (Φ a) (Ψ a)) → FromExist (P i) Ψ.
Proof.
rewrite /FromExist /MakeMonPredAt=><- H. setoid_rewrite <- H. by unseal.
rewrite /FromExist /MakeMonPredAt=><- H. setoid_rewrite <- H.
by rewrite monPred_at_exist.
Qed.
Global Instance into_exist_monPred_at {A} P (Φ : A → monPred) (Ψ : A → PROP) i :
IntoExist P Φ → (∀ a, MakeMonPredAt i (Φ a) (Ψ a)) → IntoExist (P i) Ψ.
Proof.
rewrite /IntoExist /MakeMonPredAt=>-> H. setoid_rewrite <- H. by unseal.
rewrite /IntoExist /MakeMonPredAt=>-> H. setoid_rewrite <- H.
by rewrite monPred_at_exist.
Qed.
Global Instance foram_forall_monPred_at_plainly i P Φ :
(∀ i, MakeMonPredAt i P (Φ i)) →
FromForall (bi_plainly P i) (λ j, bi_plainly (Φ j)).
Proof. rewrite /FromForall /MakeMonPredAt=>H. unseal. do 3 f_equiv. rewrite H //. Qed.
Proof.
rewrite /FromForall /MakeMonPredAt=>H. rewrite monPred_at_plainly.
by setoid_rewrite H.
Qed.
Global Instance into_forall_monPred_at_plainly i P Φ :
(∀ i, MakeMonPredAt i P (Φ i)) →
IntoForall (bi_plainly P i) (λ j, bi_plainly (Φ j)).
Proof. rewrite /IntoForall /MakeMonPredAt=>H. unseal. do 3 f_equiv. rewrite H //. Qed.
Proof.
rewrite /IntoForall /MakeMonPredAt=>H. rewrite monPred_at_plainly.
by setoid_rewrite H.
Qed.
Global Instance from_forall_monPred_at_all P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → FromForall (monPred_all P i) Φ.
Proof.
rewrite /FromForall /MakeMonPredAt=>H. setoid_rewrite <- H. by unseal.
rewrite /FromForall /MakeMonPredAt monPred_at_all=>H. by setoid_rewrite <- H.
Qed.
Global Instance into_forall_monPred_at_all P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → IntoForall (monPred_all P i) Φ.
Proof.
rewrite /IntoForall /MakeMonPredAt=>H. setoid_rewrite <- H. by unseal.
rewrite /IntoForall /MakeMonPredAt monPred_at_all=>H. by setoid_rewrite <- H.
Qed.
Global Instance from_exist_monPred_at_ex P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → FromExist (monPred_ex P i) Φ.
Proof.
rewrite /FromExist /MakeMonPredAt=>H. setoid_rewrite <- H. by unseal.
rewrite /FromExist /MakeMonPredAt monPred_at_ex=>H. by setoid_rewrite <- H.
Qed.
Global Instance into_exist_monPred_at_ex P (Φ : I → PROP) i :
(∀ i, MakeMonPredAt i P (Φ i)) → IntoExist (monPred_ex P i) Φ.
Proof.
rewrite /IntoExist /MakeMonPredAt=>H. setoid_rewrite <- H. by unseal.
rewrite /IntoExist /MakeMonPredAt monPred_at_ex=>H. by setoid_rewrite <- H.
Qed.
Global Instance from_forall_monPred_at {A} P (Φ : A → monPred) (Ψ : A → PROP) i :
FromForall P Φ → (∀ a, MakeMonPredAt i (Φ a) (Ψ a)) → FromForall (P i) Ψ.
Proof.
rewrite /FromForall /MakeMonPredAt=><- H. setoid_rewrite <- H. by unseal.
rewrite /FromForall /MakeMonPredAt=><- H. setoid_rewrite <- H.
by rewrite monPred_at_forall.
Qed.
Global Instance into_forall_monPred_at {A} P (Φ : A → monPred) (Ψ : A → PROP) i :
IntoForall P Φ → (∀ a, MakeMonPredAt i (Φ a) (Ψ a)) → IntoForall (P i) Ψ.
Proof.
rewrite /IntoForall /MakeMonPredAt=>-> H. setoid_rewrite <- H. by unseal.
rewrite /IntoForall /MakeMonPredAt=>-> H. setoid_rewrite <- H.
by rewrite monPred_at_forall.
Qed.
(* FIXME : there are two good ways to frame under a call to
......@@ -308,13 +316,14 @@ Global Instance frame_monPred_at p P Q 𝓠 R i j :
Proof.
rewrite /Frame /MakeMonPredAt /bi_affinely_if /bi_persistently_if
/IsBiIndexRel=> Hij <- <-.
by destruct p; rewrite Hij; unseal.
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 → Frame p 𝓡 (P i) 𝓠.
Proof.
rewrite /Frame /MakeMonPredAt /bi_affinely_if /bi_persistently_if=> <- <-.
by destruct p; unseal.
destruct p; by rewrite monPred_at_sep ?monPred_at_affinely
?monPred_at_persistently monPred_at_embed.
Qed.
Global Instance from_modal_monPred_at i P Q 𝓠 :
......@@ -337,16 +346,15 @@ Proof. by rewrite /AddModal !monPred_bupd_embed. Qed.
Global Instance elim_modal_at_bupd_goal `{BUpdFacts PROP} 𝓟 𝓟' Q Q' i :
ElimModal 𝓟 𝓟' (|==> Q i) (|==> Q' i) →
ElimModal 𝓟 𝓟' ((|==> Q) i) ((|==> Q') i).
Proof. by rewrite /ElimModal !monPred_bupd_at. Qed.
Proof. by rewrite /ElimModal !monPred_at_bupd. Qed.
Global Instance elim_modal_at_bupd_hyp `{BUpdFacts PROP} P 𝓟' 𝓠 𝓠' i:
ElimModal (|==> P i) 𝓟' 𝓠 𝓠' →
ElimModal ((|==> P) i) 𝓟' 𝓠 𝓠'.
Proof. by rewrite /ElimModal monPred_bupd_at. Qed.
Proof. by rewrite /ElimModal monPred_at_bupd. Qed.
Global Instance add_modal_at_bupd_goal `{BUpdFacts PROP} 𝓟 𝓟' Q i :
AddModal 𝓟 𝓟' (|==> Q i)%I → AddModal 𝓟 𝓟' ((|==> Q) i).
Proof. by rewrite /AddModal !monPred_bupd_at. Qed.
Proof. by rewrite /AddModal !monPred_at_bupd. Qed.
End bi.
(* When P and/or Q are evars when doing typeclass search on [IntoWand
......@@ -381,36 +389,40 @@ Implicit Types i j : I.
Global Instance is_except_0_monPred_at i P :
IsExcept0 P → IsExcept0 (P i).
Proof. rewrite /IsExcept0=>- [/(_ i)]. by unseal. Qed.
Proof. rewrite /IsExcept0=>- [/(_ i)]. by rewrite monPred_at_except_0. Qed.
Global Instance make_monPred_at_except_0 i P 𝓠 :
MakeMonPredAt i P 𝓠 → MakeMonPredAt i (◇ P)%I (◇ 𝓠)%I.
Proof. rewrite /MakeMonPredAt=><-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_except_0=><-. Qed.
Global Instance make_monPred_at_later i P 𝓠 :
MakeMonPredAt i P 𝓠 → MakeMonPredAt i (▷ P)%I (▷ 𝓠)%I.
Proof. rewrite /MakeMonPredAt=><-. by unseal. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_later=><-. Qed.
Global Instance make_monPred_at_laterN i n P 𝓠 :
MakeMonPredAt i P 𝓠 → MakeMonPredAt i (▷^n P)%I (▷^n 𝓠)%I.
Proof. rewrite /MakeMonPredAt=> <-. elim n=>//= ? <-. by unseal. Qed.
Proof. rewrite /MakeMonPredAt=> <-. elim n=>//= ? <-. by rewrite monPred_at_later. Qed.
Global Instance make_monPred_at_fupd `{FUpdFacts PROP} i E1 E2 P 𝓟 :
MakeMonPredAt i P 𝓟 → MakeMonPredAt i (|={E1,E2}=> P)%I (|={E1,E2}=> 𝓟)%I.
Proof. by rewrite /MakeMonPredAt monPred_fupd_at=> <-. Qed.
Proof. by rewrite /MakeMonPredAt monPred_at_fupd=> <-. Qed.
Global Instance into_except_0_monPred_at_fwd i P Q 𝓠 :
IntoExcept0 P Q → MakeMonPredAt i Q 𝓠 → IntoExcept0 (P i) 𝓠.
Proof. rewrite /IntoExcept0 /MakeMonPredAt=> -> <-. by unseal. Qed.
Proof. rewrite /IntoExcept0 /MakeMonPredAt=> -> <-. by rewrite monPred_at_except_0. Qed.
Global Instance into_except_0_monPred_at_bwd i P 𝓟 Q :
IntoExcept0 P Q → MakeMonPredAt i P 𝓟 → IntoExcept0 𝓟 (Q i).
Proof. rewrite /IntoExcept0 /MakeMonPredAt=> H <-. rewrite H. by unseal. Qed.
Proof. rewrite /IntoExcept0 /MakeMonPredAt=> H <-. by rewrite H monPred_at_except_0. Qed.
Global Instance into_later_monPred_at i n P Q 𝓠 :
IntoLaterN n P Q → MakeMonPredAt i Q 𝓠 → IntoLaterN n (P i) 𝓠.
Proof.
rewrite /IntoLaterN /MakeMonPredAt=> -> <-. elim n=>//= ? <-. by unseal.
rewrite /IntoLaterN /MakeMonPredAt=> -> <-. elim n=>//= ? <-.
by rewrite monPred_at_later.
Qed.
Global Instance from_later_monPred_at i n P Q 𝓠 :
FromLaterN n P Q → MakeMonPredAt i Q 𝓠 → FromLaterN n (P i) 𝓠.
Proof. rewrite /FromLaterN /MakeMonPredAt=> <- <-. elim n=>//= ? ->. by unseal. Qed.
Proof.
rewrite /FromLaterN /MakeMonPredAt=> <- <-. elim n=>//= ? ->.
by rewrite monPred_at_later.
Qed.
Global Instance elim_modal_embed_fupd_goal `{FUpdFacts PROP} E1 E2 E3 P P' 𝓠 𝓠' :
ElimModal P P' (|={E1,E3}=> ⎡𝓠⎤)%I (|={E2,E3}=> ⎡𝓠'⎤)%I →
......@@ -428,13 +440,13 @@ Proof. by rewrite /AddModal !monPred_fupd_embed. Qed.
Global Instance elim_modal_at_fupd_goal `{FUpdFacts PROP} E1 E2 E3 𝓟 𝓟' Q Q' i :
ElimModal 𝓟 𝓟' (|={E1,E3}=> Q i) (|={E2,E3}=> Q' i) →
ElimModal 𝓟 𝓟' ((|={E1,E3}=> Q) i) ((|={E2,E3}=> Q') i).
Proof. by rewrite /ElimModal !monPred_fupd_at. Qed.
Proof. by rewrite /ElimModal !monPred_at_fupd. Qed.
Global Instance elim_modal_at_fupd_hyp `{FUpdFacts PROP} E1 E2 P 𝓟' 𝓠 𝓠' i :
ElimModal (|={E1,E2}=> P i) 𝓟' 𝓠 𝓠' →
ElimModal ((|={E1,E2}=> P) i) 𝓟' 𝓠 𝓠'.
Proof. by rewrite /ElimModal monPred_fupd_at. Qed.
Proof. by rewrite /ElimModal monPred_at_fupd. Qed.
Global Instance add_modal_at_fupd_goal `{FUpdFacts PROP} E1 E2 𝓟 𝓟' Q i :
AddModal 𝓟 𝓟' (|={E1,E2}=> Q i) → AddModal 𝓟 𝓟' ((|={E1,E2}=> Q) i).
Proof. by rewrite /AddModal !monPred_fupd_at. Qed.
Proof. by rewrite /AddModal !monPred_at_fupd. Qed.
End sbi.
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