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.

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.
f763f227 · Jacques-Henri Jourdan · a6483223 · f763f227 · f763f227 · f763f227
Commit f763f227 authored 7 years ago by Jacques-Henri Jourdan
--- a/theories/bi/embedding.v
+++ b/theories/bi/embedding.v
 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.

--- a/theories/bi/monpred.v
+++ b/theories/bi/monpred.v
@@ -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_pure (φ : Prop) : monPred := tc_opaque ⎡⌜φ⌝⎤%I.
-Definition monPred_emp : monPred := ⎡emp⎤%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.
+  apply bi.equiv_spec. split.
-  - intros n P Q HPQ. split=>/= i. by repeat f_equiv.
+  - apply bi.forall_intro=>?. apply (bi.f_equiv (flip monPred_at _)).
-  - intros P. split=>/= i. apply bi.forall_intro=>?. rewrite bi.pure_impl_forall.
+  - by rewrite -{2}(sig_monPred_sig P) -{2}(sig_monPred_sig Q)
-    apply bi.forall_intro=><-. apply bupd_intro.
+               -bi.f_equiv -bi.sig_equivI !bi.ofe_fun_equivI.
-  - 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.
 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.

--- a/theories/proofmode/monpred.v
+++ b/theories/proofmode/monpred.v
 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.
+Global Instance make_monPred_at_embed i 𝓟 : MakeMonPredAt i ⎡𝓟⎤ 𝓟.
-Proof. rewrite /MakeMonPredAt. by unseal. Qed.
+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.
+  setoid_rewrite <-EQ. split.
-  - move=>[/= /bi.forall_intro //].
+  - move=>[H]. apply bi.forall_intro=>i. rewrite -H. by rewrite monPred_at_emp.
-  - move=>HP. split=>i. rewrite /= HP bi.forall_elim //.
+  - 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 :
+Global Instance into_pure_monPred_at P φ i : IntoPure P φ → IntoPure (P i) φ.
-  IntoPure P φ → IntoPure (P i) φ.
+Proof. rewrite /IntoPure=>->. by rewrite monPred_at_pure. Qed.
-Proof. rewrite /IntoPure=>->. by unseal. Qed.
+Global Instance from_pure_monPred_at P φ i : FromPure P φ → FromPure (P i) φ.
-Global Instance from_pure_monPred_at P φ i :
+Proof. rewrite /FromPure=><-. by rewrite monPred_at_pure. Qed.
-  FromPure P φ → FromPure (P i) φ.
+Global Instance into_pure_monPred_in i j : @IntoPure PROP (monPred_in i j) (i ⊑ j).
-Proof. rewrite /FromPure=><-. by unseal. Qed.
+Proof. by rewrite /IntoPure monPred_at_in. Qed.
-Global Instance into_pure_monPred_in i j :
+Global Instance from_pure_monPred_in i j : @FromPure PROP (monPred_in i j) (i ⊑ j).
-  @IntoPure PROP (monPred_in i j) (i ⊑ j).
+Proof. by rewrite /FromPure monPred_at_in. Qed.
-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_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.
+  rewrite /IntoPersistent /MakeMonPredAt  =>-[/(_ i) ?] <-.
-  unseal=>-[/(_ i) ?] <-. by destruct p.
+  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=><-.
+  rewrite /FromAlways /MakeMonPredAt /bi_persistently_if /bi_affinely_if=><- /=.
-  by destruct a, pe=><-; try unseal.
+  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.