Rename `always` → `persistently` (the persistent modality).

theories/base_logic/base_logic.v

@@ -11,7 +11,7 @@ End uPred.
 Hint Resolve pure_intro.
 Hint Resolve or_elim or_intro_l' or_intro_r' : I.
 Hint Resolve and_intro and_elim_l' and_elim_r' : I.
-Hint Resolve always_mono : I.
+Hint Resolve persistently_mono : I.
 Hint Resolve sep_elim_l' sep_elim_r' sep_mono : I.
 Hint Immediate True_intro False_elim : I.
 Hint Immediate iff_refl internal_eq_refl' : I.

theories/base_logic/big_op.v

@@ -117,11 +117,11 @@ Section list.
     ▷^n ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, ▷^n Φ k x).
   Proof. apply (big_opL_commute _). Qed.
 
-  Lemma big_sepL_always Φ l :
+  Lemma big_sepL_persistently Φ l :
     (□ [∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, □ Φ k x).
   Proof. apply (big_opL_commute _). Qed.
 
-  Lemma big_sepL_always_if p Φ l :
+  Lemma big_sepL_persistently_if p Φ l :
     □?p ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, □?p Φ k x).
   Proof. apply (big_opL_commute _). Qed.
 
@@ -134,7 +134,7 @@ Section list.
       apply impl_intro_l, pure_elim_l=> ?; by apply big_sepL_lookup. }
     revert Φ HΦ. induction l as [|x l IH]=> Φ HΦ.
     { rewrite big_sepL_nil; auto with I. }
-    rewrite big_sepL_cons. rewrite -always_and_sep_l; apply and_intro.
+    rewrite big_sepL_cons. rewrite -persistently_and_sep_l; apply and_intro.
     - by rewrite (forall_elim 0) (forall_elim x) pure_True // True_impl.
     - rewrite -IH. apply forall_intro=> k; by rewrite (forall_elim (S k)).
   Qed.
@@ -143,10 +143,10 @@ Section list.
     □ (∀ k x, ⌜l !! k = Some x⌝ → Φ k x → Ψ k x) ∧ ([∗ list] k↦x ∈ l, Φ k x)
     ⊢ [∗ list] k↦x ∈ l, Ψ k x.
   Proof.
-    rewrite always_and_sep_l. do 2 setoid_rewrite always_forall.
-    setoid_rewrite always_impl; setoid_rewrite always_pure.
+    rewrite persistently_and_sep_l. do 2 setoid_rewrite persistently_forall.
+    setoid_rewrite persistently_impl; setoid_rewrite persistently_pure.
     rewrite -big_sepL_forall -big_sepL_sepL. apply big_sepL_mono; auto=> k x ?.
-    by rewrite -always_wand_impl always_elim wand_elim_l.
+    by rewrite -persistently_wand_impl persistently_elim wand_elim_l.
   Qed.
 
   Global Instance big_sepL_nil_persistent Φ :
@@ -307,11 +307,11 @@ Section gmap.
     ▷^n ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ ([∗ map] k↦x ∈ m, ▷^n Φ k x).
   Proof. apply (big_opM_commute _). Qed.
 
-  Lemma big_sepM_always Φ m :
+  Lemma big_sepM_persistently Φ m :
     (□ [∗ map] k↦x ∈ m, Φ k x) ⊣⊢ ([∗ map] k↦x ∈ m, □ Φ k x).
   Proof. apply (big_opM_commute _). Qed.
 
-  Lemma big_sepM_always_if p Φ m :
+  Lemma big_sepM_persistently_if p Φ m :
     □?p ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ ([∗ map] k↦x ∈ m, □?p Φ k x).
   Proof. apply (big_opM_commute _). Qed.
 
@@ -323,7 +323,7 @@ Section gmap.
     { apply forall_intro=> k; apply forall_intro=> x.
       apply impl_intro_l, pure_elim_l=> ?; by apply big_sepM_lookup. }
     induction m as [|i x m ? IH] using map_ind; [rewrite ?big_sepM_empty; auto|].
-    rewrite big_sepM_insert // -always_and_sep_l. apply and_intro.
+    rewrite big_sepM_insert // -persistently_and_sep_l. apply and_intro.
     - rewrite (forall_elim i) (forall_elim x) lookup_insert.
       by rewrite pure_True // True_impl.
     - rewrite -IH. apply forall_mono=> k; apply forall_mono=> y.
@@ -336,10 +336,10 @@ Section gmap.
     □ (∀ k x, ⌜m !! k = Some x⌝ → Φ k x → Ψ k x) ∧ ([∗ map] k↦x ∈ m, Φ k x)
     ⊢ [∗ map] k↦x ∈ m, Ψ k x.
   Proof.
-    rewrite always_and_sep_l. do 2 setoid_rewrite always_forall.
-    setoid_rewrite always_impl; setoid_rewrite always_pure.
+    rewrite persistently_and_sep_l. do 2 setoid_rewrite persistently_forall.
+    setoid_rewrite persistently_impl; setoid_rewrite persistently_pure.
     rewrite -big_sepM_forall -big_sepM_sepM. apply big_sepM_mono; auto=> k x ?.
-    by rewrite -always_wand_impl always_elim wand_elim_l.
+    by rewrite -persistently_wand_impl persistently_elim wand_elim_l.
   Qed.
 
   Global Instance big_sepM_empty_persistent Φ :
@@ -460,10 +460,10 @@ Section gset.
     ▷^n ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ▷^n Φ y).
   Proof. apply (big_opS_commute _). Qed.
 
-  Lemma big_sepS_always Φ X : □ ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, □ Φ y).
+  Lemma big_sepS_persistently Φ X : □ ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, □ Φ y).
   Proof. apply (big_opS_commute _). Qed.
 
-  Lemma big_sepS_always_if q Φ X :
+  Lemma big_sepS_persistently_if q Φ X :
     □?q ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, □?q Φ y).
   Proof. apply (big_opS_commute _). Qed.
 
@@ -475,7 +475,7 @@ Section gset.
       apply impl_intro_l, pure_elim_l=> ?; by apply big_sepS_elem_of. }
     induction X as [|x X ? IH] using collection_ind_L.
     { rewrite big_sepS_empty; auto. }
-    rewrite big_sepS_insert // -always_and_sep_l. apply and_intro.
+    rewrite big_sepS_insert // -persistently_and_sep_l. apply and_intro.
     - by rewrite (forall_elim x) pure_True ?True_impl; last set_solver.
     - rewrite -IH. apply forall_mono=> y. apply impl_intro_l, pure_elim_l=> ?.
       by rewrite pure_True ?True_impl; last set_solver.
@@ -484,10 +484,10 @@ Section gset.
   Lemma big_sepS_impl Φ Ψ X :
     □ (∀ x, ⌜x ∈ X⌝ → Φ x → Ψ x) ∧ ([∗ set] x ∈ X, Φ x) ⊢ [∗ set] x ∈ X, Ψ x.
   Proof.
-    rewrite always_and_sep_l always_forall.
-    setoid_rewrite always_impl; setoid_rewrite always_pure.
+    rewrite persistently_and_sep_l persistently_forall.
+    setoid_rewrite persistently_impl; setoid_rewrite persistently_pure.
     rewrite -big_sepS_forall -big_sepS_sepS. apply big_sepS_mono; auto=> x ?.
-    by rewrite -always_wand_impl always_elim wand_elim_l.
+    by rewrite -persistently_wand_impl persistently_elim wand_elim_l.
   Qed.
 
   Global Instance big_sepS_empty_persistent Φ : Persistent ([∗ set] x ∈ ∅, Φ x).
@@ -571,10 +571,10 @@ Section gmultiset.
     ▷^n ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, ▷^n Φ y).
   Proof. apply (big_opMS_commute _). Qed.
 
-  Lemma big_sepMS_always Φ X : □ ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, □ Φ y).
+  Lemma big_sepMS_persistently Φ X : □ ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, □ Φ y).
   Proof. apply (big_opMS_commute _). Qed.
 
-  Lemma big_sepMS_always_if q Φ X :
+  Lemma big_sepMS_persistently_if q Φ X :
     □?q ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, □?q Φ y).
   Proof. apply (big_opMS_commute _). Qed.
 

theories/base_logic/lib/fractional.v

@@ -51,7 +51,7 @@ Section fractional.
   (** Fractional and logical connectives *)
   Global Instance persistent_fractional P :
     Persistent P → Fractional (λ _, P).
-  Proof. intros HP q q'. by apply uPred.always_sep_dup. Qed.
+  Proof. intros HP q q'. by apply uPred.persistently_sep_dup. Qed.
 
   Global Instance fractional_sep Φ Ψ :
     Fractional Φ → Fractional Ψ → Fractional (λ q, Φ q ∗ Ψ q)%I.
@@ -134,7 +134,7 @@ Section fractional.
     AsFractional P Φ (q1 + q2) → AsFractional P1 Φ q1 → AsFractional P2 Φ q2 →
     IntoAnd p P P1 P2.
   Proof.
-    (* TODO: We need a better way to handle this boolean here; always
+    (* TODO: We need a better way to handle this boolean here; persistently
        applying mk_into_and_sep (which only works after introducing all
        assumptions) is rather annoying.
        Ideally, it'd not even be possible to make the mistake that
@@ -148,7 +148,7 @@ Section fractional.
   Proof. intros. apply mk_into_and_sep. rewrite [P]fractional_half //. Qed.
 
   (* The instance [frame_fractional] can be tried at all the nodes of
-     the proof search. The proof search then fails almost always on
+     the proof search. The proof search then fails almost persistently on
      [AsFractional R Φ r], but the slowdown is still noticeable.  For
      that reason, we factorize the three instances that could have been
      defined for that purpose into one. *)
@@ -179,6 +179,6 @@ Section fractional.
     - rewrite fractional=><-<-. by rewrite assoc.
     - rewrite fractional=><-<-=>_.
       by rewrite (comm _ Q (Φ q0)) !assoc (comm _ (Φ _)).
-    - move=>-[-> _]->. by rewrite uPred.always_if_elim -fractional Qp_div_2.
+    - move=>-[-> _]->. by rewrite uPred.persistently_if_elim -fractional Qp_div_2.
   Qed.
 End fractional.

theories/base_logic/lib/iprop.v

@@ -83,7 +83,7 @@ Class subG (Σ1 Σ2 : gFunctors) := in_subG i : { j | Σ1 i = Σ2 j }.
 
 (** Avoid trigger happy type class search: this line ensures that type class
 search is only triggered if the arguments of [subG] do not contain evars. Since
-instance search for [subG] is restrained, instances should always have [subG] as
+instance search for [subG] is restrained, instances should persistently have [subG] as
 their first parameter to avoid loops. For example, the instances [subG_authΣ]
 and [auth_discrete] otherwise create a cycle that pops up arbitrarily. *)
 Hint Mode subG + + : typeclass_instances.

theories/base_logic/lib/own.v

@@ -102,7 +102,7 @@ Proof. apply wand_intro_r. by rewrite -own_op own_valid. Qed.
 Lemma own_valid_3 γ a1 a2 a3 : own γ a1 -∗ own γ a2 -∗ own γ a3 -∗ ✓ (a1 ⋅ a2 ⋅ a3).
 Proof. do 2 apply wand_intro_r. by rewrite -!own_op own_valid. Qed.
 Lemma own_valid_r γ a : own γ a ⊢ own γ a ∗ ✓ a.
-Proof. apply: uPred.always_entails_r. apply own_valid. Qed.
+Proof. apply: uPred.persistently_entails_r. apply own_valid. Qed.
 Lemma own_valid_l γ a : own γ a ⊢ ✓ a ∗ own γ a.
 Proof. by rewrite comm -own_valid_r. Qed.
 

theories/base_logic/lib/viewshifts.v

@@ -81,5 +81,5 @@ Lemma vs_alloc N P : ▷ P ={↑N}=> inv N P.
 Proof. iIntros "!# HP". by iApply inv_alloc. Qed.
 
 Lemma wand_fupd_alt E1 E2 P Q : (P ={E1,E2}=∗ Q) ⊣⊢ ∃ R, R ∗ (P ∗ R ={E1,E2}=> Q).
-Proof. rewrite uPred.wand_alt. by setoid_rewrite <-uPred.always_wand_impl. Qed.
+Proof. rewrite uPred.wand_alt. by setoid_rewrite <-uPred.persistently_wand_impl. Qed.
 End vs.

theories/base_logic/primitive.v

@@ -97,16 +97,16 @@ Definition uPred_wand {M} := unseal uPred_wand_aux M.
 Definition uPred_wand_eq :
   @uPred_wand = @uPred_wand_def := seal_eq uPred_wand_aux.
 
-Program Definition uPred_always_def {M} (P : uPred M) : uPred M :=
+Program Definition uPred_persistently_def {M} (P : uPred M) : uPred M :=
   {| uPred_holds n x := P n (core x) |}.
 Next Obligation.
   intros M; naive_solver eauto using uPred_mono, @cmra_core_monoN.
 Qed.
 Next Obligation. naive_solver eauto using uPred_closed, @cmra_core_validN. Qed.
-Definition uPred_always_aux : seal (@uPred_always_def). by eexists. Qed.
-Definition uPred_always {M} := unseal uPred_always_aux M.
-Definition uPred_always_eq :
-  @uPred_always = @uPred_always_def := seal_eq uPred_always_aux.
+Definition uPred_persistently_aux : seal (@uPred_persistently_def). by eexists. Qed.
+Definition uPred_persistently {M} := unseal uPred_persistently_aux M.
+Definition uPred_persistently_eq :
+  @uPred_persistently = @uPred_persistently_def := seal_eq uPred_persistently_aux.
 
 Program Definition uPred_later_def {M} (P : uPred M) : uPred M :=
   {| uPred_holds n x := match n return _ with 0 => True | S n' => P n' x end |}.
@@ -176,7 +176,7 @@ Notation "∀ x .. y , P" :=
 Notation "∃ x .. y , P" :=
   (uPred_exist (λ x, .. (uPred_exist (λ y, P)) ..)%I)
   (at level 200, x binder, y binder, right associativity) : uPred_scope.
-Notation "□ P" := (uPred_always P)
+Notation "□ P" := (uPred_persistently P)
   (at level 20, right associativity) : uPred_scope.
 Notation "▷ P" := (uPred_later P)
   (at level 20, right associativity) : uPred_scope.
@@ -198,7 +198,7 @@ Notation "P -∗ Q" := (P ⊢ Q)
 Module uPred.
 Definition unseal_eqs :=
   (uPred_pure_eq, uPred_and_eq, uPred_or_eq, uPred_impl_eq, uPred_forall_eq,
-  uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_always_eq,
+  uPred_exist_eq, uPred_internal_eq_eq, uPred_sep_eq, uPred_wand_eq, uPred_persistently_eq,
   uPred_later_eq, uPred_ownM_eq, uPred_cmra_valid_eq, uPred_bupd_eq).
 Ltac unseal := rewrite !unseal_eqs /=.
 
@@ -295,13 +295,13 @@ Proof.
 Qed.
 Global Instance later_proper' :
   Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_later M) := ne_proper _.
-Global Instance always_ne : NonExpansive (@uPred_always M).
+Global Instance persistently_ne : NonExpansive (@uPred_persistently M).
 Proof.
   intros n P1 P2 HP.
   unseal; split=> n' x; split; apply HP; eauto using @cmra_core_validN.
 Qed.
-Global Instance always_proper :
-  Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_always M) := ne_proper _.
+Global Instance persistently_proper :
+  Proper ((⊣⊢) ==> (⊣⊢)) (@uPred_persistently M) := ne_proper _.
 Global Instance ownM_ne : NonExpansive (@uPred_ownM M).
 Proof.
   intros n a b Ha.
@@ -422,22 +422,22 @@ Proof.
 Qed.
 
 (* Always *)
-Lemma always_mono P Q : (P ⊢ Q) → □ P ⊢ □ Q.
+Lemma persistently_mono P Q : (P ⊢ Q) → □ P ⊢ □ Q.
 Proof. intros HP; unseal; split=> n x ? /=. by apply HP, cmra_core_validN. Qed.
-Lemma always_elim P : □ P ⊢ P.
+Lemma persistently_elim P : □ P ⊢ P.
 Proof.
   unseal; split=> n x ? /=.
   eauto using uPred_mono, @cmra_included_core, cmra_included_includedN.
 Qed.
-Lemma always_idemp_2 P : □ P ⊢ □ □ P.
+Lemma persistently_idemp_2 P : □ P ⊢ □ □ P.
 Proof. unseal; split=> n x ?? /=. by rewrite cmra_core_idemp. Qed.
 
-Lemma always_forall_2 {A} (Ψ : A → uPred M) : (∀ a, □ Ψ a) ⊢ (□ ∀ a, Ψ a).
+Lemma persistently_forall_2 {A} (Ψ : A → uPred M) : (∀ a, □ Ψ a) ⊢ (□ ∀ a, Ψ a).
 Proof. by unseal. Qed.
-Lemma always_exist_1 {A} (Ψ : A → uPred M) : (□ ∃ a, Ψ a) ⊢ (∃ a, □ Ψ a).
+Lemma persistently_exist_1 {A} (Ψ : A → uPred M) : (□ ∃ a, Ψ a) ⊢ (∃ a, □ Ψ a).
 Proof. by unseal. Qed.
 
-Lemma always_and_sep_l_1 P Q : □ P ∧ Q ⊢ □ P ∗ Q.
+Lemma persistently_and_sep_l_1 P Q : □ P ∧ Q ⊢ □ P ∗ Q.
 Proof.
   unseal; split=> n x ? [??]; exists (core x), x; simpl in *.
   by rewrite cmra_core_l cmra_core_idemp.
@@ -475,7 +475,7 @@ Proof.
   intros [|n'] x' ????; [|done].
   eauto using uPred_closed, uPred_mono, cmra_included_includedN.
 Qed.
-Lemma always_later P : □ ▷ P ⊣⊢ ▷ □ P.
+Lemma persistently_later P : □ ▷ P ⊣⊢ ▷ □ P.
 Proof. by unseal. Qed.
 
 (* Own *)
@@ -489,7 +489,7 @@ Proof.
     by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1)
       -(assoc op _ a2) (comm op z1) -Hy1 -Hy2.
 Qed.
-Lemma always_ownM_core (a : M) : uPred_ownM a ⊢ □ uPred_ownM (core a).
+Lemma persistently_ownM_core (a : M) : uPred_ownM a ⊢ □ uPred_ownM (core a).
 Proof.
   split=> n x /=; unseal; intros Hx. simpl. by apply cmra_core_monoN.
 Qed.
@@ -512,7 +512,7 @@ Lemma cmra_valid_intro {A : cmraT} (a : A) : ✓ a → uPred_valid (M:=M) (✓ a
 Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed.
 Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ ✓{0} a → ✓ a ⊢ False.
 Proof. unseal=> Ha; split=> n x ??; apply Ha, cmra_validN_le with n; auto. Qed.
-Lemma always_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ □ ✓ a.
+Lemma persistently_cmra_valid_1 {A : cmraT} (a : A) : ✓ a ⊢ □ ✓ a.
 Proof. by unseal. Qed.
 Lemma cmra_valid_weaken {A : cmraT} (a b : A) : ✓ (a ⋅ b) ⊢ ✓ a.
 Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed.

theories/program_logic/hoare.v

@@ -37,7 +37,7 @@ Global Instance ht_proper E :
 Proof. solve_proper. Qed.
 Lemma ht_mono E P P' Φ Φ' e :
   (P ⊢ P') → (∀ v, Φ' v ⊢ Φ v) → {{ P' }} e @ E {{ Φ' }} ⊢ {{ P }} e @ E {{ Φ }}.
-Proof. by intros; apply always_mono, wand_mono, wp_mono. Qed.
+Proof. by intros; apply persistently_mono, wand_mono, wp_mono. Qed.
 Global Instance ht_mono' E :
   Proper (flip (⊢) ==> eq ==> pointwise_relation _ (⊢) ==> (⊢)) (ht E).
 Proof. solve_proper. Qed.

theories/program_logic/weakestpre.v

@@ -283,7 +283,7 @@ Section proofmode_classes.
     ElimModal (|={E}=> P) P (WP e @ E {{ Φ }}) (WP e @ E {{ Φ }}).
   Proof. by rewrite /ElimModal fupd_frame_r wand_elim_r fupd_wp. Qed.
 
-  (* lower precedence, if possible, it should always pick elim_upd_fupd_wp *)
+  (* lower precedence, if possible, it should persistently pick elim_upd_fupd_wp *)
   Global Instance elim_modal_fupd_wp_atomic E1 E2 e P Φ :
     atomic e →
     ElimModal (|={E1,E2}=> P) P

theories/proofmode/class_instances.v

@@ -11,17 +11,17 @@ Implicit Types P Q R : uPred M.
 
 (* FromAssumption *)
 Global Instance from_assumption_exact p P : FromAssumption p P P | 0.
-Proof. destruct p; by rewrite /FromAssumption /= ?always_elim. Qed.
+Proof. destruct p; by rewrite /FromAssumption /= ?persistently_elim. Qed.
 Global Instance from_assumption_False p P : FromAssumption p False P | 1.
-Proof. destruct p; rewrite /FromAssumption /= ?always_pure; apply False_elim. Qed.
+Proof. destruct p; rewrite /FromAssumption /= ?persistently_pure; apply False_elim. Qed.
 
-Global Instance from_assumption_always_r P Q :
+Global Instance from_assumption_persistently_r P Q :
   FromAssumption true P Q → FromAssumption true P (□ Q).
-Proof. rewrite /FromAssumption=><-. by rewrite always_always. Qed.
+Proof. rewrite /FromAssumption=><-. by rewrite persistently_persistently. Qed.
 
-Global Instance from_assumption_always_l p P Q :
+Global Instance from_assumption_persistently_l p P Q :
   FromAssumption p P Q → FromAssumption p (□ P) Q.
-Proof. rewrite /FromAssumption=><-. by rewrite always_elim. Qed.
+Proof. rewrite /FromAssumption=><-. by rewrite persistently_elim. Qed.
 Global Instance from_assumption_later p P Q :
   FromAssumption p P Q → FromAssumption p P (▷ Q)%I.
 Proof. rewrite /FromAssumption=>->. apply later_intro. Qed.
@@ -48,8 +48,8 @@ Global Instance into_pure_cmra_valid `{CmraDiscrete A} (a : A) :
   @IntoPure M (✓ a) (✓ a).
 Proof. by rewrite /IntoPure discrete_valid. Qed.
 
-Global Instance into_pure_always P φ : IntoPure P φ → IntoPure (□ P) φ.
-Proof. rewrite /IntoPure=> ->. by rewrite always_pure. Qed.
+Global Instance into_pure_persistently P φ : IntoPure P φ → IntoPure (□ P) φ.
+Proof. rewrite /IntoPure=> ->. by rewrite persistently_pure. Qed.
 
 Global Instance into_pure_pure_and (φ1 φ2 : Prop) P1 P2 :
   IntoPure P1 φ1 → IntoPure P2 φ2 → IntoPure (P1 ∧ P2) (φ1 ∧ φ2).
@@ -66,7 +66,7 @@ Proof. rewrite /FromPure /IntoPure pure_impl. by intros -> ->. Qed.
 Global Instance into_pure_pure_wand (φ1 φ2 : Prop) P1 P2 :
   FromPure P1 φ1 → IntoPure P2 φ2 → IntoPure (P1 -∗ P2) (φ1 → φ2).
 Proof.
-  rewrite /FromPure /IntoPure pure_impl always_impl_wand. by intros -> ->.
+  rewrite /FromPure /IntoPure pure_impl persistently_impl_wand. by intros -> ->.
 Qed.
 
 Global Instance into_pure_exist {X : Type} (Φ : X → uPred M) (φ : X → Prop) :
@@ -97,8 +97,8 @@ Proof.
   rewrite -cmra_valid_intro //. auto with I.
 Qed.
 
-Global Instance from_pure_always P φ : FromPure P φ → FromPure (□ P) φ.
-Proof. rewrite /FromPure=> <-. by rewrite always_pure. Qed.
+Global Instance from_pure_persistently P φ : FromPure P φ → FromPure (□ P) φ.
+Proof. rewrite /FromPure=> <-. by rewrite persistently_pure. Qed.
 Global Instance from_pure_later P φ : FromPure P φ → FromPure (▷ P) φ.
 Proof. rewrite /FromPure=> ->. apply later_intro. Qed.
 Global Instance from_pure_laterN n P φ : FromPure P φ → FromPure (▷^n P) φ.
@@ -113,7 +113,7 @@ Global Instance from_pure_pure_and (φ1 φ2 : Prop) P1 P2 :
 Proof. rewrite /FromPure pure_and. by intros -> ->. Qed.
 Global Instance from_pure_pure_sep (φ1 φ2 : Prop) P1 P2 :
   FromPure P1 φ1 → FromPure P2 φ2 → FromPure (P1 ∗ P2) (φ1 ∧ φ2).
-Proof. rewrite /FromPure pure_and always_and_sep_l. by intros -> ->. Qed.
+Proof. rewrite /FromPure pure_and persistently_and_sep_l. by intros -> ->. Qed.
 Global Instance from_pure_pure_or (φ1 φ2 : Prop) P1 P2 :
   FromPure P1 φ1 → FromPure P2 φ2 → FromPure (P1 ∨ P2) (φ1 ∨ φ2).
 Proof. rewrite /FromPure pure_or. by intros -> ->. Qed.
@@ -123,7 +123,7 @@ Proof. rewrite /FromPure /IntoPure pure_impl. by intros -> ->. Qed.
 Global Instance from_pure_pure_wand (φ1 φ2 : Prop) P1 P2 :
   IntoPure P1 φ1 → FromPure P2 φ2 → FromPure (P1 -∗ P2) (φ1 → φ2).
 Proof.
-  rewrite /FromPure /IntoPure pure_impl always_impl_wand. by intros -> ->.
+  rewrite /FromPure /IntoPure pure_impl persistently_impl_wand. by intros -> ->.
 Qed.
 
 Global Instance from_pure_exist {X : Type} (Φ : X → uPred M) (φ : X → Prop) :
@@ -140,10 +140,10 @@ Proof.
 Qed.
 
 (* IntoPersistent *)
-Global Instance into_persistent_always_trans p P Q :
+Global Instance into_persistent_persistently_trans p P Q :
   IntoPersistent true P Q → IntoPersistent p (□ P) Q | 0.
-Proof. rewrite /IntoPersistent /==> ->. by rewrite always_if_always. Qed.
-Global Instance into_persistent_always P : IntoPersistent true P P | 1.
+Proof. rewrite /IntoPersistent /==> ->. by rewrite persistently_if_persistently. Qed.
+Global Instance into_persistent_persistently P : IntoPersistent true P P | 1.
 Proof. done. Qed.
 Global Instance into_persistent_persistent P :
   Persistent P → IntoPersistent false P P | 100.
@@ -258,9 +258,9 @@ Global Instance from_later_sep n P1 P2 Q1 Q2 :
   FromLaterN n P1 Q1 → FromLaterN n P2 Q2 → FromLaterN n (P1 ∗ P2) (Q1 ∗ Q2).
 Proof. intros ??; red. by rewrite laterN_sep; apply sep_mono. Qed.
 
-Global Instance from_later_always n P Q :
+Global Instance from_later_persistently n P Q :
   FromLaterN n P Q → FromLaterN n (□ P) (□ Q).
-Proof. by rewrite /FromLaterN -always_laterN=> ->. Qed.
+Proof. by rewrite /FromLaterN -persistently_laterN=> ->. Qed.
 
 Global Instance from_later_forall {A} n (Φ Ψ : A → uPred M) :
   (∀ x, FromLaterN n (Φ x) (Ψ x)) → FromLaterN n (∀ x, Φ x) (∀ x, Ψ x).
@@ -278,19 +278,19 @@ Global Instance wand_weaken_later p P Q P' Q' :
   WandWeaken p P Q P' Q' → WandWeaken' p P Q (▷ P') (▷ Q').
 Proof.
   rewrite /WandWeaken' /WandWeaken=> ->.
-  by rewrite always_if_later -later_wand -later_intro.
+  by rewrite persistently_if_later -later_wand -later_intro.
 Qed.
 Global Instance wand_weaken_laterN p n P Q P' Q' :
   WandWeaken p P Q P' Q' → WandWeaken' p P Q (▷^n P') (▷^n Q').
 Proof.
   rewrite /WandWeaken' /WandWeaken=> ->.
-  by rewrite always_if_laterN -laterN_wand -laterN_intro.
+  by rewrite persistently_if_laterN -laterN_wand -laterN_intro.
 Qed.
 Global Instance bupd_weaken_laterN p P Q P' Q' :
   WandWeaken false P Q P' Q' → WandWeaken' p P Q (|==> P') (|==> Q').
 Proof.
   rewrite /WandWeaken' /WandWeaken=> ->.
-  apply wand_intro_l. by rewrite always_if_elim bupd_wand_r.
+  apply wand_intro_l. by rewrite persistently_if_elim bupd_wand_r.
 Qed.
 
 Global Instance into_wand_wand p P P' Q Q' :
@@ -310,16 +310,16 @@ Proof. rewrite /WandWeaken /IntoWand=> <-. apply and_elim_r', impl_wand. Qed.
 Global Instance into_wand_forall {A} p (Φ : A → uPred M) P Q x :
   IntoWand p (Φ x) P Q → IntoWand p (∀ x, Φ x) P Q.
 Proof. rewrite /IntoWand=> <-. apply forall_elim. Qed.
-Global Instance into_wand_always p R P Q :
+Global Instance into_wand_persistently p R P Q :
   IntoWand p R P Q → IntoWand p (□ R) P Q.
-Proof. rewrite /IntoWand=> ->. apply always_elim. Qed.
+Proof. rewrite /IntoWand=> ->. apply persistently_elim. Qed.
 
 Global Instance into_wand_later p R P Q :
   IntoWand p R P Q → IntoWand p (▷ R) (▷ P) (▷ Q).
-Proof. rewrite /IntoWand=> ->. by rewrite always_if_later -later_wand. Qed.
+Proof. rewrite /IntoWand=> ->. by rewrite persistently_if_later -later_wand. Qed.
 Global Instance into_wand_laterN p n R P Q :
   IntoWand p R P Q → IntoWand p (▷^n R) (▷^n P) (▷^n Q).
-Proof. rewrite /IntoWand=> ->. by rewrite always_if_laterN -laterN_wand. Qed.
+Proof. rewrite /IntoWand=> ->. by rewrite persistently_if_laterN -laterN_wand. Qed.
 
 Global Instance into_wand_bupd R P Q :
   IntoWand false R P Q → IntoWand false (|==> R) (|==> P) (|==> Q).
@@ -340,18 +340,18 @@ Global Instance from_and_sep P1 P2 : FromAnd false (P1 ∗ P2) P1 P2 | 100.
 Proof. done. Qed.
 Global Instance from_and_sep_persistent_l P1 P2 :
   Persistent P1 → FromAnd true (P1 ∗ P2) P1 P2 | 9.
-Proof. intros. by rewrite /FromAnd always_and_sep_l. Qed.
+Proof. intros. by rewrite /FromAnd persistently_and_sep_l. Qed.
 Global Instance from_and_sep_persistent_r P1 P2 :
   Persistent P2 → FromAnd true (P1 ∗ P2) P1 P2 | 10.
-Proof. intros. by rewrite /FromAnd always_and_sep_r. Qed.
+Proof. intros. by rewrite /FromAnd persistently_and_sep_r. Qed.
 
 Global Instance from_and_pure p φ ψ : @FromAnd M p ⌜φ ∧ ψ⌝ ⌜φ⌝ ⌜ψ⌝.
 Proof. apply mk_from_and_and. by rewrite pure_and. Qed.
-Global Instance from_and_always p P Q1 Q2 :
+Global Instance from_and_persistently p P Q1 Q2 :
   FromAnd false P Q1 Q2 → FromAnd p (□ P) (□ Q1) (□ Q2).
 Proof.
   intros. apply mk_from_and_and.
-  by rewrite always_and_sep_l' -always_sep -(from_and _ P).
+  by rewrite persistently_and_sep_l' -persistently_sep -(from_and _ P).
 Qed.
 Global Instance from_and_later p P Q1 Q2 :
   FromAnd p P Q1 Q2 → FromAnd p (▷ P) (▷ Q1) (▷ Q2).
@@ -387,7 +387,7 @@ Proof. by rewrite /FromAnd big_sepL_cons. Qed.
 Global Instance from_and_big_sepL_cons_persistent {A} (Φ : nat → A → uPred M) x l :
   Persistent (Φ 0 x) →
   FromAnd true ([∗ list] k ↦ y ∈ x :: l, Φ k y) (Φ 0 x) ([∗ list] k ↦ y ∈ l, Φ (S k) y).
-Proof. intros. by rewrite /FromAnd big_opL_cons always_and_sep_l. Qed.
+Proof. intros. by rewrite /FromAnd big_opL_cons persistently_and_sep_l. Qed.
 
 Global Instance from_and_big_sepL_app {A} (Φ : nat → A → uPred M) l1 l2 :
   FromAnd false ([∗ list] k ↦ y ∈ l1 ++ l2, Φ k y)
@@ -397,7 +397,7 @@ Global Instance from_sep_big_sepL_app_persistent {A} (Φ : nat → A → uPred M
   (∀ k y, Persistent (Φ k y)) →
   FromAnd true ([∗ list] k ↦ y ∈ l1 ++ l2, Φ k y)
     ([∗ list] k ↦ y ∈ l1, Φ k y) ([∗ list] k ↦ y ∈ l2, Φ (length l1 + k) y).
-Proof. intros. by rewrite /FromAnd big_opL_app always_and_sep_l. Qed.
+Proof. intros. by rewrite /FromAnd big_opL_app persistently_and_sep_l. Qed.
 
 (* FromOp *)
 (* TODO: Worst case there could be a lot of backtracking on these instances,
@@ -431,17 +431,17 @@ Global Instance into_and_and P Q : IntoAnd true (P ∧ Q) P Q.
 Proof. done. Qed.
 Global Instance into_and_and_persistent_l P Q :
   Persistent P → IntoAnd false (P ∧ Q) P Q.
-Proof. intros; by rewrite /IntoAnd /= always_and_sep_l. Qed.
+Proof. intros; by rewrite /IntoAnd /= persistently_and_sep_l. Qed.
 Global Instance into_and_and_persistent_r P Q :
   Persistent Q → IntoAnd false (P ∧ Q) P Q.
-Proof. intros; by rewrite /IntoAnd /= always_and_sep_r. Qed.
+Proof. intros; by rewrite /IntoAnd /= persistently_and_sep_r. Qed.
 
 Global Instance into_and_pure p φ ψ : @IntoAnd M p ⌜φ ∧ ψ⌝ ⌜φ⌝ ⌜ψ⌝.
-Proof. apply mk_into_and_sep. by rewrite pure_and always_and_sep_r. Qed.
-Global Instance into_and_always p P Q1 Q2 :
+Proof. apply mk_into_and_sep. by rewrite pure_and persistently_and_sep_r. Qed.
+Global Instance into_and_persistently p P Q1 Q2 :
   IntoAnd true P Q1 Q2 → IntoAnd p (□ P) (□ Q1) (□ Q2).
 Proof.
-  rewrite /IntoAnd=>->. destruct p; by rewrite ?always_and always_and_sep_r.
+  rewrite /IntoAnd=>->. destruct p; by rewrite ?persistently_and persistently_and_sep_r.
 Qed.
 Global Instance into_and_later p P Q1 Q2 :
   IntoAnd p P Q1 Q2 → IntoAnd p (▷ P) (▷ Q1) (▷ Q2).
@@ -471,9 +471,9 @@ Proof. rewrite /IsApp=>->. apply mk_into_and_sep. by rewrite big_sepL_app. Qed.
 
 (* Frame *)
 Global Instance frame_here p R : Frame p R R True.
-Proof. by rewrite /Frame right_id always_if_elim. Qed.
+Proof. by rewrite /Frame right_id persistently_if_elim. Qed.
 Global Instance frame_here_pure p φ Q : FromPure Q φ → Frame p ⌜φ⌝ Q True.
-Proof. rewrite /FromPure /Frame=> ->. by rewrite always_if_elim right_id. Qed.
+Proof. rewrite /FromPure /Frame=> ->. by rewrite persistently_if_elim right_id. Qed.
 
 Class MakeSep (P Q PQ : uPred M) := make_sep : P ∗ Q ⊣⊢ PQ.
 Global Instance make_sep_true_l P : MakeSep True P P.
@@ -488,7 +488,7 @@ Global Instance frame_sep_persistent_l R P1 P2 Q1 Q2 Q' :
   Frame true R (P1 ∗ P2) Q' | 9.