more consistent lemma naming

Fixes #177

theories/bi/derived_laws_bi.v

@@ -2,6 +2,13 @@ From iris.bi Require Export derived_connectives.
 From iris.algebra Require Import monoid.
 From stdpp Require Import hlist.
 
+(** Naming schema for lemmas about modalities:
+    M1_into_M2: M1 P ⊢ M2 P
+    M1_M2_elim: M1 (M2 P) ⊣⊢ M1 P
+    M1_elim_M2: M1 (M2 P) ⊣⊢ M2 P
+    M1_M2: M1 (M2 P) ⊣⊢ M2 (M1 P)
+*)
+
 Module bi.
 Import interface.bi.
 Section bi_derived.
@@ -605,7 +612,7 @@ Proof. by rewrite /bi_absorbingly !assoc (comm _ P). Qed.
 Lemma absorbingly_sep_lr P Q : <absorb> P ∗ Q ⊣⊢ P ∗ <absorb> Q.
 Proof. by rewrite absorbingly_sep_l absorbingly_sep_r. Qed.
 
-Lemma affinely_absorbingly `{!BiPositive PROP} P : <affine> <absorb> P ⊣⊢ <affine> P.
+Lemma affinely_absorbingly_elim `{!BiPositive PROP} P : <affine> <absorb> P ⊣⊢ <affine> P.
 Proof.
   apply (anti_symm _), affinely_mono, absorbingly_intro.
   by rewrite /bi_absorbingly affinely_sep affinely_True_emp affinely_emp left_id.
@@ -714,7 +721,7 @@ Global Instance persistently_flip_mono' :
   Proper (flip (⊢) ==> flip (⊢)) (@bi_persistently PROP).
 Proof. intros P Q; apply persistently_mono. Qed.
 
-Lemma absorbingly_persistently P : <absorb> <pers> P ⊣⊢ <pers> P.
+Lemma absorbingly_elim_persistently P : <absorb> <pers> P ⊣⊢ <pers> P.
 Proof.
   apply (anti_symm _), absorbingly_intro.
   by rewrite /bi_absorbingly comm persistently_absorbing.
@@ -776,16 +783,16 @@ Proof.
 Qed.
 Lemma persistently_and_emp_elim P : emp ∧ <pers> P ⊢ P.
 Proof. by rewrite comm persistently_and_sep_elim_emp right_id and_elim_r. Qed.
-Lemma persistently_elim_absorbingly P : <pers> P ⊢ <absorb> P.
+Lemma persistently_into_absorbingly P : <pers> P ⊢ <absorb> P.
 Proof.
   rewrite -(right_id True%I _ (<pers> _)%I) -{1}(left_id emp%I _ True%I).
   by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim comm.
 Qed.
 Lemma persistently_elim P `{!Absorbing P} : <pers> P ⊢ P.
-Proof. by rewrite persistently_elim_absorbingly absorbing_absorbingly. Qed.
+Proof. by rewrite persistently_into_absorbingly absorbing_absorbingly. Qed.
 
 Lemma persistently_idemp_1 P : <pers> <pers> P ⊢ <pers> P.
-Proof. by rewrite persistently_elim_absorbingly absorbingly_persistently. Qed.
+Proof. by rewrite persistently_into_absorbingly absorbingly_elim_persistently. Qed.
 Lemma persistently_idemp P : <pers> <pers> P ⊣⊢ <pers> P.
 Proof. apply (anti_symm _); auto using persistently_idemp_1, persistently_idemp_2. Qed.
 
@@ -795,7 +802,7 @@ Proof. intros <-. apply persistently_idemp_2. Qed.
 Lemma persistently_pure φ : <pers> ⌜φ⌝ ⊣⊢ ⌜φ⌝.
 Proof.
   apply (anti_symm _).
-  { by rewrite persistently_elim_absorbingly absorbingly_pure. }
+  { by rewrite persistently_into_absorbingly absorbingly_pure. }
   apply pure_elim'=> Hφ.
   trans (∀ x : False, <pers> True : PROP)%I; [by apply forall_intro|].
   rewrite persistently_forall_2. auto using persistently_mono, pure_intro.
@@ -824,7 +831,7 @@ Proof.
   by rewrite persistently_and_sep_assoc (comm bi_and) persistently_and_emp_elim.
 Qed.
 
-Lemma persistently_affinely P : <pers> <affine> P ⊣⊢ <pers> P.
+Lemma persistently_affinely_elim P : <pers> <affine> P ⊣⊢ <pers> P.
 Proof.
   by rewrite /bi_affinely persistently_and -persistently_True_emp
              persistently_pure left_id.
@@ -842,7 +849,7 @@ Proof. by rewrite -persistently_and_sep persistently_and -and_sep_persistently.
 Lemma persistently_sep `{BiPositive PROP} P Q : <pers> (P ∗ Q) ⊣⊢ <pers> P ∗ <pers> Q.
 Proof.
   apply (anti_symm _); auto using persistently_sep_2.
-  rewrite -persistently_affinely affinely_sep -and_sep_persistently. apply and_intro.
+  rewrite -persistently_affinely_elim affinely_sep -and_sep_persistently. apply and_intro.
   - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
   - by rewrite (affinely_elim_emp P) left_id affinely_elim.
 Qed.
@@ -858,7 +865,8 @@ Qed.
 Lemma persistently_alt_fixpoint' P :
   <pers> P ⊣⊢ <affine> P ∗ <pers> P.
 Proof.
-  rewrite -{1}persistently_affinely {1}persistently_alt_fixpoint persistently_affinely //.
+  rewrite -{1}persistently_affinely_elim {1}persistently_alt_fixpoint
+          persistently_affinely_elim //.
 Qed.
 
 Lemma persistently_wand P Q : <pers> (P -∗ Q) ⊢ <pers> P -∗ <pers> Q.
@@ -946,7 +954,7 @@ Proof. rewrite /bi_intuitionistically affinely_elim_emp //. Qed.
 Lemma intuitionistically_intro' P Q : (□ P ⊢ Q) → □ P ⊢ □ Q.
 Proof.
   intros <-.
-  by rewrite /bi_intuitionistically persistently_affinely persistently_idemp.
+  by rewrite /bi_intuitionistically persistently_affinely_elim persistently_idemp.
 Qed.
 
 Lemma intuitionistically_emp : □ emp ⊣⊢ emp.
@@ -973,11 +981,11 @@ Lemma intuitionistically_sep `{BiPositive PROP} P Q : □ (P ∗ Q) ⊣⊢ □ P
 Proof. by rewrite /bi_intuitionistically -affinely_sep -persistently_sep. Qed.
 
 Lemma intuitionistically_idemp P : □ □ P ⊣⊢ □ P.
-Proof. by rewrite /bi_intuitionistically persistently_affinely persistently_idemp. Qed.
+Proof. by rewrite /bi_intuitionistically persistently_affinely_elim persistently_idemp. Qed.
 
-Lemma intuitionistically_persistently_1 P : □ P ⊢ <pers> P.
+Lemma intuitionistically_into_persistently_1 P : □ P ⊢ <pers> P.
 Proof. rewrite /bi_intuitionistically affinely_elim //. Qed.
-Lemma intuitionistically_persistently_persistently P : □ <pers> P ⊣⊢ □ P.
+Lemma intuitionistically_persistently_elim P : □ <pers> P ⊣⊢ □ P.
 Proof. rewrite /bi_intuitionistically persistently_idemp //. Qed.
 
 Lemma intuitionistic_intuitionistically P :
@@ -992,8 +1000,8 @@ Proof.
   - rewrite and_elim_l //.
   - apply persistently_and_emp_elim.
 Qed.
-Lemma intuitionistically_affinely_affinely P : □ <affine> P ⊣⊢ □ P.
-Proof. rewrite /bi_intuitionistically persistently_affinely //. Qed.
+Lemma intuitionistically_affinely_elim P : □ <affine> P ⊣⊢ □ P.
+Proof. rewrite /bi_intuitionistically persistently_affinely_elim //. Qed.
 
 Lemma persistently_and_intuitionistically_sep_l P Q : <pers> P ∧ Q ⊣⊢ □ P ∗ Q.
 Proof.
@@ -1038,7 +1046,7 @@ Qed.
 Section bi_affine_intuitionistically.
   Context `{BiAffine PROP}.
 
-  Lemma intuitionistically_persistently P : □ P ⊣⊢ <pers> P.
+  Lemma intuitionistically_into_persistently P : □ P ⊣⊢ <pers> P.
   Proof. rewrite /bi_intuitionistically affine_affinely //. Qed.
 End bi_affine_intuitionistically.
 
@@ -1205,10 +1213,11 @@ Proof. intros. rewrite -persistent_and_sep_1; auto. Qed.
 Lemma persistent_entails_r P Q `{!Persistent Q} : (P ⊢ Q) → P ⊢ P ∗ Q.
 Proof. intros. rewrite -persistent_and_sep_1; auto. Qed.
 
-Lemma absorbingly_intuitionistically P : <absorb> □ P ⊣⊢ <pers> P.
+Lemma absorbingly_intuitionistically_into_persistently P :
+  <absorb> □ P ⊣⊢ <pers> P.
 Proof.
   apply (anti_symm _).
-  - by rewrite intuitionistically_persistently_1 absorbingly_persistently.
+  - by rewrite intuitionistically_into_persistently_1 absorbingly_elim_persistently.
   - rewrite -{1}(idemp bi_and (<pers> _)%I) persistently_and_intuitionistically_sep_r.
     by rewrite {1} (True_intro (<pers> _)%I).
 Qed.
@@ -1216,13 +1225,13 @@ Qed.
 Lemma persistent_absorbingly_affinely_2 P `{!Persistent P} :
   P ⊢ <absorb> <affine> P.
 Proof.
-  rewrite {1}(persistent P) -absorbingly_intuitionistically.
+  rewrite {1}(persistent P) -absorbingly_intuitionistically_into_persistently.
   by rewrite intuitionistically_affinely.
 Qed.
 Lemma persistent_absorbingly_affinely P `{!Persistent P, !Absorbing P} :
   <absorb> <affine> P ⊣⊢ P.
 Proof.
-  by rewrite -(persistent_persistently P) absorbingly_intuitionistically.
+  by rewrite -(persistent_persistently P) absorbingly_intuitionistically_into_persistently.
 Qed.
 
 Lemma persistent_and_sep_assoc P `{!Persistent P, !Absorbing P} Q R :
@@ -1309,7 +1318,7 @@ Proof. intros. by rewrite /Absorbing absorbingly_wand !absorbing -absorbingly_in
 Global Instance absorbingly_absorbing P : Absorbing (<absorb> P).
 Proof. rewrite /bi_absorbingly. apply _. Qed.
 Global Instance persistently_absorbing P : Absorbing (<pers> P).
-Proof. by rewrite /Absorbing absorbingly_persistently. Qed.
+Proof. by rewrite /Absorbing absorbingly_elim_persistently. Qed.
 Global Instance persistently_if_absorbing P p :
   Absorbing P → Absorbing (<pers>?p P).
 Proof. intros; destruct p; simpl; apply _. Qed.

theories/bi/derived_laws_sbi.v

@@ -121,7 +121,7 @@ Qed.
 Lemma persistently_internal_eq {A : ofeT} (a b : A) : <pers> (a ≡ b) ⊣⊢ a ≡ b.
 Proof.
   apply (anti_symm (⊢)).
-  { by rewrite persistently_elim_absorbingly absorbingly_internal_eq. }
+  { by rewrite persistently_into_absorbingly absorbingly_internal_eq. }
   apply (internal_eq_rewrite' a b (λ b, <pers> (a ≡ b))%I); auto.
   rewrite -(internal_eq_refl emp%I a). apply persistently_emp_intro.
 Qed.

theories/bi/plainly.v

@@ -121,16 +121,16 @@ Proof. intros P Q; apply plainly_mono. Qed.
 Lemma affinely_plainly_elim P : <affine> ■ P ⊢ P.
 Proof. by rewrite plainly_elim_persistently /bi_affinely persistently_and_emp_elim. Qed.
 
-Lemma persistently_plainly P : <pers> ■ P ⊣⊢ ■ P.
+Lemma persistently_elim_plainly P : <pers> ■ P ⊣⊢ ■ P.
 Proof.
   apply (anti_symm _).
-  - by rewrite persistently_elim_absorbingly /bi_absorbingly comm plainly_absorb.
+  - by rewrite persistently_into_absorbingly /bi_absorbingly comm plainly_absorb.
   - by rewrite {1}plainly_idemp_2 plainly_elim_persistently.
 Qed.
-Lemma persistently_if_plainly P p : <pers>?p ■ P ⊣⊢ ■ P.
-Proof. destruct p; last done. exact: persistently_plainly. Qed.
+Lemma persistently_if_elim_plainly P p : <pers>?p ■ P ⊣⊢ ■ P.
+Proof. destruct p; last done. exact: persistently_elim_plainly. Qed.
 
-Lemma plainly_persistently P : ■ <pers> P ⊣⊢ ■ P.
+Lemma plainly_persistently_elim P : ■ <pers> P ⊣⊢ ■ P.
 Proof.
   apply (anti_symm _).
   - rewrite -{1}(left_id True%I bi_and (■ _)%I) (plainly_emp_intro True%I).
@@ -139,22 +139,22 @@ Proof.
   - by rewrite {1}plainly_idemp_2 (plainly_elim_persistently P).
 Qed.
 
-Lemma absorbingly_plainly P : <absorb> ■ P ⊣⊢ ■ P.
-Proof. by rewrite -(persistently_plainly P) absorbingly_persistently. Qed.
+Lemma absorbingly_elim_plainly P : <absorb> ■ P ⊣⊢ ■ P.
+Proof. by rewrite -(persistently_elim_plainly P) absorbingly_elim_persistently. Qed.
 
 Lemma plainly_and_sep_elim P Q : ■ P ∧ Q -∗ (emp ∧ P) ∗ Q.
 Proof. by rewrite plainly_elim_persistently persistently_and_sep_elim_emp. Qed.
 Lemma plainly_and_sep_assoc P Q R : ■ P ∧ (Q ∗ R) ⊣⊢ (■ P ∧ Q) ∗ R.
-Proof. by rewrite -(persistently_plainly P) persistently_and_sep_assoc. Qed.
+Proof. by rewrite -(persistently_elim_plainly P) persistently_and_sep_assoc. Qed.
 Lemma plainly_and_emp_elim P : emp ∧ ■ P ⊢ P.
 Proof. by rewrite plainly_elim_persistently persistently_and_emp_elim. Qed.
-Lemma plainly_elim_absorbingly P : ■ P ⊢ <absorb> P.
-Proof. by rewrite plainly_elim_persistently persistently_elim_absorbingly. Qed.
+Lemma plainly_into_absorbingly P : ■ P ⊢ <absorb> P.
+Proof. by rewrite plainly_elim_persistently persistently_into_absorbingly. Qed.
 Lemma plainly_elim P `{!Absorbing P} : ■ P ⊢ P.
 Proof. by rewrite plainly_elim_persistently persistently_elim. Qed.
 
 Lemma plainly_idemp_1 P : ■ ■ P ⊢ ■ P.
-Proof. by rewrite plainly_elim_absorbingly absorbingly_plainly. Qed.
+Proof. by rewrite plainly_into_absorbingly absorbingly_elim_plainly. Qed.
 Lemma plainly_idemp P : ■ ■ P ⊣⊢ ■ P.
 Proof. apply (anti_symm _); auto using plainly_idemp_1, plainly_idemp_2. Qed.
 
@@ -213,15 +213,15 @@ Proof.
   by rewrite plainly_and_sep_assoc (comm bi_and) plainly_and_emp_elim.
 Qed.
 
-Lemma plainly_affinely P : ■ <affine> P ⊣⊢ ■ P.
+Lemma plainly_affinely_elim P : ■ <affine> P ⊣⊢ ■ P.
 Proof. by rewrite /bi_affinely plainly_and -plainly_True_emp plainly_pure left_id. Qed.
 
 Lemma intuitionistically_plainly_elim P : □ ■ P -∗ □ P.
 Proof. rewrite intuitionistically_affinely plainly_elim_persistently //. Qed.
 Lemma intuitionistically_plainly P : □ ■ P -∗ ■ □ P.
 Proof.
-  rewrite /bi_intuitionistically plainly_affinely affinely_elim.
-  rewrite persistently_plainly plainly_persistently. done.
+  rewrite /bi_intuitionistically plainly_affinely_elim affinely_elim.
+  rewrite persistently_elim_plainly plainly_persistently_elim. done.
 Qed.
 
 Lemma and_sep_plainly P Q : ■ P ∧ ■ Q ⊣⊢ ■ P ∗ ■ Q.
@@ -236,7 +236,7 @@ Proof. by rewrite -plainly_and_sep plainly_and -and_sep_plainly. Qed.
 Lemma plainly_sep `{BiPositive PROP} P Q : ■ (P ∗ Q) ⊣⊢ ■ P ∗ ■ Q.
 Proof.
   apply (anti_symm _); auto using plainly_sep_2.
-  rewrite -(plainly_affinely (_ ∗ _)%I) affinely_sep -and_sep_plainly. apply and_intro.
+  rewrite -(plainly_affinely_elim (_ ∗ _)%I) affinely_sep -and_sep_plainly. apply and_intro.
   - by rewrite (affinely_elim_emp Q) right_id affinely_elim.
   - by rewrite (affinely_elim_emp P) left_id affinely_elim.
 Qed.
@@ -260,7 +260,7 @@ Lemma impl_wand_plainly_2 P Q : (■ P -∗ Q) ⊢ (■ P → Q).
 Proof. apply impl_intro_l. by rewrite plainly_and_sep_l_1 wand_elim_r. Qed.
 
 Lemma impl_wand_affinely_plainly P Q : (■ P → Q) ⊣⊢ (<affine> ■ P -∗ Q).
-Proof. by rewrite -(persistently_plainly P) impl_wand_intuitionistically. Qed.
+Proof. by rewrite -(persistently_elim_plainly P) impl_wand_intuitionistically. Qed.
 
 Lemma persistently_wand_affinely_plainly P Q :
   (<affine> ■ P -∗ <pers> Q) ⊢ <pers> (<affine> ■ P -∗ Q).
@@ -357,11 +357,11 @@ Lemma impl_persistent P Q :
   Absorbing P → Plain P → Persistent Q → Persistent (P → Q).
 Proof.
   intros. by rewrite /Persistent {2}(plain P) -persistently_impl_plainly
-                     -(persistent Q) (plainly_elim_absorbingly P) absorbing.
+                     -(persistent Q) (plainly_into_absorbingly P) absorbing.
 Qed.
 
 Global Instance plainly_persistent P : Persistent (■ P).
-Proof. by rewrite /Persistent persistently_plainly. Qed.
+Proof. by rewrite /Persistent persistently_elim_plainly. Qed.
 
 Global Instance wand_persistent P Q :
   Plain P → Persistent Q → Absorbing Q → Persistent (P -∗ Q).
@@ -428,7 +428,7 @@ Qed.
 Global Instance impl_plain P Q : Absorbing P → Plain P → Plain Q → Plain (P → Q).
 Proof.
   intros. by rewrite /Plain {2}(plain P) -plainly_impl_plainly -(plain Q)
-                     (plainly_elim_absorbingly P) absorbing.
+                     (plainly_into_absorbingly P) absorbing.
 Qed.
 Global Instance wand_plain P Q :
   Plain P → Plain Q → Absorbing Q → Plain (P -∗ Q).
@@ -445,7 +445,7 @@ Global Instance plainly_plain P : Plain (■ P).
 Proof. by rewrite /Plain plainly_idemp. Qed.
 Global Instance persistently_plain P : Plain P → Plain (<pers> P).
 Proof.
-  rewrite /Plain=> HP. rewrite {1}HP plainly_persistently persistently_plainly //.
+  rewrite /Plain=> HP. rewrite {1}HP plainly_persistently_elim persistently_elim_plainly //.
 Qed.
 Global Instance affinely_plain P : Plain P → Plain (<affine> P).
 Proof. rewrite /bi_affinely. apply _. Qed.
@@ -468,7 +468,7 @@ Qed.
 
 Lemma plainly_alt P : ■ P ⊣⊢ <affine> P ≡ emp.
 Proof.
-  rewrite -plainly_affinely. apply (anti_symm (⊢)).
+  rewrite -plainly_affinely_elim. apply (anti_symm (⊢)).
   - rewrite -prop_ext. apply plainly_mono, and_intro; apply wand_intro_l.
     + by rewrite affinely_elim_emp left_id.
     + by rewrite left_id.

theories/proofmode/class_instances_bi.v

@@ -64,13 +64,13 @@ Global Instance from_assumption_persistently_l_true P Q :
   FromAssumption true P Q → KnownLFromAssumption true (<pers> P) Q.
 Proof.
   rewrite /KnownLFromAssumption /FromAssumption /= =><-.
-  rewrite intuitionistically_persistently_persistently //.
+  rewrite intuitionistically_persistently_elim //.
 Qed.
 Global Instance from_assumption_persistently_l_false `{BiAffine PROP} P Q :
   FromAssumption true P Q → KnownLFromAssumption false (<pers> P) Q.
 Proof.
   rewrite /KnownLFromAssumption /FromAssumption /= =><-.
-  by rewrite intuitionistically_persistently.
+  by rewrite intuitionistically_into_persistently.
 Qed.
 Global Instance from_assumption_affinely_l_true p P Q :
   FromAssumption p P Q → KnownLFromAssumption p (<affine> P) Q.
@@ -195,7 +195,7 @@ Global Instance from_pure_persistently P a φ :
   FromPure true P φ → FromPure a (<pers> P) φ.
 Proof.
   rewrite /FromPure=> <- /=.
-  by rewrite persistently_affinely affinely_if_elim persistently_pure.
+  by rewrite persistently_affinely_elim affinely_if_elim persistently_pure.
 Qed.
 Global Instance from_pure_affinely_true P φ :
   FromPure true P φ → FromPure true (<affine> P) φ.
@@ -206,7 +206,7 @@ Proof. rewrite /FromPure /= affine_affinely //. Qed.
 Global Instance from_pure_intuitionistically_true P φ :
   FromPure true P φ → FromPure true (□ P) φ.
 Proof.
-  rewrite /FromPure=><- /=. rewrite intuitionistically_affinely_affinely.
+  rewrite /FromPure=><- /=. rewrite intuitionistically_affinely_elim.
   rewrite {1}(persistent ⌜φ⌝%I) //.
 Qed.
 
@@ -233,7 +233,7 @@ Proof.
   rewrite /IntoPersistent /= =><-.
   destruct p; simpl;
     eauto using persistently_mono, intuitionistically_elim,
-    intuitionistically_persistently_1.
+    intuitionistically_into_persistently_1.
 Qed.
 Global Instance into_persistent_embed `{BiEmbed PROP PROP'} p P Q :
   IntoPersistent p P Q → IntoPersistent p ⎡P⎤ ⎡Q⎤ | 0.
@@ -259,7 +259,9 @@ Global Instance from_modal_intuitionistically P :
 Proof. by rewrite /FromModal. Qed.
 Global Instance from_modal_intuitionistically_affine_bi P :
   BiAffine PROP → FromModal modality_persistently (□ P) (□ P) P | 0.
-Proof. intros. by rewrite /FromModal /= intuitionistically_persistently. Qed.
+Proof.
+  intros. by rewrite /FromModal /= intuitionistically_into_persistently.
+Qed.
 
 Global Instance from_modal_absorbingly P :
   FromModal modality_id (<absorb> P) (<absorb> P) P.
@@ -356,7 +358,7 @@ Global Instance into_wand_intuitionistically p q R P Q :
 Proof. by rewrite /IntoWand intuitionistically_elim. Qed.
 Global Instance into_wand_persistently_true q R P Q :
   IntoWand true q R P Q → IntoWand true q (<pers> R) P Q.
-Proof. by rewrite /IntoWand /= intuitionistically_persistently_persistently. Qed.
+Proof. by rewrite /IntoWand /= intuitionistically_persistently_elim. Qed.
 Global Instance into_wand_persistently_false q R P Q :
   Absorbing R → IntoWand false q R P Q → IntoWand false q (<pers> R) P Q.
 Proof. intros ?. by rewrite /IntoWand persistently_elim. Qed.
@@ -494,7 +496,7 @@ Global Instance into_and_affinely p P Q1 Q2 :
   IntoAnd p P Q1 Q2 → IntoAnd p (<affine> P) (<affine> Q1) (<affine> Q2).
 Proof.
   rewrite /IntoAnd. destruct p; simpl.
-  - rewrite -affinely_and !intuitionistically_affinely_affinely //.
+  - rewrite -affinely_and !intuitionistically_affinely_elim //.
   - intros ->. by rewrite affinely_and.
 Qed.
 Global Instance into_and_intuitionistically p P Q1 Q2 :
@@ -509,7 +511,7 @@ Global Instance into_and_persistently p P Q1 Q2 :
   IntoAnd p (<pers> P) (<pers> Q1) (<pers> Q2).
 Proof.
   rewrite /IntoAnd /=. destruct p; simpl.
-  - rewrite -persistently_and !intuitionistically_persistently_persistently //.
+  - rewrite -persistently_and !intuitionistically_persistently_elim //.
   - intros ->. by rewrite persistently_and.
 Qed.
 Global Instance into_and_embed `{BiEmbed PROP PROP'} p P Q1 Q2 :
@@ -746,7 +748,7 @@ Qed.
 Global Instance from_forall_intuitionistically `{BiAffine PROP} {A} P (Φ : A → PROP) :
   FromForall P Φ → FromForall (□ P) (λ a, □ (Φ a))%I.
 Proof.
-  rewrite /FromForall=> <-. setoid_rewrite intuitionistically_persistently.
+  rewrite /FromForall=> <-. setoid_rewrite intuitionistically_into_persistently.
   by rewrite persistently_forall.
 Qed.
 Global Instance from_forall_persistently {A} P (Φ : A → PROP) :

theories/proofmode/class_instances_sbi.v

@@ -36,7 +36,7 @@ Global Instance from_assumption_plainly_l_false `{BiPlainly PROP, BiAffine PROP}
   FromAssumption true P Q → KnownLFromAssumption false (■ P) Q.
 Proof.
   rewrite /KnownLFromAssumption /FromAssumption /= =><-.
-  rewrite plainly_elim_persistently intuitionistically_persistently //.
+  rewrite plainly_elim_persistently intuitionistically_into_persistently //.
 Qed.
 
 (* FromPure *)

theories/proofmode/coq_tactics.v

@@ -375,7 +375,7 @@ Lemma env_spatial_is_nil_intuitionistically Δ :
 Proof.
   intros. unfold of_envs; destruct Δ as [? []]; simplify_eq/=.
   rewrite !right_id /bi_intuitionistically {1}affinely_and_r persistently_and.
-  by rewrite persistently_affinely persistently_idemp persistently_pure.
+  by rewrite persistently_affinely_elim persistently_idemp persistently_pure.
 Qed.
 
 Lemma envs_lookup_envs_delete Δ i p P :
@@ -616,7 +616,7 @@ Proof.
     + rewrite -(affine_affinely P) (_ : P = <pers>?false P)%I //
               (into_persistent _ P) wand_elim_r //.
     + rewrite (_ : P = <pers>?false P)%I // (into_persistent _ P).
-      by rewrite -{1}absorbingly_intuitionistically
+      by rewrite -{1}absorbingly_intuitionistically_into_persistently
         absorbingly_sep_l wand_elim_r HQ.
 Qed.
 
@@ -673,7 +673,7 @@ Proof.
   - rewrite -(affine_affinely P) (_ : P = <pers>?false P)%I //
             (into_persistent _ P) wand_elim_r //.
   - rewrite (_ : P = <pers>?false P)%I // (into_persistent _ P).
-    by rewrite -{1}absorbingly_intuitionistically
+    by rewrite -{1}absorbingly_intuitionistically_into_persistently
       absorbingly_sep_l wand_elim_r HQ.
 Qed.
 Lemma tac_wand_intro_pure Δ P φ Q R :
@@ -769,7 +769,7 @@ Lemma tac_specialize_assert_pure Δ Δ' j q R P1 P2 φ Q :
 Proof.
   rewrite envs_entails_eq=> ????? <-. rewrite envs_simple_replace_singleton_sound //=.
   rewrite -intuitionistically_if_idemp into_wand /= -(from_pure _ P1) /bi_intuitionistically.
-  rewrite pure_True //= persistently_affinely persistently_pure
+  rewrite pure_True //= persistently_affinely_elim persistently_pure
           affinely_True_emp affinely_emp.
   by rewrite emp_wand wand_elim_r.
 Qed.
@@ -787,7 +787,7 @@ Proof.
   rewrite -(idemp bi_and (of_envs (envs_delete _ _ _ _))).
   rewrite {2}envs_simple_replace_singleton_sound' //; simpl.
   rewrite {1}HP1 (into_absorbingly P1') (persistent_persistently_2 P1).
-  rewrite absorbingly_persistently persistently_and_intuitionistically_sep_l assoc.
+  rewrite absorbingly_elim_persistently persistently_and_intuitionistically_sep_l assoc.
   rewrite -intuitionistically_if_idemp -intuitionistically_idemp.
   rewrite (intuitionistically_intuitionistically_if q).
   by rewrite intuitionistically_if_sep_2 into_wand wand_elim_l wand_elim_r.
@@ -804,7 +804,7 @@ Proof.
   rewrite envs_entails_eq => ? HR ? Hpos ? <-. rewrite -(idemp bi_and (of_envs Δ)) {1}HR.
   rewrite envs_replace_singleton_sound //; destruct q; simpl.
   - by rewrite (_ : R = <pers>?false R)%I // (into_persistent _ R)
-      absorbingly_persistently sep_elim_r persistently_and_intuitionistically_sep_l wand_elim_r.
+      absorbingly_elim_persistently sep_elim_r persistently_and_intuitionistically_sep_l wand_elim_r.
   - by rewrite (absorbing_absorbingly R) (_ : R = <pers>?false R)%I //
        (into_persistent _ R) sep_elim_r persistently_and_intuitionistically_sep_l wand_elim_r.
 Qed.
@@ -848,7 +848,7 @@ Proof.
   rewrite /IntoIH envs_entails_eq. intros HP ? HPQ.
   rewrite (env_spatial_is_nil_intuitionistically Δ) //.
   rewrite -(idemp bi_and (□ (of_envs Δ))%I) {1}HP // HPQ.
-  rewrite {1}intuitionistically_persistently_1 intuitionistically_elim impl_elim_r //.
+  rewrite {1}intuitionistically_into_persistently_1 intuitionistically_elim impl_elim_r //.
 Qed.
 
 Lemma tac_assert Δ Δ' j P Q :
@@ -1407,6 +1407,6 @@ Proof.
   rewrite -(löb (<pers> Q)%I) later_persistently. apply impl_intro_l.
   rewrite envs_app_singleton_sound //; simpl; rewrite HQ.
   rewrite persistently_and_intuitionistically_sep_l -{1}intuitionistically_idemp.
-  rewrite intuitionistically_sep_2 wand_elim_r intuitionistically_persistently_1 //.
+  rewrite intuitionistically_sep_2 wand_elim_r intuitionistically_into_persistently_1 //.
 Qed.
 End sbi_tactics.

theories/proofmode/frame_instances.v

@@ -229,8 +229,8 @@ Global Instance frame_persistently R P Q Q' :
 Proof.
   rewrite /Frame /MakePersistently=> <- <- /=.
   rewrite -persistently_and_intuitionistically_sep_l.
-  by rewrite -persistently_sep_2 -persistently_and_sep_l_1 persistently_affinely
-              persistently_idemp.
+  by rewrite -persistently_sep_2 -persistently_and_sep_l_1
+     persistently_affinely_elim persistently_idemp.
 Qed.
 
 Global Instance frame_exist {A} p R (Φ Ψ : A → PROP) :