Merge commit '92c7' into gen_proofmode

CHANGELOG.md

@@ -7,6 +7,7 @@ Coq development, but not every API-breaking change is listed.  Changes marked
 
 Changes in and extensions of the theory:
 
+* [#] Add new modality: ■ ("plainly").
 * [#] Camera morphisms have to be homomorphisms, not just monotone functions.
 * [#] A proof that `f` has a fixed point if `f^k` is contractive.
 * Constructions for least and greatest fixed points over monotone predicates

ProofMode.md

@@ -121,6 +121,13 @@ Modalities
   an instance of the `ElimModal` type class. Instances include: later, except 0,
   basic update and fancy update.
 
+The persistence and plainness modalities
+----------------------------------------
+
+- `iAlways` : introduce a persistence or plainness modality and the spatial
+  context. In case of a plainness modality, the tactic will prune all persistent
+  hypotheses that are not plain.
+
 The later modality
 ------------------
 
@@ -197,6 +204,7 @@ _introduction patterns_:
 - `[ipat1|ipat2]` : disjunction elimination.
 - `[]` : false elimination.
 - `%` : move the hypothesis to the pure Coq context (anonymously).
+- `->` and `<-` : rewrite using a pure Coq equality
 - `# ipat` : move the hypothesis to the persistent context.
 - `> ipat` : eliminate a modality (if the goal permits).
 
@@ -207,10 +215,8 @@ appear at the top level:
   Items of the selection pattern can be prefixed with `$`, which cause them to
   be framed instead of cleared.
 - `!%` : introduce a pure goal (and leave the proof mode).
-- `!#` : introduce an persistence or plainness modality and clear the spatial
-  context. In case of a plainness modality, it will prune all persistent
-  hypotheses that are not plain.
-- `!>` : introduce a modality.
+- `!#` : introduce an persistence or plainness modality (by calling `iAlways`).
+- `!>` : introduce a modality (by calling `iModIntro`).
 - `/=` : perform `simpl`.
 - `//` : perform `try done` on all goals.
 - `//=` : syntactic sugar for `/= //`

_CoqProject

@@ -78,7 +78,7 @@ theories/heap_lang/lib/ticket_lock.v
 theories/heap_lang/lib/counter.v
 theories/heap_lang/proofmode.v
 theories/heap_lang/adequacy.v
-theories/proofmode/strings.v
+theories/proofmode/base.v
 theories/proofmode/tokens.v
 theories/proofmode/coq_tactics.v
 theories/proofmode/environments.v

theories/base_logic/upred.v

@@ -248,6 +248,9 @@ Definition uPred_wand {M} := unseal uPred_wand_aux M.
 Definition uPred_wand_eq :
   @uPred_wand = @uPred_wand_def := seal_eq uPred_wand_aux.
 
+(* Equivalently, this could be `∀ y, P n y`.  That's closer to the intuition
+   of "embedding the step-indexed logic in Iris", but the two are equivalent
+   because Iris is afine.  The following is easier to work with. *)
 Program Definition uPred_plainly_def {M} (P : uPred M) : uPred M :=
   {| uPred_holds n x := P n ε |}.
 Solve Obligations with naive_solver eauto using uPred_closed, ucmra_unit_validN.

theories/bi/interface.v

@@ -112,6 +112,9 @@ Section bi_mixin.
     bi_mixin_prop_ext P Q : bi_plainly ((P → Q) ∧ (Q → P)) ⊢
       bi_internal_eq (OfeT PROP prop_ofe_mixin) P Q;
 
+    (* The following two laws are very similar, and indeed they hold
+       not just for □ and ■, but for any modality defined as
+       `M P n x := ∀ y, R x y → P n y`. *)
     bi_mixin_persistently_impl_plainly P Q :
       (bi_plainly P → bi_persistently Q) ⊢ bi_persistently (bi_plainly P → Q);
     bi_mixin_plainly_impl_plainly P Q :

theories/proofmode/strings.v → theories/proofmode/base.v

-From stdpp Require Import strings.
-From iris.algebra Require Import base.
+From stdpp Require Export strings.
+From iris.algebra Require Export base.
 From Coq Require Import Ascii.
 Set Default Proof Using "Type".
 
+(* Directions of rewrites *)
+Inductive direction := Left | Right.
+
+(* Some specific versions of operations on strings for the proof mode. We need
+those so that we can make [cbv] unfold just them, but not the actual operations
+that may appear in users' proofs. *)
 Local Notation "b1 && b2" := (if b1 then b2 else false) : bool_scope.
 
 Lemma lazy_andb_true (b1 b2 : bool) : b1 && b2 = true ↔ b1 = true ∧ b2 = true.

theories/proofmode/coq_tactics.v

 From iris.bi Require Export bi.
 From iris.bi Require Import tactics.
-From iris.proofmode Require Export environments classes.
+From iris.proofmode Require Export base environments classes.
 From stdpp Require Import stringmap hlist.
 Set Default Proof Using "Type".
 Import bi.
@@ -111,12 +111,12 @@ Fixpoint envs_split_go {PROP}
      if p then envs_split_go js Δ1 Δ2 else
      envs_split_go js Δ1' (envs_snoc Δ2 false j P)
   end.
-(* if [lr = true] then [result = (remaining hyps, hyps named js)] and
-   if [lr = false] then [result = (hyps named js, remaining hyps)] *)
-Definition envs_split {PROP} (lr : bool)
+(* if [d = Right] then [result = (remaining hyps, hyps named js)] and
+   if [d = Left] then [result = (hyps named js, remaining hyps)] *)
+Definition envs_split {PROP} (d : direction)
     (js : list string) (Δ : envs PROP) : option (envs PROP * envs PROP) :=
   '(Δ1,Δ2) ← envs_split_go js Δ (envs_clear_spatial Δ);
-  if lr then Some (Δ1,Δ2) else Some (Δ2,Δ1).
+  if d is Right then Some (Δ1,Δ2) else Some (Δ2,Δ1).
 
 (* Coq versions of the tactics *)
 Section bi_tactics.
@@ -216,11 +216,11 @@ Proof.
   apply wand_intro_l; destruct p; simpl.
   - apply and_intro; [apply pure_intro|].
     + destruct Hwf; constructor; simpl; eauto using Esnoc_wf.
-      intros j; destruct (strings.string_beq_reflect j i); naive_solver.
+      intros j; destruct (string_beq_reflect j i); naive_solver.
     + by rewrite affinely_persistently_and and_sep_affinely_persistently assoc.
   - apply and_intro; [apply pure_intro|].
     + destruct Hwf; constructor; simpl; eauto using Esnoc_wf.
-      intros j; destruct (strings.string_beq_reflect j i); naive_solver.
+      intros j; destruct (string_beq_reflect j i); naive_solver.
     + solve_sep_entails.
 Qed.
 
@@ -380,8 +380,8 @@ Proof.
     - rewrite envs_lookup_envs_delete_ne // envs_lookup_snoc_ne //. eauto. }
   rewrite (envs_snoc_sound Δ2 false j P) /= ?wand_elim_r; eauto.
 Qed.
-Lemma envs_split_sound Δ lr js Δ1 Δ2 :
-  envs_split lr js Δ = Some (Δ1,Δ2) → Δ ⊢ Δ1 ∗ Δ2.
+Lemma envs_split_sound Δ d js Δ1 Δ2 :
+  envs_split d js Δ = Some (Δ1,Δ2) → Δ ⊢ Δ1 ∗ Δ2.
 Proof.
   rewrite /envs_split=> ?. rewrite -(idemp bi_and Δ).
   rewrite {2}envs_clear_spatial_sound.
@@ -392,7 +392,7 @@ Proof.
   destruct (envs_split_go _ _) as [[Δ1' Δ2']|] eqn:HΔ; [|done].
   apply envs_split_go_sound in HΔ as ->; last first.
   { intros j P. by rewrite envs_lookup_envs_clear_spatial=> ->. }
-  destruct lr; simplify_eq; solve_sep_entails.
+  destruct d; simplify_eq; solve_sep_entails.
 Qed.
 
 Global Instance envs_Forall2_refl (R : relation PROP) :
@@ -705,11 +705,11 @@ Proof.
     by rewrite into_wand /= assoc wand_elim_r wand_elim_r.
 Qed.
 
-Lemma tac_specialize_assert Δ Δ' Δ1 Δ2' j q lr js R P1 P2 P1' Q :
+Lemma tac_specialize_assert Δ Δ' Δ1 Δ2' j q neg js R P1 P2 P1' Q :
   envs_lookup_delete j Δ = Some (q, R, Δ') →
   IntoWand q false R P1 P2 →
   ElimModal P1' P1 Q Q →
-  ('(Δ1,Δ2) ← envs_split lr js Δ';
+  ('(Δ1,Δ2) ← envs_split (if neg is true then Right else Left) js Δ';
     Δ2' ← envs_app false (Esnoc Enil j P2) Δ2;
     Some (Δ1,Δ2')) = Some (Δ1,Δ2') → (* does not preserve position of [j] *)
   (Δ1 ⊢ P1') → (Δ2' ⊢ Q) → Δ ⊢ Q.
@@ -819,9 +819,9 @@ Proof.
   by rewrite -{1}persistently_idemp !affinely_persistently_elim impl_elim_r.
 Qed.
 
-Lemma tac_assert Δ Δ1 Δ2 Δ2' lr js j P P' Q :
+Lemma tac_assert Δ Δ1 Δ2 Δ2' neg js j P P' Q :
   ElimModal P' P Q Q →
-  envs_split lr js Δ = Some (Δ1,Δ2) →
+  envs_split (if neg is true then Right else Left) js Δ = Some (Δ1,Δ2) →
   envs_app false (Esnoc Enil j P) Δ2 = Some Δ2' →
   (Δ1 ⊢ P') → (Δ2' ⊢ Q) → Δ ⊢ Q.
 Proof.
@@ -829,8 +829,8 @@ Proof.
   rewrite (envs_app_singleton_sound Δ2) //; simpl. by rewrite HP HQ.
 Qed.
 
-Lemma tac_assert_persistent Δ Δ1 Δ2 Δ' lr js j P P' Q :
-  envs_split lr js Δ = Some (Δ1,Δ2) →
+Lemma tac_assert_persistent Δ Δ1 Δ2 Δ' neg js j P P' Q :
+  envs_split (if neg is true then Right else Left) js Δ = Some (Δ1,Δ2) →
   Persistent P →
   FromAffinely P' P →
   envs_app false (Esnoc Enil j P') Δ = Some Δ' →
@@ -885,28 +885,28 @@ Proof.
 Qed.
 
 (** * Rewriting *)
-Lemma tac_rewrite Δ i p Pxy (lr : bool) Q :
+Lemma tac_rewrite Δ i p Pxy d Q :
   envs_lookup i Δ = Some (p, Pxy) →
   ∀ {A : ofeT} (x y : A) (Φ : A → PROP),
     IntoInternalEq Pxy x y →
-    (Q ⊣⊢ Φ (if lr then y else x)) →
+    (Q ⊣⊢ Φ (if d is Left then y else x)) →
     NonExpansive Φ →
-    (Δ ⊢ Φ (if lr then x else y)) → Δ ⊢ Q.
+    (Δ ⊢ Φ (if d is Left then x else y)) → Δ ⊢ Q.
 Proof.
   intros ? A x y ? HPxy -> ?; apply internal_eq_rewrite'; auto.
   rewrite {1}envs_lookup_sound //.
   rewrite HPxy affinely_persistently_if_elim sep_elim_l.
-  destruct lr; auto using internal_eq_sym.
+  destruct d; auto using internal_eq_sym.
 Qed.
 
-Lemma tac_rewrite_in Δ i p Pxy j q P (lr : bool) Q :
+Lemma tac_rewrite_in Δ i p Pxy j q P d Q :
   envs_lookup i Δ = Some (p, Pxy) →
   envs_lookup j Δ = Some (q, P) →
   ∀ {A : ofeT} Δ' (x y : A) (Φ : A → PROP),
     IntoInternalEq Pxy x y →
-    (P ⊣⊢ Φ (if lr then y else x)) →
+    (P ⊣⊢ Φ (if d is Left then y else x)) →
     NonExpansive Φ →
-    envs_simple_replace j q (Esnoc Enil j (Φ (if lr then x else y))) Δ = Some Δ' →
+    envs_simple_replace j q (Esnoc Enil j (Φ (if d is Left then x else y))) Δ = Some Δ' →
     (Δ' ⊢ Q) → Δ ⊢ Q.
 Proof.
   intros ?? A Δ' x y Φ HPxy HP ?? <-.
@@ -914,7 +914,7 @@ Proof.
   rewrite (envs_simple_replace_singleton_sound _ _ j) //; simpl.
   rewrite HP HPxy (affinely_persistently_if_elim _ (_ ≡ _)%I) sep_elim_l.
   rewrite persistent_and_affinely_sep_r -assoc. apply wand_elim_r'.
-  rewrite -persistent_and_affinely_sep_r. apply impl_elim_r'. destruct lr.
+  rewrite -persistent_and_affinely_sep_r. apply impl_elim_r'. destruct d.
   - apply (internal_eq_rewrite x y (λ y, □?q Φ y -∗ Δ')%I). solve_proper.
   - rewrite internal_eq_sym.
     eapply (internal_eq_rewrite y x (λ y, □?q Φ y -∗ Δ')%I). solve_proper.
@@ -926,9 +926,9 @@ Lemma tac_and_split Δ P Q1 Q2 :
 Proof. intros. rewrite -(from_and P). by apply and_intro. Qed.
 
 (** * Separating conjunction splitting *)
-Lemma tac_sep_split Δ Δ1 Δ2 lr js P Q1 Q2 :
+Lemma tac_sep_split Δ Δ1 Δ2 d js P Q1 Q2 :
   FromSep P Q1 Q2 →
-  envs_split lr js Δ = Some (Δ1,Δ2) →
+  envs_split d js Δ = Some (Δ1,Δ2) →
   (Δ1 ⊢ Q1) → (Δ2 ⊢ Q2) → Δ ⊢ P.
 Proof. intros ?? HQ1 HQ2. rewrite envs_split_sound //. by rewrite HQ1 HQ2. Qed.
 
@@ -972,25 +972,25 @@ Qed.
 (* Using this tactic, one can destruct a (non-separating) conjunction in the
 spatial context as long as one of the conjuncts is thrown away. It corresponds
 to the principle of "external choice" in linear logic. *)
-Lemma tac_and_destruct_choice Δ Δ' i p (lr : bool) j P P1 P2 Q :
+Lemma tac_and_destruct_choice Δ Δ' i p d j P P1 P2 Q :
   envs_lookup i Δ = Some (p, P) →
   (if p then IntoAnd p P P1 P2 : Type else
     TCOr (IntoAnd p P P1 P2) (TCAnd (IntoSep P P1 P2)
-      (if lr then TCOr (Affine P2) (Absorbing Q)
+      (if d is Left then TCOr (Affine P2) (Absorbing Q)
        else TCOr (Affine P1) (Absorbing Q)))) →
-  envs_simple_replace i p (Esnoc Enil j (if lr then P1 else P2)) Δ = Some Δ' →
+  envs_simple_replace i p (Esnoc Enil j (if d is Left then P1 else P2)) Δ = Some Δ' →
   (Δ' ⊢ Q) → Δ ⊢ Q.
 Proof.
   intros ? HP ? HQ. rewrite envs_simple_replace_singleton_sound //; simpl.
   destruct p.
-  { rewrite (into_and _ P) HQ. destruct lr; simpl.
+  { rewrite (into_and _ P) HQ. destruct d; simpl.
     - by rewrite and_elim_l wand_elim_r.
     - by rewrite and_elim_r wand_elim_r. }
   destruct HP as [?|[??]].
-  { rewrite (into_and _ P) HQ. destruct lr; simpl.
+  { rewrite (into_and _ P) HQ. destruct d; simpl.
     - by rewrite and_elim_l wand_elim_r.
     - by rewrite and_elim_r wand_elim_r. }
-  rewrite (into_sep P) HQ. destruct lr; simpl.
+  rewrite (into_sep P) HQ. destruct d; simpl.
   - by rewrite (comm _ P1) -assoc wand_elim_r sep_elim_r.
   - by rewrite -assoc wand_elim_r sep_elim_r.
 Qed.

theories/proofmode/environments.v

 From stdpp Require Export strings.
-From iris.proofmode Require Import strings.
+From iris.proofmode Require Import base.
 From iris.algebra Require Export base.
 From stdpp Require Import stringmap.
 Set Default Proof Using "Type".

theories/proofmode/intro_patterns.v

 From stdpp Require Export strings.
-From iris.proofmode Require Import tokens sel_patterns.
+From iris.proofmode Require Import base tokens sel_patterns.
 Set Default Proof Using "Type".
 
 Inductive intro_pat :=
@@ -11,6 +11,7 @@ Inductive intro_pat :=
   | IPureElim : intro_pat
   | IAlwaysElim : intro_pat → intro_pat
   | IModalElim : intro_pat → intro_pat
+  | IRewrite : direction → intro_pat
   | IPureIntro : intro_pat
   | IAlwaysIntro : intro_pat
   | IModalIntro : intro_pat
@@ -84,6 +85,7 @@ Fixpoint parse_go (ts : list token) (k : stack) : option stack :=
   | TPure :: ts => parse_go ts (SPat IPureElim :: k)
   | TAlways :: ts => parse_go ts (SAlwaysElim :: k)
   | TModal :: ts => parse_go ts (SModalElim :: k)
+  | TArrow d :: ts => parse_go ts (SPat (IRewrite d) :: k)
   | TPureIntro :: ts => parse_go ts (SPat IPureIntro :: k)
   | TAlwaysIntro :: ts => parse_go ts (SPat IAlwaysIntro :: k)
   | TModalIntro :: ts => parse_go ts (SPat IModalIntro :: k)

theories/proofmode/sel_patterns.v

 From stdpp Require Export strings.
-From iris.proofmode Require Import tokens.
+From iris.proofmode Require Import base tokens.
 Set Default Proof Using "Type".
 
 Inductive sel_pat :=

theories/proofmode/tactics.v

 From iris.proofmode Require Import coq_tactics.
-From iris.proofmode Require Import intro_patterns spec_patterns sel_patterns.
+From iris.proofmode Require Import base intro_patterns spec_patterns sel_patterns.
 From iris.bi Require Export bi.
 From iris.proofmode Require Export classes notation.
 From iris.proofmode Require Import class_instances.
 From stdpp Require Import stringmap hlist.
-From iris.proofmode Require Import strings.
 Set Default Proof Using "Type".
 
 Declare Reduction env_cbv := cbv [
@@ -748,7 +747,7 @@ Tactic Notation "iSplit" :=
 Tactic Notation "iSplitL" constr(Hs) :=
   iStartProof;
   let Hs := words Hs in
-  eapply tac_sep_split with _ _ false Hs _ _; (* (js:=Hs) *)
+  eapply tac_sep_split with _ _ Left Hs _ _; (* (js:=Hs) *)
     [apply _ ||
      let P := match goal with |- FromSep _ ?P _ _ => P end in
      fail "iSplitL:" P "not a separating conjunction"
@@ -760,7 +759,7 @@ Tactic Notation "iSplitL" constr(Hs) :=
 Tactic Notation "iSplitR" constr(Hs) :=
   iStartProof;
   let Hs := words Hs in
-  eapply tac_sep_split with _ _ true Hs _ _; (* (js:=Hs) *)
+  eapply tac_sep_split with _ _ Right Hs _ _; (* (js:=Hs) *)
     [apply _ ||
      let P := match goal with |- FromSep _ ?P _ _ => P end in
      fail "iSplitR:" P "not a separating conjunction"
@@ -784,8 +783,8 @@ Local Tactic Notation "iAndDestruct" constr(H) "as" constr(H1) constr(H2) :=
      fail "iAndDestruct: cannot destruct" P
     |env_reflexivity || fail "iAndDestruct:" H1 "or" H2 " not fresh"|].
 
-Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(lr) constr(H') :=
-  eapply tac_and_destruct_choice with _ H _ lr H' _ _ _;
+Local Tactic Notation "iAndDestructChoice" constr(H) "as" constr(d) constr(H') :=
+  eapply tac_and_destruct_choice with _ H _ d H' _ _ _;
     [env_reflexivity || fail "iAndDestructChoice:" H "not found"
     |env_cbv; apply _ ||
      let P := match goal with |- TCOr (IntoAnd _ ?P _ _) _ => P end in
@@ -898,12 +897,14 @@ Tactic Notation "iDestructHyp" constr(H) "as" constr(pat) :=
     | IFrame => iFrame Hz
     | IName ?y => iRename Hz into y
     | IList [[]] => iExFalso; iExact Hz
-    | IList [[?pat1; IDrop]] => iAndDestructChoice Hz as true Hz; go Hz pat1
-    | IList [[IDrop; ?pat2]] => iAndDestructChoice Hz as false Hz; go Hz pat2
+    | IList [[?pat1; IDrop]] => iAndDestructChoice Hz as Left Hz; go Hz pat1
+    | IList [[IDrop; ?pat2]] => iAndDestructChoice Hz as Right Hz; go Hz pat2
     | IList [[?pat1; ?pat2]] =>
        let Hy := iFresh in iAndDestruct Hz as Hz Hy; go Hz pat1; go Hy pat2
     | IList [[?pat1];[?pat2]] => iOrDestruct Hz as Hz Hz; [go Hz pat1|go Hz pat2]
     | IPureElim => iPure Hz as ?
+    | IRewrite Right => iPure Hz as ->
+    | IRewrite Left => iPure Hz as <-
     | IAlwaysElim ?pat => iPersistent Hz; go Hz pat
     | IModalElim ?pat => iModCore Hz; go Hz pat
     | _ => fail "iDestruct:" pat "invalid"
@@ -1689,8 +1690,8 @@ Local Tactic Notation "iRewriteCore" constr(lr) open_constr(lem) :=
       |iRewriteFindPred
       |intros ??? ->; reflexivity|lazy beta; iClear Heq]).
 
-Tactic Notation "iRewrite" open_constr(lem) := iRewriteCore false lem.
-Tactic Notation "iRewrite" "-" open_constr(lem) := iRewriteCore true lem.
+Tactic Notation "iRewrite" open_constr(lem) := iRewriteCore Right lem.
+Tactic Notation "iRewrite" "-" open_constr(lem) := iRewriteCore Left lem.
 
 Local Tactic Notation "iRewriteCore" constr(lr) open_constr(lem) "in" constr(H) :=
   iPoseProofCore lem as true true (fun Heq =>
@@ -1705,9 +1706,9 @@ Local Tactic Notation "iRewriteCore" constr(lr) open_constr(lem) "in" constr(H)
       |env_reflexivity|lazy beta; iClear Heq]).
 
 Tactic Notation "iRewrite" open_constr(lem) "in" constr(H) :=
-  iRewriteCore false lem in H.
+  iRewriteCore Right lem in H.
 Tactic Notation "iRewrite" "-" open_constr(lem) "in" constr(H) :=
-  iRewriteCore true lem in H.
+  iRewriteCore Left lem in H.
 
 Ltac iSimplifyEq := repeat (
   iMatchHyp (fun H P => match P with ⌜_ = _⌝%I => iDestruct H as %? end)

theories/proofmode/tokens.v

-From stdpp Require Export strings.
+From iris.proofmode Require Import base.
 Set Default Proof Using "Type".
 
 Inductive token :=
@@ -25,7 +25,8 @@ Inductive token :=
   | TForall : token
   | TAll : token
   | TMinus : token
-  | TSep : token.
+  | TSep : token
+  | TArrow : direction → token.
 
 Inductive state :=
   | SName : string → state
@@ -64,6 +65,8 @@ Fixpoint tokenize_go (s : string) (k : list token) (kn : state) : list token :=
   | String "/" (String "=" s) => tokenize_go s (TSimpl :: cons_state kn k) SNone
   | String "*" (String "*" s) => tokenize_go s (TAll :: cons_state kn k) SNone
   | String "*" s => tokenize_go s (TForall :: cons_state kn k) SNone
+  | String "-" (String ">" s) => tokenize_go s (TArrow Right :: cons_state kn k) SNone
+  | String "<" (String "-" s) => tokenize_go s (TArrow Left :: cons_state kn k) SNone
   | String "-" s => tokenize_go s (TMinus :: cons_state kn k) SNone
   | String (Ascii.Ascii false true false false false true true true) (* unicode ∗ *)
       (String (Ascii.Ascii false false false true false false false true)

theories/tests/proofmode.v

@@ -235,6 +235,10 @@ Proof.
   iIntros ([]).
 Qed.
 
+Lemma test_iIntros_rewrite P (x1 x2 x3 x4 : nat) :
+  x1 = x2 → (⌜ x2 = x3 ⌝ ∗ ⌜ x3 ≡ x4 ⌝ ∗ P) -∗ ⌜ x1 = x4 ⌝ ∗ P.
+Proof. iIntros (?) "(-> & -> & $)"; auto. Qed.
+
 Lemma test_iNext_affine P Q :
   bi_affinely (▷ (Q ≡ P)) -∗ bi_affinely (▷ Q) -∗ bi_affinely (▷ P).
 Proof. iIntros "#HPQ HQ !#". iNext. by iRewrite "HPQ" in "HQ". Qed.
@@ -242,4 +246,5 @@ Proof. iIntros "#HPQ HQ !#". iNext. by iRewrite "HPQ" in "HQ". Qed.
 Lemma test_iAlways P Q R :
   □ P -∗ bi_persistently Q → R -∗ bi_persistently (bi_affinely (bi_affinely P)) ∗ □ Q.
 Proof. iIntros "#HP #HQ HR". iSplitL. iAlways. done. iAlways. done. Qed.
+
 End tests.