Better notations for raw view shifts.

algebra/upred.v

@@ -310,8 +310,12 @@ Notation "▷ P" := (uPred_later P)
   (at level 20, right associativity) : uPred_scope.
 Infix "≡" := uPred_eq : uPred_scope.
 Notation "✓ x" := (uPred_valid x) (at level 20) : uPred_scope.
-Notation "|==r> Q" := (uPred_rvs Q)
-  (at level 99, Q at level 200, format "|==r>  Q") : uPred_scope.
+Notation "|=r=> Q" := (uPred_rvs Q)
+  (at level 99, Q at level 200, format "|=r=>  Q") : uPred_scope.
+Notation "P =r=> Q" := (P ⊢ |=r=> Q)
+  (at level 99, Q at level 200, only parsing) : C_scope.
+Notation "P =r=★ Q" := (P -★ |=r=> Q)%I
+  (at level 99, Q at level 200, format "P  =r=★  Q") : uPred_scope.
 
 Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P → Q) ∧ (Q → P))%I.
 Instance: Params (@uPred_iff) 1.
@@ -1167,27 +1171,27 @@ Proof.
 Qed.
 
 (* Viewshifts *)
-Lemma rvs_intro P : P ⊢ |==r> P.
+Lemma rvs_intro P : P =r=> P.
 Proof.
   unseal. split=> n x ? HP k yf ?; exists x; split; first done.
   apply uPred_closed with n; eauto using cmra_validN_op_l.
 Qed.
-Lemma rvs_mono P Q : (P ⊢ Q) → (|==r> P) ⊢ |==r> Q.
+Lemma rvs_mono P Q : (P ⊢ Q) → (|=r=> P) =r=> Q.
 Proof.
   unseal. intros HPQ; split=> n x ? HP k yf ??.
   destruct (HP k yf) as (x'&?&?); eauto.
   exists x'; split; eauto using uPred_in_entails, cmra_validN_op_l.
 Qed.
-Lemma rvs_timeless P : TimelessP P → ▷ P ⊢ |==r> P.
+Lemma rvs_timeless P : TimelessP P → ▷ P =r=> P.
 Proof.
   unseal. rewrite timelessP_spec=> HP.
   split=> -[|n] x ? HP' k yf ??; first lia.
   exists x; split; first done.
   apply HP, uPred_closed with n; eauto using cmra_validN_le.
 Qed.
-Lemma rvs_trans P : (|==r> |==r> P) ⊢ |==r> P.
+Lemma rvs_trans P : (|=r=> |=r=> P) =r=> P.
 Proof. unseal; split; naive_solver. Qed.
-Lemma rvs_frame_r P R : (|==r> P) ★ R ⊢ |==r> P ★ R.
+Lemma rvs_frame_r P R : (|=r=> P) ★ R =r=> P ★ R.
 Proof.
   unseal; split; intros n x ? (x1&x2&Hx&HP&?) k yf ??.
   destruct (HP k (x2 ⋅ yf)) as (x'&?&?); eauto.
@@ -1197,7 +1201,7 @@ Proof.
   apply uPred_closed with n; eauto 3 using cmra_validN_op_l, cmra_validN_op_r.
 Qed.
 Lemma rvs_ownM_updateP x (Φ : M → Prop) :
-  x ~~>: Φ → uPred_ownM x ⊢ |==r> ∃ y, ■ Φ y ∧ uPred_ownM y.
+  x ~~>: Φ → uPred_ownM x =r=> ∃ y, ■ Φ y ∧ uPred_ownM y.
 Proof.
   unseal=> Hup; split=> -[|n] x2 ? [x3 Hx] [|k] yf ??; try lia.
   destruct (Hup (S k) (Some (x3 ⋅ yf))) as (y&?&?); simpl in *.
@@ -1211,17 +1215,15 @@ Global Instance rvs_mono' : Proper ((⊢) ==> (⊢)) (@uPred_rvs M).
 Proof. intros P Q; apply rvs_mono. Qed.
 Global Instance rvs_flip_mono' : Proper (flip (⊢) ==> flip (⊢)) (@uPred_rvs M).
 Proof. intros P Q; apply rvs_mono. Qed.
-Lemma rvs_strip_rvs P Q : (P ⊢ |==r> Q) → (|==r> P) ⊢ (|==r> Q).
-Proof. move=>->. by rewrite rvs_trans. Qed.
-Lemma rvs_frame_l R Q : (R ★ |==r> Q) ⊢ |==r> R ★ Q.
+Lemma rvs_frame_l R Q : (R ★ |=r=> Q) =r=> R ★ Q.
 Proof. rewrite !(comm _ R); apply rvs_frame_r. Qed.
-Lemma rvs_wand_l P Q : (P -★ Q) ★ (|==r> P) ⊢ |==r> Q.
+Lemma rvs_wand_l P Q : (P -★ Q) ★ (|=r=> P) =r=> Q.
 Proof. by rewrite rvs_frame_l wand_elim_l. Qed.
-Lemma rvs_wand_r P Q : (|==r> P) ★ (P -★ Q) ⊢ |==r> Q.
+Lemma rvs_wand_r P Q : (|=r=> P) ★ (P -★ Q) =r=> Q.
 Proof. by rewrite rvs_frame_r wand_elim_r. Qed.
-Lemma rvs_sep P Q : (|==r> P) ★ (|==r> Q) ⊢ |==r> P ★ Q.
+Lemma rvs_sep P Q : (|=r=> P) ★ (|=r=> Q) =r=> P ★ Q.
 Proof. by rewrite rvs_frame_r rvs_frame_l rvs_trans. Qed.
-Lemma rvs_ownM_update x y : x ~~> y → uPred_ownM x ⊢ |==r> uPred_ownM y.
+Lemma rvs_ownM_update x y : x ~~> y → uPred_ownM x ⊢ |=r=> uPred_ownM y.
 Proof.
   intros; rewrite (rvs_ownM_updateP _ (y =)); last by apply cmra_update_updateP.
   by apply rvs_mono, exist_elim=> y'; apply pure_elim_l=> ->.

proofmode/classes.v

@@ -17,7 +17,7 @@ Global Instance from_assumption_always_r P Q :
   FromAssumption true P Q → FromAssumption true P (□ Q).
 Proof. rewrite /FromAssumption=><-. by rewrite always_always. Qed.
 Global Instance from_assumption_rvs p P Q :
-  FromAssumption p P Q → FromAssumption p P (|==r> Q)%I.
+  FromAssumption p P Q → FromAssumption p P (|=r=> Q)%I.
 Proof. rewrite /FromAssumption=>->. apply rvs_intro. Qed.
 
 Class IntoPure (P : uPred M) (φ : Prop) := into_pure : P ⊢ ■ φ.
@@ -38,7 +38,7 @@ Global Instance from_pure_eq {A : cofeT} (a b : A) : FromPure (a ≡ b) (a ≡ b
 Proof. intros ->. apply eq_refl. Qed.
 Global Instance from_pure_valid {A : cmraT} (a : A) : FromPure (✓ a) (✓ a).
 Proof. intros ?. by apply valid_intro. Qed.
-Global Instance from_pure_rvs P φ : FromPure P φ → FromPure (|==r> P) φ.
+Global Instance from_pure_rvs P φ : FromPure P φ → FromPure (|=r=> P) φ.
 Proof. intros ??. by rewrite -rvs_intro (from_pure P). Qed.
 
 Class IntoPersistentP (P Q : uPred M) := into_persistentP : P ⊢ □ Q.
@@ -111,7 +111,7 @@ Proof. apply and_elim_r', impl_wand. Qed.
 Global Instance into_wand_always R P Q : IntoWand R P Q → IntoWand (□ R) P Q.
 Proof. rewrite /IntoWand=> ->. apply always_elim. Qed.
 Global Instance into_wand_rvs R P Q :
-  IntoWand R P Q → IntoWand R (|==r> P) (|==r> Q) | 100.
+  IntoWand R P Q → IntoWand R (|=r=> P) (|=r=> Q) | 100.
 Proof. rewrite /IntoWand=>->. apply wand_intro_l. by rewrite rvs_wand_r. Qed.
 
 Class FromAnd (P Q1 Q2 : uPred M) := from_and : Q1 ∧ Q2 ⊢ P.
@@ -144,7 +144,7 @@ Global Instance from_sep_later P Q1 Q2 :
   FromSep P Q1 Q2 → FromSep (▷ P) (▷ Q1) (▷ Q2).
 Proof. rewrite /FromSep=> <-. by rewrite later_sep. Qed.
 Global Instance from_sep_rvs P Q1 Q2 :
-  FromSep P Q1 Q2 → FromSep (|==r> P) (|==r> Q1) (|==r> Q2).
+  FromSep P Q1 Q2 → FromSep (|=r=> P) (|=r=> Q1) (|=r=> Q2).
 Proof. rewrite /FromSep=><-. apply rvs_sep. Qed.
 
 Global Instance from_sep_ownM (a b : M) :
@@ -293,7 +293,7 @@ Global Instance frame_forall {A} R (Φ Ψ : A → uPred M) :
   (∀ a, Frame R (Φ a) (Ψ a)) → Frame R (∀ x, Φ x) (∀ x, Ψ x).
 Proof. rewrite /Frame=> ?. by rewrite sep_forall_l; apply forall_mono. Qed.
 
-Global Instance frame_rvs R P Q : Frame R P Q → Frame R (|==r> P) (|==r> Q).
+Global Instance frame_rvs R P Q : Frame R P Q → Frame R (|=r=> P) (|=r=> Q).
 Proof. rewrite /Frame=><-. by rewrite rvs_frame_l. Qed.
 
 Class FromOr (P Q1 Q2 : uPred M) := from_or : Q1 ∨ Q2 ⊢ P.
@@ -301,7 +301,7 @@ Global Arguments from_or : clear implicits.
 Global Instance from_or_or P1 P2 : FromOr (P1 ∨ P2) P1 P2.
 Proof. done. Qed.
 Global Instance from_or_rvs P Q1 Q2 :
-  FromOr P Q1 Q2 → FromOr (|==r> P) (|==r> Q1) (|==r> Q2).
+  FromOr P Q1 Q2 → FromOr (|=r=> P) (|=r=> Q1) (|=r=> Q2).
 Proof. rewrite /FromOr=><-. apply or_elim; apply rvs_mono; auto with I. Qed.
 
 Class IntoOr P Q1 Q2 := into_or : P ⊢ Q1 ∨ Q2.
@@ -318,7 +318,7 @@ Global Arguments from_exist {_} _ _ {_}.
 Global Instance from_exist_exist {A} (Φ: A → uPred M): FromExist (∃ a, Φ a) Φ.
 Proof. done. Qed.
 Global Instance from_exist_rvs {A} P (Φ : A → uPred M) :
-  FromExist P Φ → FromExist (|==r> P) (λ a, |==r> Φ a)%I.
+  FromExist P Φ → FromExist (|=r=> P) (λ a, |=r=> Φ a)%I.
 Proof.
   rewrite /FromExist=><-. apply exist_elim=> a. by rewrite -(exist_intro a).
 Qed.
@@ -337,7 +337,7 @@ Proof. rewrite /IntoExist=> HP. by rewrite HP always_exist. Qed.
 
 Class TimelessElim (Q : uPred M) :=
   timeless_elim `{!TimelessP P} : ▷ P ★ (P -★ Q) ⊢ Q.
-Global Instance timeless_elim_rvs Q : TimelessElim (|==r> Q).
+Global Instance timeless_elim_rvs Q : TimelessElim (|=r=> Q).
 Proof.
   intros P ?. by rewrite (rvs_timeless P) rvs_frame_r wand_elim_r rvs_trans.
 Qed.

proofmode/coq_tactics.v

@@ -465,7 +465,7 @@ Class IntoAssert (P : uPred M) (Q : uPred M) (R : uPred M) :=
 Global Arguments into_assert _ _ _ {_}.
 Lemma into_assert_default P Q : IntoAssert P Q P.
 Proof. by rewrite /IntoAssert wand_elim_r. Qed.
-Global Instance to_assert_rvs P Q : IntoAssert P (|==r> Q) (|==r> P).
+Global Instance to_assert_rvs P Q : IntoAssert P (|=r=> Q) (|=r=> P).
 Proof. by rewrite /IntoAssert rvs_frame_r wand_elim_r rvs_trans. Qed.
 
 Lemma tac_specialize_assert Δ Δ' Δ1 Δ2' j q lr js R P1 P2 P1' Q :