Use checkmark for valid.

iris/agree.v

@@ -19,7 +19,7 @@ Global Instance agree_validN : ValidN (agree A) := λ n x,
 Lemma agree_valid_le (x : agree A) n n' :
   agree_is_valid x n → n' ≤ n → agree_is_valid x n'.
 Proof. induction 2; eauto using agree_valid_S. Qed.
-Global Instance agree_valid : Valid (agree A) := λ x, ∀ n, validN n x.
+Global Instance agree_valid : Valid (agree A) := λ x, ∀ n, ✓{n} x.
 Global Instance agree_equiv : Equiv (agree A) := λ x y,
   (∀ n, agree_is_valid x n ↔ agree_is_valid y n) ∧
   (∀ n, agree_is_valid x n → x n ={n}= y n).
@@ -100,7 +100,7 @@ Proof.
   * by intros x y n; rewrite agree_includedN.
 Qed.
 Lemma agree_op_inv (x y1 y2 : agree A) n :
-  validN n x → x ={n}= y1 ⋅ y2 → y1 ={n}= y2.
+  ✓{n} x → x ={n}= y1 ⋅ y2 → y1 ={n}= y2.
 Proof. by intros [??] Hxy; apply Hxy. Qed.
 Global Instance agree_extend : CMRAExtend (agree A).
 Proof.
@@ -113,12 +113,12 @@ Program Definition to_agree (x : A) : agree A :=
 Solve Obligations with done.
 Global Instance to_agree_ne n : Proper (dist n ==> dist n) to_agree.
 Proof. intros x1 x2 Hx; split; naive_solver eauto using @dist_le. Qed.
-Lemma agree_car_ne (x y : agree A) n : validN n x → x ={n}= y → x n ={n}= y n.
+Lemma agree_car_ne (x y : agree A) n : ✓{n} x → x ={n}= y → x n ={n}= y n.
 Proof. by intros [??] Hxy; apply Hxy. Qed.
-Lemma agree_cauchy (x : agree A) n i : n ≤ i → validN i x → x n ={n}= x i.
+Lemma agree_cauchy (x : agree A) n i : n ≤ i → ✓{i} x → x n ={n}= x i.
 Proof. by intros ? [? Hx]; apply Hx. Qed.
 Lemma agree_to_agree_inj (x y : agree A) a n :
-  validN n x → x ={n}= to_agree a ⋅ y → x n ={n}= a.
+  ✓{n} x → x ={n}= to_agree a ⋅ y → x n ={n}= a.
 Proof.
   by intros; transitivity ((to_agree a ⋅ y) n); [by apply agree_car_ne|].
 Qed.

iris/auth.v

@@ -45,15 +45,15 @@ Qed.
 Instance auth_empty `{Empty A} : Empty (auth A) := Auth ∅ ∅.
 Instance auth_valid `{Equiv A, Valid A, Op A} : Valid (auth A) := λ x,
   match authoritative x with
-  | Excl a => own x ≼ a ∧ valid a
-  | ExclUnit => valid (own x)
+  | Excl a => own x ≼ a ∧ ✓ a
+  | ExclUnit => ✓ (own x)
   | ExclBot => False
   end.
 Arguments auth_valid _ _ _ _ !_ /.
 Instance auth_validN `{Dist A, ValidN A, Op A} : ValidN (auth A) := λ n x,
   match authoritative x with
-  | Excl a => own x ≼{n} a ∧ validN n a
-  | ExclUnit => validN n (own x)
+  | Excl a => own x ≼{n} a ∧ ✓{n} a
+  | ExclUnit => ✓{n} (own x)
   | ExclBot => n = 0
   end.
 Arguments auth_validN _ _ _ _ _ !_ /.
@@ -76,9 +76,9 @@ Proof.
   intros [[z1 Hz1] [z2 Hz2]]; exists (Auth z1 z2); split; auto.
 Qed.
 Lemma authoritative_validN `{CMRA A} n (x : auth A) :
-  validN n x → validN n (authoritative x).
+  ✓{n} x → ✓{n} (authoritative x).
 Proof. by destruct x as [[]]. Qed.
-Lemma own_validN `{CMRA A} n (x : auth A) : validN n x → validN n (own x).
+Lemma own_validN `{CMRA A} n (x : auth A) : ✓{n} x → ✓{n} (own x).
 Proof. destruct x as [[]]; naive_solver eauto using cmra_valid_includedN. Qed.
 Instance auth_cmra `{CMRA A} : CMRA (auth A).
 Proof.

iris/cmra.v

@@ -2,6 +2,7 @@ Require Export iris.ra iris.cofe.
 
 Class ValidN (A : Type) := validN : nat → A → Prop.
 Instance: Params (@validN) 3.
+Notation "✓{ n }" := (validN n) (at level 1, format "✓{ n }").
 
 Definition includedN `{Dist A, Op A} (n : nat) (x y : A) := ∃ z, y ={n}= x ⋅ z.
 Notation "x ≼{ n } y" := (includedN n x y)
@@ -14,32 +15,32 @@ Class CMRA A `{Equiv A, Compl A, Unit A, Op A, Valid A, ValidN A, Minus A} := {
   cmra_cofe :> Cofe A;
   cmra_op_ne n x :> Proper (dist n ==> dist n) (op x);
   cmra_unit_ne n :> Proper (dist n ==> dist n) unit;
-  cmra_valid_ne n :> Proper (dist n ==> impl) (validN n);
+  cmra_valid_ne n :> Proper (dist n ==> impl) (✓{n});
   cmra_minus_ne n :> Proper (dist n ==> dist n ==> dist n) minus;
   (* valid *)
-  cmra_valid_0 x : validN 0 x;
-  cmra_valid_S n x : validN (S n) x → validN n x;
-  cmra_valid_validN x : valid x ↔ ∀ n, validN n x;
+  cmra_valid_0 x : ✓{0} x;
+  cmra_valid_S n x : ✓{S n} x → ✓{n} x;
+  cmra_valid_validN x : ✓ x ↔ ∀ n, ✓{n} x;
   (* monoid *)
   cmra_associative : Associative (≡) (⋅);
   cmra_commutative : Commutative (≡) (⋅);
   cmra_unit_l x : unit x ⋅ x ≡ x;
   cmra_unit_idempotent x : unit (unit x) ≡ unit x;
   cmra_unit_preserving n x y : x ≼{n} y → unit x ≼{n} unit y;
-  cmra_valid_op_l n x y : validN n (x ⋅ y) → validN n x;
+  cmra_valid_op_l n x y : ✓{n} (x ⋅ y) → ✓{n} x;
   cmra_op_minus n x y : x ≼{n} y → x ⋅ y ⩪ x ={n}= y
 }.
 Class CMRAExtend A `{Equiv A, Dist A, Op A, ValidN A} :=
   cmra_extend_op n x y1 y2 :
-    validN n x → x ={n}= y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z ={n}= (y1,y2) }.
+    ✓{n} x → x ={n}= y1 ⋅ y2 → { z | x ≡ z.1 ⋅ z.2 ∧ z ={n}= (y1,y2) }.
 
 Class CMRAMonotone
     `{Dist A, Op A, ValidN A, Dist B, Op B, ValidN B} (f : A → B) := {
   includedN_preserving n x y : x ≼{n} y → f x ≼{n} f y;
-  validN_preserving n x : validN n x → validN n (f x)
+  validN_preserving n x : ✓{n} x → ✓{n} (f x)
 }.
 
-Hint Extern 0 (validN 0 _) => apply cmra_valid_0.
+Hint Extern 0 (✓{0} _) => apply cmra_valid_0.
 
 (** Bundeled version *)
 Structure cmraT := CMRAT {
@@ -73,12 +74,12 @@ Coercion cmra_cofeC (A : cmraT) : cofeT := CofeT A.
 Canonical Structure cmra_cofeC.
 
 (** Updates *)
-Definition cmra_updateP `{Op A, ValidN A} (x : A) (P : A → Prop) :=
-  ∀ z n, validN n (x ⋅ z) → ∃ y, P y ∧ validN n (y ⋅ z).
+Definition cmra_updateP `{Op A, ValidN A} (x : A) (P : A → Prop) := ∀ z n,
+  ✓{n} (x ⋅ z) → ∃ y, P y ∧ ✓{n} (y ⋅ z).
 Instance: Params (@cmra_updateP) 3.
 Infix "⇝:" := cmra_updateP (at level 70).
 Definition cmra_update `{Op A, ValidN A} (x y : A) := ∀ z n,
-  validN n (x ⋅ z) → validN n (y ⋅ z).
+  ✓{n} (x ⋅ z) → ✓{n} (y ⋅ z).
 Infix "⇝" := cmra_update (at level 70).
 Instance: Params (@cmra_update) 3.
 
@@ -94,9 +95,9 @@ Proof.
   symmetry; apply cmra_op_minus, Hxy.
 Qed.
 
-Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (validN n) | 1.
+Global Instance cmra_valid_ne' : Proper (dist n ==> iff) (✓{n}) | 1.
 Proof. by split; apply cmra_valid_ne. Qed.
-Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (validN n) | 1.
+Global Instance cmra_valid_proper : Proper ((≡) ==> iff) (✓{n}) | 1.
 Proof. by intros n x1 x2 Hx; apply cmra_valid_ne', equiv_dist. Qed.
 Global Instance cmra_ra : RA A.
 Proof.
@@ -110,19 +111,19 @@ Proof.
   * intros x y [z Hz]; apply equiv_dist; intros n.
     by apply cmra_op_minus; exists z; rewrite Hz.
 Qed.
-Lemma cmra_valid_op_r x y n : validN n (x ⋅ y) → validN n y.
+Lemma cmra_valid_op_r x y n : ✓{n} (x ⋅ y) → ✓{n} y.
 Proof. rewrite (commutative _ x); apply cmra_valid_op_l. Qed.
-Lemma cmra_valid_le x n n' : validN n x → n' ≤ n → validN n' x.
+Lemma cmra_valid_le x n n' : ✓{n} x → n' ≤ n → ✓{n'} x.
 Proof. induction 2; eauto using cmra_valid_S. Qed.
 Global Instance ra_op_ne n : Proper (dist n ==> dist n ==> dist n) (⋅).
 Proof.
   intros x1 x2 Hx y1 y2 Hy.
   by rewrite Hy, (commutative _ x1), Hx, (commutative _ y2).
 Qed.
-Lemma cmra_unit_valid x n : validN n x → validN n (unit x).
+Lemma cmra_unit_valid x n : ✓{n} x → ✓{n} (unit x).
 Proof. rewrite <-(cmra_unit_l x) at 1; apply cmra_valid_op_l. Qed.
 Lemma cmra_op_timeless `{!CMRAExtend A} x1 x2 :
-  validN 1 (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2).
+  ✓{1} (x1 ⋅ x2) → Timeless x1 → Timeless x2 → Timeless (x1 ⋅ x2).
 Proof.
   intros ??? z Hz.
   destruct (cmra_extend_op 1 z x1 x2) as ([y1 y2]&Hz'&?&?); auto; simpl in *.
@@ -159,9 +160,9 @@ Proof.
   * intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2).
     by rewrite (associative _), <-Hy, <-Hz.
 Qed.
-Lemma cmra_valid_includedN x y n : validN n y → x ≼{n} y → validN n x.
+Lemma cmra_valid_includedN x y n : ✓{n} y → x ≼{n} y → ✓{n} x.
 Proof. intros Hyv [z Hy]; rewrite Hy in Hyv; eauto using cmra_valid_op_l. Qed.
-Lemma cmra_valid_included x y n : validN n y → x ≼ y → validN n x.
+Lemma cmra_valid_included x y n : ✓{n} y → x ≼ y → ✓{n} x.
 Proof. rewrite cmra_included_includedN; eauto using cmra_valid_includedN. Qed.
 Lemma cmra_included_dist_l x1 x2 x1' n :
   x1 ≼ x2 → x1' ={n}= x1 → ∃ x2', x1' ≼ x2' ∧ x2' ={n}= x2.
@@ -203,7 +204,7 @@ Section discrete.
   Existing Instances discrete_dist discrete_compl.
   Let discrete_cofe' : Cofe A := discrete_cofe.
   Instance discrete_validN : ValidN A := λ n x,
-    match n with 0 => True | S n => valid x end.
+    match n with 0 => True | S n => ✓ x end.
   Instance discrete_cmra : CMRA A.
   Proof.
     split; try by (destruct ra; auto).
@@ -224,13 +225,12 @@ Section discrete.
   Qed.
   Definition discreteRA : cmraT := CMRAT A.
   Lemma discrete_updateP (x : A) (P : A → Prop) `{!Inhabited (sig P)} :
-    (∀ z, valid (x ⋅ z) → ∃ y, P y ∧ valid (y ⋅ z)) → x ⇝: P.
+    (∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z)) → x ⇝: P.
   Proof.
     intros Hvalid z [|n]; [|apply Hvalid].
     by destruct (_ : Inhabited (sig P)) as [[y ?]]; exists y.
   Qed.
-  Lemma discrete_update (x y : A) :
-    (∀ z, valid (x ⋅ z) → valid (y ⋅ z)) → x ⇝ y.
+  Lemma discrete_update (x y : A) : (∀ z, ✓ (x ⋅ z) → ✓ (y ⋅ z)) → x ⇝ y.
   Proof. intros Hvalid z [|n]; [done|apply Hvalid]. Qed.
 End discrete.
 Arguments discreteRA _ {_ _ _ _ _ _}.
@@ -241,9 +241,9 @@ Instance prod_empty `{Empty A, Empty B} : Empty (A * B) := (∅, ∅).
 Instance prod_unit `{Unit A, Unit B} : Unit (A * B) := λ x,
   (unit (x.1), unit (x.2)).
 Instance prod_valid `{Valid A, Valid B} : Valid (A * B) := λ x,
-  valid (x.1) ∧ valid (x.2).
+  ✓ (x.1) ∧ ✓ (x.2).
 Instance prod_validN `{ValidN A, ValidN B} : ValidN (A * B) := λ n x,
-  validN n (x.1) ∧ validN n (x.2).
+  ✓{n} (x.1) ∧ ✓{n} (x.2).
 Instance prod_minus `{Minus A, Minus B} : Minus (A * B) := λ x y,
   (x.1 ⩪ y.1, x.2 ⩪ y.2).
 Lemma prod_included `{Equiv A, Equiv B, Op A, Op B} (x y : A * B) :

iris/cmra_maps.v

@@ -4,9 +4,9 @@ Require Import prelude.stringmap prelude.natmap.
 
 (** option *)
 Instance option_valid `{Valid A} : Valid (option A) := λ mx,
-  match mx with Some x => valid x | None => True end.
+  match mx with Some x => ✓ x | None => True end.
 Instance option_validN `{ValidN A} : ValidN (option A) := λ n mx,
-  match mx with Some x => validN n x | None => True end.
+  match mx with Some x => ✓{n} x | None => True end.
 Instance option_unit `{Unit A} : Unit (option A) := fmap unit.
 Instance option_op `{Op A} : Op (option A) := union_with (λ x y, Some (x ⋅ y)).
 Instance option_minus `{Minus A} : Minus (option A) :=
@@ -34,7 +34,7 @@ Proof.
   * by destruct 1; constructor; apply (_ : Proper (dist n ==> _) _).
   * destruct 1 as [[?|] [?|]| |]; unfold validN, option_validN; simpl;
      intros ?; auto using cmra_valid_0;
-     eapply (_ : Proper (dist _ ==> impl) (validN _)); eauto.
+     eapply (_ : Proper (dist _ ==> impl) (✓{_})); eauto.
   * by destruct 1; inversion_clear 1; constructor;
       repeat apply (_ : Proper (dist _ ==> _ ==> _) _).
   * intros [x|]; unfold validN, option_validN; auto using cmra_valid_0.
@@ -82,8 +82,8 @@ Section map.
   Existing Instances map_dist map_compl map_cofe.
   Instance map_op `{Op A} : Op (M A) := merge op.
   Instance map_unit `{Unit A} : Unit (M A) := fmap unit.
-  Instance map_valid `{Valid A} : Valid (M A) := λ m, ∀ i, valid (m !! i).
-  Instance map_validN `{ValidN A} : ValidN (M A) := λ n m, ∀ i, validN n (m!!i).
+  Instance map_valid `{Valid A} : Valid (M A) := λ m, ∀ i, ✓ (m !! i).
+  Instance map_validN `{ValidN A} : ValidN (M A) := λ n m, ∀ i, ✓{n} (m!!i).
   Instance map_minus `{Minus A} : Minus (M A) := merge minus.
   Lemma lookup_op `{Op A} m1 m2 i : (m1 ⋅ m2) !! i = m1 !! i ⋅ m2 !! i.
   Proof. by apply lookup_merge. Qed.
@@ -116,7 +116,7 @@ Section map.
     * by intros n m1 m2 Hm ? i; rewrite <-(Hm i).
     * intros n m1 m1' Hm1 m2 m2' Hm2 i.
       by rewrite !lookup_minus, (Hm1 i), (Hm2 i).
-    * intros m i; apply cmra_valid_0.
+    * by intros m i.
     * intros n m Hm i; apply cmra_valid_S, Hm.
     * intros m; split; [by intros Hm n i; apply cmra_valid_validN|].
       intros Hm i; apply cmra_valid_validN; intros n; apply Hm.

iris/dra.v

@@ -26,13 +26,11 @@ Proof.
     split; [|intros; transitivity y]; tauto.
 Qed.
 Instance validity_valid_proper `{Equiv A} (P : A → Prop) :
-  Proper ((≡) ==> iff) (valid : validity P → Prop).
+  Proper ((≡) ==> iff) (✓ : validity P → Prop).
 Proof. intros ?? [??]; naive_solver. Qed.
 
-Local Notation V := valid.
-
 Definition dra_included `{Equiv A, Valid A, Disjoint A, Op A} := λ x y,
-  ∃ z, y ≡ x ⋅ z ∧ V z ∧ x ⊥ z.
+  ∃ z, y ≡ x ⋅ z ∧ ✓ z ∧ x ⊥ z.
 Instance: Params (@dra_included) 4.
 Local Infix "≼" := dra_included.
 
@@ -44,21 +42,21 @@ Class DRA A `{Equiv A, Valid A, Unit A, Disjoint A, Op A, Minus A} := {
   dra_disjoint_proper :> ∀ x, Proper ((≡) ==> impl) (disjoint x);
   dra_minus_proper :> Proper ((≡) ==> (≡) ==> (≡)) minus;
   (* validity *)
-  dra_op_valid x y : V x → V y → x ⊥ y → V (x ⋅ y);
-  dra_unit_valid x : V x → V (unit x);
-  dra_minus_valid x y : V x → V y → x ≼ y → V (y ⩪ x);
+  dra_op_valid x y : ✓ x → ✓ y → x ⊥ y → ✓ (x ⋅ y);
+  dra_unit_valid x : ✓ x → ✓ (unit x);
+  dra_minus_valid x y : ✓ x → ✓ y → x ≼ y → ✓ (y ⩪ x);
   (* monoid *)
   dra_associative :> Associative (≡) (⋅);
-  dra_disjoint_ll x y z : V x → V y → V z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z;
-  dra_disjoint_move_l x y z : V x → V y → V z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z;
+  dra_disjoint_ll x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ z;
+  dra_disjoint_move_l x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → x ⊥ y ⋅ z;
   dra_symmetric :> Symmetric (@disjoint A _);
-  dra_commutative x y : V x → V y → x ⊥ y → x ⋅ y ≡ y ⋅ x;
-  dra_unit_disjoint_l x : V x → unit x ⊥ x;
-  dra_unit_l x : V x → unit x ⋅ x ≡ x;
-  dra_unit_idempotent x : V x → unit (unit x) ≡ unit x;
-  dra_unit_preserving x y : V x → V y → x ≼ y → unit x ≼ unit y;
-  dra_disjoint_minus x y : V x → V y → x ≼ y → x ⊥ y ⩪ x;
-  dra_op_minus x y : V x → V y → x ≼ y → x ⋅ y ⩪ x ≡ y
+  dra_commutative x y : ✓ x → ✓ y → x ⊥ y → x ⋅ y ≡ y ⋅ x;
+  dra_unit_disjoint_l x : ✓ x → unit x ⊥ x;
+  dra_unit_l x : ✓ x → unit x ⋅ x ≡ x;
+  dra_unit_idempotent x : ✓ x → unit (unit x) ≡ unit x;
+  dra_unit_preserving x y : ✓ x → ✓ y → x ≼ y → unit x ≼ unit y;
+  dra_disjoint_minus x y : ✓ x → ✓ y → x ≼ y → x ⊥ y ⩪ x;
+  dra_op_minus x y : ✓ x → ✓ y → x ≼ y → x ⋅ y ⩪ x ≡ y
 }.
 
 Section dra.
@@ -72,14 +70,14 @@ Proof.
   * by rewrite Hy, (symmetry_iff (⊥) x1), (symmetry_iff (⊥) x2), Hx.
   * by rewrite <-Hy, (symmetry_iff (⊥) x2), (symmetry_iff (⊥) x1), <-Hx.
 Qed.
-Lemma dra_disjoint_rl x y z : V x → V y → V z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y.
+Lemma dra_disjoint_rl x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y.
 Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed.
-Lemma dra_disjoint_lr x y z : V x → V y → V z → x ⊥ y → x ⋅ y ⊥ z → y ⊥ z.
+Lemma dra_disjoint_lr x y z : ✓ x → ✓ y → ✓ z → x ⊥ y → x ⋅ y ⊥ z → y ⊥ z.
 Proof.
   intros ????. rewrite dra_commutative by done. by apply dra_disjoint_ll.
 Qed.
 Lemma dra_disjoint_move_r x y z :
-  V x → V y → V z → y ⊥ z → x ⊥ y ⋅ z → x ⋅ y ⊥ z.
+  ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⋅ y ⊥ z.
 Proof.
   intros; symmetry; rewrite dra_commutative by eauto using dra_disjoint_rl.
   apply dra_disjoint_move_l; auto; by rewrite dra_commutative.
@@ -87,20 +85,20 @@ Qed.
 Hint Immediate dra_disjoint_move_l dra_disjoint_move_r.
 Hint Unfold dra_included.
 
-Notation T := (validity (valid : A → Prop)).
-Lemma validity_valid_car_valid (z : T) : V z → V (validity_car z).
+Notation T := (validity (✓ : A → Prop)).
+Lemma validity_valid_car_valid (z : T) : ✓ z → ✓ (validity_car z).
 Proof. apply validity_prf. Qed.
 Hint Resolve validity_valid_car_valid.
 Program Instance validity_unit : Unit T := λ x,
-  Validity (unit (validity_car x)) (V x) _.
+  Validity (unit (validity_car x)) (✓ x) _.
 Solve Obligations with naive_solver auto using dra_unit_valid.
 Program Instance validity_op : Op T := λ x y,
   Validity (validity_car x ⋅ validity_car y)
-           (V x ∧ V y ∧ validity_car x ⊥ validity_car y) _.
+           (✓ x ∧ ✓ y ∧ validity_car x ⊥ validity_car y) _.
 Solve Obligations with naive_solver auto using dra_op_valid.
 Program Instance validity_minus : Minus T := λ x y,
   Validity (validity_car x ⩪ validity_car y)
-           (V x ∧ V y ∧ validity_car y ≼ validity_car x) _.
+           (✓ x ∧ ✓ y ∧ validity_car y ≼ validity_car x) _.
 Solve Obligations with naive_solver auto using dra_minus_valid.
 Instance validity_ra : RA T.
 Proof.
@@ -135,7 +133,7 @@ Proof.
 Qed.
 Definition validityRA : cmraT := discreteRA T.
 Definition validity_update (x y : validityRA) :
-  (∀ z, V x → V z → validity_car x ⊥ z → V y ∧ validity_car y ⊥ z) → x ⇝ y.
+  (∀ z, ✓ x → ✓ z → validity_car x ⊥ z → ✓ y ∧ validity_car y ⊥ z) → x ⇝ y.
 Proof.
   intros Hxy; apply discrete_update.
   intros z (?&?&?); split_ands'; try eapply Hxy; eauto.

iris/excl.v

@@ -114,7 +114,7 @@ Proof.
 Qed.
 
 (* Updates *)
-Lemma excl_update {A} (x : A) y : valid y → Excl x ⇝ y.
+Lemma excl_update {A} (x : A) y : ✓ y → Excl x ⇝ y.
 Proof. by destruct y; intros ? [?| |]. Qed.
 
 (* Functor *)

iris/logic.v

@@ -8,7 +8,7 @@ Structure uPred (M : cmraT) : Type := IProp {
   uPred_ne x1 x2 n : uPred_holds n x1 → x1 ={n}= x2 → uPred_holds n x2;
   uPred_0 x : uPred_holds 0 x;
   uPred_weaken x1 x2 n1 n2 :
-    uPred_holds n1 x1 → x1 ≼ x2 → n2 ≤ n1 → validN n2 x2 → uPred_holds n2 x2
+    uPred_holds n1 x1 → x1 ≼ x2 → n2 ≤ n1 → ✓{n2} x2 → uPred_holds n2 x2
 }.
 Arguments uPred_holds {_} _ _ _.
 Hint Resolve uPred_0.
@@ -16,9 +16,9 @@ Add Printing Constructor uPred.
 Instance: Params (@uPred_holds) 3.
 
 Instance uPred_equiv (M : cmraT) : Equiv (uPred M) := λ P Q, ∀ x n,
-  validN n x → P n x ↔ Q n x.
+  ✓{n} x → P n x ↔ Q n x.
 Instance uPred_dist (M : cmraT) : Dist (uPred M) := λ n P Q, ∀ x n',
-  n' ≤ n → validN n' x → P n' x ↔ Q n' x.
+  n' ≤ n → ✓{n'} x → P n' x ↔ Q n' x.
 Program Instance uPred_compl (M : cmraT) : Compl (uPred M) := λ c,
   {| uPred_holds n x := c n n x |}.
 Next Obligation. by intros M c x y n ??; simpl in *; apply uPred_ne with x. Qed.
@@ -72,7 +72,7 @@ Qed.
 
 (** logical entailement *)
 Instance uPred_entails {M} : SubsetEq (uPred M) := λ P Q, ∀ x n,
-  validN n x → P n x → Q n x.
+  ✓{n} x → P n x → Q n x.
 
 (** logical connectives *)
 Program Definition uPred_const {M} (P : Prop) : uPred M :=
@@ -89,12 +89,12 @@ Program Definition uPred_or {M} (P Q : uPred M) : uPred M :=
 Solve Obligations with naive_solver eauto 2 using uPred_ne, uPred_weaken.
 Program Definition uPred_impl {M} (P Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∀ x' n',
-       x ≼ x' → n' ≤ n → validN n' x' → P n' x' → Q n' x' |}.
+       x ≼ x' → n' ≤ n → ✓{n'} x' → P n' x' → Q n' x' |}.
 Next Obligation.
   intros M P Q x1' x1 n1 HPQ Hx1 x2 n2 ????.
   destruct (cmra_included_dist_l x1 x2 x1' n1) as (x2'&?&Hx2); auto.
   assert (x2' ={n2}= x2) as Hx2' by (by apply dist_le with n1).
-  assert (validN n2 x2') by (by rewrite Hx2'); rewrite <-Hx2'.
+  assert (✓{n2} x2') by (by rewrite Hx2'); rewrite <-Hx2'.
   eauto using uPred_weaken, uPred_ne.
 Qed.
 Next Obligation. intros M P Q x1 x2 [|n]; auto with lia. Qed.
@@ -134,7 +134,7 @@ Qed.
 
 Program Definition uPred_wand {M} (P Q : uPred M) : uPred M :=
   {| uPred_holds n x := ∀ x' n',
-       n' ≤ n → validN n' (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}.
+       n' ≤ n → ✓{n'} (x ⋅ x') → P n' x' → Q n' (x ⋅ x') |}.
 Next Obligation.
   intros M P Q x1 x2 n1 HPQ Hx x3 n2 ???; simpl in *.
   rewrite <-(dist_le _ _ _ _ Hx) by done; apply HPQ; auto.
@@ -172,7 +172,7 @@ Next Obligation.
   exists (a' ⋅ x2). by rewrite (associative op), <-(dist_le _ _ _ _ Hx1), Hx.
 Qed.
 Program Definition uPred_valid {M : cmraT} (a : M) : uPred M :=
-  {| uPred_holds n x := validN n a |}.
+  {| uPred_holds n x := ✓{n} a |}.
 Solve Obligations with naive_solver eauto 2 using cmra_valid_le, cmra_valid_0.
 
 Delimit Scope uPred_scope with I.
@@ -196,11 +196,12 @@ Notation "∃ x .. y , P" :=
 Notation "▷ P" := (uPred_later P) (at level 20) : uPred_scope.
 Notation "□ P" := (uPred_always P) (at level 20) : uPred_scope.
 Infix "≡" := uPred_eq : uPred_scope.
+Notation "✓" := uPred_valid (at level 1) : uPred_scope.
 
 Definition uPred_iff {M} (P Q : uPred M) : uPred M := ((P → Q) ∧ (Q → P))%I.
 Infix "↔" := uPred_iff : uPred_scope.
 
-Class TimelessP {M} (P : uPred M) := timelessP x n : validN 1 x → P 1 x → P n x.
+Class TimelessP {M} (P : uPred M) := timelessP x n : ✓{1} x → P 1 x → P n x.
 
 Section logic.
 Context {M : cmraT}.
@@ -687,7 +688,7 @@ Proof.
 Qed.
 Lemma uPred_own_empty `{Empty M, !RAEmpty M} : True%I ⊆ uPred_own ∅.
 Proof. intros x [|n] ??; [done|]. by  exists x; rewrite (left_id _ _). Qed.
-Lemma uPred_own_valid (a : M) : uPred_own a ⊆ uPred_valid a.
+Lemma uPred_own_valid (a : M) : uPred_own a ⊆ (✓ a)%I.
 Proof.
   intros x n Hv [a' Hx]; simpl; rewrite Hx in Hv; eauto using cmra_valid_op_l.
 Qed.

iris/ra.v

@@ -2,6 +2,7 @@ Require Export prelude.collections prelude.relations prelude.fin_maps.
 
 Class Valid (A : Type) := valid : A → Prop.
 Instance: Params (@valid) 2.
+Notation "✓" := valid (at level 1).
 
 Class Unit (A : Type) := unit : A → A.
 Instance: Params (@unit) 2.
@@ -26,7 +27,7 @@ Class RA A `{Equiv A, Valid A, Unit A, Op A, Minus A} : Prop := {
   ra_equivalence :> Equivalence ((≡) : relation A);
   ra_op_proper x :> Proper ((≡) ==> (≡)) (op x);
   ra_unit_proper :> Proper ((≡) ==> (≡)) unit;
-  ra_valid_proper :> Proper ((≡) ==> impl) valid;
+  ra_valid_proper :> Proper ((≡) ==> impl) ✓;
   ra_minus_proper :> Proper ((≡) ==> (≡) ==> (≡)) minus;
   (* monoid *)
   ra_associative :> Associative (≡) (⋅);
@@ -34,18 +35,18 @@ Class RA A `{Equiv A, Valid A, Unit A, Op A, Minus A} : Prop := {
   ra_unit_l x : unit x ⋅ x ≡ x;
   ra_unit_idempotent x : unit (unit x) ≡ unit x;
   ra_unit_preserving x y : x ≼ y → unit x ≼ unit y;
-  ra_valid_op_l x y : valid (x ⋅ y) → valid x;
+  ra_valid_op_l x y : ✓ (x ⋅ y) → ✓ x;
   ra_op_minus x y : x ≼ y → x ⋅ y ⩪ x ≡ y
 }.
 Class RAEmpty A `{Equiv A, Valid A, Op A, Empty A} : Prop := {
-  ra_empty_valid : valid ∅;
+  ra_empty_valid : ✓ ∅;
   ra_empty_l :> LeftId (≡) ∅ (⋅)
 }.
 
 Class RAMonotone
     `{Equiv A, Op A, Valid A, Equiv B, Op B, Valid B} (f : A → B) := {
   included_preserving x y : x ≼ y → f x ≼ f y;
-  valid_preserving x : valid x → valid (f x)
+  valid_preserving x : ✓ x → ✓ (f x)
 }.
 
 (** Big ops *)
@@ -60,11 +61,11 @@ Instance: Params (@big_opM) 4.
 
 (** Updates *)
 Definition ra_update_set `{Op A, Valid A} (x : A) (P : A → Prop) :=
-  ∀ z, valid (x ⋅ z) → ∃ y, P y ∧ valid (y ⋅ z).
+  ∀ z, ✓ (x ⋅ z) → ∃ y, P y ∧ ✓ (y ⋅ z).
 Instance: Params (@ra_update_set) 3.
 Infix "⇝:" := ra_update_set (at level 70).
 Definition ra_update `{Op A, Valid A} (x y : A) := ∀ z,
-  valid (x ⋅ z) → valid (y ⋅ z).