change naming conventions for LocalUpdates: L is the function, Lv the validity

algebra/auth.v

@@ -157,12 +157,12 @@ Proof.
   split; [by rewrite Ha' left_id associative; apply cmra_includedN_l|done].
 Qed.
 
-Lemma auth_local_update f `{!LocalUpdate P f} a a' :
-  P a → ✓ (f a') →
-  ● a' ⋅ ◯ a ~~> ● f a' ⋅ ◯ f a.
+Lemma auth_local_update L `{!LocalUpdate Lv L} a a' :
+  Lv a → ✓ (L a') →
+  ● a' ⋅ ◯ a ~~> ● L a' ⋅ ◯ L a.
 Proof.
   intros. apply auth_update=>n af ? EQ; split; last done.
-  by rewrite EQ (local_updateN f) // -EQ.
+  by rewrite EQ (local_updateN L) // -EQ.
 Qed.
 
 Lemma auth_update_op_l a a' b :
@@ -173,15 +173,15 @@ Lemma auth_update_op_r a a' b :
 Proof. rewrite -!(commutative _ b); apply auth_update_op_l. Qed.
 
 (* This does not seem to follow from auth_local_update.
-   The trouble is that given ✓ (f a ⋅ a'), P a
+   The trouble is that given ✓ (L a ⋅ a'), Lv a
    we need ✓ (a ⋅ a'). I think this should hold for every local update,
    but adding an extra axiom to local updates just for this is silly. *)
-Lemma auth_local_update_l f `{!LocalUpdate P f} a a' :
-  P a → ✓ (f a ⋅ a') →
-  ● (a ⋅ a') ⋅ ◯ a ~~> ● (f a ⋅ a') ⋅ ◯ f a.
+Lemma auth_local_update_l L `{!LocalUpdate Lv L} a a' :
+  Lv a → ✓ (L a ⋅ a') →
+  ● (a ⋅ a') ⋅ ◯ a ~~> ● (L a ⋅ a') ⋅ ◯ L a.
 Proof.
   intros. apply auth_update=>n af ? EQ; split; last done.
-  by rewrite -(local_updateN f) // EQ -(local_updateN f) // -EQ.
+  by rewrite -(local_updateN L) // EQ -(local_updateN L) // -EQ.
 Qed.
 
 End cmra.

algebra/cmra.v

@@ -137,9 +137,9 @@ Class CMRAMonotone {A B : cmraT} (f : A → B) := {
 }.
 
 (** * Local updates *)
-Class LocalUpdate {A : cmraT} (P : A → Prop) (f : A → A) := {
-  local_update_ne n :> Proper (dist n ==> dist n) f;
-  local_updateN n x y : P x → ✓{n} (x ⋅ y) → f (x ⋅ y) ≡{n}≡ f x ⋅ y
+Class LocalUpdate {A : cmraT} (Lv : A → Prop) (L : A → A) := {
+  local_update_ne n :> Proper (dist n ==> dist n) L;
+  local_updateN n x y : Lv x → ✓{n} (x ⋅ y) → L (x ⋅ y) ≡{n}≡ L x ⋅ y
 }.
 Arguments local_updateN {_ _} _ {_} _ _ _ _ _.
 
@@ -322,13 +322,13 @@ Section identity.
 End identity.
 
 (** ** Local updates *)
-Global Instance local_update_proper P (f : A → A) :
-  LocalUpdate P f → Proper ((≡) ==> (≡)) f.
+Global Instance local_update_proper Lv (L : A → A) :
+  LocalUpdate Lv L → Proper ((≡) ==> (≡)) L.
 Proof. intros; apply (ne_proper _). Qed.
 
-Lemma local_update f `{!LocalUpdate P f} x y :
-  P x → ✓ (x ⋅ y) → f (x ⋅ y) ≡ f x ⋅ y.
-Proof. by rewrite equiv_dist=>?? n; apply (local_updateN f). Qed.
+Lemma local_update L `{!LocalUpdate Lv L} x y :
+  Lv x → ✓ (x ⋅ y) → L (x ⋅ y) ≡ L x ⋅ y.
+Proof. by rewrite equiv_dist=>?? n; apply (local_updateN L). Qed.
 
 Global Instance local_update_op x : LocalUpdate (λ _, True) (op x).
 Proof. split. apply _. by intros n y1 y2 _ _; rewrite associative. Qed.

algebra/fin_maps.v

@@ -288,14 +288,14 @@ End freshness.
 
 (* Deallocation is not a local update. The trouble is that if we own {[ i ↦ x ]},
    then the frame could always own "unit x", and prevent deallocation. *)
-Global Instance map_alter_update `{!LocalUpdate P f} i :
-  LocalUpdate (λ m, ∃ x, m !! i = Some x ∧ P x) (alter f i).
+Global Instance map_alter_update `{!LocalUpdate Lv L} i :
+  LocalUpdate (λ m, ∃ x, m !! i = Some x ∧ Lv x) (alter L i).
 Proof.
   split; first apply _.
   intros n m1 m2 (x&Hix&?) Hm j; destruct (decide (i = j)) as [->|].
   - rewrite lookup_alter !lookup_op lookup_alter Hix /=.
     move: (Hm j); rewrite lookup_op Hix.
-    case: (m2 !! j)=>[y|] //=; constructor. by apply (local_updateN f).
+    case: (m2 !! j)=>[y|] //=; constructor. by apply (local_updateN L).
   - by rewrite lookup_op !lookup_alter_ne // lookup_op.
 Qed.
 End properties.