more mask-related fancy update lemmas

theories/bi/updates.v

@@ -224,23 +224,38 @@ Section fupd_derived.
   Qed.
   (** How to apply an arbitrary mask-changing view shift when having
       an arbitrary mask. *)
-  Lemma fupd_mask_frame E E1 E2 P :
-    E1 ⊆ E → (|={E1,E2}=> P) -∗ (|={E,E2 ∪ (E ∖ E1)}=> P).
+  Lemma fupd_mask_frame E E' E1 E2 P :
+    E1 ⊆ E → E' = E2 ∪ (E ∖ E1) →
+    (|={E1,E2}=> P) -∗ (|={E,E'}=> P).
   Proof.
-    intros ?. rewrite (fupd_mask_frame_r _ _ (E ∖ E1)); last set_solver.
+    intros ? ->. rewrite (fupd_mask_frame_r _ _ (E ∖ E1)); last set_solver.
     assert (E = E1 ∪ E ∖ E1) as <-; last done.
     apply union_difference_L. done.
   Qed.
-  Lemma fupd_mask_frame_diff E E1 E2 P :
+  Lemma fupd_mask_frame_diff_open E E1 E2 P :
     (* E2 ⊆ E1 is needed bcause otherwise the [E ∖ E2] could remove
        more invariants from E than it did from E1. *)
-    E2 ⊆ E1 → E1 ⊆ E → (|={E1,E1∖E2}=> P) -∗ (|={E,E∖E2}=> P).
+    E1 ⊆ E → E2 ⊆ E1 → (|={E1,E1∖E2}=> P) -∗ (|={E,E∖E2}=> P).
   Proof.
-    intros ? HE. rewrite (fupd_mask_frame E); last done.
+    intros HE ?. rewrite (fupd_mask_frame E); [|done..].
     assert (E1 ∖ E2 ∪ E ∖ E1 = E ∖ E2) as <-; last done.
     apply (anti_symm (⊆)); first set_solver.
     rewrite {1}(union_difference_L _ _ HE). set_solver.
   Qed.
+  Lemma fupd_mask_frame_diff_close E E1 E2 P :
+    (* E2 ⊆ E1 is needed bcause otherwise the [E ∖ E2] could remove
+       more invariants from E than it did from E1. *)
+    E1 ⊆ E → E2 ⊆ E1 → (|={E1∖E2,E1}=> P) -∗ (|={E∖E2,E}=> P).
+  Proof.
+    intros HE ?. rewrite (fupd_mask_frame (E ∖ E2)); [|set_solver..].
+    assert (E = E1 ∪ E ∖ E2 ∖ (E1 ∖ E2)) as <-; last done.
+    apply (anti_symm (⊆)); last set_solver.
+    rewrite {1}(union_difference_L _ _ HE). set_solver.
+  Qed.
+
+  Lemma fupd_mask_same E E1 P :
+    E = E1 → (|={E}=> P) -∗ (|={E,E1}=> P).
+  Proof. intros <-. done. Qed.
 
   Lemma fupd_sep E P Q : (|={E}=> P) ∗ (|={E}=> Q) ={E}=∗ P ∗ Q.
   Proof. by rewrite fupd_frame_r fupd_frame_l fupd_trans. Qed.