Add nclose N ⊆ E → nclose (N .@ x) ⊆ E to ndisj hints.

base_logic/lib/namespaces.v

@@ -22,33 +22,38 @@ Definition nclose_eq : @nclose = @nclose_def := proj2_sig nclose_aux.
 Infix ".@" := ndot (at level 19, left associativity) : C_scope.
 Notation "(.@)" := ndot (only parsing) : C_scope.
 
-Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _).
-Proof. intros N1 x1 N2 x2; rewrite !ndot_eq=> ?; by simplify_eq. Qed.
-Lemma nclose_nroot : nclose nroot = ⊤.
-Proof. rewrite nclose_eq. by apply (sig_eq_pi _). Qed.
-Lemma encode_nclose N : encode N ∈ nclose N.
-Proof.
-  rewrite nclose_eq.
-  by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _).
-Qed.
-Lemma nclose_subseteq `{Countable A} N x : nclose (N .@ x) ⊆ nclose N.
-Proof.
-  intros p; rewrite nclose_eq /nclose !ndot_eq !elem_coPset_suffixes.
-  intros [q ->]. destruct (list_encode_suffix N (ndot_def N x)) as [q' ?].
-  { by exists [encode x]. }
-  by exists (q ++ q')%positive; rewrite <-(assoc_L _); f_equal.
-Qed.
-Lemma ndot_nclose `{Countable A} N x : encode (N .@ x) ∈ nclose N.
-Proof. apply nclose_subseteq with x, encode_nclose. Qed.
-Lemma nclose_infinite N : ¬set_finite (nclose N).
-Proof. rewrite nclose_eq. apply coPset_suffixes_infinite. Qed.
-
 Instance ndisjoint : Disjoint namespace := λ N1 N2, nclose N1 ⊥ nclose N2.
 
-Section ndisjoint.
+Section namespace.
   Context `{Countable A}.
   Implicit Types x y : A.
 
+  Global Instance ndot_inj : Inj2 (=) (=) (=) (@ndot A _ _).
+  Proof. intros N1 x1 N2 x2; rewrite !ndot_eq=> ?; by simplify_eq. Qed.
+
+  Lemma nclose_nroot : nclose nroot = ⊤.
+  Proof. rewrite nclose_eq. by apply (sig_eq_pi _). Qed.
+  Lemma encode_nclose N : encode N ∈ nclose N.
+  Proof.
+    rewrite nclose_eq.
+    by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _).
+  Qed.
+
+  Lemma nclose_subseteq N x : nclose (N .@ x) ⊆ nclose N.
+  Proof.
+    intros p; rewrite nclose_eq /nclose !ndot_eq !elem_coPset_suffixes.
+    intros [q ->]. destruct (list_encode_suffix N (ndot_def N x)) as [q' ?].
+    { by exists [encode x]. }
+    by exists (q ++ q')%positive; rewrite <-(assoc_L _); f_equal.
+  Qed.
+  Lemma nclose_subseteq' E N x : nclose N ⊆ E → nclose (N .@ x) ⊆ E.
+  Proof. intros. etrans; eauto using nclose_subseteq. Qed.
+
+  Lemma ndot_nclose N x : encode (N .@ x) ∈ nclose N.
+  Proof. apply nclose_subseteq with x, encode_nclose. Qed.
+  Lemma nclose_infinite N : ¬set_finite (nclose N).
+  Proof. rewrite nclose_eq. apply coPset_suffixes_infinite. Qed.
+
   Lemma ndot_ne_disjoint N x y : x ≠ y → N .@ x ⊥ N .@ y.
   Proof.
     intros Hxy a. rewrite !nclose_eq !elem_coPset_suffixes !ndot_eq.
@@ -65,7 +70,7 @@ Section ndisjoint.
   Lemma ndisj_subseteq_difference N E F :
     E ⊥ nclose N → E ⊆ F → E ⊆ F ∖ nclose N.
   Proof. set_solver. Qed.
-End ndisjoint.
+End namespace.
 
 (* The hope is that registering these will suffice to solve most goals
 of the form [N1 ⊥ N2] and those of the form [N1 ⊆ E ∖ N2 ∖ .. ∖ Nn]. *)
@@ -73,5 +78,6 @@ Hint Resolve ndisj_subseteq_difference : ndisj.
 Hint Extern 0 (_ ⊥ _) => apply ndot_ne_disjoint; congruence : ndisj.
 Hint Resolve ndot_preserve_disjoint_l : ndisj.
 Hint Resolve ndot_preserve_disjoint_r : ndisj.
+Hint Extern 1 (_ ⊆ _) => apply nclose_subseteq' : ndisj.
 
 Ltac solve_ndisj := solve [eauto with ndisj].