Move namespace stuff to separate file.

Now that there is more of it, it deserves its own place :).

_CoqProject

@@ -70,6 +70,7 @@ program_logic/ghost_ownership.v
 program_logic/saved_prop.v
 program_logic/auth.v
 program_logic/sts.v
+program_logic/namespaces.v
 heap_lang/heap_lang.v
 heap_lang/tactics.v
 heap_lang/wp_tactics.v

program_logic/invariants.v

 From algebra Require Export base.
-From prelude Require Export countable co_pset.
 From program_logic Require Import ownership.
-From program_logic Require Export pviewshifts weakestpre.
+From program_logic Require Export namespaces pviewshifts weakestpre.
 Import uPred.
-
 Local Hint Extern 100 (@eq coPset _ _) => solve_elem_of.
 Local Hint Extern 100 (@subseteq coPset _ _) => solve_elem_of.
 Local Hint Extern 100 (_ ∉ _) => solve_elem_of.
 Local Hint Extern 99 ({[ _ ]} ⊆ _) => apply elem_of_subseteq_singleton.
 
-Definition namespace := list positive.
-Definition nnil : namespace := nil.
-Definition ndot `{Countable A} (N : namespace) (x : A) : namespace :=
-  encode x :: N.
-Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N).
-
-Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _).
-Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed.
-Lemma nclose_nnil : nclose nnil = coPset_all.
-Proof. by apply (sig_eq_pi _). Qed.
-Lemma encode_nclose N : encode N ∈ nclose N.
-Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed.
-Lemma nclose_subseteq `{Countable A} N x : nclose (ndot N x) ⊆ nclose N.
-Proof.
-  intros p; rewrite /nclose !elem_coPset_suffixes; intros [q ->].
-  destruct (list_encode_suffix N (ndot 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 (ndot N x) ∈ nclose N.
-Proof. apply nclose_subseteq with x, encode_nclose. Qed.
-
-Instance ndisjoint : Disjoint namespace := λ N1 N2,
-  ∃ N1' N2', N1' `suffix_of` N1 ∧ N2' `suffix_of` N2 ∧
-             length N1' = length N2' ∧ N1' ≠ N2'.
-
-Section ndisjoint.
-  Context `{Countable A}.
-  Implicit Types x y : A.
-
-  Global Instance ndisjoint_comm : Comm iff ndisjoint.
-  Proof. intros N1 N2. rewrite /disjoint /ndisjoint; naive_solver. Qed.
-
-  Lemma ndot_ne_disjoint N (x y : A) : x ≠ y → ndot N x ⊥ ndot N y.
-  Proof. intros Hxy. exists (ndot N x), (ndot N y); naive_solver. Qed.
-
-  Lemma ndot_preserve_disjoint_l N1 N2 x : N1 ⊥ N2 → ndot N1 x ⊥ N2.
-  Proof.
-    intros (N1' & N2' & Hpr1 & Hpr2 & Hl & Hne). exists N1', N2'.
-    split_ands; try done; []. by apply suffix_of_cons_r.
-  Qed.
-
-  Lemma ndot_preserve_disjoint_r N1 N2 x : N1 ⊥ N2 → N1 ⊥ ndot N2 x .
-  Proof. rewrite ![N1 ⊥ _]comm. apply ndot_preserve_disjoint_l. Qed.
-
-  Lemma ndisj_disjoint N1 N2 : N1 ⊥ N2 → nclose N1 ∩ nclose N2 = ∅.
-  Proof.
-    intros (N1' & N2' & [N1'' ->] & [N2'' ->] & Hl & Hne).
-    apply elem_of_equiv_empty_L=> p; unfold nclose.
-    rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]].
-    rewrite !list_encode_app !assoc in Hq.
-    by eapply Hne, list_encode_suffix_eq.
-  Qed.
-End ndisjoint.
-
 (** Derived forms and lemmas about them. *)
 Definition inv {Λ Σ} (N : namespace) (P : iProp Λ Σ) : iProp Λ Σ :=
   (∃ i, ■ (i ∈ nclose N) ∧ ownI i P)%I.

program_logic/namespaces.v

+From prelude Require Export countable co_pset.
+From algebra Require Export base.
+
+Definition namespace := list positive.
+Definition nnil : namespace := nil.
+Definition ndot `{Countable A} (N : namespace) (x : A) : namespace :=
+  encode x :: N.
+Coercion nclose (N : namespace) : coPset := coPset_suffixes (encode N).
+
+Instance ndot_inj `{Countable A} : Inj2 (=) (=) (=) (@ndot A _ _).
+Proof. by intros N1 x1 N2 x2 ?; simplify_equality. Qed.
+Lemma nclose_nnil : nclose nnil = coPset_all.
+Proof. by apply (sig_eq_pi _). Qed.
+Lemma encode_nclose N : encode N ∈ nclose N.
+Proof. by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _). Qed.
+Lemma nclose_subseteq `{Countable A} N x : nclose (ndot N x) ⊆ nclose N.
+Proof.
+  intros p; rewrite /nclose !elem_coPset_suffixes; intros [q ->].
+  destruct (list_encode_suffix N (ndot 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 (ndot N x) ∈ nclose N.
+Proof. apply nclose_subseteq with x, encode_nclose. Qed.
+
+Instance ndisjoint : Disjoint namespace := λ N1 N2,
+  ∃ N1' N2', N1' `suffix_of` N1 ∧ N2' `suffix_of` N2 ∧
+             length N1' = length N2' ∧ N1' ≠ N2'.
+
+Section ndisjoint.
+  Context `{Countable A}.
+  Implicit Types x y : A.
+
+  Global Instance ndisjoint_comm : Comm iff ndisjoint.
+  Proof. intros N1 N2. rewrite /disjoint /ndisjoint; naive_solver. Qed.
+
+  Lemma ndot_ne_disjoint N (x y : A) : x ≠ y → ndot N x ⊥ ndot N y.
+  Proof. intros Hxy. exists (ndot N x), (ndot N y); naive_solver. Qed.
+
+  Lemma ndot_preserve_disjoint_l N1 N2 x : N1 ⊥ N2 → ndot N1 x ⊥ N2.
+  Proof.
+    intros (N1' & N2' & Hpr1 & Hpr2 & Hl & Hne). exists N1', N2'.
+    split_ands; try done; []. by apply suffix_of_cons_r.
+  Qed.
+
+  Lemma ndot_preserve_disjoint_r N1 N2 x : N1 ⊥ N2 → N1 ⊥ ndot N2 x .
+  Proof. rewrite ![N1 ⊥ _]comm. apply ndot_preserve_disjoint_l. Qed.
+
+  Lemma ndisj_disjoint N1 N2 : N1 ⊥ N2 → nclose N1 ∩ nclose N2 = ∅.
+  Proof.
+    intros (N1' & N2' & [N1'' ->] & [N2'' ->] & Hl & Hne).
+    apply elem_of_equiv_empty_L=> p; unfold nclose.
+    rewrite elem_of_intersection !elem_coPset_suffixes; intros [[q ->] [q' Hq]].
+    rewrite !list_encode_app !assoc in Hq.
+    by eapply Hne, list_encode_suffix_eq.
+  Qed.
+End ndisjoint.
\ No newline at end of file