From prelude Require Export countable co_pset. From algebra Require Export base. Definition namespace := list positive. Definition nroot : 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_eq. Qed. Lemma nclose_nroot : nclose nroot = ⊤. 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. (* This tactic solves goals about inclusion and disjointness of masks (i.e., coPsets) with set_solver, taking disjointness of namespaces into account. *) (* TODO: This tactic is by far now yet as powerful as it should be. For example, given N1 ⊥ N2, it should be able to solve nclose (ndot N1 x) ∩ N2 ≡ ∅. It should also solve (ndot N x) ∩ (ndot N y) ≡ ∅ if x ≠ y is in the context or follows from [discriminate]. *) Ltac set_solver_ndisj := repeat match goal with (* TODO: Restrict these to have type namespace *) | [ H : (?N1 ⊥ ?N2) |-_ ] => apply ndisj_disjoint in H end; set_solver. (* TODO: restrict this to match only if this is ⊆ of coPset *) Hint Extern 500 (_ ⊆ _) => set_solver_ndisj : ndisj. (* The hope is that registering these will suffice to solve most goals of the form N1 ⊥ N2. TODO: Can this prove x ≠ y if discriminate can? *) Hint Resolve ndot_ne_disjoint : ndisj. Hint Resolve ndot_preserve_disjoint_l : ndisj. Hint Resolve ndot_preserve_disjoint_r : ndisj.