Commit 28e72e3c authored by Robbert Krebbers's avatar Robbert Krebbers

Add namespaces from Iris.

See the discussion at https://gitlab.mpi-sws.org/FP/iris-coq/merge_requests/116.
parent cd1ed3dd
...@@ -41,4 +41,5 @@ theories/hlist.v ...@@ -41,4 +41,5 @@ theories/hlist.v
theories/sorting.v theories/sorting.v
theories/infinite.v theories/infinite.v
theories/nat_cancel.v theories/nat_cancel.v
theories/namespaces.v
From stdpp Require Export countable coPset.
Set Default Proof Using "Type".
Definition namespace := list positive.
Instance namespace_eq_dec : EqDecision namespace := _.
Instance namespace_countable : Countable namespace := _.
Typeclasses Opaque namespace.
Definition nroot : namespace := nil.
Definition ndot_def `{Countable A} (N : namespace) (x : A) : namespace :=
encode x :: N.
Definition ndot_aux : seal (@ndot_def). by eexists. Qed.
Definition ndot {A A_dec A_count}:= unseal ndot_aux A A_dec A_count.
Definition ndot_eq : @ndot = @ndot_def := seal_eq ndot_aux.
Definition nclose_def (N : namespace) : coPset := coPset_suffixes (encode N).
Definition nclose_aux : seal (@nclose_def). by eexists. Qed.
Instance nclose : UpClose namespace coPset := unseal nclose_aux.
Definition nclose_eq : @nclose = @nclose_def := seal_eq nclose_aux.
Notation "N .@ x" := (ndot N x)
(at level 19, left associativity, format "N .@ x") : stdpp_scope.
Notation "(.@)" := ndot (only parsing) : stdpp_scope.
Instance ndisjoint : Disjoint namespace := λ N1 N2, nclose N1 ## nclose N2.
Section namespace.
Context `{Countable A}.
Implicit Types x y : A.
Implicit Types N : namespace.
Implicit Types E : coPset.
Global Instance ndot_inj : Inj2 (=) (=) (=) (@ndot A _ _).
Proof. intros N1 x1 N2 x2; rewrite !ndot_eq; naive_solver. Qed.
Lemma nclose_nroot : nroot = (:coPset).
Proof. rewrite nclose_eq. by apply (sig_eq_pi _). Qed.
Lemma encode_nclose N : encode N (N:coPset).
Proof.
rewrite nclose_eq.
by apply elem_coPset_suffixes; exists xH; rewrite (left_id_L _ _).
Qed.
Lemma nclose_subseteq N x : N.@x (N : coPset).
Proof.
intros p. unfold up_close. rewrite !nclose_eq, !ndot_eq.
unfold nclose_def, ndot_def; rewrite !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 : N E N.@x E.
Proof. intros. etrans; eauto using nclose_subseteq. Qed.
Lemma ndot_nclose N x : encode (N.@x) (N:coPset).
Proof. apply nclose_subseteq with x, encode_nclose. Qed.
Lemma nclose_infinite N : ¬set_finite ( N : coPset).
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. unfold up_close. rewrite !nclose_eq, !ndot_eq.
unfold nclose_def, ndot_def; rewrite !elem_coPset_suffixes.
intros [qx ->] [qy Hqy].
revert Hqy. by intros [= ?%encode_inj]%list_encode_suffix_eq.
Qed.
Lemma ndot_preserve_disjoint_l N E x : N ## E N.@x ## E.
Proof. intros. pose proof (nclose_subseteq N x). set_solver. Qed.
Lemma ndot_preserve_disjoint_r N E x : E ## N E ## N.@x.
Proof. intros. by apply symmetry, ndot_preserve_disjoint_l. Qed.
Lemma ndisj_subseteq_difference N E F : E ## N E F E F N.
Proof. set_solver. Qed.
Lemma namespace_subseteq_difference_l E1 E2 E3 : E1 E3 E1 E2 E3.
Proof. set_solver. Qed.
Lemma ndisj_difference_l E N1 N2 : N2 (N1 : coPset) E N1 ## N2.
Proof. set_solver. Qed.
End namespace.
(* The hope is that registering these will suffice to solve most goals
of the forms:
- [N1 ## N2]
- [↑N1 ⊆ E ∖ ↑N2 ∖ .. ∖ ↑Nn]
- [E1 ∖ ↑N1 ⊆ E2 ∖ ↑N2 ∖ .. ∖ ↑Nn] *)
Create HintDb ndisj.
Hint Resolve ndisj_subseteq_difference : ndisj.
Hint Extern 0 (_ ## _) => apply ndot_ne_disjoint; congruence : ndisj.
Hint Resolve ndot_preserve_disjoint_l ndot_preserve_disjoint_r : ndisj.
Hint Resolve nclose_subseteq' ndisj_difference_l : ndisj.
Hint Resolve namespace_subseteq_difference_l | 100 : ndisj.
Hint Resolve (empty_subseteq (A:=positive) (C:=coPset)) : ndisj.
Hint Resolve (union_least (A:=positive) (C:=coPset)) : ndisj.
Ltac solve_ndisj :=
repeat match goal with
| H : _ _ _ |- _ => apply union_subseteq in H as [??]
end;
solve [eauto with ndisj].
Hint Extern 1000 => solve_ndisj : solve_ndisj.
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