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From iris.algebra Require Import functions gmap proofmode_classes.
From iris.base_logic.lib Require Export iprop.
Set Default Proof Using "Type".
Import uPred.
(** The class [inG Σ A] expresses that the CMRA [A] is in the list of functors
[Σ]. This class is similar to the [subG] class, but written down in terms of
individual CMRAs instead of (lists of) CMRA *functors*. This additional class is
needed because Coq is otherwise unable to solve type class constraints due to
higher-order unification problems. *)
Class inG (Σ : gFunctors) (A : cmraT) :=
InG { inG_id : gid Σ; inG_prf : A = gFunctors_lookup Σ inG_id (iPrePropO Σ) _ }.
Arguments inG_id {_ _} _.
(** We use the mode [-] for [Σ] since there is always a unique [Σ]. We use the
mode [!] for [A] since we can have multiple [inG]s for different [A]s, so we do
not want Coq to pick one arbitrarily. *)
Hint Mode inG - ! : typeclass_instances.
Lemma subG_inG Σ (F : gFunctor) : subG F Σ → inG Σ (F (iPrePropO Σ) _).
Proof. move=> /(_ 0%fin) /= [j ->]. by exists j. Qed.
(** This tactic solves the usual obligations "subG ? Σ → {in,?}G ? Σ" *)
Ltac solve_inG :=
(* Get all assumptions *)
intros;
(* Unfold the top-level xΣ. We need to support this to be a function. *)
lazymatch goal with
| H : subG (?xΣ _ _ _ _) _ |- _ => try unfold xΣ in H
| H : subG (?xΣ _ _ _) _ |- _ => try unfold xΣ in H
| H : subG (?xΣ _ _) _ |- _ => try unfold xΣ in H
| H : subG (?xΣ _) _ |- _ => try unfold xΣ in H
| H : subG ?xΣ _ |- _ => try unfold xΣ in H
end;
(* Take apart subG for non-"atomic" lists *)
repeat match goal with
| H : subG (gFunctors.app _ _) _ |- _ => apply subG_inv in H; destruct H
end;
(* Try to turn singleton subG into inG; but also keep the subG for typeclass
resolution -- to keep them, we put them onto the goal. *)
repeat match goal with
| H : subG _ _ |- _ => move:(H); (apply subG_inG in H || clear H)
end;
(* Again get all assumptions *)
intros;
(* We support two kinds of goals: Things convertible to inG;
and records with inG and typeclass fields. Try to solve the
first case. *)
try done;
(* That didn't work, now we're in for the second case. *)
split; (assumption || by apply _).
(** * Definition of the connective [own] *)
Definition iRes_singleton {Σ A} {i : inG Σ A} (γ : gname) (a : A) : iResUR Σ :=
discrete_fun_singleton (inG_id i) {[ γ := cmra_transport inG_prf a ]}.
Instance: Params (@iRes_singleton) 4 := {}.
Definition own_def `{!inG Σ A} (γ : gname) (a : A) : iProp Σ :=
uPred_ownM (iRes_singleton γ a).
Definition own_aux : seal (@own_def). by eexists. Qed.
Definition own {Σ A i} := own_aux.(unseal) Σ A i.
Definition own_eq : @own = @own_def := own_aux.(seal_eq).
Instance: Params (@own) 4 := {}.
Typeclasses Opaque own.
(** * Properties about ghost ownership *)
Section global.
Context `{Hin: !inG Σ A}.
Implicit Types a : A.
(** ** Properties of [iRes_singleton] *)Global Instance iRes_singleton_ne γ :
NonExpansive (@iRes_singleton Σ A _ γ).
Proof. by intros n a a' Ha; apply discrete_fun_singleton_ne; rewrite Ha. Qed.
Lemma iRes_singleton_op γ a1 a2 :
iRes_singleton γ (a1 ⋅ a2) ≡ iRes_singleton γ a1 ⋅ iRes_singleton γ a2.
Proof.
by rewrite /iRes_singleton discrete_fun_op_singleton op_singleton cmra_transport_op.
Qed.
(** ** Properties of [own] *)
Global Instance own_ne γ : NonExpansive (@own Σ A _ γ).
Proof. rewrite !own_eq. solve_proper. Qed.
Global Instance own_proper γ :
Proper ((≡) ==> (⊣⊢)) (@own Σ A _ γ) := ne_proper _.
Lemma own_op γ a1 a2 : own γ (a1 ⋅ a2) ⊣⊢ own γ a1 ∗ own γ a2.
Proof. by rewrite !own_eq /own_def -ownM_op iRes_singleton_op. Qed.
Lemma own_mono γ a1 a2 : a2 ≼ a1 → own γ a1 ⊢ own γ a2.
Proof. move=> [c ->]. by rewrite own_op sep_elim_l. Qed.
Global Instance own_mono' γ : Proper (flip (≼) ==> (⊢)) (@own Σ A _ γ).
Proof. intros a1 a2. apply own_mono. Qed.
Lemma own_valid γ a : own γ a ⊢ ✓ a.
Proof.
rewrite !own_eq /own_def ownM_valid /iRes_singleton.
rewrite discrete_fun_validI (forall_elim (inG_id Hin)) discrete_fun_lookup_singleton.
rewrite gmap_validI (forall_elim γ) lookup_singleton option_validI.
(* implicit arguments differ a bit *)
by trans (✓ cmra_transport inG_prf a : iProp Σ)%I; last destruct inG_prf.
Qed.
Lemma own_valid_2 γ a1 a2 : own γ a1 -∗ own γ a2 -∗ ✓ (a1 ⋅ a2).
Proof. apply wand_intro_r. by rewrite -own_op own_valid. Qed.
Lemma own_valid_3 γ a1 a2 a3 : own γ a1 -∗ own γ a2 -∗ own γ a3 -∗ ✓ (a1 ⋅ a2 ⋅ a3).
Proof. do 2 apply wand_intro_r. by rewrite -!own_op own_valid. Qed.
Lemma own_valid_r γ a : own γ a ⊢ own γ a ∗ ✓ a.
Proof. apply: bi.persistent_entails_r. apply own_valid. Qed.
Lemma own_valid_l γ a : own γ a ⊢ ✓ a ∗ own γ a.
Proof. by rewrite comm -own_valid_r. Qed.
Global Instance own_timeless γ a : Discrete a → Timeless (own γ a).
Proof. rewrite !own_eq /own_def; apply _. Qed.
Global Instance own_core_persistent γ a : CoreId a → Persistent (own γ a).
Proof. rewrite !own_eq /own_def; apply _. Qed.
Lemma later_own γ a : ▷ own γ a -∗ ◇ (∃ b, own γ b ∧ ▷ (a ≡ b)).
Proof.
rewrite own_eq /own_def later_ownM. apply exist_elim=> r.
assert (NonExpansive (λ r : iResUR Σ, r (inG_id Hin) !! γ)).
{ intros n r1 r2 Hr. f_equiv. by specialize (Hr (inG_id Hin)). }
rewrite (f_equiv (λ r : iResUR Σ, r (inG_id Hin) !! γ) _ r).
rewrite {1}/iRes_singleton discrete_fun_lookup_singleton lookup_singleton.
rewrite option_equivI. case Hb: (r (inG_id _) !! γ)=> [b|]; last first.
{ by rewrite and_elim_r /sbi_except_0 -or_intro_l. }
rewrite -except_0_intro -(exist_intro (cmra_transport (eq_sym inG_prf) b)).
apply and_mono.
- rewrite /iRes_singleton. assert (∀ {A A' : cmraT} (Heq : A' = A) (a : A),
cmra_transport Heq (cmra_transport (eq_sym Heq) a) = a) as ->
by (by intros ?? ->).
apply ownM_mono=> /=.
exists (discrete_fun_insert (inG_id _) (delete γ (r (inG_id Hin))) r).
intros i'. rewrite discrete_fun_lookup_op.
destruct (decide (i' = inG_id Hin)) as [->|?].
+ rewrite discrete_fun_lookup_insert discrete_fun_lookup_singleton.
intros γ'. rewrite lookup_op. destruct (decide (γ' = γ)) as [->|?].
* by rewrite lookup_singleton lookup_delete Hb.
* by rewrite lookup_singleton_ne // lookup_delete_ne // left_id.
+ rewrite discrete_fun_lookup_insert_ne //.
by rewrite discrete_fun_lookup_singleton_ne // left_id.
- apply later_mono. by assert (∀ {A A' : cmraT} (Heq : A' = A) (a' : A') (a : A),
cmra_transport Heq a' ≡ a ⊢@{iPropI Σ}
a' ≡ cmra_transport (eq_sym Heq) a) as -> by (by intros ?? ->).
Qed.
(** ** Allocation *)
(* TODO: This also holds if we just have ✓ a at the current step-idx, as Iris
assertion. However, the map_updateP_alloc does not suffice to show this. *)
Lemma own_alloc_strong_dep (f : gname → A) (P : gname → Prop) :
pred_infinite P →
(∀ γ, P γ → ✓ (f γ)) →
(|==> ∃ γ, ⌜P γ⌝ ∧ own γ (f γ))%I.
Proof.
intros HP Ha.
rewrite -(bupd_mono (∃ m, ⌜∃ γ, P γ ∧ m = iRes_singleton γ (f γ)⌝ ∧ uPred_ownM m)%I).
- rewrite /uPred_valid /bi_emp_valid (ownM_unit emp).
eapply bupd_ownM_updateP, (discrete_fun_singleton_updateP_empty (inG_id Hin)).
+ eapply alloc_updateP_strong_dep'; first done.
intros i _ ?. eapply cmra_transport_valid, Ha. done.
+ naive_solver.
- apply exist_elim=>m; apply pure_elim_l=>-[γ [Hfresh ->]].
by rewrite !own_eq /own_def -(exist_intro γ) pure_True // left_id.
Qed.
Lemma own_alloc_cofinite_dep (f : gname → A) (G : gset gname) :
(∀ γ, γ ∉ G → ✓ (f γ)) → (|==> ∃ γ, ⌜γ ∉ G⌝ ∧ own γ (f γ))%I.
Proof.
intros Ha.
apply (own_alloc_strong_dep f (λ γ, γ ∉ G))=> //.
apply (pred_infinite_set (C:=gset gname)).
intros E. set (i := fresh (G ∪ E)).
exists i. apply not_elem_of_union, is_fresh.
Qed.
Lemma own_alloc_dep (f : gname → A) :
(∀ γ, ✓ (f γ)) → (|==> ∃ γ, own γ (f γ))%I.
Proof.
intros Ha. rewrite /uPred_valid /bi_emp_valid (own_alloc_cofinite_dep f ∅) //; [].
apply bupd_mono, exist_mono=>?. eauto using and_elim_r.
Qed.
Lemma own_alloc_strong a (P : gname → Prop) :
pred_infinite P →
✓ a → (|==> ∃ γ, ⌜P γ⌝ ∧ own γ a)%I.
Proof. intros HP Ha. eapply own_alloc_strong_dep with (f := λ _, a); eauto. Qed.
Lemma own_alloc_cofinite a (G : gset gname) :
✓ a → (|==> ∃ γ, ⌜γ ∉ G⌝ ∧ own γ a)%I.
Proof. intros Ha. eapply own_alloc_cofinite_dep with (f := λ _, a); eauto. Qed.
Lemma own_alloc a : ✓ a → (|==> ∃ γ, own γ a)%I.
Proof. intros Ha. eapply own_alloc_dep with (f := λ _, a); eauto. Qed.
(** ** Frame preserving updates *)
Lemma own_updateP P γ a : a ~~>: P → own γ a ==∗ ∃ a', ⌜P a'⌝ ∧ own γ a'.
Proof.
intros Ha. rewrite !own_eq.
rewrite -(bupd_mono (∃ m, ⌜∃ a', m = iRes_singleton γ a' ∧ P a'⌝ ∧ uPred_ownM m)%I).
- eapply bupd_ownM_updateP, discrete_fun_singleton_updateP;
first by (eapply singleton_updateP', cmra_transport_updateP', Ha).
naive_solver.
- apply exist_elim=>m; apply pure_elim_l=>-[a' [-> HP]].
rewrite -(exist_intro a'). by apply and_intro; [apply pure_intro|].
Qed.
Lemma own_update γ a a' : a ~~> a' → own γ a ==∗ own γ a'.
Proof.
intros; rewrite (own_updateP (a' =.)); last by apply cmra_update_updateP.
by apply bupd_mono, exist_elim=> a''; apply pure_elim_l=> ->.
Qed.
Lemma own_update_2 γ a1 a2 a' :
a1 ⋅ a2 ~~> a' → own γ a1 -∗ own γ a2 ==∗ own γ a'.
Proof. intros. apply wand_intro_r. rewrite -own_op. by apply own_update. Qed.
Lemma own_update_3 γ a1 a2 a3 a' : a1 ⋅ a2 ⋅ a3 ~~> a' → own γ a1 -∗ own γ a2 -∗ own γ a3 ==∗ own γ a'.
Proof. intros. do 2 apply wand_intro_r. rewrite -!own_op. by apply own_update. Qed.
End global.
Arguments own_valid {_ _} [_] _ _.
Arguments own_valid_2 {_ _} [_] _ _ _.
Arguments own_valid_3 {_ _} [_] _ _ _ _.
Arguments own_valid_l {_ _} [_] _ _.
Arguments own_valid_r {_ _} [_] _ _.
Arguments own_updateP {_ _} [_] _ _ _ _.
Arguments own_update {_ _} [_] _ _ _ _.
Arguments own_update_2 {_ _} [_] _ _ _ _ _.
Arguments own_update_3 {_ _} [_] _ _ _ _ _ _.
Lemma own_unit A `{!inG Σ (A:ucmraT)} γ : (|==> own γ (ε:A))%I.
Proof.
rewrite /uPred_valid /bi_emp_valid (ownM_unit emp) !own_eq /own_def.
apply bupd_ownM_update, discrete_fun_singleton_update_empty.
apply (alloc_unit_singleton_update (cmra_transport inG_prf ε)); last done.
- apply cmra_transport_valid, ucmra_unit_valid.
- intros x; destruct inG_prf. by rewrite left_id.
Qed.
(** Big op class instances *)
Instance own_cmra_sep_homomorphism `{!inG Σ (A:ucmraT)} :
WeakMonoidHomomorphism op uPred_sep (≡) (own γ).
Proof. split; try apply _. apply own_op. Qed.
(** Proofmode class instances *)
Section proofmode_classes.
Context `{!inG Σ A}.
Implicit Types a b : A.
Global Instance into_sep_own γ a b1 b2 :
IsOp a b1 b2 → IntoSep (own γ a) (own γ b1) (own γ b2).
Proof. intros. by rewrite /IntoSep (is_op a) own_op. Qed.
Global Instance into_and_own p γ a b1 b2 :
IsOp a b1 b2 → IntoAnd p (own γ a) (own γ b1) (own γ b2).
Proof. intros. by rewrite /IntoAnd (is_op a) own_op sep_and. Qed.
Global Instance from_sep_own γ a b1 b2 :
IsOp a b1 b2 → FromSep (own γ a) (own γ b1) (own γ b2).
Proof. intros. by rewrite /FromSep -own_op -is_op. Qed.
Global Instance from_and_own_persistent γ a b1 b2 :
IsOp a b1 b2 → TCOr (CoreId b1) (CoreId b2) →
FromAnd (own γ a) (own γ b1) (own γ b2).
Proof.
intros ? Hb. rewrite /FromAnd (is_op a) own_op.
destruct Hb; by rewrite persistent_and_sep.
Qed.
End proofmode_classes.