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Commit 152920f9 authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan
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Cleanned up perm_incl.v. Proved new ways of proving such permission inclusions.

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perm_incl.v
+ 80
− 50
View file @ 152920f9
......@@ -2,43 +2,106 @@ From iris.algebra Require Import upred_big_op.
From iris.program_logic Require Import thread_local.
From lrust Require Export type perm.
Section perm_incl.
Import Perm.
Section defs.
Context `{heapG Σ, lifetimeG Σ, thread_localG Σ}.
(* Definitions *)
Definition perm_incl (ρ1 ρ2 : perm) :=
∀ tid, ρ1 tid ={⊤}=> ρ2 tid.
Lemma perm_incl_top ρ : perm_incl ρ ⊤.
Proof. iIntros (tid) "H". eauto. Qed.
Global Instance perm_equiv : Equiv perm :=
λ ρ1 ρ2, perm_incl ρ1 ρ2 ∧ perm_incl ρ2 ρ1.
Class Duplicable (ρ : @perm Σ) :=
perm_incl_duplicable : perm_incl ρ (ρ ★ ρ).
End defs.
Infix "⇒" := perm_incl (at level 95, no associativity) : C_scope.
Notation "(⇒)" := perm_incl (only parsing) : C_scope.
Global Instance perm_incl_refl : Reflexive perm_incl.
Proof. iIntros (? tid) "H". eauto. Qed.
Notation "ρ1 ⇔ ρ2" := (equiv (A:=perm) ρ1%P ρ2%P)
(at level 95, no associativity) : C_scope.
Notation "(⇔)" := (equiv (A:=perm)) (only parsing) : C_scope.
Section props.
Context `{heapG Σ, lifetimeG Σ, thread_localG Σ}.
(* Properties *)
Global Instance perm_incl_preorder : PreOrder (⇒).
Proof.
split.
- iIntros (? tid) "H". eauto.
- iIntros (??? H12 H23 tid) "H". iVs (H12 with "H") as "H". by iApply H23.
Qed.
Global Instance perm_incl_trans : Transitive perm_incl.
Global Instance perm_equiv_equiv : Equivalence (⇔).
Proof.
iIntros (??? H12 H23 tid) "H". iVs (H12 with "H") as "H". by iApply H23.
split.
- by split.
- by intros x y []; split.
- intros x y z [] []. split; by transitivity y.
Qed.
Lemma perm_incl_frame_l ρ ρ1 ρ2 :
perm_incl ρ1 ρ2 → perm_incl (ρ ★ ρ1) (ρ ★ ρ2).
Global Instance perm_incl_proper :
Proper ((⇔) ==> (⇔) ==> iff) (⇒).
Proof. intros ??[??]??[??]; split; intros ?; by simplify_order. Qed.
Lemma perm_incl_top ρ : ρ ⇒ ⊤.
Proof. iIntros (tid) "H". eauto. Qed.
Lemma perm_incl_frame_l ρ ρ1 ρ2 : ρ1 ⇒ ρ2 → ρ ★ ρ1 ⇒ ρ ★ ρ2.
Proof. iIntros (Hρ tid) "[$?]". by iApply Hρ. Qed.
Lemma perm_incl_frame_r ρ ρ1 ρ2 :
perm_incl ρ1 ρ2 → perm_incl (ρ1 ★ ρ) (ρ2 ★ ρ).
ρ1 ⇒ ρ2 → ρ1 ★ ρ ⇒ ρ2 ★ ρ.
Proof. iIntros (Hρ tid) "[?$]". by iApply Hρ. Qed.
End perm_incl.
Lemma perm_incl_exists_intro {A} P x : P x ⇒ ∃ x : A, P x.
Proof. iIntros (tid) "H!==>". by iExists x. Qed.
Section duplicable.
Global Instance perm_sep_assoc : Assoc (⇔) sep.
Proof. intros ???; split. by iIntros (tid) "[$[$$]]". by iIntros (tid) "[[$$]$]". Qed.
Context `{heapG Σ, lifetimeG Σ, thread_localG Σ}.
Global Instance perm_sep_comm : Comm (⇔) sep.
Proof. intros ??; split; by iIntros (tid) "[$$]". Qed.
Class Duplicable (ρ : perm) :=
perm_incl_duplicable : perm_incl ρ (ρ ★ ρ).
Global Instance perm_top_right_id : RightId (⇔) ⊤ sep.
Proof. intros ρ; split. by iIntros (tid) "[? _]". by iIntros (tid) "$". Qed.
Global Instance perm_top_left_id : LeftId (⇔) ⊤ sep.
Proof. intros ρ. by rewrite comm right_id. Qed.
Lemma perm_tok_plus κ q1 q2 :
tok κ q1 ★ tok κ q2 ⇔ tok κ (q1 + q2).
Proof.
rewrite /tok /sep /=; split; intros tid; rewrite -lft_own_op;
iIntros "[[$$]H]!==>". by iDestruct "H" as "[$?]". by auto.
Qed.
Lemma perm_lftincl_refl κ : ⊤ ⇒ κ ⊑ κ.
Proof. iIntros (tid) "_!==>". iApply lft_incl_refl. Qed.
Lemma perm_lftincl_trans κ1 κ2 κ3 : κ1 ⊑ κ2 ★ κ2 ⊑ κ3 ⇒ κ1 ⊑ κ3.
Proof. iIntros (tid) "[#?#?]!==>". iApply lft_incl_trans. auto. Qed.
Lemma perm_incl_share v κ ty :
v ◁ &uniq{κ} ty ⇒ v ◁ &shr{κ} ty.
Proof.
iIntros (tid) "Huniq". iDestruct "Huniq" as (l) "[% Hown]".
(* FIXME : we nee some tokens here. *)
iAssert (∃ q, [κ]{q})%I as "Htok". admit.
iDestruct "Htok" as (q) "Htok".
iVs (ty.(ty_share) _ lrustN with "Hown Htok") as "[Hown _]".
admit. set_solver. iVsIntro. iExists _. iSplit. done. done.
Admitted.
Lemma has_type_dup v ty : ty.(ty_dup) → Duplicable (v ◁ ty).
Proof. iIntros (Hdup tid) "H!==>". by iApply ty_dup_dup. Qed.
Proof. iIntros (Hdup tid) "H". by iApply ty_dup_dup. Qed.
Global Instance lft_incl_dup κ κ' : Duplicable (κ ⊑ κ').
Proof. iIntros (tid) "#H!==>". by iSplit. Qed.
......@@ -53,39 +116,6 @@ Section duplicable.
Global Instance top_dup : Duplicable ⊤.
Proof. iIntros (tid) "_!==>". by iSplit. Qed.
End duplicable.
End props.
Hint Extern 0 (Duplicable (_ ◁ _)) => apply has_type_dup; exact I.
Section perm_equiv.
Context `{heapG Σ, lifetimeG Σ, thread_localG Σ}.
Definition perm_equiv ρ1 ρ2 := perm_incl ρ1 ρ2 ∧ perm_incl ρ2 ρ1.
Global Instance perm_equiv_refl : Reflexive perm_equiv.
Proof. by split. Qed.
Global Instance perm_equiv_trans : Transitive perm_equiv.
Proof. intros x y z [] []. split; by transitivity y. Qed.
Global Instance perm_equiv_sym : Symmetric perm_equiv.
Proof. by intros x y []; split. Qed.
Global Instance perm_incl_proper :
Proper (perm_equiv ==> perm_equiv ==> iff) perm_incl.
Proof. intros ??[??]??[??]; split; eauto using perm_incl_trans. Qed.
Global Instance sep_assoc : Assoc perm_equiv Perm.sep.
Proof. intros ???; split. by iIntros (tid) "[$[$$]]". by iIntros (tid) "[[$$]$]". Qed.
Global Instance sep_comm : Comm perm_equiv Perm.sep.
Proof. intros ??; split; by iIntros (tid) "[$$]". Qed.
Global Instance top_right_id : RightId perm_equiv ⊤ Perm.sep.
Proof. intros ρ; split. by iIntros (tid) "[? _]". by iIntros (tid) "$". Qed.
Global Instance top_left_id : LeftId perm_equiv ⊤ Perm.sep.
Proof. intros ρ. by rewrite comm right_id. Qed.
End perm_equiv.
\ No newline at end of file
type.v
+ 11
− 10
View file @ 152920f9
......@@ -22,7 +22,8 @@ Section defs.
ty_shr_persistent κ tid N l : PersistentP (ty_shr κ tid N l);
ty_size_eq tid vl : ty_own tid vl ⊢ length vl = ty_size;
ty_dup_dup tid vl : ty_dup → ty_own tid vl ⊢ ty_own tid vl ★ ty_own tid vl;
ty_dup_dup tid vl E :
ty_dup → ty_own tid vl ={E}=> ty_own tid vl ★ ty_own tid vl;
(* The mask for starting the sharing does /not/ include the
namespace N, for allowing more flexibility for the user of
this type (typically for the [own] type). AFAIK, there is no
......@@ -59,7 +60,7 @@ Section defs.
(∃ vl, (&frac{κ} λ q, l ↦★{q} vl) ★ ▷ st.(st_own) tid vl)%I
|}.
Next Obligation. apply st_size_eq. Qed.
Next Obligation. iIntros (st tid vl _) "H". eauto. Qed.
Next Obligation. iIntros (st tid vl E _) "H". eauto. Qed.
Next Obligation.
intros st E N κ l tid q ??. iIntros "Hmt Htok".
iVs (lft_borrow_exists with "Hmt Htok") as (vl) "[Hmt Htok]". set_solver.
......@@ -105,7 +106,7 @@ Section types.
{| ty_size := 0; ty_dup := true;
ty_own tid vl := False%I; ty_shr κ tid N l := False%I |}.
Next Obligation. iIntros (tid vl) "[]". Qed.
Next Obligation. iIntros (???) "[]". Qed.
Next Obligation. iIntros (????) "[]". Qed.
Next Obligation.
iIntros (????????) "Hb Htok".
iVs (lft_borrow_exists with "Hb Htok") as (vl) "[Hb Htok]". set_solver.
......@@ -341,15 +342,15 @@ Section types.
subst. by iApply (list_ty_type_eq with "Hown").
Qed.
Next Obligation.
induction tyl as [|ty tyq IH]; iIntros (tid vl Hfa) "H";
induction tyl as [|ty tyq IH]; iIntros (tid vl E Hfa) "H";
iDestruct "H" as ([|vl0 vlq]) "(%&#Hlen&Hown)"; subst vl;
iDestruct "Hlen" as %Hlen; inversion Hlen; simpl in *.
- iSplit; iExists []; by repeat iSplit.
- iIntros "!==>". iSplit; iExists []; by repeat iSplit.
- apply andb_prop_elim in Hfa. destruct Hfa as [Hfah Hfaq].
iDestruct (big_sepL_cons with "Hown") as "[Hh Hq]".
iDestruct (ty_dup_dup with "Hh") as "[Hh1 Hh2]". done.
iDestruct (IH tid (concat vlq) Hfaq with "[Hq]") as "[Hq1 Hq2]". by eauto.
iSplitL "Hh1 Hq1".
iVs (ty_dup_dup with "Hh") as "[Hh1 Hh2]". done.
iVs (IH tid (concat vlq) _ Hfaq with "[Hq]") as "[Hq1 Hq2]". by eauto.
iVsIntro. iSplitL "Hh1 Hq1".
+ iDestruct "Hq1" as (vllq) "(%&%&?)". iExists (vl0::vllq).
rewrite big_sepL_cons/=. iFrame. iSplit; iPureIntro; congruence.
+ iDestruct "Hq2" as (vllq) "(%&%&?)". iExists (vl0::vllq).
......@@ -443,9 +444,9 @@ Section types.
simpl. by iDestruct (sum_size_eq with "Hown") as %->.
Qed.
Next Obligation.
iIntros (n tyl Hn tid vl Hdup%Is_true_eq_true) "Hown".
iIntros (n tyl Hn tid vl E Hdup%Is_true_eq_true) "Hown".
iDestruct "Hown" as (i vl') "[% Hown]". subst.
iDestruct ((nth i tyl bot).(ty_dup_dup) with "Hown") as "[Hown1 Hown2]".
iVs ((nth i tyl bot).(ty_dup_dup) with "Hown") as "[Hown1 Hown2]".
- edestruct nth_in_or_default as [| ->]; last done.
rewrite ->forallb_forall in Hdup. auto using Is_true_eq_left.
- iSplitR "Hown1"; eauto.
......
type_incl.v
+ 43
− 29
View file @ 152920f9
......@@ -19,9 +19,7 @@ Section ty_incl.
ty2.(ty_shr) κ tid E l ★ ty1.(ty_size) = ty2.(ty_size)).
Global Instance ty_incl_refl ρ : Reflexive (ty_incl ρ).
Proof.
iIntros (ty tid) "_!==>". iSplit; iIntros "!#"; eauto.
Qed.
Proof. iIntros (ty tid) "_!==>". iSplit; iIntros "!#"; eauto. Qed.
Lemma ty_incl_trans ρ θ ty1 ty2 ty3 :
ty_incl ρ ty1 ty2 → ty_incl θ ty2 ty3 → ty_incl (ρ ★ θ) ty1 ty3.
......@@ -37,11 +35,15 @@ Section ty_incl.
Qed.
Lemma ty_incl_weaken ρ θ ty1 ty2 :
perm_incl ρ θ → ty_incl θ ty1 ty2 → ty_incl ρ ty1 ty2.
ρ ⇒ θ → ty_incl θ ty1 ty2 → ty_incl ρ ty1 ty2.
Proof. iIntros (Hρθ Hρ' tid) "H". iVs (Hρθ with "H"). by iApply Hρ'. Qed.
Global Instance ty_incl_trans' ρ: Duplicable ρ → Transitive (ty_incl ρ).
Proof. intros ??????. eauto using ty_incl_weaken, ty_incl_trans. Qed.
Global Instance ty_incl_preorder ρ: Duplicable ρ → PreOrder (ty_incl ρ).
Proof.
split.
- apply _.
- intros ?????. eauto using ty_incl_weaken, ty_incl_trans.
Qed.
Lemma ty_incl_bot ρ ty : ty_incl ρ ! ty.
Proof. iIntros (tid) "_!==>". iSplit; iIntros "!#*/=[]". Qed.
......@@ -197,30 +199,42 @@ Section ty_incl.
(* FIXME : idem : cannot combine the fractured borrow. *)
Admitted.
(* Lemma ty_incl_cont_frame {n} ρ ρ1 ρ2 : *)
(* Duplicable ρ → (∀ vl : vec val n, perm_incl (ρ ★ ρ2 vl) (ρ1 vl)) → *)
(* ty_incl ρ (cont ρ1) (cont ρ2). *)
(* Proof. *)
(* iIntros (? Hρ1ρ2 tid) "Hρ". *)
(* iVs (inv_alloc lrustN _ (ρ tid) with "[Hρ]") as "#INV". by auto. *)
(* iIntros "!==>". iSplit; iIntros "!#*H"; last by auto. *)
(* iDestruct "H" as (f xl e ?) "[% Hwp]". subst. iExists _, _, _, _. iSplit. done. *)
(* iIntros (vl) "Htl Hown". *)
(* iApply pvs_wp. iInv lrustN as "Hρ" "Hclose". *)
(* iSplit; last by iIntros "{Hρ}!#*_"; auto. *)
(* (* FIXME : I do not know how to prove this. Should we put this *)
(* under a view modality and then put Hρ in an invariant? *) *)
(* admit. *)
(* Admitted. *)
(* Lemma ty_incl_fn_frame {n} ρ ρ1 ρ2 : *)
(* Duplicable ρ → (∀ vl : vec val n, perm_incl (ρ ★ ρ2 vl) (ρ1 vl)) → *)
(* ty_incl ρ (fn ρ1) (fn ρ2). *)
(* (* FIXME : idem. *) *)
(* admit. *)
(* Admitted. *)
Lemma ty_incl_cont {n} ρ ρ1 ρ2 :
Duplicable ρ → (∀ vl : vec val n, ρ ★ ρ2 vl ⇒ ρ1 vl) →
ty_incl ρ (cont ρ1) (cont ρ2).
Proof.
iIntros (? Hρ1ρ2 tid) "Hρ".
iVs (inv_alloc lrustN _ (ρ tid) with "[Hρ]") as "#INV". by auto.
iIntros "!==>". iSplit; iIntros "!#*H"; last by auto.
iDestruct "H" as (f) "[% Hwp]". subst. iExists _. iSplit. done.
iIntros (vl) "Htl Hown".
iApply pvs_wp. iInv lrustN as "Hρ" "Hclose".
(* FIXME : we need some kind of "Invariant of duplicable
propositions" that we can open several times. *)
admit.
Admitted.
Lemma ty_incl_fn {n} ρ ρ1 ρ2 :
Duplicable ρ → (∀ vl : vec val n, ρ ★ ρ2 vl ⇒ ρ1 vl) →
ty_incl ρ (fn ρ1) (fn ρ2).
(* FIXME : idem. *)
admit.
Admitted.
Lemma ty_incl_fn_cont {n} ρ ρf : ty_incl ρ (fn ρf) (cont (n:=n) ρf).
Proof.
iIntros (tid) "_!==>". iSplit; iIntros "!#*H"; last by iSplit.
iDestruct "H" as (f) "[%#H]". subst. iExists _. iSplit. done.
iIntros (vl) "Hρf Htl". iApply "H". by iFrame.
Qed.
(* TODO : forall, when we will have figured out the right definition. *)
Lemma ty_incl_perm_incl ρ ty1 ty2 v :
ty_incl ρ ty1 ty2 → ρ ★ v ◁ ty1 ⇒ v ◁ ty2.
Proof.
iIntros (Hincl tid) "[Hρ Hty1]". iVs (Hincl with "Hρ") as "[#Hownincl _]".
by iApply "Hownincl".
Qed.
End ty_incl.
\ No newline at end of file
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