ticket_lock.v 10.4 KB
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From iris.proofmode Require Import tactics.
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From iris.algebra Require Export auth gset excl.
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From iris.base_logic Require Import auth.
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From iris_logrel Require Export logrel examples.lock examples.counter.
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Definition wait_loop: val :=
  rec: "wait_loop" "x" "lk" :=
    if: "x" = !(Fst "lk")
      then #() (* my turn *)
      else "wait_loop" "x" "lk".

Definition newlock : val :=
  λ: <>, ((* owner *) ref #0, (* next *) ref #0).

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Definition acquire : val := λ: "lk",
  let: "n" := FG_increment (Snd "lk") #() in
  wait_loop "n" "lk".
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Definition release : val :=
  λ: "lk", (Fst "lk") <- !(Fst "lk") + #1.

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Definition LockType : type := ref TNat × ref TNat.
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Hint Unfold LockType : typeable.

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Lemma newlock_type Γ : typed Γ newlock (Unit  LockType).
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Proof. solve_typed. Qed.

Hint Resolve newlock_type : typeable.

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Lemma acquire_type Γ : typed Γ acquire (LockType  TUnit).
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Proof.
  unlock acquire wait_loop.
  econstructor; cbn; solve_typed.
  econstructor; cbn; solve_typed.
  econstructor; cbn; solve_typed.
  econstructor; cbn; solve_typed.
Qed.
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Hint Resolve acquire_type : typeable.

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Lemma release_type Γ : typed Γ release (LockType  TUnit).
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Proof. solve_typed. Qed.

Hint Resolve release_type : typeable.

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Definition lockT : type := : (Unit  TVar 0) × (TVar 0  Unit) × (TVar 0  Unit).
Lemma ticket_lock_typed Γ : typed Γ (Pack (newlock, acquire, release)) lockT.
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Proof.
  apply TPack with LockType.
  asimpl. solve_typed.
Qed.

Class tlockG Σ :=
  tlock_G :> authG Σ (gset_disjUR nat).
Definition tlockΣ : gFunctors :=
  #[ authΣ (gset_disjUR nat) ].

Definition lockPool := gset ((loc * loc * gname) * loc).
Definition lockPoolR := gsetUR ((loc * loc * gname) * loc).

Class lockPoolG Σ :=
  lockPool_inG :> authG Σ lockPoolR.
Section refinement.
  Context `{logrelG Σ, tlockG Σ, lockPoolG Σ}.

  Definition lockInv (lo ln : loc) (γ : gname) (l' : loc) : iProp Σ :=
    ( (o n : nat) (b : bool), lo ↦ᵢ #o  ln ↦ᵢ #n
    own γ ( GSet (seq_set 0 n))  l' ↦ₛ #b
    if b then own γ ( GSet {[ o ]}) else True)%I.

  Definition lockPoolInv (P : lockPool) : iProp Σ :=
    ([ set] rs  P, match rs with
                     | ((lo, ln, γ), l') => lockInv lo ln γ l'
                     end)%I.

  Definition moduleInv γp : iProp Σ :=
    ( (P : lockPool), own γp ( P)  lockPoolInv P)%I.

  Program Definition lockInt (γp : gname) := λne vv,
    ( (lo ln : loc) (γ : gname) (l' : loc),
        vv.1 = (#lo, #ln)%V  vv.2 = #l'⌝
       own γp ( {[(lo, ln, γ), l']}))%I.
  Next Obligation. solve_proper. Qed.

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  Instance lockInt_persistent γp ww : Persistent (lockInt γp ww).
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  Proof. apply _. Qed.

  Lemma lockPool_open_later (P : lockPool) (lo ln : loc) (γ : gname) (l' : loc) :
    (lo, ln, γ, l')  P 
     lockPoolInv P -
    ( lockInv lo ln γ l')   (lockInv lo ln γ l' - lockPoolInv P).
  Proof.
    iIntros (Hrin) "Hreg".
    rewrite /lockPoolInv.
    iDestruct (big_sepS_elem_of_acc _ P _ with "Hreg") as "[Hrs Hreg]"; first apply Hrin.
    by iFrame.
  Qed.

  Lemma lockPool_lookup γp (P : lockPool) x :
    own γp ( P) -
    own γp ( {[ x ]}) -
    x  P.
  Proof.
    iIntros "Ho Hrs".
    iDestruct (own_valid_2 with "Ho Hrs") as %Hfoo.
    iPureIntro.
    apply auth_valid_discrete in Hfoo.
    simpl in Hfoo. destruct Hfoo as [Hfoo _].
    revert Hfoo. rewrite left_id.
    by rewrite gset_included elem_of_subseteq_singleton.
  Qed.

  Lemma lockPool_excl (P : lockPool) (lo ln : loc) γ l' (v : val) :
    lockPoolInv P - lo ↦ᵢ v - (lo, ln, γ, l')  P.
  Proof.
    rewrite /lockPoolInv.
    iIntros "HP Hlo".
    iAssert ((lo, ln, γ, l')  P  False)%I as %Hbaz.
    {
      iIntros (HP).
      rewrite (big_sepS_elem_of _ P _ HP).
      iDestruct "HP" as (a b c) "(Hlo' & Hln & ?)".
      iDestruct (mapsto_valid_2 with "Hlo' Hlo") as %Hfoo;
      compute in Hfoo; contradiction.
    }
    iPureIntro. eauto.
  Qed.

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  Definition N := logrelN.@"locked".

  (* Allocating a new lock *)
  Lemma newlock_refinement Δ Γ γp:
    inv N (moduleInv γp) -
    {(lockInt γp :: Δ); ⤉Γ}  newlock log lock.newlock : (Unit  TVar 0).
  Proof.
    iIntros "#Hinv".
    unlock newlock.
    iApply bin_log_related_arrow_val; eauto.
    { by unlock lock.newlock. }
    iAlways. iIntros (? ?) "/= [% %]"; simplify_eq.
    rel_let_l.
    rel_alloc_l as lo "Hlo".
    rel_apply_r bin_log_related_newlock_r.
    { solve_ndisj. }
    iIntros (l') "Hl'".
    rel_alloc_l_atomic.
    iInv N as (P) "[>HP Hpool]" "Hcl".
    iModIntro. iNext.
    iIntros (ln) "Hln".
    iMod (own_alloc ( (GSet )   (GSet ))) as (γ) "[Hγ Hγ']".
    { by rewrite -auth_both_op. }
    iMod (own_update with "HP") as "[HP Hls]".
    { eapply auth_update_alloc.
      eapply (gset_local_update _ _ ({[(lo, ln, γ, l')]}  P)).
      apply union_subseteq_r. }
    iDestruct (lockPool_excl _ lo ln γ l' with "Hpool Hlo") as %Hnew.
    iMod ("Hcl" with "[-Hls]") as "_".
    { iNext. iExists _; iFrame.
      rewrite /lockPoolInv.
      rewrite big_sepS_insert; last assumption.
      iFrame. iExists _,_,_. iFrame. simpl. iFrame. }
    rel_vals. iModIntro.
    rewrite -gset_op_union.
    iDestruct "Hls" as "[#Hls _]".
    iAlways. iExists _,_,_,_. iFrame "Hls". eauto.
  Qed.

  (* Acquiring a lock *)
  Lemma wait_loop_refinement Δ Γ γp (lo ln : loc) γ (l' : loc) (m : nat) :
    inv N (moduleInv γp) -
    own γp ( {[(lo, ln), γ, l']}) - (* two locks are linked *)
    own γ ( GSet {[m]}) - (* the ticket *)
    {(lockInt γp :: Δ); ⤉Γ} 
      wait_loop #m (#lo, #ln) log lock.acquire #l' : TUnit.
  Proof.
    iIntros "#Hinv #Hls Hticket".
    unlock wait_loop.
    rel_rec_l.
    iLöb as "IH".
    rel_let_l. rel_proj_l.
    rel_load_l_atomic.
    iInv N as (P) "[>HP Hpool]" "Hcl".
    iDestruct (lockPool_lookup with "HP Hls") as %Hls.
    iDestruct (lockPool_open_later with "Hpool") as "[Hlk Hpool]"; first apply Hls.
    rewrite {1}/lockInv.
    iDestruct "Hlk" as (o n' b) "(>Hlo & >Hln & Hseq & Hl' & Hrest)".
    iModIntro. iExists _; iFrame; iNext.
    iIntros "Hlo".
    rel_op_l.
    case_decide; subst; rel_if_l.
    (* Whether the ticket is called out *)
    - destruct b.
      { iDestruct (own_valid_2 with "Hticket Hrest") as %?%gset_disj_valid_op.
        set_solver. }
      rel_apply_r (bin_log_related_acquire_r with "Hl'").
      { solve_ndisj. }
      iIntros "Hl'".
      iMod ("Hcl" with "[-]") as "_".
      { iNext. iExists P; iFrame.
        iApply "Hpool". iExists _,_,_; iFrame. }
      iApply bin_log_related_unit.
    - iMod ("Hcl" with "[-Hticket]") as "_".
      { iNext. iExists P; iFrame.
        iApply "Hpool". iExists _,_,_; by iFrame. }
      rel_rec_l.
      by iApply "IH".
  Qed.

  Lemma acquire_refinement Δ Γ γp :
    inv N (moduleInv γp) -
    {(lockInt γp :: Δ); ⤉Γ}  acquire log lock.acquire : (TVar 0  Unit).
  Proof.
    iIntros "#Hinv".
    unlock acquire; simpl.
    iApply bin_log_related_arrow_val; eauto.
    { by unlock lock.acquire. }
    iAlways. iIntros (? ?) "/= #Hl".
    iDestruct "Hl" as (lo ln γ l') "(% & % & Hls)". simplify_eq.
    rel_let_l. repeat rel_proj_l.
    (* rel_rec_l. (* TODO: cannot find the reduct *) *)
    rel_bind_l (FG_increment _ #()).
    rel_rec_l.
    rel_apply_l (bin_log_FG_increment_logatomic _ (fun n => own γ ( GSet (seq_set 0 n)))%I True%I); first done.
    iAlways.
    iInv N as (P) "[>HP Hpool]" "Hcl".
    iDestruct (lockPool_lookup with "HP Hls") as %Hls.
    iDestruct (lockPool_open_later with "Hpool") as "[Hlk Hpool]"; first apply Hls.
    rewrite {1}/lockInv.
    iDestruct "Hlk" as (o n b) "(>Hlo & >Hln & >Hseq & Hl' & Hrest)".
    iModIntro. iExists _; iFrame.
    iSplit.
    - iDestruct 1 as (m) "[Hln ?]".
      iApply ("Hcl" with "[-]").
      iNext. iExists P; iFrame.
      iApply "Hpool". iExists _,_,_; by iFrame.
    - iIntros (m) "[Hln Hseq] _".
      iMod (own_update with "Hseq") as "[Hseq Hticket]".
      { eapply auth_update_alloc.
        eapply (gset_disj_alloc_empty_local_update _ {[ m ]}).
        apply (seq_set_S_disjoint 0). }
      rewrite -(seq_set_S_union_L 0).
      iMod ("Hcl" with "[-Hticket]") as "_".
      { iNext. iExists P; iFrame.
        iApply "Hpool". iExists _,_,_; by iFrame. }
      simpl. rel_let_l.
      by iApply wait_loop_refinement.
  Qed.

  (* Releasing the lock *)
  Lemma release_refinement Δ Γ γp :
    inv N (moduleInv γp) -
    {(lockInt γp :: Δ); ⤉Γ}  release log lock.release : (TVar 0  Unit).
  Proof.
    iIntros "#Hinv".
    unlock release.
    iApply bin_log_related_arrow_val; eauto.
    { by unlock lock.release. }
    iAlways. iIntros (? ?) "/= #Hl".
    iDestruct "Hl" as (lo ln γ l') "(% & % & Hls)". simplify_eq.
    rel_let_l. repeat rel_proj_l.
    rel_load_l_atomic.
    iInv N as (P) "[>HP Hpool]" "Hcl".
    iDestruct (lockPool_lookup with "HP Hls") as %Hls.
    iDestruct (lockPool_open_later with "Hpool") as "[Hlk Hpool]"; first apply Hls.
    rewrite {1}/lockInv.
    iDestruct "Hlk" as (o n b) "(>Hlo & >Hln & ?)".
    iModIntro. iExists _; iFrame; iNext.
    iIntros "Hlo".
    iMod ("Hcl" with "[-]") as "_".
    { iNext. iExists P; iFrame.
      iApply "Hpool". iExists _,_,_; iFrame. }
    rel_op_l.
    rel_store_l_atomic. clear Hls n b P.
    iInv N as (P) "[>HP Hpool]" "Hcl".
    iDestruct (lockPool_lookup with "HP Hls") as %Hls.
    iDestruct (lockPool_open_later with "Hpool") as "[Hlk Hpool]"; first apply Hls.
    rewrite {1}/lockInv.
    iDestruct "Hlk" as (o' n b) "(>Hlo & >Hln & Hseq & Hl' & Hrest)".
    iModIntro. iExists _; iFrame; iNext.
    iIntros "Hlo".
    rel_apply_r (bin_log_related_release_r with "Hl'").
    { solve_ndisj. }
    iIntros "Hl'".
    iMod ("Hcl" with "[-]") as "_".
    { iNext. iExists P; iFrame.
      iApply "Hpool". iExists _,_,_. by iFrame. }
    iApply bin_log_related_unit.
  Qed.

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  Lemma ticket_lock_refinement Γ :
    Γ  Pack (newlock, acquire, release)
      log
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        Pack (lock.newlock, lock.acquire, lock.release) : lockT.
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  Proof.
    iIntros (Δ).
    iMod (own_alloc ( ( : lockPoolR))) as (γp) "HP"; first done.
    iMod (inv_alloc N _ (moduleInv γp) with "[HP]") as "#Hinv".
    { iNext. iExists . iFrame. by rewrite /lockPoolInv big_sepS_empty. }
    iApply (bin_log_related_pack _ (lockInt γp)).
    repeat iApply bin_log_related_pair.
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    - by iApply newlock_refinement.
    - by iApply acquire_refinement.
    - by iApply release_refinement.
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  Qed.
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End refinement.