ticket_lock.v 9.73 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.
From iris_logrel Require Export logrel examples.lock.

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).

Definition acquire : val :=
  rec: "acquire" "lk" :=
    let: "n" := !(Snd "lk") in
    if: CAS (Snd "lk") "n" ("n" + #1)
      then wait_loop "n" "lk"
      else "acquire" "lk".

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. solve_typed. Qed.

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 lockτ : type := : (Unit  TVar 0) × (TVar 0  Unit) × (TVar 0  Unit).
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Lemma ticket_lock_typed Γ : typed Γ (Pack (newlock, acquire, release)) lockτ.
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.

  Lemma ticket_lock_refinement Γ :
    Γ  Pack (newlock, acquire, release)
      log
        Pack (lock.newlock, lock.acquire, lock.release) : lockτ.
  Proof.
    iIntros (Δ).
    pose (N:=logrelN.@"locked").
    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.
    - (* Allocating a new lock *)
      unlock newlock lock.newlock.
      iApply bin_log_related_arrow_val; eauto.
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      iAlways. iIntros (? ?) "/= [% %]"; simplify_eq.
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      rel_let_l. rel_let_r.
      rel_alloc_r as l' "Hl'".
      rel_alloc_l as lo "Hlo".
      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.
    - (* Acquire *)
      unlock acquire.
      iApply bin_log_related_arrow_val; eauto.
      { by unlock lock.acquire. }
      iAlways. iIntros (? ?) "/= #Hl".
      iDestruct "Hl" as (lo ln γ l') "(% & % & Hls)". simplify_eq.
      iLöb as "IH".
      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 "Hln".
      iMod ("Hcl" with "[-]") as "_".
      { iNext. iExists P; iFrame.
        iApply "Hpool". iExists _,_,_; iFrame. }
      rel_let_l. rel_proj_l. rel_op_l.
      clear Hls o b P.
      rel_cas_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.
      iSplit; iIntros (?); iNext; iIntros "Hln"; rel_if_l.
      + iMod ("Hcl" with "[-]") as "_".
        { iNext. iExists P; iFrame.
          iApply "Hpool". iExists _,_,_; by iFrame. }
        iApply "IH".
      + simplify_eq.
        iMod (own_update with "Hseq") as "[Hseq Hticket]".
        { eapply auth_update_alloc.
          eapply (gset_disj_alloc_empty_local_update _ {[ n ]}).
          apply (seq_set_S_disjoint 0). }
        rewrite -(seq_set_S_union_L 0).
        rewrite Nat.add_1_r.
        iMod ("Hcl" with "[-Hticket]") as "_".
        { iNext. iExists P; iFrame.
          iApply "Hpool". iExists _,_,_; by iFrame. }
        iClear "IH".
        unlock wait_loop.
        rel_rec_l.
        iLöb as "IH".
        rel_let_l. rel_proj_l.
        rel_load_l_atomic. clear Hls P o b.
        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".
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        rel_op_l.
        case_decide; subst; rel_if_l.
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        (* The ticket is called out *)
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        * unlock lock.acquire.
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          rel_rec_r.
          destruct b.
          { iDestruct (own_valid_2 with "Hticket Hrest") as %?%gset_disj_valid_op.
            set_solver. }
          rel_cas_suc_r. rel_if_r.
          iMod ("Hcl" with "[-]") as "_".
          { iNext. iExists P; iFrame.
            iApply "Hpool". iExists _,_,_; by iFrame. }
          iApply bin_log_related_unit.

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        * iMod ("Hcl" with "[-Hticket]") as "_".
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          { iNext. iExists P; iFrame.
            iApply "Hpool". iExists _,_,_; by iFrame. }
          rel_rec_l.
          by iApply "IH".
    - (* Release *)
      unlock release lock.release.
      iApply bin_log_related_arrow_val; eauto.
      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_let_r. rel_store_r.
      iMod ("Hcl" with "[-]") as "_".
      { iNext. iExists P; iFrame.
        iApply "Hpool". iExists _,_,_; iFrame. }
      iApply bin_log_related_unit.
  Qed.
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End refinement.