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atomic_persists.v
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atomic_persists.v
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From iris.proofmode Require Import tactics.
From iris.algebra Require Import excl.
From Perennial.program_logic Require Import staged_invariant.
From self.base Require Import primitive_laws.
From self.lang Require Import lang.
From self.high Require Import dprop.
From self.lang Require Import notation lang.
From self.algebra Require Import view.
From self.base Require Import primitive_laws class_instances crash_borrow.
From self.high Require Import proofmode wpc_proofmode.
From self.high Require Import crash_weakestpre modalities weakestpre
weakestpre_na weakestpre_at recovery_weakestpre protocol crash_borrow no_buffer
abstract_state_instances locations protocol if_rec.
From self.high.modalities Require Import fence no_buffer.
From self.examples Require Import spin_lock crash_lock.
Section program.
Definition leftProg (x y : loc) lock : expr :=
with_lock lock (
#x <-_NA #true ;;
#y <-_NA #true ;;
Flush #x ;;
Flush #y ;;
Fence
).
Definition rightProg (x y z : loc) lock : expr :=
with_lock lock (
if: !_NA #x = #true
then
#z <-_NA #true
else #()
).
Definition prog (x y z : loc) : expr :=
let: "lock" := mk_lock #() in
Fork (leftProg x y "lock") ;; rightProg y x z "lock".
Definition recovery (x y z : loc) : expr :=
if: !_NA #z = #true
then
assert: !_NA #x = #true ;;
assert: !_NA #y = #true
else #().
End program.
Section spec.
Context `{nvmG Σ, lockG Σ}.
Context `{!stagedG Σ}.
Context (x y z : loc).
Program Definition prot_bool : LocationProtocol bool :=
{| p_inv (b : bool) v := ⌜ v = #b ⌝%I;
p_bumper b := b; |}.
Global Instance prot_bool_conditions : ProtocolConditions prot_bool.
Proof. split; try apply _. iIntros. by iApply post_crash_flush_pure. Qed.
Definition prot_z :=
{| p_inv (b : bool) (v : val) :=
if b
then
⌜ v = #true ⌝ ∗ flush_lb x prot_bool true ∗ flush_lb y prot_bool true
else ⌜ v = #false ⌝;
p_bumper := id; |}%I.
Global Instance prot_z_conditions : ProtocolConditions prot_z.
Proof.
split; try apply _.
iIntros ([|] ?) "H !>"; simpl; last done.
rewrite -2!persist_lb_to_flush_lb.
iDestruct "H" as "($ & [$ _] & [$ _])".
Qed.
Definition lock_res : dProp Σ :=
∃ ss b,
"#pLbX" ∷ persist_lb x prot_bool false ∗
"#pLbY" ∷ persist_lb y prot_bool false ∗
"#fLbX" ∷ flush_lb x prot_bool b ∗
"#fLbY" ∷ flush_lb y prot_bool b ∗
"xPts" ∷ x ↦_{prot_bool} (ss ++ [b]) ∗
"yPts" ∷ y ↦_{prot_bool} (ss ++ [b]).
Definition lock_condition : dProp Σ :=
(* [x] and [y] *)
(∃ (xss yss : list bool) (bx byy : bool) bx' by',
"#pLbX" ∷ persist_lb x prot_bool bx' ∗
"#pLbY" ∷ persist_lb y prot_bool by' ∗
"xPts" ∷ x ↦_{prot_bool} (xss ++ [bx]) ∗
"yPts" ∷ y ↦_{prot_bool} (yss ++ [byy])).
Lemma lock_condition_impl bx' by' bx byy xss yss :
persist_lb x prot_bool bx' -∗
persist_lb y prot_bool by' -∗
x ↦_{prot_bool} (xss ++ [bx]) -∗
y ↦_{prot_bool} (yss ++ [byy]) -∗
<PC> lock_condition.
Proof.
iIntros "flX flY xPts yPts".
iCrashIntro.
iDestruct "flX" as "[pLbX (% & ? & xCrashed)]".
iDestruct "flY" as "[pLbY (% & ? & yCrashed)]".
iDestruct (crashed_in_if_rec with "xCrashed xPts")
as (??) "(? & crashedIn2 & xPts)".
iDestruct (crashed_in_if_rec with "yCrashed yPts")
as (??) "(? & crashedInY & yPts)".
repeat iExists _.
iFrameF "pLbX".
iFrameF "pLbY".
iFrame "xPts".
iFrame "yPts".
Qed.
Ltac solve_lc :=
try iModIntro (|={_}=> _)%I;
iDestruct (lock_condition_impl with "pLbX pLbY xPts yPts") as "$".
Definition crash_condition : dProp Σ :=
(∃ (zss : list bool) (bz : bool) bz',
"#zPerLb" ∷ persist_lb z prot_z bz' ∗
"zPts" ∷ z ↦_{prot_z} (zss ++ [bz])).
Lemma crash_condition_impl_alt zss bz b :
persist_lb z prot_z b -∗
z ↦_{prot_z} (zss ++ [bz]) -∗
<PC> crash_condition.
Proof.
iIntros "zPerLb zPts".
iCrashIntro.
iDestruct "zPerLb" as "[zPerLb (% & ? & crashedIn)]".
iDestruct (crashed_in_if_rec with "crashedIn zPts")
as (??) "(? & crashedIn2 & zPts)".
simpl.
iExists _, _, _.
iFrame "zPts".
iApply (crashed_in_persist_lb with "crashedIn2").
Qed.
Lemma crash_condition_impl bz b :
persist_lb z prot_z b -∗
z ↦_{prot_z} [bz] -∗
<PC> crash_condition.
Proof. iApply (crash_condition_impl_alt [] bz b). Qed.
Ltac solve_cc :=
try iModIntro (|={_}=> _)%I;
iDestruct (crash_condition_impl with "zPerLb zPts") as "$".
Ltac solve_cc_alt :=
try iModIntro (|={_}=> _)%I;
iDestruct (crash_condition_impl_alt with "zPerLb zPts") as "$".
Global Instance buffer_free_view_objective (P : dProp Σ) :
ViewObjective P → BufferFree P.
Proof.
intros ?.
rewrite /IntoNoBuffer.
iModel.
rewrite no_buffer_at.
iApply view_objective_at.
Qed.
Lemma prog_spec :
pre_borrow_d -∗
persist_lb x prot_bool false -∗
persist_lb y prot_bool false -∗
persist_lb z prot_z false -∗
x ↦_{prot_bool} [false] -∗
y ↦_{prot_bool} [false] -∗
z ↦_{prot_z} [false] -∗
WPC prog x y z @ ⊤
{{ v, True }}
{{ (<PC> crash_condition) ∗ <PC> lock_condition }}.
Proof.
iIntros "pb #xPerLb #yPerLb #zPerLb xPts yPts zPts".
rewrite /prog.
eassert (
(let: "lock" := mk_lock #() in
Fork (leftProg x y "lock");; rightProg y x z "lock")%E
=
fill _ (mk_lock #())
) as ->.
{
reshape_expr (
(let: "lock" := mk_lock #() in
Fork (leftProg x y "lock");; rightProg y x z "lock")%E)
ltac:(fun K e' => apply (@eq_refl _ (fill K e'))).
}
iApply (
newlock_crash_spec (nroot .@ "lock") _ lock_res (<PC> lock_condition)%I
_
(λ _, True)%I
(<PC> crash_condition)
with "[xPts yPts] [] [-]").
{ iExists [], _. simpl.
rewrite -2!persist_lb_to_flush_lb.
iFrame "xPerLb yPerLb xPts yPts". }
{ iModIntro.
iNamed 1.
iApply (lock_condition_impl with "pLbX pLbY xPts yPts"). }
iSplit; first solve_cc.
iIntros (lk γ) "#isLock".
wpc_pures; first solve_cc.
wpc_bind (Fork _)%E.
iApply (wpc_fork with "[]").
- (* Show safety of the forked off thread. I.e., the _left_ thread. *)
iNext.
wpc_apply (with_lock_spec with "[$isLock]"); first done.
iSplit; first done.
setoid_rewrite (left_id (True%I) _ _).
iNamed 1.
(* write to [x] *)
wpc_bind (_ <-_NA _)%E.
iApply wpc_atomic_no_mask.
iSplit.
{ solve_lc. }
iApply (wp_store_na _ prot_bool _ _ b true with "[$xPts]"); eauto.
{ rewrite last_snoc. done. }
{ destruct b; done. }
iNext. iIntros "xPts".
iSplit.
{ solve_lc. }
iModIntro.
wpc_pures. { solve_lc. }
(* write to [y] *)
wpc_bind (_ <-_NA _)%E.
iApply wpc_atomic_no_mask.
iSplit. { solve_lc. }
iApply (wp_store_na _ prot_bool _ _ b true with "[$yPts]"); eauto.
{ rewrite last_snoc. done. }
{ destruct b; done. }
iNext. iIntros "yPts".
iSplit. { solve_lc. }
iModIntro.
(* flushes *)
wpc_pures. { solve_lc. }
wpc_bind (Flush _)%E.
iApply wpc_atomic_no_mask.
iSplit. { solve_lc. }
iApply (wp_flush_na with "xPts").
iNext.
iIntros "(xPts & #xLowerBound & ?)".
iSplit. { solve_lc. }
iModIntro.
wpc_pures.
{ solve_lc. }
wpc_bind (Flush _)%E.
iApply wpc_atomic_no_mask.
iSplit. { solve_lc. }
iApply (wp_flush_na with "yPts").
iNext.
iIntros "(yPts & #yLowerBound & ?)".
iSplit. { solve_lc. }
iModIntro.
wpc_pures.
{ solve_lc. }
iApply wpc_atomic_no_mask. iSplit. { solve_lc. }
wp_apply wp_fence.
iModIntro. iModIntro.
iSplit.
{ solve_lc. }
iModIntro. rewrite left_id.
iExists _, true.
iFrameF "pLbX". iFrameF "pLbY". iFrameF "xLowerBound".
iFrameF "yLowerBound". iFrameF "xPts". iFrame "yPts".
- (* verify the _right_ thread. *)
iSplit; first solve_cc.
iNext.
wpc_pure1 _; first solve_cc.
wpc_pure1 _; first solve_cc.
wpc_apply (with_lock_spec with "[$isLock zPts]"); first done.
iSplit; first solve_cc.
iNamed 1.
wpc_bind (!_NA _)%E.
iApply wpc_atomic_no_mask.
iSplit.
{ solve_lc. solve_cc. }
iApply (wp_load_na with "[$yPts]").
{ apply last_snoc. }
{ iModIntro.
iIntros (?). iIntros "#H". iFrame "H". rewrite right_id. iApply "H". }
iNext. iIntros (?) "[yPts ->]".
iSplit. { solve_lc. solve_cc. }
iModIntro.
destruct b.
* wpc_pures. { solve_lc. solve_cc. }
iApply wpc_atomic_no_mask.
iSplit. { solve_lc. solve_cc. }
iApply (wp_store_na _ prot_z _ _ _ true with "[$zPts]"); eauto.
{ simpl. iSplitPure; first done.
iFrame "fLbX". done. }
iNext. iIntros "zPts".
iSplit.
{ solve_lc. solve_cc_alt. }
iModIntro.
solve_cc_alt.
iExists _, true.
iFrameF "xPerLb". iFrameF "yPerLb". iFrameF "fLbX". iFrameF "fLbY".
iFrameF "xPts". iFrame "yPts".
* wpc_pures.
{ solve_lc. solve_cc. }
iModIntro.
rewrite right_id.
solve_cc.
iExists _, false.
iFrameF "xPerLb". iFrameF "yPerLb". iFrameF "fLbX". iFrameF "fLbY".
iFrameF "xPts". iFrame "yPts".
Qed.
Instance if_else_persistent {PROP : bi} (b : bool) (P Q : PROP) :
Persistent P →
Persistent Q →
Persistent (if b then P else Q).
Proof. intros ??. destruct b; done. Qed.
Lemma recovery_prog_spec s E :
crash_condition ∗ lock_condition -∗
WPC recovery x y z @ s; E
{{ _, True }}
{{ (<PC> crash_condition) ∗ <PC> lock_condition }}.
Proof.
iIntros "[A B]".
iNamed "A".
iNamed "B".
rewrite /recovery.
(* load [z] *)
wpc_bind (!_NA _)%E.
iApply wpc_atomic_no_mask.
iSplit. { solve_cc_alt. solve_lc. }
iApply (wp_load_na with "[$zPts]").
{ apply last_snoc. }
{ iIntros "!>" (?) "#H". iFrame "H". rewrite right_id. iApply "H". }
iNext. iIntros (?) "[zPts #H]".
iSplit. { solve_cc_alt. solve_lc. }
iModIntro.
simpl.
destruct bz.
- iDestruct "H" as (->) "[flushX flushY]".
rewrite /assert.
wpc_pures. { solve_cc_alt. solve_lc. }
iDestruct (mapsto_na_store_lb_incl with "[] xPts") as %incl.
{ iApply flush_lb_to_store_lb. iApply "flushX". }
inversion incl.
iDestruct (mapsto_na_store_lb_incl with "[] yPts") as %inclY.
{ iApply flush_lb_to_store_lb. iApply "flushY". }
inversion inclY.
wpc_bind (!_NA _)%E.
iApply wpc_atomic_no_mask.
iSplit. { solve_lc. solve_cc_alt. }
iApply (wp_load_na with "[$xPts]").
{ apply last_snoc. }
{ iModIntro.
iIntros (?). iIntros "#H". iFrame "H". rewrite right_id. iApply "H". }
iNext. iIntros (?) "[xPts ->]".
iSplit. { solve_lc. solve_cc_alt. }
iModIntro.
wpc_pures. { solve_cc_alt. solve_lc. }
wpc_bind (!_NA _)%E.
iApply wpc_atomic_no_mask.
iSplit. { solve_lc. solve_cc_alt. }
iApply (wp_load_na with "[$yPts]").
{ apply last_snoc. }
{ iModIntro.
iIntros (?). iIntros "#H". iFrame "H". rewrite right_id. iApply "H". }
iNext. iIntros (?) "[yPts ->]".
iSplit. { solve_lc. solve_cc_alt. }
iModIntro.
wpc_pures. { solve_cc_alt. solve_lc. }
iModIntro. done.
- iDestruct "H" as %->.
wpc_pures. { solve_cc_alt. solve_lc. }
iModIntro. done.
Qed.
End spec.