Commit 5e4112d3 authored by Robbert Krebbers's avatar Robbert Krebbers

Shorter barrier proof that uses auth instead of sts.

parent 79d26fe9
......@@ -9,7 +9,6 @@
theories/barrier/proof.v
theories/barrier/specification.v
theories/barrier/barrier.v
theories/barrier/protocol.v
theories/barrier/example_client.v
theories/barrier/example_joining_existentials.v
......
From iris.program_logic Require Export weakestpre.
From iris.base_logic Require Import invariants lib.saved_prop.
From iris.heap_lang Require Export lang.
From stdpp Require Import functions.
From iris.base_logic Require Import lib.saved_prop lib.sts.
From iris.heap_lang Require Import proofmode.
From iris.algebra Require Import auth gset.
From iris_examples.barrier Require Export barrier.
From iris_examples.barrier Require Import protocol.
Set Default Proof Using "Type".
(** The CMRAs/functors we need. *)
Class barrierG Σ := BarrierG {
barrier_stsG :> stsG Σ sts;
barrier_inG :> inG Σ (authR (gset_disjUR gname));
barrier_savedPropG :> savedPropG Σ;
}.
Definition barrierΣ : gFunctors := #[stsΣ sts; savedPropΣ].
Definition barrierΣ : gFunctors :=
#[ GFunctor (authRF (gset_disjUR gname)); savedPropΣ ].
Instance subG_barrierΣ {Σ} : subG barrierΣ Σ barrierG Σ.
Proof. solve_inG. Qed.
......@@ -20,181 +20,130 @@ Proof. solve_inG. Qed.
(** Now we come to the Iris part of the proof. *)
Section proof.
Context `{!heapG Σ, !barrierG Σ} (N : namespace).
Implicit Types I : gset gname.
Definition ress (P : iProp Σ) (I : gset gname) : iProp Σ :=
( Ψ : gname iProp Σ,
(P - [ set] i I, Ψ i) [ set] i I, saved_prop_own i (Ψ i))%I.
Coercion state_to_val (s : state) : val :=
match s with State Low _ => #false | State High _ => #true end.
Arguments state_to_val !_ / : simpl nomatch.
Definition state_to_prop (s : state) (P : iProp Σ) : iProp Σ :=
match s with State Low _ => P | State High _ => True%I end.
Arguments state_to_prop !_ _ / : simpl nomatch.
Definition barrier_inv (l : loc) (P : iProp Σ) (s : state) : iProp Σ :=
(l s ress (state_to_prop s P) (state_I s))%I.
Definition barrier_ctx (γ : gname) (l : loc) (P : iProp Σ) : iProp Σ :=
sts_ctx γ N (barrier_inv l P).
Definition send (l : loc) (P : iProp Σ) : iProp Σ :=
( γ, barrier_ctx γ l P sts_ownS γ low_states {[ Send ]})%I.
Definition barrier_inv (l : loc) (γ : gname) (P : iProp Σ) : iProp Σ :=
( (b : bool) (γsps : gset gname),
l #b
own γ ( (GSet γsps))
((if b then True else P) -
([ set] γsp γsps, R, saved_prop_own γsp R R)))%I.
Definition recv (l : loc) (R : iProp Σ) : iProp Σ :=
( γ P Q i,
barrier_ctx γ l P sts_ownS γ (i_states i) {[ Change i ]}
saved_prop_own i Q (Q - R))%I.
( γ P R' γsp,
inv N (barrier_inv l γ P)
(R' - R)
own γ ( GSet {[ γsp ]})
saved_prop_own γsp R')%I.
Global Instance barrier_ctx_persistent (γ : gname) (l : loc) (P : iProp Σ) :
Persistent (barrier_ctx γ l P).
Proof. apply _. Qed.
Definition send (l : loc) (P : iProp Σ) : iProp Σ :=
( γ, inv N (barrier_inv l γ P))%I.
(** Setoids *)
Global Instance ress_ne n : Proper (dist n ==> (=) ==> dist n) ress.
Proof. solve_proper. Qed.
Global Instance state_to_prop_ne s :
NonExpansive (state_to_prop s).
Proof. solve_proper. Qed.
Global Instance barrier_inv_ne n l :
Proper (dist n ==> eq ==> dist n) (barrier_inv l).
Proof. solve_proper. Qed.
Global Instance barrier_ctx_ne γ l : NonExpansive (barrier_ctx γ l).
Instance barrier_inv_ne l γ : NonExpansive (barrier_inv l γ).
Proof. solve_proper. Qed.
Global Instance send_ne l : NonExpansive (send l).
Proof. solve_proper. Qed.
Global Instance recv_ne l : NonExpansive (recv l).
Proof. solve_proper. Qed.
(** Helper lemmas *)
Lemma ress_split i i1 i2 Q R1 R2 P I :
i I i1 I i2 I i1 i2
saved_prop_own i Q - saved_prop_own i1 R1 - saved_prop_own i2 R2 -
(Q - R1 R2) - ress P I -
ress P ({[i1;i2]} I {[i]}).
Proof.
iIntros (????) "#HQ #H1 #H2 HQR"; iDestruct 1 as (Ψ) "[HPΨ HΨ]".
iDestruct (big_opS_delete _ _ i with "HΨ") as "[#HΨi HΨ]"; first done.
iExists (<[i1:=R1]> (<[i2:=R2]> Ψ)). iSplitL "HQR HPΨ".
- iPoseProof (saved_prop_agree with "HQ HΨi") as "#Heq".
iNext. iRewrite "Heq" in "HQR". iIntros "HP". iSpecialize ("HPΨ" with "HP").
iDestruct (big_opS_delete _ _ i with "HPΨ") as "[HΨ HPΨ]"; first done.
iDestruct ("HQR" with "HΨ") as "[HR1 HR2]".
rewrite -assoc_L !big_opS_fn_insert'; [|abstract set_solver ..].
by iFrame.
- rewrite -assoc_L !big_opS_fn_insert; [|abstract set_solver ..]. eauto.
Qed.
(** Actual proofs *)
Lemma newbarrier_spec (P : iProp Σ) :
{{{ True }}} newbarrier #() {{{ l, RET #l; recv l P send l P }}}.
Proof.
iIntros (Φ) "_ HΦ".
rewrite -wp_fupd /newbarrier /=. wp_lam. wp_alloc l as "Hl".
iIntros (Φ) "_ HΦ". iApply wp_fupd. wp_lam. wp_alloc l as "Hl".
iApply ("HΦ" with "[> -]").
iMod (saved_prop_alloc P) as (γ) "#?".
iMod (sts_alloc (barrier_inv l P) _ N (State Low {[ γ ]}) with "[-]")
as (γ') "[#? Hγ']"; eauto.
{ iNext. rewrite /barrier_inv /=. iFrame.
iExists (const P). rewrite !big_opS_singleton /=. eauto. }
iAssert (barrier_ctx γ' l P)%I as "#?".
{ done. }
iAssert (sts_ownS γ' (i_states γ) {[Change γ]}
sts_ownS γ' low_states {[Send]})%I with "[> -]" as "[Hr Hs]".
{ iApply sts_ownS_op; eauto using i_states_closed, low_states_closed.
- set_solver.
- iApply (sts_own_weaken with "Hγ'");
auto using sts.closed_op, i_states_closed, low_states_closed;
abstract set_solver. }
iModIntro. iSplitL "Hr".
- iExists γ', P, P, γ. iFrame. auto.
- rewrite /send. auto.
iMod (saved_prop_alloc P) as (γsp) "#Hsp".
iMod (own_alloc ( GSet {[ γsp ]} GSet {[ γsp ]})) as (γ) "[H● H◯]".
{ by apply auth_both_valid. }
iMod (inv_alloc N _ (barrier_inv l γ P) with "[Hl H●]") as "#Hinv".
{ iExists false, {[ γsp ]}. iIntros "{$Hl $H●} !> HP".
rewrite big_sepS_singleton; eauto. }
iModIntro; iSplitL "H◯".
- iExists γ, P, P, γsp. iFrame; auto.
- by iExists γ.
Qed.
Lemma signal_spec l P :
{{{ send l P P }}} signal #l {{{ RET #(); True }}}.
Proof.
rewrite /signal /=.
iIntros (Φ) "[Hs HP] HΦ". iDestruct "Hs" as (γ) "[#Hsts Hγ]". wp_lam.
iMod (sts_openS (barrier_inv l P) _ _ γ with "[Hγ]")
as ([p I]) "(% & [Hl Hr] & Hclose)"; eauto.
destruct p; [|done]. wp_store.
iSpecialize ("HΦ" with "[#]") => //. iFrame "HΦ".
iMod ("Hclose" $! (State High I) ( : propset token) with "[-]"); last done.
iSplit; [iPureIntro; by eauto using signal_step|].
rewrite /barrier_inv /ress /=. iNext. iFrame "Hl".
iDestruct "Hr" as (Ψ) "[Hr Hsp]"; iExists Ψ; iFrame "Hsp".
iNext. iIntros "_"; by iApply "Hr".
iIntros (Φ) "[Hs HP] HΦ". iDestruct "Hs" as (γ) "#Hinv". wp_lam.
iInv N as ([] γsps) "(>Hl & H● & HRs)".
{ wp_store. iModIntro. iSplitR "HΦ"; last by iApply "HΦ".
iExists true, γsps. iFrame. }
wp_store. iDestruct ("HRs" with "HP") as "HRs".
iModIntro. iSplitR "HΦ"; last by iApply "HΦ".
iExists true, γsps. iFrame; eauto.
Qed.
Lemma wait_spec l P:
{{{ recv l P }}} wait #l {{{ RET #(); P }}}.
Proof.
rename P into R.
iIntros (Φ) "Hr HΦ"; iDestruct "Hr" as (γ P Q i) "(#Hsts & Hγ & #HQ & HQR)".
iIntros (Φ) "HR HΦ". iDestruct "HR" as (γ P R' γsp) "(#Hinv & HR & H◯ & #Hsp)".
iLöb as "IH". wp_rec. wp_bind (! _)%E.
iMod (sts_openS (barrier_inv l P) _ _ γ with "[Hγ]")
as ([p I]) "(% & [Hl Hr] & Hclose)"; eauto.
wp_load. destruct p.
- iMod ("Hclose" $! (State Low I) {[ Change i ]} with "[Hl Hr]") as "Hγ".
{ iSplit; first done. rewrite /barrier_inv /=. by iFrame. }
iAssert (sts_ownS γ (i_states i) {[Change i]})%I with "[> Hγ]" as "Hγ".
{ iApply (sts_own_weaken with "Hγ"); eauto using i_states_closed. }
iModIntro. wp_if.
iApply ("IH" with "Hγ [HQR] [HΦ]"); auto.
- (* a High state: the comparison succeeds, and we perform a transition and
return to the client *)
iDestruct "Hr" as (Ψ) "[HΨ Hsp]".
iDestruct (big_opS_delete _ _ i with "Hsp") as "[#HΨi Hsp]"; first done.
iAssert ( (Ψ i [ set] j I {[i]}, Ψ j))%I with "[HΨ]" as "[HΨ HΨ']".
{ iNext. iApply (big_opS_delete _ _ i); first done. by iApply "HΨ". }
iMod ("Hclose" $! (State High (I {[ i ]})) with "[HΨ' Hl Hsp]").
{ iSplit; [iPureIntro; by eauto using wait_step|].
rewrite /barrier_inv /=. iNext. iFrame "Hl". iExists Ψ; iFrame. auto. }
iPoseProof (saved_prop_agree with "HQ HΨi") as "#Heq".
iModIntro. wp_if.
iApply "HΦ". iApply "HQR". by iRewrite "Heq".
iInv N as ([] γsps) "(>Hl & >H● & HRs)"; last first.
{ wp_load. iModIntro. iSplitL "Hl H● HRs".
{ iExists false, γsps. iFrame. }
by wp_apply ("IH" with "[$] [$]"). }
iSpecialize ("HRs" with "[//]"). wp_load.
iDestruct (own_valid_2 with "H● H◯")
as %[Hvalid%gset_disj_included%elem_of_subseteq_singleton _]%auth_both_valid.
iDestruct (big_sepS_delete with "HRs") as "[HR'' HRs]"; first done.
iDestruct "HR''" as (R'') "[#Hsp' HR'']".
iDestruct (saved_prop_agree with "Hsp Hsp'") as "#Heq".
iMod (own_update_2 with "H● H◯") as "H●".
{ apply (auth_update_dealloc _ _ (GSet (γsps {[ γsp ]}))).
apply gset_disj_dealloc_local_update. }
iIntros "!>". iSplitL "Hl H● HRs".
{ iDestruct (bi.later_intro with "HRs") as "HRs".
iModIntro. iExists true, (γsps {[ γsp ]}). iFrame; eauto. }
wp_if. iApply "HΦ". iApply "HR". by iRewrite "Heq".
Qed.
Lemma recv_split E l P1 P2 :
N E recv l (P1 P2) ={E}= recv l P1 recv l P2.
Proof.
rename P1 into R1; rename P2 into R2.
iIntros (?). iDestruct 1 as (γ P Q i) "(#Hsts & Hγ & #HQ & HQR)".
iMod (sts_openS (barrier_inv l P) _ _ γ with "[Hγ]")
as ([p I]) "(% & [Hl Hr] & Hclose)"; eauto.
iMod (saved_prop_alloc_cofinite I) as (i1) "[% #Hi1]".
iMod (saved_prop_alloc_cofinite (I {[i1]}))
as (i2) "[Hi2' #Hi2]"; iDestruct "Hi2'" as %Hi2.
rewrite ->not_elem_of_union, elem_of_singleton in Hi2; destruct Hi2.
iMod ("Hclose" $! (State p ({[i1; i2]} I {[i]}))
{[Change i1; Change i2 ]} with "[-]") as "Hγ".
{ iSplit; first by eauto using split_step.
rewrite /barrier_inv /=. iNext. iFrame "Hl".
by iApply (ress_split with "HQ Hi1 Hi2 HQR"). }
iAssert (sts_ownS γ (i_states i1) {[Change i1]}
sts_ownS γ (i_states i2) {[Change i2]})%I with "[> -]" as "[Hγ1 Hγ2]".
{ iApply sts_ownS_op; eauto using i_states_closed, low_states_closed.
- abstract set_solver.
- iApply (sts_own_weaken with "Hγ");
eauto using sts.closed_op, i_states_closed.
abstract set_solver. }
iModIntro; iSplitL "Hγ1".
- iExists γ, P, R1, i1. iFrame; auto.
- iExists γ, P, R2, i2. iFrame; auto.
iIntros (?). iDestruct 1 as (γ P R' γsp) "(#Hinv & HR & H◯ & #Hsp)".
iInv N as (b γsps) "(>Hl & >H● & HRs)".
iDestruct (own_valid_2 with "H● H◯")
as %[Hvalid%gset_disj_included%elem_of_subseteq_singleton _]%auth_both_valid.
iMod (own_update_2 with "H● H◯") as "H●".
{ apply (auth_update_dealloc _ _ (GSet (γsps {[ γsp ]}))).
apply gset_disj_dealloc_local_update. }
set (γsps' := γsps {[γsp]}).
iMod (saved_prop_alloc_cofinite γsps' R1) as (γsp1 Hγsp1) "#Hsp1".
iMod (saved_prop_alloc_cofinite (γsps' {[ γsp1 ]}) R2)
as (γsp2 [? ?%not_elem_of_singleton]%not_elem_of_union) "#Hsp2".
iMod (own_update _ _ ( _ ( GSet {[ γsp1 ]} (GSet {[ γsp2 ]})))
with "H●") as "(H● & H◯1 & H◯2)".
{ rewrite -auth_frag_op gset_disj_union; last set_solver.
apply auth_update_alloc, (gset_disj_alloc_empty_local_update _ {[ γsp1; γsp2 ]}).
set_solver. }
iModIntro. iSplitL "HR Hl HRs H●".
{ iModIntro. iExists b, ({[γsp1; γsp2]} γsps').
iIntros "{$Hl $H●} HP". iSpecialize ("HRs" with "HP").
iDestruct (big_sepS_delete with "HRs") as "[HR'' HRs]"; first done.
iDestruct "HR''" as (R'') "[#Hsp' HR'']".
iDestruct (saved_prop_agree with "Hsp Hsp'") as "#Heq".
iAssert ( R')%I with "[HR'']" as "HR'"; [iNext; by iRewrite "Heq"|].
iDestruct ("HR" with "HR'") as "[HR1 HR2]".
iApply big_sepS_union; [set_solver|iFrame "HRs"].
iApply big_sepS_union; [set_solver|].
iSplitL "HR1"; rewrite big_sepS_singleton; eauto. }
iModIntro; iSplitL "H◯1".
- iExists γ, P, R1, γsp1. iFrame; auto.
- iExists γ, P, R2, γsp2. iFrame; auto.
Qed.
Lemma recv_weaken l P1 P2 : (P1 - P2) - recv l P1 - recv l P2.
Proof.
iIntros "HP". iDestruct 1 as (γ P Q i) "(#Hctx&Hγ&Hi&HP1)".
iExists γ, P, Q, i. iFrame "Hctx Hγ Hi".
iNext. iIntros "HQ". by iApply "HP"; iApply "HP1".
iIntros "HP". iDestruct 1 as (γ P R' i) "(#Hinv & HR & H◯)".
iExists γ, P, R', i. iIntros "{$Hinv $H◯} !> HQ". iApply "HP". by iApply "HR".
Qed.
Lemma recv_mono l P1 P2 : (P1 P2) recv l P1 recv l P2.
Proof. iIntros (HP) "H". iApply (recv_weaken with "[] H"). iApply HP. Qed.
End proof.
Typeclasses Opaque barrier_ctx send recv.
Typeclasses Opaque send recv.
From iris.algebra Require Export sts.
From iris.base_logic Require Import lib.own.
From stdpp Require Export gmap.
Set Default Proof Using "Type".
(** The STS describing the main barrier protocol. Every state has an index-set
associated with it. These indices are actually [gname], because we use them
with saved propositions. *)
Inductive phase := Low | High.
Record state := State { state_phase : phase; state_I : gset gname }.
Add Printing Constructor state.
Inductive token := Change (i : gname) | Send.
Global Instance stateT_inhabited: Inhabited state := populate (State Low ).
Global Instance Change_inj : Inj (=) (=) Change.
Proof. by injection 1. Qed.
Inductive prim_step : relation state :=
| ChangeI p I2 I1 : prim_step (State p I1) (State p I2)
| ChangePhase I : prim_step (State Low I) (State High I).
Definition tok (s : state) : propset token :=
{[ t | i, t = Change i i state_I s ]}
(if state_phase s is High then {[ Send ]} else ).
Global Arguments tok !_ /.
Canonical Structure sts := sts.Sts prim_step tok.
(* The set of states containing some particular i *)
Definition i_states (i : gname) : propset state := {[ s | i state_I s ]}.
(* The set of low states *)
Definition low_states : propset state := {[ s | state_phase s = Low ]}.
Lemma i_states_closed i : sts.closed (i_states i) {[ Change i ]}.
Proof.
split; first (intros [[] I]; set_solver).
(* If we do the destruct of the states early, and then inversion
on the proof of a transition, it doesn't work - we do not obtain
the equalities we need. So we destruct the states late, because this
means we can use "destruct" instead of "inversion". *)
intros s1 s2 Hs1 [T1 T2 Hdisj Hstep'].
inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
destruct Htrans as [[] ??|]; done || set_solver.
Qed.
Lemma low_states_closed : sts.closed low_states {[ Send ]}.
Proof.
split; first (intros [??]; set_solver).
intros s1 s2 Hs1 [T1 T2 Hdisj Hstep'].
inversion_clear Hstep' as [? ? ? ? Htrans _ _ Htok].
destruct Htrans as [[] ??|]; done || set_solver.
Qed.
(* Proof that we can take the steps we need. *)
Lemma signal_step I : sts.steps (State Low I, {[Send]}) (State High I, ).
Proof. apply rtc_once. constructor; first constructor; set_solver. Qed.
Lemma wait_step i I :
i I
sts.steps (State High I, {[ Change i ]}) (State High (I {[ i ]}), ).
Proof.
intros. apply rtc_once.
constructor; first constructor; [set_solver..|].
apply elem_of_equiv=>-[j|]; last set_solver.
destruct (decide (i = j)); set_solver.
Qed.
Lemma split_step p i i1 i2 I :
i I i1 I i2 I i1 i2
sts.steps
(State p I, {[ Change i ]})
(State p ({[i1; i2]} I {[i]}), {[ Change i1; Change i2 ]}).
Proof.
intros. apply rtc_once. constructor; first constructor.
- destruct p; set_solver.
- destruct p; set_solver.
- apply elem_of_equiv=> /= -[j|]; last set_solver.
set_unfold; rewrite !(inj_iff Change).
assert (Change j match p with Low => : propset token | High => {[Send]} end False)
as -> by (destruct p; set_solver).
destruct (decide (i1 = j)) as [->|]; first naive_solver.
destruct (decide (i2 = j)) as [->|]; first naive_solver.
destruct (decide (i = j)) as [->|]; naive_solver.
Qed.
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