proof.v 8.73 KB
Newer Older
1 2 3
From iris.program_logic Require Export weakestpre.
From iris.heap_lang Require Export lang.
From iris.heap_lang.lib.barrier Require Export barrier.
4
From iris.prelude Require Import functions.
5
From iris.base_logic Require Import big_op lib.saved_prop lib.sts.
Robbert Krebbers's avatar
Robbert Krebbers committed
6
From iris.heap_lang Require Import proofmode.
Ralf Jung's avatar
Ralf Jung committed
7
From iris.heap_lang.lib.barrier Require Import protocol.
8
Set Default Proof Using "Type*".
9

10
(** The CMRAs/functors we need. *)
11 12
(* Not bundling heapG, as it may be shared with other users. *)
Class barrierG Σ := BarrierG {
13 14
  barrier_stsG :> stsG Σ sts;
  barrier_savedPropG :> savedPropG Σ idCF;
15
}.
16
Definition barrierΣ : gFunctors := #[stsΣ sts; savedPropΣ idCF].
17

18 19
Instance subG_barrierΣ {Σ} : subG barrierΣ Σ  barrierG Σ.
Proof. intros [? [? _]%subG_inv]%subG_inv. split; apply _. Qed.
20 21 22

(** Now we come to the Iris part of the proof. *)
Section proof.
23
Context `{!heapG Σ, !barrierG Σ} (N : namespace).
Robbert Krebbers's avatar
Robbert Krebbers committed
24
Implicit Types I : gset gname.
25

26 27
Definition ress (P : iProp Σ) (I : gset gname) : iProp Σ :=
  ( Ψ : gname  iProp Σ,
28
     (P - [ set] i  I, Ψ i)  [ set] i  I, saved_prop_own i (Ψ i))%I.
29 30

Coercion state_to_val (s : state) : val :=
31
  match s with State Low _ => #false | State High _ => #true end.
Robbert Krebbers's avatar
Robbert Krebbers committed
32
Arguments state_to_val !_ / : simpl nomatch.
33

34
Definition state_to_prop (s : state) (P : iProp Σ) : iProp Σ :=
35
  match s with State Low _ => P | State High _ => True%I end.
Robbert Krebbers's avatar
Robbert Krebbers committed
36
Arguments state_to_prop !_ _ / : simpl nomatch.
37

38
Definition barrier_inv (l : loc) (P : iProp Σ) (s : state) : iProp Σ :=
39
  (l  s  ress (state_to_prop s P) (state_I s))%I.
40

41
Definition barrier_ctx (γ : gname) (l : loc) (P : iProp Σ) : iProp Σ :=
42
  sts_ctx γ N (barrier_inv l P).
43

44
Definition send (l : loc) (P : iProp Σ) : iProp Σ :=
45
  ( γ, barrier_ctx γ l P  sts_ownS γ low_states {[ Send ]})%I.
46

47
Definition recv (l : loc) (R : iProp Σ) : iProp Σ :=
48
  ( γ P Q i,
49 50
    barrier_ctx γ l P  sts_ownS γ (i_states i) {[ Change i ]} 
    saved_prop_own i Q   (Q - R))%I.
51

52
Global Instance barrier_ctx_persistent (γ : gname) (l : loc) (P : iProp Σ) :
53
  PersistentP (barrier_ctx γ l P).
54 55 56 57
Proof. apply _. Qed.

Typeclasses Opaque barrier_ctx send recv.

58
(** Setoids *)
59 60 61 62
Global Instance ress_ne n : Proper (dist n ==> (=) ==> dist n) ress.
Proof. solve_proper. Qed.
Global Instance state_to_prop_ne n s :
  Proper (dist n ==> dist n) (state_to_prop s).
63
Proof. solve_proper. Qed.
64
Global Instance barrier_inv_ne n l :
65 66
  Proper (dist n ==> eq ==> dist n) (barrier_inv l).
Proof. solve_proper. Qed.
67
Global Instance barrier_ctx_ne n γ l : Proper (dist n ==> dist n) (barrier_ctx γ l).
68
Proof. solve_proper. Qed. 
69
Global Instance send_ne n l : Proper (dist n ==> dist n) (send l).
70
Proof. solve_proper. Qed.
71
Global Instance recv_ne n l : Proper (dist n ==> dist n) (recv l).
72
Proof. solve_proper. Qed.
73 74

(** Helper lemmas *)
75
Lemma ress_split i i1 i2 Q R1 R2 P I :
76
  i  I  i1  I  i2  I  i1  i2 
77 78 79
  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]}).
80
Proof.
81
  iIntros (????) "#HQ #H1 #H2 HQR"; iDestruct 1 as (Ψ) "[HPΨ HΨ]".
82
  iDestruct (big_sepS_delete _ _ i with "HΨ") as "[#HΨi HΨ]"; first done.
Robbert Krebbers's avatar
Robbert Krebbers committed
83
  iExists (<[i1:=R1]> (<[i2:=R2]> Ψ)). iSplitL "HQR HPΨ".
84
  - iPoseProof (saved_prop_agree i Q (Ψ i) with "[#]") as "Heq"; first by iSplit.
85 86 87
    iNext. iRewrite "Heq" in "HQR". iIntros "HP". iSpecialize ("HPΨ" with "HP").
    iDestruct (big_sepS_delete _ _ i with "HPΨ") as "[HΨ HPΨ]"; first done.
    iDestruct ("HQR" with "HΨ") as "[HR1 HR2]".
88
    rewrite -assoc_L !big_sepS_fn_insert'; [|abstract set_solver ..].
89
    by iFrame.
90
  - rewrite -assoc_L !big_sepS_fn_insert; [|abstract set_solver ..]. eauto.
91
Qed.
92 93

(** Actual proofs *)
94
Lemma newbarrier_spec (P : iProp Σ) :
95
  {{{ True }}} newbarrier #() {{{ l, RET #l; recv l P  send l P }}}.
96
Proof.
97
  iIntros (Φ) "HΦ".
98
  rewrite -wp_fupd /newbarrier /=. wp_seq. wp_alloc l as "Hl".
99 100 101
  iApply ("HΦ" with ">[-]").
  iMod (saved_prop_alloc (F:=idCF) P) as (γ) "#?".
  iMod (sts_alloc (barrier_inv l P) _ N (State Low {[ γ ]}) with "[-]")
102
    as (γ') "[#? Hγ']"; eauto.
103
  { iNext. rewrite /barrier_inv /=. iFrame.
104
    iExists (const P). rewrite !big_sepS_singleton /=. eauto. }
Robbert Krebbers's avatar
Robbert Krebbers committed
105 106
  iAssert (barrier_ctx γ' l P)%I as "#?".
  { rewrite /barrier_ctx. by repeat iSplit. }
107
  iAssert (sts_ownS γ' (i_states γ) {[Change γ]}
108
     sts_ownS γ' low_states {[Send]})%I with ">[-]" as "[Hr Hs]".
Robbert Krebbers's avatar
Robbert Krebbers committed
109
  { iApply sts_ownS_op; eauto using i_states_closed, low_states_closed.
110 111
    - set_solver.
    - iApply (sts_own_weaken with "Hγ'");
Robbert Krebbers's avatar
Robbert Krebbers committed
112
        auto using sts.closed_op, i_states_closed, low_states_closed;
113
        abstract set_solver. }
114
  iModIntro. rewrite /recv /send. iSplitL "Hr".
115
  - iExists γ', P, P, γ. iFrame. auto.
116
  - auto.
117 118
Qed.

119
Lemma signal_spec l P :
120
  {{{ send l P  P }}} signal #l {{{ RET #(); True }}}.
121
Proof.
122
  rewrite /signal /send /barrier_ctx /=.
123
  iIntros (Φ) "[Hs HP] HΦ". iDestruct "Hs" as (γ) "[#Hsts Hγ]". wp_let.
124
  iMod (sts_openS (barrier_inv l P) _ _ γ with "[Hγ]")
125
    as ([p I]) "(% & [Hl Hr] & Hclose)"; eauto.
126 127
  destruct p; [|done]. wp_store.
  iSpecialize ("HΦ" with "[#]") => //. iFrame "HΦ".
128
  iMod ("Hclose" $! (State High I) ( : set token) with "[-]"); last done.
Robbert Krebbers's avatar
Robbert Krebbers committed
129
  iSplit; [iPureIntro; by eauto using signal_step|].
130
  rewrite {2}/barrier_inv /ress /=. iNext. iFrame "Hl".
131
  iDestruct "Hr" as (Ψ) "[Hr Hsp]"; iExists Ψ; iFrame "Hsp".
132
  iNext. iIntros "_"; by iApply "Hr".
133 134
Qed.

135
Lemma wait_spec l P:
136
  {{{ recv l P }}} wait #l {{{ RET #(); P }}}.
137
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
138
  rename P into R; rewrite /recv /barrier_ctx.
139
  iIntros (Φ) "Hr HΦ"; iDestruct "Hr" as (γ P Q i) "(#Hsts & Hγ & #HQ & HQR)".
140
  iLöb as "IH". wp_rec. wp_bind (! _)%E.
141
  iMod (sts_openS (barrier_inv l P) _ _ γ with "[Hγ]")
142 143
    as ([p I]) "(% & [Hl Hr] & Hclose)"; eauto.
  wp_load. destruct p.
144
  - iMod ("Hclose" $! (State Low I) {[ Change i ]} with "[Hl Hr]") as "Hγ".
145
    { iSplit; first done. rewrite {2}/barrier_inv /=. by iFrame. }
146
    iAssert (sts_ownS γ (i_states i) {[Change i]})%I with ">[Hγ]" as "Hγ".
147
    { iApply (sts_own_weaken with "Hγ"); eauto using i_states_closed. }
148
    iModIntro. wp_if.
Ralf Jung's avatar
Ralf Jung committed
149
    iApply ("IH" with "Hγ [HQR] [HΦ]"); auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
150 151
  - (* a High state: the comparison succeeds, and we perform a transition and
    return to the client *)
152
    iDestruct "Hr" as (Ψ) "[HΨ Hsp]".
153
    iDestruct (big_sepS_delete _ _ i with "Hsp") as "[#HΨi Hsp]"; first done.
154
    iAssert ( Ψ i   [ set] j  I  {[i]}, Ψ j)%I with "[HΨ]" as "[HΨ HΨ']".
Robbert Krebbers's avatar
Robbert Krebbers committed
155
    { iNext. iApply (big_sepS_delete _ _ i); first done. by iApply "HΨ". }
156
    iMod ("Hclose" $! (State High (I  {[ i ]})) ( : set token) with "[HΨ' Hl Hsp]").
157
    { iSplit; [iPureIntro; by eauto using wait_step|].
158
      rewrite {2}/barrier_inv /=. iNext. iFrame "Hl". iExists Ψ; iFrame. auto. }
159
    iPoseProof (saved_prop_agree i Q (Ψ i) with "[#]") as "Heq"; first by auto.
160
    iModIntro. wp_if.
161
    iApply "HΦ". iApply "HQR". by iRewrite "Heq".
162 163
Qed.

164
Lemma recv_split E l P1 P2 :
165
  N  E  recv l (P1  P2) ={E}= recv l P1  recv l P2.
166
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
167
  rename P1 into R1; rename P2 into R2. rewrite {1}/recv /barrier_ctx.
168
  iIntros (?). iDestruct 1 as (γ P Q i) "(#Hsts & Hγ & #HQ & HQR)".
169
  iMod (sts_openS (barrier_inv l P) _ _ γ with "[Hγ]")
170
    as ([p I]) "(% & [Hl Hr] & Hclose)"; eauto.
171 172
  iMod (saved_prop_alloc_strong (R1: %CF (iProp Σ)) I) as (i1) "[% #Hi1]".
  iMod (saved_prop_alloc_strong (R2: %CF (iProp Σ)) (I  {[i1]}))
173
    as (i2) "[Hi2' #Hi2]"; iDestruct "Hi2'" as %Hi2.
Robbert Krebbers's avatar
Robbert Krebbers committed
174
  rewrite ->not_elem_of_union, elem_of_singleton in Hi2; destruct Hi2.
175
  iMod ("Hclose" $! (State p ({[i1; i2]}  I  {[i]}))
176 177
                   {[Change i1; Change i2 ]} with "[-]") as "Hγ".
  { iSplit; first by eauto using split_step.
178
    rewrite {2}/barrier_inv /=. iNext. iFrame "Hl".
179
    by iApply (ress_split with "HQ Hi1 Hi2 HQR"). }
180
  iAssert (sts_ownS γ (i_states i1) {[Change i1]}
181
     sts_ownS γ (i_states i2) {[Change i2]})%I with ">[-]" as "[Hγ1 Hγ2]".
182 183 184 185 186
  { 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. }
187
  iModIntro; iSplitL "Hγ1"; rewrite /recv /barrier_ctx.
188 189
  - iExists γ, P, R1, i1. iFrame; auto.
  - iExists γ, P, R2, i2. iFrame; auto.
190 191
Qed.

192
Lemma recv_weaken l P1 P2 : (P1 - P2) - recv l P1 - recv l P2.
193
Proof.
194
  rewrite /recv. iIntros "HP". iDestruct 1 as (γ P Q i) "(#Hctx&Hγ&Hi&HP1)".
195
  iExists γ, P, Q, i. iFrame "Hctx Hγ Hi".
196
  iNext. iIntros "HQ". by iApply "HP"; iApply "HP1".
197
Qed.
198

199
Lemma recv_mono l P1 P2 : (P1  P2)  recv l P1  recv l P2.
200
Proof. iIntros (HP) "H". iApply (recv_weaken with "[] H"). iApply HP. Qed.
201
End proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
202 203

Typeclasses Opaque barrier_ctx send recv.