proof.v 8.21 KB
Newer Older
Ralf Jung's avatar
Ralf Jung committed
1 2 3
From iris.program_logic Require Export weakestpre.
From iris.heap_lang Require Export lang.
From stdpp Require Import functions.
Ralf Jung's avatar
Ralf Jung committed
4
From iris.base_logic Require Import lib.saved_prop lib.sts.
Ralf Jung's avatar
Ralf Jung committed
5
From iris.heap_lang Require Import proofmode.
Ralf Jung's avatar
Ralf Jung committed
6 7
From iris_examples.barrier Require Export barrier.
From iris_examples.barrier Require Import protocol.
Ralf Jung's avatar
Ralf Jung committed
8 9 10 11 12
Set Default Proof Using "Type".

(** The CMRAs/functors we need. *)
Class barrierG Σ := BarrierG {
  barrier_stsG :> stsG Σ sts;
13
  barrier_savedPropG :> savedPropG Σ;
Ralf Jung's avatar
Ralf Jung committed
14
}.
15
Definition barrierΣ : gFunctors := #[stsΣ sts; savedPropΣ].
Ralf Jung's avatar
Ralf Jung committed
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

Instance subG_barrierΣ {Σ} : subG barrierΣ Σ  barrierG Σ.
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 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.

Global Instance barrier_ctx_persistent (γ : gname) (l : loc) (P : iProp Σ) :
52
  Persistent (barrier_ctx γ l P).
Ralf Jung's avatar
Ralf Jung committed
53 54 55 56 57 58 59 60 61 62 63 64
Proof. apply _. Qed.

(** 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).
65
Proof. solve_proper. Qed.
Ralf Jung's avatar
Ralf Jung committed
66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94
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".
95
  rewrite -wp_fupd /newbarrier /=. wp_lam. wp_alloc l as "Hl".
Ralf Jung's avatar
Ralf Jung committed
96
  iApply ("HΦ" with "[> -]").
97
  iMod (saved_prop_alloc P) as (γ) "#?".
Ralf Jung's avatar
Ralf Jung committed
98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119
  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.
Qed.

Lemma signal_spec l P :
  {{{ send l P  P }}} signal #l {{{ RET #(); True }}}.
Proof.
  rewrite /signal /=.
120
  iIntros (Φ) "[Hs HP] HΦ". iDestruct "Hs" as (γ) "[#Hsts Hγ]". wp_lam.
Ralf Jung's avatar
Ralf Jung committed
121 122 123 124
  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".
Robbert Krebbers's avatar
Robbert Krebbers committed
125
  iMod ("Hclose" $! (State High I) ( : propset token) with "[-]"); last done.
Ralf Jung's avatar
Ralf Jung committed
126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150
  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".
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)".
  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.
Ralf Jung's avatar
Ralf Jung committed
151
    iAssert ( (Ψ i  [ set] j  I  {[i]}, Ψ j))%I with "[HΨ]" as "[HΨ HΨ']".
Ralf Jung's avatar
Ralf Jung committed
152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167
    { 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".
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.
168 169
  iMod (saved_prop_alloc_cofinite I) as (i1) "[% #Hi1]".
  iMod (saved_prop_alloc_cofinite (I  {[i1]}))
Ralf Jung's avatar
Ralf Jung committed
170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200
    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.
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".
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.