adequacy.v 6.99 KB
Newer Older
1
2
From iris.program_logic Require Export weakestpre.
From iris.algebra Require Import gmap auth agree gset coPset upred_big_op.
3
From iris.program_logic Require Import wsat.
4
From iris.proofmode Require Import tactics.
5
Import uPred.
Robbert Krebbers's avatar
Robbert Krebbers committed
6

7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Record adequate {Λ} (e1 : expr Λ) (σ1 : state Λ) (φ : val Λ  Prop) := {
  adequate_result t2 σ2 v2 :
   rtc step ([e1], σ1) (of_val v2 :: t2, σ2)  φ v2;
  adequate_safe t2 σ2 e2 :
   rtc step ([e1], σ1) (t2, σ2) 
   e2  t2  (is_Some (to_val e2)  reducible e2 σ2)
}.

Theorem adequate_tp_safe {Λ} (e1 : expr Λ) t2 σ1 σ2 φ :
  adequate e1 σ1 φ 
  rtc step ([e1], σ1) (t2, σ2) 
  Forall (λ e, is_Some (to_val e)) t2   t3 σ3, step (t2, σ2) (t3, σ3).
Proof.
  intros Had ?.
  destruct (decide (Forall (λ e, is_Some (to_val e)) t2)) as [|Ht2]; [by left|].
  apply (not_Forall_Exists _), Exists_exists in Ht2; destruct Ht2 as (e2&?&He2).
23
  destruct (adequate_safe e1 σ1 φ Had t2 σ2 e2) as [?|(e3&σ3&efs&?)];
24
25
26
    rewrite ?eq_None_not_Some; auto.
  { exfalso. eauto. }
  destruct (elem_of_list_split t2 e2) as (t2'&t2''&->); auto.
27
  right; exists (t2' ++ e3 :: t2'' ++ efs), σ3; econstructor; eauto.
28
29
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
30
Section adequacy.
31
Context `{irisG Λ Σ}.
32
Implicit Types e : expr Λ.
33
34
35
Implicit Types P Q : iProp Σ.
Implicit Types Φ : val Λ  iProp Σ.
Implicit Types Φs : list (val Λ  iProp Σ).
Robbert Krebbers's avatar
Robbert Krebbers committed
36

37
38
Notation world σ := (wsat  ownE   ownP_auth σ)%I.

39
Notation wptp t := ([ list] ef  t, WP ef {{ _, True }})%I.
40

41
42
Lemma wp_step e1 σ1 e2 σ2 efs Φ :
  prim_step e1 σ1 e2 σ2 efs 
43
  world σ1  WP e1 {{ Φ }} =r=>  |=r=>  (world σ2  WP e2 {{ Φ }}  wptp efs).
44
45
46
47
Proof.
  rewrite {1}wp_unfold /wp_pre. iIntros (Hstep) "[(Hw & HE & Hσ) [H|[_ H]]]".
  { iDestruct "H" as (v) "[% _]". apply val_stuck in Hstep; simplify_eq. }
  rewrite pvs_eq /pvs_def.
48
  iVs ("H" $! σ1 with "Hσ [Hw HE]") as ">(Hw & HE & _ & H)"; first by iFrame.
49
  iVsIntro; iNext.
50
  iVs ("H" $! e2 σ2 efs with "[%] [Hw HE]")
51
    as ">($ & $ & $ & $)"; try iFrame; eauto.
52
53
54
55
56
57
58
59
Qed.

Lemma wptp_step e1 t1 t2 σ1 σ2 Φ :
  step (e1 :: t1,σ1) (t2, σ2) 
  world σ1  WP e1 {{ Φ }}  wptp t1
  =r=>  e2 t2', t2 = e2 :: t2'   |=r=>  (world σ2  WP e2 {{ Φ }}  wptp t2').
Proof.
  iIntros (Hstep) "(HW & He & Ht)".
60
61
  destruct Hstep as [e1' σ1' e2' σ2' efs [|? t1'] t2' ?? Hstep]; simplify_eq/=.
  - iExists e2', (t2' ++ efs); iSplitR; first eauto.
62
    rewrite big_sepL_app. iFrame "Ht". iApply wp_step; try iFrame; eauto.
63
  - iExists e, (t1' ++ e2' :: t2' ++ efs); iSplitR; first eauto.
64
    rewrite !big_sepL_app !big_sepL_cons big_sepL_app.
65
66
67
68
69
70
71
72
73
    iDestruct "Ht" as "($ & He' & $)"; iFrame "He".
    iApply wp_step; try iFrame; eauto.
Qed.

Lemma wptp_steps n e1 t1 t2 σ1 σ2 Φ :
  nsteps step n (e1 :: t1, σ1) (t2, σ2) 
  world σ1  WP e1 {{ Φ }}  wptp t1 
  Nat.iter (S n) (λ P, |=r=>  P) ( e2 t2',
    t2 = e2 :: t2'  world σ2  WP e2 {{ Φ }}  wptp t2').
Robbert Krebbers's avatar
Robbert Krebbers committed
74
Proof.
75
76
77
78
  revert e1 t1 t2 σ1 σ2; simpl; induction n as [|n IH]=> e1 t1 t2 σ1 σ2 /=.
  { inversion_clear 1; iIntros "?"; eauto 10. }
  iIntros (Hsteps) "H". inversion_clear Hsteps as [|?? [t1' σ1']].
  iVs (wptp_step with "H") as (e1' t1'') "[% H]"; first eauto; simplify_eq.
79
  iVsIntro; iNext; iVs "H" as ">?". by iApply IH.
Robbert Krebbers's avatar
Robbert Krebbers committed
80
Qed.
81
82
83
84

Instance rvs_iter_mono n : Proper (() ==> ()) (Nat.iter n (λ P, |=r=>  P)%I).
Proof. intros P Q HP. induction n; simpl; do 2?f_equiv; auto. Qed.

85
86
87
88
89
90
91
Lemma rvs_iter_frame_l n R Q :
  R  Nat.iter n (λ P, |=r=>  P) Q  Nat.iter n (λ P, |=r=>  P) (R  Q).
Proof.
  induction n as [|n IH]; simpl; [done|].
  by rewrite rvs_frame_l {1}(later_intro R) -later_sep IH.
Qed.

92
93
94
95
Lemma wptp_result n e1 t1 v2 t2 σ1 σ2 φ :
  nsteps step n (e1 :: t1, σ1) (of_val v2 :: t2, σ2) 
  world σ1  WP e1 {{ v,  φ v }}  wptp t1 
  Nat.iter (S (S n)) (λ P, |=r=>  P) ( φ v2).
Robbert Krebbers's avatar
Robbert Krebbers committed
96
Proof.
Ralf Jung's avatar
Ralf Jung committed
97
98
  intros. rewrite wptp_steps //.
  rewrite (Nat_iter_S_r (S n)). apply rvs_iter_mono.
99
100
  iDestruct 1 as (e2 t2') "(% & (Hw & HE & _) & H & _)"; simplify_eq.
  iDestruct (wp_value_inv with "H") as "H". rewrite pvs_eq /pvs_def.
101
  iVs ("H" with "[Hw HE]") as ">(_ & _ & $)"; iFrame; auto.
Robbert Krebbers's avatar
Robbert Krebbers committed
102
Qed.
Ralf Jung's avatar
Ralf Jung committed
103

104
105
Lemma wp_safe e σ Φ :
  world σ  WP e {{ Φ }} =r=>   (is_Some (to_val e)  reducible e σ).
106
Proof.
107
108
  rewrite wp_unfold /wp_pre. iIntros "[(Hw&HE&Hσ) [H|[_ H]]]".
  { iDestruct "H" as (v) "[% _]"; eauto 10. }
109
110
  rewrite pvs_eq. iVs ("H" with "* Hσ [-]") as ">(?&?&%&?)"; first by iFrame.
  eauto 10.
111
Qed.
Ralf Jung's avatar
Ralf Jung committed
112

113
114
115
116
Lemma wptp_safe n e1 e2 t1 t2 σ1 σ2 Φ :
  nsteps step n (e1 :: t1, σ1) (t2, σ2)  e2  t2 
  world σ1  WP e1 {{ Φ }}  wptp t1 
  Nat.iter (S (S n)) (λ P, |=r=>  P) ( (is_Some (to_val e2)  reducible e2 σ2)).
Robbert Krebbers's avatar
Robbert Krebbers committed
117
Proof.
118
  intros ? He2. rewrite wptp_steps //; rewrite (Nat_iter_S_r (S n)). apply rvs_iter_mono.
119
120
  iDestruct 1 as (e2' t2') "(% & Hw & H & Htp)"; simplify_eq.
  apply elem_of_cons in He2 as [<-|?]; first (iApply wp_safe; by iFrame "Hw H").
121
  iApply wp_safe. iFrame "Hw". by iApply (big_sepL_elem_of with "Htp").
Robbert Krebbers's avatar
Robbert Krebbers committed
122
Qed.
123

Robbert Krebbers's avatar
Robbert Krebbers committed
124
Lemma wptp_invariance n e1 e2 t1 t2 σ1 σ2 I φ Φ :
125
126
  nsteps step n (e1 :: t1, σ1) (t2, σ2) 
  (I ={,}=>  σ', ownP σ'   φ σ') 
Robbert Krebbers's avatar
Robbert Krebbers committed
127
  I  world σ1  WP e1 {{ Φ }}  wptp t1 
128
129
  Nat.iter (S (S n)) (λ P, |=r=>  P) ( φ σ2).
Proof.
130
  intros ? HI. rewrite wptp_steps //.
131
132
133
134
135
136
  rewrite (Nat_iter_S_r (S n)) rvs_iter_frame_l. apply rvs_iter_mono.
  iIntros "[HI H]".
  iDestruct "H" as (e2' t2') "(% & (Hw&HE&Hσ) & _)"; subst.
  rewrite pvs_eq in HI;
    iVs (HI with "HI [Hw HE]") as "> (_ & _ & H)"; first by iFrame.
  iDestruct "H" as (σ2') "[Hσf %]".
137
  iDestruct (ownP_agree σ2 σ2' with "[-]") as %<-. by iFrame. eauto.
138
Qed.
139
End adequacy.
Ralf Jung's avatar
Ralf Jung committed
140

141
142
143
Theorem wp_adequacy Σ `{irisPreG Λ Σ} e σ φ :
  ( `{irisG Λ Σ}, ownP σ  WP e {{ v,  φ v }}) 
  adequate e σ φ.
Ralf Jung's avatar
Ralf Jung committed
144
Proof.
145
146
147
148
149
  intros Hwp; split.
  - intros t2 σ2 v2 [n ?]%rtc_nsteps.
    eapply (adequacy (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
    rewrite Nat_iter_S. iVs (iris_alloc σ) as (?) "(?&?&?&Hσ)".
    iVsIntro. iNext. iApply wptp_result; eauto.
150
    iFrame. iSplitL. by iApply Hwp. by iApply big_sepL_nil.
151
152
153
154
  - intros t2 σ2 e2 [n ?]%rtc_nsteps ?.
    eapply (adequacy (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
    rewrite Nat_iter_S. iVs (iris_alloc σ) as (?) "(Hw & HE & Hσ & Hσf)".
    iVsIntro. iNext. iApply wptp_safe; eauto.
155
    iFrame "Hw HE Hσ". iSplitL. by iApply Hwp. by iApply big_sepL_nil.
Robbert Krebbers's avatar
Robbert Krebbers committed
156
Qed.
157

Robbert Krebbers's avatar
Robbert Krebbers committed
158
159
Theorem wp_invariance Σ `{irisPreG Λ Σ} e σ1 t2 σ2 I φ Φ :
  ( `{irisG Λ Σ}, ownP σ1 ={}=> I  WP e {{ Φ }}) 
160
161
162
163
  ( `{irisG Λ Σ}, I ={,}=>  σ', ownP σ'   φ σ') 
  rtc step ([e], σ1) (t2, σ2) 
  φ σ2.
Proof.
164
  intros Hwp HI [n ?]%rtc_nsteps.
165
166
167
168
  eapply (adequacy (M:=iResUR Σ) _ (S (S (S n)))); iIntros "".
  rewrite Nat_iter_S. iVs (iris_alloc σ1) as (?) "(Hw & HE & ? & Hσ)".
  rewrite pvs_eq in Hwp.
  iVs (Hwp _ with "Hσ [Hw HE]") as ">(? & ? & ? & ?)"; first by iFrame.
169
  iVsIntro. iNext. iApply wptp_invariance; eauto. iFrame. by iApply big_sepL_nil.
170
Qed.