1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
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
52
53
54
55
56
57
58
59
60
61
62
63
64
65
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
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
|
function RightOpen(n: int) : [int]bool;
function RightClosed(n: int) : [int]bool;
axiom (forall x: int, y: int :: RightOpen(x)[y] <==> y < x);
axiom (forall x: int, y: int :: RightClosed(x)[y] <==> y <= x);
type X;
function {:builtin "MapConst"} mapconstbool(bool): [X]bool;
const nil: X;
var {:qed} t: int;
var {:qed} s: int;
var {:qed} cs: X;
var {:qed} T: [int]bool;
function {:builtin "MapConst"} MapConstBool(bool) : [X]bool;
function {:inline} {:linear "tid"} TidCollector(x: X) : [X]bool
{
MapConstBool(false)[x := true]
}
function {:inline} {:linear "tid"} TidSetCollector(x: [X]bool) : [X]bool
{
x
}
function {:inline} Inv1(tickets: [int]bool, ticket: int): (bool)
{
tickets == RightOpen(ticket)
}
function {:inline} Inv2(tickets: [int]bool, ticket: int, lock: X): (bool)
{
if (lock == nil) then tickets == RightOpen(ticket) else tickets == RightClosed(ticket)
}
procedure Allocate({:linear "tid"} xls':[X]bool) returns ({:linear "tid"} xls: [X]bool, {:linear "tid"} xl: X);
ensures {:phase 1} {:phase 2} xl != nil;
procedure {:yields} {:entrypoint} main({:linear "tid"} xls':[X]bool)
requires {:phase 3} xls' == mapconstbool(true);
{
var {:linear "tid"} tid: X;
var {:linear "tid"} xls: [X]bool;
yield;
call xls := Init(xls');
par Yield1() | Yield2() | Yield();
while (*)
invariant {:phase 1} Inv1(T, t);
invariant {:phase 2} Inv2(T, s, cs);
{
call xls, tid := Allocate(xls);
async call Customer(tid);
}
}
procedure {:yields} {:stable} Customer({:linear "tid"} tid': X)
requires {:phase 1} Inv1(T, t);
requires {:phase 2} tid' != nil && Inv2(T, s, cs);
requires {:phase 3} true;
{
var {:linear "tid"} tid: X;
tid := tid';
while (*)
invariant {:phase 1} Inv1(T, t);
invariant {:phase 2} tid != nil && Inv2(T, s, cs);
{
par Yield1() | Yield2() | Yield();
call tid := Enter(tid);
par Yield1() | Yield2();
call tid := Leave(tid);
par Yield1() | Yield2() | Yield();
}
}
procedure {:yields} Enter({:linear "tid"} tid': X) returns ({:linear "tid"} tid: X)
requires {:phase 1} Inv1(T, t);
ensures {:phase 1} Inv1(T,t);
requires {:phase 2} tid' != nil && Inv2(T, s, cs);
ensures {:phase 2} tid != nil && Inv2(T, s, cs);
ensures {:right 2} |{ A: tid := tid'; havoc t, T; assume tid != nil && cs == nil; cs := tid; return true; }|;
{
var m: int;
par Yield1() | Yield2();
tid := tid';
call tid, m := GetTicketAbstract(tid);
par Yield1();
call tid := WaitAndEnter(tid, m);
par Yield1() | Yield2();
}
procedure {:yields} GetTicketAbstract({:linear "tid"} tid': X) returns ({:linear "tid"} tid: X, m: int)
requires {:phase 1} Inv1(T, t);
ensures {:phase 1} Inv1(T, t);
ensures {:right 1,2} |{ A: tid := tid'; havoc m, t; assume !T[m]; T[m] := true; return true; }|;
{
par Yield1();
tid := tid';
call tid, m := GetTicket(tid);
par Yield1();
}
procedure {:yields} {:stable} Yield()
{
}
procedure {:yields} {:stable} Yield2()
requires {:phase 2} Inv2(T, s, cs);
ensures {:phase 2} Inv2(T, s, cs);
ensures {:both 2} |{ A: return true; }|;
{
}
procedure {:yields} {:stable} Yield1()
requires {:phase 1} Inv1(T, t);
ensures {:phase 1} Inv1(T,t);
ensures {:both 1} |{ A: return true; }|;
{
}
procedure {:yields} Init({:linear "tid"} xls':[X]bool) returns ({:linear "tid"} xls:[X]bool);
ensures {:atomic 0} |{ A: assert xls' == mapconstbool(true); xls := xls'; cs := nil; t := 0; s := 0; T := RightOpen(0); return true; }|;
procedure {:yields} GetTicket({:linear "tid"} tid': X) returns ({:linear "tid"} tid: X, m: int);
ensures {:atomic 0,1} |{ A: tid := tid'; m := t; t := t + 1; T[m] := true; return true; }|;
procedure {:yields} WaitAndEnter({:linear "tid"} tid': X, m:int) returns ({:linear "tid"} tid: X);
ensures {:atomic 0,2} |{ A: tid := tid'; assume m <= s; cs := tid; return true; }|;
procedure {:yields} Leave({:linear "tid"} tid': X) returns ({:linear "tid"} tid: X);
ensures {:atomic 0} |{ A: assert cs == tid'; assert tid' != nil; tid := tid'; s := s + 1; cs := nil; return true; }|;
|
