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
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
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
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
|
datatype List<T> = Nil | Cons(T, List<T>);
class Node {
var data: int;
var next: Node;
function Repr(list: List<int>): bool
reads *;
decreases list;
{ match list
case Nil =>
next == null
case Cons(d,cdr) =>
data == d && next != null && next.Repr(cdr)
}
method Init()
modifies this;
ensures Repr(Nil);
{
next := null;
}
method Add(d: int, L: List<int>) returns (r: Node)
requires Repr(L);
ensures r != null && r.Repr(Cons(d, L));
{
r := new Node;
r.data := d;
r.next := this;
}
}
class AnotherNode {
var data: int;
var next: AnotherNode;
function Repr(n: AnotherNode, list: List<int>): bool
reads *;
decreases list;
{ match list
case Nil =>
n == null
case Cons(d,cdr) =>
n != null && n.data == d && Repr(n.next, cdr)
}
method Create() returns (n: AnotherNode)
ensures Repr(n, Nil);
{
n := null;
}
method Add(n: AnotherNode, d: int, L: List<int>) returns (r: AnotherNode)
requires Repr(n, L);
ensures Repr(r, Cons(d, L));
{
r := new AnotherNode;
r.data := d;
r.next := n;
}
}
method TestAllocatednessAxioms(a: List<Node>, b: List<Node>, c: List<AnotherNode>)
{
var n := new Node;
var p := n;
match a {
case Nil =>
case Cons(x, tail) => assert x != n; p := x;
}
match b {
case Nil =>
case Cons(x, tail) =>
match tail {
case Nil =>
case Cons(y, more) =>
assert y != n;
assert y != p; // error: if p is car(a), then it and y may very well be equal
}
}
match c {
case Nil =>
case Cons(x, tail) =>
match tail {
case Nil =>
case Cons(y, more) =>
var o: object := y;
assert p != null ==> p != o; // follows from well-typedness
}
}
}
class NestedMatchExpr {
function Cadr<T>(a: List<T>, default: T): T
{
match a
case Nil => default
case Cons(x,t) =>
match t
case Nil => default
case Cons(y,tail) => y
}
// CadrAlt is the same as Cadr, but it writes its two outer cases in the opposite order
function CadrAlt<T>(a: List<T>, default: T): T
{
match a
case Cons(x,t) => (
match t
case Nil => default
case Cons(y,tail) => y)
case Nil => default
}
method TestNesting0()
{
var x := 5;
var list := Cons(3, Cons(6, Nil));
assert Cadr(list, x) == 6;
match (list) {
case Nil => assert false;
case Cons(h,t) => assert Cadr(t, x) == 5;
}
}
method TestNesting1(a: List<NestedMatchExpr>)
ensures Cadr(a, this) == CadrAlt(a, this);
{
match (a) {
case Nil =>
case Cons(x,t) =>
match (t) {
case Nil =>
case Cons(y,tail) =>
}
}
}
}
// ------------------- datatype destructors ---------------------------------------
datatype XList = XNil | XCons(Car: int, Cdr: XList);
method Destructors0(d: XList) {
Lemma_AllCases(d);
if {
case d.XNil? =>
assert d == XNil;
case d.XCons? =>
var hd := d.Car;
var tl := d.Cdr;
assert d == XCons(hd, tl);
}
}
method Destructors1(d: XList) {
match (d) {
case XNil =>
assert d.XNil?;
case XCons(hd,tl) =>
assert d.XCons?;
}
}
method Destructors2(d: XList) {
// this method gets it backwards
match (d) {
case XNil =>
assert d.XCons?; // error
case XCons(hd,tl) =>
assert d.XNil?; // error
}
}
ghost method Lemma_AllCases(d: XList)
ensures d.XNil? || d.XCons?;
{
match (d) {
case XNil =>
case XCons(hd,tl) =>
}
}
method InjectivityTests(d: XList)
requires d != XNil;
{
match (d) {
case XCons(a,b) =>
match (d) {
case XCons(x,y) =>
assert a == x && b == y;
}
assert a == d.Car;
assert b == d.Cdr;
assert d == XCons(d.Car, d.Cdr);
}
}
// -------------
method FwdBug(f: Fwd, initialized: bool)
requires !f.FwdCons?;
{
match (f) {
case FwdNil =>
// Syntactically, there is a missing case here, but the verifier checks that this is still cool.
// There was once a bug in Dafny, where this had caused an ill-defined Boogie program.
}
if (!initialized) { // There was once a Dafny parsing bug with this line
}
}
function FwdBugFunction(f: Fwd): bool
requires !f.FwdCons?;
{
match f
case FwdNil => true
// Syntactically, there is a missing case here, but the verifier checks that this is still cool.
// There was once a bug in Dafny, where this had caused an ill-defined Boogie program.
}
datatype Fwd = FwdNil | FwdCons(int, Fwd);
|
