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
|
// --------------------------------------------------
module CoRecursion {
codatatype Stream<T> = More(head: T, rest: Stream);
function AscendingChain(n: int): Stream<int>
{
More(n, AscendingChain(n+1))
}
function AscendingChainAndRead(n: nat): Stream<int>
reads this; // with a reads clause, this function is not a co-recusvie function
{
More(n, AscendingChainAndRead(n+1)) // error: cannot prove termination
}
function AscendingChainAndPostcondition(n: nat): Stream<int>
ensures false; // with an ensures clause, this function is not a co-recusvie function
{
More(n, AscendingChainAndPostcondition(n+1)) // error: cannot prove termination
}
datatype List<T> = Nil | Cons(T, List);
function Prefix(n: nat, s: Stream): List
{
if n == 0 then Nil else
Cons(s.head, Prefix(n-1, s.rest))
}
}
// --------------------------------------------------
module CoRecursionNotUsed {
codatatype Stream<T> = More(T, Stream);
function F(s: Stream, n: nat): Stream
decreases n, true;
{
G(s, n)
}
function G(s: Stream, n: nat): Stream
decreases n, false;
{
if n == 0 then s else Tail(F(s, n-1))
}
function Tail(s: Stream): Stream
{
match s case More(hd, tl) => tl
}
function Diverge(n: nat): nat
{
Diverge(n) // error: cannot prove termination
}
}
// --------------------------------------------------
module EqualityIsSuperDestructive {
codatatype Stream<T> = Cons(head: T, tail: Stream)
function F(s: Stream<int>): Stream<int>
{
// Co-recursive calls are not allowed in arguments of equality, so the following call to
// F(s) is a recursive call.
if Cons(1, F(s)) == Cons(1, Cons(1, s)) // error: cannot prove termination
then Cons(2, s) else Cons(1, s)
}
lemma Lemma(s: Stream<int>)
{
// The following three assertions follow from the definition of F, so F had better
// generate some error (which it does -- the recursive call to F in F does not terminate).
assert F(s) == Cons(1, s);
assert F(s) == Cons(2, s);
assert false;
}
}
// --------------------------------------------------
module MixRecursiveAndCorecursive {
codatatype Stream<T> = Cons(head: T, tail: Stream)
function F(n: nat): Stream<int>
{
if n == 0 then
Cons(0, F(5)) // error: cannot prove termination -- by itself, this would look like a properly guarded co-recursive call...
else
F(n - 1).tail // but the fact that this recursive call is not tail recursive means that call in the 'then' branch is not
// allowed to be a co-recursive
}
// same thing but with some mutual recursion going on
function G(n: nat): Stream<int>
{
if n == 0 then
Cons(0, H(5)) // error: cannot prove termination
else
H(n)
}
function H(n: nat): Stream<int>
requires n != 0;
decreases n, 0;
{
G(n-1).tail
}
// but if all the recursive calls are tail recursive, then all is cool
function X(n: nat): Stream<int>
{
if n == 0 then
Cons(0, Y(5)) // error: cannot prove termination
else
Y(n)
}
function Y(n: nat): Stream<int>
requires n != 0;
decreases n, 0;
{
X(n-1)
}
}
// --------------------------------------------------
module FunctionSCCsWithMethods {
codatatype Stream<T> = Cons(head: T, tail: Stream)
lemma M(n: nat)
decreases n, 0;
{
if n != 0 {
var p := Cons(10, F(n-1));
}
}
function F(n: nat): Stream<int>
decreases n;
{
M(n);
// the following call to F is not considered co-recursive, because the SCC contains a method
Cons(5, F(n)) // error: cannot prove termination
}
function G(): Stream<int>
{
Lemma();
H()
}
function H(): Stream<int>
decreases 0;
{
// the following call to G is not considered co-recursive, because the SCC contains a method
Cons(5, G()) // error: cannot prove termination
}
lemma Lemma()
decreases 1;
{
var h := H();
}
}
// --------------------------------------------------
module AutomaticPrefixingOfCoClusterDecreasesClauses {
codatatype Stream<T> = Cons(head: T, tail: Stream)
// The following three functions will verify automatically
function H(): Stream<int> // automatic: decreases 1;
{
F(true)
}
function F(b: bool): Stream<int> // automatic: decreases 0, b;
{
if b then Cons(5, G()) else Cons(7, H())
}
function G(): Stream<int> // automatic: decreases 1;
{
F(false)
}
// In the following, A gets a default decreases clause of 1, because
// the only recursive call to A is a self-call. B, on the other
// hand, has a mutually recursive call from A, and therefore it gets
// a decreases clause of 0.
function A(n: nat): Stream<int> // automatic: decreases 1, n;
{
if n < 100 then
B(n) // the automatic decreases clauses take care of the termination of this call
else
A(n - 1) // termination proved on account of decreasing 1,n
}
function B(n: nat): Stream<int> // automatic: decreases 0, n;
{
if n < 100 then
Cons(n, A(n + 102)) // co-recursive call, so no termination check
else
B(n - 1) // termination proved on account of decreasing 0,n
}
}
|
