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
|
method test1()
{
var ms: multiset<int> := multiset{1,1,1};
var ms2: multiset<int> := multiset{3};
assert 1 in ms;
assert forall i :: i != 1 ==> i !in ms; // 1 is the only thing in ms.
assert ((ms - multiset{1}) - multiset {1}) != multiset{}; // it has more than 2 ones
assert ((ms - multiset{1}) - multiset {1}) - multiset{1} == multiset{}; // and exactly three
assert ms2 - ms == ms2; // set difference works correctly.
assert ms - multiset{1} == multiset{1,1};
assert !(multiset{1} !! multiset{1});
assert exists m :: !(m !! multiset{1});
assert forall m :: m !! multiset{};
assert forall s :: (s == set x: int | x in ms :: x) ==> s == {1};
}
method test2(ms: multiset<int>)
{
var s := set x | x in ms :: x; // seems to be a reasonable conversion
assert forall x :: x in s <==> x in ms;
assert ms !! multiset{};
}
method test3(s: set<int>)
{
assert forall x :: x in s <==> x in multiset(s);
}
method test4(sq: seq<int>, a: array<int>)
requires a != null;
modifies a;
{
assert sq == sq[..|sq|];
assert sq == sq[0..];
assert sq == sq[..];
assert a.Length >= 0;
var s := a[..];
}
method test5()
{
assert multiset({1,1}) == multiset{1};
assert multiset([1,1]) == multiset{1,1};
}
method test6(a: array<int>, n: int, e: int)
requires a != null && 0 <= n < a.Length;
modifies a;
ensures multiset(a[..n+1]) == multiset(a[..n]) + multiset{e};
{
a[n] := e;
assert a[..n+1] == a[..n] + [e];
}
method test7(a: array<int>, i: int, j: int)
requires a != null && 0 <= i < j < a.Length;
modifies a;
ensures old(multiset(a[..])) == multiset(a[..]);
ensures a[j] == old (a[i]) && a[i] == old(a[j]);
ensures forall k :: 0 <= k < a.Length && k !in {i, j} ==> a[k] == old(a[k]);
{
ghost var s := a[..i] + [a[i]] + a[i+1 .. j] + [a[j]] + a[j+1..];
assert a[..] == s;
a[i], a[j] := a[j], a[i];
assert a[..] == a[..i] + [a[i]] + a[i+1 .. j] + [a[j]] + a[j+1..];
assert s == a[..i] + [old(a[i])] + a[i+1 .. j] + [old(a[j])] + a[j+1..];
}
method test8(a: array<int>, i: int, j: int)
requires a != null && 0 <= i < j < a.Length;
modifies a;
ensures old(multiset(a[..])) == multiset(a[..]);
ensures a[j] == old (a[i]) && a[i] == old(a[j]);
ensures forall k :: 0 <= k < a.Length && k !in {i, j} ==> a[k] == old(a[k]);
{
a[i], a[j] := a[j], a[i];
}
method test9(a: array<int>, i: int, j: int, limit: int)
requires a != null && 0 <= i < j < limit <= a.Length;
modifies a;
ensures multiset(a[0..limit]) == old(multiset(a[0..limit]));
ensures a[j] == old (a[i]) && a[i] == old(a[j]);
ensures forall k :: 0 <= k < limit && k !in {i, j} ==> a[k] == old(a[k]);
{
a[i], a[j] := a[j], a[i];
}
method test10(s: seq<int>)
requires |s| > 4;
{
assert multiset( s[3 := 2] ) == multiset(s) - multiset{s[3]} + multiset{2};
assert multiset( (s[2 := 1])[3 := 2] ) == (((multiset(s) - multiset{s[2]}) + multiset{1}) - multiset{s[3]}) + multiset{2};
assert multiset( (s[2 := s[3]])[3 := s[2]] ) == (((multiset(s) - multiset{s[2]}) + multiset{s[3]}) - multiset{s[3]}) + multiset{s[2]};
}
method test11(a: array<int>, n: int, c: int)
requires a != null && 0 <= c < n <= a.Length;
modifies a;
ensures multiset(a[c..n-1]) == multiset(a[c..n]) - multiset{a[n-1]};
{
}
// ----------- the following tests mostly have to do with sets -------------
class BoxTests<G> {
// ---------- sets ----------
ghost method LemmaSet0<T>(a: set<T>, b: set<T>)
requires forall x :: x in a ==> x in b;
ensures a <= b;
{
}
ghost method LemmaSet1(a: set<G>, b: set<G>)
requires forall x :: x in a ==> x in b;
ensures a <= b;
{
}
ghost method LemmaSet2(a: set<int>, b: set<int>)
requires forall x :: x in a ==> x in b;
ensures a <= b;
{
}
ghost method LemmaSet3(a: set<object>, b: set<object>)
requires forall x :: x in a ==> x in b;
ensures a <= b;
{
}
ghost method LemmaSet4(a: set<bool>, b: set<bool>)
requires forall x :: x in a ==> x in b;
ensures a <= b;
{
}
ghost method Lemma_NonEmpty(x : set<int>, y : set<int>)
requires x * y == {};
ensures x !! y;
{
assert forall k :: k !in (x*y);
}
// ---------- sequences (don't require any special tricks in the encoding) ----------
ghost method LemmaSeq(a: seq<int>, b: seq<int>)
requires |a| <= |b| && forall i :: 0 <= i < |a| ==> a[i] == b[i];
ensures a <= b;
{
}
// ---------- multisets ----------
ghost method LemmaMultiset0<T>(a: multiset<T>, b: multiset<T>)
requires forall x :: x in a ==> x in b;
ensures a <= b; // error: this property does not hold for multisets
{
}
ghost method LemmaMultiset1(a: multiset<int>, b: multiset<int>)
requires forall x :: x in a ==> x in b;
ensures a <= b; // error: this property does not hold for multisets
{
}
}
|
