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
|
class SumOfCubes {
static use function SumEmUp(n: int, m: int): int
requires 0 <= n && n <= m;
decreases m - n;
{
if m == n then 0 else n*n*n + SumEmUp(n+1, m)
}
static method Socu(n: int, m: int) returns (r: int)
requires 0 <= n && n <= m;
ensures r == SumEmUp(n, m);
{
call a := SocuFromZero(m);
call b := SocuFromZero(n);
r := a - b;
call Lemma0(n, m);
}
static method SocuFromZero(k: int) returns (r: int)
requires 0 <= k;
ensures r == SumEmUp(0, k);
{
call g := Gauss(k);
r := g * g;
call Lemma1(k);
}
ghost static method Lemma0(n: int, m: int)
requires 0 <= n && n <= m;
ensures SumEmUp(n, m) == SumEmUp(0, m) - SumEmUp(0, n);
{
var k := n;
while (k < m)
invariant n <= k && k <= m;
invariant SumEmDown(0, n) + SumEmDown(n, k) == SumEmDown(0, k);
decreases m - k;
{
k := k + 1;
}
call Lemma3(0, n);
call Lemma3(n, k);
call Lemma3(0, k);
}
static use function GSum(k: int): int
requires 0 <= k;
decreases k;
{
if k == 0 then 0 else GSum(k-1) + k-1
}
ghost static method Gauss(k: int) returns (r: int)
requires 0 <= k;
ensures r == GSum(k);
{
r := k * (k - 1) / 2;
call Lemma2(k);
}
ghost static method Lemma1(k: int)
requires 0 <= k;
ensures SumEmUp(0, k) == GSum(k) * GSum(k);
{
var i := 0;
while (i < k)
invariant i <= k;
invariant SumEmDown(0, i) == GSum(i) * GSum(i);
decreases k - i;
{
call Lemma2(i);
i := i + 1;
}
call Lemma3(0, k);
}
ghost static method Lemma2(k: int)
requires 0 <= k;
ensures 2 * GSum(k) == k * (k - 1);
{
var i := 0;
while (i < k)
invariant i <= k;
invariant 2 * GSum(i) == i * (i - 1);
decreases k - i;
{
i := i + 1;
}
}
static use function SumEmDown(n: int, m: int): int
requires 0 <= n && n <= m;
decreases m;
{
if m == n then 0 else SumEmDown(n, m-1) + (m-1)*(m-1)*(m-1)
}
ghost static method Lemma3(n: int, m: int)
requires 0 <= n && n <= m;
ensures SumEmUp(n, m) == SumEmDown(n, m);
{
var k := n;
while (k < m)
invariant n <= k && k <= m;
invariant SumEmUp(n, m) == SumEmDown(n, k) + SumEmUp(k, m);
decreases m - k;
{
k := k + 1;
}
}
}
|
