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
|
// A version of Turing's additive factorial program [Dr. A. Turing, "Checking a large routine",
// In "Report of a Conference of High Speed Automatic Calculating Machines", pp. 67-69, 1949].
function Factorial(n: nat): nat
{
if n == 0 then 1 else n * Factorial(n-1)
}
method AdditiveFactorial(n: nat) returns (u: nat)
ensures u == Factorial(n);
{
u := 1;
var r := 0;
while (r < n)
invariant 0 <= r <= n;
invariant u == Factorial(r);
{
var v := u;
var s := 1;
while (s <= r)
invariant 1 <= s <= r+1;
invariant u == s * Factorial(r);
{
u := u + v;
s := s + 1;
}
r := r + 1;
}
}
// Hoare's FIND program [C.A.R. Hoare, "Proof of a program: FIND", CACM 14(1): 39-45, 1971].
// The proof annotations here are not the same as in Hoare's article.
// In Hoare's words:
// This program operates on an array A[1:N], and a value of f (1 <= f <= N).
// Its effect is to rearrange the elements of A in such a way that:
// forall p,q (1 <= p <= f <= q <= N ==> A[p] <= A[f] <= A[q]).
//
// Here, we use 0-based indices, so we would say:
// This method operates on an array A[0..N], and a value of f (0 <= f < N).
// Its effect is to rearrange the elements of A in such a way that:
// forall p,q :: 0 <= p <= f <= q < N ==> A[p] <= A[f] <= A[q]).
method FIND(A: array<int>, N: int, f: int)
requires A != null && A.Length == N;
requires 0 <= f < N;
modifies A;
ensures forall p,q :: 0 <= p <= f <= q < N ==> A[p] <= A[q];
{
var m, n := 0, N-1;
while (m < n)
invariant 0 <= m <= f <= n < N;
invariant forall p,q :: 0 <= p < m <= q < N ==> A[p] <= A[q];
invariant forall p,q :: 0 <= p <= n < q < N ==> A[p] <= A[q];
{
var r, i, j := A[f], m, n;
while (i <= j)
invariant m <= i && j <= n;
invariant -1 <= j && i <= N;
invariant i <= j ==> exists g :: i <= g < N && r <= A[g];
invariant i <= j ==> exists g :: 0 <= g <= j && A[g] <= r;
invariant forall p :: 0 <= p < i ==> A[p] <= r;
invariant forall q :: j < q < N ==> r <= A[q];
// the following two invariants capture (and follow from) the fact that the array is not modified outside the [m:n] range
invariant forall p,q :: 0 <= p < m <= q < N ==> A[p] <= A[q];
invariant forall p,q :: 0 <= p <= n < q < N ==> A[p] <= A[q];
// the following invariant is used to prove progress of the outer loop
invariant (i==m && j==n && r==A[f]) || (m<i && j<n);
{
ghost var firstIteration := i==m && j==n;
while (A[i] < r)
invariant m <= i <= N && (firstIteration ==> i <= f);
invariant exists g :: i <= g < N && r <= A[g];
invariant exists g :: 0 <= g <= j && A[g] <= r;
invariant forall p :: 0 <= p < i ==> A[p] <= r;
decreases j - i;
{ i := i + 1; }
while (r < A[j])
invariant 0 <= j <= n && (firstIteration ==> f <= j);
invariant exists g :: i <= g < N && r <= A[g];
invariant exists g :: 0 <= g <= j && A[g] <= r;
invariant forall q :: j < q < N ==> r <= A[q];
decreases j;
{ j := j - 1; }
assert A[j] <= r <= A[i];
if (i <= j) {
var w := A[i]; A[i] := A[j]; A[j] := w; // swap A[i] and A[j] (which may be referring to the same location)
assert A[i] <= r <= A[j];
i, j := i + 1, j - 1;
}
}
if (f <= j) {
n := j;
} else if (i <= f) {
m := i;
} else {
break; // Hoare used a goto
}
}
}
|
