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// Bubble Sort, where the specification says the output is a permutation of
// the input.
// Introduce a constant 'N' and postulate that it is non-negative
const N: int;
axiom 0 <= N;
// Declare a map from integers to integers. In the procedure below, 'a' will be
// treated as an array of 'N' elements, indexed from 0 to less than 'N'.
var a: [int]int;
// This procedure implements Bubble Sort. One of the postconditions says that,
// in the final state of the procedure, the array is sorted. The other
// postconditions say that the final array is a permutation of the initial
// array. To write that part of the specification, the procedure returns that
// permutation mapping. That is, out-parameter 'perm' injectively maps the
// numbers [0..N) to [0..N), as stated by the second and third postconditions.
// The final postcondition says that 'perm' describes how the elements in
// 'a' moved: what is now at index 'i' used to be at index 'perm[i]'.
// Note, the specification says nothing about the elements of 'a' outside the
// range [0..N). Moreover, Boogie does not prove that the program will terminate.
procedure BubbleSort() returns (perm: [int]int)
modifies a;
// array is sorted
ensures (forall i, j: int :: 0 <= i && i <= j && j < N ==> a[i] <= a[j]);
// perm is a permutation
ensures (forall i: int :: 0 <= i && i < N ==> 0 <= perm[i] && perm[i] < N);
ensures (forall i, j: int :: 0 <= i && i < j && j < N ==> perm[i] != perm[j]);
// the final array is that permutation of the input array
ensures (forall i: int :: 0 <= i && i < N ==> a[i] == old(a)[perm[i]]);
{
var n, p, tmp: int;
n := 0;
while (n < N)
invariant n <= N;
invariant (forall i: int :: 0 <= i && i < n ==> perm[i] == i);
{
perm[n] := n;
n := n + 1;
}
while (true)
invariant 0 <= n && n <= N;
// array is sorted from n onwards
invariant (forall i, k: int :: n <= i && i < N && 0 <= k && k < i ==> a[k] <= a[i]);
// perm is a permutation
invariant (forall i: int :: 0 <= i && i < N ==> 0 <= perm[i] && perm[i] < N);
invariant (forall i, j: int :: 0 <= i && i < j && j < N ==> perm[i] != perm[j]);
// the current array is that permutation of the input array
invariant (forall i: int :: 0 <= i && i < N ==> a[i] == old(a)[perm[i]]);
{
n := n - 1;
if (n < 0) {
break;
}
p := 0;
while (p < n)
invariant p <= n;
// array is sorted from n+1 onwards
invariant (forall i, k: int :: n+1 <= i && i < N && 0 <= k && k < i ==> a[k] <= a[i]);
// perm is a permutation
invariant (forall i: int :: 0 <= i && i < N ==> 0 <= perm[i] && perm[i] < N);
invariant (forall i, j: int :: 0 <= i && i < j && j < N ==> perm[i] != perm[j]);
// the current array is that permutation of the input array
invariant (forall i: int :: 0 <= i && i < N ==> a[i] == old(a)[perm[i]]);
// a[p] is at least as large as any of the first p elements
invariant (forall k: int :: 0 <= k && k < p ==> a[k] <= a[p]);
{
if (a[p+1] < a[p]) {
tmp := a[p]; a[p] := a[p+1]; a[p+1] := tmp;
tmp := perm[p]; perm[p] := perm[p+1]; perm[p+1] := tmp;
}
p := p + 1;
}
}
}
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