soft-an01/selectionsort.dfy

predicate sorted(s: seq<int>)
{
  forall i,j: int :: 0 <= i < j < |s| ==> s[i] <= s[j]
}

predicate sameElements(a: seq<int>, b: seq<int>)
{
  multiset(a) == multiset(b)
}

method SelectionSort(a: array<int>) 
  modifies a
  ensures sorted(a[..])
  ensures sameElements(a[..], old(a[..]))
{
  if (a.Length == 0) {
    return;
  }
  
  var s := 0;

  while (s < a.Length - 1) 
  invariant sameElements(a[..], old(a[..]))
  invariant forall i,j: int :: 0 <= i < s <= j < a.Length ==> a[i] <= a[j]
  invariant s <= a.Length - 1
  invariant s >= 1 ==> sorted(a[0..s])
  decreases a.Length - s
  {
    var min: int := s;
    var i: int := s + 1;

    while i < a.Length
    invariant sameElements(a[..], old(a[..]))
    invariant s <= min < a.Length
    invariant s + 1 <= i <= a.Length
    invariant forall j: int :: s <= j < i ==> a[min] <= a[j]
    decreases a.Length - i
    {
      if (a[i] < a[min]) {
        min := i;
      }
      i := i + 1;
    }

    a[s], a[min] := a[min], a[s];
    s := s + 1;
  }
}