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Dafny Task 3 complete

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Aadi Desai 2022-11-28 10:26:36 +00:00
parent e9a2bf5354
commit ba694b4e41
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1 changed files with 31 additions and 38 deletions
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dafny/submission.dfy
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@ -93,32 +93,36 @@ method binarysearch_iter(A:array<int>, v:int) returns (result:bool)
method partition(A:array<int>, lo:int, hi:int) returns (pivot:int)
requires 0 <= lo < hi <= A.Length;
ensures 0 <= lo <= pivot < hi <= A.Length;
requires hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
ensures hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
requires 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
ensures 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
ensures forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k];
ensures forall k :: lo <= k < pivot ==> A[k] <= A[pivot];
ensures forall k :: pivot <= k < hi ==> A[pivot] <= A[k];
ensures forall k :: lo <= k < pivot ==> A[k] < A[pivot];
ensures forall k :: pivot < k < hi ==> A[pivot] <= A[k];
modifies A;
{
pivot := lo;
var i := lo+1;
while i < hi
invariant 0 <= lo <= pivot < i <= hi
invariant forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k]
invariant forall k :: (k==pivot) ==> A[k] == old(A[lo])
invariant forall k :: (lo <= k <= pivot) ==> A[k] <= A[pivot]
invariant forall k :: (pivot <= k < i) ==> A[k] >= A[pivot]
decreases hi - i
invariant 0 <= lo <= pivot < i <= hi;
invariant forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k];
invariant forall k :: (lo <= k < pivot) ==> A[k] < A[pivot];
invariant forall k :: (pivot < k < i) ==> A[pivot] <= A[k];
invariant hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
invariant 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
{
if A[i] < A[pivot] {
var j := i-1;
var tmp := A[i];
A[i] := A[j];
while pivot < j
invariant forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k]
invariant forall k :: (k==pivot) ==> A[k] == old(A[lo])
invariant forall k :: (pivot <= k <= i) ==> A[k] >= A[pivot]
invariant forall k :: (lo <= k <= pivot) ==> A[k] <= A[pivot]
invariant A[pivot] > tmp
decreases j
invariant forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k];
invariant forall k :: (pivot < k <= i) ==> A[pivot] <= A[k];
invariant forall k :: (lo <= k < pivot) ==> A[k] < A[pivot];
invariant A[pivot] > tmp;
invariant hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
invariant 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
{
A[j+1] := A[j];
j := j-1;
@ -131,53 +135,42 @@ method partition(A:array<int>, lo:int, hi:int) returns (pivot:int)
}
}
predicate sorted_seq(A:seq<int>)
predicate sorted_between(A: array<int>, lo: int, hi: int)
reads A;
requires 0 <= lo <= hi <= A.Length;
{
forall m,n :: 0 <= m < n < |A| ==> A[m] <= A[n]
forall m, n :: lo <= m < n < hi ==> A[m] <= A[n]
}
method quicksort_between(A:array<int>, lo:int, hi:int)
requires A.Length == 2;
requires 0 <= lo <= hi <= A.Length;
requires hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
ensures hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
requires 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
ensures 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
ensures forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k];
// old(A[k]) evaluates to the value of A[k] on entry to the method, so this
// checkes that only values in A[lo..hi] can be modified by this method
// ensures sorted_seq(A[lo..hi]);
ensures sorted_between(A, lo, hi);
modifies A;
decreases hi - lo;
{
if lo+1 >= hi { return; }
var pivot := partition(A, lo, hi);
assert forall a :: a in A[lo..pivot] ==> a <= A[pivot];
assert forall a :: a in A[pivot..hi] ==> A[pivot] <= a;
quicksort_between(A, lo, pivot);
// assert sorted_seq(A[lo..pivot]);
// assert sorted_seq(A[pivot..hi]);
assert forall a :: a in A[lo..pivot] ==> a <= A[pivot];
assert forall a :: a in A[pivot..hi] ==> A[pivot] <= a;
quicksort_between(A, pivot+1, hi);
// assert sorted_seq(A[lo..pivot]);
// assert sorted_seq(A[pivot..hi]);
// assert forall a :: a in A[lo..pivot] ==> a <= A[pivot];
// assert forall a :: a in A[pivot..hi] ==> A[pivot] <= a;
}
method quicksort(A:array<int>)
modifies A;
requires A.Length == 2;
// ensures sorted(A);
ensures sorted(A);
{
quicksort_between(A, 0, A.Length);
}
method Main() {
// var A:array<int> := new int[7] [4,0,1,9,7,1,2];
// print "Before: ", A[0], A[1], A[2], A[3], A[4], A[5], A[6], "\n";
// quicksort(A);
// print "After: ", A[0], A[1], A[2], A[3], A[4], A[5], A[6], "\n";
var A:array<int> := new int[2] [1040, -197];
print "Before: ", A[0], " ", A[1], "\n";
var A:array<int> := new int[7] [4,0,1,9,7,1,2];
print "Before: ", A[0], A[1], A[2], A[3], A[4], A[5], A[6], "\n";
quicksort(A);
print "After: ", A[0], " ", A[1], "\n";
print "After: ", A[0], A[1], A[2], A[3], A[4], A[5], A[6], "\n";
}