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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) method partition(A:array<int>, lo:int, hi:int) returns (pivot:int)
requires 0 <= lo < hi <= A.Length; requires 0 <= lo < hi <= A.Length;
ensures 0 <= lo <= pivot < 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 :: (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 :: lo <= k < pivot ==> A[k] < A[pivot];
ensures forall k :: pivot <= k < hi ==> A[pivot] <= A[k]; ensures forall k :: pivot < k < hi ==> A[pivot] <= A[k];
modifies A; modifies A;
{ {
pivot := lo; pivot := lo;
var i := lo+1; var i := lo+1;
while i < hi while i < hi
invariant 0 <= lo <= pivot < 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 :: (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 :: (lo <= k <= pivot) ==> A[k] <= A[pivot] invariant forall k :: (pivot < k < i) ==> A[pivot] <= A[k];
invariant forall k :: (pivot <= k < i) ==> A[k] >= A[pivot] invariant hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
decreases hi - i invariant 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
{ {
if A[i] < A[pivot] { if A[i] < A[pivot] {
var j := i-1; var j := i-1;
var tmp := A[i]; var tmp := A[i];
A[i] := A[j]; A[i] := A[j];
while pivot < j while pivot < j
invariant forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k] 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[pivot] <= A[k];
invariant forall k :: (pivot <= k <= i) ==> A[k] >= A[pivot] invariant forall k :: (lo <= k < pivot) ==> A[k] < A[pivot];
invariant forall k :: (lo <= k <= pivot) ==> A[k] <= A[pivot] invariant A[pivot] > tmp;
invariant A[pivot] > tmp invariant hi < A.Length ==> forall k :: lo <= k < hi ==> A[k] < A[hi];
decreases j invariant 0 < lo ==> forall k :: lo <= k < hi ==> A[lo-1] <= A[k];
{ {
A[j+1] := A[j]; A[j+1] := A[j];
j := j-1; 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) method quicksort_between(A:array<int>, lo:int, hi:int)
requires A.Length == 2;
requires 0 <= lo <= hi <= A.Length; 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]; 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 // 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 // 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; modifies A;
decreases hi - lo; decreases hi - lo;
{ {
if lo+1 >= hi { return; } if lo+1 >= hi { return; }
var pivot := partition(A, lo, hi); 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); 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); 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>) method quicksort(A:array<int>)
modifies A; modifies A;
requires A.Length == 2; ensures sorted(A);
// ensures sorted(A);
{ {
quicksort_between(A, 0, A.Length); quicksort_between(A, 0, A.Length);
} }
method Main() { method Main() {
// var A:array<int> := new int[7] [4,0,1,9,7,1,2]; 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"; 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";
quicksort(A); 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";
} }