supleed2/ELEC70056-HSV-CW1
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Dafny Task 3 mostly complete

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Aadi Desai 2022-11-21 18:33:48 +00:00
parent 88e407c8f7
commit e9a2bf5354
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GPG key ID: CFFFE425830EF4D9
1 changed files with 48 additions and 16 deletions
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dafny/submission.dfy
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@ -76,8 +76,7 @@ method binarysearch_iter(A:array<int>, v:int) returns (result:bool)
var hi := A.Length; var hi := A.Length;
while lo < hi while lo < hi
invariant 0 <= lo <= hi <= A.Length; invariant 0 <= lo <= hi <= A.Length;
invariant forall j :: 0 <= j < lo ==> A[j] != v; invariant forall j :: (0 <= j < lo || hi <= j < A.Length) ==> A[j] != v;
invariant forall j :: hi <= j < A.Length ==> A[j] != v;
decreases hi - lo; decreases hi - lo;
{ {
var mid := (lo + hi) / 2; var mid := (lo + hi) / 2;
@ -92,16 +91,21 @@ 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 ensures 0 <= lo <= pivot < hi <= A.Length;
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];
modifies A ensures forall k :: lo <= k < pivot ==> A[k] <= A[pivot];
ensures forall k :: pivot <= k < hi ==> A[pivot] <= A[k];
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 :: (pivot <= k < i) ==> A[k] >= A[pivot]
decreases hi - i decreases hi - i
{ {
if A[i] < A[pivot] { if A[i] < A[pivot] {
@ -110,6 +114,10 @@ method partition(A:array<int>, lo:int, hi:int) returns (pivot:int)
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[k] >= A[pivot]
invariant forall k :: (lo <= k <= pivot) ==> A[k] <= A[pivot]
invariant A[pivot] > tmp
decreases j decreases j
{ {
A[j+1] := A[j]; A[j+1] := A[j];
@ -123,29 +131,53 @@ method partition(A:array<int>, lo:int, hi:int) returns (pivot:int)
} }
} }
predicate sorted_seq(A:seq<int>)
{
forall m,n :: 0 <= m < n < |A| ==> 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 0 <= lo <= hi <= A.Length requires A.Length == 2;
ensures forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k] requires 0 <= lo <= hi <= A.Length;
modifies A ensures forall k :: (0 <= k < lo || hi <= k < A.Length) ==> old(A[k]) == A[k];
decreases hi - lo // 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]);
modifies A;
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);
{ {
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], // print "Before: ", A[0], A[1], A[2], A[3], A[4], A[5], A[6], "\n";
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], A[2], A[3], print "After: ", A[0], " ", A[1], "\n";
A[4], A[5], A[6], "\n";
} }