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// RUN: %dafny /compile:0 /dprint:"%t.dprint" "%s" > "%t"
// RUN: %diff "%s.expect" "%t"
method FindZero(a: array<int>) returns (r: int)
requires a != null && forall i :: 0 <= i < a.Length ==> 0 <= a[i];
requires forall i {:nowarn} :: 0 <= i && i+1 < a.Length ==> a[i]-1 <= a[i+1];
ensures 0 <= r ==> r < a.Length && a[r] == 0;
ensures r < 0 ==> forall i :: 0 <= i < a.Length ==> a[i] != 0;
{
var n := 0;
while (n < a.Length)
invariant forall i :: 0 <= i < n && i < a.Length ==> a[i] != 0;
{
if (a[n] == 0) { r := n; return; }
Lemma(a, n, a[n]);
n := n + a[n];
}
r := -1;
}
lemma Lemma(a: array<int>, k: int, m: int)
requires a != null && forall i :: 0 <= i < a.Length ==> 0 <= a[i];
requires forall i {:nowarn} :: 0 <= i && i+1 < a.Length ==> a[i]-1 <= a[i+1];
requires 0 <= k;
requires k < a.Length ==> m <= a[k];
ensures forall i :: k <= i < k+m && i < a.Length ==> a[i] != 0;
decreases m;
{
if (0 < m && k < a.Length) {
assert a[k] != 0;
Lemma(a, k+1, m-1);
}
}
// -----------------------------------------------------------------
method FindZero_GhostLoop(a: array<int>) returns (r: int)
requires a != null && forall i :: 0 <= i < a.Length ==> 0 <= a[i];
requires forall i {:nowarn} :: 0 <= i && i+1 < a.Length ==> a[i]-1 <= a[i+1];
ensures 0 <= r ==> r < a.Length && a[r] == 0;
ensures r < 0 ==> forall i :: 0 <= i < a.Length ==> a[i] != 0;
{
var n := 0;
while (n < a.Length)
invariant forall i :: 0 <= i < n && i < a.Length ==> a[i] != 0;
{
if (a[n] == 0) { return n; }
ghost var m := n;
while (m < n + a[n])
invariant m <= n + a[n] && m < a.Length;
invariant n + a[n] - m <= a[m];
invariant forall i :: 0 <= i < m && i < a.Length ==> a[i] != 0;
{
m := m + 1;
if (m == a.Length) { break; }
}
n := n + a[n];
}
return -1;
}
// -----------------------------------------------------------------
method FindZero_Assert(a: array<int>) returns (r: int)
requires a != null && forall i :: 0 <= i < a.Length ==> 0 <= a[i];
requires forall i {:nowarn} :: 0 <= i-1 && i < a.Length ==> a[i-1]-1 <= a[i];
ensures 0 <= r ==> r < a.Length && a[r] == 0;
ensures r < 0 ==> forall i :: 0 <= i < a.Length ==> a[i] != 0;
{
var n := 0;
while (n < a.Length)
invariant forall i :: 0 <= i < n && i < a.Length ==> a[i] != 0;
{
if (a[n] == 0) { return n; }
assert forall m {:induction} :: n <= m < n + a[n] && m < a.Length ==> n+a[n]-m <= a[m];
n := n + a[n];
}
return -1;
}
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