verification - 如何证明 Dafny count < size

Question

我目前正在学习达夫尼。我完全被引理弄糊涂了，我不知道如何使用它。该教程没有那么有用。如果我想证明 count(a) <= |a| 我应该怎么做。感谢您的帮助。

function count(a: seq<bool>): nat
ensures count(a) <= |a|;
{
   if |a| == 0 then 0 else
   (if a[0] then 1 else 0) + count(a[1..])
}

Answer 1

你已经证明了！你写了你想要的属性作为函数的后置条件，Dafny 毫无怨言地验证它。而已。

您还可以使用引理来证明该属性。这是一个例子：

function count(a: seq<bool>): nat
{
  if |a| == 0 then 0 else
  (if a[0] then 1 else 0) + count(a[1..])
}

lemma CountProperty(a: seq<bool>)
  ensures count(a) <= |a|
{
}

再一次，Dafny 验证了引理，没有发出任何抱怨，所以你已经证明了！

假设 Dafny 总是会自动为您证明事情是不正确的。因此，学习如何手动编写证明也是一个好主意。这是此属性的手动证明。为了确保 Dafny 不会尝试自动进行归纳，我使用了一个指令将其关闭（从而使我们的生活比 Dafny 通常会更难）：

lemma {:induction false} CountProperty(a: seq<bool>)
  ensures count(a) <= |a|
{
  // Let's consider two cases, just like the definition of "count"
  // considers two cases.
  if |a| == 0 {
    // In this case, we have:
    assert count(a) == 0 && |a| == 0;
    // so the postcondition follows easily.
  } else {
    // By the definition of "count", we have:
    assert count(a) == (if a[0] then 1 else 0) + count(a[1..]);
    // We know an upper bound on the first term of the addition:
    assert (if a[0] then 1 else 0) <= 1;
    // We can also obtain an upper bound on the second term by
    // calling the lemma recursively.  We do that here:
    CountProperty(a[1..]);
    // The call we just did gives us the following property:
    assert count(a[1..]) <= |a[1..]|;
    // Putting these two upper bounds together, we have:
    assert count(a) <= 1 + |a[1..]|;
    // We're almost done.  We just need to relate |a[1..]| to |a|.
    // This is easy:
    assert |a[1..]| == |a| - 1;
    // By the last two assertions, we now have:
    assert count(a) <= 1 + |a| - 1;
    // which is the postcondition we have to prove.
  }
}

编写这样的证明的更好方法是使用经过验证的计算，Dafny 称之为“计算语句”：

lemma {:induction false} CountProperty(a: seq<bool>)
  ensures count(a) <= |a|
{
  if |a| == 0 {
    // trivial
  } else {
    calc {
      count(a);
    ==  // def. count
      (if a[0] then 1 else 0) + count(a[1..]);
    <=  // left term is bounded by 1
      1 + count(a[1..]);
    <=  { CountProperty(a[1..]); }  // induction hypothesis gives a bound for the right term
      1 + |a[1..]|;
    ==  { assert |a[1..]| == |a| - 1; }
      |a|;
    }
  }
}

我希望这能让你开始。

安全编程，

鲁斯坦