oop - 类型 - 子类型关系。似乎有些不清楚

Question

我正在阅读有关面向对象的编程语言的类(class)的一些幻灯片，并进入了类型子类型定义:

Barbara Liskov，“数据抽象和层次结构”，SIGPLAN 通知，
23,5(1988 年 5 月):

What is wanted here is something like the following substitution property: If for each object o_s of type S there is an object o_T of type T such that for all programs P
defined in terms of T, the behavior of P is unchanged when o_S is substituted for o_T then S is a subtype of T

然后是一个例子:

Point = { x:Integer, y:Integer }
PositivePoint = { x:Positive, y:Positive }
where Positive = { k:Integer | k > 0 }

Can we say that PositivePoint ≤ Point?

Yes, because an element of type PositivePoint may always replace an element of type Point in a program defined in Point terms!

现在......对我来说，它似乎应该完全相反:Point ≤ PositivePoint 因为我不能在使用带有负坐标的 Point 的程序中使用 PositivePoint，而我可以相反。

我怀疑语法是否是 Type ≤ Sub-type或 Sub-Type ≤ Type ，但声明似乎更清楚，那有什么问题呢？

编辑

为了让事情更容易，问题是:
你能说PositivePoint是 Point 的子类型?
为什么？

第二次编辑

我在这里报告我在评论中写的内容，希望它能让我的问题更清楚:

Suppose that the program has to draw a square map from Point (-100, -100) to Point (100, 100). What would happen if you use type PositivePoint? Would the program's behavior be unchanged? It would not. This "unchanged behavior" is the only thing I don't get. If the definition of subtype was simply inheriting and overriding from an other type it would be ok, but it doesn't seem to be the case.

Answer 1

Liskov 是正确的，PositivePoint ≤ Point，因为 PositivePoint 是 Point 的细化。任何使用 Point 的代码也必须能够使用 PositivePoint，因为无论如何 Point 的坐标总是有可能是正的。反之则不然，因为使用 PositivePoint 的代码可能会在坐标始终为正的假设下运行，用 Point 替换 PositivePoint 会破坏该假设。

请注意，她并不是说 PositivePoint 可以替换 Point，只是在需要 Point 的地方可以使用 PositivePoint。