Overview

- what is Subtyping?
- what is Liskov Substitution Principle?
- what's the problem?
- how to solve this mistake of type system

### What is Subtyping?

A common misunderstanding is OOP must have **subtyping** and **generic** at the same time, but they are independent features, **generic** is just a type-level function with takes a type, generates a type. **Subtyping** is a relationship between type, when we say `A <: B`

, which means `A`

is a subtype of `B`

. In Java, we use concrete syntax `extends`

or `implements`

.

### What is Liskov Substitution Principle(LSP)?

**LSP** said

if S is a subtype of T, then objects of type T in a program may be replaced with objects of type S without altering the program

### What's the problem

Square is a special Rectangle where all four sides are equal in length. Thus we might write down:

```
public class Rectangle {
double x, y;
Rectangle(double initX, double initY) { x = initX; y = initY; }
public void setX(double x) { this.x = x; }
public void setY(double y) { this.y = y; }
}
class Square extends Rectangle {
Square(double initX, double initY) throws SquareException {
super(initX, initY);
if (initX != initY) {
throw new SquareException(initX, initY);
}
}
}
```

Ok, so we can have following program:

```
Square s = new Square(2, 2);
Rectangle r = s;
r.setX(10);
```

Oops, `s`

is not a square anymore, this correctly follows **LSP** but incorrect generally.

### Solution

Here is a solution: refinement type. By using refinement type, we can claim `type Square = Rectangle when (x = y)`

. Then problem solved, when programming if we can't proof `x = y`

this predicate then compiler won't think that is a `Square`

but a `Rectangle`

. Since `Square`

is a `Rectangle`

, method in `Rectangle`

still can be using by `Square`

, thus we didn't lose our power. However, program like `s.setX(v)`

, would need to prove `v = y`

, to prove it is a `Square`

. Here we have two choices:

- change
`s`

type to`Rectangle`

, the problem of this solution is if we write`r = s; r.setX(v)`

, compiler might not work for this(this is also why I'm designing controllable-refinement), mutable semantic always needs more effort. - throw compile error that
`Square`

constraint was broke by the statement.

Choose one and work with it, that's all, have a nice day.

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