# alstruct

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代数的構造を表すトレイトや、それにかかるパッチを提供するモジュール。

* member

## BinOp

```python
BinOp Op: Kind 2 = Subsume Op(Self, Self.ReturnTypeOf Op), Additional: {
    .ReturnTypeof = TraitType -> Type
}

Nat <: BinOp Add
assert Nat.ReturnTypeof(Add) == Nat
assert Nat.ReturnTypeof(Sub) == Int
assert Nat.ReturnTypeof(Mul) == Nat
assert Nat.ReturnTypeof(Div) == Positive Ratio
```

## SemiGroup

```python
SemiGroup Op: Kind 2 = Op(Self, Self)

IntIsSemiGroupAdd = Patch Int, Impl := SemiGroup Add

Int <: SemiGroup Add
```

## Functor

```python
## * Identity law: x.map(id) == x
## * Composition law: x.map(f).map(g) == x.map(f.then g)
Functor = Trait {
    .map|T, U: Type| = (Self(T), T -> U) -> Self U
}
```

## Applicative

```python
## * Identity law: x.app(X.pure(id)) == x
Applicative = Subsume Functor, Additional: {
    .pure|T: Type| = T -> Self T
    .app|T, U: Type| = (Self(T), Self(T -> U)) -> Self U
}
```

## Monad

```python
Monad = Subsume Applicative, Additional: {
    .bind|T, U: Type| = (Self(T), T -> Self U) -> Self U
}
```