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| // 定义零：作为自然数的起点 | |
| struct Zero; | |
| // 定义后继：Succ<N> 表示自然数 N 的后继（即 N + 1） | |
| struct Succ<N>(N); | |
| // 为自然数定义特性 | |
| trait Natural { | |
| // 获取数值（运行时） | |
| fn value() -> usize; | |
| // 类型级别的值表示（编译时） | |
| type Value: NaturalValue; | |
| } | |
| // 类型级别的值表示 | |
| trait NaturalValue { | |
| const VALUE: usize; | |
| } | |
| // 零的实现 | |
| impl Natural for Zero { | |
| fn value() -> usize { 0 } | |
| type Value = ZeroValue; | |
| } | |
| struct ZeroValue; | |
| impl NaturalValue for ZeroValue { | |
| const VALUE: usize = 0; | |
| } | |
| // 后继的实现 | |
| impl<N: Natural> Natural for Succ<N> { | |
| fn value() -> usize { N::value() + 1 } | |
| type Value = SuccValue<N::Value>; | |
| } | |
| struct SuccValue<V: NaturalValue>(V); | |
| impl<V: NaturalValue> NaturalValue for SuccValue<V> { | |
| const VALUE: usize = V::VALUE + 1; | |
| } | |
| // 类型别名，方便使用 | |
| type One = Succ<Zero>; | |
| type Two = Succ<One>; | |
| type Three = Succ<Two>; | |
| type Four = Succ<Three>; | |
| // 加法：N + M = R | |
| trait Add<M: Natural> { | |
| type Result: Natural; | |
| } | |
| // 零加任何数等于该数 | |
| impl<M: Natural> Add<M> for Zero { | |
| type Result = M; | |
| } | |
| // Succ<N> + M = Succ<N + M> | |
| impl<N: Natural, M: Natural> Add<M> for Succ<N> | |
| where | |
| N: Add<M>, | |
| { | |
| type Result = Succ<<N as Add<M>>::Result>; | |
| } | |
| // 乘法：N * M = R | |
| trait Mul<M: Natural> { | |
| type Result: Natural; | |
| } | |
| // 零乘任何数等于零 | |
| impl<M: Natural> Mul<M> for Zero { | |
| type Result = Zero; | |
| } | |
| // Succ<N> * M = (N * M) + M | |
| impl<N: Natural, M: Natural> Mul<M> for Succ<N> | |
| where | |
| N: Mul<M>, | |
| <N as Mul<M>>::Result: Add<M>, | |
| { | |
| type Result = <<N as Mul<M>>::Result as Add<M>>::Result; | |
| } | |
| // 相等性证明：证明 N == M | |
| trait Eq<M: Natural> {} | |
| // 零等于零 | |
| impl Eq<Zero> for Zero {} | |
| // 如果 N == M，则 Succ<N> == Succ<M> | |
| impl<N: Natural, M: Natural> Eq<Succ<M>> for Succ<N> | |
| where | |
| N: Eq<M>, | |
| {} | |
| // 函数：验证两个自然数相加的结果 | |
| fn verify_addition<N: Natural, M: Natural, R: Natural>() | |
| where | |
| N: Add<M, Result = R>, // 类型级别验证：N + M = R | |
| { | |
| println!("{} + {} = {}", N::value(), M::value(), R::value()); | |
| } | |
| // 函数：验证两个自然数相乘的结果 | |
| fn verify_multiplication<N: Natural, M: Natural, R: Natural>() | |
| where | |
| N: Mul<M, Result = R>, // 类型级别验证：N * M = R | |
| { | |
| println!("{} * {} = {}", N::value(), M::value(), R::value()); | |
| } | |
| // 函数：验证两个自然数相等 | |
| fn verify_equality<N: Natural, M: Natural>() | |
| where | |
| N: Eq<M>, // 类型级别验证：N == M | |
| { | |
| println!("{} == {}", N::value(), M::value()); | |
| } | |
| fn main() { | |
| // 这些调用会在编译时验证类型级别的计算 | |
| // 如果计算不正确，会导致编译错误 | |
| // 验证加法 | |
| verify_addition::<One, Two, Three>(); // 1 + 2 = 3 | |
| verify_addition::<Two, Two, Four>(); // 2 + 2 = 4 | |
| // 验证乘法 | |
| verify_multiplication::<Two, Two, Four>(); // 2 * 2 = 4 | |
| verify_multiplication::<Two, Three, Six>(); // 2 * 3 = 6 | |
| // 验证相等性 | |
| verify_equality::<Two, Two>(); // 2 == 2 | |
| // 下面的代码会导致编译错误，因为 2 + 2 ≠ 5 | |
| // verify_addition::<Two, Two, Five>(); // 编译错误! | |
| // 下面的代码会导致编译错误，因为 3 ≠ 2 | |
| // verify_equality::<Three, Two>(); // 编译错误! | |
| } | |
| // 定义更多的自然数 | |
| type Five = Succ<Four>; | |
| type Six = Succ<Five>; | |
| type Seven = Succ<Six>; | |
| type Eight = Succ<Seven>; | |
| type Nine = Succ<Eight>; | |
| type Ten = Succ<Nine>; |
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