所以：有什么意义？

所以与合适的证明项相比似乎有严重的缺点： oh 上的模式匹配不会为您提供任何信息，您可以使用该信息进行另一种类型检查（是否？） - 这意味着因此值无法有效参与交互式证明。与 Disjoint 的计算有用性相对比，它表示为一个证明列表，其中 s'中的每列都不会出现在 s 中。

我不相信规格 So （不相交s s'）比不相交s s <更简单 - 您必须定义布尔不相交函数没有来自类型检查器的帮助 - 无论如何不相交当你想操纵其中包含的证据时为它自己支付。

我也怀疑所以在构建产品。为了给出 So（）的所有值，你仍然需要在 s 和 s'来满足它们实际上是不相交的类型检查器。放弃由此产生的证据似乎是一种浪费。

因此对于代码的作者和用户来说似乎都很笨拙在其中部署。 '所以'，在什么情况下我想要用那么？

<如果你已经有一个 b：Bool ，你可以把它变成命题：所以b ，这是一个bit比 b≡true 短。有时（我不记得任何实际的情况），没有必要打扰一个正确的数据类型，这个快速解决方案就足够了。

如果 n 在规范形式（即 suc（suc（suc ... zero））），则 n <=？ 3 可以由编译器检查，不需要证明。在实际的Agda中，它是

我在这个答案（它是 {_：False（m≟0）} there）。我想如果没有这个简单的定义，就不可能写出这里描述的机器的可用版本：

其中 T 在Agda的标准库中是 So 。

另外，在实例参数因此 -as-a-data-type可以用作所以 -as-a-constraint：

What is the intended purpose of the So type? Transliterating into Agda:

So lifts a Boolean proposition up to a logical one. Oury and Swierstra's introductory paper The Power of Pi gives an example of a relational algebra indexed by the tables' columns. Taking the product of two tables requires that they have different columns, for which they use So:

I'm used to constructing evidence terms for the things I want to prove about my programs. It seems more natural to construct a logical relation on Schemas to ensure disjointedness:

So seems to have serious disadvantages compared to a "proper" proof-term: pattern matching on oh doesn't give you any information with which you could make another term type-check (Does it?) - which would mean So values can't usefully participate in interactive proving. Contrast this with the computational usefulness of Disjoint, which is represented as a list of proofs that each column in s' doesn't appear in s.

I don't really believe that the specification So (disjoint s s') is simpler to write than Disjoint s s' - you have to define the Boolean disjoint function without help from the type-checker - and in any case Disjoint pays for itself when you want to manipulate the evidence contained therein.

I am also sceptical that So saves effort when you're constructing a Product. In order to give a value of So (disjoint s s'), you still have to do enough pattern matching on s and s' to satisfy the type checker that they are in fact disjoint. It seems like a waste to discard the evidence thus generated.

So seems unwieldy for both authors and users of code in which it's deployed. 'So', under what circumstances would I want to use So?

If you already have a b : Bool, you can turn it into proposition: So b, which is a bit shorther than b ≡ true. Sometimes (I don't remember any actual case) there is no need to bother with a proper data type, and this quick solution is enough.

So seems to have serious disadvantages compared to a "proper" proof-term: pattern matching on oh doesn't give you any information with which you could make another term type-check. As a corollary, So values can't usefully participate in interactive proving. Contrast this with the computational usefulness of Disjoint, which is represented as a list of proofs that each column in s' doesn't appear in s.

So does give you the same information as Disjoint — you just need to extract it. Basically, if there is no inconsistency between disjoint and Disjoint, then you should be able to write a function So (disjoint s) -> Disjoint s using pattern matching, recursion and impossible cases elimination.

However, if you tweak the definition a bit:

So becomes a really useful data type, because x : So true immediately reduces to tt due to the eta-rule for ⊤. This allows to use So like a constraint: in pseudo-Haskell we could write

and if n is in canonical form (i.e. suc (suc (suc ... zero))), then n <=? 3 can be checked by the compiler and no proofs are needed. In actual Agda it is

I used this trick in this answer (it is {_ : False (m ≟ 0)} there). And I guess it would be impossible to write a usable version of the machinery decribed here without this simple definition:

where T is So in the Agda's standard library.

Also, in the presence of instance arguments So-as-a-data-type can be used as So-as-a-constraint: