Make Zero and One extensional traits

[qryxip]

Makes Zero and One extensional traits for Sum and Product, respectively.

As I said in rust-jp.slack.com 3 years ago IIRC, we might as well regard the "zero value" and "one value" described in the documentation as identity elements.

• std::iter::Iterator::sum
• std::iter::Iterator::product

The implementations should be still inlined.

Affects #102.

[TonalidadeHidrica]

Will you need to implement Sum trait (Sum::sum function) when you want to use the additive segtree for a custom type, if we adopt this PR?

[qryxip]

Yes - instead of impling the Zero from this crate.

struct S(i32); impl Add for S { type Output = Self; fn add(self, rhs: Self) -> Self::Output { Self(self.0 + rhs.0) } } impl Sum for S { fn sum<I: Iterator<Item = Self>>(iter: I) -> Self { iter.fold(Self(0), Add::add) // ^^^^^^^ // additive identity here } }

// or: struct S(i32); //impl Add for S { // type Output = Self; // // fn add(self, rhs: Self) -> Self::Output { // Self(self.0 + rhs.0) // } //} impl Sum for S { // only used for `Zero` from ac-library-rs fn sum<I: Iterator<Item = Self>>(iter: I) -> Self { iter.fold(Self(0), |_, _| unreachable!()) // ^^^^^^^ // additive identity here } }

Sum and Product are free from ac-library-rs, so personal libraries for non-AtCoder competitive programming services should easily adopt.

[qryxip]

↑ The second code example was not necessary.

[TonalidadeHidrica]

I don't think it is a good idea. "If you want to use + monoid, implement Zero trait." --- that's straightforward. But, "if you want to use + monoid, implement Sum trait." --- that requires an additional effort and not intuitive for me. How do you think?