Extension of constructed generic type in Swift

This can be achieved using protocol extensions (See The Swift Programming Language: Protocols for more information). In Swift 3:

To sum just Ints you could do:

extension Sequence where Iterator.Element == Int {
    var sum: Int {
        return reduce(0, +)
    }
}

Usage:

let nums = [1, 2, 3, 4]
print(nums.sum) // Prints: "10"

Or, for something more generic you could what @Wes Campaigne suggested and create an Addable protocol:

protocol Addable {
    init()
    func + (lhs: Self, rhs: Self) -> Self
}

extension Int   : Addable {}
extension Double: Addable {}
extension String: Addable {}
...

Next, extend Sequence to add sequences of Addable elements:

extension Sequence where Iterator.Element: Addable {
    var sum: Iterator.Element {
        return reduce(Iterator.Element(), +)
    }
}

Usage:

let doubles = [1.0, 2.0, 3.0, 4.0]
print(doubles.sum) // Prints: "10.0"

let strings = ["a", "b", "c"]
print(strings.sum) // Prints: "abc"

Managed to get something working in an extensible, generic fashion without abusing the type system too badly, however it has some limitations.

protocol Addable {
    func +(lhs: Self, rhs: Self) -> Self
    class var identity: Self { get }
}

extension Int : Addable {
    static var identity: Int { get { return 0 } }
}

extension String : Addable {
    static var identity: String { get { return "" } }
}

extension Array {
    func sum<U : Addable>() -> U? {
        let s: U? = U.identity
        return self.sum(s)
    }

    func sum<U : Addable>(start: U?) -> U? {
        return reduce(start) { lhs, rhs in
            switch (lhs, rhs) {
            case (.Some(let left), let right as U):
                return left + right
            default:
                return nil
            }
        }
    }
}

Specifically: with this solution, type inferencing won't work on the no-parameter sum() method, so you have to either annotate the expected return type or give it a starting value (from which it can infer the type).

Note also that this returns a value of Optional type: if for any reason a sum of the expected type cannot be computed from the array, it returns nil.

To illustrate:

let int_array = Array(1...10)

let x: Int? = int_array.sum() // result: {Some 55}
let x2 = int_array.sum(0) // result: {Some 55}
let x3 = int_array.sum() // Compiler error because it can't infer type


let string_array = ["a", "b", "c"]

let y: String? = string_array.sum() // result: {Some "abc"}
let y2 = string_array.sum("") // result: {Some "abc"}

let y3: Int? = string_array.sum() // result: nil  (can't cast String to Int)
let y4 = string_array.sum(0) // result: nil  (can't cast String to Int)


let double_array = [1.3, 4.2, 2.1]

let z = double_array.sum(0.0) // Compiler error because we haven't extended Double to be Addable