Having ExpressibleByStringInterpolation refine ExpressibleByStringLiteral makes 
sense. I think there's a more powerful alternative design you should also 
consider. If the protocol looked like this:

protocol ExpressibleByStringInterpolation: ExpressibleByStringLiteral {
  associatedtype LiteralSegment: ExpressibleByStringLiteral
  associatedtype InterpolatedSegment
  init(forStringInterpolation: Void)

  mutating func append(literalSegment: LiteralSegment)
  mutating func append(interpolatedSegment: InterpolatedSegment)
}

Then an interpolation expression like this in `Thingy` type context:

"foo \(bar) bas \(zim: 1, zang: 2)\n"

could desugar to something like:

{
  var x = Thingy(forStringInterpolation: ())
  // Literal segments get appended using append(literalSegment: "literal")
  x.append(literalSegment: "foo ")
  // \(...) segments are arguments to a InterpolatedSegment constructor
  x.append(interpolatedSegment: Thingy.InterpolatedSegment(bar))
  x.append(literalSegment: " bas ")
  x.append(interpolatedSegment: Thingy.InterpolatedSegment(zim: 1, zang: 2))

  return x
}()

This design should be more efficient, since there's no temporary array of 
segments that needs to be formed for a variadic argument, you don't need to 
homogenize everything to Self type up front, and the string can be built up 
in-place. It also provides means to address problems 3 and 4, since the 
InterpolatedSegment associated type can control what types it's initializable 
from, and can provide initializers with additional arguments for formatting or 
other purposes.

-Joe
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