#include of a system header from a modularized project header file gives "Include of non-modular header inside framework module..." error

I am trying to embed libarchive in a Swift-based framework I am building.

I created a module map to privately include `archive.h` (inside module `libarchive` near the end)

framework module X
{
requires objc, blocks, objc_arc
umbrella header "X.h"

private header "XPrivate.h"
private header "x.h"

use Darwin.C

module libarchive
{
private header "archive.h"
export *
}
}

Inside `archive.h`, the following line is causing the error:

#include <inttypes.h>

I don't understand why this include isn't being mapped to the correct inttypes module in my SDK.

There is a module Darwin.C.inttypes is defined inside my SDK, here:

/Applications/Xcode.app/.../MacOSX10.12.sdk/usr/include/module.modulemap

Can someone tell me how to get the `#include` in `archive.h` to map to the declared module?

Pretend to be a system framework by changing to ```framework module X [system]```.

Not the correct solution but it seems to work.