c++ - Inheriting typedefs?

We can simplify your failing example down to:

template 
struct Base { using type = T; };


template 
struct Derived : Base
{
    type mem; // error: 'type' does not name a type
};

The issue is that type here is a dependent name. It depends on T. There is no guarantee that for a given T there isn't some specialization of Base which does not name type. As such, base templates of class templates are not part of the standard name lookup, so you have to qualify it:

Base::type mem;

Although now we run afoul of the rule which indicates that dependent names are not assumed to be types unless explicitly states as such, so you need:

typename Base::type mem;

None of the other cases presented in the OP rely upon unqualified lookup of a dependent name.

To return to the specific problem, this doesn't compile:

static result_type apply(input_type i) {

because result_type and input_type are dependent types and so must be qualified and prefixed with typename:

static typename FuncPtrTypes::result_type apply(typename FuncPtrTypes::input_type i) {

Or, if you prefer, you can simply bring both names in with a using-declaration:

 using typename FuncPtrTypes::input_type;
 using typename FuncPtrTypes::result_type;

 static result_type apply(input_type i) {