Functions within modules within Compile

I think I answered my own question via CompilePrint (just learned about this); will post for posterity:

ClearAll(f1, f2, f3, f4, f5, f6, f7, f8)
f3 := Function(y, y^4);
c1 = Compile({x},
   m1 = Module({f1},
     f1 = Function(y, y^2);
     f1(x)
     );
   Module({},
    {Function(y, y^3)(x), f3(x)}
    )
   );
c1(2)
m1
Needs("CompiledFunctionTools`")
CompilePrint(c1)

1   V17 = MainEvaluate(                                                 
  2
Function({x}, m1 = Module({f1}, f1 = Function(y, y ); f1(x)))( R0))
2   R2 = R0
3   R1 = Power( R2, I0)
4   R2 = MainEvaluate( Hold(f3)( R0))
5   T(R1)0 = {R1, R2}
6   Return

It looks like the versions that I thought were “working” were just running in non-compiled mode (not just the assignments, either — the whole step). I’m used to getting an error/warning in such cases, but I had to look at CompilePrint to see it this time.

I guess you just have to use anonymous functions exclusively. It looks like BOTH the methods in my original post AND defining the Function outside of Compile altogether (as I indicated I was unsure-about in my original post) cause MainEvaluate() to be used…

I could’ve sworn I’d come across other examples on StackOverflow where a function was defined outside of Compile and then used within it, but I’ll have to educate myself further.