I think a header of that size might be overkill. I built a 9x12 lean-to woodshed and used two sistered 2x6s for the 12-foot header [I do have an third support 4x4 post in the middle, though]. This was at the advice of my brother-in-law, who has been a carpenter for 25 years. And I second the notion that you should be using 2x6s rather than 2x4s for the rafters. I did mine every two feet on center.