Decoction and BeerSmith

I entered my recipe into BeerSmith and picked double decoction mash.

The instructions are:
Add 7.76qts of water at 127
hold at 122 for 35 minutes
Decoct 2.5qts of mash and boil it
Hold at 147 for 20 minutes
Decoct 1.5 qts of mash and boil it
Hold at 158 for 20 minutes
Heat to 168 and hold for 10 minutes.
Sparge with 2.2 gallons of 168 water

Does these seem right? After searching here and looking at How To Brew again, it doesn't seem right to me.

From HTB, I would do a protein rest at 122 and hold. Then pull 1/3 of the mash and heat to 147 and hold. Boil that and then put enough into the mash tun to bring it to 158 and hold. Then take another 1/3 or so and heat to 160 and hold. Bring that to a boil and add it all back into the mash tun for mash out.

From BeerSmith, it seems like it wants me to do the protein rest then pull 2.5qts and boil it then let it rest at 147. To me that seems backwards. Wouldn't I want to bring it to 147 to rest then boil?

I could be way off here but here goes... I have only done two decoction mashes in my brewing "career". The purpose of holding the decoction at 147 for an amount of time before boiling would be to have some conversion occur before you boil it. I'm not sure if it is necessary. The two times I have performed a decoction mash I did what beersmith told me to and I did not end up with any conversion problems.
Beersmith is telling you to pull 2.5qt. of the mash and boil it then return the decoction to the main mash to bring the temp of the main mash up to 147. HTB is saying to pull the decoction, heat the decoction to 147, hold then boil. Then when the decoction is boiling add it back to the main mash to bring the main mash up to 147. Does that make sense? I feel like I got a bit wordy with my explanation. Anyway, I hope this helps.

Cheers,
Brandon

Thanks. That does make sense now. I just couldn't see why they were so different but I understand now. Time to try this out

I will also add here that those hold times are in parallel with you doing the decoction part. That is, basically as soon as you dough in at 122F, you want to pull out the thick decoction and start heating it to saccharification temp, hold 15 minutes, bring to boil, etc. so that by the time you add it back to the main mash, you will probably already be at or past the 35 minute hold time they say there.

So basically 5.25qts will be held at 122 while I'm heating the 2.5qts to a boil. After 35 minutes are up I dump it all back in to let it sit at 147 for 20 minutes but I take 1.5 qts of that and bring it to a boil and so on?

Or did I mess that up completely? hahaha