Dominance reduced at Liberty. Filly solves a problem

I've spent two days playing with Filly trying to overcome the dominance issues. She was trying to bite at every opportunity and reared a couple of times. Being fairly small still she is not too intimidating when she does this and gets backed up a long way after each attempt. However the problem was not resolving itself, so taking the idea that repeating the same actions time and again and expecting a different result is a sign of madness I changed tack.
Now I must admit to being a bit of a risk taker at heart, so in light of this I removed the lead rope and started playing at liberty. I hoped that the removal of one form of pressure would enhance the partnership side of the relationship. To start with she then advanced on me rearing slightly, for which she got a sharp tap and the shoulder. This startled her somewhat, and she settled down to a bout of licking and chewing. Now this could have been good or bad news as she digested the fact that she was now "free". I waited patiently but pensively to see what would happen next.
She approached me with the nicest look on her face and I had my old Filly back ! In fact liberty was better than ever, with no tendency to leave for the gate at all.
We moved around the school with "stick to me" game. I found that if I turned into her she again tried to nip me, but I countered this with a rapid forequarter yield through 360 degrees to continue walking in the original direction. To achieve the yield I looked stern, turned sharply into her and rhythmically moved the carrot stick towards her neck. If she turned quickly enough there was no contact, any defiance in the turn and she got tapped on the neck with the stick handle. Left brain horses particularly seem to resent yielding the forequarter as in the horse world it is a sign of submission.
There is a world of difference between asking for a yield in this way and merely hitting the horse. When asking for the yield my intention is NOT to touch her, but if she is slow in moving out of the way of the approaching stick then it does make contact. The difference is CHOICE. If you hit a horse then you make contact irrespective of what the horse does, if you ask for a yield the horse does not get touched if it yields away quickly enough. Obviously a long response time is given in the early stages of training, which reduces as they get better.
We then moved onto weave and figure 8 at liberty with very few problems at all. We even managed the aforementioned squeeze game, and football. Circling was good, but a bit closer than I would really like, but at least this means the draw is good and there is no desire to leave.

After playing last night I had 15 minutes to kill before the pub opened so decided to spend it with Filly in the stable. I had made up a hay net in which I had buried pieces of carrot and apple and carefully watched to see what she would do. What I was not prepared for was her ability to solve the problem of the next treat being around the back of the net. Very carefully, using her nose, she rotated the net until the morsel faced her then picked it out. This may not sound that impressive but to me it showed a problem solving ability I had not expected. Lets consider the logical steps required
1) Notice that morsel is round the back of the net
2) Decide to rotate the net in order to get to it ( an interesting spatial awareness problem)
3) Realise that by flicking the net with her nose in a particular way she could get the net to turn (this involved moving her nose/mouth away from the desired morsel so showed a degree of planning)
4) Stopping the rotation at the correct point
5) Eat morsel

If anyone has further insight into horse problem solving of this sort please leave a comment as I would be fascinated to hear about it. Also, if you have any experience of how liberty exercises can solve dominance issues then please do comment.