MU Puzzle - The Puzzle

The Puzzle

Suppose there are the symbols M, I, and U which can be combined to produce strings of symbols called "words". The MU puzzle asks one to start with the "axiomatic" word MI and transform it into the word MU using in each step one of the following transformation rules:

1. Add a U to the end of any string ending in I. For example: MI to MIU.
2. Double any string after the M (that is, change Mx, to Mxx). For example: MIU to MIUIU.
3. Replace any III with a U. For example: MUIIIU to MUUU.
4. Remove any UU. For example: MUUU to MU.

Using these four rules is it possible to change MI into MU in a finite number of steps?

The production rules can be written in a more schematic way. Suppose x and y behave as variables (standing for strings of symbols). Then the production rules can be written as:

1. xI → xIU
2. Mx → Mxx
3. xIIIy → xUy
4. xUUy → xy

Is it possible to obtain the word MU using these rules?