..With this assumptions the proof looks simpler and I'm worried that I could
..be making the wrong assumptions.
..
You can assume these. Actually, you do not need to assume some of it.
The lemma uses only a smallish subset of available information.
..2.)In Lemma 12 shouldn't we assume that e_1 and e_2 share an initial segment
..e'?
..
Intuitively no. But remember the empty path from any state to itself
is an execution.
..3.)Proof by using contradiction would be a good strategy?
..
..
You mean for Theorem 13. Yes.
BW