I'm doing some research on game semantics using these notes. Currently I'm trying to prove that the definition of composite-strategy is indeed a strategy.
I have already proved all the conditions of strategies except that it is prefix-closed(for even-length prefixes) and respects the alternation rule.
Can somebody give me a hint on approaching this two specific conditions? I have already tried many times but all my attempts led to unfinished proofs with a lot of cases (e.g. 10-12) and after proving most of those I couldn't finish the others, so I believe I'm approaching the problem from the wrong point of view.
For the definitions please refer to Chapter 3 of the notes linked above, not to the articles of, say, Abramsky and Hyland/Ong, since the definitions are slightly different(e.g strategies contains only even-length strings, while Hyland/Ong admit odd-length strings). If you refer to these you should also give some hint as to how to apply the techniques to the definitions in the notes.
Also note that the missing conditions that I'm trying to prove are all about proving that $u \upharpoonright A, C$ is a legal position for $A \multimap C$, which, in Ong's article, is something assumed in their definition of composite strategy.