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I was not able to prove the termination clause for the containsKey method using KeY alone. It seemed that Z3 would have been able to prove the remaining goals, but I thought we should rather stay within KeY.
Therefore I changed the decreases clause in containsKey such that it does not use % any more. Now the proof can be conducted with some interaction. Mainly instantiating "i_0/2".
It seems to make proofs more complicated that we sometimes refer to the array index ai and sometimes to the entry index ei with 2·ei = ai and then have to make use of ai/2 even though this does not occur itself.
If have committed my proof using MU_ as filename prefix since I do not know to which version which proof refers. I did not want to destroy anything.
I believe for get and the other contains-functions the procedure would be the same.