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Yes.

There's no reason to have two nonnegative integers, you can just use one integer xi-yi. Then you care about whether the sum is zero. The language K of things which sum to zero is recognized by a push down automata -- the stack is always just a bunch of +1 tokens or -1 tokens corresponding to the current sum. Since K is recognized by a push down automata, it is context free.

The language you are interested in is K intersect L intersect K. The intersection of a regular language and a context free language is always context free.
show/hide this revision's text 1

Yes.

There's no reason to have two nonnegative integers, you can just use one integer xi-yi. Then you care about whether the sum is zero. The language K of things which sum to zero is recognized by a push down automata -- the stack is always just a bunch of +1 tokens or -1 tokens corresponding to the current sum. Since K is recognized by a push down automata, it is context free.

The language you are interested in is K intersect L. The intersection of a regular language and a context free language is always context free.