Suppose $L_1, L_2, ..., L_k$ are recursively enumerable languages forming a partition of $\Sigma^*$.
How do I show that each $L_i$ are recursive ?
I see that for $x \in \Sigma^*$, $x$ belongs to exactly one $L_i$.
My idea is to run every Turing machine simultaneosly, stopping as soon as one of them accepts $x$ (must happen at some definite time by assumption).
What theory can I use to make my idea rigorous ? I've considered an non-deterministic TM, but also a $k$-tape TM, but none seems to suit my needs ?
Can someone help ?