What is the pumping lemma for recursively enumerable languages?

What is the pumping lemma for recursively enumerable languages?

The pumping lemma for recursively enumerable languages states that for any recursively enumerable language L, there exists a pumping length p such that any string s in L with length greater than or equal to p can be divided into five parts, s = uvwxy, satisfying the following conditions:

1. For each i ≥ 0, the string uv^iwx^iy is also in L.
2. The lengths of v and x combined, |vx|, is greater than 0.
3. The length of uvwxy is less than or equal to p.

In other words, the pumping lemma guarantees that there exists a substring within a string in a recursively enumerable language that can be repeated or removed while still remaining in the language.