Define $L_min$ to be the language of all minimal Turing machines, in some standard encoding. (A Turing Maching is minimal if it has the shortest encoding among all the TMs recognizing the same language.) Sipser gives a slick proof that $L_min$ is not a Recursively Enumerable (RE) language. The argument goes like this: suppose to the contrary that $L_min$ is in RE, with some enumerator $E$. Define the Turing machine $B$, which obtains its own description via the recursion theorem, waits until $E$ generates a program $C$ that is longer than $B$, and then simulates the behavior of $C$. The contradiction results from the assumption that $E$ only generates minimal programs and the construction of $B$ as a program that's shorter than some "minimal" program.

Now I want to show that $L_min$ is not in coRE (meaning that its complement is not in RE). Any ideas?

Define $L_{min}$ to be the language of all minimal Turing machines, in some standard encoding. (A Turing Maching is minimal if it has the shortest encoding among all the TMs recognizing the same language.) Sipser gives a slick proof that $L_{min}$ is not a Recursively Enumerable (RE) language. The argument goes like this: suppose to the contrary that $L_{min}$ is in RE, with some enumerator $E$. Define the Turing machine $B$, which obtains its own description via the recursion theorem, waits until $E$ generates a program $C$ that is longer than $B$, and then simulates the behavior of $C$. The contradiction results from the assumption that $E$ only generates minimal programs and the construction of $B$ as a program that's shorter than some "minimal" program.

Now I want to show that $L_{min}$ is not in coRE (meaning that its complement is not in RE). Any ideas?