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I have the following two languages: $$ L_{U}\triangleq\left\{ \langle M,x\rangle\,:\,M\text{ accepts }x\right\} ,L_{D}\triangleq\left\{ \langle M\rangle\,:\,M\text{ accepts }\langle M\rangle\right\} $$ I'm trying to prove that $L_{D}\leq L_{U}$. For that I need to show that there exists $f\,:\,\Sigma^{*}\to\Sigma^{*}$ total, can be calculated by a Turing machine and follows: $\forall x\in\Sigma^*$, $x\in L_D$ iff $f(x)\in L_U$.

Let there be $f\,:\,\Sigma^{*}\to\Sigma^{*}$ so $f\left(\langle M\rangle\right)=\langle M,M\rangle$. I have proved that $f$ is total and also follows the last condition. All that left to do is to prove that $f$ can be calculated by a Turing machine. How should I describe the steps of machine $M_f$ on $\langle M\rangle$?

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It is the turing machine that clones its input. If you want a description of this machine's operation, you may find this useful.

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