Using diagonal argument to prove that $H(x) = \mu y T(x,x,y)$ has no total computable extension
(This answer assumes my conjectural interpretation of the predicate $T$.)
The function $H(x)$ is given by
$$
H(x) = \begin{cases} t &amp; \text{if program $x$ run on input $x$ halts after $t$ steps} \\ \bot &amp; \text{if program $x$ never halts when run on input $x$} \end{cases}
$$

Hint: Given a total computable function extending $H$, show how to solve the halting problem. There is no need to use diagonalization – this is already taken care of in the proof that the halting problem is undecidable.