I'm stuck on this problem:
Given $f:\mathbb{N} \rightarrow \mathbb{N}$ a partial recursive function that is also injective and total. Prove that the function $f^{-1}:\mathbb{N} \rightarrow \mathbb{N}$ $$f^{-1} = \left\{\begin{array}{rcl} x & \mbox{if} & f(x) = y \\ \uparrow & \mbox{otherwise}\end{array}\right.$$ is partial recursive.
So to prove that a function is partial recursive I have to show that I can build it from basic recursive functions (successor, projection and constant) by applying composition, primitive recursion and then minimization.
I don't know what form $f$ has, but I think I can use it to construct $f^{-1}$. The only problem is that I don't know what to write...