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Flaw in my NP = CoNP Proof?

I have this very simple "proof" for NP = CoNP and I think I did something wrongly somewhere, but I cannot find what is wrong. Can someone help me out? Let A be some problem in NP, and let M be the ...
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Show $x^y$ is a primitive recursive function

As this thread title gives away I need to prove $x^y$ to be a primitive recursive function. So mathematically speaking, I think the following are the recursion equations, well aware that I am ...
Show $H_1(x)$ is partially computable
I need to show that $H_1(x)$ defined as follows is partially computable. \begin{equation} H_1(x)= \begin{cases} 1 \;\;\;\;\;\text{ if } \Phi(x,x) \downarrow \\ \uparrow \;\;\;\;\; \text{ otherwise} \...
Let $G= (V,E)$ be a given directed weighted graph, and $s,t$ two specified nodes, so that there is no negative cycle reachable from $s$, and $t$ is reachable from $s$. We're looking for the shortest ...