Hoare Logic: Consequence Rule

Question

I can substitute $R$ in the first tuple which gives me $x$ $>$ $2$, but I don't know what $S$ is. What is $S$ in this case and how do you arrive at $x$ $>$ $2$ $\rightarrow$ $x$ $>$ $0$?

Answer 1

$S$ is $x>0$. Here, it is assumed that $x>2 \implies x>0$ is true, it is not argued for the truth of this in your screenshot. You can reach to by the axioms of arithmetic, such a proof depends on the system of arithmetic you are using.