Write down either the synthetic or the recursive definitions of ordinal addition and multiplication. Using your definitions, give proofs or counterexamples for the following statements:

(i) For all $\alpha, \beta$ and $\gamma$, we have $\alpha \cdot(\beta+\gamma)=\alpha \cdot \beta+\alpha \cdot \gamma$.

(ii) For all $\alpha, \beta$ and $\gamma$, we have $(\alpha+\beta) \cdot \gamma=\alpha \cdot \gamma+\beta \cdot \gamma$.

(iii) For all $\alpha$ and $\beta$ with $\beta>0$, there exist $\gamma$ and $\delta$ with $\delta<\beta$ and $\alpha=\beta \cdot \gamma+\delta$.

(iv) For all $\alpha$ and $\beta$ with $\beta>0$, there exist $\gamma$ and $\delta$ with $\delta<\beta$ and $\alpha=\gamma \cdot \beta+\delta$.

(v) For every $\alpha$, either there exists a cofinal map $f: \omega \rightarrow \alpha$ (that is, one such that $\left.\alpha=\bigcup\left\{f(n)^{+} \mid n \in \omega\right\}\right)$, or there exists $\beta$ such that $\alpha=\omega_{1} \cdot \beta .$