(1)

\(\alpha\)を\(\beta\)で割った商は\(\left\lfloor \frac{\alpha}{\beta}\right\rfloor \)、余りは\(\mod\left(\alpha,\beta\right)\)なので与式は成り立つ。

(2)

\begin{align*} \alpha & =-\beta\left\lfloor \frac{\alpha}{-\beta}\right\rfloor +\mod\left(\alpha,-\beta\right)\\ & =\beta\left\lceil \frac{\alpha}{\beta}\right\rceil +\mod\left(\alpha,-\beta\right) \end{align*}

(3)

(1)で\(\beta=1\)を代入すればいい。

(4)

(2)で\(\beta=1\)を代入すればいい。

(5)

\begin{align*} \alpha & =\alpha-\gamma+\gamma\\ & =\beta\left\lfloor \frac{\alpha-\gamma}{\beta}\right\rfloor +\mod\left(\alpha-\gamma,\beta\right)+\gamma\\ & =\beta\left\lfloor \frac{\alpha-\gamma}{\beta}\right\rfloor +\mod\left(\alpha,\beta,\gamma\right) \end{align*}

(6)

\begin{align*} \alpha & =\alpha-\gamma+\gamma\\ & =\beta\left\lceil \frac{\alpha-\gamma}{\beta}\right\rceil +\mod\left(\alpha-\gamma,-\beta\right)+\gamma\\ & =\beta\left\lceil \frac{\alpha-\gamma}{\beta}\right\rceil +\mod\left(\alpha,-\beta,\gamma\right) \end{align*}