(1)

\begin{align*} \left\lceil x\right\rceil -1 & =x-\mod\left(x,-1\right)-1\\ & =x+\mod\left(-x,1\right)-1\\ & =x+\left|\sgn\mod\left(x,1\right)\right|-\mod\left(x,1\right)-1\\ & =x-\mod\left(x,1\right)-\delta_{0,\mod\left(x,1\right)}\\ & \leq x-\mod\left(x,1\right)\\ & =\left\lfloor x\right\rfloor \\ & \leq x\\ & \leq x-\mod\left(x,-1\right)\\ & =\left\lceil x\right\rceil \\ & =x+\mod\left(-x,1\right)\\ & =x-\mod\left(x,1\right)+\left|\sgn\mod\left(x,1\right)\right|\\ & \leq x-\mod\left(x,1\right)+1\\ & =\left\lfloor x\right\rfloor +1 \end{align*}

(2)

\begin{align*} -\left\lfloor -z\right\rfloor & =-\left(-z-\mod\left(-z,1\right)\right)\\ & =z+\mod\left(-z,1\right)\\ & =z-\mod\left(z,-1\right)\\ & =\left\lceil z\right\rceil \end{align*}

(3)

\begin{align*} -\left\lceil -z\right\rceil & =-\left(-z-\mod\left(-z,-1\right)\right)\\ & =z+\mod\left(-z,-1\right)\\ & =z-\mod\left(z,1\right)\\ & =\left\lfloor z\right\rfloor \end{align*}

(4)

\begin{align*} \left\lfloor \left\lfloor z\right\rfloor \right\rfloor & =\left\lfloor \left\lfloor z\right\rfloor +\mod\left(z,1\right)\right\rfloor \\ & =\left\lfloor z\right\rfloor \end{align*}

(5)

\begin{align*} \left\lceil \left\lceil z\right\rceil \right\rceil & =\left\lceil \left\lceil z\right\rceil +\mod\left(z,-1\right)\right\rceil \\ & =\left\lceil z\right\rceil \end{align*}