Sum of Ceilings not less than Ceiling of Sum

Theorem
Let $\left \lceil {x} \right \rceil$ be the ceiling function.

Then:
 * $\left \lceil {x} \right \rceil + \left \lceil {y} \right \rceil \ge \left \lceil {x + y} \right \rceil$

The equality holds:
 * $\left \lceil {x} \right \rceil + \left \lceil {y} \right \rceil = \left \lceil {x + y} \right \rceil$

either:
 * $x \in \Z$ or $y \in \Z$

or:
 * $x \bmod 1 + y \bmod 1 > 1$

where $x \bmod 1$ denotes the modulo operation.

Proof
From the definition of the modulo operation, we have that:
 * $x = \left \lfloor {x}\right \rfloor + \left({x \bmod 1}\right)$

from which we obtain:
 * $x = \left \lceil {x}\right \rceil - \left[{x \notin \Z}\right] + \left({x \bmod 1}\right)$

where $\left[{x \notin \Z}\right]$ uses Iverson's convention.

We have that:
 * $x \notin \Z \implies x \bmod 1 > 0$

As $0 \le x \bmod 1 < 1$ it follows that:
 * $\left[{x \notin \Z}\right] \ge x \bmod 1$

Hence the inequality.

The equality holds :
 * $\left \lceil {\left({x \bmod 1}\right) + \left({y \bmod 1}\right)} \right \rceil = \left[{x \notin \Z}\right] + \left[{y \notin \Z}\right]$

that is, one of the following holds:
 * $x \in \Z$, in which case $x \bmod 1 = 0$
 * $y \in \Z$, in which case $y \bmod 1 = 0$
 * both $x, y \in \Z$, in which case $\left({x \bmod 1}\right) + \left({y \bmod 1}\right) = 0$
 * both $x, y \notin \Z$ and $\left({x \bmod 1}\right) + \left({y \bmod 1}\right) > 1$.