Sum of Ceilings not less than Ceiling of Sum

Theorem
Let $\ceiling x$ be the ceiling function.

Then:
 * $\ceiling x + \ceiling y \ge \ceiling {x + y}$

The equality holds:
 * $\ceiling x + \ceiling y = \ceiling {x + y}$

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:
 * $x = \floor x + \paren {x \bmod 1}$

from which we obtain:
 * $x = \ceiling x - \sqbrk {x \notin \Z} + \paren {x \bmod 1}$

where $\sqbrk {x \notin \Z}$ 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:
 * $\sqbrk {x \notin \Z} \ge x \bmod 1$

Hence the inequality.

The equality holds :
 * $\ceiling {\paren {x \bmod 1} + \paren {y \bmod 1} } = \sqbrk {x \notin \Z} + \sqbrk {y \notin \Z}$

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 $\paren {x \bmod 1} + \paren {y \bmod 1} = 0$
 * both $x, y \notin \Z$ and $\paren {x \bmod 1} + \paren {y \bmod 1} > 1$.