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Floor and Ceiling Rules

Proposition: For every x in R and i in Z, we have:

floor(x+i) = floor(x)+i,

ceiling(x+i) = ceiling(x)+i.

x < i	<=>	floor(x) < i,
i < x	<=>	i < ceiling(x),
x <= i	<=>	ceiling(x) <= i,
i <= x	<=>	i <= floor(x).

May shift integer terms out of a floor or ceiling; may get rid of floor and ceiling under some assumptions.