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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.