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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.

Author: Wolfgang Schreiner
Last Modification: December 14, 1999

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