Are There Proofs for Floor and Ceiling Properties in Computer Science?

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Some floor and ceiling properties like
ceil(lg(n+1))=floor(lg(n)+1)
and "Some properties of ceiling function" section under "http://en.wikipedia.org/wiki/Floor_function"

and summation properties like
"Identities" section under "http://en.wikipedia.org/wiki/Sum"
prove to me very useful for computer science study.

But I am interesting of their proof. Does any mathematics book or computer science book has provide proofs for the properties mentioned above?
 
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Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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