We are asking the question How can we establish whether two functional expressions are equivalent (interchangeable)?
You can’t just test across a few results/arguments; there may be an infinite number of possible arguments.
Instead, you can use maths style proving like natural induction.
However a caveat to bear in mind; beware when doing integer division and shifting around dividers. For example this causes problems when doing integer division:
m / d + n / d = (m + n) / d
There is a similar law which does hold for + and /:
m / d + n = (m + n * d) / d (+n).(/d) = (/d).(+n x d)
The second line states ‘adding
n after dividing by
d is the same as dividing by
d after adding
n and multiplying by
Never forget the undefined case when you are proving equivalence