To prove the ideas while solving?

While solving, in many problems we use some basic mathematical idea of our own or some kind of theorem. Is there any branch that deals with proving these things? Just and just for example to prove Euclidean Algorithm for finding GCD (A,B).

I think to prove them is very important while solving problems so I want to know some resources which deals these in great detail!! PLS help.

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watch CP algorithms tutorials they are the best for this purpose