There is a very nice paper of Wagon, which can serve as a sort of case study. The paper presents fourteen different proofs of the following theorem.

Theorem. If a rectangle $R$ is tiled by rectangles, each of which has at least one integer side, then $R$ itself has at least one integer side.

If you have not thought about the problem, you may want to think about it before reading the paper. At first glance some of the proofs certainly appear different. For example, there is a proof using a complex double integral, and another which uses Sperner's lemma.

In fact, all fourteen proofs are shown to be different by considering generalizations of the problem. It turns out that no two of the fourteen proofs work for the same set of generalizations. I do not know if this can be formalized in general.

The paper contains an amusing Appendix titled Apendix to justify that the proofs are different, listing the generalizations that each proof works for.

There is a very nice paper of Wagon, which can serve as a sort of case study. The paper presents fourteen different proofs of the following theorem.

Theorem. If a rectangle $R$ is tiled by rectangles, each of which has at least one integer side, then $R$ itself has at least one integer side.

If you have not thought about the problem, you may want to think about it before reading the paper. At first glance some of the proofs certainly appear different. For example, there is a proof using a complex double integral, and another which uses Sperner's lemma.

In fact, all fourteen proofs are shown to be different by considering generalizations of the problem. It turns out that no two of the fourteen proofs work for the same set of generalizations. I do not know if this can be formalized in general.

The paper contains an amusing Appendix titled Appendix to justify that the proofs are different, listing the generalizations that each proof works for.