Proof by exhaustion

This article is about the type of mathematical proof. For the method of calculating limits, see Method of exhaustion.
"Proof by cases" redirects here. For the concept in propositional logic, see Disjunction elimination.

Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which a statement is established by dividing the argument into several distinct cases and proving that the statement holds in each case. The cases must be collectively exhaustive, ensuring that all possible situations are considered.

The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof by exhaustion method can be used whenever the number of cases is finite. However, because most mathematical sets are infinite, this method is rarely used to derive general mathematical results.

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