Proof by exhaustion

Proof by exhaustion, also known as proof by cases, proof by case analysis, perfect induction, or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases and each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion contains two stages:

In the Curry–Howard isomorphism, proof by exhaustion and case analysis are related to ML-style pattern matching.

To prove that every integer that is a perfect cube is a multiple of 9, or is 1 more than a multiple of 9, or is 1 less than a multiple of 9.

Proof:
Each cube number is the cube of some integer n. Every integer n is either a multiple of 3, or 1 more or 1 less than a multiple of 3. So these 3 cases are exhaustive:

Mathematicians prefer to avoid proofs by exhaustion with large numbers of cases, as they seem inelegant. An illustration of how such proofs might be inelegant is to prove that every year in which the modern Summer Olympic Games is held is divisible by 4.

Proof: the first modern Summer Olympics were held in 1896, and then every 4 years thereafter (neglecting years in which the games were not held due to World War I and World War II). Since 1896 = 474 * 4 is divisible by 4, the next Olympics would be in year 474 * 4 + 4 = (474 + 1) * 4, which is also divisible by four, and so on (this is a proof by mathematical induction). Therefore the statement is proved.

The statement can also be proved by exhaustion by listing out every year in which the Summer Olympics were held, and checking that every one of them can be divided by four. With 28 total Summer Olympics as of 2016, this is a proof by exhaustion with 28 cases. In addition to being more elegant, the proof by mathematical induction also proves the statement indefinitely into the future, while after each new Summer Olympics the proof by exhaustion will require an extra case.

There is no upper limit to the number of cases allowed in a proof by exhaustion. Sometimes there are only two or three cases. Sometimes there may be thousands or even millions. For example, rigorously solving an endgame puzzle in chess might involve considering a very large number of possible positions in the game tree of that problem.

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