What is "proof by exhaustion"

n. (context logic English) The indirect verification or falsification of a statement by the verification or falsification of each of the finite number of cases which arise therefrom.

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:

1. A proof that the cases are exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases.
2. A proof of each of the cases.

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