Editing Proof by exhaustion (section)

{{Short description|Type of mathematical proof}}
{{about|the type of mathematical proof|the method of calculating limits|Method of exhaustion}}
{{redirect|Brute force method|similarly named methods in other disciplines|Brute force (disambiguation)}}
{{redirect|Proof by cases|the concept in propositional logic|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 the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds.<ref>{{citation|last1=Reid|first1=D. A|last2=Knipping|first2=C|year=2010|title=Proof in Mathematics Education: Research, Learning, and Teaching|publisher=Sense Publishers|page=133|isbn=978-9460912443}}.</ref> This is a method of [[direct proof]]. A proof by exhaustion typically contains two stages:

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

The prevalence of digital [[computer]]s 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 system]]s 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.<ref>{{Cite book|title=Discrete mathematics with applications|last=S.|first=Epp, Susanna|date=2011-01-01|publisher=Brooks/Cole|isbn=978-0495391326|oclc=970542319}}</ref>

In the [[Curry–Howard isomorphism]], proof by exhaustion and case analysis are related to ML-style [[pattern matching]].{{Citation needed|date=March 2017}}