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Constructive proof
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{{short description|Method of proof in mathematics}} In [[mathematics]], a '''constructive proof''' is a method of [[mathematical proof|proof]] that demonstrates the existence of a [[mathematical object]] by creating or providing a method for creating the object. This is in contrast to a '''non-constructive proof''' (also known as an '''existence proof''' or [[existence theorem|''pure existence theorem'']]), which proves the existence of a particular kind of object without providing an example. For avoiding confusion with the stronger concept that follows, such a constructive proof is sometimes called an '''effective proof'''. A '''constructive proof''' may also refer to the stronger concept of a proof that is valid in [[constructive mathematics]]. [[Constructivism (mathematics)|Constructivism]] is a mathematical philosophy that rejects all proof methods that involve the existence of objects that are not explicitly built. This excludes, in particular, the use of the [[law of the excluded middle]], the [[axiom of infinity]], and the [[axiom of choice]]. Constructivism also induces a different meaning for some terminology (for example, the term "or" has a stronger meaning in constructive mathematics than in classical).<ref>{{Citation|last1=Bridges|first1=Douglas|title=Constructive Mathematics|date=2018|url=https://plato.stanford.edu/archives/sum2018/entries/mathematics-constructive/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Summer 2018|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-25|last2=Palmgren|first2=Erik}}</ref> Some non-constructive proofs show that if a certain proposition is false, a contradiction ensues; consequently the proposition must be true ([[proof by contradiction]]). However, the [[principle of explosion]] (''ex falso quodlibet'') has been accepted in some varieties of constructive mathematics, including [[intuitionism]]. Constructive proofs can be seen as defining certified mathematical [[algorithm]]s: this idea is explored in the [[Brouwer–Heyting–Kolmogorov interpretation]] of [[constructive logic]], the [[Curry–Howard correspondence]] between proofs and programs, and such logical systems as [[Per Martin-Löf]]'s [[intuitionistic type theory]], and [[Thierry Coquand]] and [[Gérard Huet]]'s [[calculus of constructions]].
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