Constructive proof

A proof that establishes a claim by explicitly producing an object with the required property. Its method can often be used to compute or identify a witness, not merely establish that one exists.

Constructive existence theorem: A theorem asserting that an object exists together with a method for producing one. It states the kind of existence claim a constructive proof fulfills directly.
Predicate (mathematical logic): A statement with one or more variables that becomes true or false when values are assigned. The produced object must make the claim's predicate true.
Constructive algorithm: An algorithm that explicitly builds an output satisfying a specified mathematical condition. A constructive proof can yield this kind of procedure as part of its argument.
Nonconstructive proof: A proof of existence that does not explicitly produce an object satisfying the claim. It establishes existence without meeting the defining demand of a constructive proof.
Witness (mathematics): An object or value that demonstrates an existential claim is true. Producing this object is the defining step in an existential constructive proof.
Proof: A finite argument establishing that a mathematical statement follows from accepted assumptions. Constructive proof is one way of establishing a mathematical statement.
Proof mining: The extraction of quantitative or computational information from mathematical proofs, often nonconstructive ones. It seeks effective content that may be hidden inside proofs without explicit witnesses.
Proof by contradiction: A proof that derives a contradiction from assuming a statement is false. A contradiction can rule out nonexistence without identifying an object.
Existential quantification: A logical quantifier asserting that at least one object in a domain has a stated property. A constructive proof of an existential statement supplies an instance satisfying its predicate.
Intuitionistic logic: A logical system that does not generally accept the law of excluded middle as a proof principle. Its interpretation of existence connects proving a claim with constructing evidence for it.

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