Beweisfunktion

Beweisfunktion is a term used in logic and type theory to denote a function that, given a proposition, yields a proof of that proposition or a verification certificate. In constructive or computational settings, a Beweisfunktion does not merely state that something is true; it provides an explicit witness or construction that demonstrates the truth of the proposition.

In constructive logic and in type theory, proofs are treated as mathematical objects. By the Curry-Howard correspondence, a proposition corresponds to a type and a proof to a term of that type. A Beweisfunktion, in this view, can be seen as a function that, for any proposition, returns a corresponding proof term. In proof assistants such as Coq, Agda or Lean, this corresponds to programs that, when executed, produce a proof object that can be checked by the system.

Examples illustrate the idea. For a conjunction, a Beweisfunktion that takes the statement “P and Q” would return a pair consisting of a proof of P and a proof of Q. For implication, a Beweisfunktion from “If P then Q” would take a proof of P and return a proof of Q. More generally, for universal statements, the Beweisfunktion would construct a method that, given any valid instance, provides a proof of the instance.

Applications of Beweisfunktionen include verified software, formalized mathematics, and certified certificates that ensure correctness. Limitations arise in non-constructive or classical logics, where existence statements may not yield computable witnesses. Related concepts include proof terms, the Curry-Howard correspondence, and proof assistants.