The proof of completeness for propositional logic is a constructive one, so a computer program is suggested by the proof. We prove the completeness theorem for Łukasiewicz’ axioms directly, and translate the proof into the functional languages SML and Haskell. In this paper we consider this proof as a program. The program produces enormous proof trees, but it is, we contend, as good a proof of completeness as the standard mathematical proofs.The real value of the exercise is the further evidence it provides that typed, functional languages can clearly express the complex abstractions of mathematics.
Stansifer, Ryan, "Completeness of Propositional Logic as a Program" (2001). Electrical Engineering and Computer Science Faculty Publications. 133.