Tableau-based theorem proving and synthesis of λ-terms in the intuitionistic logic

Abstract

Because of its constructive aspect, the intuitionistic logic plays an important role in the context of the programming paradigm ”programming by proving”. Programs are expressed by λ-terms which can be seen as compact representations of natural deduction proofs. We are presenting a tableau calculus for the first-order intuitionistic logic which allows to synthesize λ-terms. The calculus is obtained from the tableau calculus for the classical logic by extending its rules by λ-terms. In each rule application and closing of tableau branches, λ-terms are synthesized by unification. Particularly, a new λ-term construct (implicit case analysis) is introduced for the the disjunction rules.

In contrary to existing approachs based on the natural deduction calculus, our calculus is very appropriate for automatic reasoning. We implemented the calculus in Prolog. A strategy which is similar to model elimination has been built in. Several formulas (including program synthesis problems) have been proven automatically.