Balanced Formulas, Minimal Formulas and Their Proofs
Sachio Hirokawa, 佐千男 廣川 · Kyushu University Institutional Repository (QIR) (Kyushu University) · 1991
According to the formulas-as-types notion, an implicational formula can be identified with a type of a $ lambda $-term which represents a proof of the formula in implicational fragment of intuitionistic logic. A formula is balanced iff no type variable occurs more than twice in it. It is known that balanced formulas have unique proofs. In this paper, it is shown that closed $ lamda $-terms in $ \\beta $-normal form having balanced types are BCK-$ lamda $-terms in which each variable occurs at most once. A formula is BCK-minimal iff it is BCK-provable and it is not a non-trivial substitution instance of other BCK-provable formula. It is also shown that the set BCK-minimal formulas is identical to the set of principal type-schemes of BCK-$ lamda $-terms in $ \\beta $-$ eta $-normal form.