A Proof Description Language and Its Reduction System

Abstract

The normalization of a natural deduction proof of a closed existential formula $\exists xA$ gives a term $t$ and a proof of $A_x[t]$. This allows us to regard a proof as a program (Goad [9] [10] etc.). But it is not always necessary to completely normalize the given proof to obtain $t$. We analyze the situation by introducing the notions called minimal $I$-reduct, proper reduction etc.; in a word, we define the normal order of proof reduction and study its proof-theoretical property. Then, we present an experimental proof-checker-reducer system that actually uses those principles. In designing a proof-checker (or rather a proof description language), we focussed our attention on the readability of proofs.