The normalization of a natural deduction proof of a closed existential formula ∃_xA_ gives a term t and a proof of Ax[t]. This allows us to regard a proof as a program (Goad   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.
Cite this article
Masamichi Hagiya, A Proof Description Language and Its Reduction System. Publ. Res. Inst. Math. Sci. 19 (1983), no. 1, pp. 237–261