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 [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.
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