Acknowledgements: the Ideas and Comments Of

16 negative occurrences of various connectives and quantiiers. Herbrandization will always work for a class of formulas that does not admit an LJ impermutability R1/R2, where an existential quantiier can govern the principal formula of R2, and the principal formula of R1 contains a universally quantiied subformula. For example, Herbrandization fails for the class of hereditary Harrop formulas MNPS91] since the unprovable formula ((8x: p(x)) q) (9y: (p(y) q)) has the Herbrandized form (p(c) q) (p(u) q) which is easily seen to be intuitionistically provable with c instantiating Herbrand variable u. This counterexample arises from the (`)=(``) impermutability permitted by the class of hereditary Harrop formulas. 6 Conclusion Proof search in the intuitionistic sequent calculus is relevant to logic programming MNPS91]. We have presented a notion of Herbrand functions that is appropriate for proof search in the intuitionistic sequent calculus. A proof search procedure was described and proved to be sound and complete. This procedure has been implemented in Common Lisp. The implementation optimizes Search in some small ways but there are a number of deeper optimizations that remain to be implemented. The extent to which Search can be optimized without destroying the transparency of the proof search deserves further exploration. Owre have contributed to this presentation. Sol Feferman's course on proof theory provided the initial stimulus for this work. References BB74] W.W. Bledsoe and P. Bruell. A man-machine theorem-proving system.