Improved efficiency of code generated for < predicate on natural numbers.
\begin{thebibliography}{10}
\bibitem{andrews86}
Andrews, P.~B.,
\newblock {\em An Introduction to Mathematical Logic and Type Theory: To Truth
Through Proof},
\newblock Academic Press, 1986
\bibitem{basin91}
Basin, D., Kaufmann, M.,
\newblock The {Boyer-Moore} prover and {Nuprl}: An experimental comparison,
\newblock In {\em Logical Frameworks}, G.~Huet, G.~Plotkin, Eds. Cambridge
Univ. Press, 1991, pp.~89--119
\bibitem{boyer86}
Boyer, R., Lusk, E., McCune, W., Overbeek, R., Stickel, M., Wos, L.,
\newblock Set theory in first-order logic: Clauses for {G\"odel's} axioms,
\newblock {\em J. Auto. Reas. {\bf 2}}, 3 (1986), 287--327
\bibitem{bm88book}
Boyer, R.~S., Moore, J.~S.,
\newblock {\em A Computational Logic Handbook},
\newblock Academic Press, 1988
\bibitem{camilleri92}
Camilleri, J., Melham, T.~F.,
\newblock Reasoning with inductively defined relations in the {HOL} theorem
prover,
\newblock Tech. Rep. 265, Comp. Lab., Univ. Cambridge, Aug. 1992
\bibitem{charniak80}
Charniak, E., Riesbeck, C.~K., McDermott, D.~V.,
\newblock {\em Artificial Intelligence Programming},
\newblock Lawrence Erlbaum Associates, 1980
\bibitem{church40}
Church, A.,
\newblock A formulation of the simple theory of types,
\newblock {\em J. Symb. Logic {\bf 5}\/} (1940), 56--68
\bibitem{coen92}
Coen, M.~D.,
\newblock {\em Interactive Program Derivation},
\newblock PhD thesis, University of Cambridge, 1992,
\newblock Computer Laboratory Technical Report 272
\bibitem{constable86}
{Constable et al.}, R.~L.,
\newblock {\em Implementing Mathematics with the Nuprl Proof Development
System},
\newblock Prentice-Hall, 1986
\bibitem{davey&priestley}
Davey, B.~A., Priestley, H.~A.,
\newblock {\em Introduction to Lattices and Order},
\newblock Cambridge Univ. Press, 1990
\bibitem{dawson90}
Dawson, W.~M.,
\newblock {\em A Generic Logic Environment},
\newblock PhD thesis, Imperial College, London, 1990
\bibitem{debruijn72}
de~Bruijn, N.~G.,
\newblock Lambda calculus notation with nameless dummies, a tool for automatic
formula manipulation, with application to the {Church-Rosser Theorem},
\newblock {\em Indag. Math. {\bf 34}\/} (1972), 381--392
\bibitem{devlin79}
Devlin, K.~J.,
\newblock {\em Fundamentals of Contemporary Set Theory},
\newblock Springer, 1979
\bibitem{coq}
{Dowek et al.}, G.,
\newblock The {Coq} proof assistant user's guide,
\newblock Technical Report 134, INRIA-Rocquencourt, 1991
\bibitem{dummett}
Dummett, M.,
\newblock {\em Elements of Intuitionism},
\newblock Oxford University Press, 1977
\bibitem{dyckhoff}
Dyckhoff, R.,
\newblock Contraction-free sequent calculi for intuitionistic logic,
\newblock {\em J. Symb. Logic {\bf 57}}, 3 (1992), 795--807
\bibitem{felty91a}
Felty, A.,
\newblock A logic program for transforming sequent proofs to natural deduction
proofs,
\newblock In {\em Extensions of Logic Programming\/} (1991),
P.~Schroeder-Heister, Ed., Springer, pp.~157--178,
\newblock LNAI 475
\bibitem{felty93}
Felty, A.,
\newblock Implementing tactics and tacticals in a higher-order logic
programming language,
\newblock {\em J. Auto. Reas. {\bf 11}}, 1 (1993), 43--82
\bibitem{frost93}
Frost, J.,
\newblock A case study of co-induction in {Isabelle HOL},
\newblock Tech. Rep. 308, Comp. Lab., Univ. Cambridge, Aug. 1993
\bibitem{OBJ}
Futatsugi, K., Goguen, J., Jouannaud, J.-P., Meseguer, J.,
\newblock Principles of {OBJ2},
\newblock In {\em Princ. Prog. Lang.\/} (1985), pp.~52--66
\bibitem{gallier86}
Gallier, J.~H.,
\newblock {\em Logic for Computer Science: Foundations of Automatic Theorem
Proving},
\newblock Harper \& Row, 1986
\bibitem{mgordon-hol}
Gordon, M. J.~C., Melham, T.~F.,
\newblock {\em Introduction to {HOL}: A Theorem Proving Environment for Higher
Order Logic},
\newblock Cambridge Univ. Press, 1993
\bibitem{halmos60}
Halmos, P.~R.,
\newblock {\em Naive Set Theory},
\newblock Van Nostrand, 1960
\bibitem{harper-jacm}
Harper, R., Honsell, F., Plotkin, G.,
\newblock A framework for defining logics,
\newblock {\em J.~ACM {\bf 40}}, 1 (1993), 143--184
\bibitem{haskell-tutorial}
Hudak, P., Fasel, J.~H.,
\newblock A gentle introduction to {Haskell},
\newblock {\em {SIGPLAN} {\bf 27}}, 5 (May 1992)
\bibitem{haskell-report}
Hudak, P., Jones, S.~P., Wadler, P.,
\newblock Report on the programming language {Haskell}: A non-strict, purely
functional language,
\newblock {\em {SIGPLAN} {\bf 27}}, 5 (May 1992),
\newblock Version 1.2
\bibitem{huet75}
Huet, G.~P.,
\newblock A unification algorithm for typed $\lambda$-calculus,
\newblock {\em Theoretical Comput. Sci. {\bf 1}\/} (1975), 27--57
\bibitem{huet78}
Huet, G.~P., Lang, B.,
\newblock Proving and applying program transformations expressed with
second-order patterns,
\newblock {\em Acta Inf. {\bf 11}\/} (1978), 31--55
\bibitem{mural}
Jones, C.~B., Jones, K.~D., Lindsay, P.~A., Moore, R.,
\newblock {\em Mural: A Formal Development Support System},
\newblock Springer, 1991
\bibitem{alf}
Magnusson, L., {Nordstr\"om}, B.,
\newblock The {ALF} proof editor and its proof engine,
\newblock In {\em Types for Proofs and Programs: International Workshop {TYPES
'93}\/} (published 1994), Springer, pp.~213--237,
\newblock LNCS 806
\bibitem{mw81}
Manna, Z., Waldinger, R.,
\newblock Deductive synthesis of the unification algorithm,
\newblock {\em Sci. Comput. Programming {\bf 1}}, 1 (1981), 5--48
\bibitem{martin-nipkow}
Martin, U., Nipkow, T.,
\newblock Ordered rewriting and confluence,
\newblock In {\em 10th Conf. Auto. Deduct.\/} (1990), M.~E. Stickel, Ed.,
Springer, pp.~366--380,
\newblock LNCS 449
\bibitem{martinlof84}
Martin-L\"of, P.,
\newblock {\em Intuitionistic type theory},
\newblock Bibliopolis, 1984
\bibitem{melham89}
Melham, T.~F.,
\newblock Automating recursive type definitions in higher order logic,
\newblock In {\em Current Trends in Hardware Verification and Automated Theorem
Proving}, G.~Birtwistle, P.~A. Subrahmanyam, Eds. Springer, 1989,
pp.~341--386
\bibitem{miller-mixed}
Miller, D.,
\newblock Unification under a mixed prefix,
\newblock {\em J. Symb. Comput. {\bf 14}}, 4 (1992), 321--358
\bibitem{milner-coind}
Milner, R., Tofte, M.,
\newblock Co-induction in relational semantics,
\newblock {\em Theoretical Comput. Sci. {\bf 87}\/} (1991), 209--220
\bibitem{nipkow-prehofer}
Nipkow, T., Prehofer, C.,
\newblock Type checking type classes,
\newblock In {\em 20th Princ. Prog. Lang.\/} (1993), ACM Press, pp.~409--418,
\newblock Revised version to appear in \bgroup\em J. Func. Prog.\egroup
\bibitem{noel}
{No\"el}, P.,
\newblock Experimenting with {Isabelle} in {ZF} set theory,
\newblock {\em J. Auto. Reas. {\bf 10}}, 1 (1993), 15--58
\bibitem{nordstrom90}
{Nordstr\"om}, B., Petersson, K., Smith, J.,
\newblock {\em Programming in {Martin-L\"of}'s Type Theory. An Introduction},
\newblock Oxford University Press, 1990
\bibitem{paulin92}
Paulin-Mohring, C.,
\newblock Inductive definitions in the system {Coq}: Rules and properties,
\newblock Research Report 92-49, LIP, Ecole Normale Sup\'erieure de Lyon, Dec.
1992
\bibitem{paulson85}
Paulson, L.~C.,
\newblock Verifying the unification algorithm in {LCF},
\newblock {\em Sci. Comput. Programming {\bf 5}\/} (1985), 143--170
\bibitem{paulson87}
Paulson, L.~C.,
\newblock {\em Logic and Computation: Interactive proof with Cambridge LCF},
\newblock Cambridge Univ. Press, 1987
\bibitem{paulson89}
Paulson, L.~C.,
\newblock The foundation of a generic theorem prover,
\newblock {\em J. Auto. Reas. {\bf 5}}, 3 (1989), 363--397
\bibitem{paulson-COLOG}
Paulson, L.~C.,
\newblock A formulation of the simple theory of types (for {Isabelle}),
\newblock In {\em COLOG-88: International Conference on Computer Logic\/}
(Tallinn, 1990), P.~Martin-L\"of, G.~Mints, Eds., Estonian Academy of
Sciences, Springer,
\newblock LNCS 417
\bibitem{paulson700}
Paulson, L.~C.,
\newblock {Isabelle}: The next 700 theorem provers,
\newblock In {\em Logic and Computer Science}, P.~Odifreddi, Ed. Academic
Press, 1990, pp.~361--386
\bibitem{paulson91}
Paulson, L.~C.,
\newblock {\em {ML} for the Working Programmer},
\newblock Cambridge Univ. Press, 1991
\bibitem{paulson-coind}
Paulson, L.~C.,
\newblock Co-induction and co-recursion in higher-order logic,
\newblock Tech. Rep. 304, Comp. Lab., Univ. Cambridge, July 1993
\bibitem{paulson-fixedpt}
Paulson, L.~C.,
\newblock A fixedpoint approach to implementing (co)inductive definitions,
\newblock Tech. Rep. 320, Comp. Lab., Univ. Cambridge, Dec. 1993
\bibitem{paulson-set-I}
Paulson, L.~C.,
\newblock Set theory for verification: {I}. {From} foundations to functions,
\newblock {\em J. Auto. Reas. {\bf 11}}, 3 (1993), 353--389
\bibitem{paulson-set-II}
Paulson, L.~C.,
\newblock Set theory for verification: {II}. {Induction} and recursion,
\newblock Tech. Rep. 312, Comp. Lab., Univ. Cambridge, 1993
\bibitem{paulson-final}
Paulson, L.~C.,
\newblock A concrete final coalgebra theorem for {ZF} set theory,
\newblock Tech. rep., Comp. Lab., Univ. Cambridge, 1994
\bibitem{pelletier86}
Pelletier, F.~J.,
\newblock Seventy-five problems for testing automatic theorem provers,
\newblock {\em J. Auto. Reas. {\bf 2}\/} (1986), 191--216,
\newblock Errata, JAR 4 (1988), 235--236
\bibitem{plaisted90}
Plaisted, D.~A.,
\newblock A sequent-style model elimination strategy and a positive refinement,
\newblock {\em J. Auto. Reas. {\bf 6}}, 4 (1990), 389--402
\bibitem{quaife92}
Quaife, A.,
\newblock Automated deduction in {von Neumann-Bernays-G\"odel} set theory,
\newblock {\em J. Auto. Reas. {\bf 8}}, 1 (1992), 91--147
\bibitem{sawamura92}
Sawamura, H., Minami, T., Ohashi, K.,
\newblock Proof methods based on sheet of thought in {EUODHILOS},
\newblock Research Report IIAS-RR-92-6E, International Institute for Advanced
Study of Social Information Science, Fujitsu Laboratories, 1992
\bibitem{suppes72}
Suppes, P.,
\newblock {\em Axiomatic Set Theory},
\newblock Dover, 1972
\bibitem{takeuti87}
Takeuti, G.,
\newblock {\em Proof Theory}, 2nd~ed.,
\newblock North Holland, 1987
\bibitem{thompson91}
Thompson, S.,
\newblock {\em Type Theory and Functional Programming},
\newblock Addison-Wesley, 1991
\bibitem{principia}
Whitehead, A.~N., Russell, B.,
\newblock {\em Principia Mathematica},
\newblock Cambridge Univ. Press, 1962,
\newblock Paperback edition to *56, abridged from the 2nd edition (1927)
\bibitem{wos-bledsoe}
Wos, L.,
\newblock Automated reasoning and {Bledsoe's} dream for the field,
\newblock In {\em Automated Reasoning: Essays in Honor of {Woody Bledsoe}},
R.~S. Boyer, Ed. Kluwer Academic Publishers, 1991, pp.~297--342
\end{thebibliography}