Higher-Order Annotated Terms for Proof Search

A notion of embedding appropriate to higher-order syntax is described. This provides a representation of annotated formulae in terms of the difference between pairs of formulae. We define substitution and unification for such annotated terms. Using this representation of annotated terms, the proof search guidance technique of rippling can be extended to higher-order theorems. We illustrate this with two selected examples using our implementation of these ideas in λProlog.

[1]  Furio Honsell,et al.  A framework for defining logics , 1993, JACM.

[2]  Tobias Nipkow,et al.  Higher-order critical pairs , 1991, [1991] Proceedings Sixth Annual IEEE Symposium on Logic in Computer Science.

[3]  Dale Miller,et al.  A Logic Programming Language with Lambda-Abstraction, Function Variables, and Simple Unification , 1991, J. Log. Comput..

[4]  Alan Bundy,et al.  The Use of Explicit Plans to Guide Inductive Proofs , 1988, CADE.

[5]  Toby Walsh,et al.  Coloured Rippling: An Extension of a Theorem Proving Heuristic , 1994, ECAI.

[6]  Amy P. Felty,et al.  A Logic Programming Approach to Implementing Higher-Order Term Rewriting , 1991, ELP.

[7]  Harold T. Hodes,et al.  The | lambda-Calculus. , 1988 .

[8]  Dieter Hutter,et al.  A colored version of the λ-calculus , 1997 .

[9]  Hubert Comon-Lundh,et al.  About the Theory of Tree Embedding , 1993, TAPSOFT.

[10]  Lawrence C. Paulson,et al.  Natural Deduction as Higher-Order Resolution , 1986, J. Log. Program..

[11]  Rance Cleaveland,et al.  Implementing mathematics with the Nuprl proof development system , 1986 .

[12]  Toby Walsh,et al.  A Calculus for Rippling , 1994, CTRS.

[13]  Jason Gallagher,et al.  The use of proof plans in tactic synthesis , 1993 .

[14]  Dieter Hutter,et al.  A Colored Version of the Lambda-Calculus , 1997, CADE.

[15]  F. Honsell,et al.  A Framework for De ning LogicsRobert Harper , 1987 .

[16]  Frank van Harmelen,et al.  Rippling: A Heuristic for Guiding Inductive Proofs , 1993, Artif. Intell..

[17]  Frank van Harmelen,et al.  The Oyster-Clam System , 1990, CADE.

[18]  Toby Walsh,et al.  Termination Orderings for Rippling , 1994, CADE.