Combining Algebra and Higher-Order Types

We study the higher-order rewrite/equational proof systems obtained by adding the simply typed lambda calculus to algebraic rewrite/equational proof systems. We show that if a many-sorted algebraic rewrite system has the Church-Rosser property, then the corresponding higher-order rewrite system which adds simply typed s-reduction has the Church-Rosser property too. This result is relevant to parallel implementations of functional programming languages. We also show that provability in the higher-order equational proof system obtained by adding the simply typed s and η axioms to some many-sorted algebraic proof system is effectively reducible to provability in that algebraic proof system. This effective reduction also establishes transformations between higher-order and algebraic equational proofs, transformations which can be useful in automated deduction. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-88-21. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/617 COMBINING ALGEBRA AND HIGHER-ORDER TYPES Val Breazu-Tannen MS-CIS-88-21 LlNC LAB 107 Department of Computer and Information Science School of Engineering and Applied Science University of Pennsylvania Philadelphia, PA 191 04