Layer Systems for Proving Confluence

We introduce layer systems for proving generalizations of the modularity of confluence for first-order rewrite systems. Layer systems specify how terms can be divided into layers. We establish structural conditions on those systems that imply confluence. Our abstract framework covers known results like many-sorted persistence, layer-preservation and currying. We present a counterexample to an extension of the former to order-sorted rewriting and derive new sufficient conditions for the extension to hold

Polymorphic Rewriting Conserves Algebraic Strong Normalization and Confluence

We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R is strongly normalizing (terminating, noetherian), then R + β + η + type-β + type-η rewriting of mixed terms is also strongly normalizing. We obtain this results using a technique which generalizes Girard\u27s candidats de reductibilité , introduced in the original proof of strong normalization for the polymorphic lambda calculus. We also show that if a many-sorted algebraic rewrite system R has the Church-Rosser property (is confluent), then R + β + type-β + type-η rewriting of mixed terms has the Church- Rosser property too. Combining the two results, we conclude that if R is canonical (complete) on algebraic terms, then R + β + type-β + type-η is canonical on mixed terms. η reduction does not commute with a1gebraic reduction, in general. However, using long β- normal forms, we show that if R is canonical then R + β + η + type-β + type-η convertibility is still decidable

Polymorphic Rewriting Conserves Algebraic Confluence

We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R has the Church-Rosser property (is confluent), then R + β + type-β + type-η rewriting of mixed terms has the Church-Rosser property too. η reduction does not commute with algebraic reduction, in general. However, using long normal forms, we show that if R is canonical (confluent and strongly normalizing) then equational provability from R + β + η + type-β + type-η is still decidable

Polymorphic Rewriting Conserves Algebraic Strong Normalization

We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R is strongly normalizing (terminating, noetherian), then R + β + η + type-η rewriting of mixed terms is also strongly normalizing. The result is obtained using a technique which generalizes Girard\u27s candidats de reductibilité , introduced in the original proof of strong normalization for the polymorphic lambda calculus

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 ß-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 ß 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

07401 Abstracts Collection -- Deduction and Decision Procedures

From 01.10. to 05.10.2007, the Dagstuhl Seminar 07401 ``Deduction and Decision Procedures\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper

Nonfree datatypes in Isabelle/HOL: animating a many-sorted metatheory

Datatypes freely generated by their constructors are well supported in mainstream proof assistants. Algebraic specification languages offer more expressive datatypes on axiomatic means: nonfree datatypes generated from constructors modulo equations. We have implemented an Isabelle/HOL package for nonfree datatypes, without compromising foundations. The use of the package, and its nonfree iterator in particular, is illustrated with examples: bags, polynomials and λ-terms modulo α-equivalence. The many-sorted metatheory of nonfree datatypes is formalized as an ordinary Isabelle theory and is animated by the package into user-specified instances. HOL lacks a type of types, so we employ an ad hoc construction of a universe embedding the relevant parameter types