Combining second order matching and first order E-matching

We propose an algorithm for combining second order matching and first order matching in an algebraic first order theory E. This algorithm has the flavor of the higher order E-unification algorithmof Nipkow and Qian, but relies on the classical second order matching algorithm of Huet and Lang instead of higher order unification. Since matching is simpler than unification, we are able to prove the termination of our algorithm when the algebraic theory E respects some conditions. We show that it is possible to preserve the termination when we relax some of these conditions by adapting the previous algorithm. It allows us to use AC1, ACI and ACI1 for example. These algebraic theories are the more useful for our purpose (recognizing logical or functional schemata). We have implemented our algorithm for the AC and AC1 theories and we show examples of possible applications

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

E-Unification for Second-Order Abstract Syntax

Higher-order unification (HOU) concerns unification of (extensions of) ?-calculus and can be seen as an instance of equational unification (E-unification) modulo ??-equivalence of ?-terms. We study equational unification of terms in languages with arbitrary variable binding constructions modulo arbitrary second-order equational theories. Abstract syntax with general variable binding and parametrised metavariables allows us to work with arbitrary binders without committing to ?-calculus or use inconvenient and error-prone term encodings, leading to a more flexible framework. In this paper, we introduce E-unification for second-order abstract syntax and describe a unification procedure for such problems, merging ideas from both full HOU and general E-unification. We prove that the procedure is sound and complete

Untyped Confluence in Dependent Type Theories

International audienceWe investigate techniques based on van Oostrom's decreasing diagrams that reduce confluence proofs to the checking of critical pairs in the absence of termination properties, which are useful in dependent type calculi to prove confluence on untyped terms. These techniques are applied to a complex example originating from practice: a faithful encoding, in an extension of LF with rewrite rules on objects and types, of a subset of the calculus of inductive constructions with a cumulative hierarchy of predicative universes above Prop. The rules may be first-order or higher-order, plain or modulo, non-linear on the right or on the left. Variables which occur non-linearly in lefthand sides of rules must take their values in confined types: in our example, the natural numbers. The first-order rules are assumed to be terminating and confluent modulo some theory: in our example, associativity, commutativity and identity. Critical pairs involving higher-order rules must satisfy van Oostrom's decreasing diagram condition wrt their indexes taken as labels

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

Proceedings of Sixth International Workshop on Unification

Swiss National Science Foundation; Austrian Federal Ministry of Science and Research; Deutsche Forschungsgemeinschaft (SFB 314); Christ Church, Oxford; Oxford University Computing Laborator

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

Introduction to Milestones in Interactive Theorem Proving

On March 8, 2018, Tobias Nipkow celebrated his sixtieth birthday. In anticipation of the occasion, in January 2016, two of his former students, Gerwin Klein and Jasmin Blanchette, and one of his former postdocs, Andrei Popescu, approached the editorial board of the Journal of Automated Reasoning with a proposal to publish a surprise Festschrift issue in his honor. The e-mail was sent to twenty-six members of the board, leaving out one, for reasons that will become clear in a moment. It is a sign of the love and respect that Tobias commands from his colleagues that within two days every recipient of the e-mail had responded favorably and enthusiastically to the proposal

Higher Order Unification Revisited: Complete Sets of Transformations

In this paper, we reexamine the problem of general higher-order unification and develop an approach based on the method of transformations on systems of terms which has its roots in Herbrand\u27s thesis, and which was developed by Martelli and Montanari in the context of first-order unification. This method provides an abstract and mathematically elegant means of analyzing the invariant properties of unification in various settings by providing a clean separation of the logical issues from the specification of procedural information. Our major contribution is three-fold. First, we have extended the Herbrand- Martelli-Montanari method of transformations on systems to higher-order unification and pre-unification; second, we have used this formalism to provide a more direct proof of the completeness of a method for higher-order unification than has previously been available; and, finally, we have shown the completeness of the strategy of eager variable elimination. In addition, this analysis provides another justification of the design of Huet\u27s procedure, and shows how its basic principles work in a more general setting. Finally, it is hoped that this presentation might form a good introduction to higher-order unification for those readers unfamiliar with the field