A Type Checker for a Logical Framework with Union and Intersection Types

• Main Authors: Stolze, Claude, Liquori, Luigi
• Other Authors: Institut de Recherche en Informatique Fondamentale (IRIF (UMR_8243)), Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), Logical Time for Formal Embedded System Design (KAIROS), Inria Sophia Antipolis - Méditerranée (CRISAM), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-COMmunications, Réseaux, systèmes Embarqués et Distribués (Laboratoire I3S - COMRED), Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)-Laboratoire d'Informatique, Signaux, et Systèmes de Sophia Antipolis (I3S), Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)-Université Nice Sophia Antipolis (1965 - 2019) (UNS)-Centre National de la Recherche Scientifique (CNRS)-Université Côte d'Azur (UniCA)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2020
• Subjects: 2012 ACM Subject Classification Theory of computation → Lambda calculus; Theory of computation → Proof theory Keywords and phrases Intersection types; Union types; Dependent types; Subtyping; Type checker; Refiner; ∆-Framework; [INFO.INFO-CL]Computer Science [cs]/Computation and Language [cs.CL]; [INFO.INFO-LO]Computer Science [cs]/Logic in Computer Science [cs.LO]; Lambda; Beluga; Beluga*
• Online Access: https://hal.science/hal-02573605; https://hal.science/hal-02573605v1/document; https://hal.science/hal-02573605v1/file/article.pdf; https://doi.org/10.4230/LIPIcs.FSCD.2020

International audience We present the syntax, semantics, typing, subtyping, unification, refinement, and REPL of Bull, a prototype theorem prover based on the ∆-Framework, i.e. a fully-typed Logical Framework à la Edinburgh LF decorated with union and intersection types, as described in previous papers by the authors. Bull also implements a subtyping algorithm for the Type Theory Ξ of Barbanera-Dezani-de'Liguoro. Bull has a command-line interface where the user can declare axioms, terms, and perform computations and some basic terminal-style features like error pretty-printing, subexpressions highlighting, and file loading. Moreover, it can typecheck a proof or normalize it. These terms can be incomplete, therefore the typechecking algorithm uses unification to try to construct the missing subterms. Bull uses the syntax of Berardi's Pure Type Systems to improve the compactness and the modularity of the kernel. Abstract and concrete syntax are mostly aligned and similar to the concrete syntax of Coq. Bull uses a higher-order unification algorithm for terms, while typechecking and partial type inference are done by a bidirectional refinement algorithm, similar to the one found in Matita and Beluga. The refinement can be split into two parts: the essence refinement and the typing refinement. Binders are implemented using commonly-used de Bruijn indices. We have defined a concrete language syntax that will allow user to write ∆-terms. We have defined the reduction rules and an evaluator. We have implemented from scratch a refiner which does partial typechecking and type reconstruction. We have experimented Bull with classical examples of the intersection and union literature, such as the ones formalized by Pfenning with his Refinement Types in LF and by Pierce. We hope that this research vein could be useful to experiment, in a proof theoretical setting, forms of polymorphism alternatives to Girard's parametric one.