Programming and proving with classical types

The propositions-as-types correspondence is ordinarily presen- ted as linking the metatheory of typed λ-calculi and the proof theory of intuitionistic logic. Griffin observed that this correspondence could be extended to classical logic through the use of control operators. This observation set off a flurry of further research, leading to the development of Parigot’s λμ-calculus. In this work, we use the λμ-calculus as the foundation for a system of proof terms for classical first-order logic. In particular, we define an extended call-by-value λμ-calculus with a type system in correspondence with full classical logic. We extend the language with polymorphic types, add a host of data types in ‘direct style’, and prove several metatheoretical properties. All of our proofs and definitions are mechanised in Isabelle/HOL, and we automatically obtain an inter- preter for a system of proof terms cum programming language—called μML—using Isabelle’s code generation mechanism. Atop our proof terms, we build a prototype LCF-style interactive theorem prover—called μTP— for classical first-order logic, capable of synthesising μML programs from completed tactic-driven proofs. We present example closed μML programs with classical tautologies for types, including some inexpressible as closed programs in the original λμ-calculus, and some example tactic-driven μTP proofs of classical tautologies

A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions

The paper describes the refinement algorithm for the Calculus of (Co)Inductive Constructions (CIC) implemented in the interactive theorem prover Matita. The refinement algorithm is in charge of giving a meaning to the terms, types and proof terms directly written by the user or generated by using tactics, decision procedures or general automation. The terms are written in an "external syntax" meant to be user friendly that allows omission of information, untyped binders and a certain liberal use of user defined sub-typing. The refiner modifies the terms to obtain related well typed terms in the internal syntax understood by the kernel of the ITP. In particular, it acts as a type inference algorithm when all the binders are untyped. The proposed algorithm is bi-directional: given a term in external syntax and a type expected for the term, it propagates as much typing information as possible towards the leaves of the term. Traditional mono-directional algorithms, instead, proceed in a bottom-up way by inferring the type of a sub-term and comparing (unifying) it with the type expected by its context only at the end. We propose some novel bi-directional rules for CIC that are particularly effective. Among the benefits of bi-directionality we have better error message reporting and better inference of dependent types. Moreover, thanks to bi-directionality, the coercion system for sub-typing is more effective and type inference generates simpler unification problems that are more likely to be solved by the inherently incomplete higher order unification algorithms implemented. Finally we introduce in the external syntax the notion of vector of placeholders that enables to omit at once an arbitrary number of arguments. Vectors of placeholders allow a trivial implementation of implicit arguments and greatly simplify the implementation of primitive and simple tactics

A formalization of multi-tape Turing machines

We discuss the formalization, in the Matita Theorem Prover, of basic results on multi-tapes Turing machines, up to the existence of a (certified) Universal Machine, and propose it as a natural benchmark for comparing different interactive provers and assessing the state of the art in the mechanization of formal reasoning. The work is meant to be a preliminary step towards the creation of a formal repository in Complexity Theory, and is a small piece in our long-term Reverse Complexity program, aiming to a comfortable, machine independent axiomatization of the field

Foundational Extensible Corecursion

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under well-behaved operations, including constructors. Corecursive functions that are well behaved can be registered as such, thereby increasing the corecursor's expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

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

We present the syntax, semantics, and typing rules of Bull, a prototype theorem prover based on the Delta-Framework, i.e. a fully-typed lambda-calculus decorated with union and intersection types, as described in previous papers by the authors. Bull also implements a subtyping algorithm for the Type Theory Xi 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 the user to write Delta-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. 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

Foundational extensible corecursion: a proof assistant perspective

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under “friendly” operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor’s expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Foundational extensible corecursion: a proof assistant perspective

This paper presents a formalized framework for defining corecursive functions safely in a total setting, based on corecursion up-to and relational parametricity. The end product is a general corecursor that allows corecursive (and even recursive) calls under “friendly” operations, including constructors. Friendly corecursive functions can be registered as such, thereby increasing the corecursor’s expressiveness. The metatheory is formalized in the Isabelle proof assistant and forms the core of a prototype tool. The corecursor is derived from first principles, without requiring new axioms or extensions of the logic

Foundational (co)datatypes and (co)recursion for higher-order logic

We describe a line of work that started in 2011 towards enriching Isabelle/HOL's language with coinductive datatypes, which allow infinite values, and with a more expressive notion of inductive datatype than previously supported by any system based on higher-order logic. These (co)datatypes are complemented by definitional principles for (co)recursive functions and reasoning principles for (co)induction. In contrast with other systems offering codatatypes, no additional axioms or logic extensions are necessary with our approach

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

International audienceWe 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