LF _P

Abstract

The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates. This is accomplished by defining lock type constructors, which are a sort of ⋄-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP , one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and substructural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincaré Principle. We investigate and characterize the metatheoretical properties of the calculus underpinning LFP : strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are wellbehaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments.

Plugging-in proof development environments usingLocksinLF

“…Building a universal proof metalanguage where different tools and formalisms can be "plugged in" and "glued together" is a long standing goal that has been extensively explored in a vast and inspiring literature on Logical Frameworks by (Barthe et al 2003;Pfenning and Schürmann 1999;Watkins et al 2002;Schack-Nielsen and Schürmann 2008;Cousineau and Dowek 2007;Boespflug et al 2012;Nanevski et al 2008;Pientka et al 2008;Pientka et al 2010;Honsell et al 2007;Honsell et al 2012;Honsell 2013;Wang and Chaudhuri 2015;Battel and Felty 2015). The clear-cutting monadic structure and properties of the lock/unlock mechanism go back to Moggi's notion of computational monads (Moggi 1989) and our system can be seen as a generalization, to a family of dependent lax operators, of Moggi's partial λ-calculus (Moggi 1988) and of the work carried out in Mendler 1991) (which is also the original source of the term "lax").…”

“…Building a universal proof metalanguage, where different tools and formalisms can be 'plugged in' and 'glued together' is a long standing goal that has been extensively explored in a vast and inspiring literature on logical frameworks by Barthe et al (2003); Battel and Felty (2015); Boespflug et al (2012); Cousineau and Dowek (2007); Honsell et al (2007Honsell et al ( , 2012; Honsell (2013); Nanevski et al (2008); Pfenning and Schürmann (1999); Pientka et al (2008Pientka et al ( , 2010; Schack-Nielsen and Schürmann (2008); Wang and Chaudhuri (2015); Watkins et al (2002). Recent work by Chicani et al on the Foundational Proof Certificates (FPC) (Blanco et al 2017;Chihani et al 2015;Chihani and Miller 2016) and Dedukti (Boespflug et al 2012;Cousineau and Dowek 2007) is very promising.…”

“…This work is an extended version of Honsell et al (2015) and is part of an ongoing research programme, (Honsell et al 2012;Honsell 2013;Honsell et al 2014, aiming to define a simple Universal Meta-language for Logics, extending the constructive type theory LF, that can support the effects of plugging-in and integrating different proof development environments.…”

Gluing together Proof Environments: Canonical extensions of LF Type Theories featuring Locks

“…This latter syntax is more suitable in implementations because it simplifies the notation. Following [18], we stick to the typeful syntax because it allows for a more direct comparison with non-canonical systems. This, however, is technically immaterial.…”

“…In recent years, the authors have introduced in a series of papers [18,16,21,20] various extensions of the Constructive Type Theory LF, with the goal of defining a simple Universal Meta-language that can support the effect of gluing together, i.e. interconnecting, different type systems and proof development environments.…”