An open logical framework

Abstract

Abstract. The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. 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 sub-structural 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 or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP . We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP : strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments. Moreover, we provide a canonical presentation of LFP , based on a suitable extension of the notion of βη-long normal form, allowing for smooth formulations of adequacy statements.

A Definitional Implementation of the Lax Logical Framework LLFP in Coq, for Supporting Fast and Loose Reasoning

“…Definition 1 (Well-behaved predicates, [10]) A finite set of predicates {P i } i∈I is well-behaved if each P in the set satisfies the following conditions:…”

“…Our encoding could be, actually "should be", as "shallow" as possible so that we may be able to delegate to Coq's metalanguage not only all of LLF P metalanguage, but moreover, reduce inhabitation-search in LLF P to proof-search in Coq. We achieve this by exploiting the fact that Coq is a conservative extension of the dependent constructive type theory of LF [7] which underpins the type system of LLF P , [10]. We simulate/implement, therefore, in Coq the mechanism of lock-types, and use Coq both as the host system and as the oracle for external propositions.…”

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

“…Atomic Family rules The judgements Σ sig, and ⊢ Σ Γ, and Γ ⊢ Σ K are as in Section 2.1 of [19], whereas the remaining ones are peculiar to the canonical style. Informally, the judgment Γ ⊢ Σ M ⇐ σ uses σ to check the type of the canonical term M, while the judgment Γ ⊢ Σ A ⇒ σ uses the type information contained in the atomic term A and Γ to synthesize σ .…”

“…For lack of space we omit proofs, but these follow the standard patterns in [14,19]. We start by studying the basic properties of hereditary substitution and the type system.…”

“…, # » L n are vectors of locks, if ξ appears only in contexts that are of the form . Assume that all predicates in LLF P are well-behaved, according to Definition 2.1 [19]. For any predicate P in LLF P , we define a corresponding predicate in CLLF P with:…”