$\mathsf{LLF}_{\cal P}$: a logical framework for modeling external evidence, side conditions, and proof irrelevance using monads

Abstract

We extend the constructive dependent type theory of the Logical Framework LF with monadic, dependent type constructors indexed with predicates over judgements, called Locks. These monads capture various possible proof attitudes in establishing the judgment of the object logic encoded by an LF type. Standard examples are factoring-out the verification of a constraint or delegating it to an external oracle, or supplying some non-apodictic epistemic evidence, or simply discarding the proof witness of a precondition deeming it irrelevant. This new framework, called Lax Logical Framework, LLFP , is a conservative extension of LF, and hence it is the appropriate metalanguage for dealing formally with side-conditions in rules or external evidence in logical systems. LLFP arises once the monadic nature of the lock type-constructor, L P M,σ [•], introduced by the authors in a series of papers, together with Marina Lenisa, is fully exploited. The nature of the lock monads permits to utilize the very Lock destructor, U P M,σ [•], in place of Moggi's monadic letT , thus simplifying the equational theory. The rules for U P M,σ [•] permit also the removal of the monad once the constraint is satisfied. We derive the meta-theory of LLFP by a novel indirect method based on the encoding of LLFP in LF. We discuss encodings in LLFP of call-by-value λ-calculi, Hoare's Logic, and Fitch-Prawitz Naive Set Theory.

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

“…In this section, following the standard pattern and conventions of [7], we introduce the syntax and the rules of LLF P , see [11] for more details. In Figure 1, we give the syntactic categories of LLF P , namely signatures, contexts, kinds, families (i.e.…”

“…We could give a slightly deeper implementation which, following [11], would yield a more perspicuous rendering of the monadic nature of Locks.…”

“…The case studies in LLF P , carried out in this paper, namely call-by-value λ -calculus and branch prediction for URM machines (see [5]) suggest natural extensions of LLF P itself, for expressing nested lock-types. This was already envisaged in [11]. Furthermore, when the predicate in the lock-type is decidable, the case-study on branch prediction suggests to consider the encoding of alternatives as a sort of sum type.…”

“…The Lax Logical Framework LLF P is a conservative extension of LF. It was introduced in [11] with the goal of factoring-out, postponing, or delegating to external tools the verification of those time-consuming judgments, which are "morally" proof-irrelevant. This system was the final step of a series of papers stemming from [7,8], aiming at integrating different sources of evidence in a unique Logical Framework.…”

“…The L P N,σ [•] constructor was introduced as the appropriate type constructor for expressing inhabitability up-to. It turned out to be smoothly expressible as a monad, see [11], for details.…”