Untyped Confluence in Dependent Type Theories

International audienceWe investigate techniques based on van Oostrom's decreasing diagrams that reduce confluence proofs to the checking of critical pairs in the absence of termination properties, which are useful in dependent type calculi to prove confluence on untyped terms. These techniques are applied to a complex example originating from practice: a faithful encoding, in an extension of LF with rewrite rules on objects and types, of a subset of the calculus of inductive constructions with a cumulative hierarchy of predicative universes above Prop. The rules may be first-order or higher-order, plain or modulo, non-linear on the right or on the left. Variables which occur non-linearly in lefthand sides of rules must take their values in confined types: in our example, the natural numbers. The first-order rules are assumed to be terminating and confluent modulo some theory: in our example, associativity, commutativity and identity. Critical pairs involving higher-order rules must satisfy van Oostrom's decreasing diagram condition wrt their indexes taken as labels

CoqMTU: a higher-order type theory with a predicative hierarchy of universes parametrized by a decidable first-order theory

International audienceWe study a complex type theory, a Calculus of Inductive Constructions with a predicative hierarchy of universes and a first-order theory T built in its conversion relation. The theory T is specified abstractly, by a set of constructors, a set of defined symbols, axioms expressing that constructors are free and defined symbols completely defined, and a generic elimination principle relying on crucial properties of first-order structures satisfying the axioms. We first show that CoqMTU enjoys all basic meta-theoretical properties of such calculi, confluence, subject reduction and strong normalization when restricted to weak-elimination, implying the decidability of type-checking in this case as well as consistency. The case of strong elimination is left open

Subtype Universes

We introduce a new concept called a subtype universe, which is a collection of subtypes of a particular type. Amongst other things, subtype universes can model bounded quantification without undecidability. Subtype universes have applications in programming, formalisation and natural language semantics. Our construction builds on coercive subtyping, a system of subtyping that preserves canonicity. We prove Strong Normalisation, Subject Reduction and Logical Consistency for our system via transfer from its parent system UTT[?]. We discuss the interaction between subtype universes and other sorts of universe and compare our construction to previous work on Power types

Confluence in UnTyped Higher-Order Theories by means of Critical Pairs

User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. We develop here techniques that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. The present paper concentrates on the case where rewrite rules are left-linear and critical pairs can be joined without using beta-rewrite steps. The other two cases will be addressed in forthcoming papers

A Type-Theoretic Analysis of Modular Specifications

We study the problem of representing a modular specification language in a type-theory based theorem prover. Our goals are: to provide mechanical support for reasoning about specifications and about the specification language itself; to clarify the semantics of the specification language by formalising them fully; to augment the specification language with a programming language in a setting where they are both part of the same formal environment, allowing us to define a formal implementation relationship between the two. Previous work on similar issues has given rise to a dichotomy between "shallow" and "deep" embedding styles when representing one language within another. We show that the expressiveness of type theory, and the high degree of reflection that it permits, allow us to develop embedding techniques which lie between the "shallow" and "deep" extremes. We consider various possible embedding strategies and then choose one of them to explore more fully. As our object of study we choose a fragment of the Z specification language, which we encode in the type theory UTT, as implemented in the LEGO proof-checker. We use the encoding to study some of the operations on schemas provided by Z. One of our main concerns is whether it is possible to reason about Z specifications at the level of these operations. We prove some theorems about Z showing that, within certain constraints, this kind of reasoning is indeed possible. We then show how these metatheorems can be used to carry out formal reasoning about Z specifications. For this we make use of an example taken from the Z Reference Manual (ZRM). Finally, we exploit the fact that type theory provides a programming language as well as a logic to define a notion of implementation for Z specifications. We illustrate this by encoding some example programs taken from the ZRM

Relational reasoning for effects and handlers

This thesis studies relational reasoning techniques for FRANK, a strict functional language supporting algebraic effects and their handlers, within a general, formalised approach for completely characterising observational equivalence. Algebraic effects and handlers are an emerging paradigm for representing computational effects where primitive operations, which give rise to an effect, are primary, and given semantics through their interpretation by effect handlers. FRANK is a novel point in the design space because it recasts effect handling as part of a generalisation of call-by-value function application. Furthermore, FRANK generalises unary effect handlers to the n-ary notion of multihandlers, supporting more elegant expression of certain handlers. There have been recent efforts to develop sound reasoning principles, with respect to observational equivalence, for languages supporting effects and handlers. Such techniques support powerful equational reasoning about code, such as substitution of equivalent sub-terms (‘equals for equals’) in larger programs. However, few studies have considered a complete characterisation of observational equivalence, and its implications for reasoning techniques. Furthermore, there has been no account of reasoning principles for FRANK programs. Our first contribution is a formal reconstruction of a general proof technique, triangulation, for proving completeness results for observational equivalence. The technique brackets observational equivalence between two structural relations, a logical and an applicative notion. We demonstrate the triangulation proof method for a pure simply-typed λ-calculus. We show that such results are readily formalisable in an implementation of type theory, specifically AGDA, using state-of-the-art technology for dealing with syntaxes with binding. Our second contribution is a calculus, ELLA, capturing the essence of FRANK’s novel design. In particular, ELLA supports binary handlers and generalises function application to incorporate effect handling. We extend our triangulation proof technique to this new setting, completely characterising observational equivalence for this calculus. We report on our partial progress in formalising our extension to ELLA in AGDA. Our final contribution is the application of sound reasoning principles, inspired by existing literature, to a variety of ELLA programs, including a proof of associativity for a canonical pipe multihandler. Moreover, we show how leveraging completeness leads, in certain instances, to simpler proofs of observational equivalence