New Equations for Neutral Terms: A Sound and Complete Decision Procedure, Formalized

The definitional equality of an intensional type theory is its test of type compatibility. Today's systems rely on ordinary evaluation semantics to compare expressions in types, frustrating users with type errors arising when evaluation fails to identify two `obviously' equal terms. If only the machine could decide a richer theory! We propose a way to decide theories which supplement evaluation with `$\nu$-rules', rearranging the neutral parts of normal forms, and report a successful initial experiment. We study a simple -calculus with primitive fold, map and append operations on lists and develop in Agda a sound and complete decision procedure for an equational theory enriched with monoid, functor and fusion laws

Typing with Leftovers - A mechanization of Intuitionistic Multiplicative-Additive Linear Logic

We start from an untyped, well-scoped lambda-calculus and introduce a bidirectional typing relation corresponding to a Multiplicative-Additive Intuitionistic Linear Logic. We depart from typical presentations to adopt one that is well-suited to the intensional setting of Martin-Löf Type Theory. This relation is based on the idea that a linear term consumes some of the resources available in its context whilst leaving behind leftovers which can then be fed to another program. Concretely, this means that typing derivations have both an input and an output context. This leads to a notion of weakening (the extra resources added to the input context come out unchanged in the output one), a rather direct proof of stability under substitution, an analogue of the frame rule of separation logic showing that the state of unused resources can be safely ignored, and a proof that typechecking is decidable. Finally, we demonstrate that this alternative formalization is sound and complete with respect to a more traditional representation of Intuitionistic Linear Logic. The work has been fully formalised in Agda, commented source files are provided as additional material available at https://github.com/gallais/typing-with-leftovers

Builtin types viewed as inductive families

This research was funded by the Engineering and Physical Sciences Research Council (grant number EP/T007265/1).State of the art optimisation passes for dependently typed languages can help erase the redundant information typical of invariant-rich data structures and programs. These automated processes do not dramatically change the structure of the data, even though more efficient representations could be available. Using Quantitative Type Theory, we demonstrate how to define an invariant-rich, typechecking time data structure packing an efficient runtime representation together with runtime irrelevant invariants. The compiler can then aggressively erase all such invariants during compilation. Unlike other approaches, the complexity of the resulting representation is entirely predictable, we do not require both representations to have the same structure, and yet we are able to seamlessly program as if we were using the high-level structure.Publisher PD

Views of pi : definition and computation

We study several formal proofs and algorithms related to the number pi in the context of Coq's standard library.  In particular, we clarify the relation between roots of the cosine function and the limit of the alternated series whose terms are the inverse of odd natural numbers (known as Leibnitz' formula).We give a formal description of the arctangent function and its expansion as a power series.  We then study other possible descriptions of pi, first as the surface of the unit disk, second as the limit of perimeters of regular polygons with an increasing number of sides.In a third section, we concentrate on techniques to effectively compute approximations of pi in the proof assistant by relying on rational numbers and decimal representations

Scoped and typed staging by evaluation

Using a dependently typed host language, we give a well scoped-and-typed by construction presentation of a minimal two level simply typed calculus with a static and a dynamic stage. The staging function partially evaluating the parts of a term that are static is obtained by a model construction inspired by normalisation by evaluation. We then go on to demonstrate how this minimal language can be extended to provide additional metaprogramming capabilities, and to define a higher order functional language evaluating to digital circuit descriptions

Seamless, correct, and generic programming over serialised data

In typed functional languages, one can typically only manipulate data in a type-safe manner if it first has been deserialised into an in-memory tree represented as a graph of nodes-as-structs and subterms-as-pointers. We demonstrate how we can use QTT as implemented in \idris{} to define a small universe of serialised datatypes, and provide generic programs allowing users to process values stored contiguously in buffers. Our approach allows implementors to prove the full functional correctness by construction of the IO functions processing the data stored in the buffer

Builtin types viewed as inductive families

State of the art optimisation passes for dependently typed languages can help erase the redundant information typical of invariant-rich data structures and programs. These automated processes do not dramatically change the structure of the data, even though more efficient representations could be available. Using Quantitative Type Theory as implemented in Idris 2, we demonstrate how to define an invariant-rich, typechecking-time data structure packing an efficient runtime representation together with runtime irrelevant invariants. The compiler can then aggressively erase all such invariants during compilation. Unlike other approaches, the complexity of the resulting representation is entirely predictable, we do not require both representations to have the same structure, and yet we are able to seamlessly program as if we were using the high-level structure

Generic level polymorphic N-ary functions

Agda's standard library struggles in various places with n-ary functions and relations. It introduces congruence and substitution operators for functions of arities one and two, and provides users with convenient combinators for manipulating indexed families of arity exactly one. After a careful analysis of the kinds of problems the unifier can easily solve, we design a unifier-friendly representation of n-ary functions. This allows us to write generic programs acting on n-ary functions which automatically reconstruct the representation of their inputs' types by unification. In particular, we can define fully level polymorphic n-ary versions of congruence, substitution and the combinators for indexed families, all requiring minimal user input

Scoped and typed staging by evaluation

Using a dependently typed host language, we give a well scoped-and-typed by construction presentation of a minimal two level simply typed calculus with a static and a dynamic stage. The staging function partially evaluating the part of a term that are static is obtained by a model construction inspired by normalisation by evaluation. We then go on to demonstrate how this minimal language can be extended to provide additional metaprogramming capabilities, and to define a higher order functional language evaluating to digital circuit descriptions

Frex: dependently-typed algebraic simplification

We present an extensible, mathematically-structured algebraic simplification library design. We structure the library using universal algebraic concepts: a free algebra -- fral -- and a free extension -- frex -- of an algebra by a set of variables. The library's dependently-typed API guarantees simplification modules, even user-defined ones, are terminating, sound, and complete with respect to a well-specified class of equations. Completeness offers intangible benefits in practice -- our main contribution is the novel design. Cleanly separating between the interface and implementation of simplification modules provides two new modularity axes. First, simplification modules share thousands of lines of infrastructure code dealing with term-representation, pretty-printing, certification, and macros/reflection. Second, new simplification modules can reuse existing ones. We demonstrate this design by developing simplification modules for monoid varieties: ordinary, commutative, and involutive. We implemented this design in the new Idris2 dependently-typed programming language, and in Agda