On the exp-log normal form of types: Decomposing extensional equality and representing terms compactly

International audienceLambda calculi with algebraic data types lie at the core of functional programming languages and proof assistants, but concealat least two fundamental theoretical problems already in the presence of the simplest non-trivial data type, the sum type. First,we do not know of an explicit and implemented algorithm for deciding the beta-eta-equality of terms---and this in spite of the firstdecidability results proven two decades ago. Second, it is not clear how to decide when two types are essentially the same,i.e. isomorphic, in spite of the meta-theoretic results on decidability of the isomorphism.In this paper, we present the exp-log normal form of types---derived from the representation of exponential polynomials via the unaryexponential and logarithmic functions---that any type built from arrows, products, and sums, can be isomorphically mapped to. The type normal form can be used as a simple heuristic for deciding type isomorphism, thanks to the fact that it is a systematic application of the high-school identities.We then show that the type normal form allows to reduce the standard beta-eta equational theory of the lambda calculus to a specializedversion of itself, while preserving the completeness of equality on terms. We end by describing an alternative representation of normal terms of the lambda calculus with sums, together with a Coq-implementedconverter into/from our new term calculus. The difference with the only other previously implemented heuristic for deciding interestinginstances of eta-equality by Balat, Di Cosmo, and Fiore, is that we exploit the type information of terms substantially and this oftenallows us to obtain a canonical representation of terms without performing sophisticated term analyses