Book of Abstracts - IRIT

Mendler-style Recursion Schemes for
Mixed-Variant Datatypes
Ki Yung Ahn 1 , Tim Sheard 1 and Marcelo Fiore 2
1 Portland State University ∗
2 University of Cambridge
The context of our work is the Nax project. Our goal is to develop a language system, called
Nax, which supports the merits of both functional programming languages and formal reasoning
systems based on the Curry–Howard correspondence. Our approach towards these goals is to
design an appropriate foundational calculus [3] that extends F ω [5] (or Fix ω [1] similarly) to justify
the theory of Mendler-style recursion schemes [6] with term-indexed datatypes.
In this abstract, we outline a paper that will 1○ discuss the advantages of the Mendler
style, 2○ report that we can define an evaluator for the simply-typed HOAS using Mendlerstyle
iteration with syntactic inverses (msfit), and 3○ propose a new recursion scheme (work
in progress) whose termination relies on the invariants specified by size measures on indices.
Advantages of the Mendler style include allowing arbitrary definition of recursive datatypes,
while still ensuring well-behaved use by providing a rich set of principled eliminators. Certain
concepts, such as HOAS, are most succinctly defined as mixed-variant datatypes, which are
unfortunately, outlawed in many existing reasoning systems (e.g., Coq, Agda). One is forced
to devise clever encodings [4], to use concepts like HOAS within such systems.
In functional programming languages, for instance, in Haskell, a HOAS for the untyped
λ-calculus can be defined as data Exp = Abs (Exp -> Exp) | App Exp Exp . Even if we assume all
functions embedded in Abs are non-recursive, evaluating HOAS may still cause problems for
logical reasoning, since the untyped λ-calculus has diverging terms. However, there are many
well-behaved (i.e., terminating) computations on Exp, such as converting an HOAS expression
to first-order syntax. Ahn and Sheard [2] formalized a Mendler-style recursion scheme (msfit,
a.k.a. msfcata) that captures these well-behaved computations.
If the datatype Exp had indexes to assert invariants of well-formed expressions, we could rely
on these invariants to write even more expressive programs, such as a terminating well-typed
evaluator. Discussion around this idea will constitute the latter parts of the paper.
A simply-typed HOAS evaluator can be defined using msfit at kind * -> *. Since msfit
terminates for any datatype, we are also proving that the evaluation of the simply-typed λ-
calculus always terminates just by defining eval : Exp t -> Id t in Nax, as below. We wonder
eval has similarities to other normalization strategies like NbE [7].
data E : (* -> *) -> (* -> *) where -- the "deriving fixpoint Exp" defines
Abs : (r a -> r b) -> E r (a -> b) -- abs f = In[* -> *] (Abs f)
App : E r (a -> b) -> E r a -> E r b -- app f e = In[* -> *] (App f e)
deriving fixpoint Exp -- synonym Exp t = Mu[* -> *] E t
data Id a = MkId a -- the identity type
unId (MkId x) = x -- destructor of Id
eval e = msfit { t . Id t } e with
call inv (App f x) = MkId (unId(call f) (unId(call x)))
call inv (Abs f) = MkId (\v -> unId(call (f (inv (MkId v)))))
∗ supported by NSF grant 0910500.
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