Guided by Tarksi’s fixpoint theorem in order theory, we show how to derive monotone recursive types with constant-time roll and unroll operations within Cedille, an impredicative, constructive, and logically consistent pure typed lambda calculus. This derivation takes place within the preorder on Cedille types induced by type inclusions, a notion which is expressible within the theory itself. As applications, we use monotone recursive types to generically derive two recursive representations of data in lambda calculus, the Parigot and Scott encoding. For both encodings, we prove induction and examine the computational and extensional properties of their destructor, iterator, and primitive recursor in Cedille. For our Scott encoding in particular, we translate into Cedille a construction due to Lepigre and Raffalli (2019) that equips Scott naturals with primitive recursion, then extend this construction to derive a generic induction principle. This allows us to give efficient and provably unique (up to function extensionality) solutions for the iteration and primitive recursion schemes for Scott-encoded data.

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This paper describes an implementation of Harper's continuation-based regular-expression matcher as a strong functional program in Cedille; i.e., Cedille statically confirms termination of the program on all inputs. The approach uses neither dependent types nor termination proofs. Instead, a particular interface dubbed a recursion universe is provided by Cedille, and the language ensures that all programs written against this interface terminate. Standard polymorphic typing is all that is needed to check the code against the interface. This answers a challenge posed by Bove, Krauss, and Sozeau. CCS Concepts: • Software and its engineering → Functional languages; Recursion; • Theory of computation → Regular languages.

We present spine-local type inference, a partial type inference system for inferring omi ed type annotations for System F terms based on local type inference. Local type inference relies on bidirectional inference rules to propagate type information into and out of adjacent nodes of the AST and restricts type-argument inference to occur only within a single node. Spine-local inference relaxes the restriction on type-argument inference by allowing it to occur only within an application spine and improves upon it by using contextual type-argument inference. As our goal is to explore the design space of local type inference, we show that, relative to other variants, spine-local type inference enables desirable features such as first-class curried applications, partial type applications, and the ability to infer types for some terms not otherwise possible. Our approach enjoys usual properties of a bidirectional system of having a specification for our inference algorithm and predictable requirements for typing annotations, and in particular maintains some the advantages of local type inference such as a relatively simple implementation and a tendency to produce good-quality error messages when type inference fails. CCS CONCEPTS•So ware and its engineering → Language features; KEYWORDS bidirectional typechecking, polymorphism, type errors