Efficient Mendler-Style Lambda-Encodings in Cedille

It is common to model inductive datatypes as least fixed points of functors. We show that within the Cedille type theory we can relax functoriality constraints and generically derive an induction principle for Mendler-style lambda-encoded inductive datatypes, which arise as least fixed points of covariant schemes where the morphism lifting is defined only on identities. Additionally, we implement a destructor for these lambda-encodings that runs in constant-time. As a result, we can define lambda-encoded natural numbers with an induction principle and a constant-time predecessor function so that the normal form of a numeral requires only linear space. The paper also includes several more advanced examples

Course-of-Value Induction in Cedille

In the categorical setting, histomorphisms model a course-of-value recursion scheme that allows functions to be defined using arbitrary previously computed values. In this paper, we use the Calculus of Dependent Lambda Eliminations (CDLE) to derive a lambda-encoding of inductive datatypes that admits course-of-value induction. Similar to course-of-value recursion, course-of-value induction gives access to inductive hypotheses at arbitrary depth of the inductive arguments of a function. We show that the derived course-of-value datatypes are well-behaved by proving Lambek's lemma and characterizing the computational behavior of the induction principle. Our work is formalized in the Cedille programming language and also includes several examples of course-of-value functions

Generic Zero-Cost Reuse for Dependent Types

Dependently typed languages are well known for having a problem with code reuse. Traditional non-indexed algebraic datatypes (e.g. lists) appear alongside a plethora of indexed variations (e.g. vectors). Functions are often rewritten for both non-indexed and indexed versions of essentially the same datatype, which is a source of code duplication. We work in a Curry-style dependent type theory, where the same untyped term may be classified as both the non-indexed and indexed versions of a datatype. Many solutions have been proposed for the problem of dependently typed reuse, but we exploit Curry-style type theory in our solution to not only reuse data and programs, but do so at zero-cost (without a runtime penalty). Our work is an exercise in dependently typed generic programming, and internalizes the process of zero-cost reuse as the identity function in a Curry-style theory.Comment: Additional section on Generic Relational Reus

Monotone recursive types and recursive data representations in Cedille

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 type theory. As applications, we use monotone recursive types to generically derive two recursive representations of data in the lambda calculus, the Parigot and Scott encoding, together with constant-time destructors, a recursion scheme, and the standard induction principle