Recursive types and bounded quantification are prominent features in many modern programming languages, such as Java, C#, Scala, or TypeScript. Unfortunately, the interaction between recursive types, bounded quantification, and subtyping has shown to be problematic in the past. Consequently, defining a simple foundational calculus that combines those features and has desirable properties, such as decidability, transitivity of subtyping, conservativity, and a sound and complete algorithmic formulation, has been a long-time challenge.

This paper shows how to extend $F_{\le}$ with iso-recursive types in a new calculus called $F_{\le}^{\mu}$. $F_{\le}$ is a well-known polymorphic calculus with bounded quantification. In $F_{\le}^{\mu}$, we add iso-recursive types and correspondingly extend the subtyping relation with iso-recursive subtyping using the recently proposed nominal unfolding rules. In addition, we use so-called structural folding/unfolding rules for typing iso-recursive expressions, inspired by the structural unfolding rule proposed by Abadi et al. (1996). The structural rules add expressive power to the more conventional folding/unfolding rules in the literature, and they enable additional applications. We present several results, including: type soundness; transitivity; the conservativity of $F_{\le}^{\mu}$ over $F_{\le}$; and a sound and complete algorithmic formulation of $F_{\le}^{\mu}$. We study two variants of $F_{\le}^{\mu}$. The first one uses an extension of the $\textrm{kernel}~F_{\le}$ (a well-known decidable variant of $F_{\le}$). This extension accepts equivalent rather than equal bounds and is shown to preserve decidable subtyping. The second variant employs the $\textrm{full}~F_{\le}$ rule for bounded quantification and has undecidable subtyping. Moreover, we also study an extension of the kernel version of $F_{\le}^{\mu}$, called $F_{\le\ge}^{\mu\wedge}$, with a form of intersection types and lower bounded quantification. All the properties from the kernel version of $F_{\le}^{\mu}$ are preserved in $F_{\le\ge}^{\mu\wedge}$. All the results in this paper have been formalized in the Coq theorem prover.

Experience in teaching functional programming (FP) on a relational basis has led the author to focus on a graphical style of expression and reasoning in which a geometric construct shines: the (semi) commutative square. In the classroom this is termed the “magic square” (MS), since virtually everything that we do in logic, FP, database modeling, formal semantics and so on fits in some MS geometry. The sides of each magic square are binary relations and the square itself is a comparison of two paths, each involving two sides. MSs compose and have a number of useful properties. Among several examples given in the paper ranging over different application domains, free-theorem MSs are shown to be particularly elegant and productive. Helped by a little bit of Galois connections, a generic, induction-free theory for ${\mathsf{foldr}}$ and $\mathsf{foldl}$ is given, showing in particular that ${\mathsf{foldl} \, {{s}}{}\mathrel{=}\mathsf{foldr}{({flip} \unicode{x005F}{s})}{}}$ holds under conditions milder than usually advocated.

Functional programmers have many things for which to thank the late David Turner: design decisions he made in his languages SASL, KRC, and Miranda over the last 50 years are still influential and inspirational now. In particular, Turner was a strong advocate of lazy evaluation and of list comprehensions. As an illustration of these techniques, he popularized a one-line recursive “sieve” to generate the infinite list of prime numbers.

Turner called this algorithm The Sieve of Eratosthenes. In a lovely paper called “The Genuine Sieve of Eratosthenes”, Melissa O’Neill argued that Turner’s program is not in fact a faithful implementation of the algorithm, and gave a detailed presentation using priority queues of the real thing. She included a variation by Richard Bird, which uses only lists but makes clever use of circular programming. Bird describes his circular program again in his textbook “Thinking Functionally with Haskell”, and sets its proof of correctness as an exercise. In particular, why is this circular program productive? Unfortunately, Bird’s hint for a solution is incorrect. So what should a proof look like?

One of the last projects Turner worked on was the notion of “Total Functional Programming”. He observed that most programs are already structurally recursive or corecursive, therefore guaranteed respectively terminating or productive; he conjectured that “with more practice we will find this is always true”. We explore Bird’s circular Sieve of Eratosthenes as a challenge problem for Turner’s Total Functional Programming.

This paper reports on the experiences of using an early assessment intervention, specifically employing a Use-Modify-Create scaffold, to teach first-year undergraduate functional programming. The particular intervention that was trialled was the use of an early assessment instrument, in which students had to use code given to them, or slightly modify it, to achieve certain goals. The intended outcome was that the students would thus engage earlier with the functional language, enabling them to be better prepared for the second piece of assessment, where they create code to solve given problems. This intervention showed promise: the difference between a student’s score on the Create assignment improved by an average of 9% in the year after the intervention was implemented, a small effect.

Fenwick trees, also known as binary indexed trees are a clever solution to the problem of maintaining a sequence of values while allowing both updates and range queries in sublinear time. Their implementation is concise and efficient—but also somewhat baffling, consisting largely of nonobvious bitwise operations on indices. We begin with segment trees, a much more straightforward, easy-to-verify, purely functional solution to the problem, and use equational reasoning to explain the implementation of Fenwick trees as an optimized variant, making use of a Haskell EDSL for operations on infinite two’s complement binary numbers.

One can perform equational reasoning about computational effects with a purely functional programming language thanks to monads. Even though equational reasoning for effectful programs is desirable, it is not yet mainstream. This is partly because it is difficult to maintain pencil-and-paper proofs of large examples. We propose a formalization of a hierarchy of effects using monads in the Coq proof assistant that makes monadic equational reasoning practical. Our main idea is to formalize the hierarchy of effects and algebraic laws as interfaces like it is done when formalizing hierarchy of algebras in dependent-type theory. Thanks to this approach, we clearly separate equational laws from models. We can then take advantage of the sophisticated rewriting capabilities of Coq and build libraries of lemmas to achieve concise proofs of programs. We can also use the resulting framework to leverage on Coq’s mathematical theories and formalize models of monads. In this article, we explain how we formalize a rich hierarchy of effects (nondeterminism, state, probability, etc.), how we mechanize examples of monadic equational reasoning from the literature, and how we apply our framework to the design of equational laws for a subset of ML with references.