Abstract 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.

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 presents an extension of kernel ‍ F ≤ , called F ≤ µ , with iso-recursive types. F ≤ is a well-known polymorphic calculus with bounded quantification. In F ≤ µ we add iso-recursive types, and correspondingly extend the subtyping relation with iso-recursive subtyping using the recently proposed nominal unfolding rules. We also add two smaller extensions to F ≤ . The first one is a generalization of the kernel ‍ F ≤ rule for bounded quantification that accepts equivalent rather than equal bounds. The second extension is the use of so-called structural folding/unfolding rules, inspired by the structural unfolding rule proposed by Abadi, Cardelli, and Viswanathan [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 and decidability of subtyping; the conservativity of F ≤ µ over F ≤ ; and a sound and complete algorithmic formulation of F ≤ µ . Moreover, we study an extension of F ≤ µ , called F ≤≥ µ , which includes lower bounded quantification in addition to the conventional (upper) bounded quantification of F ≤ . All the results in this paper have been formalized in the Coq theorem prover.

The Amber rules are well-known and widely used for subtyping iso-recursive types. They were first briefly and informally introduced in 1985 by Cardelli in a manuscript describing the Amber language. Despite their use over many years, important aspects of the metatheory of the iso-recursive style Amber rules have not been studied in depth or turn out to be quite challenging to formalize. This article aims to revisit the problem of subtyping iso-recursive types. We start by introducing a novel declarative specification for Amber-style iso-recursive subtyping. Informally, the specification states that two recursive types are subtypes if all their finite unfoldings are subtypes . The Amber rules are shown to have equivalent expressive power to this declarative specification. We then show two variants of sound , complete and decidable algorithmic formulations of subtyping with respect to the declarative specification, which employ the idea of double unfoldings . Compared to the Amber rules, the double unfolding rules have the advantage of: (1) being modular; (2) not requiring reflexivity to be built in; (3) leading to an easy proof of transitivity of subtyping; and (4) being easily applicable to subtyping relations that are not antisymmetric (such as subtyping relations with record types). This work sheds new insights on the theory of subtyping iso-recursive types, and the new rules based on double unfoldings have important advantages over the original Amber rules involving recursive types. All results are mechanically formalized in the Coq theorem prover.

The Amber rules are well-known and widely used for subtyping iso-recursive types. They were first briefly and informally introduced in 1985 by Cardelli in a manuscript describing the Amber language. Despite their use over many years, important aspects of the metatheory of the iso-recursive style Amber rules have not been studied in depth or turn out to be quite challenging to formalize. This paper aims to revisit the problem of subtyping iso-recursive types. We start by introducing a novel declarative specification that we believe captures the “spirit” of Amber-style iso-recursive subtyping. Informally, the specification states that two recursive types are subtypes if all their finite unfoldings are subtypes . The Amber rules are shown to be sound with respect to this declarative specification. We then derive a sound , complete and decidable algorithmic formulation of subtyping that employs a novel double unfolding rule. Compared to the Amber rules, the double unfolding rule has the advantage of: 1) being modular; 2) not requiring reflexivity to be built in; and 3) leading to an easy proof of transitivity of subtyping. This work sheds new insights on the theory of subtyping iso-recursive types, and the new double unfolding rule has important advantages over the original Amber rules for both implementations and metatheoretical studies involving recursive types. All results are mechanically formalized in the Coq theorem prover. As far as we know, this is the first comprehensive treatment of iso-recursive subtyping dealing with unrestricted recursive types in a theorem prover.