September 17, 2020 Software Open Access

Zhou, Yaoda; Dos Santos Oliveira, Bruno Cesar; Zhao, Jinxu

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        <foaf:name>Dos Santos Oliveira, Bruno Cesar</foaf:name>
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    <dct:title>Revisiting Iso-Recursive Subtyping</dct:title>
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    <dct:description>&lt;p&gt;The Amber rules are well-known and widely used for subtyping&lt;/p&gt; &lt;p&gt;iso-recursive types. They were first briefly and informally introduced&lt;/p&gt; &lt;p&gt;in 1985 by Cardelli in a manuscript describing the Amber&lt;/p&gt; &lt;p&gt;language.&lt;/p&gt; &lt;p&gt;Despite their use over many years, important aspects of the metatheory of the iso-recursive&lt;/p&gt; &lt;p&gt;style Amber rules have not been studied in depth or turn out to be&lt;/p&gt; &lt;p&gt;quite challenging to formalize.&lt;/p&gt; &lt;p&gt;&amp;nbsp;&lt;/p&gt; &lt;p&gt;This paper aims to revisit the problem of subtyping iso-recursive&lt;/p&gt; &lt;p&gt;types. We start by introducing a novel declarative specification&lt;/p&gt; &lt;p&gt;that we believe captures the ``spirit&amp;#39;&amp;#39; of Amber-style&lt;/p&gt; &lt;p&gt;iso-recursive subtyping. Informally, the specification states that&lt;/p&gt; &lt;p&gt;two recursive types are subtypes \emph{if all their finite&lt;/p&gt; &lt;p&gt;unfoldings are subtypes}. The Amber rules are shown to be sound&lt;/p&gt; &lt;p&gt;with respect to this declarative specification. We then derive a&lt;/p&gt; &lt;p&gt;\emph{sound}, \emph{complete} and \emph{decidable} algorithmic&lt;/p&gt; &lt;p&gt;formulation of subtyping that employs a novel \emph{double&lt;/p&gt; &lt;p&gt;unfolding} rule. Compared to the Amber rules, the double&lt;/p&gt; &lt;p&gt;unfolding rule has the advantage of: 1) being modular; 2)&lt;/p&gt; &lt;p&gt;not requiring reflexivity to be built in; and 3) leading to&lt;/p&gt; &lt;p&gt;an easy proof of transitivity of subtyping. This work&lt;/p&gt; &lt;p&gt;sheds new insights on the theory of subtyping iso-recursive types,&lt;/p&gt; &lt;p&gt;and the new double unfolding rule has important advantages over&lt;/p&gt; &lt;p&gt;the original Amber rules for both implementations and&lt;/p&gt; &lt;p&gt;metatheoretical studies involving recursive types. All results&lt;/p&gt; &lt;p&gt;are mechanically formalized in the Coq theorem prover. As far as&lt;/p&gt; &lt;p&gt;we know, this is the first comprehensive treatment of iso-recursive&lt;/p&gt; &lt;p&gt;subtyping dealing with unrestricted recursive types in a theorem prover.&lt;/p&gt;</dct:description>
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