About a Proof Pearl: A Purported Solution to a POPLMARK Challenge Problem that is Not One
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The POPLMARK Challenge comprises a set of problems intended to measure the strength of reasoning systems in the realm of mechanizing programming language meta-theory at the time the challenge was enunciated. Included in the collection is the exercise of demonstrating transitivity of subtyping for a specific algorithmic formulation of subtyping for an extension of System F. The challenge represented by this problem derives from the fact that, for the given formulation, subtyping must be proved simultaneously with another property called narrowing. In a paper published as a proof pearl, Brigitte Pientka claimed to have presented a solution to the problem in which "the full power of parametric and higher-order judgments" is exploited to "get the narrowing lemma for free." We show this claim to be inaccurate. In particular, we show that the simplification is in substantial part the result of changing the formulation of the subtyping relation in a way that modifies the challenge rather than the outcome of the manner in which the argument is mechanized.
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