Anti-symmetry of higher-order subtyping and equality by subtyping

  • Authors:

  • Adriana Compagnoni;Healfdene Goguen

  • Affiliations:

  • Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA Email: abc@cs.stevens.edu;AT&T Labs, 180 Park Ave., Florham Park, NJ 07932, USA Email: hhg@att.com

  • Venue:
  • Mathematical Structures in Computer Science
  • Year:
  • 2006

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Abstract

This paper gives the first proof that the subtyping relation of a higher-order lambda calculus, ${\cal F}^{\omega}_{\leq}$, is anti-symmetric, establishing in the process that the subtyping relation is a partial order – reflexive, transitive, and anti-symmetric up to $\beta$-equality. While a subtyping relation is reflexive and transitive by definition, anti-symmetry is a derived property. The result, which may seem obvious to the non-expert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping, and the logical relation used to prove its equivalence with the declarative presentation of ${\cal F}^{\omega}_{\leq}$, offers a powerful new technology to solve the problem: of particular importance is our extended rule for the well-formedness of types with head variables. The paper also gives a presentation of ${\cal F}^{\omega}_{\leq}$ without a relation for $\beta$-equality, which is apparently the first such, and shows its equivalence with the traditional presentation.