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Reduction and unification in lambda calculi with a general notion of subtype

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Reduction, equality, and unification are studied for a family of simply typed λ-calculi with subtypes. The subtype relation is required to relate base types only to base types and to satisfy some order-theoretic conditions. Constants are required to have a least type, that is, ‘no overloading’. We define the usual β and a subtype-dependent η-reduction. These are related to a typed equality relation and shown to be confluent in a certain sense. We present a generic algorithm for preunification modulo βη-conversion and an arbitrary subtype relation. Furthermore it is shown that unification with respect to any subtype relation is universal.

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Qian, Z., Nipkow, T. Reduction and unification in lambda calculi with a general notion of subtype. J Autom Reasoning 12, 389–406 (1994). https://doi.org/10.1007/BF00885767

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Key words

  • simply typed λ-calculi
  • subtypes
  • reduction
  • higher-order unification