On the power of coercion abstraction

Abstract

Erasable coercions in System F-eta, also known as retyping functions, are well-typed eta-expansions of the identity. They may change the type of terms without changing their behavior and can thus be erased before reduction. Coercions in F-eta can model subtyping of known types and some displacement of quantifiers, but not subtyping assumptions nor certain forms of delayed type instantiation. We generalize F-eta by allowing abstraction over retyping functions. We follow a general approach where computing with coercions can be seen as computing in the lambda-calculus but keeping track of which parts of terms are coercions. We obtain a language where coercions do not contribute to the reduction but may block it and are thus not erasable. We recover erasable coercions by choosing a weak reduction strategy and restricting coercion abstraction to value-forms or by restricting abstraction to coercions that are polymorphic in their domain or codomain. The latter variant subsumes F-eta, F-sub, and MLF in a unified framework.

Safe zero-cost coercions for Haskell

“…Such unidirectional version of Coercible amounts to explicit inclusive subtyping and is more complicated than our current symmetric system: For example, the lifting rule would have to take variance into account: For a type constructor T, does Coercible (T a) (T b) require Coercible a b, or Coercible b a, or both, or neither? Furthermore, we would have to adapt our internal language, FC, to work with explicit subtyping proofs (Crary, 2000;Rémy & Yakobowski, 2010;Cretin & Rémy, 2012).…”