Monotone ( Co ) Inductive Types and Fixed - Point Types
نویسنده
چکیده
We study ve extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model xed-points of monotone operators. Building on work by H. Geuvers concerning the relation between term rewrite systems for least pre-xed-points and greatest post-xed-points of positive type schemes (i. e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are type-respecting and reduction-preserving em-beddings even between systems of monotone (co)inductive types and non-interleaving positive xed-point types (which are essentially those retract types). The reduction relation considered is-and-reduction for system F plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the xed-point types. Monotonicity is not reduced to the syntactic restriction of only positive occurrences of the type variable in when forming the inductive type or the coinductive type. Instead only a \monotonicity witness" which is a term of type 88:(!) ! ! := ] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be \interleaved" and not only nested.
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