Strict Ideal Completions of the Lambda Calculus

We present a family of infinitary lambda calculi with varying strictness. This family of calculi corresponds to the infinitary lambda calculi of Kennaway et al. but instead of metric completion our calculi are based on ideal completion. We show that each of our calculi conservatively extends the corresponding metric-based calculus. The extension that our calculi provide is characterised in the form of ’–-rules’: β-reduction in the metric calculi extended by these –-rules, also know as Böhm reduction, coincides with β-reduction in the corresponding ideal completion calculi. Thus, our calculi allow reasoning over partially converging infinite reductions without the need for additional rules, while their total fragments still coincide with the metric calculi. 1998 ACM Subject Classification F.4.2, F.1.1