Checking algorithms for Pure Type Systems

We have presented efficient syntax directed presentations of two subclasses of PTS:
the semi-full systems, via the ⊢ _sdsf relation

the functional systems, via the ⊢_f relation

The only remaining defect in these presentations lies in the possible failure of tests for conversion in the application rule. Thus for normalizing functional and semi-full systems, everything has been said.

For non-functional systems the situation is less clear. We know of no a priori bound on the amount of reduction necessary to correctly type λ-abstractions, so we must be content with the collective completeness of the family of syntax directed systems ⊢_sd−n.

We have made little impact on the Expansion Postponement problem, which we leave as future work. We can however bask in the relative peace of mind gained from the machine-checked presentation of most (i.e. those not concerning schematic judgments) of the above results.