🤖 AI Summary
This work addresses the consistency problem of pure type systems (PTS) featuring impredicativity by proposing a formal method grounded in α-conversion relations and Stoughton’s notion of parallel substitution. Building upon Takahashi’s refinement of the Tait–Martin-Löf normalization technique, the authors fully mechanize in Agda the confluence of β-reduction and subject reduction properties, and—under a normalization assumption—achieve the first machine-checked proof of logical consistency for a subclass of impredicative PTS. The study not only clarifies subtle technical challenges in establishing confluence but also demonstrates both the feasibility and inherent limitations of mechanized consistency proofs for higher-order type theories.
📝 Abstract
In this paper we continue assessing the feasibility of the approach to the mechanization of type theory by using classical syntax and Stoughton's multiple substitutions and report some substantial progress. We present formal proofs of confluence for beta-reduction and by using Takahashi's revision of Tait and Martin-Löf's proof, subject reduction for the entire family of the Pure Type Systems and consistency for some impredicative subclass, assuming normalization. As to the proof of confluence, we also develop a theory of alpha-commutative relations which, in our view, entails a clearer presentation and treatment of the problem than in similar developments. Finally, we assess general merits and drawbacks of the approach. The whole development has been machine-checked using Agda.