Decidability of Type-Checking in the Calculus of Algebraic Constructions with Size Annotations

Abstract

Since Val Tannen’s pioneering work on the combination of simply-typed λ-calculus and first-order rewriting [11], many authors have contributed to this subject by extending it to richer typed λ-calculi and rewriting paradigms, culminating in the Calculus of Algebraic Constructions. These works provide theoretical foundations for type-theoretic proof assistants where functions and predicates are defined by oriented higher-order equations. This kind of definitions subsumes usual inductive definitions, is easier to write and provides more automation.

On the other hand, checking that such user-defined rewrite rules, when combined with β-reduction, are strongly normalizing and confluent, and preserve the decidability of type-checking, is more difficult. Most termination criteria rely on the term structure. In a previous work, we extended to dependent types and higher-order rewriting, the notion of “sized types” studied by several authors in the simpler framework of ML-like languages, and proved that it preserves strong normalization.

The main contribution of the present paper is twofold. First, we prove that, in the Calculus of Algebraic Constructions with size annotations, the problems of type inference and type-checking are decidable, provided that the sets of constraints generated by size annotations are satisfiable and admit most general solutions. Second, we prove the latter properties for a size algebra rich enough for capturing usual induction-based definitions and much more.