Definitions by rewriting in the Calculus of Constructions

TL;DR

This paper introduces syntactic conditions that ensure strong normalization and logical consistency in the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with higher-order rewrite rules, enhancing formal proof development.

Contribution

It generalizes rewriting at the predicate level within the Calculus of Constructions, broadening its applicability and automation capabilities in formal proofs.

Findings

Established conditions for strong normalization

Proved logical consistency of the extended calculus

Enhanced automation in formal proof development

Abstract

This paper presents general syntactic conditions ensuring the strong normalization and the logical consistency of the Calculus of Algebraic Constructions, an extension of the Calculus of Constructions with functions and predicates defined by higher-order rewrite rules. On the one hand, the Calculus of Constructions is a powerful type system in which one can formalize the propositions and natural deduction proofs of higher-order logic. On the other hand, rewriting is a simple and powerful computation paradigm. The combination of both allows, among other things, to develop formal proofs with a reduced size and more automation compared with more traditional proof assistants. The main novelty is to consider a general form of rewriting at the predicate-level which generalizes the strong elimination of the Calculus of Inductive Constructions.