Building Decision Procedures in the Calculus of Inductive Constructions

TL;DR

This paper introduces a new version of the calculus of inductive constructions that integrates decision procedures into deduction, maintaining key properties like confluence and normalization, to enhance proof assistant capabilities.

Contribution

It presents a novel extension of the calculus of inductive constructions that incorporates arbitrary decision procedures without losing essential logical properties.

Findings

Confluence is preserved in the extended calculus.

Subject reduction remains valid with decision procedures.

Strong normalization and consistency are maintained.

Abstract

It is commonly agreed that the success of future proof assistants will rely on their ability to incorporate computations within deduction in order to mimic the mathematician when replacing the proof of a proposition P by the proof of an equivalent proposition P' obtained from P thanks to possibly complex calculations. In this paper, we investigate a new version of the calculus of inductive constructions which incorporates arbitrary decision procedures into deduction via the conversion rule of the calculus. The novelty of the problem in the context of the calculus of inductive constructions lies in the fact that the computation mechanism varies along proof-checking: goals are sent to the decision procedure together with the set of user hypotheses available from the current context. Our main result shows that this extension of the calculus of constructions does not compromise its main properties: confluence, subject reduction, strong normalization and consistency are all preserved.