# Definitions by Rewriting in the Calculus of Constructions

Computing Research Repository, no. 1 (2006): 9-9

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Abstract

Abstract: The main novelty of this paper is to consider an extension of the Calculus of Constructions where predicates can be defined with a general form of rewrite rules. We prove the strong normalization of the reduction relation generated by the β-rule and the user-defined rules under some general syntactic conditions including conflue...More

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Introduction

- This work aims at defining an expressive language allowing to specify and prove mathematical properties in which functions and predicates can be defined by rewrite rules, enabling the automatic proof of equational problems.

The Calculus of Constructions. - The main contribution of the work is a strong normalization result for the Calculus of Constructions extended with, at the predicatelevel, user-defined rewrite rules satisfying some general admissibility conditions.
- The authors assume given a set R of rewrite rules defining the symbols in F .

Highlights

- This work aims at defining an expressive language allowing to specify and prove mathematical properties in which functions and predicates can be defined by rewrite rules, enabling the automatic proof of equational problems.

The Calculus of Constructions - Following Martin-Lof’s theory of types [24], Coquand and Paulin-Mohring defined an extension of CC with inductive types and their associated induction principles as first-class objects : the Calculus of Inductive Constructions (CIC) [26] which is the basis of the proof-assistant Coq [17]
- The main contribution of our work is a strong normalization result for the Calculus of Constructions extended with, at the predicatelevel, user-defined rewrite rules satisfying some general admissibility conditions. We show that these conditions are satisfied by a sub-system of CIC with strong elimination [26] and the Natural Deduction Modulo [13] a large class of equational theories
- We can define simplification rules on propositions that are not definable in CIC
- We consider general syntactic conditions, including confluence, that ensure the strong normalization of the calculus. These conditions are fulfilled by two important systems : a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq [17], and the Natural Deduction Modulo [12, 13] a large class of equational theories

Results

- The symbols 0 : int, s : int → int, p : int → int, + : int → int → int and × : int → int → int defined by the rules s(p(x)) → x, p(s(x)) → x and others for + and × are all constructors of the type int of integers.
- The constructors of primitive predicates, defined by usual first-order rules, are shown to be strongly normalizing since the combination of first-order rewriting with →β preserves strong normalization [8].
- The authors present an extension of the General Schema defined in [5] to deal with type-level rewriting, the main novelty of the paper.
- The General Schema is based on Tait and Girard’s computability predicate technique [19] for proving the strong normalization of the -typed λ-calculus and system F.
- The simplicity condition in (A3) extends to the case of rewriting the restriction in CIC of strong elimination to “small” inductive types, that is, to the types whose constructors have no predicate-arguments except the parameters of the type.
- The main difficulty is to define an interpretation for predicate symbols that is invariant by reduction, a condition required by the type conversion rule.
- When a predicate symbol is defined by a rule whose right-hand side contains quantifiers, its combination with β may not preserve normalization.
- The authors require that left-hand sides are made of free symbols and that at most one rule can apply at the top of a term.

Conclusion

- As a combination of the two previous applications, the work shows that the extension of CIC− with userdefined rewrite rules, even at the predicate-level, is sound if these rules follow the admissibility conditions.
- These conditions are fulfilled by two important systems : a sub-system of the Calculus of Inductive Constructions which is the basis of the proof assistant Coq [17], and the Natural Deduction Modulo [12, 13] a large class of equational theories.
- The authors expect to extend this work with rewriting modulo some useful equational theories like associativity and commutativity, and by allowing η-reductions in the type conversion rule.

Reference

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