On the Power of Simple Diagrams

  title={On the Power of Simple Diagrams},
  author={Roberto Di Cosmo},
In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systematically confluence and/or normalization (or decidability… 

Isomorphisms of simple inductive types through extensional rewriting

  • D. Chemouil
  • Mathematics
    Mathematical Structures in Computer Science
  • 2005
The notion of a faithful copy of an inductive type and a corresponding conversion relation that also preserves the good properties of the calculus are defined.

Some Algebraic Structures in Lambda-Calculus with Inductive Types

This paper is part of a research project where methods to extend the computational content of various systems of typed λ-calculus adding new reductions by presenting new results concerning representation of finite sets as inductive types and related algebraic structures.

Eta-Expansions in Dependent Type Theory - The Calculus of Constructions

This paper applies η-expansions to the Calculus of Constructions to discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long βη-normal form and deduce the decidability of β- equality, typeability and type inhabitation as corollaries.

An insertion operator preserving infinite reduction sequences

  • D. Chemouil
  • Mathematics
    Mathematical Structures in Computer Science
  • 2008
An operator is introduced enabling us to insert a reduction step on such an object, and therefore to change its shape, while still preserving the ability to use the property.

Life without the Terminal Type

It is shown that categories that are cartesian closed except for the lack of a terminal object have a universal full extension to a Cartesian closed category, and categories for which the latter category is a topos are characterized.

On Modular Properties of Higher Order Extensional Lambda Calculi

We prove that confluence and strong normalisation are both modular properties for the addition of algebraic term rewriting systems to Girard's F ω equipped with either β-equality or βη-equality.

The Journal of Functional and Logic Programming the Journal of Functional and Logic Programming Reasoning about Redundant Patterns

It is shown that only layered and wildcard patterns are redundant in lw ?!, while product patterns are unnecessary in lwp ?!. Connuence of both reduction systems is proven by the composition of modular properties of the systems' extensional and nonextensional parts.

Programming with first-class modules in a core language with subtyping, singleton kinds and open existential types. (Programmer avec des modules de première classe dans un langage noyau pourvu de sous-typage, sortes singletons et types existentiels ouverts)

This thesis explains how the adjunction of three features to System Fω allows writing programs in a modular way in an explicit system a la Church, while keeping a style that is similar to ML modules.

Higher-Order Rewriting with Dependent Types

This dissertation studies a theory of Higher-order Term Rewriting for the LF calculus, on which Twelf is based, and presents applications to Milner's Action and Process Calculi, Category Theory, and Proof Theory.

A brief history of rewriting with extensionality

• A survey of confluence, decidability and normalization results for typed λ-calculi with extensional rules • A survey of the proof techniques • Some applications



Combining Algebraic Rewriting, Extensional Lambda Calculi, and Fixpoints

Expanding Extensional Polymorphism

We prove the confluence and strong normalization properties for second order lambda calculus equipped with an expansive version of η-reduction. Our proof technique, based on a simple abstract lemma

Simulating expansions without expansions

This work proves the calculus to be weakly confluent, and provides an effective mechanism to simulate expansions without expansion rules, so that the strong normalization of the calculus can be derived from that of the underlying, traditional, non extensional system.

Combining Term Rewriting and Type Assignment Systems

In this paper strong normalization and confluence are proved for λ-terms obtained by merging pure κ-terms and first order canonical term rewriting systems, in the framework of a system which extends the Coppo-Dezani intersection type assignment system.

A Concluent Reduction for the Lambda-Calculus with Surjective Pairing and Terminal Object

This work is the first treatment of the lambda calculus extended with surjective pairing and terminal object via a confluent rewriting system, and the first solution to the decidability problem of the full equational theory of Cartesian Closed Categories extended with polymorphic types.

Some Lambda Calculi with Categorial Sums and Products

The simply typed λ-calculus with primitive recursion operators and types corresponding to categorical products and coproducts is considered, and strong normalization and ground (base-type) confluence is proved for the full calculus.

Combining algebra and higher-order types

  • V. Breazu-Tannen
  • Computer Science, Mathematics
    [1988] Proceedings. Third Annual Information Symposium on Logic in Computer Science
  • 1988
The author shows that provability in the higher-order equational proof system obtained by adding the simply typed beta and eta axions to some many-sorted algebraic proof system is effectively reducible to Provability in that algebraicProof system.

Polymorphic Rewriting Conserves Algebraic Strong Normalization

Polymorphic Rewriting Conserves Algebraic Confluence

We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to