• Corpus ID: 44235193

Confluence and Normalization in Reduction Systems Lecture Notes

  title={Confluence and Normalization in Reduction Systems Lecture Notes},
  author={Gert Smolka},
We study confluence and normalization in abstract reduction systems and apply the results to combinatory logic and an abstract version of the lambda beta calculus. For both systems we obtain the Church-Rosser property and a complete normalization strategy. For an abstract weak call-by-value lambda calculus we obtain uniform confluence, which ensures that for a given term all maximal reduction chains have the same length. The development is based on constructive type theory with inductive… 

Formal verification of the equivalence of system F and the pure type system L2

A formal proof of the equivalence of two different variants of System F where expressions are separated into distinct syntactic classes of types and terms is developed and an extended notion of context morphism lemmas as a structured proof method for this setting is developed.

A Type Theory with Computational Assumptions

An important limitation of the current work is that it is possible to introduce new computational assumptions, but not to discharge them (except on the meta-level as the authors do in Sect. 6.4), so a better approach would be to not abstract over individual rewrite rules but instead abstract simultaneously over a set of values and a setof rewrite rules on these values.

Mechanising syntax with binders in Coq

The topic is investigated from three angles: formal systems with binders based on both pure and scoped de Bruijn algebras together with basic syntactic rewriting lemmas and automation, and concise, transparent, and accessible mechanised proofs for a variety of case studies refined to de bruijn substitutions.

The taming of the rew: a type theory with computational assumptions

This paper introduces Rewriting Type Theory (RTT), a type theory where it is possible to add computational assumptions in the form of rewrite rules, and provides a framework where confluence of user-defined rewrite rules can be checked modularly and automatically and where adding new rewrite rules is guaranteed to preserve subject reduction.

Coq Coq correct! verification of type checking and erasure for Coq, in Coq

This paper presents the first implementation of a type checker for the kernel of Coq (without the module system and template polymorphism), which is proven correct in Coq with respect to its formal specification and axiomatisation of part of its metatheory.

Coq Coq Codet!

This work reports on the ongoing work on verifying a reasonably large subset of Coq’s kernel in Coq, as part of the MetaCoq project, and implements a verified, sound type checker for PCUIC.

Language Support for Programming High-Performance Code

Sierra: an extension for C++ that facilitates portable and effective SIMD programming and AnyDSL, a framework that allows to embed a so-called domain-specific language (DSL) into a host language.

Over then Under Tangles

Brilliant wrong ideas should not be buried and forgotten. Instead, they should be mined for the gold that lies underneath the layer of wrong. In this paper we explain how "over then under tangles"



A Confluent Relational Calculus for Higher-Order Programming with Constraints

The ρ-calculus is presented, a relational calculus parametrized with a logical constraint system, and it is proved that all maximal derivations issuing from a given expression have equal length.

Confluent Reductions: Abstract Properties and Applications to Term Rewriting Systems

This paper gives new results, and presents old ones, concerning ChurchRosser theorems for rewrmng systems, depending solely on axioms for a binary relatton called reduction, and how these criteria yield new methods for the mechanizaUon of equattonal theories.

Term rewriting and all that

Lambda-Calculus and Combinators: An Introduction

This long-awaited new version of this book is thoroughly revised and offers a fully up-to-date account of the subject, with the same authoritative exposition on combinatory logic and lambda-calculus.

Uniform confluence in concurrent computation

This paper investigates concurrent programs that are uniformly confluent and their relation to eager and lazy functional programs and proves a folk theorem, namely that the call- by-need complexity of a functional program is smaller than its call-by-value complexity.

Constructions: A Higher Order Proof System for Mechanizing Mathematics

  • T. CoquandG. Huet
  • Mathematics, Computer Science
    European Conference on Computer Algebra
  • 1985
We present an extensive set of mathematical propositions and proofs in order to demonstrate the power of expression of the theory of constructions.

On Theories with a Combinatorial Definition of "Equivalence"

The name "combinatorial theory" is often given to branches of mathematics in which the central concept is an equivalence relation defined by means of certain "allowed transformations" or "moves." A

Combinatory Logic, Volume I

Parallel reductions in λ-calculus (revised version)

The weak lambda calculus as a reasonable machine