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TypeBased Termination of Recursive Definitions
, 2002
"... This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the techn ..."
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This article The purpose of this paper is to introduce b, a simply typed calculus that supports typebased recursive definitions. Although heavily inspired from previous work by Giménez (Giménez 1998) and closely related to recent work by Amadio and Coupet (Amadio and CoupetGrimal 1998), the technical machinery behind our system puts a slightly different emphasis on the interpretation of types. More precisely, we formalize the notion of typebased termination using a restricted form of type dependency (a.k.a. indexed types), as popularized by (Xi and Pfenning 1998; Xi and Pfenning 1999). This leads to a simple and intuitive system which is robust under several extensions, such as mutually inductive datatypes and mutually recursive function definitions; however, such extensions are not treated in the paper
Lambda Calculus: A Case for Inductive Definitions
, 2000
"... These lecture notes intend to introduce to the subject of lambda calculus and types. A special focus is on the use of inductive denitions. The ultimate goal of the course is an advanced treatment of inductive types. Contents 1 Overview 2 2 Introduction to Inductive Denitions 4 3 Lambda Calculus 13 ..."
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These lecture notes intend to introduce to the subject of lambda calculus and types. A special focus is on the use of inductive denitions. The ultimate goal of the course is an advanced treatment of inductive types. Contents 1 Overview 2 2 Introduction to Inductive Denitions 4 3 Lambda Calculus 13 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Pure Untyped Lambda Calculus . . . . . . . . . . . . . . . . . . 15 4 Conuence 19 5 Weak and Strong Normalization 27 6 Simple and Intersection Types 33 6.1 SimplyTyped Lambda Calculus . . . . . . . . . . . . . . . . . . 34 6.2 Lambda Calculus with Intersection Types . . . . . . . . . . . . . 36 6.3 Strong Normalization of Typable Terms . . . . . . . . . . . . . . 39 6.4 Typability of Strongly Normalizing Terms . . . . . . . . . . . . . 41 7 Parametric Polymorphism 41 7.1 Strong Normalization of Typable Terms . . . . . . . . . . . . . . 44 7.1.1 Saturated Sets . . . . . . . . . . . . . . . . . . . . . ....
Parigot's Second Order λμCalculus and Inductive Types
, 2001
"... . A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their ass ..."
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. A new proof of strong normalization of Parigot's (second order) calculus is given by a reductionpreserving embedding into system F (second order polymorphic calculus). The main idea is to use the least stable supertype for any type. These nonstrictly positive inductive types and their associated iteration principle are available in system F, and allow to give a translation vaguely related to CPS translations (corresponding to the Kolmogorov embedding of classical logic into intuitionistic logic). However, they simulate Parigot's reductions whereas CPS translations hide them. As a major advantage, this embedding does not use the idea of reducing stability (:: ! ) to that for atomic formulae. Therefore, it even extends to noninterleaving positive xedpoint types. As a nontrivial application, strong normalization of calculus, extended by primitive recursion on monotone inductive types, is established. 1 Introduction calculus [12] essentially is the extension of nat...