Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics

Dybjer, Peter

doi:10.1017/cbo9780511569807.012

Cited by 79 publications

(75 citation statements)

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“…Dependent types come in families [20], indexed by data. A standard example is the family of vectors indexed by length: data n : Nat X : ⋆ Vec n X where vnil : Vec zero X a : A as : Vec n X vcons a as : Vec (suc n) X Here Vec n X is the type of vectors, length n, of items of type X .…”

Section: Data-type Familiesmentioning

confidence: 99%

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Generic Programming with Dependent Types

Altenkirch

McBride

Morris

2007

Datatype-Generic Programming

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Section: Data-type Familiesmentioning

confidence: 99%

“…However, the latter has become possible, thanks to the recent extension of (ghc) Haskell with a type-indexed variant of inductive families [20], the so-called 'Generalized Algebraic Data Types' [38]. Of course, in Epigram, these two kinds of collection are readily interchangeable.…”

Section: Universes and Representation Typesmentioning

confidence: 99%

Generic Programming with Dependent Types

Altenkirch

McBride

Morris

2007

Datatype-Generic Programming

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“…We mean the strictly positive inductive families of datatypes from Luo's UTT [18]. Modulo technical details, these correspond to the inductive families found in the Alf system [10], the Coq system [26] and Epigram [25]. The story we tell here is by no means specific to any one presentation.…”

Section: Inductive Datatypesmentioning

confidence: 99%

“…In this paper, we show how to equip inductive families of datatypes [10] with useful standard constructions. When you declare an inductive datatype, you expect the following to be available 'for free':…”

Section: Introductionmentioning

confidence: 99%

A Few Constructions on Constructors

McBride

Goguen

McKinna

2006

Lecture Notes in Computer Science

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Abstract. We present four constructions for standard equipment which can be generated for every inductive datatype: case analysis, structural recursion, no confusion, acyclicity. Our constructions follow a two-level approach-they require less work than the standard techniques which inspired them [11,8]. Moreover, given a suitably heterogeneous notion of equality, they extend without difficulty to inductive families of datatypes. These constructions are vital components of the translation from dependently typed programs in pattern matching style [7] to the equivalent programs expressed in terms of induction principles [21] and as such play a crucial behind-the-scenes rôle in Epigram [25].

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“…We give a novel consistency proof by lexicographic path induction of an intuitionistic theory of inductive definitions, based on the system by Martin-Löf [ML71] that inspired the definition of inductive types in type theory [Dyb91,PM93], and several sequent calculi in the programming languages literature [MM00,MT03,Bro06,GMN08]. The consistency of Heyting Arithmetic follows as a simple corollary, and the weak normalization of Gödel's T follows via a structural logical relation [SS08].…”

Section: Introductionmentioning

confidence: 99%

Lexicographic Path Induction

Sarnat

Schürmann

2009

Lecture Notes in Computer Science

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Abstract. Programming languages theory is full of problems that reduce to proving the consistency of a logic, such as the normalization of typed lambda-calculi, the decidability of equality in type theory, equivalence testing of traces in security, etc. Although the principle of transfinite induction is routinely employed by logicians in proving such theorems, it is rarely used by programming languages researchers who often prefer alternatives such as proofs by logical relations and model theoretic constructions. In this paper we harness the well-foundedness of the lexicographic path ordering to derive an induction principle that combines the comfort of structural induction with the expressive strength of transfinite induction. Using lexicographic path induction, we give a consistency proof of Martin-Löf's intuitionistic theory of inductive definitions. The consistency of Heyting arithmetic follows directly, and weak normalization for Gödel's T follows indirectly; both have been formalized in a prototypical extension of Twelf.

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Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics

Cited by 79 publications

References 0 publications

Generic Programming with Dependent Types

Generic Programming with Dependent Types

A Few Constructions on Constructors

Lexicographic Path Induction

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