## A proof of strong normalisation using domain theory (2006)

### Cached

### Download Links

- [arnaud.spiwack.free.fr]
- [hal.inria.fr]
- [hal.archives-ouvertes.fr]
- [arxiv.org]
- [arxiv.org]
- DBLP

### Other Repositories/Bibliography

Venue: | In LICS’06 |

Citations: | 13 - 1 self |

### BibTeX

@INPROCEEDINGS{Coquand06aproof,

author = {Thierry Coquand and Chalmers Tekniska Högskola},

title = {A proof of strong normalisation using domain theory},

booktitle = {In LICS’06},

year = {2006},

pages = {307--316},

publisher = {IEEE CS Press}

}

### OpenURL

### Abstract

U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics is. The goal of this paper is to show that, using ideas from the theory of intersection types [2, 6, 7, 21] and Martin-Löf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type theory. As an example, we show in some details how it can be used to prove strong normalization for Martin-Löf dependent type theory extended with bar recursion, and with some form of proof-irrelevance. 1

### Citations

397 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...significantly by U. Berger [11, 12], who used instead a Arnaud Spiwack Ecole Normale Supérieure de Cachan Arnaud.Spiwack@dptinfo.ens-cachan.fr modification of Plotkin’s computational adequacy theorem =-=[22]-=-, and could prove strong normalisation. In a way, the idea is to replace infinite terms by elements of a domain interpretation. This domain has the property that a term is strongly normalisable if its... |

216 | A filter lambda model and the completeness of type assignment. J.Symbolic Logic, 48:931--940 - Barendregt, Coppo, et al. - 1983 |

103 | Complete restrictions of the Intersection Type Discipline
- Bakel
- 1992
(Show Context)
Citation Context ...he construction of a domain having the property that a term is strongly normalizing if its semantics iss. The goal of this paper is to show ¡£¢ that, using ideas from the theory of intersection types =-=[2, 6, 7, 21]-=- and Martin-Löf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give... |

98 | Martin-Löf Type Theory
- Nordström, Petersson, et al.
- 2000
(Show Context)
Citation Context ... , but also ���� ��������������£����� � . This illustrates the ¡ fact � that can be thought of as the semantics of a top level “error” element. 4 Application to Type Theory 4.1 Typing rules We follow =-=[19]-=- and present dependent type theory in a Logical Framework extended with some constants. We have three syntactical §���§ cat, for terms ¦¨§�©�§ ����� ����� egories, for types � and for contexts � �����... |

84 | Raamsdonk: Combinatory reduction systems: Introduction and survey
- Klop, Oostrom, et al.
- 1993
(Show Context)
Citation Context ...reduces in one step to M ′ by β,ι-reduction and M =β,ι M ′ if M, M ′ are convertible by β,ι conversion. It follows from our hypothesis on our system of reduction rules that β,ι-reduction is confluent =-=[14]-=-. We write → (M) for the set of terms M ′ such that M → M ′ . We work with a given set of constants, that are listed in section 3, but our arguments are general and make use only of the fact that the ... |

83 | On equivalence and canonical forms in the LF type theory
- Harper, Pfenning
(Show Context)
Citation Context ...ts are also given in the appendix. The system is designed in such a way that the following lemmas can be directly proved by induction on derivation. For a detailed metatheory of a similar system, see =-=[16]-=-. � If is a substitution, we � ��� ��� write to express that we have � � � ��� for all � � � in ��� . � Lemma 10 If � correct � and ��� ��� and ��� then � � ��� . � 2In this presentation, we � conside... |

78 |
Interpretation of analysis by means of constructive functionals of finite types
- Kreisel
- 1959
(Show Context)
Citation Context ...called bar recursion. With this new schema, he was able to give an interpretation of Analysis, extending Gödel’s Dialectica interpretation of Arithmetic, and completing preliminary results of Kreisel =-=[17]-=-. Tait proved a normalisation theorem for Spector’s bar recursion, by embedding it in a system with infinite terms [27]. In [9], an alternative form of bar recursion was introduced. This allowed to gi... |

74 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ...ome details how it can be used to prove strong normalization for Martin-Löf dependent type theory extended with bar recursion, and with some form of proof-irrelevance. 1 Introduction In 1961, Spector =-=[25]-=- presented an extension of Gödel’s system ¤ by a new schema of definition called bar recursion. With this new schema, he was able to give an interpretation of Analysis, extending Gödel’s Dialectica in... |

72 |
Foundations of Constructive Mathematics: Metamathematical Studies
- Beeson
- 1985
(Show Context)
Citation Context ...� � § � � and so � � ¡�¢ holds. If � � � � ����� � then � � � ¡£¢ ands�s��� � then � � � ¡£¢ so that ��� ¡ � � ����� � . This shows § � � . ��� The next theorem has a subtle proof, but it is standard =-=[1, 8, 24]-=-. The main idea is to define § the by an inductive process, using Lemma 7 � to ensure the consistency of this definition. pair ¤ Theorem 17 The filter model D of UPL can be extended to a model of our ... |

68 | Subtyping dependent types
- Aspinall, Compagnoni
- 1996
(Show Context)
Citation Context ...� � � � � ������� D � by § � � ��� implies � ��¡ � � � � ����� � � � ��¡ � � � � � ����� ��� � § � if and only if � ��¡ � � if and only if � � � ¡ � � � for all � � � These constructions are standard =-=[5]-=-. � � � Definition 7 A PER model of our type theory consists of a ¤ § � pair ¤ � ������� with D � � � and is such that 1. Set � ¤ ������� D � 2. if � ��� � ¤ and � ��¡ � ��¡ � � � ��¡ � � � plies Fun ... |

62 |
Frege structures and the notion of proposition, truth and set
- Aczel
- 1980
(Show Context)
Citation Context ...� � § � � and so � � ¡�¢ holds. If � � � � ����� � then � � � ¡£¢ ands�s��� � then � � � ¡£¢ so that ��� ¡ � � ����� � . This shows § � � . ��� The next theorem has a subtle proof, but it is standard =-=[1, 8, 24]-=-. The main idea is to define § the by an inductive process, using Lemma 7 � to ensure the consistency of this definition. pair ¤ Theorem 17 The filter model D of UPL can be extended to a model of our ... |

36 |
A type assignment for the strongly normalizable terms
- Pottinger
- 1980
(Show Context)
Citation Context ...he construction of a domain having the property that a term is strongly normalizing if its semantics iss. The goal of this paper is to show ¡£¢ that, using ideas from the theory of intersection types =-=[2, 6, 7, 21]-=- and Martin-Löf’s domain interpretation of type theory [18], we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give... |

28 |
Semantics of Type Theory
- Streicher
- 1991
(Show Context)
Citation Context ...ets D such � � � that �s¡�¢ and � � � if . It is then direct that totality predicates are closed under arbitrary non empty intersections. By working in the D-set model instead of the PER model over D =-=[26, 3]-=-, one should be able to get also strong normalisation theorems for various impredicative type theories extended with bar recursion. For proving normalisation for predicative type systems, the use of t... |

26 | Modified bar recursion and classical dependent choice - Berger, Oliva - 2005 |

23 | Closure under alpha-conversion
- Pollack
- 1994
(Show Context)
Citation Context ...f P. Martin-Löf [15]. We note however that the following judgement is derivable Set��� � ��� Set ������� ������ ��� � � ����� ��� � ��� ��� while it is not in the substitution calculus (as noticed in =-=[20]-=-).srules for contexts rules for types rules for terms correct � � � � � � � ����� type equality rule � ��� ������������� � Set ��� ��������������� � ¦�� Set ��� El ¦ ��� ������������� ��� � � § � � � ... |

21 | A Logical Framework with Dependently Typed Record
- Coquand, Takeyama
- 2005
(Show Context)
Citation Context ...eorem 19 � � If � then is strongly normalisis strongly normalisable. able. If � ¦�� � then ¦ �s4.3 Decidability properties In order to get decidability of conversion, we use a technique introduced in =-=[14]-=- and first define ¨ the ¨�� -expansion in a syntactical way. ¦ Set ¨ ¦ ¦ ¨�� El ¡ ����¦ ¦ ¡ Fun ��� ��¦ ¡ ¨�� Lemma 20 If ��� ¦�� � ¥ ��� ¨������©¨�� � ��� ¦ � �©¨�� � ��� � then ��� ¦ ¡ ¨£� ¦�� The i... |

15 |
Normal form theorem for bar recursive functions of finite type
- Tait
- 1971
(Show Context)
Citation Context ...isation using domain theory Thierry Coquand Chalmers Tekniska Högskola Gothenburg coquand@cs.chalmers.se Abstract U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion =-=[27]-=-, see also [9], replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics iss. The goal of this paper is ... |

12 |
Inductive Types and Strong Normalization
- Constructions
- 1993
(Show Context)
Citation Context ...ets D such � � � that �s¡�¢ and � � � if . It is then direct that totality predicates are closed under arbitrary non empty intersections. By working in the D-set model instead of the PER model over D =-=[26, 3]-=-, one should be able to get also strong normalisation theorems for various impredicative type theories extended with bar recursion. For proving normalisation for predicative type systems, the use of t... |

11 |
Lecture note on the domain interpretation of type theory
- Martin-Löf
- 1983
(Show Context)
Citation Context ...gly normalizing if its semantics iss. The goal of this paper is to show ¡£¢ that, using ideas from the theory of intersection types [2, 6, 7, 21] and Martin-Löf’s domain interpretation of type theory =-=[18]-=-, we can in turn simplify U. Berger’s argument in the construction of such a domain model. We think that our domain model can be used to give modular proofs of strong normalization for various type th... |

9 | Continuous functionals of dependent and transfinite types. In Models and computability
- Berger
- 1997
(Show Context)
Citation Context ...anguage with an extra � element that plays the role of a top-level error. A natural extension of this work would be also to state and prove a density theorem for our denotational semantics, following =-=[13]-=-. The first step would be to define when a formal neighborhood is of a given type. In [6, 21], for ¥ untyped -calculus without constants, it is proved that a term ¦ is strongly normalizing if and only... |

8 |
Domains and lambda-calculi. Cambridge Tracts
- Amadio, Curien
- 1998
(Show Context)
Citation Context ... � ��������� � or of the form and this defines a partition of . Furthermore the following “continuity condition” holds: if finite set and then the set is not empty and . Similar results are proved in =-=[4, 2, 7, 6, 18]-=-. For the proof one can introduce the set of neighborhood in “normal form” by the grammar � ����� ������� § ��� ��� ¡ ��������������������� � � ��� ¡ � ��������� � ����������� � � � � ����� � � and de... |

8 |
An intuitionistic theory of types. In twenty-five years of constructive type theory. Oxford Logic Guides
- Martin-Löf
- 1998
(Show Context)
Citation Context ...ft The goal of this section is to prove strong normalisation for dependent type theory extended with Spector’s double negation shift [23]. The version of type theory we present is close to the one in =-=[17]-=-: we have a type of natural numbers Nat : U, where U is an universe. It is shown in [17], using the propositions-as-types principle, how to represent intuitionistic higher-order arithmetic in type the... |

7 | A computational interpretation of open induction - Berger |

6 |
Strong normalization for applied lambda calculi
- Berger
- 2005
(Show Context)
Citation Context ...r [9] presented also a normalisation proof for this new schema, but this proof, which used Tait’s method of introducing infinite terms, was quite complex. It was simplified significantly by U. Berger =-=[11, 12]-=-, who used instead a modification of Plotkin’s computational adequacy theorem [19], and could prove strong normalisation. In a way, the idea is to replace infinite terms by elements of a domain interp... |

4 |
Lectures on a mathematical theory of computation. Theoretical foundations of programming methodology
- Scott
- 1981
(Show Context)
Citation Context ...es we use the formal equality relation � � defined to be ¡ ����� and ����� . We � let be the set of neighbourhoods quotiented by the formal equality. The terminology “formal neighborhoods” comes from =-=[17, 23, 18]-=-. ��§������ ¡������ � ������� � ��� � � � � ����� ��� ¡ � � � ������� � � � ��� � ������� � ��¡ � � � ������� � � ��� ��� ¡ � � � � � � ��� � � � ����� ¡ � � � � � ������� � � ��� � � ����� � � � ��� ... |

3 |
On the computational content of the axiom of choice
- Coquand
- 1998
(Show Context)
Citation Context ...omain theory Thierry Coquand Chalmers Tekniska Högskola Gothenburg coquand@cs.chalmers.se Abstract U. Berger, [11] significantly simplified Tait’s normalisation proof for bar recursion [27], see also =-=[9]-=-, replacing Tait’s introduction of infinite terms by the construction of a domain having the property that a term is strongly normalizing if its semantics iss. The goal of this paper is to show ¡£¢ th... |

2 |
A proof-irrelevant model of Martin-Lf’s logical framework
- Fridlender
- 2005
(Show Context)
Citation Context ...Lemma 10 If � correct � and ��� ��� and ��� then � � ��� . � 2In this presentation, we � consider -terms up � to - conversion. This system is quite close to the substitution calculus of P. Martin-Löf =-=[15]-=-. We note however that the following judgement is derivable Set��� � ��� Set ������� ������ ��� � � ����� ��� � ��� ��� while it is not in the substitution calculus (as noticed in [20]).srules for con... |

2 |
Combinators and classes. ¥ - calculus and computer science theory
- Scott
- 1975
(Show Context)
Citation Context ...� � § � � and so � � ¡�¢ holds. If � � � � ����� � then � � � ¡£¢ ands�s��� � then � � � ¡£¢ so that ��� ¡ � � ����� � . This shows § � � . ��� The next theorem has a subtle proof, but it is standard =-=[1, 8, 24]-=-. The main idea is to define § the by an inductive process, using Lemma 7 � to ensure the consistency of this definition. pair ¤ Theorem 17 The filter model D of UPL can be extended to a model of our ... |

2 |
A Proof-Irrelevant Type Theory. Unpublished manuscript
- Werner
- 2003
(Show Context)
Citation Context ...n has the strong normalisation property. To illustrate further the modularity of this approach, we show the strong normalisation property when adding some form of proof-irrelevance to our type theory =-=[28]-=-. 2 An Untyped Programming Language Our programming language is untyped ¥ - calculus extended with constants, and has the following syntax. ¡���� ¥ ��� ¦ � ¦�© ���£��� ¦¨§�©���� There are two kinds of... |

2 |
Syntactical Normalization for Intersection Types with Term Rewriting Rules
- Abel
- 2007
(Show Context)
Citation Context ...s by Andreas Abel, it seems likely that Theorem 2.11 has a purely combinatorial proof, similar in complexity to the one for simply typed λ-calculus. He gave such a proof for a reasonable subsystem in =-=[1]-=-. A natural extension of this work would be also to state and prove a density theorem for our denotational semantics, following [13]. The first step would be to define when a formal neighbourhood is o... |

2 |
Combinators and classes. λ-calculus and computer science theory
- Scott
- 1975
(Show Context)
Citation Context ... a typical example of an inductive-recursive definition: we define simulatenously the subset T and the function El on this subset. The justification of such a definition is subtle, but it is standard =-=[2, 8, 22]-=-. It can be checked by induction that T ∈ TP(D) and El(A) ∈ TP(D) if A ∈ T. The next subsection proves that [h] ∈ El ([A]) if ⊢ h:A is a typing rule for a constant h. 4.3. Strong normalisation via tot... |

1 |
Continuous Semantics for Strong Normalisation
- Berger
- 2005
(Show Context)
Citation Context ...r [9] presented also a normalisation proof for this new schema, but this proof, which used Tait’s method of introducing infinite terms, was quite complex. It was simplified significantly by U. Berger =-=[11, 12]-=-, who used instead a Arnaud Spiwack Ecole Normale Supérieure de Cachan Arnaud.Spiwack@dptinfo.ens-cachan.fr modification of Plotkin’s computational adequacy theorem [22], and could prove strong normal... |

1 |
Strong Normalization as Safe Interaction. Logic
- Riba
- 2007
(Show Context)
Citation Context ...lence between M is strongly normalizing and [M ] �=⊥ holds. Additionally, Colin Riba showed this result for a system where the neighbourhoods are closed by union but were the rewrite rules are weaker =-=[20]-=-. Most of our results hold without the hypotheses that the rewrite rules are mutually disjoint. We only have to change the typing rules for a constant f in Figure 2 by the uniform rule: Γ ⊢M f : U1 → ... |