## Infinite Objects in Type Theory (0)

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Citations: | 87 - 2 self |

### BibTeX

@MISC{Coquand_infiniteobjects,

author = {Thierry Coquand},

title = {Infinite Objects in Type Theory},

year = {}

}

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### Abstract

. We show that infinite objects can be constructively understood without the consideration of partial elements, or greatest fixedpoints, through the explicit consideration of proof objects. We present then a proof system based on these explanations. According to this analysis, the proof expressions should have the same structure as the program expressions of a pure functional lazy language: variable, constructor, application, abstraction, case expressions, and local let expressions. 1 Introduction The usual explanation of infinite objects relies on the use of greatest fixed-points of monotone operators, whose existence is justified by the impredicative proof of Tarski's fixed point theorem. The proof theory of such infinite objects, based on the so called co-induction principle, originally due to David Park [21] and explained with this name for instance in the paper [18], reflects this explanation. Constructively, to rely on such impredicative methods is somewhat unsatisfactory (see fo...

### Citations

3426 | Communication and Concurrency - Milner - 1989 |

1418 | A Calculus of Communicating Systems - Milner - 1980 |

699 |
Concurrency and automata on infinite sequences
- Park
- 1981
(Show Context)
Citation Context ...existence is justified by the impredicative proof of Tarski fixed point theorem. The proof theory of such infinite objects, based on the so called co-induction principle, originally due to David Park =-=[18]-=- and explained with this name for instance in the paper [15], reflects this explanation. Constructively, to rely on such impredicative methods is somewhat unsatisfactory and this paper is a tentative ... |

353 |
Intuitionistic Type Theory
- Martin-Löf
- 1984
(Show Context)
Citation Context ...pproach suggests an alternative reasoning system. In particular, the notion of constructors, or introduction rules, keeps the fundamental importance it has for proof system about well-founded objects =-=[13]-=-, while it appears as a derived notion in proof systems based on co-induction (where this notion is secondary to the notion of destructors, or elimination rules). As a consequence, the strong normalis... |

272 |
Programming in Martin-Lof ’s Type Theory, An Introduction
- Nordstrom, Petersson, et al.
- 1990
(Show Context)
Citation Context ...esentation 1.1 Type Theory of Well-Founded Objects We recall briefly some basic notions of type theory of well-founded objects, that will be important for the extension to infinite objects. The books =-=[17, 13]-=- contain more detailed explanations, and the reference [3] describes the addition of case expressions and pattern-matching. 1.1.1 Semantics A(n inductive) set A is defined by its constructors. A close... |

188 |
An introduction to inductive definitions
- Aczel
- 1977
(Show Context)
Citation Context ...easoning about infinite objects. 1.3 Reformulation with rule sets In this section we express in an abstract way how one can understand inductively a greatest fixed-point. We follow the terminology of =-=[1]-=-. We start with a set U of atoms and a set 8 of rules, that are pairs (X; x) such that X ` U and x 2 U: We write 8 : X 7! x to mean that (X; x) 2 8: An element (X; x) 2 8 is called a rule of conclusio... |

149 | Intuitionistic type theory," Bibliopolis - Martin-Lof - 1984 |

143 |
Elements of Intuitionism
- Dummett
- 1977
(Show Context)
Citation Context ...ify sets and propositions. The constructors can be interpreted as introduction rules, and a closed proof of the proposition A is a well-founded proof tree built out of introduction rules. It is clear =-=[13, 7]-=-, that, besides terms purely built out of constructors, one needs also to consider noncanonical expressions. The addition of such expressions is done in such a way however that any closed term of a cl... |

96 |
Inductive definitions, semantics and abstract interpretation
- Cousot, Cousot
- 1992
(Show Context)
Citation Context ...s in relational (or natural) semantics [15]. The user introduces new sets, predicates, relations defined by their introduction rule. We remark that, in practice and probably because it is clearer, in =-=[12, 4]-=-, the lazy relations are not given by their elimination rules, but by their introduction rules. When one wants to prove a result, or builds a noncanonical function, one first gives to it a name and a ... |

45 | Mechanizing coinduction and corecursion in higher-order logic
- Paulson
- 1997
(Show Context)
Citation Context ... lazy) objects. Since one important application we have in mind is the mechanisation of reasoning about programs and processes, we analyse in our formalism some concrete examples from the litterature =-=[19, 15]-=-. One possible way of reading this paper is to read the proof principle described in the subsection 1.2.4, and then to look at the examples in the second section. The first section contains motivation... |

41 | Structured type theory
- Coquand, Coquand
- 1999
(Show Context)
Citation Context ...iefly some basic notions of type theory of well-founded objects, that will be important for the extension to infinite objects. The books [17, 13] contain more detailed explanations, and the reference =-=[3]-=- describes the addition of case expressions and pattern-matching. 1.1.1 Semantics A(n inductive) set A is defined by its constructors. A closed term of type A can be thought of as a well-founded tree,... |

20 |
Verifiable Programming
- Dahl
- 1992
(Show Context)
Citation Context ...ition may be recursive, but, using the semantics of a term as a well-founded tree, we can ensure that the recursive calls are well-founded and justify in such a way this recursivity. We notice, as in =-=[6]-=-, that there is a simple syntactical check that ensures this: there exists a lexicographic ordering of the arguments of f; such that all recursive calls are well-founded for the lexicographic extensio... |

20 | Programming with broadcasts - Prasad - 1993 |

11 | Innite objects in type theory - Mendler, Panangaden, et al. - 1986 |

10 | An intensional characterization of the largest bisimulation - Hallnas - 1987 |

6 | Non-Well-Founded Sets,” CSLI - Aczel - 1988 |

5 |
A semantics for locally bottom-avoiding choice
- Hughes, Moran
- 1992
(Show Context)
Citation Context ...form a partition of A: In the present intuitionistic framework, one cannot expect in general to have a proof of (x : A)[Acc(x) + Inf(x)]: In particular, we cannot derive in our system some results of =-=[12]-=-, which establish the equivalence of two notions of divergence using the fact that an element either diverges or converges. It seems quite interesting to investigate this problem more in detail from a... |

4 | Logical Reflection and Formalism - Lorenzen - 1958 |

2 |
Realisability Interpretation of Coinductive Definitions and Program Synthesis with Streams
- Tatsuta
- 1992
(Show Context)
Citation Context ...)) when f(x) is of the form inr(y): Hence, we have a representation of co-recursion over the lazy set C: This indicates how one can develop a realisability semantics of co-induction with streams (see =-=[23]-=-), in such a way that an element of a coinductive type is interpreted by a productive element. 2.5 Fairness We introduce an inductively defined predicate Event 1 on C; such that Event 1 (x; y) means t... |

2 | Constructive Definition of Certain Sets of Numbers - Lorenzen, Myhill |

1 |
Non-Well-Founded Set Theory CSLI
- Aczel
- 1988
(Show Context)
Citation Context ...f soundness of a type inference system analysed in [15]. This corresponds to using the present version of type theory with possibly infinite objects instead of Peter Aczel non-well-founded set theory =-=[2]-=-. We suppose given a set of constants and we introduce EXP : Set constexp : (CONST)EXP lambda : (IDENT;EXP)EXP app : (EXP;EXP)EXP var : (IDENT)EXP fix : (IDENT;IDENT;EXP)EXP VAL : Set ENV : Set constv... |

1 |
Inductive Families. To appear in Formal Aspects of Computing
- Dybjer
- 1993
(Show Context)
Citation Context ...infinite), imposes strong restriction on the type of the constructors. Thus, we cannot have a set X with a constructor of type ((X)X)X or of type (((X)N)N)X: However, a condition of strict positivity =-=[8]-=- on the type of the constructors is enough to ensure that we can think of elements as trees built out of constructors. The Ackerman function A : (N)(N)N defined by the equation A(0; n) = s(n); A(s(m);... |

1 |
An Intensional Characterization of the Largest Bissimulation
- Hallnas
- 1987
(Show Context)
Citation Context ...tions with the work of C. Raffalli [20], which presents ideas that seem similar in the logical system AF2 should be precised. One main point of this paper, which goes back to the work of Lars Hallnas =-=[10]-=-, is that the infinitary notions that seem necessary in dealing with infinite objects, typically the use of greatest fixed-point or infinite ordinals, can be avoided altogether by explicit considerati... |

1 |
On the syntax of infinite objects: an extension of Martin-Lof's theory of expressions
- Hallnas
- 1989
(Show Context)
Citation Context ...essarilly well-founded objects, that relies directly on the semantics of an object as a not necessarilly well-founded tree built out of constructors. 2 This definition can be extracted from the paper =-=[11], where th-=-e notion of "convergence" corresponds to our notion of productivity. 1.2.3 A key example At this point, the basic difficulty is to find a way of defining functions that ensures that any inst... |

1 |
Fixed points and type systems (Abstract) proceeding of the third
- Raffalli
- 1992
(Show Context)
Citation Context ...ossibility of declaring and proving local lemmas (that can be themselves recursively defined) corresponds to the addition of a local let construct. Conclusion Connections with the work of C. Raffalli =-=[20]-=-, which presents ideas that seem similar in the logical system AF2 should be precised. One main point of this paper, which goes back to the work of Lars Hallnas [10], is that the infinitary notions th... |

1 | On Backtracking and Greatest Fixpoints Formal Description of Programming Concepts - Roever - 1978 |

1 |
On the productivity of recursive list functions
- Sijtsma
- 1989
(Show Context)
Citation Context ...y seem too restrictive, especially in the definition of functions over infinite objects. Several programs on streams, even if they preserve productivity, do not obey in general this guarded condition =-=[22]-=-. But we think that the situation is similar to the one of well-founded objects, where the condition on structurally smaller recursive calls does not capture all usual definitions of programs defined ... |