## Monotone Inductive and Coinductive Constructors of Rank 2 (2001)

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Venue: | Proceedings of CSL 2001 |

Citations: | 10 - 4 self |

### BibTeX

@INPROCEEDINGS{Matthes01monotoneinductive,

author = {Ralph Matthes},

title = {Monotone Inductive and Coinductive Constructors of Rank 2},

booktitle = {Proceedings of CSL 2001},

year = {2001},

pages = {600--614},

publisher = {Springer Verlag}

}

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### Abstract

A generalization of positive inductive and coinductive types to monotone inductive and coinductive constructors of rank 1 and rank 2 is described. The motivation is taken from initial algebras and nal coalgebras in a functor category and the Curry-Howard-correspondence. The denition of the system as a -calculus requires an appropriate denition of monotonicity to overcome subtle problems, most notably to ensure that the (co-)inductive constructors introduced via monotonicity of the underlying constructor of rank 2 are also monotone as constructors of rank 1. The problem is solved, strong normalization shown, and the notion proven to be wide enough to cover even highly complex datatypes. 1

### Citations

252 |
Interprétation Fonctionnelle et Elimination des Coupures de l’Arithmétique d’Ordre Supérieur. Thèse d’Etat, Université de Paris VII
- Girard
- 1972
(Show Context)
Citation Context ... rank 2 while many closure properties of monotone constructors are shown in section 7. A few examples are considered in section 8. 2 System F ! Girard's system of higher-order parametric polymorphism =-=[7] is an e-=-xtension of system F where the set of types is extended to a simply-\typed" -calculus of constructors. The \types" of the constructors are called kinds and are built from the base kind (the... |

100 | Recursive types and type constraints in second-order lambda calculus - Mendler - 1987 |

81 | Monadic presentations of lambda-terms using generalized inductive types
- Altenkirch, Reus
- 1999
(Show Context)
Citation Context ...some type and produces a list of the same type. One might wonder whether there are interesting examples. It is by now well-known that de Bruijn notation indeed may be represented as a nested datatype =-=[3, 2]. How-=-ever, the question arose whether there are also examples taken from \the outside world". In [12], a nested datatype is shown which represents arbitrary square matrices over some type with element... |

72 | De Bruijn notation as a nested datatype
- Bird, Paterson
- 1999
(Show Context)
Citation Context ...some type and produces a list of the same type. One might wonder whether there are interesting examples. It is by now well-known that de Bruijn notation indeed may be represented as a nested datatype =-=[3, 2]. How-=-ever, the question arose whether there are also examples taken from \the outside world". In [12], a nested datatype is shown which represents arbitrary square matrices over some type with element... |

52 | Inductive and coinductive types with iteration and recursion
- Geuvers
- 1992
(Show Context)
Citation Context ... into the present system is carried out in all details (even with primitive recursion instead of iteration only). A more categorical motivation (still withsxed monotonicity witnesses) may be found in [5]: may be conceived as a weakly initial -algebra (for rank 2, this is carried out below). Now we also extend the system by monotone inductive constructors of rank 2, i. e., inductive constructor... |

36 | Generalised folds for nested datatypes
- Bird, Paterson
- 1999
(Show Context)
Citation Context ... to [3]'s generalized folds should be studied carefully. Can their behaviour be simulated in MICC 2 ? This is easy to see in the situation of [3] but not for the general approach to generalized folds =-=[4]-=-. Can onesnd data structures which need deeper nested inductive constructors than the example of square matrices in [12]? Is there a chance to get a similar clean view on constructors of higher rank t... |

36 |
Contracting proofs to programs
- Leivant
- 1990
(Show Context)
Citation Context ...tyle. For studies onsxed-points in -calculi this seems to be a reasonable choice, see also [15]. These rules obviously generalize the system of positive inductive types, e. g., system 2J of Leivant [9] in which is only allowed if only occurs \positively " in . In our language, all those are proven monotone inductive types. 4 Therefore, the iteration rule for 2J simply makes use of those... |

29 | From fast exponentiation to square matrices: an adventure in types
- Okasaki
- 1999
(Show Context)
Citation Context .... It is by now well-known that de Bruijn notation indeed may be represented as a nested datatype [3, 2]. However, the question arose whether there are also examples taken from \the outside world"=-=. In [12]-=-, a nested datatype is shown which represents arbitrary square matrices over some type with elements accessible in logarithmic time. It uses type constructors of rank 2. On the theoretical side, [1] s... |

28 | Beta-eta equality for coproducts
- Ghani
- 1995
(Show Context)
Citation Context ... 1 + 2 ! and ( 1 ! ) ( 2 ! ) are. However, in applications we would need to refer to full extensional equality for sum types (even beyond permutative conversions) which is still decidable [6] but dicult to treat since it leaves the realm of rewriting. Since our focus is on intensional equality as expressed by term rewriting, the slightly less elegant denition had to be made. First, we ex... |

26 |
Extentions of System F by Iteration and Primitive Recursion on Monotone Inductive Types
- Matthes
- 1998
(Show Context)
Citation Context ...ven number to the left of !. Positivity is replaced by proven monotonicity: a term inhabiting the type expressing that FF is monotone. For rank 1 (inductive types) this has been studied at length in [=-=1-=-0]|with the obvious notion of monotonicity expressible in system F. The main technical contribution is the denition of rank 2 monotonicity which will be justied at many places in this article. The nex... |

14 | Fixed Points and Extensionality in Typed Functional Programming Languages
- Howard
- 1992
(Show Context)
Citation Context ...oof. Thesrst 9 properties are quite common since they (except for number 6) form the basis of the denition of canonical monotonicity witnesses for positive inductive and coinductive types (see e. g. [=-=9, 8, -=-5, 10]). Rules 11 and 12 are proved bysrst establishing 8 (:(FF)smon) and 8 (:(FF)smon), using the assumptions. For this to work, one has to make use of theorem 1. The other rules are more or less ada... |

4 |
De Inductives en Theorie des Types d'Ordre Superieur. Habilitation a diriger les recherches
- Paulin-Mohring
- 1996
(Show Context)
Citation Context ...h that strong normalization of the system with iteration and coiteration can be inferred from that of the impredicative system, see e. g. [5, 15]. This also worked easily for monotone inductive types =-=[13, 10-=-]. Hence, it does not come as a surprise that MICC 2 embeds into F ! . Theorem 2. There is an embedding 0 of MICC 2 into F ! , i. e., for every constructorsX of kind in MICC 2 , there is a constructo... |

3 |
Representations of order function types as terminal coalgebras
- Altenkirch
- 2001
(Show Context)
Citation Context ... [12], a nested datatype is shown which represents arbitrary square matrices over some type with elements accessible in logarithmic time. It uses type constructors of rank 2. On the theoretical side, =-=[1-=-] studies even coinductive type constructors of rank 2 with considerable nesting. The present paper intends to give a thoroughly justied general framework for the description of terminating algorithms... |

1 | Category theory as coherently constructive lattice theory: An illustration
- Backhouse, Bijsterveld
- 1994
(Show Context)
Citation Context ...lows the latticetheoretic understanding of inductive denitions. But there are also the terms and the term rewrite rule of iteration. Viewing category theory as coherently constructive lattice theory [=-=3]-=-, we can justify the iteration rule by reference to the concept of a weakly initial algebra as has been done for inductive types in [5] mentioned earlier. We will do this only rather sloppily. The \ca... |