## An arithmetical proof of the strong normalization for the λ-calculus with recursive equations on types (2007)

### BibTeX

@MISC{David07anarithmetical,

author = {René David and Karim Nour},

title = {An arithmetical proof of the strong normalization for the λ-calculus with recursive equations on types},

year = {2007}

}

### OpenURL

### Abstract

### Citations

1115 | The Lambda Calculus: Its Syntax and Semantics - Barendregt - 1981 |

847 | A formulation of the simple theory of types - Church - 1940 |

518 | Lambda calculi with types - Barendregt - 1992 |

471 |
The calculus of constructions
- Coquand, Huet
- 1988
(Show Context)
Citation Context ...F for intuitionistic logic and Parigot’s λµ-calculus for classical logic, many others, more and more powerful, type systems were introduced. For example, the calculus of constructions (Coquand & Huet =-=[7]-=-) and, more generally, the Pure Type Systems. It is also worth here to mention the system T T R of Parigot [33] where some types are defined as the least fixed point of an operator. This system was in... |

352 | Proofs and Types - Girard, Lafont, et al. - 1989 |

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...stems correspond to intuitionistic logic. Other systems correspond to classical logic. There, again, new constructors for terms are introduced. This is, for example, the case of Parigot’s λµ-calculus =-=[35]-=-. ⋆ Université de Savoie, Campus Scientifique, 73376 Le Bourget du Lac, France. Email : {david, nour}@univ-savoie.frSince the introduction of Girard system F for intuitionistic logic and Parigot’s λµ... |

213 | Intensional Interpretations of Functionals of Finite Type - Tait - 1967 |

183 | The Lambda Calculus
- Barendregt
- 1984
(Show Context)
Citation Context ...udy here other kinds of extension of the simply typed λ-calculus, i.e. systems where equations on types are allowed. These types are usually called recursive types. For more details see, for example, =-=[3]-=-. They are present in many languages and are intended to be able to be unfolded recursively to match other types. The subject reduction and the decidability of type assignment are preserved but the st... |

123 | Lambda-calcul types et modèles - Krivine - 1990 |

94 | Inductive Types and Type Constraints in the Second-Order Lambda Calculus - Mendler - 1991 |

75 | Proofs of strong normalization for second order classical natural deduction - Parigot - 1997 |

55 |
Programming with proofs
- Krivine, Parigot
- 1990
(Show Context)
Citation Context ... type systems were introduced. For example, the calculus of constructions (Coquand & Huet [7]) and, more generally, the Pure Type Systems. It is also worth here to mention the system T T R of Parigot =-=[33]-=- where some types are defined as the least fixed point of an operator. This system was introduced, not to represent more functions, but to represent more algorithms. For example, to be able to represe... |

44 |
Classical Logic, Storage Operators and Second-Order lambda-Calculus
- Krivine
- 1994
(Show Context)
Citation Context ... λ-calculus from which the strong normalization of the λµ-calculus can be deduced from the one of the λ-calculus. This translation, quite different from the CPS translations, has been used by Krivine =-=[26]-=- to code the λµ-calculus with second order types in the λC-calculus. With recursive equations, we do not have to add the constant aX since we can use the equation X ≈ ¬¬X. We give here, without proof,... |

34 | Proofs of Strong Normalisation for Second Order Classical Natural Deduction - Parigot - 1997 |

33 | Short proofs of normalization for the simplytyped lambda-calculus, permutative conversions and Gödel’s - Joachimski, Matthes - 2003 |

26 |
Normalization without reducibility
- David
- 2001
(Show Context)
Citation Context ...-calculus) with recursive equations on types satisfying Mendler’s condition. This proof is an extension of the one given by the first author for the simply typed λ-calculus. It can be found either in =-=[8]-=- (where it appears among many other things) or as a simple unpublished note on the web page of the first author [9]. Apparently, proof methods similar to that used here were independently invented by ... |

26 | A short proof of the strong normalization of the simply typed λµ-calculus
- David, Nour
- 2003
(Show Context)
Citation Context ...given by the first author for the simply typed λ-calculus. It can be found either in [8] (where it appears among many other things) or as a simple unpublished note on the web page of the first author =-=[9]-=-. Apparently, proof methods similar to that used here were independently invented by several authors (Levy, van Daalen, Valentini and others). The proof for the λµ-calculus is an extension of the ones... |

26 | Tiuryn, A New Characterization of Lambda Definability - Jung, A - 1993 |

23 | The Typed lambda-Calculus is not Elementary Recursive. FOCS - Statman - 1977 |

22 | Lambda definability and logical relations - Plotkin |

19 | On representation of data in lambda calculus - Parigot - 1989 |

18 |
On the Relation between the Lambda-Mu-Calculus and the Syntactic Theory
- Groote
- 1994
(Show Context)
Citation Context ...n will be called a µ-substitution whereas the (usual) substitution M[x := N] will be called a λ-substitution. Remarks – Note that we adopt here a more liberal syntax (also called de Groote’s calculus =-=[13]-=-) than in the original calculus since we do not ask that a µα is immediately followed by a (β M) (denoted [β]M in Parigot’s notation). – We also have changed Parigot’s typing notations. Instead of wri... |

17 | Arithmetical proofs of strong normalization results for symmetric λ-calculi
- David, Nour
(Show Context)
Citation Context ...ly, proof methods similar to that used here were independently invented by several authors (Levy, van Daalen, Valentini and others). The proof for the λµ-calculus is an extension of the ones given in =-=[11]-=- or [12]. The paper is organized as follows. In section 2 we define the simply typed λcalculus with recursive equations on types. To help the reader and show the main ideas, we first give, in section ... |

15 | A CPS-Translation of the λµ-Calculus
- Groote
- 1994
(Show Context)
Citation Context ...n of the λµ-calculus into the λ-calculus The strong normalization of a typed λµ-calculus can be deduced from the one of the corresponding typed λ-calculus by using CPS translations. See, for example, =-=[14]-=- for such a translation. There is another, somehow simpler, way of doing such a translation. Add, for each atomic type X, a constant aX of type ¬¬X → X. Using these constants, it is not difficult to g... |

12 | An Inverse of the Evaluation Functional for Typed lambda-calculus - Berger, Schwichtenberg - 1991 |

9 | Cartesian Closed Categories and Typed Lambda-calculi - Lambek - 1985 |

9 | λ-definable functionals and βη-conversion - Statman - 1983 |

8 | The Undecidability of λ-definability - Loader - 2001 |

7 | Recursive types and the subject reduction theorem - Statman - 1994 |

6 | The Undecidability of Unification - Huet - 1973 |

6 |
Recursive Types and Type
- Mendler
- 1987
(Show Context)
Citation Context ...ondition is to accept the equation X = F (where F is a type containing the variable X) only when the variable X is positive in F . For a set {Xi = Fi / i ∈ I} of mutually recursive equations, Mendler =-=[29]-=- has given a very simple and natural condition that ensures the strong normalization of the system. He also showed that the given condition is necessary to have the strong normalization. His proof is ... |

6 | A proof of strongly uniform termination for Gödel’s T by methods from local predicativity. Archive fot Mathematical Logic 36 - Weiermann - 1997 |

4 |
A new type assignment for lambda terms Archiv
- Coppo, Dezani
- 1978
(Show Context)
Citation Context ...gly normalizing since having two equations (for example X ≈ ¬¬X and X ≈ F ) is problematic. 6 Remarks and open questions 1. The proof of the strong normalization of the system D of intersection types =-=[6]-=- is exactly the same as the one for simple types. Is it possible to extend our proof to such systems with equations ? Note that the sort of constraints that must be given on the equations is not so cl... |

4 |
Quelques propriétés du typage des fonctions des entiers dans les entiers
- Doyen
- 1995
(Show Context)
Citation Context ... Schwichtenberg’s result and to determine the class of functions that are represented in such systems and, in particular, to see whether or not they allow to represent more functions. Note that Doyen =-=[15]-=- and Fortune & all [16] have given extensions of Schwichtenberg’s result. Here is an example of function that cannot be typed (of the good type) in the simply typed λ-calculus. Let Nat = (X → X) → (X ... |

4 |
Leivant and M.O’Donnell. Simple and Second order Types Structures
- Fortune, D
- 1983
(Show Context)
Citation Context ...t and to determine the class of functions that are represented in such systems and, in particular, to see whether or not they allow to represent more functions. Note that Doyen [15] and Fortune & all =-=[16]-=- have given extensions of Schwichtenberg’s result. Here is an example of function that cannot be typed (of the good type) in the simply typed λ-calculus. Let Nat = (X → X) → (X → X) and Bool = Y → (Y ... |

4 | Equality between functionals. Logic Coll’73 - Friedman - 1975 |

4 | Über eine bisher noch nicht benütztz Erweiterung des finiten Standpunkts - Gödel - 1958 |

4 | The undecidability of the 2nd order unification problem - Goldfarb - 1981 |

4 |
Un algorithme non typable dans le système F
- Krivine
- 1987
(Show Context)
Citation Context ... M λz1 (y M λz0)) where M = λx λy (y x) has been introduced by B.Maurey. It is easy to see that, for every n, m ∈ N, the term (Inf ˜m ñ) reduces to 1 if m ≤ n and to 0 otherwise. Krivine has shown in =-=[24]-=- that the type Nat → Nat → Bool cannot be given to Inf in system F but, by adding the equation X ≈ (X → Bool) → Bool, it becomes typable. Our example uses the same ideas. Let ≈ be the congruence gener... |

4 |
Functions definable in the simply-typed lambda calculus
- Schwichtenberg
- 1976
(Show Context)
Citation Context ...f the form λfλx(f (f ... (f x))), are codes for the integers. They are the only terms (in normal form) of type (o → o) → (o → o). Thus, functions on the integers can be represented but Schwichtenberg =-=[38]-=- has shown that very few functions are so. He showed that the extended polynomials (i.e. polynomials with positive coefficients together with a conditional operator) are the only functions that can be... |