### Citations

387 | λµ-Calculus: An algorithmic interpretation of classical natural deduction - Parigot - 1992 |

112 |
Inductive types and type constraints in the second-order lambda calculus
- Mendler
- 1991
(Show Context)
Citation Context ...ding behaves correctly with the computation, via the Curry-Howard correspondence, has never been analyzed. This proof is much simpler than the existing ones 1 . It also presents a new result. Mendler =-=[11]-=- has shown that strong normalization is preserved if, on types, we allow some equations satisfying natural (and necessary) conditions. Mendler’s result concerned the implicative fragment of intuitioni... |

89 |
Proofs of strong normalization for second order classical natural deduction
- Parigot
- 1997
(Show Context)
Citation Context ...of the λµ-calculus into the λ-calculus We give here a translation of the simply typed λµ-calculus into the simply typed λcalculus. This translation is a simplified version of Parigot’s translation in =-=[15]-=-. His translation uses both a translation of types (by replacing each atomic formula A by ¬¬A) and a translation of terms. But it is known that, in the implicative fragment of propositional logic, it ... |

34 | R.: Short proofs of normalization for the simply-typed lambdacalculus, permutative conversions and Gödel’s T. Archive for Mathematical Logic 42 - Joachimski, Matthes - 2003 |

30 | Nour: A short proof of the strong normalization of the simply typed λµ-calculus, Schedae Informaticae - David, K - 2003 |

29 | Confluence en λµ-calcul - Py - 1998 |

26 | A short proof of the strong normalization of classical natural deduction with disjunction. The
- David, Nour
- 2003
(Show Context)
Citation Context ...stic logic but his proof contains an error as Matthes pointed out in [8]. Nakazawa and Tatsuta corrected de Groote’s proof in [12] by using the notion of augmentations. Syntactical proofs. We gave in =-=[4]-=- a direct and syntactical proof of strong normalization. The proof is based on a substitution lemma which stipulates that replacing in a strongly normalizable deduction an hypothesis by another strong... |

25 |
K.: A Semantical Proof of the Strong Normalization Theorem for full
- Nour, Saber
(Show Context)
Citation Context ...is lemma (as given in [4]) works, it is not complete and (as pointed out by Matthes in a private communication) it also contains some errors. Semantical proofs. K. Saber and the second author gave in =-=[13]-=- a semantical proof of this result by using the notion of saturated sets. This proof is a generalization of Parigot’s strong normalization result of the λµ-calculus with the types 1of Girard’s system... |

22 |
P.: Strong Normalization for Classical Natural Deduction with Disjunction
- Groote
(Show Context)
Citation Context ...ponds to a product and ∨ to a co-product, i.e. a case of) and it is thus useful to have them as primitive. In this paper, we study the typed λµ →∧∨-calculus. This calculus, introduced by de Groote in =-=[7]-=-, is an extension of Parigot’s λµ-calculus. It is the computational counterpart of classical natural deduction with →, ∧ and ∨. Three notions of conversions are necessary in order to have the sub-form... |

9 | Church-Rosser property of a simple reduction for full first order classical natural deduction, submitted - Andou |

8 |
Simple saturated sets for disjunction and second-order existential quantification
- Tatsuta
(Show Context)
Citation Context ...ormalization result of the λµ-calculus with the types 1of Girard’s system F by using reducibility candidates. This proof uses the technical lemma of [4] concerning commutative reductions. In [9] and =-=[17]-=-, R. Matthes and Tastuta give another semantical proofs by using a (more complex) concept of saturated sets. This paper presents a new proof of the strong normalization of the simply typed λµ →∧∨ -cal... |

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 =-=[10]-=- 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. Mendler’s res... |

6 |
Strong normalization of classical natural deduction with disjunctions
- Nakazawa, Tatsuta
(Show Context)
Citation Context ...anslation into the simply typed λ-calculus i.e. the implicative intuitionistic logic but his proof contains an error as Matthes pointed out in [8]. Nakazawa and Tatsuta corrected de Groote’s proof in =-=[12]-=- by using the notion of augmentations. Syntactical proofs. We gave in [4] a direct and syntactical proof of strong normalization. The proof is based on a substitution lemma which stipulates that repla... |

5 | Non-strictly positive fixed-points for classical natural deduction
- Matthes
(Show Context)
Citation Context ...strong normalization result of the λµ-calculus with the types 1of Girard’s system F by using reducibility candidates. This proof uses the technical lemma of [4] concerning commutative reductions. In =-=[9]-=- and [17], R. Matthes and Tastuta give another semantical proofs by using a (more complex) concept of saturated sets. This paper presents a new proof of the strong normalization of the simply typed λµ... |

4 | Stabilization—an alternative to double-negation translation for classical natural deduction
- Matthes
- 2006
(Show Context)
Citation Context ...normalization of the typed λµ →∧∨-calculus using a CPS-translation into the simply typed λ-calculus i.e. the implicative intuitionistic logic but his proof contains an error as Matthes pointed out in =-=[8]-=-. Nakazawa and Tatsuta corrected de Groote’s proof in [12] by using the notion of augmentations. Syntactical proofs. We gave in [4] a direct and syntactical proof of strong normalization. The proof is... |

1 | Normalization properties of symmetric logical calculi - Battyanyi - 2007 |

1 |
An arithmetical proof of the strong normalization for the lambda-calculus with recursive equations on types
- David, Nour
- 2007
(Show Context)
Citation Context ...ization of S µ ≈ Let ≈ be the congruence generated by a set F of types of T . If ≈ is good, then the system S≈ is strongly normalizTheorem 7.1 (Mendler) ing. Proof See [10] for the original proof and =-=[5]-=- for an arithmetical one. □ Lemma 7.1 If Γ ⊢ µ S ≈ M : A, then Γ⋄ ⊢Sc ≈ M ⋄ : A. Proof By induction on the typing Γ ⊢ µ S ≈ Theorem 7.2 M : A. □ If ≈ is good, then the system S µ ≈ is strongly normali... |

1 | Short proofs of strong normalization
- Wojdyga
(Show Context)
Citation Context ...ction between named and un-named terms is forgotten. In this calculus, µα is not necessarily followed by [β]. We also write (α M) instead of [α]M. 1 Recently, we have been aware of a paper by Wojdyga =-=[18]-=- who uses the same kind of translations but where all the atomic types are collapsed to ⊥. Our translation allows us to extend trivially Mendler’s result whereas the one of Wojdyga, of course, does no... |

1 | version 1 - hal-00385206 - 2009 |