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## A short proof of the Strong Normalization of Classical Natural Deduction with Disjunction (2003)

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Venue: | Journal of symbolic Logic |

Citations: | 26 - 16 self |

### Citations

387 |
λµ-Calculus: An algorithmic interpretation of classical natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...s can be seen as types for functional programming languages and correct programs can be extracted from proofs. For that reason various systems have been studied in the last decades (see, for example, =-=[2, 4, 5, 12, 14, 15, 17, 22]-=-) both for intuitionistic and classical logic. The connectives ∧ and ∨ have a functional counter-part (∧ corresponds to a product and ∨ to a co-product, i.e. a case of) and it is thus useful to have t... |

256 | Natural Deduction, a Proof-Theoretical Study. Almqvist and Wiksell - Prawitz - 1965 |

209 |
A new constructive logic: Classical logic
- Girard
- 1991
(Show Context)
Citation Context ...s can be seen as types for functional programming languages and correct programs can be extracted from proofs. For that reason various systems have been studied in the last decades (see, for example, =-=[2, 4, 5, 12, 14, 15, 17, 22]-=-) both for intuitionistic and classical logic. The connectives ∧ and ∨ have a functional counter-part (∧ corresponds to a product and ∨ to a co-product, i.e. a case of) and it is thus useful to have t... |

154 | Ideas and Results in Proof Theory. in - Prawitz - 1970 |

115 |
Rèduction correcte et optimales dans le lambda calcul
- Levy
- 1978
(Show Context)
Citation Context ...on of the one given by Matthes in [11]. After this paper had been written we were told by Curien and some others that this kind of technique was already present in van Daalen (see [27]) and Levy (see =-=[13]-=-). The same idea is used in [10] to give a short proof of the strong normalization of the simply typed λµ-calculus of [17]. Apart the fact that this proof is direct (i.e. uses no translation into an o... |

89 |
A symmetric lambda calculus for classical program extraction, Information and computation
- Barbanera, Berardi
- 1996
(Show Context)
Citation Context ...s can be seen as types for functional programming languages and correct programs can be extracted from proofs. For that reason various systems have been studied in the last decades (see, for example, =-=[2, 4, 5, 12, 14, 15, 17, 22]-=-) both for intuitionistic and classical logic. The connectives ∧ and ∨ have a functional counter-part (∧ corresponds to a product and ∨ to a co-product, i.e. a case of) and it is thus useful to have t... |

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

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
(Show Context)
Citation Context ...full classical logic to intuitionistic logic with → as the only connective, i.e. the simply typed λ-calculus. A very elegant and direct proof of the strong normalization of the full logic is given in =-=[11]-=- but only the intuitionistic case is given. We give here another proof of de Groote’s result. This proof is based on a proof of the strong normalization of the simply typed λ-calculus due to the first... |

30 |
Normalization without reducibility
- David
(Show Context)
Citation Context ... intuitionistic case is given. We give here another proof of de Groote’s result. This proof is based on a proof of the strong normalization of the simply typed λ-calculus due to the first author (see =-=[8]-=-) which, itself, is a simplification of the one given by Matthes in [11]. After this paper had been written we were told by Curien and some others that this kind of technique was already present in va... |

30 | Nour: A short proof of the strong normalization of the simply typed λµ-calculus, Schedae Informaticae
- David, K
- 2003
(Show Context)
Citation Context ...n [11]. After this paper had been written we were told by Curien and some others that this kind of technique was already present in van Daalen (see [27]) and Levy (see [13]). The same idea is used in =-=[10]-=- to give a short proof of the strong normalization of the simply typed λµ-calculus of [17]. Apart the fact that this proof is direct (i.e. uses no translation into an other system whose strong normali... |

27 |
Sørensen: The λ△ -calculus
- Rehof, H
- 1994
(Show Context)
Citation Context |

26 |
A Computational Analysis of Girard’s Translation and LC
- Murthy
- 1992
(Show Context)
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25 |
K.: A Semantical Proof of the Strong Normalization Theorem for full
- Nour, Saber
(Show Context)
Citation Context ...ument of the proof is an induction on the complexity of the cut-formula) we believe that our technique is quite general and may be used in other circumstances. A crucial lemma of our proof is used in =-=[16]-=- to give a semantical proof of the strong normalization. Finally [9] uses 1the same technique to give an elementary proof of the strong normalization of a typed λ-calculus with explicit substitutions... |

22 |
P.: Strong Normalization for Classical Natural Deduction with Disjunction
- Groote
(Show Context)
Citation Context ...∧ and ∨ have a functional counter-part (∧ corresponds to a product and ∨ to a co-product, i.e. a case of) and it is thus useful to have them as primitive. Until very recently (see the introduction of =-=[7]-=- for a brief history), no proof of the strong normalization of the cut-elimination procedure was known for full logic. In [7], de Groote gives such a proof by using a CPS-style transformation from ful... |

21 |
Finding computational content in classical proofs
- Constable, Murthy
- 1991
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Citation Context |

21 | On the semantics of classical disjunction - Pym, Ritter |

21 | On the intuitionistic force of classical search - Ritter, Pym, et al. |

18 |
Classical logic, storage operators and 2nd order lambda-calculus Ann. Pure and Applied Logic 68
- Krivine
- 1994
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14 | Proof-terms for classical and intuitionistic resolution - Ritter, Pym, et al. - 2000 |

9 |
Church-Rosser property of a simple reduction for full first order classical natural deduction, submitted
- Andou
(Show Context)
Citation Context ... A2 Γ1, x1 : A1 ⊢ N1 : C Γ2, x2 : A2 ⊢ N2 : C Γ, Γ1, Γ2 ⊢ (M [x1.N1, x2.N2]) : C Γ, a : ¬A ⊢ M : ⊥ Γ ⊢ µaM : A absi Figure 1. Γ, a : ¬A ⊢ M : A Γ ⊢ (a M) : ⊥ This coding is essentially the same as in =-=[1]-=- and [7]. We have adopted the notations of [1] which are also used by [11]: what is written πiM in [7] is written (M πi) here and δ(M, x1.N1, x2.N2) in [7] is written (M [x1.N1, x2.N2]) here. These no... |

6 | Strong Normalisation Proofs for Cut-Elimination in Gentzen’s Sequent Calculi
- Bittar
- 1999
(Show Context)
Citation Context ...en, theorem 3.1 needs the difficult theorem 3.2. For the same reason (the rule ∨e), the proof of theorem 4.1 needs a rather complex induction: we use a 5-tuple of integers. Note that E. Tahhan Bittar =-=[3]-=- has given a proof of the strong normalization of the sequent calculus by using essentially the same 5-tuple of integers. Remark It is also for simplicity of proofs that, in the totality of this secti... |

6 | On the strong normalization of natural deduction with permutation-conversions - Groote - 2005 |

3 | On the proof theory of the intermediate logic MH - Seldin - 1986 |

3 |
Daalen The language theory of Automath
- van
- 1977
(Show Context)
Citation Context ...f, is a simplification of the one given by Matthes in [11]. After this paper had been written we were told by Curien and some others that this kind of technique was already present in van Daalen (see =-=[27]-=-) and Levy (see [13]). The same idea is used in [10] to give a short proof of the strong normalization of the simply typed λµ-calculus of [17]. Apart the fact that this proof is direct (i.e. uses no t... |

2 | Strong Normalization of the Typed λws-Calculus
- David, Guillaume
- 2003
(Show Context)
Citation Context ...la) we believe that our technique is quite general and may be used in other circumstances. A crucial lemma of our proof is used in [16] to give a semantical proof of the strong normalization. Finally =-=[9]-=- uses 1the same technique to give an elementary proof of the strong normalization of a typed λ-calculus with explicit substitutions which, from the logical point of view, correspond to explicit cuts ... |

1 | Normalization theorems for full first-order classical natural deduction - Stalmarck - 1991 |