## On the Correspondence Between Proofs and λ-Terms (1995)

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Venue: | Cahiers Du Centre de Logique |

Citations: | 6 - 2 self |

### BibTeX

@INPROCEEDINGS{Gallier95onthe,

author = {Jean Gallier},

title = {On the Correspondence Between Proofs and λ-Terms},

booktitle = {Cahiers Du Centre de Logique},

year = {1995},

pages = {55--138}

}

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### Abstract

Abstract. The correspondence between natural deduction proofs and λ-terms is presented and discussed. A variant of the reducibility method is presented, and a general theorem for establishing properties of typed (first-order) λ-terms is proved. As a corollary, we obtain a simple proof of the Church-Rosser property, and of the strong normalization property, for the typed λ-calculus associated with the system of (intuitionistic) first-order natural deduction, including all the connectors

### Citations

462 |
The formulae-as-types notion of construction
- Howard
- 1980
(Show Context)
Citation Context ...the remarkably insightful observation that certain typed combinators can be viewed as representations of proofs (in a Hilbert system) of certain propositions. Building up on this observation, Howard (=-=[12]-=-, 1969) described a general correspondence between propositions and types, proofs in natural deduction and certain typed λ-terms, and proof normalization and β-reduction. This correspondence, usually ... |

272 | Investigations into logical deduction - Gentzen - 1969 |

248 |
Interprétation Fonctionnelle et Élimination des Coupures de l’Arithmétique d’Ordre Supérieur
- Girard
- 1972
(Show Context)
Citation Context ...lization steps correspond to reduction steps in a certain typed λ-calculus, one can translate properties of λ-terms in terms of properties of proofs. In fact, Girard did just that (Girard [8] (1971), =-=[9]-=- (1972)), but he proved a much stronger result, namely strong normalization for higher-order (intuitionistic) logic. A similar proof also appears in Stenlund [19]. Prawitz ([17], 1971) also uses a var... |

231 |
Une extension de l’interpretation de Gödel, à l’analyse, et son application l’élimination des coupures dans l’analyse et la théorie des types
- Girard
- 1971
(Show Context)
Citation Context ... proof normalization steps correspond to reduction steps in a certain typed λ-calculus, one can translate properties of λ-terms in terms of properties of proofs. In fact, Girard did just that (Girard =-=[8]-=- (1971), [9] (1972)), but he proved a much stronger result, namely strong normalization for higher-order (intuitionistic) logic. A similar proof also appears in Stenlund [19]. Prawitz ([17], 1971) als... |

221 |
Intensional interpretations of functionals of finite type I
- Tait
- 1967
(Show Context)
Citation Context ...l form is unique. A few years later, Prawitz ([17], 1971) showed that in fact, every reduction sequence terminates, a property also called strong normalization. Sometimes between 1965 and 1967, Tait (=-=[20]-=-) proved that β-reduction in the simply-typed λ-calculus is strongly normalizing. For this, he used a method usually known as reducibility or computability. The word computability already having a mea... |

173 | Introduction to Combinators and Calculus - Hindley, Seldin - 1986 |

148 |
Proofs and Types, volume 7 of Cambridge Tracts
- Girard, Lafont, et al.
- 1989
(Show Context)
Citation Context ...umed that the reader has a certain familiarity with logic and the lambda calculus. If the reader does not feel sufficiently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor =-=[7]-=- or Gallier [4] for background on logic, and Barendregt [1], Hindley and Seldin [11], or Krivine [14] for 4background on the lambda calculus. For an in-depth study of constructivism in mathematics, w... |

129 |
Ideas and results in proof theory
- Prawitz
- 1971
(Show Context)
Citation Context ...ery clearly what redundancies are in systems of natural deduction, and he proved that every proof can be reduced to a normal form. Furthermore, this normal form is unique. A few years later, Prawitz (=-=[17]-=-, 1971) showed that in fact, every reduction sequence terminates, a property also called strong normalization. Sometimes between 1965 and 1967, Tait ([20]) proved that β-reduction in the simply-typed ... |

120 |
Introduction to Combinators and λ-calculus
- Hindley, Seldin
- 1986
(Show Context)
Citation Context ...If the reader does not feel sufficiently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor [7] or Gallier [4] for background on logic, and Barendregt [1], Hindley and Seldin =-=[11]-=-, or Krivine [14] for 4background on the lambda calculus. For an in-depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [22]. 2 Natural Deduction, Simply-Typed λ... |

102 | Geometry of interaction i: Interpretation of system f - Girard - 1989 |

67 | A theory of types
- Martin-Lof
- 1971
(Show Context)
Citation Context ...itz (1971)] Reduction in λ→,×,+,⊥ (specified in Definition 3.3) is confluent. Equivalently, conversion in λ→,×,+,⊥ is Church-Rosser. A proof can be given by adapting the method of Tait and Martin-Löf =-=[15]-=- using a form of parallel reduction (see also Barendregt [1], Hindley and Seldin [11], or Stenlund [19]). We will give another proof in section 8. Theorem 3.5 [Strong normalization, Prawitz (1971)] Re... |

63 |
Natural Deduction, a Proof-theoretical Study
- Prawitz
- 1965
(Show Context)
Citation Context ...of can be reduced to some normal form using a specific strategy, but there may be more than one normal form, and certain normalization strategies may not terminate. About thirty years later, Prawitz (=-=[16]-=-, 1965) reconsidered the issue of proof normalization, but in the framework of natural deduction rather than the framework of sequent calculi. 1 Prawitz explained very clearly what redundancies are in... |

52 |
A Realizability Interpretation of the Theory of Species
- Tait
- 1975
(Show Context)
Citation Context ...n section 8. Theorem 3.5 [Strong normalization, Prawitz (1971)] Reduction in λ→,×,+,⊥ (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait’s reducibility method [20], =-=[21]-=-, as done in Girard [8] (1971), [9] (1972) (see also Gallier [5]). We will give another proof in section 8. 4 First-Order Quantifiers We extend the system N ⊃,∧,∨,⊥ i to deal with the quantifiers. for... |

36 |
The Lambda-Calculus. Volume 103
- Barendregt
- 1984
(Show Context)
Citation Context ...nd the lambda calculus. If the reader does not feel sufficiently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor [7] or Gallier [4] for background on logic, and Barendregt =-=[1]-=-, Hindley and Seldin [11], or Krivine [14] for 4background on the lambda calculus. For an in-depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [22]. 2 Natural ... |

36 | On Girard’s “Candidats de Réductibilite
- Gallier
- 1990
(Show Context)
Citation Context ...eduction in λ→,×,+,⊥ (as in Definition 3.3) is strongly normalizing. A proof can be given by adapting Tait’s reducibility method [20], [21], as done in Girard [8] (1971), [9] (1972) (see also Gallier =-=[5]-=-). We will give another proof in section 8. 4 First-Order Quantifiers We extend the system N ⊃,∧,∨,⊥ i to deal with the quantifiers. for intuitionistic firstDefinition 4.1 The axioms and inference rul... |

31 | Logical Relations and the Typed -Calculus - Statman - 1985 |

26 | On Girard's "candidats de reductibilit'es - Gallier - 1990 |

23 |
Logical relations and the typed λ-calculus
- Statman
- 1985
(Show Context)
Citation Context ...n also be used to prove confluence or other properties does not seem to be as well known. Statman showed that various properties of the simply-typed λ-calculus can be obtained using logical relations =-=[18]-=-, but John Mitchell seems to be one of the first who realized that reducibility can be used to prove more general properties than strong normalization. The general idea is that if a unary predicate P ... |

14 | Church-Rosser theorem for typed functional systems - Koletsos |

10 |
types et modèles. Etudes et recherches en informatique
- Lambda-calcul
- 1990
(Show Context)
Citation Context ...s not feel sufficiently comfortable with these topics, we suggest consulting Girard, Lafont, Taylor [7] or Gallier [4] for background on logic, and Barendregt [1], Hindley and Seldin [11], or Krivine =-=[14]-=- for 4background on the lambda calculus. For an in-depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [22]. 2 Natural Deduction, Simply-Typed λ-Calculus We firs... |

5 |
Constructive Logics. Part I: A Tutorial on
- Gallier
- 1993
(Show Context)
Citation Context ...that if the above propositions have a proof in normal form, this leads to a contradiction. Another even simpler method is to use cut-free Gentzen systems. The interested reader is referred to Gallier =-=[3]-=-. The typed λ-calculus λ →,×,+,⊥ corresponding to N ⊃,∧,∨,⊥ i is given in the following definition. Definition 3.2 The typed λ-calculus λ →,×,+,⊥ is defined by the following rules. Γ, x: A ⊲ x: A with... |