## Embedding pure type systems in the lambda-Pi-calculus modulo (2007)

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Venue: | TLCA |

Citations: | 19 - 5 self |

### BibTeX

@INPROCEEDINGS{Cousineau07embeddingpure,

author = {Denis Cousineau and Gilles Dowek},

title = {Embedding pure type systems in the lambda-Pi-calculus modulo},

booktitle = { TLCA},

year = {2007},

pages = {102--117},

publisher = {Springer}

}

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### Abstract

The lambda-Pi-calculus allows to express proofs of minimal predicate logic. It can be extended, in a very simple way, by adding computation rules. This leads to the lambda-Pi-calculus modulo. We show in this paper that this simple extension is surprisingly expressive and, in particular, that all functional Pure Type Systems, such as the system F, or the Calculus of Constructions, can be embedded in it. And, moreover, that this embedding is conservative under termination hypothesis.

### Citations

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(Show Context)
Citation Context ...al predicate logic. To express proofs in a theory T , we can declare a variable for each axiom of T and consider proofs-terms containing such free variables, this is the idea of the Logical Framework =-=[11]-=-. However, when considering such open terms most benefits of termination, such as the existence of empty types, are lost. An alternative is to replace axioms by rewrite rules, moving from predicate lo... |

471 |
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Citation Context ...c type theory used as a logical framework [13]. A second way to extend the λΠ-calculus is to add typing rules, in particular to allow polymorphic typing. We get this way the Calculus of Constructions =-=[4]-=- that allows to express proofs of simple type theory and more generally the Pure Type Systems [2, 15, 1]. These two kinds of extensions of the λΠ-calculus are somewhat redundant. For instance, simple ... |

238 |
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Citation Context ...itself takes a natural number as an argument and returns a type. Thus, its type is nat ⇒ T ype, i.e. Πx : nat T ype. The terms T ype, nat ⇒ T ype, ... cannot have type T ype, because Girard’s paradox =-=[10]-=- could then be expressed in the system, thus we introduce a new symbol Kind to type such terms. To form terms, like Πx : nat T ype, whose type is Kind, we need a rule expressing that the symbol T ype ... |

141 |
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Citation Context ...λΠ-calculus modulo using an appropriate rewrite system. This rewrite system is inspired both by the expression of simple type theory in Deduction modulo and by the mechanisms of universes à la Tarski =-=[12]-=- of Intuitionistic type theory. In particular, this work extends Palmgren’s construction of an impredicative universe in type theory [14].s1 The λΠ-calculus The λΠ-calculus is a dependently typed lamb... |

98 | Martin-Löf Type Theory
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(Show Context)
Citation Context ...ulo [3]. We get this way the λΠ-calculus modulo. This idea of extending the dependently typed lambda-calculus with rewrite rules is also that of Intuitionistic type theory used as a logical framework =-=[13]-=-. A second way to extend the λΠ-calculus is to add typing rules, in particular to allow polymorphic typing. We get this way the Calculus of Constructions [4] that allows to express proofs of simple ty... |

76 | Theorem proving modulo
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(Show Context)
Citation Context ...ogorov interpretation and the Curry-de Bruijn-Howard correspondence. It can be extended in several ways to express proofs of some theory. A first solution is to express the theory in Deduction modulo =-=[7, 9]-=-, i.e. to orient the axioms as rewrite rules and to extend the λΠ-calculus to express proofs in Deduction modulo [3]. We get this way the λΠ-calculus modulo. This idea of extending the dependently typ... |

63 |
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(Show Context)
Citation Context ...typing rules, in particular to allow polymorphic typing. We get this way the Calculus of Constructions [4] that allows to express proofs of simple type theory and more generally the Pure Type Systems =-=[2, 15, 1]-=-. These two kinds of extensions of the λΠ-calculus are somewhat redundant. For instance, simple type theory can be expressed in deduction modulo [8], hence the proofs of this theory can be expressed i... |

45 | Proof Normalization Modulo
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(Show Context)
Citation Context ...ogorov interpretation and the Curry-de Bruijn-Howard correspondence. It can be extended in several ways to express proofs of some theory. A first solution is to express the theory in Deduction modulo =-=[7, 9]-=-, i.e. to orient the axioms as rewrite rules and to extend the λΠ-calculus to express proofs in Deduction modulo [3]. We get this way the λΠ-calculus modulo. This idea of extending the dependently typ... |

36 | Definitions by rewriting in the Calculus of Constructions
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(Show Context)
Citation Context ...fs of some theory. A first solution is to express the theory in Deduction modulo [7, 9], i.e. to orient the axioms as rewrite rules and to extend the λΠ-calculus to express proofs in Deduction modulo =-=[3]-=-. We get this way the λΠ-calculus modulo. This idea of extending the dependently typed lambda-calculus with rewrite rules is also that of Intuitionistic type theory used as a logical framework [13]. A... |

30 | HOL-λσ an intentional first-order expression of higher-order logic
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(Show Context)
Citation Context ...ory and more generally the Pure Type Systems [2, 15, 1]. These two kinds of extensions of the λΠ-calculus are somewhat redundant. For instance, simple type theory can be expressed in deduction modulo =-=[8]-=-, hence the proofs of this theory can be expressed in the λΠ-calculus modulo. But they can also be expressed in the Calculus of Constructions. This suggests to relate and compare these two ways to ext... |

25 |
Towards a mathematical analysis of the Coquand-Huet calculus of constructions and the other systems in Barendregt’s cube
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(Show Context)
Citation Context ...typing rules, in particular to allow polymorphic typing. We get this way the Calculus of Constructions [4] that allows to express proofs of simple type theory and more generally the Pure Type Systems =-=[2, 15, 1]-=-. These two kinds of extensions of the λΠ-calculus are somewhat redundant. For instance, simple type theory can be expressed in deduction modulo [8], hence the proofs of this theory can be expressed i... |

25 |
Een nadere bewijstheoretische analyse van GSTT’s
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(Show Context)
Citation Context ...typing rules, in particular to allow polymorphic typing. We get this way the Calculus of Constructions [4] that allows to express proofs of simple type theory and more generally the Pure Type Systems =-=[2, 15, 1]-=-. These two kinds of extensions of the λΠ-calculus are somewhat redundant. For instance, simple type theory can be expressed in deduction modulo [8], hence the proofs of this theory can be expressed i... |

8 |
Strong Normalization of the Dual Classical Sequent Calculus
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(Show Context)
Citation Context .../x)A −→�� (N ′ /x)A ′ and (N/x)B −→�� (N ′ /x)B ′ . Therefore, by (η2), (N/x)M = εs3 ( ˙ Π 〈s1,s2,s3〉 (N/x)A (N/x)B) −→�� Πy : (εs1 (N ′ /x)A ′ ) (εs2 ((N ′ /x)B ′ y)) = (N ′ /x)M ′ . Then, following =-=[6]-=-, we associate, to each term t of λΠP , a term t † , obtained by reducing in parallel all its βR-redices. Definition 10. Let t be a term of λΠP . We define, by induction on the structure of t, the ter... |

2 |
On universes in type theory, Twenty five years of constructive type theory
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(Show Context)
Citation Context ...eduction modulo and by the mechanisms of universes à la Tarski [12] of Intuitionistic type theory. In particular, this work extends Palmgren’s construction of an impredicative universe in type theory =-=[14]-=-.s1 The λΠ-calculus The λΠ-calculus is a dependently typed lambda-calculus that permits to construct types depending on terms, for instance a type array n, of arrays of size n, that depends on a term ... |

1 | Un plongement conservatif des Pure Type Systems dans le lambda Pi modulo, Master Parisien de Recherche en Informatique - Cousineau - 2006 |