## Proof Pearl: de Bruijn Terms Really Do Work

Citations: | 6 - 1 self |

### BibTeX

@MISC{Norrish_proofpearl:,

author = {Michael Norrish and René Vestergaard},

title = {Proof Pearl: de Bruijn Terms Really Do Work},

year = {}

}

### OpenURL

### Abstract

Abstract. Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λ-calculus (à la Huet, Nipkow and Shankar) is isomorphic to an α-quotiented λ-calculus. In order to establish the link, we introduce an “index-carrying ” abstraction mechanism over de Bruijn terms, and consider it alongside a simplified substitution mechanism. Relating the new notions to those of the α-quotiented and the proper de Bruijn formalisms draws on techniques from the theory of nominal sets. 1

### Citations

1119 |
The Lambda Calculus: Its Syntax and Semantics
- Barendregt
- 1984
(Show Context)
Citation Context ...roperties of de Bruijn terms. In particular, we can confirm that de Bruijn terms are a model for the λcalculus, and can do so in an elegant way. Bywayofcontrast,seeforexample Appendix C of Barendregt =-=[2]-=-, where the proofs are given as “implicit in de Bruijn [4]”; or the pioneering work by Shankar [15], where mechanised proofs of complex theorems establish a connection between unquotiented λ-terms and... |

303 | Lambda calculus notation with nameless dummies, a tool for automatic formal manipulation with application to the Church-Rosser theorem
- Bruijn
- 1972
(Show Context)
Citation Context ...o those of the α-quotiented and the proper de Bruijn formalisms draws on techniques from the theory of nominal sets. 1 Introduction Implementors and theorists alike have long been in de Bruijn’s debt =-=[4]-=-. Representing λ-terms without names is a relatively straightforward way of implementing binders internally in, e.g., theorem-proving systems such as Isabelle and HOL4. Complementing this, the same id... |

289 |
The Lambda Calculus – Its Syntax and Semantics, volume 103
- Barendregt
- 1984
(Show Context)
Citation Context ...ies of de Bruijn terms. In particular, we can confirm that de Bruijn terms are a model for the λcalculus, and can do so in an elegant way. By way of contrast, see for example Appendix C of Barendregt =-=[2]-=-, where the proofs are given as “implicit in de Bruijn [4]”; or the pioneering work by Shankar [15], where mechanised proofs of complex theorems establish a connection between unquotiented λ-terms and... |

209 |
Feys R. Combinatory logic
- Curry
- 1958
(Show Context)
Citation Context ... result in the space of relationships between ten different formalisations of the λ-calculus, following Figure 1. λC is Curry’s unquotiented λ-terms using variable names and fresh-naming substitution =-=[3]-=- (plus explicit α [7]). λHi α , due to Hindley [7], is a quotient construction saying that an equivalence class reduces to another if there are two syntactic terms that witness the reduction, in this ... |

81 | Nominal Techniques in Isabelle/HOL
- Urban
(Show Context)
Citation Context ... in its own right. In the near future, we hope to consider, for example, the twosorted calculus of McKinna and Pollack [10], and the weak HOAS style construction of the λ-calculus in Urban and Tasson =-=[16]-=-. Finally, we hope this work will go some way towards bringing about the day when reasoning about de Bruijn terms is never again necessary. Acknowledgements NICTA is funded by the Australian Governmen... |

53 | Five axioms of alpha conversion
- Gordon, Melham
- 1996
(Show Context)
Citation Context ...d from the syntactic to the quotient type, as studied in Homeier [8]. dB2 is the set of well-formed, two-sorted de Bruijn terms in Gordon [5]. λG α is the type derived from this, in Gordon and Melham =-=[6]-=-. λN α is a quotient of the first order syntax, with substitution defined directly on the quotiented level, as in Norrish [12]. dB is the type of pure de Bruijn terms (with indices representing free a... |

53 | Some lambda calculus and type theory formalized
- McKinna, Pollack
- 1999
(Show Context)
Citation Context ...e results in Figure 1, which we believe to be a valuable summary of existing work in its own right. In the near future, we hope to consider, for example, the twosorted calculus of McKinna and Pollack =-=[10]-=-, and the weak HOAS style construction of the λ-calculus in Urban and Tasson [16]. Finally, we hope this work will go some way towards bringing about the day when reasoning about de Bruijn terms is ne... |

45 | Alpha-Structural Recursion and Induction
- Pitts
(Show Context)
Citation Context ...tution is defined with reference to the substitution function in dB2 . In λN α , substitution is defined directly on Λα, using its “native” notion of primitive recursion, as in Norrish [12] and Pitts =-=[14]-=-. 2.2 de Bruijn’s Anonymous Abstraction, using Indices The type of de Bruijn terms is the free algebra generated by the recursion equation dB ∼ = N + dB ×dB + dB (2) We will use constructors dV, dAPP,... |

39 | More Church-Rosser proofs (in Isabelle/HOL
- Nipkow
(Show Context)
Citation Context ...uijn terms in their systems’ internals, but it does now seem as if it is possible to mechanise important results about syntax without needing to prove “painful” and “non-obvious” lemmas about indices =-=[11]-=-. The emerging formalisation technologies are appealing for a range of reasons, but here we focus on just two: i) it is now possible to check that multiple formalisations do actually correspond and ii... |

38 |
A Mechanical Proof of the Church-Rosser Theorem
- Shankar
- 1988
(Show Context)
Citation Context ...lculus, and can do so in an elegant way. By way of contrast, see for example Appendix C of Barendregt [2], where the proofs are given as “implicit in de Bruijn [4]”; or the pioneering work by Shankar =-=[15]-=-, where mechanised proofs of complex theorems establish a connection between unquotiented λ-terms and de Bruijn terms. In fact, there are a number of isomorphism results already extant in the literatu... |

31 |
A mechanisation of name-carrying syntax up to alpha-conversion
- Gordon
- 1994
(Show Context)
Citation Context ...rry’s set-up but with substitution rather than reduction lifted from the syntactic to the quotient type, as studied in Homeier [8]. dB2 is the set of well-formed, two-sorted de Bruijn terms in Gordon =-=[5]-=-. λG α is the type derived from this, in Gordon and Melham [6]. λN α is a quotient of the first order syntax, with substitution defined directly on the quotiented level, as in Norrish [12]. dB is the ... |

24 |
The Church-Rosser property and a result of combinatory logic. Dissertation
- Hindley
- 1964
(Show Context)
Citation Context ...of relationships between ten different formalisations of the λ-calculus, following Figure 1. λC is Curry’s unquotiented λ-terms using variable names and fresh-naming substitution [3] (plus explicit α =-=[7]-=-). λHi α , due to Hindley [7], is a quotient construction saying that an equivalence class reduces to another if there are two syntactic terms that witness the reduction, in this case in λC . λV is si... |

7 |
Residual theory in lambda-calculus: a formal development
- Huet
- 1994
(Show Context)
Citation Context ...titution defined directly on the quotiented level, as in Norrish [12]. dB is the type of pure de Bruijn terms (with indices representing free and bound variables), as mechanised in Shankar [15], Huet =-=[9]-=- and Nipkow [11]. Finally, dB ′ is an intermediate formalism that we define here. It uses constructors that are not injective, hence the oval node: conceptually, dB ′ is syntactic sugar for dB that lo... |

6 | A proof of the Church-Rosser theorem for the lambda calculus in higher order logic
- Homeier
(Show Context)
Citation Context ...rd [17]. λV α is the “Hindley-quotient” of λV . λHo α is a quotient of Curry’s set-up but with substitution rather than reduction lifted from the syntactic to the quotient type, as studied in Homeier =-=[8]-=-. dB2 is the set of well-formed, two-sorted de Bruijn terms in Gordon [5]. λG α is the type derived from this, in Gordon and Melham [6]. λN α is a quotient of the first order syntax, with substitution... |

6 | The Primitive Proof Theory of the λ-Calculus
- Vestergaard
- 2003
(Show Context)
Citation Context ...reduces to another if there are two syntactic terms that witness the reduction, in this case in λC . λV is similar to λC but uses renamingfree substitution (and explicit α), as studied in Vestergaard =-=[17]-=-. λV α is the “Hindley-quotient” of λV . λHo α is a quotient of Curry’s set-up but with substitution rather than reduction lifted from the syntactic to the quotient type, as studied in Homeier [8]. dB... |

3 |
Mechanising λ-calculus using a classical first order theory of terms with permutations
- Norrish
- 2006
(Show Context)
Citation Context ... figure are underpinned by an epimorphism (i.e., (3) plus surjectivity) by construction [7, 17]. In the remaining “raw-to-quotient” cases, property (3) plus surjectivity have been verified by Norrish =-=[12]-=- (λV -to-λG α , λV -to-λN α ) and Homeier [8] (λC-to-λHo α , implicitly). Vestergaard has mechanised the isomorphism at the base of the figure, between λV and λC (including their respective α-relation... |

1 |
Norrish and René Vestergaard. Structural preservation and reflection of diagrams
- Michael
- 2006
(Show Context)
Citation Context ...een established that the α-equivalences of λ V and λ C coincide, this means that the isomorphism of λ V and λ C implies that all the range types, i.e., dB and the λ X α , are pairwise isomorphic. (In =-=[13]-=-, we further show that a broad class of properties is also shared between the elements of the “raw” syntax and the range types.) As noted, the λ V -to-λ V α and λ C -to-λ Hi α α-quotients of the figur... |

1 |
Mechanized metatheory for the masses: the POPLMARK
- Aydemir, Bohannon, et al.
- 2005
(Show Context)
Citation Context ...on on the structure of t. The result first needs inc0 (incn π) =incn+1(inc0 π) to be shown. This latter is just the sort of tedious result that the POPLmark authors inveigh against so convincingly in =-=[1]-=-. (Note that it is also an equality on the representing lists of pairs, not just extensionally in terms of the permutations’ effect on strings.) ⊓⊔ This lemma leads to important results about sub, dLA... |

1 |
R.: Structural preservation and reflection of diagrams
- Norrish, Vestergaard
(Show Context)
Citation Context ...een established that the α-equivalences of λ V and λ C coincide, this means that the isomorphism of λ V and λ C implies that all the range types, i.e., dB and the λ X α , are pairwise isomorphic. (In =-=[13]-=-, we further show that a broad class of properties is also shared between the elements of the “raw” syntax and the range types.) As noted, the λV -to-λV α and λC-to-λHi α α-quotients of the figure are... |