## A Modal Lambda Calculus with Iteration and Case Constructs (1997)

Venue: | TYPES FOR PROOFS AND PROGRAMS: INTERNATIONAL WORKSHOP, TYPES ’98, KLOSTER IRSEE |

Citations: | 8 - 1 self |

### BibTeX

@INPROCEEDINGS{Leleu97amodal,

author = {Pierre Leleu},

title = {A Modal Lambda Calculus with Iteration and Case Constructs},

booktitle = {TYPES FOR PROOFS AND PROGRAMS: INTERNATIONAL WORKSHOP, TYPES ’98, KLOSTER IRSEE},

year = {1997},

pages = {47--61},

publisher = {SpringerVerlag LNCS}

}

### OpenURL

### Abstract

An extension of the simply-typed lambda-calculus allowing iteration and case reasoning over terms defined by means of higher order abstract syntax has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operator ` ' of modal logic IS4. Here we give a new presentation of their system, with reduction rules, instead of evaluation judgments, that compute the canonical forms of terms. Our presentation is based on a modal lambda-calculus that is better from the user's point of view, is more concise and we do not impose a particular strategy of reduction during the computation. Our system enjoys the decidability of typability, soundness of typed reduction with respect to typing rules, the Church-Rosser and strong normalization properties. Finally it is a conservative extension of the simply-typed lambda-calculus.

### Citations

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Citation Context ...n proofs `# la Tait' and is inspired by [Wer94] (for the inductive part) and [Gha96] (for the j-expansion part). 3.3.1 Reducibility Candidates First we give a denition of the reducibility candidates (=-=[GLT89]-=-) adapted to our setting. Let us callsthe set of our terms, dened in Section 2.2. Denition 3.5 (Reducibility Candidates) Given a type A, the candidates of reducibility CRA are sets of pairs (\Delta; M... |

311 |
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Citation Context ... : SYNTAXE ABSTRAITE D'ORDRE SUPERIEUR, INDUCTION, RECURSION, LOGIQUE MODALE, THEORIES TYPEES A Modal Lambda Calculus with Iteration and Case Constructs 3 1 Introduction Higher order abstract syntax (=-=[PE88]-=-) is a representation technique which proves to be useful when modelizing in a logical framework a language which involves bindings of variables. By borrowing the variables of the meta level to repres... |

271 | Programming in Martin–Löf ’s Type Theory. An introduction - Nordström, Petersson, et al. - 1990 |

196 | A modal analysis of staged computation
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(Show Context)
Citation Context ...as them to mix higher order abstract syntax and induction but our typing and reduction rules are quite dioeerent. Indeed there are several presentations of modal lambda calculus IS4 ([BdP96], [PW95], =-=[DP96]-=-). We have chosen the variant by Frank Pfenning and Hao-Chi Wong ([PW95]), which has context stacks instead of simple contexts. This peculiarity creates some diOEculties in the metatheoretical study b... |

125 | Primitive Recursion for Higherorder Abstract syntax
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Citation Context ...hey are naturally derived from the behaviour of the Case and It operators with respect to reduction (sections 2.5, 2.6). Although expressed a bit dioeerently, our typing rules are similar to those in =-=[DPS96], exc-=-ept three notable dioeerences: # As noticed before, the modal core is dioeerent. We have context stacks instead of two contexts and our modal rules ("), (#) and (Pop) are expressed in a very simp... |

95 |
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Citation Context ... Calculus with Iteration and Case Constructs 41 Another interesting direction of research consists in replacing our recursive operators by operators for pattern-matching such as those used in the ALF =-=[MN94]-=- system, implementing Martin Lof's Type Theory [NPS90]. Some hints for a concrete syntax for that extension have been given in [DPS97]. F. Pfenning and C. Sch#rmann are currently working on the deniti... |

92 | Reasoning with higher-order abstract syntax in a logical framework
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Citation Context ...over, we have written reduction rules instead of evaluation inference rules. Finally, our metatheoretic proofs are much more compact and easier to read. Raymond McDowell and Dale Miller have proposed =-=[MM97]-=- a meta-logic to reason about object logics coded using higher order abstract syntax. Their approach is quite dioeerent from ours, less ambitious in a sense. They do not give a typing system, supporti... |

37 | The Virtues of Eta-expansion
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Citation Context ...pes in the types of the recursive constructors, on the other hand to the use of j-expansion. To solve the problems due to j-expansion, we benet from previous work done for the simply-typed -calculus (=-=[JG95]-=-) and for system F ([Gha96]). In the second section of the paper, we introduce our version of the modal inductive system, its syntax, its typing and reduction rules. Then in the third section, we prov... |

26 | A framework for de ning logics - Harper, Honsell, et al. - 1992 |

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Citation Context ...-term x:(x x). Now, these constructors do not dene an inductive type in usual inductive type theories like the Inductive Calculus of Constructions ([Wer94]) or the Extended Calculus of Constructions (=-=[Luo94]-=-) because of the leftmost occurence of L in the type of constructor lam. If we allowed this kind of inductive denition, we would be confronted with two serious problems. First, we would lose the one-t... |

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Citation Context ...en thoroughly studied (see [CK93], [Aka93], [JG95]). Adopting it forces us to restrict the reduction rules in some way if we still want Strong Normalization (these restrictions were rst introduced by =-=[Min79]-=-). Thus the reduction we will consider will not be a congruence (more precisely it will not be compatible with the application) and this will induce slight changes in the usual schemes of the proofs o... |

14 |
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Citation Context ...s of our system. First, we brieAEy recall our motivation. 2.1 Higher-Order Abstract Syntax The mechanics of higher order abstract syntax (HOAS) has already been exposed in many places, for example in =-=[HHP93]-=-. Let us introduce here a simple example of representation using HOAS, that will prove useful later when we illustrate the mechanism of the reduction rules. Let us assume that we work in the simply-ty... |

14 |
Programming in Martin-Lf's Type Theory: An Introduction
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Citation Context ...ell-known that a type dened by means of higher order abstract syntax cannot be dened as an inductive type in usual inductive type theories (like CCI [Wer94], [PM92], or Martin-L#f's Logical Framework =-=[NPS90]-=- for instance). In a rst step towards the resolution of this dilemma, Jo#lle Despeyroux, Frank Pfenning and Carsten Sch#rmann have presented ([DPS97]) an extension of the simply-typed -calculus with r... |

13 |
On a modal -calculus for S4
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Citation Context ...chanism as them to mix higher order abstract syntax and induction but our typing and reduction rules are quite dioeerent. Indeed there are several presentations of modal lambda calculus IS4 ([BdP96], =-=[PW95]-=-, [DP96]). We have chosen the variant by Frank Pfenning and Hao-Chi Wong ([PW95]), which has context stacks instead of simple contexts. This peculiarity creates some diOEculties in the metatheoretical... |

12 | Inductive de nitions in the system Coq - rules and properties - Paulin-Mohring - 1993 |

11 |
Inductive denitions in the system coq, rules and properties
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Citation Context ...n developing proofs. Unfortunately it is well-known that a type dened by means of higher order abstract syntax cannot be dened as an inductive type in usual inductive type theories (like CCI [Wer94], =-=[PM92]-=-, or Martin-L#f's Logical Framework [NPS90] for instance). In a rst step towards the resolution of this dilemma, Jo#lle Despeyroux, Frank Pfenning and Carsten Sch#rmann have presented ([DPS97]) an ext... |

9 |
private communication
- Hofmann
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(Show Context)
Citation Context ...2.5 Reduction rules for Case and It on a simple example Now, we turn to the semantics of our system. They are inspired by the reduction rules for Case and It that have been proposed by Martin Hofmann =-=[Hof96]-=- as a means to describe the evaluation mechanism of [DPS97]. These reduction rules are also the ones underlying the terms and induction principles presented in [DH94]. The arguments of Case and It con... |

8 | On theories with a combinatorial de nition of `equivalence - Newman - 1942 |

7 |
Higher-order syntax and induction in coq
- Despeyroux, Hirschowitz
(Show Context)
Citation Context ... been proposed by Martin Hofmann [Hof96] as a means to describe the evaluation mechanism of [DPS97]. These reduction rules are also the ones underlying the terms and induction principles presented in =-=[DH94]-=-. The arguments of Case and It constructs have dioeerent roles which are best illustrated by the reduction rules. First we show the reduction rules for Case and It in the simple setting of the example... |

7 | A logic for reasoning with higher-order abstract syntax - McDowell, Miller - 1997 |

6 | Eta Expansions in System F
- Ghani
- 1996
(Show Context)
Citation Context ...hat there is no innite sequence of reductions starting from a well-typed term. The proof follows the sketch of normalization proofs `# la Tait' and is inspired by [Wer94] (for the inductive part) and =-=[Gha96]-=- (for the j-expansion part). 3.3.1 Reducibility Candidates First we give a denition of the reducibility candidates ([GLT89]) adapted to our setting. Let us callsthe set of our terms, dened in Section ... |

5 |
On theories with a combinatorial denition of `equivalence
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- 1942
(Show Context)
Citation Context ...nt, we have proved local conAEuence (Lemma 3.4) and strong normalization (Theorem 3.32). The proof of conAEuence is a corollary of these two results (this fact is often called iNewman's Lemmaj, after =-=[New42]-=-). Theorem 3.33 (ConAEuence) If \Delta ` M ,!sN : A and \Delta ` M ,!sP : A then there is a term Q such that \Delta ` N ,!sQ : A and \Delta ` P ,!sQ : A. Proof If M j N or M j P , the result is trivia... |

3 |
A ConAEuent Reduction for the Extensional Typed -calculus
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- 1993
(Show Context)
Citation Context ...redex. It does not seem very natural since app, which is not a term in canonical form, does not represent any term at the object level. The j-expansion reduction rule has been thoroughly studied (see =-=[CK93]-=-, [Aka93], [JG95]). Adopting it forces us to restrict the reduction rules in some way if we still want Strong Normalization (these restrictions were rst introduced by [Min79]). Thus the reduction we w... |

2 |
Metatheoretic Results for a Modal -calculus
- Leleu
- 1998
(Show Context)
Citation Context ...e proofs of the metatheoretical results. The classic properties of subject reduction, conAEuence, strong normalization have already been established for a modal lambda-calculus IS4 without induction (=-=[Lel98]-=-). Here we extend these results to the recursive operators Case and It. We begin this section with the easiest result: soundness of reduction with respect to typing rules. Then, we state important lem... |

1 |
On Mints'Reduction for ccc-calculus
- Akama
- 1993
(Show Context)
Citation Context ...t does not seem very natural since app, which is not a term in canonical form, does not represent any term at the object level. The j-expansion reduction rule has been thoroughly studied (see [CK93], =-=[Aka93]-=-, [JG95]). Adopting it forces us to restrict the reduction rules in some way if we still want Strong Normalization (these restrictions were rst introduced by [Min79]). Thus the reduction we will consi... |

1 |
and Valeria de Paiva. Intuitionnistic necessity revisited
- Bierman
- 1996
(Show Context)
Citation Context ...e same mechanism as them to mix higher order abstract syntax and induction but our typing and reduction rules are quite dioeerent. Indeed there are several presentations of modal lambda calculus IS4 (=-=[BdP96]-=-, [PW95], [DP96]). We have chosen the variant by Frank Pfenning and Hao-Chi Wong ([PW95]), which has context stacks instead of simple contexts. This peculiarity creates some diOEculties in the metathe... |

1 | A Con uent Reduction for the Extensional Typed - calculus - Cosmo, Kesner - 1993 |