## Equality Between Functionals in the Presence of Coproducts (1995)

Venue: | Information and Computation |

Citations: | 9 - 1 self |

### BibTeX

@INPROCEEDINGS{Dougherty95equalitybetween,

author = {Daniel J. Dougherty and Ramesh Subrahmanyam},

title = {Equality Between Functionals in the Presence of Coproducts},

booktitle = {Information and Computation},

year = {1995},

pages = {282--291}

}

### Years of Citing Articles

### OpenURL

### Abstract

We consider the lambda-calculus obtained from the simply-typed calculus by adding products, coproducts, and a terminal type. We prove the following theorem: The equations provable in this calculus are precisely those true in any set-theoretic model with an infinite base type. 1 Introduction The model theory of the simply-typed lambda calculus, ! , has been well developed in the last two decades. For the most part, techniques and results generalize readily to the calculus when product types are added. Indeed, a categorical treatment goes more smoothly in the presence of products. But very little is known about the model theory of the simplytyped lambda calculus with coproducts for two chief reasons. First, techniques in the model theory of ! often rely heavily on the strong syntactic properties of the calculus; many of these properties fail in the presence of coproducts. Second, the natural generalizations of several key theorems in the model theory of ! fail in the setting wi...

### Citations

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Citation Context ... approach, and techniques from the theory of (first-order) term rewriting play an important role in the development. A central role is played by the method of rewriting modulo an equivalence relation =-=[Hue80]-=-; more about this in Section 7. Modifications to the standard theory are required in the presence of abstraction and fi-conversion. Related work The syntactic properties of the -calculus with coproduc... |

286 |
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Citation Context ... is a family A of non-empty sets A indexed by the types, with binary operations playing the role of application-operators. A model is a combinatorially complete extensional applicative structure. See =-=[Bar84]-=- for a detailed treatment in the untyped case. Of course Set is a model in a natural way. The usual term-model construction yields an abstract completeness theorem: \Delta has a model iff \Delta is co... |

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(Show Context)
Citation Context ...such that \Gamma 6` M = N . Then there exists a consistent finite valuation \Gamma + extending \Gamma such that \Gamma + k\GammaM 6= N . Proof. Since =) is terminating, so its multiset extension =) m =-=[DM79]-=-. We will prove the theorem by Noetherian induction (well-founded induction over the converse of a terminating relation, [Coh81]); in this case over =) m applied to the pair (M;N ). 25 ffl If at least... |

141 |
Type systems for programming languages
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Citation Context ...rom the base type of T to the base type of Set ! (recall that Set ! refers to the function hierarchy over an infinite base set) then this function lifts uniquely to a total injective logical relation =-=[Mit90]-=- from T to Set. Since 7 logical relations are guaranteed to relate the meanings of terms, we conclude that equations true in Set ! are provable. This argument does not generalize easily to the setting... |

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P.J.: Introduction to Higher-Order Categorical Logic
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Citation Context ...ith coproduct (and/or weak coproduct) types have received a lot of attention recently. The systems typically studied have been variants of the equational theory of bi-cartesian-closed categories (see =-=[LS86]-=- for the relationship between equationally defined -calculi and cartesian-closed categories.). Both [DK93] and [Dou93] show that a theory axiomatizing weak coproducts and primitive recursive functiona... |

68 |
Algebra
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Citation Context ...aM 6= N . Proof. Since =) is terminating, so its multiset extension =) m [DM79]. We will prove the theorem by Noetherian induction (well-founded induction over the converse of a terminating relation, =-=[Coh81]-=-); in this case over =) m applied to the pair (M;N ). 25 ffl If at least one of M and N is \Gamma! \Gammaw-reducible, then \Gamma 6` M \Gamma = N \Gamma and the multiset fM \Gamma ; N \Gamma g is =) m... |

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Equality between functionals
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(Show Context)
Citation Context ... only if it is provable in the equational theory presented by the axioms ABC described above. This generalizes the corresponding result for thes! -calculus, obtained by Harvey Friedman in the seminal =-=[Fri75]-=-: it is proved there that equality between simply-typed lambda terms in the full function-type hierarchy over an infinite set is completely axiomatized by fi and j. There does not appear to be an equa... |

44 | Kripke-style models for typed lambda calculus - Mitchell, Moggi - 1991 |

8 |
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Citation Context ...ong normalization for a theory with true coproducts. Okada and Scott [OS91] have presented a similar result for bi-cccs with a weak natural numbers object. We should also mention the work of Cubri'c, =-=[Cub92]-=- who adapted Friedman's completeness theorem to show that there is a faithful ccc-functor from any free cartesian-closed category into the category of Sets. Organization The paper is organized as foll... |

7 |
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Citation Context ...have been variants of the equational theory of bi-cartesian-closed categories (see [LS86] for the relationship between equationally defined -calculi and cartesian-closed categories.). Both [DK93] and =-=[Dou93]-=- show that a theory axiomatizing weak coproducts and primitive recursive functionals at higher types is strongly normalizing and confluent; the latter paper additionally shows strong normalization for... |

3 |
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(Show Context)
Citation Context ...ly studied have been variants of the equational theory of bi-cartesian-closed categories (see [LS86] for the relationship between equationally defined -calculi and cartesian-closed categories.). Both =-=[DK93]-=- and [Dou93] show that a theory axiomatizing weak coproducts and primitive recursive functionals at higher types is strongly normalizing and confluent; the latter paper additionally shows strong norma... |

3 |
Rewriting theory for uniqueness conditions: coproducts. Talk presented
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- 1991
(Show Context)
Citation Context ... and primitive recursive functionals at higher types is strongly normalizing and confluent; the latter paper additionally shows strong normalization for a theory with true coproducts. Okada and Scott =-=[OS91]-=- have presented a similar result for bi-cccs with a weak natural numbers object. We should also mention the work of Cubri'c, [Cub92] who adapted Friedman's completeness theorem to show that there is a... |

1 |
Symbolic Logic, vol 47, 17--26
- Completeness, lambda-definability
- 1982
(Show Context)
Citation Context ...owing decidability for coproduct theories, such as ABC and the theory of bi-ccc's, is difficult. Statman's 1-section theorem Let C be a class of models of the simply-typed lambda calculuss! . Statman =-=[Sta82]-=- has shown that the equations valid in this class are completely axiomatized by fi and j iff the free algebra of binary trees can be fully and faithfully embedded in some countable direct product of m... |