Extensionality in the Calculus of Constructions

Oury, Nicolas

doi:10.1007/11541868_18

Cited by 20 publications

(24 citation statements)

References 2 publications

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“…We hope to follow Oury's proof method to show that ETT is a conservative extension of OTT, and we can certainly validate the key axioms underpinning his translation of extensional derivations to intensional terms [21]. His work gives us good reason to conjecture that OTT has the full reasoning power of the corresponding extensional type theory, and hence that we really have no need to suffer the negative computational consequences of the equality reflection rule in order to obtain its logical benefits.…”

Section: Conclusion and Further Workmentioning

confidence: 91%

“…Heterogeneous equality was employed more recently by Oury to simplify Hofmann's proof, showing that an extensional variant of the Calculus of Constructions is conservative over the intensional version with extensional axioms [21]. Our present work is a heterogeneous variant of the theory for which Altenkirch constructed a setoid model.…”

Section: A Brief History Of Equalitymentioning

confidence: 96%

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Observational equality, now!

Altenkirch

McBride

Swierstra

2007

Proceedings of the 2007 Workshop on Programming Languages Meets Program Verification

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This paper has something new and positive to say about propositional equality in programming and proof systems based on the Curry-Howard correspondence between propositions and types. We have found a way to present a propositional equality type• which is substitutive, allowing us to reason by replacing equal for equal in propositions; • which reflects the observable behaviour of values rather than their construction: in particular, we have extensionalityfunctions are equal if they take equal inputs to equal outputs; • which retains strong normalisation, decidable typechecking and canonicity-the property that closed normal forms inhabiting datatypes have canonical constructors; • which allows inductive data structures to be expressed in terms of a standard characterisation of well-founded trees; • which is presented syntactically-you can implement it directly, and we are doing so-this approach stands at the core of Epigram 2; • which you can play with now: we have simulated our system by a shallow embedding in Agda 2, shipping as part of the standard examples package for that system [20].Until now, it has always been necessary to sacrifice some of these aspects. The closest attempt in the literature is Altenkirch's construction of a setoid-model for a system with canonicity and extensionality on top of an intensional type theory with proof-irrelevant propositions [4]. Our new proposal simplifies Altenkirch's construction by adopting McBride's heterogeneous approach to equality [18].

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Section: Conclusion and Further Workmentioning

confidence: 91%

Section: A Brief History Of Equalitymentioning

confidence: 96%

Observational equality, now!

Altenkirch

McBride

Swierstra

2007

Proceedings of the 2007 Workshop on Programming Languages Meets Program Verification

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“…If this succeeds, one may finally also establish Leibniz equality. This heterogeneous equality has been introduced by McBride under the name "John Major Equality" [19] (see also [6,Section 8.2.7]), and there are even extensions of that idea that try to integrate extensional reasoning into intensional type theory [21].…”

Section: Methodsmentioning

confidence: 99%

A Datastructure for Iterated Powers

Matthes

2006

Lecture Notes in Computer Science

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Abstract. Bushes are considered as the first example of a truly nested datatype, i. e., a family of datatypes indexed over all types where a constructor argument not only calls this family with a changing index but even with an index that involves the family itself. For the time being, no induction principles for these datatypes are known. However, the author has introduced with Abel and Uustalu (TCS 333(1-2), pp. 2005) iteration schemes that guarantee to define only terminating functions on those datatypes. The article uses a generalization of Bushes to n-fold self-application and shows how to define elements of these types that have a number of data entries that is obtained by iterated raising to the power of n. Moreover, the data entries are just all the n-branching trees up to a certain height. The real question is how to extract this list of trees from that complicated data structure and to prove this extraction correct. Here, we use the "refined conventional iteration" from the cited article for the extraction and describe a verification that has been formally verified inside Coq with its predicative notion of set.

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“…Two steps in the direction of integrating decision procedures into CC are Stehr's Open Calculus of Constructions (OCC) [21] and Oury's Extensional Calculus of Constructions (ECC) [17]. Implemented in Maude, OCC allows for the use of an arbitrary equational theory in conversion.…”

Section: Introductionmentioning

confidence: 99%

From Formal Proofs to Mathematical Proofs: A Safe, Incremental Way for Building in First-order Decision Procedures

Blanqui

Jouannaud

Strub

Fifth Ifip International Conference on Theoretical Computer Science – TCS 2008

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Abstract. We investigate here a new version of the Calculus of Inductive Constructions (CIC) on which the proof assistant Coq is based: the Calculus of Congruent Inductive Constructions, which truly extends CIC by building in arbitrary first-order decision procedures: deduction is still in charge of the CIC kernel, while computation is outsourced to dedicated first-order decision procedures that can be taken from the shelves provided they deliver a proof certificate. The soundness of the whole system becomes an incremental property following from the soundness of the certificate checkers and that of the kernel. A detailed example shows that the resulting style of proofs becomes closer to that of the working mathematician.

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Extensionality in the Calculus of Constructions

Cited by 20 publications

References 2 publications

Observational equality, now!

Observational equality, now!

A Datastructure for Iterated Powers

From Formal Proofs to Mathematical Proofs: A Safe, Incremental Way for Building in First-order Decision Procedures

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