Pure Type System conversion is always typable

Siles, Vincent; Herbelin, Hugo

doi:10.1017/s0956796812000044

Cited by 14 publications

(19 citation statements)

References 15 publications

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“…In [2] it is shown that these two systems are equivalent for a special family of functional PTSs. In [15] this result is generalised to arbitrary PTSs.…”

Section: Introductionmentioning

confidence: 89%

“…In [14] the result is also proven for semi-full specifications. In [15] this equivalence is generalised to arbitrary specifications. The equivalence is formulated as follows.…”

Section: λ E S: Typed Judgemental Equalitymentioning

confidence: 99%

“…The proof is hard and is given in [15]. If one tries to prove this directly, then the direction from left to right is easy by induction over the derivation of the judgement, but for the other direction, the equivalence of equality is very hard, as is described in [2].…”

Section: λ E S: Typed Judgemental Equalitymentioning

confidence: 99%

“…The way [15] proved the Theorem was to define a new variant of PTS they called Pure Type System based on Annotated Typed Reduction or PTSatr. This system is a typed version of parallel beta reduction [16].…”

Section: λ E S: Typed Judgemental Equalitymentioning

confidence: 99%

“…In particular for functional specifications every term has a unique type. We state and prove the equivalence of the two frameworks: we show that every PTS λS is equivalent to its PTS f companion λ f S. The proof proceeds by showing that every PTSe judgement can be transformed into a PTS f judgement, where PTSe are Pure Type Systems with an explicit equality judgment, but no equality-proofs recorded in the terms (It has been shown by Siles and Herbelin [15] that PTSe and PTS are equivalent frameworks). Our equivalence proof involves a number of subtle technical steps, so we have completely formalised this proof in Coq.…”

Section: Introductionmentioning

confidence: 99%

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Explicit convertibility proofs in pure type systems

Doorn

Geuvers

Wiedijk

2013

Proceedings of the Eighth ACM SIGPLAN International Workshop on Logical Frameworks &Amp; Meta-Languages: Theory &Amp; Practice

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We define type theory with explicit conversions. When type checking a term in normal type theory, the system searches for convertibility paths between types. The results of these searches are not stored in the term, and need to be reconstructed every time again. In our system, this information is also represented in the term.The system we define has the property that the type derivation of a term has exactly the same structure as the term itself. This has the consequence that there exists a natural LF encoding of such a system in which the encoded type is a dependent parameter of the type of the encoded term.For every Pure Type System we define a system in our style. We show that such a system is always equivalent to the normal system without explicit conversions (even for non-functional systems), in the sense that the typability relation can be lifted. This proof has been fully formalised in the Coq system, building on a formalisation by Vincent Siles.In our system, explicit conversions are not allowed to be removed when checking for convertibility. This means that all terms in convertibility proofs are well typed, even in the sense of our system.

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“…In [2] it is shown that these two systems are equivalent for a special family of functional PTSs. In [15] this result is generalised to arbitrary PTSs.…”

Section: Introductionmentioning

confidence: 89%

“…In [14] the result is also proven for semi-full specifications. In [15] this equivalence is generalised to arbitrary specifications. The equivalence is formulated as follows.…”

Section: λ E S: Typed Judgemental Equalitymentioning

confidence: 99%

Section: λ E S: Typed Judgemental Equalitymentioning

confidence: 99%

Section: λ E S: Typed Judgemental Equalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Explicit convertibility proofs in pure type systems

Doorn

Geuvers

Wiedijk

2013

Proceedings of the Eighth ACM SIGPLAN International Workshop on Logical Frameworks &Amp; Meta-Languages: Theory &Amp; Practice

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Verifiable Certificates for Predicate Subtyping

Gilbert¹

2019

Programming Languages and Systems

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Adding predicate subtyping to higher-order logic yields a very expressive language in which type-checking is undecidable, making the definition of a system of verifiable certificates challenging. This work presents a solution to this issue with a minimal formalization of predicate subtyping, named PVS-Core, together with a system of verifiable certificates for PVS-Core, named PVS-Cert. PVS-Cert is based on the introduction of proof terms and explicit coercions. Its design is similar to that of PTSs with dependent pairs, with the exception of the definition of conversion, which is based on a specific notion of reduction → β * , corresponding to β-reduction combined with the erasure of coercions. The use of this reduction instead of the more standard reduction → βσ allows to establish a simple correspondence between PVS-Core and PVS-Cert. On the other hand, a type-checking algorithm is designed for PVS-Cert, built on proofs of type preservation of → βσ and strong normalization of both → βσ and → β * . Combining these results, PVS-Cert judgements are used as verifiable certificates for predicate subtyping. In addition, the reduction → βσ is used to define a cut elimination procedure for predicate subtyping. This definition provides a new tool to study the properties of predicate subtyping, as illustrated with a proof of consistency.

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From Rewrite Rules to Axioms in the $$\lambda \varPi $$-Calculus Modulo Theory

Blot,

Dowek,

Traversié

et al. 2024

Lecture Notes in Computer Science

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The $$\lambda \varPi $$ λ Π -calculus modulo theory is an extension of simply typed $$\lambda $$ λ -calculus with dependent types and user-defined rewrite rules. We show that it is possible to replace the rewrite rules of a theory of the $$\lambda \varPi $$ λ Π -calculus modulo theory by equational axioms, when this theory features the notions of proposition and proof, while maintaining the same expressiveness. To do so, we introduce in the target theory a heterogeneous equality, and we build a translation that replaces each use of the conversion rule by the insertion of a transport. At the end, the theory with rewrite rules is a conservative extension of the theory with axioms.

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Pure Type System conversion is always typable

Cited by 14 publications

References 15 publications

Explicit convertibility proofs in pure type systems

Explicit convertibility proofs in pure type systems

Verifiable Certificates for Predicate Subtyping

From Rewrite Rules to Axioms in the $$\lambda \varPi $$-Calculus Modulo Theory

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