Inductive families

Dybjer, Peter

doi:10.1007/bf01211308

Cited by 138 publications

(98 citation statements)

References 19 publications

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“…Many dependently typed programming languages and proof assistants are based on variants of I ω or CC ω , often with the addition of inductive definitions (PaulinMohring 1993;Dybjer 1994). Such tools include Agda (Norell 2007 …”

Section: Definition 22 (I ω )mentioning

confidence: 99%

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Bernardy¹,

Jansson²,

Paterson³

2012

J. Funct. Prog.

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This is the unspecified version of the paper.This version of the publication may differ from the final published version. Permanent repository link: R O S S P A T E R S O NCity University, London, UK (e-mail:)ross@soi.city.ac.uk) AbstractReynolds' abstraction theorem (Reynolds, J. C. (1983) Types, abstraction and parametric polymorphism, Inf. Process. 83(1), 513-523) shows how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems (PTSs): for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic.

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Section: Definition 22 (I ω )mentioning

confidence: 99%

Proofs for free

Bernardy¹,

Jansson²,

Paterson³

2012

J. Funct. Prog.

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“…Many dependently-typed programming languages and proof assistants are based on variants of Iω or CCω, often with the addition of inductive definitions [Dybjer 1994;Paulin-Mohring 1993]. Such tools include Agda [Norell 2007 …”

Section: Definition 3 (Ccω)mentioning

confidence: 99%

Parametricity and dependent types

2010

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract Reynolds' abstraction theorem shows how a typing judgement in System F can be translated into a relational statement (in second order predicate logic) about inhabitants of the type. We obtain a similar result for a single lambda calculus (a pure type system), in which terms, types and their relations are expressed. Working within a single system dispenses with the need for an interpretation layer, allowing for an unusually simple presentation. While the unification puts some constraints on the type system (which we spell out), the result applies to many interesting cases, including dependently-typed ones. Permanent

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“…Notably, COQ lacks support for mutual inductive types where one indexes the other (e.g., a datatype simultaneously defined with an inductive predicate on that type). Our implementation adapts a standard encoding [8] of induction-recursion [17] outlined in our technical report [23] to support examples like the list in Section 3.2. Any use of this encoding somewhat complicates generated proof obligations and data structure designs.…”

Section: Methodsmentioning

confidence: 99%

Rely-guarantee references for refinement types over aliased mutable data

2013

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Reasoning about side effects and aliasing is the heart of verifying imperative programs. Unrestricted side effects through one reference can invalidate assumptions about an alias. We present a new type system approach to reasoning about safe assumptions in the presence of aliasing and side effects, unifying ideas from reference immutability type systems and rely-guarantee program logics. Our approach, rely-guarantee references, treats multiple references to shared objects similarly to multiple threads in rely-guarantee program logics. We propose statically associating rely and guarantee conditions with individual references to shared objects. Multiple aliases to a given object may coexist only if the guarantee condition of each alias implies the rely condition for all other aliases. We demonstrate that existing reference immutability type systems are special cases of rely-guarantee references.In addition to allowing precise control over state modification, rely-guarantee references allow types to depend on mutable data while still permitting flexible aliasing. Dependent types whose denotation is stable over the actions of the rely and guarantee conditions for a reference and its data will not be invalidated by any action through any alias. We demonstrate this with refinement (subset) types that may depend on mutable data. As a special case, we derive the first reference immutability type system with dependent types over immutable data.We show soundness for our approach and describe experience using rely-guarantee references in a dependently-typed monadic DSL in COQ.

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Inductive families

Cited by 138 publications

References 19 publications

Proofs for free

Proofs for free

Parametricity and dependent types

Rely-guarantee references for refinement types over aliased mutable data

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