On the strength of proof-irrelevant type theories

Werner, Benjamin

doi:10.2168/lmcs-4(3:13)2008

Cited by 16 publications

(7 citation statements)

References 26 publications

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“…This could possibly be remedied by changes to the CIC that affect the convertibility relation for strong sums (Werner 2006). (Hence, the first projection is injective.)…”

Section: Justificationmentioning

confidence: 99%

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An induction principle for nested datatypes in intensional type theory

Matthes¹

2009

J. Funct. Prog.

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Nested datatypes are families of datatypes that are indexed over all types such that the constructors may relate different family members (unlike the homogeneous lists). Moreover, the argument types of the constructors refer to indices given by expressions in which the family name may occur. Especially in this case of true nesting, termination of functions that traverse these data structures is far from being obvious. A joint paper with A. Abel and T. Uustalu (Theor. Comput. Sci., 333 (1-2), 2005, pp. 3-66) proposed iteration schemes that guarantee termination not by structural requirements but just by polymorphic typing. They are generic in the sense that no specific syntactic form of the underlying datatype "functor" is required. However, there was no induction principle for the verification of the programs thus obtained, although they are well known in the usual model of initial algebras on endofunctor categories. The new contribution is a representation of nested datatypes in intensional type theory (more specifically, in the calculus of inductive constructions) that is still generic and covers true nesting, guarantees termination of all expressible programs, and has an induction principle that allows to prove functoriality of monotonicity witnesses (maps for nested datatypes) and naturality properties of iteratively defined polymorphic functions.

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“…This could possibly be remedied by changes to the CIC that affect the convertibility relation for strong sums (Werner 2006). (Hence, the first projection is injective.)…”

Section: Justificationmentioning

confidence: 99%

“…(Hence, the first projection is injective.) This could possibly be remedied by changes to the CIC that affect the convertibility relation for strong sums (Werner 2006). • We need that any two proofs of naturality for the same parameters are equal.…”

Section: Justificationmentioning

confidence: 99%

An induction principle for nested datatypes in intensional type theory

Matthes¹

2009

J. Funct. Prog.

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“…The benefit of proof irrelevance is that now for any p, q : Lt m n, lookup A n m p v = lookup A n m q v : A; for a more detailed discussion consult Werner [56].…”

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confidence: 99%

A Modular Type-checking algorithm for Type Theory with Singleton Types and Proof Irrelevance

Abel¹,

Coquand²,

Pagano³

2011

Logical Methods in Computer Science

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Abstract. We define a logical framework with singleton types and one universe of small types. We give the semantics using a PER model; it is used for constructing a normalisation-by-evaluation algorithm. We prove completeness and soundness of the algorithm; and get as a corollary the injectivity of type constructors. Then we give the definition of a correct and complete type-checking algorithm for terms in normal form. We extend the results to proof-irrelevant propositions.

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“…Thanks to that axiom, logical arguments of programs can always be made implicit (as for False_elim and eq_elim) and thus never compared. Note that this was an important motivation for considering a proof irrelevant Calculus of Constructions [18].…”

Section: Implementation and Inductive Typesmentioning

confidence: 99%

“…Werner [18] introduces a variant of the Calculus of Constructions where objects of the Prop kind can be erased. His idea is very similar to ours since he modifies conversion so that proofs of a given proposition are always convertible.…”

Section: Related Workmentioning

confidence: 99%

The Implicit Calculus of Constructions as a Programming Language with Dependent Types

Barras

Bernardo

Lecture Notes in Computer Science

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Abstract. In this paper, we show how Miquel's Implicit Calculus of Constructions (ICC) can be used as a programming language featuring dependent types. Since this system has an undecidable type-checking, we introduce a more verbose variant, called ICC * which fixes this issue. Datatypes and program specifications are enriched with logical assertions (such as preconditions, postconditions, invariants) and programs are decorated with proofs of those assertions. The point of using ICC * rather than the Calculus of Constructions (the core formalism of the Coq proof assistant) is that all of the static information (types and proof objects) is transparent, in the sense that it does not affect the computational behavior. This is concretized by a built-in extraction procedure that removes this static information. We also illustrate the main features of ICC * on classical examples of dependently typed programs.

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On the strength of proof-irrelevant type theories

Cited by 16 publications

References 26 publications

An induction principle for nested datatypes in intensional type theory

An induction principle for nested datatypes in intensional type theory

A Modular Type-checking algorithm for Type Theory with Singleton Types and Proof Irrelevance

The Implicit Calculus of Constructions as a Programming Language with Dependent Types

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