Fixed Points of Type Constructors and Primitive Recursion

Abel, Andreas; Matthes, Ralph

doi:10.1007/978-3-540-30124-0_17

Cited by 14 publications

(22 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In these cases, our proposal is more generic but offers less comfort since it has neither advanced pattern matching nor guardedness checking. PList and Lam are strictly positive, but Bush and LamE are not even considered to be positive [19] (see [14] for a notion based on polarity that covers these examples). Since there was no system that combined the termination guarantee for recursion schemes on truly nested datatypes with a logic to reason about the defined functions, it seems only natural that examples like Bush and LamE did not receive more attention.…”

Section: Motivating Examplesmentioning

confidence: 99%

“…Neither could BtL XA . The way out is provided already by Mendler for inductive types [13] and has been generalized to nested datatypes in [14]: Express in the step term in addition that X is an approximation of µF , in the sense of an "injection" j : X ⊆ µF that is available in the body of the definition. So, we assume a constant (the minus sign indicates that it is a preliminary version)…”

Section: Bushesmentioning

confidence: 99%

“…While an extension of primitive Mendler-style recursion [13] to nested datatypes has been described earlier [14], we will present here an extension LNMRec of system LNMIt by an enriched Mendler-style recursor where the step term additionally has access to a map term for the unknown type transformation X that occurs there. By way of examples, its merits will be studied.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Recursion on Nested Datatypes in Dependent Type Theory

Matthes

Logic and Theory of Algorithms

Self Cite

View full text Add to dashboard Cite

Abstract. Nested datatypes are families of datatypes that are indexed over all types and where the datatype constructors relate different members of the family. This may be used to represent variable binding or to maintain certain invariants through typing. In dependent type theory, a major concern is the termination of all expressible programs, so that types that depend on object terms can still be type-checked mechanically. Therefore, we study iteration and recursion schemes that have this termination guarantee throughout. This is not based on syntactic criteria (recursive calls with "smaller" arguments) but just on types ("type-based termination"). An important concern are reasoning principles that are compatible with the ambient type theory, in our case induction principles. In previous work, the author has proposed an abstract description of nested datatypes together with a mapping operation (like map for lists) and an iterator on the term side and an induction principle on the logical side that could all be implemented within the Coq system (with impredicative Set that is just needed for the justification, not for the definition and the examples). For verification purposes, it is important to have naturality theorems for the obtained iterative functions. Although intensional type theory does not provide naturality in general, criteria for naturality could be established that are met in case studies on "bushes" and representations of lambda terms (also with explicit flattening). The new contribution is an extension of this abstract description to full primitive recursion and its illustration by way of examples that have been carried out in Coq. Unlike the iterative system, we do not yet have a justification within Coq.

show abstract

Section: Motivating Examplesmentioning

confidence: 99%

Section: Bushesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Recursion on Nested Datatypes in Dependent Type Theory

Matthes

Logic and Theory of Algorithms

Self Cite

View full text Add to dashboard Cite

show abstract

“…which is the equational form of the reduction behaviour established through Mendler recursion in earlier work [1].…”

mentioning

confidence: 91%

“…Its finitary version has been programmed iteratively in the subsequent journal version [3] and through primitive recursion in Mendler-style [1]. In all these cases, no attempt was made to verify properties other than termination.…”

Section: Introductionmentioning

confidence: 99%

Verification of the Redecoration Algorithm for Triangular Matrices

Matthes

Strecker

Lecture Notes in Computer Science

Self Cite

View full text Add to dashboard Cite

Abstract. Triangular matrices with a dedicated type for the diagonal elements can be profitably represented by a nested datatype, i. e., a heterogeneous family of inductive datatypes. These families are fully supported since the version 8.1 of the Coq theorem proving environment, released in 2007. Redecoration of triangular matrices has a succinct implementation in this representation, thus giving the challenge of proving it correct. This has been achieved within Coq, using also induction with measures. An axiomatic approach allowed a verification in the Isabelle theorem prover, giving insights about the differences of both systems.

show abstract

Polarized Subtyping for Sized Types

Abel¹

2006

Computer Science – Theory and Applications

Self Cite

View full text Add to dashboard Cite

We present an algorithm for deciding polarized higher-order subtyping without bounded quantification. Constructors are identified not only modulo β, but also η. We give a direct proof of completeness, without constructing a model or establishing a strong normalization theorem. Inductive and coinductive types are enriched with a notion of size and the subtyping calculus is extended to account for the arising inclusions between the sized types.

show abstract

Fixed Points of Type Constructors and Primitive Recursion

Cited by 14 publications

References 17 publications

Recursion on Nested Datatypes in Dependent Type Theory

Recursion on Nested Datatypes in Dependent Type Theory

Verification of the Redecoration Algorithm for Triangular Matrices

Polarized Subtyping for Sized Types

Contact Info

Product

Resources

About