An induction principle and pigeonhole principles for K-finite sets

Blass, Andreas

doi:10.2307/2275881

Cited by 2 publications

(4 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In constructive mathematics there is still an amount of theory about Kuratowski finite sets left untouched in our development [14,25]. Some work has already been done to prove that the decidable Kuratowski-finite sets form a topos in Theorem 4.21, but for a full proof, function spaces and the subobject classifier have to be considered as well.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Finite sets in homotopy type theory

Frumin

Geuvers

Gondelman

et al. 2018

Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs

View full text Add to dashboard Cite

We study different formalizations of finite sets in homotopy type theory to obtain a general definition that exhibits both the computational facilities and the proof principles expected from finite sets. We use higher inductive types to define the type K(A) of łfinite sets over type Až à la Kuratowski without assuming that A has decidable equality. We show how to define basic functions and prove basic properties after which we give two applications of our definition.On the foundational side, we use K to define the notions of łKuratowski-finite typež and łKuratowski-finite subobjectž, which we contrast with established notions, e.g., Bishop-finite types and enumerated types. We argue that Kuratowski-finiteness is the most general and flexible one of those and we define the usual operations on finite types and subobjects.From the computational perspective, we show how to use K(A) for an abstract interface for well-known finite set implementations such as tree-and list-like data structures. This implies that a function defined on a concrete finite sets implementation can be obtained from a function defined on the abstract finite sets K(A) and that correctness properties are inherited. Hence, HoTT is the ideal setting for data refinement. Beside this, we define bounded quantification, which lifts a decidable property on A to one on K(A).

show abstract

Section: Discussionmentioning

confidence: 99%

“…In constructive mathematics finiteness has extensively been studied [13,44,50] and Kuratowski finite sets have been studied both in a classical [29] and constructive setting [14,25]. Other definitions include Bishop-finiteness [13], enumerated sets [44], streamless sets, and Noetherian sets [12, 20, 37ś39, 44].…”

Section: Related Workmentioning

confidence: 99%