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Normalisation by evaluation for type theory, in type theory

Altenkirch, Thorsten; Kaposi, Ambrus


Ambrus Kaposi


We develop normalisation by evaluation (NBE) for dependent types based on presheaf categories. Our construction is formulated in the metalanguage of type theory using quotient inductive types. We use a typed presentation hence there are no preterms or realizers in our construction, and every construction respects the conversion relation. NBE for simple types uses a logical relation between the syntax and the presheaf interpretation. In our construction, we merge the presheaf interpretation and the logical relation into a proof-relevant logical predicate. We prove normalisation, completeness, stability and decidability of definitional equality. Most of the constructions were formalized in Agda.


Altenkirch, T., & Kaposi, A. (2017). Normalisation by evaluation for type theory, in type theory. Logical Methods in Computer Science, 13(4:1),

Journal Article Type Article
Acceptance Date Sep 4, 2017
Publication Date Oct 23, 2017
Deposit Date Oct 30, 2017
Publicly Available Date Oct 30, 2017
Journal Logical Methods in Computer Science
Print ISSN 1860-5974
Electronic ISSN 1860-5974
Publisher Logical Methods in Computer Science
Peer Reviewed Peer Reviewed
Volume 13
Issue 4:1
Keywords normalisation by evaluation; dependent types; internal type theory; logical relations; Agda
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