  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.
