Summary: Extensional Normalization in the Logical Framework with
Proof Irrelevant Equality
Department of Computer Science
We extend the Logical Framework by proof irrelevant
equality types and present an algorithm that computes
unique long normal forms. The algorithm is inspired by
normalization-by-evaluation. Equality proofs which are not
reflexivity are erased to a single object . The algorithm
decides judgmental equality, its completeness is established
by a PER model.
In intensional Martin-L®of type theory (ITT), but also
in the Calculus of Inductive Constructions which underlies
the Coq proof assistant, we distinguish between definitional
equality of terms t and t of type T, given by the judgement
t = t : T, and propositional equality which is estab-
lished by providing an inhabitant of the identity set IdT t t .