 
Summary: Normalization by Evaluation
for MartinL¨of Type Theory
with One Universe
From PER Model to Subset Model
Andreas Abel1
Institut f¨ur Informatik, LudwigMaximiliansUniversit¨at
Oettingenstr. 67, D80538 M¨unchen
Abstract
We show how to replace the PER model of the original MFPS 2007 publication by a simpler subset model
without losing any results. This observation follows from the general insight that PER semantics is strongly
preferable when one models judgemental (aka typed) equality, yet for untyped equality is has no advantage
over subset semantics.
The paper under discussion[1] constructs a model of type theory over an untyped
model D, by constructing a partial equivalence relation (PER) Type D×D which
identifies the type values in D, plus for each a Type an associated PER [a] D×D
which identifies the values of type a in D. Equal types a = a Type have equal
extensions [a] = [a ]. The purpose of a PER semantics is to model extensional
equality on values, and it defines f = f [Pi a g] iff f · d = f · d [g(d)]
for all d = d [a]. However, we have already treated equality in the term
model D; in Lemma 3.4 we show that t  t implies [[t]] [[t ]] (in particular
