 
Summary: Fundamenta Informaticae TLCA'05 153 1
IOS Press
Untyped Algorithmic Equality for MartinL¨of's Logical Framework
with Surjective Pairs
Andreas Abel C
Institut f¨ur Informatik, LudwigsMaximiliansUniversit¨at M¨unchen
abel@informatik.unimuenchen.de
Thierry Coquand
Department of Computer Science, Chalmers University of Technology
coquand@cs.chalmers.se
Abstract. MartinL¨of's Logical Framework is extended by strong types and presented via judg
mental equality with rules for extensionality and surjective pairing. Soundness of the framework
rules is proven via a generic PER model on untyped terms. An algorithmic version of the framework
is given through an untyped equality test and a bidirectional type checking algorithm. Complete
ness is proven by instantiating the PER model with equality on normal forms, which is shown
equivalent to the algorithmic equality.
1. Introduction
Central to dependent type theories is the rule of conversion: The type of an expression can be converted to
an equal type, where in intensional type theories the notion of equality between types is decidable. In the
past, research has focused on equality, and since reduction is confluent, two types are equal iff they
