Untyped Algorithmic Equality for Martin-Löf’s Logical Framework with Surjective Pairs

Abstract

Martin-Lof's Logical Framework is extended by strong Σ-types and presented via judgmental 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 βη-equality test and a bidirectional type checking algorithm. Completeness is proven by instantiating the PER model with η-equality on β-normal forms, which is shown equivalent to the algorithmic equality.