Algorithmic Conversion with Surjective Pairing: A Syntactic and Untyped Approach

In a dependent type theory with β-equivalence as its equational theory, the confluence of untyped reduction and termination immediately give us a proof of the decidability of type conversion, where the decision procedure for convertibility simply checks the equality of the β-normal forms of its inputs. This technique is not available in the presence of surjective pairing (i.e. the η-law for pairs) because βη-reduction is not confluent. In this work, we show that by adopting established syntactic techniques, we can resolve the issue with confluence caused by surjective pairing, and recover a confluence-based proof of decidability of type conversion. Compared to existing proof developments, which rely on semantic tools such as Kripke-style logical relations, our proof modularly composes a minimal semantic proof of untyped normalization and a syntactic proof of decidability. This modularity enables us to explore algorithmic conversion through syntactic methods without modifying the minimal semantic proof. We have fully mechanized our results using the Rocq theorem prover.