• Church Rosser, surjective pairing, and strong normalization
• Joinability: A transitive equational relation
  • Joinability and logical predicates
  • Joinability and typed equality

Church Rosser, surjective pairing, and strong normalization

This repository contains a mechanized proof that the lambda calculus with beta and eta rules for functions and pairs is in fact confluent for the subset of terms that are "strongly $\beta$-normalizing".

Our notion of $\beta$-strong normalization, based on Abel's POPLMark Reloaded survey, is inductively defined and captures a strict subset of the usual notion of strong normalization in the sense that ill-formed terms such as $\pi_1 \lambda x . x$ are rejected.

Joinability: A transitive equational relation

The joinability relation $a \Leftrightarrow b$, which is true exactly when $\exists c, a \leadsto_{\beta\eta} c \wedge b \leadsto_{\beta\eta} c$, is transitive on the set of strongly normalizing terms thanks to the Church-Rosser theorem proven in this repository.

Joinability and logical predicates

When building a logical predicate where adequacy ensures beta strong normalization, we can then show that two types $A \Leftrightarrow B$ share the same meaning if both are semantically well-formed. The strong normalization constraint can be extracted from adequacy so we have transitivity of $A \Leftrightarrow B$ available in the proof that the logical predicate is functional over joinable terms.

Joinability and typed equality

For systems with a type-directed equality, the same joinability relation can be used to model the equality. The fundamental theorem for the judgmental equality would take the following form: $\vdash a \equiv b \in A$ implies $\vDash a, b \in A$ and $a \Leftrightarrow b$.

Note that such a result does not immediately give us the decidability of type conversion because the algorithm of beta-eta normalization may identify more terms when eta is not type-preserving. However, I believe Coquand's algorithm can be proven correct using the method described in Goguen 2005. We have all the ingredients needed:

• Strong normalization of beta (needed for the termination metric)
• Church-Rosser for $\beta$-SN terms (the disciplined expansion of Coquand's algorithm preserves both SN and typing)