Soundness of extraction with erased matches for open programs

Prove soundness of the extraction function for open programs when erased matches for weak Σ-types are allowed, by showing that for any well-typed term of type N whose free variables are all erasable in a consistent context, evaluation produces the same natural-number result before and after extraction even though erased matches are permitted.

Background

The paper proves soundness of the erasure-based extraction for programs of natural-number type, including certain open programs, but only under the restriction that erased matches on weak Σ-types are disallowed. Allowing erased matches complicates the simulation argument for open terms, because matching on a neutral erased pair can get stuck in the source while the extracted target term may continue to evaluate.

The authors’ current proof does not cover this case; nevertheless they indicate that the result should still hold, and they point to further discussion in the subsection on erased matches. Establishing this result would remove a key limitation of their current soundness theorem.