General semiring-only solution for natrec usage constraints

Determine whether there exists a general solution, expressible using only addition, multiplication, and meet on grades, for the usage-constraint inequalities arising from the natural-number eliminator (natrec); either construct such a formulation or prove that no such formulation exists without introducing the additional operator used in the paper.

Background

Assigning grades to the natural number eliminator yields coupled inequalities that must hold to ensure subject reduction for usage. The authors discuss prior star-based approaches and explain difficulties in ensuring subject reduction and substitution. They therefore introduce a dedicated five-argument operator (the “function” for natrec) with axioms tailored to these constraints.

They explicitly note that they could not find a general solution using only the existing semiring operations (addition, multiplication) and meet, motivating the introduction of this new operator. Establishing a semiring-only characterization (or impossibility) would clarify the algebraic requirements for recursion in graded settings.