Salvaging the stratification-based proof for unique typing and inversion

Ascertain whether a stratification of the typing judgment that breaks the mutual induction between typing and definitional equality can be constructed to respect substitution—so that from Γ, x : β ⊢ᵢ e : α and Γ ⊢ⱼ e′ : β one can derive Γ ⊢_{max(i,j)} e[x ↦ e′] : α—thereby salvaging the original proof approach for Unique typing and Definitional inversion; or otherwise develop an alternative induction strategy that establishes these results.

Background

The earlier proof strategy for unique typing and definitional inversion introduced a stratified judgment ⊢ᵢ intended to break the mutual induction between typing and equality. The authors found that this stratification does not and cannot respect substitution in the required way, undermining the proof.

Although alternative routes exist for constructing soundness models that avoid unique typing, the conjectures are needed to verify the kernel’s correctness. Determining whether the stratified approach can be repaired—or identifying a workable alternative proof scheme—is explicitly left unresolved.