Standardize explicit substitutions and establish their completeness

Determine the exact formulation of explicit substitutions and establish whether that formulation is complete for normalization-by-evaluation developments in dependent type theory.

Background

The paper contrasts operational substitutions, defined as meta-level operations on raw terms, with explicit substitutions, which are included as syntactic constructs governed by definitional equalities. Although explicit substitutions can simplify semantic proofs, the paper notes that their precise formulation varies across developments—for example, some systems include a terminal-object substitution and associated laws while others do not. The authors identify both the formulation itself and its completeness as unresolved, making standardization and completeness a problem distinct from the technical results established in the report.