Syntactic model forms a strict LFDC

Prove that the well-typed syntactic fragment of the bidirectional substructural dependent type theory for left-fibred double categories, quotiented by judgmental equality, forms a strict left-fibred double category equipped with type-level weakening, function types, product types, and the corresponding structural properties.

Background

The paper presents a bidirectional type theory intended as the internal language of LFDCs, with rules designed to compute semantic denotations. While partial admissibility results for weakening and substitution are proved, the central completeness step—showing that the syntactic fragment itself carries the full strict LFDC structure—remains to be formally established.

This result would close the soundness-completeness loop by demonstrating that the syntax is not only interpretable in strict LFDCs but also itself constitutes such a structure, justifying the use of the type theory as a canonical model and enabling further metatheoretic developments.