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Relating Riley’s bunched HoTT-based linear dependent type theory to the LFDC approach

Determine whether and how Mitchell Riley’s linear dependent type theory based on bunched logic and parameterized spectra in Homotopy Type Theory is related to the LFDC-based substructural dependent type theory; specifically, ascertain whether one subsumes the other or characterize their precise relationship.

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Background

The paper situates its LFDC-based substructural dependent type theory alongside related developments, notably Riley’s bunched linear dependent type theory formulated within Homotopy Type Theory and applied to areas such as quantum certification.

Clarifying whether one framework subsumes the other, or how they inter-translate, would map the landscape of substructural dependent type theories and potentially unify different semantic and proof-theoretic approaches.

References

It remains to be seen whether and how this type theory is related to the form of substructural dependent type theory developed in this paper, i.e. whether one subsumes the other, etc.

— Foundations of Substructural Dependent Type Theory (2401.15258 - Aberlé, 27 Jan 2024) in Subsection: Toward the type theory of monoidal topoi (final paragraph)