Contravariant families in simplicial HoTT supply proof-relevant expansion, yielding directed Boolean canonicity and a binary parametricity model over reduction-aware syntax.
The general background is the standard theory of orthogonal reflective subuniverses and localizations in homotopy type theory [Christensen et al
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Directed proof-relevant logical relations in simplicial HoTT
Contravariant families in simplicial HoTT supply proof-relevant expansion, yielding directed Boolean canonicity and a binary parametricity model over reduction-aware syntax.