A domain-theoretic logical-relations technique proves injectivity and no-confusion of type constructors for non-normalising dependent type theories with η laws, including type-in-type systems, without relying on normalisation.
Algorithmic Conversion with Surjective Pairing: A Syntactic and Untyped Ap- proach
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Definitional Inversion, Without Normalisation
A domain-theoretic logical-relations technique proves injectivity and no-confusion of type constructors for non-normalising dependent type theories with η laws, including type-in-type systems, without relying on normalisation.