Abstract
We present a game semantics for dependent type theory (DTT) with \(\varPi \)-, \(\varSigma \)-, intensional \(\mathsf {Id}\)-types and finite inductive type families. The model satisfies Streicher’s criteria of intensionality and refutes function extensionality. The principle of uniqueness of identity proofs is satisfied.
The model is fully and faithfully complete at the type hierarchy built without \(\mathsf {Id}\)-types. Although definability for the hierarchy with \(\mathsf {Id}\)-types remains to be investigated, the notions of propositional equality in syntax and semantics do coincide for (open) terms of the \(\mathsf {Id}\)-free type hierarchy.
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Abramsky, S., Jagadeesan, R., Vákár, M. (2015). Games for Dependent Types. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47666-6_3
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DOI: https://doi.org/10.1007/978-3-662-47666-6_3
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