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Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended Version)

Abstract : We show that a version of Martin-Löf type theory with an extensional identity type former I, a unit type N1, Σ-types, Π-types, and a base type is a free category with families (supporting these type formers) both in a 1-and a 2-categorical sense. It follows that the underlying category of contexts is a free locally cartesian closed category in a 2-categorical sense because of a previously proved biequivalence. We show that equality in this category is undecidable by reducing it to the undecidability of convertibility in combinatory logic. Essentially the same construction also shows a slightly strengthened form of the result that equality in extensional Martin-Löf type theory with one universe is undecidable.
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https://hal.inria.fr/hal-01579415
Contributor : Pierre Clairambault <>
Submitted on : Thursday, August 31, 2017 - 10:23:48 AM
Last modification on : Thursday, June 18, 2020 - 10:18:03 AM

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Simon Castellan, Pierre Clairambault, Peter Dybjer. Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended Version). Logical Methods in Computer Science, Logical Methods in Computer Science Association, 2017, ⟨10.23638/LMCS-13(4:22)2017⟩. ⟨hal-01579415⟩

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