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Add more results for inductively defined equality type #1011

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merged 12 commits into from
Jul 3, 2023

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JonasHoefer
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@JonasHoefer JonasHoefer commented Jun 12, 2023

#1005 wants to remove the swan identity type because Agda.Builtin.Equality now has the same computational behavior. This pull request extends the Cubical.Data.Equality module, which contains some basic results for this equality type. The main changes are the following.

  1. A proof of univalence which is behaves like "Book HoTT", that is, pr₁ (idToEquiv p) x = transport id p x holds, where transport is defined in terms of pattern matching. Thus, the function which is an equivalence is the same one as in the book (see Lemma 2.10.1 and Axiom 2.10.3).
  2. The interface for is extended with the eliminator and its corresponding β and η rules.
  3. A more general result (similar to the one here) is proven, which simplifies lifting HITs to the inductively defined equality type.

@JonasHoefer JonasHoefer marked this pull request as ready for review June 12, 2023 14:58
@felixwellen felixwellen self-assigned this Jun 21, 2023
@felixwellen felixwellen merged commit 26c5795 into agda:master Jul 3, 2023
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