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Equality #13

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solomon-b opened this issue Mar 6, 2023 · 1 comment
Open

Equality #13

solomon-b opened this issue Mar 6, 2023 · 1 comment
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@solomon-b
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@MonoidMusician
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Right now we have very very underspecified equality: just the type former x y : A |- x = y and constructor refl : a = a.

MLTT adds Axiom J, which computes on refl: https://amelia.how/quick/axiom-j.html

Observational Type Theory is an extension we can bolt on:
http://www.strictlypositive.org/ott.pdf

OTT makes equality compute on the type, so that we get stuff like “equality of functions is pointwise equality” by definition. This seems like what we want, for convenience of matching on constructors in particular (e.g. finsets for constructor names like #{ .zero, .succ }).

OTT would restrict the models we have, in particular it rules out Univalence, and the question is do we care?

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@solomon-b @MonoidMusician