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Set truncation #13

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Set truncation #13

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@ice1000 ice1000 commented Feb 28, 2020

As title.

@ice1000 ice1000 added the new label Feb 28, 2020
@ice1000 ice1000 requested a review from valis Feb 28, 2020
@ice1000 ice1000 self-assigned this Feb 28, 2020
}

\instance SetTruncTruncated {A : \Type} : Truncated_-1+ 1 (SetTrunc A)
| isTruncated x y p q => path (\lam i => path (trunc x y p q i))
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@ice1000 ice1000 Feb 28, 2020

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I wish I can prove this via:

pmap path (path (trunc x y p q))

But it doesn't work, they fail with some type inferences where only one candidates available

| i, left => x
| i, right => y
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@ice1000 ice1000 Feb 28, 2020

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Two additional definitional equalities that aren't written in Agda but seem to exist.

@ice1000 ice1000 marked this pull request as ready for review Feb 28, 2020
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It's also possible to prove isSet A -> SetTrunc A = A, but I have no time.

: Path (\lam i => A.2 (p @ i)) b b' ofHLevel_-1+ n \elim p
| idp => h a b b'
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@ice1000 ice1000 Feb 28, 2020

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We have a much shorter proof rather than in Agda!

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@ice1000 ice1000 commented Mar 1, 2020

This is actually a specialization of Trunc_-1+ 1, thus useless

@ice1000 ice1000 closed this Mar 1, 2020
@ice1000 ice1000 deleted the setTrunc branch Mar 1, 2020
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