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plurals

PO
primarykey
Id
8738053
data
AcceptedAnswerId
0
AnswerCount
0
ClosedDate
CommentCount
1
CommunityOwnedDate
CreationDate
2012-01-05T05:23:00.530
FavoriteCount
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2012-01-05T05:23:00.530
LastEditDate
LastEditorUserId
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1011995
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8736995
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2
Score
37
ViewCount
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LastEditorDisplayName
text
Body
I can't explain why, but here's the proof of the cycle: Assume <code>k >= 2</code> and <code>fmap^(4k) :: (a -> b) -> F1 F2 F3 a -> F1 F2 F3 b</code>, where <code>Fx</code> stands for an unknown/arbitrary <code>Functor</code>. <code>fmap^n</code> stands for <code>fmap</code> applied to <code>n-1</code> <code>fmap</code>s, not <code>n</code>-fold iteration. The induction's start can be verified by hand or querying ghci. <pre><code>fmap^(4k+1) = fmap^(4k) fmap fmap :: (x -> y) -> F4 x -> F4 y </code></pre> unification with a -> b yields <code>a = x -> y</code>, <code>b = F4 x -> F4 y</code>, so <pre><code>fmap^(4k+1) :: F1 F2 F3 (x -> y) -> F1 F2 F3 (F4 x -> F4 y) </code></pre> Now, for <code>fmap^(4k+2)</code> we must unify <code>F1 F2 F3 (x -> y)</code> with <code>(a -> b) -> F5 a -> F5 b</code>. Thus <code>F1 = (->) (a -> b)</code> and <code>F2 F3 (x -> y)</code> must be unified with <code>F5 a -> F5 b</code>. Hence <code>F2 = (->) (F5 a)</code> and <code>F3 (x -> y) = F5 b</code>, i.e. <code>F5 = F3</code> and <code>b = x -> y</code>. The result is <pre><code>fmap^(4k+2) :: F1 F2 F3 (F4 x -> F4 y) = (a -> (x -> y)) -> F3 a -> F3 (F4 x -> F4 y) </code></pre> For <code>fmap^(4k+3)</code>, we must unify <code>a -> (x -> y)</code> with <code>(m -> n) -> F6 m -> F6 n)</code>, giving <code>a = m -> n</code>, <code>x = F6 m</code> and <code>y = F6 n</code>, so <pre><code>fmap^(4k+3) :: F3 a -> F3 (F4 x -> F4 y) = F3 (m -> n) -> F3 (F4 F6 m -> F4 F6 n) </code></pre> Finally, we must unify <code>F3 (m -> n)</code> with <code>(a -> b) -> F7 a -> F7 b</code>, so <code>F3 = (->) (a -> b)</code>, <code>m = F7 a</code> and <code>n = F7 b</code>, therefore <pre><code>fmap^(4k+4) :: F3 (F4 F6 m -> F4 F6 n) = (a -> b) -> (F4 F6 F7 a -> F4 F6 F7 b) </code></pre> and the cycle is complete. Of course the result follows from querying ghci, but maybe the derivation sheds some light on how it works.
Tags
Title
singulars
PostAcceptedAnswerId
1. This table or related slice is empty.
PostParentId
1. POFun with repeated fmap
 singulars
 PostTypePostTypeId
 PTQuestion
PostTypePostTypeId
1. PTAnswer
UserLastEditorUserId
1. This table or related slice is empty.
UserOwnerUserId
1. USDaniel Fischer
plurals
PostLinksPostIdRelatedPostId
1. This table or related slice is empty.
PostLinksRelatedPostIdPostId
1. This table or related slice is empty.
PostsAcceptedAnswerId
1. This table or related slice is empty.
PostsParentIdCreationDate
1. This table or related slice is empty.
VotesPostIdCreationDate
1. VO
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 PostPostId
 PO
 UserUserId
 This table or related slice is empty.
 VoteTypeVoteTypeId
 VTUpMod
2. VO
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 PostPostId
 PO
 UserUserId
 This table or related slice is empty.
 VoteTypeVoteTypeId
 VTUpMod
3. VO
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 PostPostId
 PO
 UserUserId
 This table or related slice is empty.
 VoteTypeVoteTypeId
 VTUpMod
CommentsPostId
1. CO+1, nice derivation!
 singulars
 PostPostId
 PO
 UserUserId
 USehird

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