[GHC] #11207: GHC cannot infer injectivity on type family operating on GHC.TypeLits' Nat, but can for equivalent type family operating on user-defined Nat kind

[GHC]

#11207: GHC cannot infer injectivity on type family operating on GHC.TypeLits' Nat,
but can for equivalent type family operating on user-defined Nat kind
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        Reporter:  duairc            |                Owner:
            Type:  bug               |               Status:  new
        Priority:  normal            |            Milestone:
       Component:  Compiler          |              Version:  7.11
      Resolution:                    |             Keywords:
Operating System:  Unknown/Multiple  |         Architecture:
                                     |  Unknown/Multiple
 Type of failure:  None/Unknown      |            Test Case:
      Blocked By:                    |             Blocking:
 Related Tickets:                    |  Differential Rev(s):
       Wiki Page:                    |
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Changes (by goldfire):

 * cc: jstolarek (added)


Comment:

 But your definitions are subtly different. In the !TypeLits case, you're
 using `-` in the RHS. `-` is not injective. So GHC quite correctly rejects
 the equation. In the corresponding equation without !TypeLits, there are
 no function calls on the right; instead, you use pattern-matching on the
 left to get what you want.

 If you implement a `-` over your unary `Nat` and write the definitions the
 same way, GHC will reject the non-!TypeLits definition, too.

 What this requires is "generalized injectivity", where we could say

 {{{
 type family (a :: Nat) - (b :: Nat) = (r :: Nat) | r a -> b, r b -> a
 }}}

 We've considered such a thing, but we have yet to work out all the
 details, if I recall correctly. In any case, if this existed, then GHC
 would be able to accept the first definition.

 Adding in Janek, who is Mr. Injective-Type-Families.

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Ticket URL: <http://ghc.haskell.org/trac/ghc/ticket/11207#comment:2>
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