Poly-kinded type family

[alice]

Hello. I’m trying to make a type family with resolving it inside type checking classes, like CmpNat. Its type signature is «type family Cmp (a :: k1) (b :: k2)  :: Ordering», so the function is poly kinded. But it seems unclear how to make its TyCon. I’ve tried to follow the CmpNat and CmpSymbol style, and also saw Any TyCon in TysWiredIn. So this is my attempt:

typeCmpTyCon :: TyCon
typeCmpTyCon =
  mkFamilyTyCon name
    (binders ++ (mkTemplateAnonTyConBinders [ input_kind1, input_kind2 ]))
    orderingKind
    Nothing
    (BuiltInSynFamTyCon ops)
    Nothing
    NotInjective
  where
  name = mkWiredInTyConName UserSyntax gHC_TYPELITS (fsLit "Cmp")
                typeCmpTyFamNameKey typeCmpTyCon
  ops = BuiltInSynFamily
    { sfMatchFam      = matchFamCmpType
    , sfInteractTop   = interactTopCmpType
    , sfInteractInert = \_ _ _ _ -> []
    }
  binders@[kv1, kv2] = mkTemplateKindTyConBinders [ liftedTypeKind, liftedTypeKind ]
  input_kind1 = mkTyVarTy (binderVar kv1)
  input_kind2 = mkTyVarTy (binderVar kv2)

ghci says this:

:kind Cmp 
Cmp :: forall {k0} {k1}. k0 -> k1 -> Ordering

But then I try to apply it to some values and get this exception:

:kind! Cmp 4 5

<interactive>:1:15: error:
    • Expected kind ‘k0’, but ‘4’ has kind ‘Nat’
    • In the first argument of ‘Cmp’, namely ‘4’
      In the type ‘Cmp 4 5’

<interactive>:1:17: error:
    • Expected kind ‘k1’, but ‘5’ has kind ‘Nat’
    • In the second argument of ‘Cmp’, namely ‘5’
      In the type ‘Cmp 4 5’

Does anyone know where I made a mistake? Any help would be appreciated.
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