As of now, the internal (what's stored inside ArithmeticCircuit) and external (what's used in MonadCircuit interface) representations coincide: there's a unique LookupTable type. However, it is a bit clunky:

• on user side, having to first call registerFunction is non-intuitive;
• on implementation side, having this GADT-inside-existential is much harder to work with than its possible simply typed counterpart with erased functor parameter.

So I propose to split this in two types:

-- | Type for MonadCircuit API
data LookupTable a f where
  ... same as before ...
  Plot :: (forall w. IsWitness a w => f w -> g w) -> LookupTable a f -> LookupTable a (f :*: g)
  -- ^ note the use of function instead of its ID

-- | Type for internal use
newtype FunctionId = FunctionId ByteString
data LookupType a -- note the absence of functor parameter
  = LTRanges (Set (a, a))
  | LTProduct (LookupType a) (LookupType a)
  | LTPlot FunctionId (LookupType a)