Typeclass constraint inside typeclass instance

Question

Hello while doing examples from Real World Haskell book i have encountered this example and i can not understand what it does mean and how it does work:
instance Num a=>Num (SymbolicManip a)
In this case i should translate to somehting like : "For Num instance of type SymbolicManip there is a constraint regarding its field of type a,which is : a being an instance of Num itself"? Can someone please tell me if i interpreted it right or explain?
Why wouldn't instance Num (SymbolicManip a) be enough?

-- The "operators" that we're going to support
data Op = Plus | Minus | Mul | Div | Pow
        deriving (Eq, Show)

{- The core symbolic manipulation type -}
data SymbolicManip a = 
          Number a           -- Simple number, such as 5
        | Arith Op (SymbolicManip a) (SymbolicManip a)
          deriving (Eq, Show)

{- SymbolicManip will be an instance of Num.  Define how the Num
operations are handled over a SymbolicManip.  This will implement things
like (+) for SymbolicManip. -}
instance Num a => Num (SymbolicManip a) where
    a + b = Arith Plus a b
    a - b = Arith Minus a b
    a * b = Arith Mul a b
    negate a = Arith Mul (Number (-1)) a
    abs a = error "abs is unimplemented"
    signum _ = error "signum is unimplemented"
    fromInteger i = Number (fromInteger i)

P.S All code is from the book (chapter 13 - subchapter-Extended Example-Numeric Types)

Try compiling the code without the Num constraint and see what happens. — Paul Johnson

AJF AJF · Accepted Answer · 2018-07-10T08:21:17

It's important to see that a SymbolicManip a cannot be an instance of Num without a also being an instance of Num So, just like when we add constraints to functions, we can add a constraint to a typeclass:

 instance Num a => Num (SymbolicManip a) where
 --       ^^^^^^^^ "As long as `a` is an instance of `Num`..."
 --                ^^^^^^^^^^^^^^^^^^^^^ "...so is `SymbolicManip a`"

We must include the Num a => constraint, because in the implementation, we use fromInteger to produce members of the type a. This is unavoidable, just like adding a Num constraint to the function example a b = 2*a + b, i.e. example :: Num a => a -> a -> a.

Here's a simpler example. Consider this type:

newtype Identity a = Identity a

Note that an Identity a could be an instance of Num, so long as a is a Num too, so, we add a constraint:

instance Num a => Num (Identity a) where
-- (...)

Typeclass constraint inside typeclass instance

2 Answers