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Build columns by prepending the first row's entries to the columns obtained from the remaining rows:The singleton case turns each entry into a one-element column. In the recursive case, transpose rows has exactly as many columns as row has entries, so zipWith (:) extends every column once. Structural induction on the number of rows proves that each output column contains the corresponding input entries in order. This uses list zip to implement the transpose, without requiring Num a.

The two required equations are

zipWith f xs ys = map (uncurry f) (zip xs ys)
zip xs ys       = zipWith (,) xs ys

Either equation defines one operation from an independently defined other operation. They do not together supply an effective recursive definition of the intended operations. Substitution makes the first output constructor depend on itself without first emitting anything, so the mutually recursive program has the bottom value as its least solution. At the level of equations they also fail to characterize the intended operations uniquely: returning [] for every call of either operation satisfies both equations, as does the intended list zip. A constructor-producing base/step definition is essential for productivity.

Addition and subtraction of formal power series are coefficientwise:

plus, minus :: Num a => [a] -> [a] -> [a]
plus  = zipWith (+)
minus = zipWith (-)

Here every valid input is an infinite coefficient list. If finite polynomial coefficient lists were admitted, they would need zero-padding before these definitions: ordinary list zip otherwise truncates to the shorter length.