Many programs can be seen as instances of a general pattern applied to a particular situation. Very often the pattern is determined by the types of the data involved. For example, in Chapter 9 the pattern of computing by recursion over a natural number is isolated as the defining characteristic of the type of natural numbers. This concept will itself emerge as an instance of the concept of type-generic, or just generic, programming.

Suppose that we have a function f of type ρ → ρ′ that transforms values of type ρ into values of type ρ′. For example, f might be the doubling function on natural numbers. We wish to extend f to a transformation from type [ρ/t]τ to type [ρ′/t]τ by applying f to various spots in the input where a value of type τ occurs to obtain a value of type ρ′, leaving the rest of the data structure alone. For example, τ might be bool × ρ, in which case f could be extended to a function of type bool × ρ → bool × ρ′ that sends the pairs ⟨a, b⟩ to the pair ⟨a, f(b)⟩.

This example glosses over a significant problem of ambiguity of the extension. Given a function f of type ρ → ρ′, it is not obvious in general how to extend it to a function mapping [ρ/t]τ to [ρ′/t]τ.