# Recursive Genereators

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## Recursive Generators

Assume you write a function that takes a recursively defined data type as input and you like to test it with QuickCheck. How would you do that?

### Example - Cartesian product of sets

We add the function 'product/2' to Erlang's sets module (in standard library) computing the Cartesian product of two sets. A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

For example,

> S1 = sets:from_list([1,2]), S2 = sets:from_list([red,white,blue]), sets:to_list(sets:product(S1,S2)). [{1,red},{1,white},{1,blue},{2,red},{2,white},{2,blue}]

In the fashion of test driven development, we start by writing down a property that should hold.

prop_relative_complement() -> ?FORALL({S1,S2},{set(int()),set(colour())}, begin equal(sets:to_list(sets:product(S1,S2)), [ {I,C} || I<-sets:to_list(S1), C<-sets:to_list(S2)]) end). %% Since lists originating from sets may have the elements in different %% orders, we need to compare that they contain the same elements. equal(L1,L2) -> (L1 -- L2) == [] andalso (L2 -- L1) == [].

Surely, we could implement the product function with the list comprehension above, but we want a more efficient implementation. The test, however, need not at all be efficient.

### Generators for data type

After formulating the property, we need to implement the generators for the data types, sets and colours in our case, since there is a standard generator for integers. Picking a colour from a predefined set of colours is simple. Creating a set needs a thought... we have to use the library functions in the set module to do so, since we do not want to rely upon the implementation details. One possibility is to simply create a list and turn it into a set:

colour() -> elements([red,white,blue,yellow,green,black]). set(Gen) -> ?LET(L,list(Gen),sets:from_list(L)).