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G is one of the roots of the equation in x x2-x-l = 0 . (In other words, G 2 -G-1 = 0.) The other root is G' = i ( l - A / 5 ) . Here we use strong induction to prove that, for all n such that n ^ 1, Gn~2 ^ F.n ^ Gn~l . The exercises ask you to prove further properties of F.n. Basis. It is important to be very careful in the proof of any theorem about the Fibonacci numbers because there are three separate cases in the definition; it is very easy to make the mistake of applying the rule F.(n+2) = F.(n+l) + F.n when n is not defined. (See Exercise 12.14 for an example). In addition, in this case, the property we have to prove is claimed to be true only for n such that n^l. (It is false when n equals 0.) A mistake that is easily made in the induction step is to assume that the property is true when n equals 0. This means that we are obliged to prove the property for the cases n - 1 and n = 2 before proceeding to the induction step. For n = 1, we have



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More on Relationship Derivation Relationship derivation can be very useful for all of the same reasons that class inheritance is useful it allows common structure and behavior to be shared in common base classes For example, if the author of the IssueState model wished to specify some rules or behavior that applied to all transitions, whether they are start transitions or normal Issue State transitions, the Transition relationship is the place to do it However, it does cause some subtleties in the generated code Furthermore, it also imposes a few restrictions on which multiplicities are allowed for the various roles Let s look at the multiplicities rst Remember that each role causes the generation of a property on the roleplayer domain class In the case of the StartElement role on the StartTransition relationship, the generated property on the StartElement C# class is called StartState and returns the associated IssueState Similarly, for the Previous role on the Transition relationship, the generated property on the StateElement C# class is called Successors and returns the associated StateElements The restrictions on multiplicity are a logical consequence of the fact that the links navigated via the base generated property (Successors) must be a superset of the links navigated via the derived generated property (StartState) This follows from the de nition of relationship derivation every link of StartTransition is also a link of Transition For example, given a StartElement, the links navigated via the StartState property must be a subset of the links navigated via the Successors property In fact, in this model, they must be the same set of links, because there is no other relationship derived from Transition that is sourced on StartState These restrictions imply that it is an error, if the multiplicity of the base role is ZeroOne or One, for the multiplicity of the derived role to be ZeroMany or OneMany Given the multiplicity of the base role, there could never be more than one link.





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Earlier, this chapter went into detail on the internals of the foreach loop This section discusses how to use iterators to create your own implementation of

the IEnumerator<T> and nongeneric IEnumerator interfaces for custom collections Iterators provide clean syntax for specifying how to iterate on data in collection classes, especially using the foreach loop The iterator allows endusers of a collection to navigate its internal structure without knowledge of that structure

G 1 - 2 < F.I < G1"1

It s also possible, although unusual, to create misleading multiplicity combinations, which will cause the DSL Tools to issue warning messages Here are a couple of examples If the multiplicity of an abstract base role is ZeroOne or One, and there are two or more derived roles that have the same roleplayer class, they must all have the multiplicity ZeroOne, because only one of them could validly be instantiated at a time If the base relationship is not abstract and has an end with a multiplicity One, then there is no point in creating a derived relationship, because we know that the base relationship has to be instantiated directly

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