We often encounter a definition of a function over the natural numbers such as

*f* (0, *m*) = 0

*f* (*n* + 1, *m*) = *f* (*n*, *m*) + *m*

Is this a *legitimate* definition? That is, *is there* really a function *that satisfies* the above two equalities for all *n* and *m*? And if so, is there *only one* such function, or is the definition ambiguous? We address this question in this section through a somewhat more general related question.

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