Definition:Partial Function Equality

Definition
Let $g: \N^k \to \N$ and $h: \N^k \to \N$ be partial functions.

We write:
 * $g \left({n_1, n_2, \ldots, n_k}\right) \approx h \left({n_1, n_2, \ldots, n_k}\right)$

iff either:
 * both $g \left({n_1, n_2, \ldots, n_k}\right)$ and $h \left({n_1, n_2, \ldots, n_k}\right)$ are defined and equal, or:
 * neither $g \left({n_1, n_2, \ldots, n_k}\right)$ nor $h \left({n_1, n_2, \ldots, n_k}\right)$ are defined.

That is, iff $g \left({n_1, n_2, \ldots, n_k}\right) = h \left({n_1, n_2, \ldots, n_k}\right)$ wherever either are defined.

Thus, $g$ is equal to $h$, and we can write $g = h$, iff:
 * $\forall x \in \N^k: g \left({x}\right) \approx h \left({x}\right)$.