Kleene equality

In mathematics, Kleene equality, or strong equality, ($\simeq$) is an equality operator on partial functions, that states that on a given argument either both functions are undefined, or both are defined and their values on that arguments are equal.

For example, if we have partial functions $f$ and $g$, $f\simeq g$ means that for every $x$:

• $f(x)$ and $g(x)$ are both defined and $f(x)=g(x)$
• or $f(x)$ and $g(x)$ are both undefined.

Some authors are using "quasi-equality", which is defined like this: $(y_1\sim y_2):\iff ((y_1\downarrow \vee y_2\downarrow )⟶y_1=y_2),$ where the down arrow means that the term on the left side of it is defined. Then it becomes possible to define the strong equality in the following way: $(f\simeq g):\iff (\mathrm{\forall }x.(f(x)\sim g(x))).$