User blog comment:P進大好きbot/Kyodaisuutan System/@comment-24923514-20181003190853/@comment-35470197-20181004013836

Ah, I see. Sorry for the ambiguity of the purpose (or the restriction) of the blog post. (I would never like to confuse others :P)

At first, \(\forall\) is a symbol called the universal quantifier, and just means "for all". Therefore \(\forall a,b,c,d,e \in \mathbb{N}_+\) in the beginning of the rule means "for all positive integers \(a,b,c,d,e\)".

In order to define a function, we need to give relations between the values of it which are sufficient to characterise it. For example, the relations \begin{eqnarray*} & & f(1) = 1 \\ & & \forall n \in \mathbb{N}_+, f(1) = f(n) \end{eqnarray*} characterise the constant function \(f(n) = 1\). A one-ruled or one-lined function roughly means a function with a single characterising relation containing single relation symbol such as \(=, <, >, \leq, \geq, \neq\) and so on.

If you know first order logic and \(\epsilon \delta\) logic, I should note that the equation containing single \(\lim\) is a syntax sugar (or more precisely, a formula in "the extension of formal language by a definition" in the sense of formal logic) of a formula which contains at most a single relation symbol.