\begin{definition}[Definition:Gentzen Proof System/Instance 1/Alpha-Rule/Notation]
Let $\mathscr G$ be instance 1 of a Gentzen proof system.
Let $\mathbf A$ be an $\alpha$-formula, with $\mathbf A_1, \mathbf A_2$ as in the table of $\alpha$-formulas.
In a tableau proof, the $\alpha$-rule can be used as follows:
{{begin-axiom}}
{{axiom|lc = '''Pool:'''
       |t  = Empty.
}}
{{axiom|lc = '''Formula:''' 
       |t  = $U_1 \cup U_2 \cup \set {\mathbf A}$.
}}
{{axiom|lc = '''Description:''' 
       |t  = $\alpha$-Rule.
}}
{{axiom|lc = '''Depends on:''' 
       |t  = The lines containing $U_1 \cup \set {\mathbf A_1}$ and $U_2 \cup \set {\mathbf A_2}$.
}}
{{axiom|lc = '''Abbreviation:''' 
       |t  = $\alpha \circ$, where $\circ$ is the binary logical connective such that $\mathbf A {{=}} \mathbf A_1 \circ \mathbf A_2$ or $\mathbf A {{=}} \neg \paren {\mathbf A_1 \circ \mathbf A_2}$, or $\neg \neg$ in the case that $\mathbf A = \neg \neg \mathbf A_1$.
}}
{{end-axiom}}
Category:Definitions/Mathematical Logic
Category:Definitions/Propositional Logic
\end{definition}