\[ \newcommand{\tr}{\Rightarrow} \newcommand{\trs}{\tr^{\!\ast}} \newcommand{\rlnm}[1]{\mathsf{(#1)}} \newcommand{\rred}[1]{\xrightarrow{#1}} \newcommand{\rreds}[1]{\mathrel{\xrightarrow{#1}\!\!^*}} \newcommand{\cl}{\mathsf{Cl}} \newcommand{\pow}{\mathcal{P}} \newcommand{\matches}{\mathrel{\mathsf{matches}}} \newcommand{\kw}[1]{\mathsf{#1}} \]

Proof Trees

A proof tree or derivation for the statement “\(R \matches w\)” is a finite tree whose nodes are labelled by matches statements, in such a way that:

1. The root is labelled “\(R \matches w\)”.

2. and, for each node in the tree, labelled by say $S \matches v$, has parents labelled $S_1 \matches w_1,\ldots,S_k \matches w_k$, then there must be some rule (X) from our set of rules for which the following is an instance:

   \[\begin{prooftree} \AxiomC{$S_1 \matches w_1 \qquad \cdots{} \qquad S_k \matches w_k$} \LeftLabel{(X)} \UnaryInfC{$S \matches v$} \end{prooftree}\]