How to create a proof tree with LaTeX?
\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
S_1 \mathbf{R_5}
\begin{cases}
S_2 \mathbf{R_2}
\begin{cases}
S_4 \mathbf{R_1} & \\
S_5 \mathbf{R_3} & S_6 \mathbf{R_4}
\end{cases} \\
S_3 \mathbf{R_6}
\end{cases}
\end{equation*}
\end{document}
I suggest to define the command
\infer{conclusion}{premise1\\premise2\\...}
as
\newcommand\infer[2]{#1\left\{\begin{array}{@{}l@{}}#2\end{array}\right.}
Moreover, we use the abbreviation
\newcommand\claim[2]{S_{#1}\;\mathbf{R}_{#2}}
to typeset the proof items.
\documentclass{article}
\newcommand\infer[2]{#1\left\{\begin{array}{@{}l@{}}#2\end{array}\right.}
\newcommand\claim[2]{S_{#1}\;\mathbf{R}_{#2}}
\begin{document}
\[\infer
{\claim15}% from
{\infer
{\claim22}% from
{\claim41
\\% and
\claim53\quad\claim64
}
\\% and
\claim36
}
\]
\end{document}
\documentclass[a4paper,10pt]{article}
\usepackage[utf8]{inputenc}
\usepackage{mathtools}
\DeclareMathOperator{\bfR}{\mathbf{R}}
\begin{document}
\begin{equation*}
S_1 \bfR_5\begin{cases}
S_2 \bfR_2 & \begin{cases}
S_4 \bfR_1 & \\
& \\
S_5 \bfR_3 & \quad S_6 \bfR_4
\end{cases} \\
S_3 \bfR_6 &
\end{cases}
\end{equation*}
\end{document}