defi-minimal/docs/uniswap-v1/equations.tex at main · starknetdev/defi-minimal · GitHub

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\documentclass{article}

% Language setting
% Replace `english' with e.g. `spanish' to change the document language
\usepackage[english]{babel}

\usepackage{witharrows}

% Set page size and margins
% Replace `letterpaper' with `a4paper' for UK/EU standard size
\usepackage[letterpaper,top=2cm,bottom=2cm,left=3cm,right=3cm,marginparwidth=1.75cm]{geometry}

% Useful packages
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage[colorlinks=true, allcolors=blue]{hyperref}

\title{Your Paper}
\author{You}

\begin{document}
\maketitle


\section{Uniswapv1}

\begin{equation}
   x*y=k \quad  \text{where:} \quad
   \begin{gathered}
   x \quad \text{is total amount of asset A} \\
   y \quad \text{is total amount of asset B} \\
   k \quad \text{is constant} \\
    \end{gathered}
\end{equation}

\begin{equation}
  (x+\Delta x)*(y-\Delta y)=k \quad  \text{where:} \quad
  \begin{gathered}
   x \quad \text{is total amount of asset A} \\
   \Delta x \quad \text{is sold amount of asset A} \\
   y \quad \text{is total amount of asset B} \\
   \Delta y \quad \text{is bought amount of asset B} \\
   k \quad \text{is constant} \\
    \end{gathered}
\end{equation}

\begin{equation}
\begin{WithArrows}
x*y = (x+\Delta x)*(y-\Delta y) \\
y-\Delta y = \dfrac{x*y}{x+\Delta x}
\end{WithArrows}
\end{equation}


\begin{equation}
\begin{WithArrows}
\Delta y &= y - \dfrac{x*y}{x+\Delta x} \\
&= \dfrac{y*(x+\Delta x)}{x+\Delta x} - \dfrac{x*y}{x+\Delta x} \\
&= \dfrac{y*x +y \Delta x - x*y}{x+\Delta x} \\
&= \dfrac{y \Delta x}{x+\Delta x}
\end{WithArrows}
\end{equation}

Let's consider that \(\Delta x_f\) is the sold amount of asset A \(\Delta x\) minus the fees\\
\begin{equation}
\begin{WithArrows}
   \Delta x_f &= \Delta  x - \Delta x*0,3\% \\
   &= \Delta x * (1-0,3\%) \\
   &= \Delta x * (\dfrac{1000}{1000}-\dfrac{3}{1000}) \\
   &= \Delta x * \dfrac{997}{1000}
 \end{WithArrows}
\end{equation}

\begin{equation}
\begin{WithArrows}
\Delta y &=  \dfrac{y \Delta x_f}{x+\Delta x_f}
\end{WithArrows}
\end{equation}


\begin{equation}
\begin{WithArrows}
\Delta y &=  \dfrac{y \Delta x_f}{x+\Delta x_f} \\
&= \dfrac{\dfrac{997*y*\Delta x}{1000}}{x+\dfrac{997*\Delta x}{1000}} \\
&= \dfrac{997 ’y \Delta x}{1000 x + 997 \Delta x}
\end{WithArrows}
\end{equation}


\end{document}