fig-shadow-proof.tex

\documentclass[tikz,border=7pt]{standalone}
\usepackage[T1]{fontenc}
\usepackage{lmodern,amsmath,amssymb}
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\exhyphenpenalty=10000
\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds}
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\definecolor{warm}{HTML}{FAF5E8}
\tikzset{every node/.style={font=\sffamily\small,text=ink},
 box/.style={draw=ink!65,rounded corners=2pt,fill=pale,align=center,inner sep=7pt},
 warmbox/.style={box,fill=warm},
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 lab/.style={font=\sffamily\footnotesize,fill=white,inner sep=2pt},
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 title/.style={font=\sffamily\bfseries\small,align=center}}
\begin{document}\begin{tikzpicture}

\node[box,text width=10.5cm] (inv) at (6,7.5) {\textbf{Induction hypothesis at step $i$}\\$\lambda(X_i)=\lambda(Y_i)=K_i$,\quad $|K_i|<N_i=|Pu_i|$};
\node[box,text width=10.5cm] (bridge) at (6,5.8) {The preceding bridge lies inside the common suffix $Pu_i$.\\Both wrapped words therefore use the same next branch.};
\node[box,text width=4.6cm] (miss) at (2.6,3.8) {\textbf{Bridge does not match $c$}\\Next receipt: $G_c(K_i)$\\$|G_c(K_i)|\le |K_i|<N_{i+1}$};
\node[box,text width=5.3cm] (match) at (9.3,3.8) {\textbf{Bridge matches $c$}\\Next receipt: $cK_ic$\\Only possible bound failure:\\$|K_i|=N_i-1$};
\node[warmbox,text width=5.3cm] (bad) at (9.3,1.5) {Then $(u_ic)^RP=cK_ic$ is an even suffix of $w^RP$ longer than $P$.\\This contradicts maximality.};
\node[box,text width=10.5cm] (next) at (6,-.7) {\textbf{Induction conclusion}\\The next receipts agree and stay shorter than the new common suffix.};
\draw[arr] (inv)--(bridge);\draw[arr] (bridge.south west)--(miss.north);\draw[arr] (bridge.south east)--(match.north);
\draw[arr] (match)--(bad);\draw[arr] (bad.south)--(next.north east);\draw[arr] (miss.south)--(2.6,.6)--(next.north west);

\end{tikzpicture}\end{document}