\documentclass[12pt,crop,varwidth]{standalone}

% hide font logs
\newcount\XeTeXtracingfonts

\usepackage{amsmath,amsfonts,bm,braket}
\usepackage{mathtools,mathabx}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage{tabularx,booktabs}
\usepackage{caption}
\usepackage{enumitem}
\usepackage{floatrow}
\usepackage{array}
\usepackage[final]{pdfpages}
\usepackage{epigraph}
\usepackage{trimclip}
\usepackage{lmodern}
\usepackage[mono=false]{libertinus}
\usepackage{xfakebold}
\usepackage{scalerel}
\usepackage{calrsfs}
\DeclareMathAlphabet{\pazocal}{OMS}{zplm}{m}{n}
\usepackage{tikz}
\usetikzlibrary{positioning,calc,hobby}
\usepackage{pgfplots}
\pgfplotsset{width=\textwidth,compat=1.9,every axis plot/.append style={line width=1pt}}
\newcommand{\verticallycenteredcdots}{\makebox(0pt,0pt)[c]{\strut $\cdots$}}

\tikzset{
	term/.style={inner sep=0pt,minimum size=0cm},
	% abs/.pic={
	% 	\node[term] (-k) {};
	% 	\node[term, below=3em of -k] (-b) {};
	% 	\node[term, left=1em of -b] (-x) {};
	% 	\coordinate (-mid) at ($(-k)!0.5!(-b)$);
	% },
	% app/.pic={
	% 	\node[term] (-k) {};
	% 	\node[term, below=3em of -k] (-f) {};
	% 	\node[term, right=1em of -k,yshift=.2em] (-a) {};
	% 	\coordinate (-mid) at ($(-k)!0.5!(-f)$);
	% },
	% dup/.pic={
	% 	\node[term] (-x3) {};
	% 	\node[term, left=1em of -x3] (-x1) {};
	% 	\node[term, right=1em of -x3] (-x2) {};
	% 	\node[term, below=3em of -x3] (-d) {};
	% 	\coordinate (-mid) at ($(-d)!0.5!(-x3)$);
	% },
	abs/.pic={
		\node[term] (-k) {};
		\node[term, below=.5 of -k] (-b) {};
		\node[term, left=.15 of -b] (-x) {};
		\coordinate (-mid) at ($(-k)!0.5!(-b)$);
	},
	app/.pic={
		\node[term] (-k) {};
		\node[term, below=.5 of -k] (-f) {};
		\node[term, right=.15 of -k] (-a) {};
		\coordinate (-mid) at ($(-k)!0.5!(-f)$);
	},
	dup/.pic={
		\node[term] (-x3) {};
		\node[term, left=.15 of -x3] (-x1) {};
		\node[term, right=.15 of -x3] (-x2) {};
		\node[term, below=.5 of -x3] (-d) {};
		\coordinate (-mid) at ($(-d)!0.5!(-x3)$);
	},
	from ik/.style={out angle=90,out=90},
	to k/.style={in angle=90,in=90},
	%
	to x/.style={in angle=90,in=90},
	from b/.style={out angle=90,out=90},
	from ix/.style={out angle=270,out=270},
	to ib/.style={in angle=-90,in=-90},
	%
	from f/.style={out angle=90,out=90},
	from a/.style={out angle=-90,out=-90},
	to if/.style={in angle=-90,in=-90},
	to ia/.style={in angle=90,in=90},
	%
	from d/.style={out angle=90,out=90},
	from ix1/.style={out angle=90,out=90},
	from ix3/.style={out angle=90,out=90}, % middle
	from ix2/.style={out angle=90,out=90},
	to id/.style={in angle=-90,in=-90},
	to x1/.style={in angle=-90,in=-90},
	to x3/.style={in angle=-90,in=-90},
	to x2/.style={in angle=-90,in=-90},
}

\begin{document}

% --- START 2026-04-09-17.md ---

% --- ENCODING ---

% \begin{tikzpicture}[node distance=5cm]
% 	\pic (A) {abs};
% 	\node at (A-x)[yshift=-.12cm] {\tiny$x$};
% 	\node at (A-b)[yshift=-.1cm] {\tiny$b$};
% 	\node at (A-k)[yshift=.22cm] {\tiny$k$};

% 	\draw[from ix, to x] (A-mid) to (A-x);
% 	\draw[from b, to ib] (A-b) to (A-mid);
% 	\draw[from ik, to k] (A-mid) to (A-k);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=5cm]
% 	\pic (A) {app};
% 	\node at (A-f)[yshift=-.12cm] {\tiny$f$};
% 	\node at (A-a)[yshift=.1cm] {\tiny$a$};
% 	\node at (A-k)[yshift=.22cm] {\tiny$k$};

% 	\draw[from f, to if] (A-f) to (A-mid);
% 	\draw[from a, to ia] (A-a) to (A-mid);
% 	\draw[from ik, to k] (A-mid) to (A-k);
% \end{tikzpicture}

% DUPLICATION
% \begin{tikzpicture}[node distance=5cm]
% 	\pic (A) {dup};
% 	\node at (A-d)[yshift=-.12cm] {\tiny$x$};
% 	\node at (A-x1)[yshift=.1cm] {\tiny$x_1$};
% 	\node at (A-x2)[yshift=.1cm] {\tiny$x_n$};
% 	\node at ([xshift=-.2cm]A-x3)[yshift=.1cm] {\tiny$x_2$};
% 	\node at ([xshift=.15cm]A-x3) {\tiny$\dots$};

% 	\draw[from ix1, to x1] (A-mid) to (A-x1);
% 	\draw[from ix2, to x2] (A-mid) to (A-x2);
% 	\draw[from ix3, to x3] (A-mid) to ([xshift=-.2cm]A-x3);
% 	\draw[from d, to id] (A-d) to (A-mid);
% \end{tikzpicture}

% BETA REDUCTION
% \begin{tabular}{rcl}
% 	\begin{tikzpicture}[baseline=(p),node distance=5cm]
% 		\pic (B) {app};
% 		\node at (B-f)[yshift=-.12cm] {};
% 		\node at (B-a)[yshift=.1cm] {\tiny$a$};
% 		\node at (B-k)[yshift=.22cm] {\tiny$k$};
% 		\draw[from f, to if] (B-f) to (B-mid);
% 		\draw[from a, to ia] (B-a) to (B-mid);
% 		\draw[from ik, to k] (B-mid) to (B-k);

% 		\pic[shift=(B-f),yshift=1.2cm] (A) {abs};
% 		\node at (A-x)[yshift=-.12cm] {\tiny$x$};
% 		\node at (A-b)[yshift=-.1cm] {\tiny$b$};
% 		\node at (A-k)[yshift=.22cm] {};
% 		\draw[from ix, to x] (A-mid) to (A-x);
% 		\draw[from b, to ib] (A-b) to (A-mid);
% 		\draw[from ik, to k] (A-mid) to (A-k);
% 		\coordinate (p) at (current bounding box.center);
% 	\end{tikzpicture} &
% 	{\tiny$\tiny\rhd$}                                  &
% 	\begin{tikzpicture}[baseline=(p),node distance=5cm]
% 		\pic (B) {app};
% 		\node at (B-f)[yshift=-.12cm] {};
% 		\node at (B-a)[yshift=.1cm] {\tiny$a$};
% 		\node at (B-k)[yshift=.22cm] {\tiny$k$};
% 		\draw[from f, to if] (B-f) to (B-mid);
% 		% \draw[from a, to ia] (B-a) to (B-mid);
% 		\draw[from ik, to k] (B-mid) to (B-k);

% 		\pic[shift=(B-f),yshift=1.2cm] (A) {abs};
% 		\node at (A-x)[yshift=-.12cm] {\tiny$x$};
% 		\node at (A-b)[yshift=-.1cm] {\tiny$b$};
% 		\node at (A-k)[yshift=.22cm] {};
% 		% \draw[from ix, to x] (A-mid) to (A-x);
% 		\draw[from b, to ib] (A-b) to (A-mid);
% 		\draw[from ik, to k] (A-mid) to (A-k);

% 		\draw[from a, to x] (B-a) to (A-x);
% 		\coordinate (p) at (current bounding box.center);
% 	\end{tikzpicture}
% \end{tabular}

% --- MACKIE IDENTITY ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (A) {abs};

% 	\draw[from ik, to k] (A-mid) to (A-k);
% 	\draw[from ix, to ib] (A-mid) to[curve through={(A-x)}] (A-mid);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (A) {abs};
% 	\pic[shift=(A-b)] (B) {app};
% 	\pic[right=.75 of B-mid] (B1) {abs};
% 	\pic[shift=(B-f)] (C) {app};
% 	\pic[right=.5 of C-mid] (C1) {abs};
% 	\pic[shift=(C-f)] (D) {app};
% 	\pic[right=.25 of D-mid] (D1) {abs};

% 	\draw[from ik, to k] (A-mid) to (A-k);
% 	\draw[from ix, to if] (A-mid) to[curve through={(A-x)}] (D-mid);
% 	\draw[from ik, to ib] (B-mid) to (A-mid);
% 	\draw[from ik, to if] (C-mid) to (B-mid);
% 	\draw[from ik, to if] (D-mid) to (C-mid);

% 	\draw[from ik, to ia] (B1-mid) to[curve through={(B1-k)}] (B-mid);
% 	\draw[from ik, to ia] (C1-mid) to[curve through={(C1-k)}] (C-mid);
% 	\draw[from ik, to ia] (D1-mid) to[curve through={(D1-k)}] (D-mid);

% 	\draw[from ix, to ib] (B1-mid) to[curve through={(B1-x)(B1-b)}] (B1-mid);
% 	\draw[from ix, to ib] (C1-mid) to[curve through={(C1-x)(C1-b)}] (C1-mid);
% 	\draw[from ix, to ib] (D1-mid) to[curve through={(D1-x)(D1-b)}] (D1-mid);
% \end{tikzpicture}

% --- MACKIE NESTED ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (A) {abs};

% 	\draw[from ik, to k] (A-mid) to (A-k);
% 	\draw[from ix, to ib] (A-mid) to[curve through={(A-x)}] (A-mid);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (A) {abs};
% 	\pic[shift=(A-b)] (B1) {abs};
% 	\pic[shift=(B1-b)] (C1) {app};

% 	\pic[right=.5 of C1-mid] (B2) {abs};
% 	\pic[shift=(B2-b)] (C2) {app};

% 	\pic[right=.5 of C2-mid] (B3) {abs};
% 	\pic[shift=(B3-b)] (C3) {app};

% 	\draw[from ik, to k] (A-mid) to (A-k);
% 	\draw[from ik, to ib] (B1-mid) to (A-mid);

% 	\draw[from ik, to ib] (C1-mid) to (B1-mid);
% 	\draw[from ik, to ib] (C2-mid) to (B2-mid);
% 	\draw[from ik, to ib] (C3-mid) to (B3-mid);

% 	\draw[from ik, to ia] (B2-mid) to (C1-mid);
% 	\draw[from ik, to ia] (B3-mid) to (C2-mid);

% 	\draw[from ix, to ia] (A-mid) to[curve through={(A-x)([yshift=-1cm]B3-mid)(C3-a)}] (C3-mid);
% 	\draw[from ix, to if] (B1-mid) to[curve through={(B1-x)}] (C1-mid);
% 	\draw[from ix, to if] (B2-mid) to[curve through={(B2-x)}] (C2-mid);
% 	\draw[from ix, to if] (B3-mid) to[curve through={(B3-x)}] (C3-mid);
% \end{tikzpicture}

% --- BRUIJN ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (A) {abs};

% 	\draw[from ik, to k] (A-mid) to (A-k);
% 	\draw[from ix, to ib] (A-mid) to[curve through={(A-x)}] (A-mid);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (A1) {abs};
% 	\pic[shift=(A1-b)] (A2) {abs};
% 	\pic[shift=(A2-b)] (A3) {abs};
% 	\pic[shift=(A3-b)] (A4) {abs};

% 	\draw[from ik, to k] (A1-mid) to (A1-k);
% 	\draw[from ik, to ib] (A2-mid) to (A1-mid);
% 	\draw[from ik, to ib] (A3-mid) to (A2-mid);
% 	\draw[from ik, to ib] (A4-mid) to (A3-mid);

% 	% \draw[from ix, to x] (A1-mid) to (A1-x);
% 	\draw[from ix, to x] (A2-mid) to (A2-x);
% 	\draw[from ix, to x] (A3-mid) to (A3-x);
% 	\draw[from ix, to x] (A4-mid) to (A4-x);
% 	\draw[from ix, to ib] (A1-mid) to[curve through={(A1-x)}] (A4-mid);
% \end{tikzpicture}

% --- CHURCH ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (S) {abs};
% 	\pic[shift=(S-b)] (Z) {abs};

% 	\draw[from ik, to k] (S-mid) to (S-k);
% 	\draw[from ix, to x] (S-mid) to (S-x);
% 	\draw[from ik, to ib] (Z-mid) to (S-mid);
% 	\draw[from ix, to ib] (Z-mid) to[curve through={(Z-x)}] (Z-mid);
% \end{tikzpicture}

% NO-DUP
% \begin{tikzpicture}[node distance=1cm]
% 	\pic (S) {abs};
% 	\pic[shift=(S-b)] (Z) {abs};
% 	\pic[shift=(Z-b)] (A1) {app};
% 	\pic[right=.5 of A1-mid] (A2) {app};
% 	\pic[right=.5 of A2-mid] (A3) {app};

% 	\draw[from ik, to k] (S-mid) to (S-k);
% 	\draw[from ik, to ib] (Z-mid) to (S-mid);
% 	\draw[from ik, to ib] (A1-mid) to (Z-mid);

% 	\draw[from ik, to ia] (A2-mid) to[curve through={(A2-k)}] (A1-mid);
% 	\draw[from ik, to ia] (A3-mid) to[curve through={(A3-k)}] (A2-mid);

% 	\draw[from ix, to if] (S-mid) to[curve through={(S-x)}] (A1-mid);
% 	\draw[from ix, to if] (S-mid) to[curve through={(S-x)}] (A2-mid);
% 	\draw[from ix, to if] (S-mid) to[curve through={(S-x)}] (A3-mid);

% 	\draw[from ix, to ia] (Z-mid) to[curve through={(Z-x)([yshift=-.3cm]A3-mid)(A3-a)}] (A3-mid);
% \end{tikzpicture}

% YES-DUP
% \begin{tikzpicture}[node distance=1cm]
% 	\pic (S) {abs};
% 	\pic[shift=(S-b)] (Z) {abs};
% 	\pic[shift=(Z-b)] (A1) {app};
% 	\pic[right=.5 of A1-mid] (A2) {app};
% 	\pic[right=.5 of A2-mid] (A3) {app};
% 	\pic[below=.75 of A2-mid] (D) {dup};

% 	\draw[from ik, to k] (S-mid) to (S-k);
% 	\draw[from ik, to ib] (Z-mid) to (S-mid);
% 	\draw[from ik, to ib] (A1-mid) to (Z-mid);

% 	\draw[from ik, to ia] (A2-mid) to[curve through={(A2-k)}] (A1-mid);
% 	\draw[from ik, to ia] (A3-mid) to[curve through={(A3-k)}] (A2-mid);

% 	\draw[from ix1, to if] (D-mid) to[curve through={(D-x1)}] (A1-mid);
% 	\draw[from ix3, to if] (D-mid) to[curve through={(D-x3)}] (A2-mid);
% 	\draw[from ix2, to if] (D-mid) to[curve through={(D-x2)}] (A3-mid);
% 	\draw[from ix, to id] (S-mid) to[curve through={(S-x)([yshift=-1.8cm]A1-mid)(D-d)}] (D-mid);

% 	\draw[from ix, to ia] (Z-mid) to[curve through={(Z-x)(A3-mid)(A3-a)}] (A3-mid);
% \end{tikzpicture}

% --- SCOTT ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (S) {abs};
% 	\pic[shift=(S-b)] (Z) {abs};

% 	\draw[from ik, to k] (S-mid) to (S-k);
% 	\draw[from ix, to x] (S-mid) to (S-x);
% 	\draw[from ik, to ib] (Z-mid) to (S-mid);
% 	\draw[from ix, to ib] (Z-mid) to[curve through={(Z-x)}] (Z-mid);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (S1) {abs};
% 	\pic[shift=(S1-b)] (Z1) {abs};
% 	\pic[shift=(Z1-b)] (A1) {app};

% 	\pic[right=.5 of A1-mid] (S2) {abs};
% 	\pic[shift=(S2-b)] (Z2) {abs};
% 	\pic[shift=(Z2-b)] (A2) {app};

% 	\pic[right=.5 of A2-mid] (S3) {abs};
% 	\pic[shift=(S3-b)] (Z3) {abs};
% 	\pic[shift=(Z3-b)] (A3) {app};

% 	\pic[right=.5 of A3-mid] (S4) {abs};
% 	\pic[shift=(S4-b)] (Z4) {abs};

% 	\draw[from ik, to k] (S1-mid) to (S1-k);
% 	\draw[from ix, to x] (Z1-mid) to (Z1-x);
% 	\draw[from ik, to ib] (Z1-mid) to (S1-mid);
% 	\draw[from ik, to ib] (A1-mid) to (Z1-mid);
% 	\draw[from ix, to if] (S1-mid) to[curve through={(S1-x)}] (A1-mid);

% 	\draw[from ik, to ia] (S2-mid) to[curve through={(S2-k)}] (A1-mid);
% 	\draw[from ix, to x] (Z2-mid) to (Z2-x);
% 	\draw[from ik, to ib] (Z2-mid) to (S2-mid);
% 	\draw[from ik, to ib] (A2-mid) to (Z2-mid);
% 	\draw[from ix, to if] (S2-mid) to[curve through={(S2-x)}] (A2-mid);

% 	\draw[from ik, to ia] (S3-mid) to[curve through={(S3-k)}] (A2-mid);
% 	\draw[from ix, to x] (Z3-mid) to (Z3-x);
% 	\draw[from ik, to ib] (Z3-mid) to (S3-mid);
% 	\draw[from ik, to ib] (A3-mid) to (Z3-mid);
% 	\draw[from ix, to if] (S3-mid) to[curve through={(S3-x)}] (A3-mid);

% 	\draw[from ik, to ia] (S4-mid) to[curve through={(S4-k)}] (A3-mid);
% 	\draw[from ix, to x] (S4-mid) to (S4-x);
% 	\draw[from ik, to ib] (Z4-mid) to (S4-mid);
% 	\draw[from ix, to ib] (Z4-mid) to[curve through={(Z4-x)}] (Z4-mid);
% \end{tikzpicture}

% --- MOGENSEN BINARY ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (Z) {abs};
% 	\pic[shift=(Z-b)] (I) {abs};
% 	\pic[shift=(I-b)] (O) {abs};

% 	\draw[from ik, to k] (Z-mid) to (Z-k);
% 	\draw[from ik, to ib] (I-mid) to (Z-mid);
% 	\draw[from ik, to ib] (O-mid) to (I-mid);
% 	\draw[from ix, to ib] (Z-mid) to[curve through={(Z-x)([xshift=-.1cm,yshift=-.2cm]O-b)}] (O-mid);
% 	\draw[from ix, to x] (I-mid) to (I-x);
% 	\draw[from ix, to x] (O-mid) to (O-x);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (Z) {abs};
% 	\pic[shift=(Z-b)] (I) {abs};
% 	\pic[shift=(I-b)] (O) {abs};
% 	\pic[shift=(O-b)] (A1) {app};
% 	\pic[right=.6 of A1-mid] (A2) {app};
% 	\pic[below=.5 of A2-mid,xshift=-.27cm] (D) {dup};

% 	\draw[from ik, to k] (Z-mid) to (Z-k);
% 	\draw[from ik, to ib] (I-mid) to (Z-mid);
% 	\draw[from ik, to ib] (O-mid) to (I-mid);
% 	\draw[from ix, to x] (O-mid) to (O-x);

% 	\draw[from ik, to ib] (A1-mid) to (O-mid);
% 	\draw[from ik, to ia] (A2-mid) to[curve through={(A2-k)}] (A1-mid);

% 	\draw[from ix1, to if] (D-mid) to[curve through={(D-x1)}] (A1-mid);
% 	\draw[from ix2, to if] (D-mid) to[curve through={(D-x2)}] (A2-mid);
% 	\draw[from ix, to id] (I-mid) to[curve through={(I-x)([yshift=-1.4cm]A1-mid)(D-d)}] (D-mid);

% 	% \draw[from ix, to ia] (Z-mid) to[curve through={([yshift=-.1cm,xshift=.05cm]Z-x)(A2-mid)(A2-a)}] (A2-mid);
% 	\draw[from ix, to ia] (Z-mid) to[curve through={(Z-x)([yshift=-.2cm,xshift=-.12cm]A2-mid)(A2-a)}] (A2-mid);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (Z) {abs};
% 	\pic[shift=(Z-b)] (I) {abs};
% 	\pic[shift=(I-b)] (O) {abs};
% 	\pic[shift=(O-b)] (A1) {app};
% 	\pic[right=.5 of A1-mid] (A2) {app};
% 	\pic[right=.5 of A2-mid] (A3) {app};
% 	\pic[right=.5 of A3-mid] (A4) {app};
% 	\pic[right=.5 of A4-mid] (A5) {app};
% 	\pic[right=.5 of A5-mid] (A6) {app};
% 	\pic[below=1.4 of A2-mid] (DO) {dup};
% 	\pic[below=1 of A5-mid] (DI) {dup};

% 	\draw[from ik, to k] (Z-mid) to (Z-k);
% 	\draw[from ik, to ib] (I-mid) to (Z-mid);
% 	\draw[from ik, to ib] (O-mid) to (I-mid);

% 	\draw[from ik, to ib] (A1-mid) to (O-mid);
% 	\draw[from ik, to ia] (A2-mid) to[curve through={(A2-k)}] (A1-mid);
% 	\draw[from ik, to ia] (A3-mid) to[curve through={(A3-k)}] (A2-mid);
% 	\draw[from ik, to ia] (A4-mid) to[curve through={(A4-k)}] (A3-mid);
% 	\draw[from ik, to ia] (A5-mid) to[curve through={(A5-k)}] (A4-mid);
% 	\draw[from ik, to ia] (A6-mid) to[curve through={(A6-k)}] (A5-mid);

% 	\draw[from ix1, to if] (DI-mid) to[curve through={(DI-x1)}] (A2-mid);
% 	\draw[from ix3, to if] (DI-mid) to[curve through={(DI-x3)}] (A4-mid);
% 	\draw[from ix2, to if] (DI-mid) to[curve through={(DI-x2)}] (A6-mid);

% 	\draw[from ix1, to if] (DO-mid) to[curve through={(DO-x1)}] (A1-mid);
% 	\draw[from ix3, to if] (DO-mid) to[curve through={(DO-x3)}] (A3-mid);
% 	\draw[from ix2, to if] (DO-mid) to[curve through={(DO-x2)}] (A5-mid);

% 	\draw[from ix, to id] (I-mid) to[curve through={(I-x)(DI-d)}] (DI-mid);
% 	\draw[from ix, to id] (O-mid) to[curve through={(O-x)(DO-d)}] (DO-mid);

% 	\draw[from ix, to ia] (Z-mid) to[curve through={(Z-x)([yshift=-2cm]A2-mid)([xshift=.1cm]DI-d)([yshift=-.2cm,xshift=.15cm]A6-a)(A6-a)}] (A6-mid);
% \end{tikzpicture}

% --- WADSWORTH 1 ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (Z) {abs};
% 	\pic[shift=(Z-b)] (S) {app};
% 	\pic[shift=(S-a),yshift=-.1cm] (A) {abs};
% 	\pic[shift=(A-b)] (B) {abs};
% 	\pic[shift=(B-b)] (C) {abs};

% 	\draw[from ik, to k] (Z-mid) to (Z-k);
% 	\draw[from ik, to ib] (S-mid) to (Z-mid);
% 	\draw[from ix, to if] (Z-mid) to[curve through={(Z-x)([xshift=-.15cm,yshift=-.1cm]S-f)}] (S-mid);

% 	\draw[from ik, to ia] (A-mid) to (S-mid);
% 	\draw[from ik, to ib] (B-mid) to (A-mid);
% 	\draw[from ik, to ib] (C-mid) to (B-mid);
% 	\draw[from ix, to x] (A-mid) to (A-x);
% 	\draw[from ix, to ib] (B-mid) to[curve through={(B-x)([xshift=-.15cm,yshift=-.1cm]C-b)}] (C-mid);
% 	\draw[from ix, to x] (C-mid) to (C-x);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (X4) {abs};
% 	\pic[shift=(X4-b)] (X3) {abs};
% 	\pic[shift=(X3-b)] (X2) {abs};
% 	\pic[shift=(X2-b)] (X1) {abs};
% 	\pic[shift=(X1-b),yshift=-.5cm] (A4) {app};
% 	\pic[shift=(A4-f)] (A3) {app};
% 	\pic[shift=(A3-f)] (A2) {app};
% 	\pic[shift=(A2-f)] (A1) {app};
% 	\pic[right=.5 of A1-mid] (B3) {app};
% 	\pic[right=.5 of B3-mid] (B2) {app};
% 	\pic[right=.5 of B2-mid] (B1) {app};
% 	\pic[shift=(B3-f),xshift=.15cm] (D3) {dup};
% 	\pic[shift=(B2-f),xshift=.15cm] (D2) {dup};
% 	\pic[shift=(B1-f),xshift=.15cm] (D1) {dup};

% 	% end, as for zero
% 	\pic[right=.5 of B1-mid,yshift=-.2cm] (A) {abs};
% 	\pic[shift=(A-b)] (B) {abs};
% 	\pic[shift=(B-b)] (C) {abs};

% 	% \draw[from ik, to k] (A-mid) to (A-k);
% 	\draw[from ik, to ib] (B-mid) to (A-mid);
% 	\draw[from ik, to ib] (C-mid) to (B-mid);
% 	\draw[from ix, to x] (A-mid) to (A-x);
% 	\draw[from ix, to ib] (B-mid) to[curve through={(B-x)([xshift=-.15cm,yshift=-.1cm]C-b)}] (C-mid);
% 	\draw[from ix, to x] (C-mid) to (C-x);

% 	% spine between abs/app
% 	\draw[from ik, to k] (X4-mid) to (X4-k);
% 	\draw[from ik, to ib] (X3-mid) to (X4-mid);
% 	\draw[from ik, to ib] (X2-mid) to (X3-mid);
% 	\draw[from ik, to ib] (X1-mid) to (X2-mid);
% 	\draw[from ik, to ib] (A4-mid) to (X1-mid);
% 	\draw[from ik, to if] (A3-mid) to (A4-mid);
% 	\draw[from ik, to if] (A2-mid) to (A3-mid);
% 	\draw[from ik, to if] (A1-mid) to (A2-mid);

% 	% from dups
% 	\draw[from ix1, to if] (D3-mid) to[curve through={(B3-f)}] (B3-mid);
% 	\draw[from ix2, to ia] (D3-mid) to[curve through={(D3-x2)}] (A2-mid);
% 	\draw[from ix1, to if] (D2-mid) to[curve through={(B2-f)}] (B2-mid);
% 	\draw[from ix2, to ia] (D2-mid) to[curve through={(D2-x2)}] (A3-mid);
% 	\draw[from ix1, to if] (D1-mid) to[curve through={(B1-f)}] (B1-mid);
% 	\draw[from ix2, to ia] (D1-mid) to[curve through={(D1-x2)}] (A4-mid);

% 	% to dups
% 	\draw[from ix, to id] (X1-mid) to[curve through={(X1-x)([xshift=-.75cm]A1-mid)}] (D1-mid);
% 	\draw[from ix, to id] (X2-mid) to[curve through={(X2-x)([xshift=-1.25cm]A1-mid)}] (D2-mid);
% 	\draw[from ix, to id] (X3-mid) to[curve through={(X3-x)([xshift=-1.75cm]A1-mid)}] (D3-mid);
% 	\draw[from ix, to if] (X4-mid) to[curve through={(X4-x)([xshift=-2cm]A1-mid)}] (A1-mid);

% 	% to args
% 	\draw[from ik, to ia] (B3-mid) to[curve through={(B3-k)}] (A1-mid);
% 	\draw[from ik, to ia] (B2-mid) to[curve through={(B2-k)}] (B3-mid);
% 	\draw[from ik, to ia] (B1-mid) to[curve through={(B1-k)}] (B2-mid);
% 	\draw[from ik, to ia] (A-mid) to[curve through={(A-k)}] (B1-mid);
% \end{tikzpicture}

% --- WADSWORTH 2 ---

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (Z) {abs};
% 	\pic[shift=(Z-b)] (S0) {abs};
% 	\pic[shift=(S0-b)] (B) {app};

% 	\draw[from ik, to k] (Z-mid) to (Z-k);
% 	\draw[from ik, to ib] (S0-mid) to (Z-mid);
% 	\draw[from ik, to ib] (B-mid) to (S0-mid);

% 	\draw[from ix, to if] (Z-mid) to[curve through={(Z-x)(B-f)}] (B-mid);
% 	\draw[from ix, to ia] (S0-mid) to[curve through={(S0-x)(B-f)(B-a)}] (B-mid);
% \end{tikzpicture}

% \begin{tikzpicture}[node distance=1cm]
% 	\pic (Z) {abs};
% 	\pic[shift=(Z-b)] (S0) {abs};
% 	\pic[shift=(S0-b)] (S1) {abs};
% 	\pic[shift=(S1-b)] (S2) {abs};
% 	\pic[shift=(S2-b)] (S3) {abs};
% 	\pic[shift=(S3-b),yshift=-.5cm] (A1) {app};
% 	\pic[shift=(A1-f)] (A2) {app};
% 	\pic[shift=(A2-f)] (A3) {app};
% 	\pic[right=1.5 of A1-mid] (B) {app};

% 	\draw[from ik, to k] (Z-mid) to (Z-k);
% 	\draw[from ik, to ib] (S0-mid) to (Z-mid);
% 	\draw[from ik, to ib] (S1-mid) to (S0-mid);
% 	\draw[from ik, to ib] (S2-mid) to (S1-mid);
% 	\draw[from ik, to ib] (S3-mid) to (S2-mid);
% 	\draw[from ik, to ib] (A1-mid) to (S3-mid);
% 	\draw[from ik, to if] (A2-mid) to (A1-mid);
% 	\draw[from ik, to if] (A3-mid) to (A2-mid);

% 	\coordinate (l0) at ([xshift=-1.75cm,yshift=-1.75cm]A3-f);
% 	\coordinate (l1) at ([xshift=-2.00cm,yshift=-1.00cm]A3-f);
% 	\coordinate (l2) at ([xshift=-1.50cm,yshift=-0.75cm]A3-f);
% 	\coordinate (l3) at ([xshift=-1.00cm,yshift=-0.50cm]A3-f);
% 	\coordinate (l4) at ([xshift=-0.50cm,yshift=-0.25cm]A3-f);
% 	\coordinate (r0) at ([xshift=2.00cm]A3-f);
% 	\coordinate (r1) at ([xshift=1.50cm]A3-f);
% 	\coordinate (r2) at ([xshift=1.00cm]A3-f);
% 	\coordinate (r3) at ([xshift=0.50cm]A3-f);

% 	% \node at (l0) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (l1) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (l2) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (l3) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (l4) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (r0) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (r1) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (r2) [circle,fill,inner sep=1.5pt] {};
% 	% \node at (r3) [circle,fill,inner sep=1.5pt] {};

% 	\draw[from ix, to if] (Z-mid) to[curve through={(Z-x)(l0)(r1)(B-f)}] (B-mid);
% 	\draw[from ix, to if] (S0-mid) to[curve through={(S0-x)(l4)(A3-f)}] (A3-mid);
% 	\draw[from ix, to ia] (S1-mid) to[curve through={(S1-x)(l3)(r3)(A3-a)}] (A3-mid);
% 	\draw[from ix, to ia] (S2-mid) to[curve through={(S2-x)(l2)(r2)(A2-a)}] (A2-mid);
% 	\draw[from ix, to ia] (S3-mid) to[curve through={(S3-x)(l1)(r0)(B-a)}] (B-mid);
% 	\draw[from ik, to ia] (B-mid) to[curve through={(B-k)}] (A1-mid);
% \end{tikzpicture}

% --- END 2026-04-09-17.md ---

\end{document}
