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LambdaCalculus with Birds !.tex
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LambdaCalculus with Birds !.tex
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% Default to the notebook output style
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\title{Lambda Calculus with the Birds !}
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\expandafter\def\csname PY@tok@vm\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
\expandafter\def\csname PY@tok@sa\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@dl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@s2\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@mb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@ch\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@cpf\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\def\PYZbs{\char`\\}
\def\PYZus{\char`\_}
\def\PYZob{\char`\{}
\def\PYZcb{\char`\}}
\def\PYZca{\char`\^}
\def\PYZam{\char`\&}
\def\PYZlt{\char`\<}
\def\PYZgt{\char`\>}
\def\PYZsh{\char`\#}
\def\PYZpc{\char`\%}
\def\PYZdl{\char`$}
\def\PYZhy{\char`\-}
\def\PYZsq{\char`\'}
\def\PYZdq{\char`\"}
\def\PYZti{\char`\~}
% for compatibility with earlier versions
\def\PYZat{@}
\def\PYZlb{[}
\def\PYZrb{]}
\makeatother
\makeatletter
\@addtoreset{section}{part}
\def\@part[#1]#2{%
\ifnum \c@secnumdepth >\m@ne
\refstepcounter{part}%
\addcontentsline{toc}{part}{\thepart\hspace{1em}#1}%
\else
\addcontentsline{toc}{part}{#1}%
\fi
{\parindent \z@ \raggedright
\interlinepenalty \@M
\normalfont\centering
\ifnum \c@secnumdepth >\m@ne
\LARGE\bfseries \partname\nobreakspace\thepart
\par\nobreak
\fi
\huge \bfseries #2%
\markboth{}{}\par}%
\nobreak
\vskip 3ex
\@afterheading}
\renewcommand\partname{Topic}
\makeatother
% Exact colors from NB
\definecolor{incolor}{rgb}{0.0, 0.0, 0.5}
\definecolor{outcolor}{rgb}{0.545, 0.0, 0.0}
% Prevent overflowing lines due to hard-to-break entities
\sloppy
% Setup hyperref package
\hypersetup{
breaklinks=true, % so long urls are correctly broken across lines
colorlinks=true,
urlcolor=urlcolor,
linkcolor=linkcolor,
citecolor=citecolor,
}
% Slightly bigger margins than the latex defaults
\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
\author{Satyajit Ghana}
\begin{document}
\maketitle
% \section{Lambda Calculus with Birds}\label{lambda-calculus-with-birds}
\part{Introduction to the Birds}
\section{Idiot}\label{idiot}
The Identity \[ \lambda a.a \]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}1}]:} \PY{n+nx}{I} \PY{o}{=} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{a}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}1}]:} [Function: I]
\end{Verbatim}
\section{Mocking Bird}\label{mocking-bird}
The Self Applicator \[ \lambda f. f f \]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}2}]:} \PY{n+nx}{M} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}2}]:} [Function: M]
\end{Verbatim}
\section{Kerstrel}\label{kerstrel}
The Truth \[ \lambda a b . a \]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}3}]:} \PY{n+nx}{K} \PY{o}{=} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{b} \PY{p}{=\PYZgt{}} \PY{n+nx}{a}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}3}]:} [Function: K]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}4}]:} \PY{n+nx}{K}\PY{p}{.}\PY{n+nx}{inspect} \PY{o}{=} \PY{p}{(}\PY{p}{)} \PY{p}{=\PYZgt{}} \PY{l+s+s1}{\PYZsq{}T / K\PYZsq{}}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}4}]:} [Function]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}5}]:} \PY{n+nx}{K}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}5}]:} T / K
\end{Verbatim}
\section{Kite}\label{kite}
The False \[ \lambda a b.b = \text{K I} = \text{C K}\]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}6}]:} \PY{n+nx}{KI} \PY{o}{=} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{b} \PY{p}{=\PYZgt{}} \PY{n+nx}{b}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}6}]:} [Function: KI]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}7}]:} \PY{n+nx}{KI}\PY{p}{.}\PY{n+nx}{inspect} \PY{o}{=} \PY{p}{(}\PY{p}{)} \PY{p}{=\PYZgt{}} \PY{l+s+s1}{\PYZsq{}F / KI\PYZsq{}}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}7}]:} [Function]
\end{Verbatim}
\section{Cardinal}\label{cardinal}
The Reverse \[ \lambda f a b. fb a \]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}8}]:} \PY{n+nx}{C} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{b} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{b}\PY{p}{)}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}8}]:} [Function: C]
\end{Verbatim}
\section{Blue Bird}\label{blue-bird}
The Composition \[ \lambda f g a. f(g a) \]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}9}]:} \PY{n+nx}{B} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{g} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{g}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}9}]:} [Function: B]
\end{Verbatim}
\section{Thrush}\label{thrush}
The Flipper \[ \lambda af. fa = \text{C I}\]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}10}]:} \PY{n+nx}{T} \PY{o}{=} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}10}]:} [Function: T]
\end{Verbatim}
\section{Vireo}\label{vireo}
The Smallest Data Structure \[ \lambda abf. fab = \text{B C T}\]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}11}]:} \PY{n+nx}{V} \PY{o}{=} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{b} \PY{p}{=\PYZgt{}} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}\PY{p}{(}\PY{n+nx}{b}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}11}]:} [Function: V]
\end{Verbatim}
\section{Black Bird}\label{black-bird}
Blue Bird for a function that takes two arguments\\
It's the composition of composition of composition
\[ \lambda fgab. f(gab) = \text{B B B}\]
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}12}]:} \PY{n+nx}{B1} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{g} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{b} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{g}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}\PY{p}{(}\PY{n+nx}{b}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}12}]:} [Function: B1]
\end{Verbatim}
\part{Birds ! Birds ! Birds !}
\section{Fun with these Birds !}\label{fun-with-these-birds}
%\chapter{Fun with these Birds}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}13}]:} \PY{n+nx}{C}\PY{p}{(}\PY{n+nx}{K}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}T\PYZsq{}}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}F\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}13}]:} 'F'
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}14}]:} \PY{n+nx}{C}\PY{p}{(}\PY{n+nx}{KI}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}T\PYZsq{}}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}F\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}14}]:} 'T'
\end{Verbatim}
Cardinal of Kerstrel is Kite\\
Cardinal of Kite is Kerstrel
\section{Church Encodings}\label{church-encodings}
\subsection{Booleans}\label{booleans}
$ \text{TRUE} = \lambda a b. a = \text{K}$
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}15}]:} \PY{n+nx}{TRUE} \PY{o}{=} \PY{n+nx}{K}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}15}]:} T / K
\end{Verbatim}
$ \text{FALSE} = \lambda ab. b = \text{KI} = \text{CK}$
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}16}]:} \PY{n+nx}{FALSE} \PY{o}{=} \PY{n+nx}{KI}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}16}]:} F / KI
\end{Verbatim}
$ \text{NOT} = \lambda p. pFT = \text{C}$
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}17}]:} \PY{n+nx}{NOT} \PY{o}{=} \PY{n+nx}{C}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}17}]:} [Function: C]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}18}]:} \PY{n+nx}{NOT}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}T\PYZsq{}}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}F\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}18}]:} 'F'
\end{Verbatim}
$ \text{AND} = \lambda p q. pqF = \lambda p q . pqp$
if p is false and it selects false, then p can select itself
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}19}]:} \PY{n+nx}{AND} \PY{o}{=} \PY{n+nx}{p} \PY{p}{=\PYZgt{}} \PY{n+nx}{q} \PY{p}{=\PYZgt{}} \PY{n+nx}{p}\PY{p}{(}\PY{n+nx}{q}\PY{p}{)}\PY{p}{(}\PY{n+nx}{p}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}19}]:} [Function: AND]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}20}]:} \PY{n+nx}{AND}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}20}]:} T / K
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}21}]:} \PY{n+nx}{AND}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}21}]:} F / KI
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}22}]:} \PY{n+nx}{AND}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}22}]:} F / KI
\end{Verbatim}
$ \text{OR} = \lambda p q. pTq = \lambda p q. p p q = \lambda p q. M q
= \text{M}$
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}23}]:} \PY{n+nx}{OR} \PY{o}{=} \PY{n+nx}{p} \PY{p}{=\PYZgt{}} \PY{n+nx}{q} \PY{p}{=\PYZgt{}} \PY{n+nx}{M}\PY{p}{(}\PY{n+nx}{p}\PY{p}{)}\PY{p}{(}\PY{n+nx}{q}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}23}]:} [Function: OR]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}24}]:} \PY{n+nx}{OR}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}24}]:} T / K
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}25}]:} \PY{n+nx}{OR}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}25}]:} T / K
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}26}]:} \PY{n+nx}{OR}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}26}]:} F / KI
\end{Verbatim}
\((\lambda pq.ppq)xy = xxy\)\\
\(M x y = xxy\)\\
\(\text{OR} = \text{M}\)
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}27}]:} \PY{n+nx}{M}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}27}]:} T / K
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}28}]:} \PY{n+nx}{M}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}28}]:} T / K
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}29}]:} \PY{n+nx}{M}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}29}]:} F / KI
\end{Verbatim}
\(\text{BEQ} = \lambda pq.pq(\text{NOT}q)\)
This is also the \texttt{XNOR} or the Equality
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}30}]:} \PY{n+nx}{BEQ} \PY{o}{=} \PY{n+nx}{p} \PY{p}{=\PYZgt{}} \PY{n+nx}{q} \PY{p}{=\PYZgt{}} \PY{n+nx}{p}\PY{p}{(}\PY{n+nx}{q}\PY{p}{)}\PY{p}{(}\PY{n+nx}{NOT}\PY{p}{(}\PY{n+nx}{q}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}30}]:} [Function: BEQ]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}31}]:} \PY{n+nx}{BEQ}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}T\PYZsq{}}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}F\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}31}]:} 'T'
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}32}]:} \PY{n+nx}{BEQ}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}T\PYZsq{}}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}F\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}32}]:} 'T'
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}33}]:} \PY{n+nx}{BEQ}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}T\PYZsq{}}\PY{p}{)}\PY{p}{(}\PY{l+s+s1}{\PYZsq{}F\PYZsq{}}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}33}]:} 'F'
\end{Verbatim}
\subsection{Numerals}\label{numerals}
\(\text{ZERO} = \lambda fa.a\)
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}34}]:} \PY{n+nx}{ZERO} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{a}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}34}]:} [Function: ZERO]
\end{Verbatim}
\(\text{ONCE} = \lambda f a . f a\)
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}35}]:} \PY{n+nx}{ONCE} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}35}]:} [Function: ONCE]
\end{Verbatim}
$ \text{TWICE} = \lambda f a . f(f a) $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}36}]:} \PY{n+nx}{TWICE} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}36}]:} [Function: TWICE]
\end{Verbatim}
$ \text{THRICE} = \lambda fa.f(f(f a) $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}37}]:} \PY{n+nx}{THRICE} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}37}]:} [Function: THRICE]
\end{Verbatim}
$ \text{FOURFOLD} = \lambda f a. f(f(f(f a))) $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}38}]:} \PY{n+nx}{FOURFOLD} \PY{o}{=} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{a} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{a}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}38}]:} [Function: FOURFOLD]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}39}]:} \PY{n+nx}{N0} \PY{o}{=} \PY{n+nx}{ZERO}
\PY{n+nx}{N1} \PY{o}{=} \PY{n+nx}{ONCE}
\PY{n+nx}{N2} \PY{o}{=} \PY{n+nx}{TWICE}
\PY{n+nx}{N3} \PY{o}{=} \PY{n+nx}{THRICE}
\PY{n+nx}{N4} \PY{o}{=} \PY{n+nx}{FOURFOLD}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}39}]:} [Function: FOURFOLD]
\end{Verbatim}
$ \text{SUCC} = \lambda n f a. f(nfa) = \lambda n f.\text{B}f(nf) $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}40}]:} \PY{n+nx}{SUCC} \PY{o}{=} \PY{n+nx}{n} \PY{p}{=\PYZgt{}} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{f}\PY{p}{(}\PY{n+nx}{n}\PY{p}{(}\PY{n+nx}{f}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}40}]:} [Function: SUCC]
\end{Verbatim}
This is the same as function composition, why not use the Blue Bird ?
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}41}]:} \PY{n+nx}{SUCC} \PY{o}{=} \PY{n+nx}{n} \PY{p}{=\PYZgt{}} \PY{n+nx}{f} \PY{p}{=\PYZgt{}} \PY{n+nx}{B}\PY{p}{(}\PY{n+nx}{f}\PY{p}{)}\PY{p}{(}\PY{n+nx}{n}\PY{p}{(}\PY{n+nx}{f}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}41}]:} [Function: SUCC]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}42}]:} \PY{n+nx}{SUCC}\PY{p}{(}\PY{n+nx}{THRICE}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}42}]:} 4
\end{Verbatim}
$ \text{ADD} = \lambda n k. n \text{ SUCC } k $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}43}]:} \PY{n+nx}{ADD} \PY{o}{=} \PY{n+nx}{n} \PY{p}{=\PYZgt{}} \PY{n+nx}{k} \PY{p}{=\PYZgt{}} \PY{n+nx}{n}\PY{p}{(}\PY{n+nx}{SUCC}\PY{p}{)}\PY{p}{(}\PY{n+nx}{k}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}43}]:} [Function: ADD]
\end{Verbatim}
Read it as n times SUCC, applied to k, same as
\texttt{SUCC(SUCC(SUCC(k)))} where SUCC is n times this example was 3
added to k
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}44}]:} \PY{n+nx}{ADD}\PY{p}{(}\PY{n+nx}{N3}\PY{p}{)}\PY{p}{(}\PY{n+nx}{N2}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}44}]:} 5
\end{Verbatim}
$ \text{MULT} = \lambda n k f. n(kf) = \lambda n k. \text{B}nk =
\text{B}$
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}45}]:} \PY{n+nx}{MULT} \PY{o}{=} \PY{n+nx}{n} \PY{p}{=\PYZgt{}} \PY{n+nx}{k} \PY{p}{=\PYZgt{}} \PY{n+nx}{B}\PY{p}{(}\PY{n+nx}{n}\PY{p}{)}\PY{p}{(}\PY{n+nx}{k}\PY{p}{)}
\PY{n+nx}{MULT} \PY{o}{=} \PY{n+nx}{B}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}45}]:} [Function: B]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}46}]:} \PY{n+nx}{MULT}\PY{p}{(}\PY{n+nx}{N3}\PY{p}{)}\PY{p}{(}\PY{n+nx}{N2}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}46}]:} 6
\end{Verbatim}
$ \text{POW} = \lambda n k. k \text{ MULT }n = \text{C I} = \text{T}$
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}47}]:} \PY{n+nx}{POW} \PY{o}{=} \PY{n+nx}{T}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}47}]:} [Function: T]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}48}]:} \PY{n+nx}{POW}\PY{p}{(}\PY{n+nx}{N3}\PY{p}{)}\PY{p}{(}\PY{n+nx}{N2}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}48}]:} 9
\end{Verbatim}
$ \text{IS0} = \lambda n. n (K F) T $
KF always gives a False, Kerstrel of False
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}49}]:} \PY{n+nx}{IS0} \PY{o}{=} \PY{n+nx}{n} \PY{p}{=\PYZgt{}} \PY{n+nx}{n}\PY{p}{(}\PY{n+nx}{K}\PY{p}{(}\PY{n+nx}{FALSE}\PY{p}{)}\PY{p}{)}\PY{p}{(}\PY{n+nx}{TRUE}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}49}]:} [Function: IS0]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}50}]:} \PY{n+nx}{IS0}\PY{p}{(}\PY{n+nx}{N0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}50}]:} T / K
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}51}]:} \PY{n+nx}{IS0}\PY{p}{(}\PY{n+nx}{N1}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}51}]:} F / KI
\end{Verbatim}
$ \text{PRED} = \lambda n.
n(\lambda g.\text{IS0}(g\text{N1})\text{I}(\text{B SUCC }
g))(\text{K N0})\text{N0}$
$ \text{V I M} $
Vireo is the smallest data structure, you box in the two arguments like
f(a)(b) and then whenever you want a value back you send in a function
to recieve eithe a or b
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}52}]:} \PY{n+nx}{vim} \PY{o}{=} \PY{n+nx}{V}\PY{p}{(}\PY{n+nx}{I}\PY{p}{)}\PY{p}{(}\PY{n+nx}{M}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}52}]:} [Function]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}53}]:} \PY{n+nx}{vim}\PY{p}{(}\PY{n+nx}{K}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}53}]:} [Function: I]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}54}]:} \PY{n+nx}{vim}\PY{p}{(}\PY{n+nx}{C}\PY{p}{(}\PY{n+nx}{K}\PY{p}{)}\PY{p}{)} \PY{c+c1}{// C(K) = KI}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}54}]:} [Function: M]
\end{Verbatim}
$ \text{PAIR} = \text{V} $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}55}]:} \PY{n+nx}{PAIR} \PY{o}{=} \PY{n+nx}{V}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}55}]:} [Function: V]
\end{Verbatim}
$ \text{FST} = \lambda p. p K $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}56}]:} \PY{n+nx}{FST} \PY{o}{=} \PY{n+nx}{p} \PY{p}{=\PYZgt{}} \PY{n+nx}{p}\PY{p}{(}\PY{n+nx}{K}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}56}]:} [Function: FST]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}57}]:} \PY{n+nx}{FST}\PY{p}{(}\PY{n+nx}{vim}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}57}]:} [Function: I]
\end{Verbatim}
$ \text{SND} = \lambda p. p(KI) $
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}58}]:} \PY{n+nx}{SND} \PY{o}{=} \PY{n+nx}{p} \PY{p}{=\PYZgt{}} \PY{n+nx}{p}\PY{p}{(}\PY{n+nx}{KI}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}58}]:} [Function: SND]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}59}]:} \PY{n+nx}{SND}\PY{p}{(}\PY{n+nx}{vim}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}59}]:} [Function: M]
\end{Verbatim}
$ \text{PHI} =
\lambda p.\text{V}(\text{SND }p)(\text{SUCC }(\text{SND})p) $
copy 2nd to 1st, and increment 2nd
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}60}]:} \PY{n+nx}{PHI} \PY{o}{=} \PY{n+nx}{p} \PY{p}{=\PYZgt{}} \PY{n+nx}{V}\PY{p}{(}\PY{n+nx}{SND}\PY{p}{(}\PY{n+nx}{p}\PY{p}{)}\PY{p}{)}\PY{p}{(}\PY{n+nx}{SUCC}\PY{p}{(}\PY{n+nx}{SND}\PY{p}{(}\PY{n+nx}{p}\PY{p}{)}\PY{p}{)}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}60}]:} [Function: PHI]
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}61}]:} \PY{n+nx}{SND}\PY{p}{(}\PY{n+nx}{PHI}\PY{p}{(}\PY{n+nx}{V}\PY{p}{(}\PY{n+nx}{M}\PY{p}{)}\PY{p}{(}\PY{n+nx}{N3}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}61}]:} 4
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}62}]:} \PY{n+nx}{FST}\PY{p}{(}\PY{n+nx}{PHI}\PY{p}{(}\PY{n+nx}{V}\PY{p}{(}\PY{n+nx}{M}\PY{p}{)}\PY{p}{(}\PY{n+nx}{N3}\PY{p}{)}\PY{p}{)}\PY{p}{)}\PY{p}{(}\PY{n+nx}{x} \PY{p}{=\PYZgt{}} \PY{n+nx}{x} \PY{o}{+} \PY{l+m+mi}{1}\PY{p}{)}\PY{p}{(}\PY{l+m+mi}{0}\PY{p}{)}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{outcolor}Out[{\color{outcolor}62}]:} 3
\end{Verbatim}
\texttt{N0\ PHI(N0,\ N0)\ =\ (N0,\ N0)}\\
\texttt{N1\ PHI(N0,\ N0)\ =\ (N0,\ N1)}\\
\texttt{N2\ PHI(N0,\ N0)\ =\ (N1,\ N2)}\\
\texttt{...}\\
\texttt{N8\ PHI(N0,\ N0)\ =\ (N7,\ N8)}
Holy Cow !, you got the predecessor working !