I would like to have some macros which have binary trees as arguments. For the purpose of these macros, a binary tree is either root or it is {branch1}{branch2} for two binary trees branch1 and branch2 and these macros should be defined recursively. My thought was that such a macro would look like
\newcommand{\macro}[1]{\bintree@induction{#1}{\macro@root{#1}}{\macro@branches{#1}}
\newcommand{\macro@root}[1]{%describes what to do in the case of a root
<some code>}
\newcommand{\macro@branches}[1]{%describes what to do in the case of branches.
%Can use {\bintree@branchone #1} and {\bintree@branchtwo #1}
%here to access the branches
<some code>}
\newcommand{\bintree@induction}[3]{
\ifx{}{\bintree@root #1}
%the binary tree is of the form {branch1}{branch2}
#3
\else
%the binary tree is of the form root
#2
\fi}
%when it finds two branches, it gobbles both of them
\newcommand{\bintree@root}{\@ifnextchar\bgroup{\@bintree@root}{}}
\newcommand{\@bintree@root}[1]{\@ifnextchar\bgroup{\@gobble}{#1}}
%To define the macro recursively, we will usually need to handle the two
%branches. Here's the mechanism that selects them.
%This selects the first branch if there is one; all else it does is irrelevant
\newcommand{\bintree@branchone}{\@ifnextchar\bgroup{\@bintree@branchone}{}}
\newcommand{\@bintree@branchone}[1]{#1\@gobble}
%Gobbles the first branch, takes the second out of its braces
\newcommand{\bintree@branchtwo}{\@ifnextchar\bgroup{\@bintree@branchtwo}{}}
\newcommand{\@bintree@branchtwo}[1]{\@ifnextchar\bgroup{\@@bintree@branchtwo}{}}
\newcommand{\@@bintree@branchtwo}[1]{#1}
But this doesn't work. I believe it's because \bintree@root #1 isn't truly empty in the case the argument #1 is of the form {branch1}{branch2}. How can I accomplish this? Below I attach a MWE where I need this. There I've found some workarounds, but I don't see how I can define a macro by induction on binary trees in general in a similar way. So my question is: how I can do the macro \bintree@induction correctly.
If anybody knows of a different approach to this problem I'd be very pleased to know about that too. I have the feeling my code is very clumsy with the overuse of all the @'s.
\documentclass{article}
\usepackage{amsmath}
\usepackage{comment}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% General induction on binary trees.
%%%% This is used for the macro \jhom@dom@wk below only. For the others
%%%% I found an ad hoc method to get things working...
\makeatletter
\newcommand{\bintree@induction}[3]{
\ifx{}{\bintree@root #1}
%the binary tree is of the form {branch1}{branch2}
#3
\else
%the binary tree is of the form root
#2
\fi}
%when it finds two branches, it gobbles both of them
\newcommand{\bintree@root}{\@ifnextchar\bgroup{\@bintree@root}{}}
\newcommand{\@bintree@root}[1]{\@ifnextchar\bgroup{\@gobble}{#1}}
%To define the macro recursively, we will usually need to handle the two
%branches. Here's the mechanism that selects them.
%This selects the first branch if there is one; all else it does is irrelevant
\newcommand{\bintree@branchone}{\@ifnextchar\bgroup{\@bintree@branchone}{}}
\newcommand{\@bintree@branchone}[1]{#1\@gobble}
%Gobbles the first branch, takes the second out of its braces
\newcommand{\bintree@branchtwo}{\@ifnextchar\bgroup{\@bintree@branchtwo}{}}
\newcommand{\@bintree@branchtwo}[1]{\@ifnextchar\bgroup{\@@bintree@branchtwo}{}}
\newcommand{\@@bintree@branchtwo}[1]{#1}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\jterm}[3]{#1 \vdash #3:#2}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%
%%%% To get a feeling of how the command works, here are a few examples.
%%%% \ctxext{A}{B} will print A.B
%%%% \ctxext{{A}{B}}{C} will print (A.B).C
%%%% \ctxext{{{A}{B}}{C}}{{D}{E}} will print ((A.B).C).(D.E)
\makeatletter
\newcommand{\ctxext}[2]{\@ctxext@ctx #1.\@ctxext@type #2}
\newcommand{\@ctxext}{\@ifnextchar\bgroup{\@@ctxext}{}}
\newcommand{\@ctxext@ctx}{\@ifnextchar\ctxext{\@ctxext@nested}{\@ifnextchar \ctxwk{\@ctxwk@nested}{\@ctxext}}}
\newcommand{\@ctxext@type}{\@ifnextchar\ctxext{\@ctxext@nested}{\@ifnextchar\subst{\@subst@nested}{\@ctxext}}}
\newcommand{\@@ctxext}[1]{\@ifnextchar\bgroup{\@ctxext@parens{#1}}{#1}}
\newcommand{\@ctxext@parens}[2]{(\ctxext{#1}{#2})}
\newcommand{\@ctxext@nested}[3]{\@ctxext@parens{#2}{#3}}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%
\makeatletter
\newcommand{\ctxwk}[2]{\langle\@ctxwk@act #1\rangle\@ctxwk@pass #2}
\newcommand{\@ctxwk}{\@ifnextchar\bgroup{\@@ctxwk}{}}
\newcommand{\@@ctxwk}[1]{\@ifnextchar\bgroup{\ctxwk{#1}}{#1}}
\newcommand{\@ctxwk@act}{\@ctxwk}
\newcommand{\@ctxwk@pass}{\@ifnextchar\ctxext{\@ctxext@nested}{\@ifnextchar\subst{\@subst@nested}{\@ctxwk}}}
\newcommand{\@ctxwk@parens}[2]{(\ctxwk{#1}{#2})}
\newcommand{\@ctxwk@nested}[3]{\@ctxwk@parens{#2}{#3}}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\makeatletter
\newcommand{\jhom}[3]{\jterm{\jhom@dom@ext{#1}}{\jhom@dom@wk{#1}{#2}}{#3}}
%%%% First define \jhom@dom@ext
\newcommand{\jhom@dom@ext}[1]{\@jhom@dom@ext #1}
\newcommand{\@jhom@dom@ext}{\@ifnextchar\bgroup{\@@jhom@dom@ext}{}}
\newcommand{\@@jhom@dom@ext}[1]{\@ifnextchar\bgroup{\@@@jhom@dom@ext{#1}}{#1}}
\newcommand{\@@@jhom@dom@ext}[2]{\ctxext{\jhom@dom@ext@nested #1}{\ctxwk{\jhom@dom@ext{#1}}{\jhom@dom@ext@nested #2}}}
%%%% The later occurences of context extension should be with parentheses
\newcommand{\jhom@dom@ext@nested}{\@ifnextchar\bgroup{\@jhom@dom@ext@nested}{}}
\newcommand{\@jhom@dom@ext@nested}[1]{\@ifnextchar\bgroup{\@@jhom@dom@ext@nested{#1}}{#1}}
\newcommand{\@@jhom@dom@ext@nested}[2]{\@ctxext@parens{\jhom@dom@ext@nested #1}{\ctxwk{\jhom@dom@ext{#1}}{\@jhom@dom@ext@nested #2}}}
%%%% Now define \jhom@dom@wk. Here I don't see how I can do something similar as
%%%% before. Hence try the more general approach that doesn't really work...
\newcommand{\jhom@dom@wk}[2]{\bintree@induction{#1}{\jhom@dom@wk@root{#1}{#2}}
{\jhom@dom@wk@branches{#1}{#2}}}
\newcommand{\jhom@dom@wk@root}[2]{\ctxwk{\bintree@root #1}{#2}}
\newcommand{\jhom@dom@wk@branches}[2]{\ctxwk{\jhom@dom@ext{\bintree@branchone #1}}{\jhom@dom@wk{\bintree@branchtwo #1}{#2}}}
\makeatother
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
The following output
\begin{align*}
& \jhom{\Gamma}{\Delta}{f}\\
& \jhom{{\Gamma_1}{\Gamma_2}}{\Delta}{f}\\
& \jhom{{{\Gamma_{11}}{\Gamma_{12}}}{\Gamma_2}}{\Delta}{f}\\
& \jhom{{\Gamma_1}{{\Gamma_{21}}{\Gamma_{22}}}}{\Delta}{f}
\end{align*}
should be the same as
\begin{align*}
& \jterm{\Gamma}{\Delta}{f}\\
& \jterm{\ctxext{\Gamma_1}{\ctxwk{\Gamma_1}{\Gamma_2}}}{\ctxwk{\Gamma_1}{{\Gamma_2}{\Delta}}}{f}\\
& \jterm{\ctxext{{\Gamma_{11}}{\ctxwk{\Gamma_{11}}{\Gamma_{12}}}}
{\ctxwk{\ctxext{\Gamma_{11}}{\ctxwk{\Gamma_{11}}{\Gamma_{12}}}}
{\Gamma_2}}}
{\ctxwk{\ctxext{\Gamma_{11}}{\ctxwk{\Gamma_{11}}{\Gamma_{12}}}}{{\Gamma_2}{\Delta}}}{f}\\
& \jterm{\ctxext{\Gamma_1}{\ctxwk{\Gamma_1}{\ctxext{\Gamma_{21}}{\ctxwk{\Gamma_{21}}{\Gamma_{22}}}}}}{\ctxwk{\Gamma_1}{{\ctxext{\Gamma_{21}}{\ctxwk{\Gamma_{21}}{\Gamma_{22}}}}{\Delta}}}{f}
\end{align*}
\end{document}
\ctxextcommand (and also with the\ctxwkmacro in my MWE) but when the macro is more involved (as with the macro\jhom@dom@wkin the MWE) I don't see how I can do it in a similar way. – Egbert Nov 04 '13 at 15:39