% This package contains macros for typesetting proof trees.

\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{ebproof}[2013/04/06 EB's proof trees]

% The structure is inspired by the bussproofs package. In particular, proofs
% are built using statements that operate on a stack of sub-proofs, in postfix
% notation.The package provides an environment prooftree that is used as
%
%   \begin{prooftree}[OPTIONS]
%     STATEMENTS
%   \end{prooftree}
%
% This environment can be used either in math mode or in text mode. It
% produces a proof tree at the current position in the text flow. The package
% also provides a starred variant |prooftree*| that typesets the proof
% centered on a line of its own; it is essentially equivalent to wrapping the
% |prooftree| environment inside a |center| environment.
%
% The basic statements for building proofs are:
%
%   \Hypo[OPTIONS]{TEXT}
%
%     Make a hypothesis, i.e. a leaf in the proof tree, with no line above.
%
%   \Infer[OPTIONS]{ARITY}[LABEL]{TEXT}
%
%     Build an inference step. The ARITY is the number of sub-proofs, it may
%     be any number including 0 (in this case there will be a line above the
%     conclusion but no sub-proof). If LABEL is present, it is used as the
%     label on the right of the inference line. The TEXT is the conclusion of
%     the rule.
%
%   \Ellipsis{LABEL}{TEXT}
%
%     Typeset vertical dots, with a label on the right, and a new conclusion.
%     No inference lines are inserted.
%
% Each proof tree has a vertical axis, used for alignment of successive steps.
% The position of the axis is deduced from the text of the conclusion in the
% above commands. If TEXT contains the alignment character & then the axis is
% set at that position position, otherwise the axis is set at the center of
% the conclusion text. For instance, successive statements like
% |\Infer1{\Gamma&\vdash\Delta}| will produce conclusions aligned on the
% |\vdash| symbol.
%
% The formatting of trees, conclusion texts and inference rules is affected by
% the following options, which can be set for a whole tree or a single
% statement, or event globally using |\setkeys{ebproof}{OPTIONS}|:
%
% General shape:
%
%   updown=BOOLEAN
%
%     If set to true, the proofs are written upside down, with the conclusion
%     of rules is set above and the premisses below.
%
%   center=BOOLEAN
%
%     If set to true, the tree produced by the |prooftree| environment will be
%     vertically centered around the text line. If set to false, the base line
%     of the tree will be the base line of the conclusion. The default value
%     is true.
%
% Format of inference rules:
%
%   rulestyle=NAME
%
%     Defines the style of rules, among the following choices:
%       - none : no inference line at all
%       - simple : a single line (this is the default style)
%       - double : two lines
%       - dashed : a dashed line
%
%   leftlabel=TEXT
%   rightlabel=TEXT
%
%     The text to use as the labels of the rules, on the left and on the right
%     of the inference line. Using the second optional argument in |\Infer| is
%     equivalent to setting the |rightlabel| option with the value of that
%     argument.
%
%   template=MACRO
%   lefttemplate=MACRO
%   righttemplate=MACRO
%
%     Defines how conclusions are formatted. The macros are arbitrary TeX
%     code, composed in horizontal mode, with the macro argument |#1| standing
%     for the actual text passed to the |\Hypo| and |\Infer| statements. The
%     |template| value is used for conclusions with no alignment mark. The
%     |lefttemplate| and |righttemplate| values are used on the left and right
%     side of the alignment mark when it is present. The default value for
%     |template| is simply |$#1$|, so that conclusions are set in math mode.
%     The default values for |lefttemplate| and |righttemplate| are similar,
%     with spacing assuming that a relation symbol is put near the alignment
%     mark, so that |\Infer1{A &\vdash B}| is spaced correctly.
%
%   leftlabeltemplate=MACRO
%   rightlabeltemplate=MACRO
%
%     These macros are used to typeset the text of labels on the left and
%     right of inference lines. The default values are |#1| so that labels are
%     set in plain text mode.
%
% Dimensions:
%
%   hsep         the horizontal separation between sub-proofs in an inference
%   rulemargin   the spacing above and below inference lines
%   rulesep      the spacing between lines in double-line inferences
%   thickness    the thickness of inference lines
%   labelsep     the spacing between an inference lines and its labels
%
% The following additional statements may be used to affect the format of the
% last proof tree on the stack:
%
%   \Alter{COMMANDS}
%
%     Modify the proof with arbitrary commands, assuming that these commands
%     do not affect the size. A typical use is |\Alter{\color{red}}|, this
%     will typeset the subproof in red.
%
%   \Delims{LEFT}{RIGHT}
%
%     Put left and right delimiters around the whole sub-proof, without
%     changing the alignment (the spacing is affected by the delimiters,
%     however). The LEFT text must contain an open occurrence of |\left| and
%     the RIGHT text must contain an occurrence of matching occurrenc of
%     |\right|. For instance, |\Delims{\left(}{\right)}| will put the
%     sub-proof between parentheses.
%
%-----------------------------------------------------------------------------

% The |xkeyval| package is used for the parameters in proof construction.
% The |ifthen| package is used for some tests.

\RequirePackage{xkeyval}
\RequirePackage{ifthen}

%%% Registers

\newdimen\ebproof@hsep
\newdimen\ebproof@rulemargin
\newdimen\ebproof@rulesep
\newdimen\ebproof@thickness
\newdimen\ebproof@labelsep
\newbox\ebproof@leftbox
\newbox\ebproof@rightbox

%%% Parameters

\define@key{ebproof}{hsep}{\setlength\ebproof@hsep{#1}}
\define@key{ebproof}{rulemargin}{\setlength\ebproof@rulemargin{#1}}
\define@key{ebproof}{rulesep}{\setlength\ebproof@rulesep{#1}}
\define@key{ebproof}{thickness}{\setlength\ebproof@thickness{#1}}
\define@key{ebproof}{labelsep}{\setlength\ebproof@labelsep{#1}}
\define@key{ebproof}{rulestyle}{%
  \expandafter\let\expandafter\ebproof@rule@do\csname ebproof@rule@#1\endcsname}
\define@boolkey{ebproof}{updown}[true]{}
\define@key{ebproof}{template}{\def\ebproof@template##1{#1}}
\define@key{ebproof}{lefttemplate}{\def\ebproof@lefttemplate##1{#1}}
\define@key{ebproof}{righttemplate}{\def\ebproof@righttemplate##1{#1}}
\define@key{ebproof}{leftlabeltemplate}{\def\ebproof@leftlabeltemplate##1{#1}}
\define@key{ebproof}{rightlabeltemplate}{\def\ebproof@rightlabeltemplate##1{#1}}
\define@key{ebproof}{leftlabel}{%
  \def\ebproof@setleftlabel{%
    \setbox\ebproof@leftbox=\hbox{\ebproof@leftlabeltemplate{#1}}}}
\define@key{ebproof}{rightlabel}{%
  \def\ebproof@setrightlabel{%
    \setbox\ebproof@rightbox=\hbox{\ebproof@rightlabeltemplate{#1}}}}
\define@boolkey{ebproof}{center}{}

% Default values:

\ebproof@hsep=1.5em
\ebproof@rulemargin=3pt
\ebproof@rulesep=2pt
\ebproof@thickness=.4pt
\ebproof@labelsep=0.5em
\KV@ebproof@updownfalse
\def\ebproof@template#1{$#1$}
\def\ebproof@lefttemplate#1{$#1\mathrel{}$}
\def\ebproof@righttemplate#1{$\mathrel{}#1$}
\def\ebproof@leftlabeltemplate#1{#1}
\def\ebproof@rightlabeltemplate#1{#1}
\def\ebproof@setleftlabel{\setbox\ebproof@leftbox=\box\voidb@x}
\def\ebproof@setrightlabel{\setbox\ebproof@rightbox=\box\voidb@x}
\KV@ebproof@centertrue

%%% Storage

% Proof trees are represented as a data structure that consists of the
% following data:
%
% - box :   the tree itself, as a box, with the base line on that of
%           the conclusion
% - left :  the distance from the left of the box to the left of the
%           conclusion
% - right : the distance from the right of the box to the right of the
%           conclusion
% - axis :  the distance from the left of the box to the vertical axis
%           of the conclusion
%
% TeX does not actually provide data structures, so we have to encode things.
% First we provide local allocators, for temporary allocation of registers in
% a group. Dimensions are initialized to 0pt.

\newcount\ebproof@localdimens
\ebproof@localdimens=1
\def\ebproof@localdimen#1{%
  \advance\ebproof@localdimens1\relax
  \expandafter\dimendef\csname#1\endcsname\ebproof@localdimens
  \csname#1\endcsname=0pt\relax}

% For boxes, the allocator must be used as
%
%   \ebproof@localbox{NAME}=\hbox{...}
%
% in order to set the value of the box.

\newcount\ebproof@localboxes
\ebproof@localboxes=1
\def\ebproof@localbox#1{%
  \advance\ebproof@localboxes1\relax
  \expandafter\chardef\csname#1\endcsname\ebproof@localboxes
  \setbox\csname#1\endcsname}

% From this we deduce an allocator for data structures. This allocator
% receives a base name |A| and defines the registers |\A@box|, |\A@left|,
% |\A@right| and |\A@axis|. The macro is used like |\ebproof@localbox|, by
% providing a value for the box.

\def\ebproof@alloc#1{%
  \ebproof@localdimen{#1@left}%
  \ebproof@localdimen{#1@right}%
  \ebproof@localdimen{#1@axis}%
  \ebproof@localbox{#1@box}}

% Logically, such structures are stored on a stack. However, TeX does not
% provide data structures, so we encode them using what we actually have. A
% stack for boxes is implemented using a global hbox |\ebproof@box@stack| that
% contains all the boxes successively, and the |\lastbox| primitive allows us
% to pop elements from there. A macro |\ebproof@stack| is used to store the
% dimensions textually: the empty stack is an empty macro, and a non-empty
% stack is represented as |{left}{right}{axis}{tail}|. We maintain a counter
% |\ebproof@level| with the number of elements on the stack, for consistency
% checks.

\newcount\ebproof@level
\newbox\ebproof@box@stack
\newbox\ebproof@box@temp

% Clear the stack.

\def\ebproof@clear{%
  \global\ebproof@level=0%
  \global\setbox\ebproof@box@stack=\box\voidb@x%
  \gdef\ebproof@stack{}}

% Push an allocated structure (by name) on the stack.

\def\ebproof@push#1{%
  \global\advance\ebproof@level1\relax
  \global\setbox\ebproof@box@stack=\hbox{%
    \unhbox\ebproof@box@stack\copy\csname#1@box\endcsname}%
  \xdef\ebproof@stack{%
    {\the\csname#1@left\endcsname}%
    {\the\csname#1@right\endcsname}%
    {\the\csname#1@axis\endcsname}%
    {\ebproof@stack}}}

% Allocate a structure and pop its value from the top of the stack.

\def\ebproof@pop#1{%
  \ifnum\ebproof@level>0\relax
    \global\advance\ebproof@level-1\relax
    \global\setbox\ebproof@box@stack=\hbox{%
      \unhbox\ebproof@box@stack
      \global\setbox\ebproof@box@temp=\lastbox}%
    \ebproof@alloc{#1}=\box\ebproof@box@temp%
    \begingroup\def\pop##1##2##3##4{\endgroup%
      \csname#1@left\endcsname=##1\relax
      \csname#1@right\endcsname=##2\relax
      \csname#1@axis\endcsname=##3\relax
      \gdef\ebproof@stack{##4}}%
    \expandafter\pop\ebproof@stack
  \else
    \PackageError{ebproof}{%
      Missing premiss in a proof tree}{}%
    \ebproof@alloc{#1}=\box\voidb@x%
  \fi}


%%% Making boxes

% Push a box with the axis in the middle.

\def\ebproof@pushsimple#1{%
  \begingroup
  \ebproof@alloc{A}=\hbox{#1}%
  \A@axis=.5\wd\A@box
  \ebproof@push{A}%
  \endgroup}

% Push a box made of two halves, with the axis between the halves.

\def\ebproof@pushsplit#1#2{%
  \begingroup
  \ebproof@alloc{A}=\hbox{#1}%
  \A@axis=\wd\A@box
  \setbox\A@box=\hbox{\unhbox\A@box#2}%
  \ebproof@push{A}%
  \endgroup}

% Join horizontally the two elements at the top of the stack.

\def\ebproof@joinh{%
  \begingroup
  \ebproof@pop{A}%
  \ebproof@pop{B}%
  \ebproof@alloc{C}=\hbox{\box\B@box \kern\ebproof@hsep \box\A@box}%
  \C@left=\B@left
  \C@right=\A@right
  \C@axis=\wd\C@box
  \advance\C@axis\B@left
  \advance\C@axis-\A@right
  \divide\C@axis2\relax
  \ebproof@push{C}%
  \endgroup}

% An $n$-ary version of the horizontal join.

\def\ebproof@joinh@multi#1{%
  \begingroup
  \countdef\c=1
  \c=#1\relax%
  \ifnum\c=0
    \ebproof@alloc{X}=\hbox{}%
    \ebproof@push{X}%
  \else
    \ebproof@joinh@loop
  \fi
  \endgroup}
\def\ebproof@joinh@loop{%
  \ifnum\c>1
    \ebproof@joinh
    \advance\c-1
    \expandafter\ebproof@joinh@loop
  \fi}

% Append the last element to the right of the previous one, without changing
% its alignment.

\def\ebproof@joinright{%
  \begingroup
  \ebproof@pop{A}%
  \ebproof@pop{B}%
  \ebproof@alloc{C}=\hbox{\box\B@box \kern\ebproof@hsep \copy\A@box}%
  \C@left=\B@left
  \C@right=\B@right
  \advance\C@right\wd\A@box
  \advance\C@right\ebproof@hsep
  \ebproof@push{C}%
  \endgroup}

% Join vertically the two elements at the top of the stack, with a horizontal
% rule of the appropriate size.

\def\ebproof@joinv{%
  \begingroup
  \ebproof@setleftlabel
  \ebproof@setrightlabel
  \ebproof@pop{A}%
  \ebproof@pop{B}%
  %
  \ebproof@alloc{C}=\box\voidb@x%
  \ebproof@localdimen{A@shift}%
  \ebproof@localdimen{B@shift}%
  \ebproof@localdimen{R@shift}%
  \ebproof@localdimen{R@raise}%
  \ebproof@localdimen{R@width}%
  \ebproof@localdimen{C@width}%
  \ebproof@localdimen{tmp}%
  %
  % The placement of the boxes and the axis of the result
  \ifdim\A@axis>\B@axis
    \A@shift=0pt%
    \B@shift=\A@axis
    \advance\B@shift-\B@axis
    \C@axis=\A@axis
  \else
    \A@shift=\B@axis
    \advance\A@shift-\A@axis
    \B@shift=0pt%
    \C@axis=\B@axis
  \fi
  % The paddings of the result
  \C@left=\A@left
  \advance\C@left\A@shift
  \C@right=\A@right
  \tmp=\wd\B@box
  \advance\tmp\B@shift
  \advance\tmp-\wd\A@box
  \advance\tmp-\A@shift
  \ifdim\tmp>0pt%
    \C@width=\wd\B@box
    \advance\C@width\B@shift
    \advance\C@right\tmp
  \else
    \C@width=\wd\A@box
    \advance\C@width\A@shift
  \fi
  % The position of the rule
  \R@shift=\A@left
  \advance\R@shift\A@shift
  \tmp=\B@left
  \advance\tmp\B@shift
  \ifdim\R@shift>\tmp
    \R@shift=\tmp
  \fi
  % The width of the rule
  \R@width=\wd\A@box
  \advance\R@width\A@shift
  \advance\R@width-\A@right
  \tmp=\wd\B@box
  \advance\tmp\B@shift
  \advance\tmp-\B@right
  \ifdim\tmp>\R@width
    \R@width=\tmp
  \fi
  \advance\R@width-\R@shift
  % Make the rule box
  \ebproof@localbox{R@box}=\vbox{%
    \hsize=\R@width
    \hrule width \R@width height 0pt\relax
    \ebproof@rule@do}%
  % Shift things if the left box is wider than |\R@shift|
  \ifvoid\ebproof@leftbox\else
    \tmp=\wd\ebproof@leftbox
    \advance\tmp\ebproof@labelsep
    \ifdim\tmp>\R@shift
      \advance\tmp-\R@shift
      \advance\A@shift\tmp
      \advance\B@shift\tmp
      \advance\C@left\tmp
      \advance\C@axis\tmp
      \advance\C@width\tmp
      \R@shift=0pt\relax
    \else
      \advance\R@shift-\tmp
    \fi
  \fi
  % Compute how the rule box must be shifted so that labels are aligned
  \ebproof@localbox{RC@box}=\hbox{$\vcenter{\copy\R@box}$}%
  \R@raise=\ht\R@box
  \advance\R@raise-\ht\RC@box
  % Make the complete rule box
  \setbox\RC@box=\hbox{%
    \ifvoid\ebproof@leftbox\else
      \copy\ebproof@leftbox
      \kern\ebproof@labelsep
    \fi
    \box\RC@box
    \ifvoid\ebproof@rightbox\else
      \kern\ebproof@labelsep
      \copy\ebproof@rightbox
    \fi}
  % Adapt the dimensions on the right if the total rule width is too large
  \tmp=\wd\RC@box
  \advance\tmp\R@shift
  \ifdim\tmp>\C@width
    \advance\tmp-\C@width
    \advance\C@right\tmp
  \fi
  % Cancel the labels' height and depth
  \setbox\RC@box=\hbox{\raise\R@raise\box\RC@box}
  \ht\RC@box=\ht\R@box
  \dp\RC@box=\dp\R@box
  % Make the box
  \ifKV@ebproof@updown
    \setbox\C@box=\vtop{%
      \moveright\A@shift\box\A@box
      \hrule height 0pt
      \moveright\R@shift\box\RC@box%
      \hrule height 0pt
      \moveright\B@shift\box\B@box}%
  \else
    \setbox\C@box=\vbox{%
      \moveright\B@shift\box\B@box
      \hrule height 0pt
      \moveright\R@shift\box\RC@box%
      \hrule height 0pt
      \moveright\A@shift\box\A@box}%
  \fi
  \ebproof@push{C}%
  \endgroup}
% To put a simple rule in a vertical join
\let\ebproof@rule@none=\relax
\def\ebproof@rule@simple{%
  \kern\ebproof@rulemargin
  \hrule height \ebproof@thickness\relax
  \kern\ebproof@rulemargin
  }
\let\ebproof@rule@do=\ebproof@rule@simple
\def\ebproof@rule@double{%
  \kern\ebproof@rulemargin
  \hrule height \ebproof@thickness
  \kern\ebproof@rulesep
  \hrule height \ebproof@thickness
  \kern\ebproof@rulemargin}
\def\ebproof@rule@dashed{%
  \kern\ebproof@rulemargin%
  \hbox to \hsize{%
    \kern-1pt%
    \leaders\hbox{\kern1pt%
      \vrule height \ebproof@thickness width 5\ebproof@thickness%
      \kern1pt}\hfill
    \kern-1pt}%
  \kern\ebproof@rulemargin}

%%% Modifying boxes

% Alter a proof with a command that does not affect the size. Typically useful
% with |\color| commands.

\def\ebproof@alter#1{%
  \begingroup
  \ebproof@pop{A}%
  \setbox\A@box=\hbox{{#1\box\A@box}}%
  \ebproof@push{A}%
  \endgroup}

% Insert |\left| and |\right| delimiters without changing the alignment

\def\ebproof@delims#1#2{%
  \begingroup
  \ebproof@pop{TREE}%
  \ebproof@localbox{@SHIFTED}=%
    \hbox{$\vcenter{\copy\TREE@box}$}%
  \ebproof@localbox{@LEFT}=%
    \hbox{$#1\vrule height \ht\@SHIFTED depth \dp\@SHIFTED width 0pt\right.$}%
  \ebproof@localbox{@RIGHT}=%
    \hbox{$\left.\vrule height \ht\@SHIFTED depth \dp\@SHIFTED width 0pt#2$}%
  \ebproof@localdimen{dy}
  \dy=\dp\@SHIFTED
  \advance\dy-\dp\TREE@box
  \ebproof@alloc{A}=%
    \hbox{\raise\dy\hbox{\copy\@LEFT\box\@SHIFTED\copy\@RIGHT}}%
  \A@left=\wd\@LEFT \advance\A@left\TREE@left
  \A@right=\wd\@RIGHT \advance\A@right\TREE@right
  \A@axis=\wd\@LEFT \advance\A@axis\TREE@axis
  \ebproof@push{A}%
  \endgroup}

%%% High-level commands

% Push a box with default formatting, using explicit alignment if the code
% contains a |&| character

\def\ebproof@hypo@parse#1&#2&#3\ebproof@hypo@stop{
  \ifthenelse{\equal{#3}{}}%
    {\ebproof@pushsimple{\ebproof@template{#1}}}%
    {\ebproof@pushsplit
      {\ebproof@lefttemplate{#1}}%
      {\ebproof@righttemplate{#2}}}}

\newcommand\ebproof@hypo[2][]{%
  {\setkeys{ebproof}{#1}\ebproof@hypo@parse#2&&\ebproof@hypo@stop}}

% Build a n-ary rule

\def\ebproof@infer{%
  \@ifnextchar[{\ebproof@infer@}{\ebproof@infer@[]}}
\def\ebproof@infer@[#1]#2{%
  \@ifnextchar[{\ebproof@infer@@{#1}{#2}}{\ebproof@infer@@{#1}{#2}[]}}
\def\ebproof@infer@@#1#2[#3]#4{%
  \ebproof@joinh@multi{#2}%
  \ebproof@hypo{#4}%
  {\setkeys{ebproof}{#1}%
   \ifthenelse{\equal{#3}{}}{}{\setkeys{ebproof}{rightlabel={#3}}}%
   \ebproof@joinv}}

% Ellipsis with vertical dots

\def\ebproof@ellipsis#1#2{%
  \ebproof@pushsimple{$\vdots$}%
  \ebproof@pushsimple{#1}%
  {\ebproof@hsep=\ebproof@labelsep\relax \ebproof@joinright}%
  {\let\ebproof@rule@do=\relax \ebproof@joinv}%
  \ebproof@hypo{#2}%
  {\let\ebproof@rule@do=\relax \ebproof@joinv}%
  }

%%% Environment interface

\ebproof@clear

\def\ebproof@begin{%
  \edef\ebproof@start@level{\the\ebproof@level}%
  \setbox1=\vbox\bgroup
  \let\Hypo=\ebproof@hypo
  \let\Infer=\ebproof@infer
  \let\Ellipsis=\ebproof@ellipsis
  \let\Alter=\ebproof@alter
  \let\Delims=\ebproof@delims}
\def\ebproof@end{%
  \egroup
  \ebproof@pop{X}%
  \ifnum\ebproof@level=\ebproof@start@level\else
    \PackageError{ebproof}{Malformed proof tree}{%
      Some hypotheses were declared but not used in this tree.}%
  \fi
  \ifKV@ebproof@center
    \hbox{$\vcenter{\hbox{\box\X@box}}$}%
  \else
    \box\X@box
  \fi
  \global\setbox\ebproof@box@temp=\box1}

\newenvironment{prooftree}[1][]{%
  \setkeys{ebproof}{#1}%
  \leavevmode\ebproof@begin
}{%
  \ebproof@end}

\newenvironment{prooftree*}[1][]{%
  \center
  \setkeys{ebproof}{#1}%
  \leavevmode\ebproof@begin
}{%
  \ebproof@end
  \endcenter}