MATH formulas in PARragraph mode Typesetting Inference Rules

Didier Rémy

Abstract: This package provides macros for displaying lists of formulas that are typeset in mixed horizontal and vertical modes. The package is two-folded.

The first part is an environment mathpar that generalizes the math display mode to allow several formulas on the same line, and several lines in the same display. The arrangement of the sequence of formulas into lines is automatic depending on the line width and on a minimum inter-formula space and line width alike words in a paragraphs (in centerline mode). A typical application is displaying a set of type inference rules.

The second par is a macro inferrule to typeset inference rules themselves. Here again, both premises and conclusions are presented as list of formulas that should be displayed in almost the same way, except that the width is not fixed in advance; and the inference rule should use no more width than necessary so that other inference rules are given a chance to appear on the same line.

Although mathpar and inferrule look similar in their specification, and are often used in combination, they are in fact completely different in their implementations.

Mathpartir is Copyright (C) 2001, 2002 INRIA. Mathpartir has been developed by Didier Rémy. Mathpartir is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your option) any later version. See the GNU General Public License for more details (http://pauillac.inria.fr/~remy/license/GPL). Mathpartir is distributed in the hope that it will be useful, but without any warranty.

2  The mathpar environment

The mathpar environment is a “paragraph mode for formulas”. It allows to typeset long list of formulas putting as many as possible on the same line:

\begin{mathpar} A-Formula \and Longer-Formula \and And \and The-Last-One \end{mathpar}
 A−Formula
 Longer−Formula
 And
 The−Last−One

Formulas are separated by \and (or equivalently by a blank line). To enforce a vertical break it sufficies to replace \and by \\.

The implementation of mathpar entirely relies on the paragraph mode for text. It starts a new paragraph, and a math formula within a paragraph, after adjusting the spacing and penalties for breaks. Then, it simply binds \and to something like \goodbreak.

3  The inferrule macro

The inferrule macro is designed to typeset inference rules. It should only1 be used in math mode (or display math mode).

The basic use of the rule is

\inferrule
{one \\ two \\ three \\ or \\ more \\ premisses}
{and \\ any \\ number \\ of \\ conclusions \\ as \\ well}


This is the rendering on a large page

 one           two           three           or           more           premisses and           any           number           of           conclusions           as           well

However, the same formula on a narrower page will automatically be typsetted like that:

 one           two           three           or           more           premisses and           any           number           of           conclusions           as           well

An inference rule is mainly composed of a premisse and a conclusion. The premisse and the conclusions are both list of formulas where the elements are separated by \\.

Note the dissymetry between typesetting of the premisses and of conclusions where lines closer to the center are fit first.

A newline can be forced by adding an empty line \\\\

\inferrule {aa \\\\ bb} {dd \\ ee \\ ff}
 aa                   bb dd           ee           ff

3.1  Single rules

Single rules are the default mode. Rules are aligned on their fraction bar, as illustrated below:

 aa           bb ee

 aa                    bb           ee ee

If the premise or the conclusion is empty, then the fraction bar is not typeset and the premise or the conclusion is centered:

\inferrule {}{aa} + \inferrule {aa \\\\ aa}{}
 aa
+
 aa                    aa

Use use { } instead of {} to get an axiom for instance:

\inferrule { }{aa} + \inferrule {aa}{ }
 aa
+
 aa

The macro \inferrule acceps a label as optional argument, which will be typeset on the top left corner of the rule:

\inferrule [yop] {aa \\ bb} {cc}
 Yop aa           bb cc

See section 3 for changing typesetting of labels. A label can also be placed next to the rule directly, since the rule is centered:

\inferrule {aa \\ bb} {cc} \quad (\textsc {Yop})
 aa           bb cc
(Yop)

3.2  Customizing presentation

By default, lines are centerred in inference rules. However, this can be changed by either \mprset{flushleft} or \mprset{center}. For instance,

$$\mprset{flushleft} \inferrule {a \\ bbb \\\\ ccc \\ dd} {dd \\ ee \\ ff}$$
@percent 0.45@percent
 a           bb            ccc           dddd e           ff           gg

Note that lines are aligned independently in the premisse and the conclusion, which are both themselves centered. In particular, left alignment will not affect a single-line premisse or conclusion.

3.3  Customizing rules

One may wish to change use rules for rewriting rule or implications, etc. There is a generic way of definition new rules by providing three parts: a tail, a body, and a head. The rule will then be built by joining all three components in this order and filling the body with leaders to extend as much as necessary. Here are examples

$$\mprset{fraction={===}} \inferrule {a \\ bbb} {cc}$$
 a           bbb cc
$$\mprset {fraction={\models=\Rightarrow}} \inferrule {a \\ bbb} {cc}$$
 a           bbb cc

The height and depth of the body are used to adjust vertical space. One, may “smash” the body to reduce the vertical space

$$\mprset {fraction={% {\scriptstyle\vdash}% {\smash-}% {\rightarrow\!\!}% }} \inferrule {a \\ bbb} {cc}\,\,$$
 a           bbb cc
$$\mprset {fraction={\cdot\cdots\cdot}} \inferrule {a \\ bbb} {cc}$$
 a           bbb cc

Since vertical skip does not take header and footer into account, which is usually better but sometimes odd, this can be adjusted explicitly:

$$\mprset {fraction={|=/}, fractionaboveskip=0.6ex, fractionbelowskip=0.4ex} \inferrule {a \\ bbb_{\downarrow}} {cc^{\T\uparrow}}$$
 a           bbb↓ cc↑

Finally, it is also possible to provide its own definition of fraction by

\def \Over #1#2{\hbox{$#1 \over #2$}} $$\mprset{myfraction=\Over} \inferrule {a \\ bbb} {cc}$$
 a           bbb cc

Customizing the horizontal skip between premisses

(default value is 2em).

$$\mprset {sep=6em} \inferrule {a \\ bbb} {cc}$$
 a           bbb cc

3.4  Tabulars in inference rules

Although you probably do not want to do that, you may still use tabular or minipages inside inference rules, but between braces, as follows:

\infer [Tabular-Rule] {some \\ math \\ and \\ {\begin{tabular}[b]{|l|r|} \hline Ugly & and \\[1ex]\hline table & text \\\hline \end{tabular}} \\ {\begin{minipage}[b]{6em} Do you really wish to do that? \end{minipage}} \\ } {some \\ conclusions}
Tabular-Rule
some          math          and
 Ugly and table text

Do you really wish to do that?

some          conclusions

3.5  Derivation trees

To help writing cascades of rules forming a derivation tree, inference rules can also be aligned on their bottom line. For this, we use the star-version:

\inferrule* {\inferrule* {aa \\ bb}{cc} \\ dd} {ee}
 aa           bb cc
dd
ee

The star version can also take an optional argument, but with a different semantics. The optional argument is parsed by the keyval package, so as to offer a set of record-like options:

 key Effect for value v before Execute v before typesetting the rule. Useful for instance to change the maximal width of the rule. width Set the width of the rule to v narrower Set the width of the rule to v times \hsize. left Put a label v on the left of the rule Left Idem, but as if the label had zero width. Right As Left, but on the right of the rule. right As left, but on the right of the rule. leftskip Cheat by (skip negative space) v on the left side. rightskip Cheat by v on the right side of the rule. sep Set the premise skip between rules to v. vdots Raise the rule by v and insert vertical dots.

Here is an example of a complex derivation:

Total
Foo
 a           a                    bb           cc           dd Bar ee ⋮ ⋮
ff           gg
hh

 XX uu           vv ww
(1)

and its code

\inferrule* [left=Total] {\inferrule* [Left=Foo] {\inferrule* [Right=Bar, leftskip=2em,rightskip=2em,vdots=1.5em] {a \\ a \\\\ bb \\ cc \\ dd} {ee} \\ ff \\ gg} {hh} \\ \inferrule* [lab=XX]{uu \\ vv}{ww}} {(1)}

3.6  Implementation

The main macro in the implementation of inference rules is the one that either premises and conclusions. The macros uses two box-registers one hbox for typesetting each line and one vbox for collecting lines. The premise appears as a list with \\ as separator. Each element is considered in turn typeset in a hbox in display math mode. Its width is compare to the space left on the current line. If the box would not fit, the current horizontal line is transferred to the vertical box and emptied. Then, the current formula can safely be added to the horizontal line (if it does not fit, nothing can be done). When moved to the vertical list, lines are aligned on their center (as if their left-part was a left overlapped). At the end the vbox is readjusted on the right.

This description works for conclusions. For premises, the elements must be processes in reverse order and the vertical list is simply built upside down.

4  Other Options

The package also defines \infer as a shortcut for \inferrule but only if it is not previously defined.

The package uses \TirName and \RefTirName to typeset labels, which can safely be redefined by the user. The former is used for defining occurrences (ie. in rule \inferrule) while the latter is used for referencing (ie. in the star-version).

The vertical space in mathpar is adjusted by \MathparLineskip. To restore the normal paragraph parameters in mathpar mode (for instance for some inner paragraph), use the command \MathparNormalpar. The environment uses \MathparBindings to rebind \\, and, and \par. You can redefine thus command to change the default bindings or add your own.

5  Examples

See the source of this documentation —the file mathpartir.tex— for full examples.

6  HEVEA compatibility

The package also redefines \hva to do nothing in mathpar environment and nor in inference rules.

In HeVeA, \and will always produce a vertical break in mathpar environment; to obtain a horizontal break, use \hva \and instead. Conversely, \\ will always produce a horizontal break in type inference rules; to obtain a vertical break, use \hva \\ instead.

For instance, by default the following code,

\begin{mathpar} \inferrule* [Left=Foo] {\inferrule* [Right=Bar,width=8em, leftskip=2em,rightskip=2em,vdots=1.5em] {a \\ a \\ bb \\ cc \\ dd} {ee} \\ ff \\ gg} {hh} \and \inferrule* [lab=XX]{uu \\ vv}{ww} \end{mathpar}

which typesets in TEX as follows,

Foo
 a           a           bb           cc           dd Bar ee ⋮ ⋮
ff           gg
hh
 XX uu           vv ww

would appear as follows with the compatible HEVEA mode:

Foo
 a           a           bb           cc           dd Bar ee
ff           gg
hh
 XX uu           vv ww

To obtain (almost) the same rendering as in TEX, it could be typed as

\begin{mathpar} \inferrule* [Left=Foo] {\inferrule* [Right=Bar,width=8em, leftskip=2em,rightskip=2em,vdots=1.5em] {a \\ a \hva \\ bb \\ cc \\ dd} {ee} \\ ff \\ gg} {hh} \hva \and \inferrule* [lab=XX]{uu \\ vv}{ww} \end{mathpar}

Actually, it would be typeset and follows with the compatible HEVEA mode:

Foo
 a           a                    bb           cc           dd Bar ee
ff           gg
hh
 XX uu           vv ww

1
Even though the basic version may work in text mode, we discourage its use in text mode; the star-version cannot be used in text-mode

This document was translated from LATEX by HEVEA.