(*<*)

theory Sugar

imports LaTeXsugar OptionalSugar

begin

(*>*)

section "Introduction"

text{* This document is for those Isabelle users who have mastered

the art of mixing \LaTeX\ text and Isabelle theories and never want to

typeset a theorem by hand anymore because they have experienced the

bliss of writing \verb!@!\verb!{thm[display]setsum_cartesian_product[no_vars]}!

and seeing Isabelle typeset it for them:

@{thm[display,eta_contract=false] setsum_cartesian_product[no_vars]}

No typos, no omissions, no sweat.

If you have not experienced that joy, read Chapter 4, \emph{Presenting

Theories}, \cite{LNCS2283} first.

If you have mastered the art of Isabelle's \emph{antiquotations},

i.e.\ things like the above \verb!@!\verb!{thm...}!, beware: in your vanity

you may be tempted to think that all readers of the stunning ps or pdf

documents you can now produce at the drop of a hat will be struck with

awe at the beauty unfolding in front of their eyes. Until one day you

come across that very critical of readers known as the ``common referee''.

He has the nasty habit of refusing to understand unfamiliar notation

like Isabelle's infamous @{text"\<lbrakk> \<rbrakk> \<Longrightarrow>"} no matter how many times you

explain it in your paper. Even worse, he thinks that using @{text"\<lbrakk>

\<rbrakk>"} for anything other than denotational semantics is a cardinal sin

that must be punished by instant rejection.

This document shows you how to make Isabelle and \LaTeX\ cooperate to

produce ordinary looking mathematics that hides the fact that it was

typeset by a machine. You merely need to load the right files:

\begin{itemize}

\item Import theory \texttt{LaTeXsugar} in the header of your own

theory.  You may also want bits of \texttt{OptionalSugar}, which you can

copy selectively into your own theory or import as a whole.  Both

theories live in \texttt{HOL/Library} and are found automatically.

\item Should you need additional \LaTeX\ packages (the text will tell

you so), you include them at the beginning of your \LaTeX\ document,

typically in \texttt{root.tex}. For a start, you should

\verb!\usepackage{amssymb}! --- otherwise typesetting

@{prop[source]"\<not>(\<exists>x. P x)"} will fail because the AMS symbol

@{text"\<nexists>"} is missing.

\end{itemize}

*}

section{* HOL syntax*}

subsection{* Logic *}

text{* 

  The formula @{prop[source]"\<not>(\<exists>x. P x)"} is typeset as @{prop"~(EX x. P x)"}.

The predefined constructs @{text"if"}, @{text"let"} and

@{text"case"} are set in sans serif font to distinguish them from

other functions. This improves readability:

\begin{itemize}

\item @{term"if b then e\<^isub>1 else e\<^isub>2"} instead of @{text"if b then e\<^isub>1 else e\<^isub>2"}.

\item @{term"let x = e\<^isub>1 in e\<^isub>2"} instead of @{text"let x = e\<^isub>1 in e\<^isub>2"}.

\item @{term"case x of True \<Rightarrow> e\<^isub>1 | False \<Rightarrow> e\<^isub>2"} instead of\\

      @{text"case x of True \<Rightarrow> e\<^isub>1 | False \<Rightarrow> e\<^isub>2"}.

\end{itemize}

*}

subsection{* Sets *}

text{* Although set syntax in HOL is already close to

standard, we provide a few further improvements:

\begin{itemize}

\item @{term"{x. P}"} instead of @{text"{x. P}"}.

\item @{term"{}"} instead of @{text"{}"}, where

 @{term"{}"} is also input syntax.

\item @{term"insert a (insert b (insert c M))"} instead of @{text"insert a (insert b (insert c M))"}.

\end{itemize}

*}

subsection{* Lists *}

text{* If lists are used heavily, the following notations increase readability:

\begin{itemize}

\item @{term"x # xs"} instead of @{text"x # xs"},

      where @{term"x # xs"} is also input syntax.

If you prefer more space around the $\cdot$ you have to redefine

\verb!\isasymcdot! in \LaTeX:

\verb!\renewcommand{\isasymcdot}{\isamath{\,\cdot\,}}!

\item @{term"length xs"} instead of @{text"length xs"}.

\item @{term"nth xs n"} instead of @{text"nth xs n"},

      the $n$th element of @{text xs}.

\item Human readers are good at converting automatically from lists to

sets. Hence \texttt{OptionalSugar} contains syntax for suppressing the

conversion function @{const set}: for example, @{prop[source]"x \<in> set xs"}

becomes @{prop"x \<in> set xs"}.

\item The @{text"@"} operation associates implicitly to the right,

which leads to unpleasant line breaks if the term is too long for one

line. To avoid this, \texttt{OptionalSugar} contains syntax to group

@{text"@"}-terms to the left before printing, which leads to better

line breaking behaviour:

@{term[display]"term\<^isub>0 @ term\<^isub>1 @ term\<^isub>2 @ term\<^isub>3 @ term\<^isub>4 @ term\<^isub>5 @ term\<^isub>6 @ term\<^isub>7 @ term\<^isub>8 @ term\<^isub>9 @ term\<^isub>1\<^isub>0"}

\end{itemize}

*}

subsection{* Numbers *}

text{* Coercions between numeric types are alien to mathematicians who

consider, for example, @{typ nat} as a subset of @{typ int}.

\texttt{OptionalSugar} contains syntax for suppressing numeric coercions such

as @{const int} @{text"::"} @{typ"nat \<Rightarrow> int"}. For example,

@{term[source]"int 5"} is printed as @{term "int 5"}. Embeddings of types

@{typ nat}, @{typ int}, @{typ real} are covered; non-injective coercions such

as @{const nat} @{text"::"} @{typ"int \<Rightarrow> nat"} are not and should not be

hidden. *}

section "Printing theorems"

subsection "Question marks"

text{* If you print anything, especially theorems, containing

schematic variables they are prefixed with a question mark:

\verb!@!\verb!{thm conjI}! results in @{thm conjI}. Most of the time

you would rather not see the question marks. There is an attribute

\verb!no_vars! that you can attach to the theorem that turns its

schematic into ordinary free variables: \verb!@!\verb!{thm conjI[no_vars]}!

results in @{thm conjI[no_vars]}.

This \verb!no_vars! business can become a bit tedious.

If you would rather never see question marks, simply put

\begin{quote}

@{ML "Unsynchronized.reset show_question_marks"}\verb!;!

\end{quote}

at the beginning of your file \texttt{ROOT.ML}.

The rest of this document is produced with this flag set to \texttt{false}.

Hint: Setting \verb!show_question_marks! to \texttt{false} only

suppresses question marks; variables that end in digits,

e.g. @{text"x1"}, are still printed with a trailing @{text".0"},

e.g. @{text"x1.0"}, their internal index. This can be avoided by

turning the last digit into a subscript: write \verb!x\<^isub>1! and

obtain the much nicer @{text"x\<^isub>1"}. *}

(*<*)ML "Unsynchronized.reset show_question_marks"(*>*)

subsection {*Qualified names*}

text{* If there are multiple declarations of the same name, Isabelle prints

the qualified name, for example @{text "T.length"}, where @{text T} is the

theory it is defined in, to distinguish it from the predefined @{const[source]

"List.length"}. In case there is no danger of confusion, you can insist on

short names (no qualifiers) by setting \verb!short_names!, typically

in \texttt{ROOT.ML}:

\begin{quote}

@{ML "Unsynchronized.set short_names"}\verb!;!

\end{quote}

*}

subsection {*Variable names\label{sec:varnames}*}

text{* It sometimes happens that you want to change the name of a

variable in a theorem before printing it. This can easily be achieved

with the help of Isabelle's instantiation attribute \texttt{where}:

@{thm conjI[where P = \<phi> and Q = \<psi>]} is the result of

\begin{quote}

\verb!@!\verb!{thm conjI[where P = \<phi> and Q = \<psi>]}!

\end{quote}

To support the ``\_''-notation for irrelevant variables

the constant \texttt{DUMMY} has been introduced:

@{thm fst_conv[where b = DUMMY]} is produced by

\begin{quote}

\verb!@!\verb!{thm fst_conv[where b = DUMMY]}!

\end{quote}

Variables that are bound by quantifiers or lambdas cannot be renamed

like this. Instead, the attribute \texttt{rename\_abs} does the

job. It expects a list of names or underscores, similar to the

\texttt{of} attribute:

\begin{quote}

\verb!@!\verb!{thm split_paired_All[rename_abs _ l r]}!

\end{quote}

produces @{thm split_paired_All[rename_abs _ l r]}.

*}

subsection "Inference rules"

text{* To print theorems as inference rules you need to include Didier

R\'emy's \texttt{mathpartir} package~\cite{mathpartir}

for typesetting inference rules in your \LaTeX\ file.

Writing \verb!@!\verb!{thm[mode=Rule] conjI}! produces

@{thm[mode=Rule] conjI}, even in the middle of a sentence.

If you prefer your inference rule on a separate line, maybe with a name,

\begin{center}

@{thm[mode=Rule] conjI} {\sc conjI}

\end{center}

is produced by

\begin{quote}

\verb!\begin{center}!\\

\verb!@!\verb!{thm[mode=Rule] conjI} {\sc conjI}!\\

\verb!\end{center}!

\end{quote}

It is not recommended to use the standard \texttt{display} option

together with \texttt{Rule} because centering does not work and because

the line breaking mechanisms of \texttt{display} and \texttt{mathpartir} can

clash.

Of course you can display multiple rules in this fashion:

\begin{quote}

\verb!\begin{center}!\\

\verb!@!\verb!{thm[mode=Rule] conjI} {\sc conjI} \\[1ex]!\\

\verb!@!\verb!{thm[mode=Rule] conjE} {\sc disjI$_1$} \qquad!\\

\verb!@!\verb!{thm[mode=Rule] disjE} {\sc disjI$_2$}!\\

\verb!\end{center}!

\end{quote}

yields

\begin{center}\small

@{thm[mode=Rule] conjI} {\sc conjI} \\[1ex]

@{thm[mode=Rule] disjI1} {\sc disjI$_1$} \qquad

@{thm[mode=Rule] disjI2} {\sc disjI$_2$}

\end{center}

The \texttt{mathpartir} package copes well if there are too many

premises for one line:

\begin{center}

@{prop[mode=Rule] "\<lbrakk> A \<longrightarrow> B; B \<longrightarrow> C; C \<longrightarrow> D; D \<longrightarrow> E; E \<longrightarrow> F; F \<longrightarrow> G;

 G \<longrightarrow> H; H \<longrightarrow> I; I \<longrightarrow> J; J \<longrightarrow> K \<rbrakk> \<Longrightarrow> A \<longrightarrow> K"}

\end{center}

Limitations: 1. Premises and conclusion must each not be longer than

the line.  2. Premises that are @{text"\<Longrightarrow>"}-implications are again

displayed with a horizontal line, which looks at least unusual.

In case you print theorems without premises no rule will be printed by the

\texttt{Rule} print mode. However, you can use \texttt{Axiom} instead:

\begin{quote}

\verb!\begin{center}!\\

\verb!@!\verb!{thm[mode=Axiom] refl} {\sc refl}! \\

\verb!\end{center}!

\end{quote}

yields

\begin{center}

@{thm[mode=Axiom] refl} {\sc refl} 

\end{center}

*}

subsection "Displays and font sizes"

text{* When displaying theorems with the \texttt{display} option, e.g.

\verb!@!\verb!{thm[display] refl}! @{thm[display] refl} the theorem is

set in small font. It uses the \LaTeX-macro \verb!\isastyle!,

which is also the style that regular theory text is set in, e.g. *}

lemma "t = t"

(*<*)oops(*>*)

text{* \noindent Otherwise \verb!\isastyleminor! is used,

which does not modify the font size (assuming you stick to the default

\verb!\isabellestyle{it}! in \texttt{root.tex}). If you prefer

normal font size throughout your text, include

\begin{quote}

\verb!\renewcommand{\isastyle}{\isastyleminor}!

\end{quote}

in \texttt{root.tex}. On the other hand, if you like the small font,

just put \verb!\isastyle! in front of the text in question,

e.g.\ at the start of one of the center-environments above.

The advantage of the display option is that you can display a whole

list of theorems in one go. For example,

\verb!@!\verb!{thm[display] foldl.simps}!

generates @{thm[display] foldl.simps}

*}

subsection "If-then"

text{* If you prefer a fake ``natural language'' style you can produce

the body of

\newtheorem{theorem}{Theorem}

\begin{theorem}

@{thm[mode=IfThen] le_trans}

\end{theorem}

by typing

\begin{quote}

\verb!@!\verb!{thm[mode=IfThen] le_trans}!

\end{quote}

In order to prevent odd line breaks, the premises are put into boxes.

At times this is too drastic:

\begin{theorem}

@{prop[mode=IfThen] "longpremise \<Longrightarrow> longerpremise \<Longrightarrow> P(f(f(f(f(f(f(f(f(f(x)))))))))) \<Longrightarrow> longestpremise \<Longrightarrow> conclusion"}

\end{theorem}

In which case you should use \texttt{IfThenNoBox} instead of

\texttt{IfThen}:

\begin{theorem}

@{prop[mode=IfThenNoBox] "longpremise \<Longrightarrow> longerpremise \<Longrightarrow> P(f(f(f(f(f(f(f(f(f(x)))))))))) \<Longrightarrow> longestpremise \<Longrightarrow> conclusion"}

\end{theorem}

*}

subsection{* Doing it yourself\label{sec:yourself}*}

text{* If for some reason you want or need to present theorems your

own way, you can extract the premises and the conclusion explicitly

and combine them as you like:

\begin{itemize}

\item \verb!@!\verb!{thm (prem 1)! $thm$\verb!}!

prints premise 1 of $thm$.

\item \verb!@!\verb!{thm (concl)! $thm$\verb!}!

prints the conclusion of $thm$.

\end{itemize}

For example, ``from @{thm (prem 2) conjI} and

@{thm (prem 1) conjI} we conclude @{thm (concl) conjI}''

is produced by

\begin{quote}

\verb!from !\verb!@!\verb!{thm (prem 2) conjI}! \verb!and !\verb!@!\verb!{thm (prem 1) conjI}!\\

\verb!we conclude !\verb!@!\verb!{thm (concl) conjI}!

\end{quote}

Thus you can rearrange or hide premises and typeset the theorem as you like.

Styles like \verb!(prem 1)! are a general mechanism explained

in \S\ref{sec:styles}.

*}

subsection "Patterns"

text {*

  In \S\ref{sec:varnames} we shows how to create patterns containing

  ``@{term DUMMY}''.

  You can drive this game even further and extend the syntax of let

  bindings such that certain functions like @{term fst}, @{term hd}, 

  etc.\ are printed as patterns. \texttt{OptionalSugar} provides the

  following:

  \begin{center}

  \begin{tabular}{l@ {~~produced by~~}l}

  @{term "let x = fst p in t"} & \verb!@!\verb!{term "let x = fst p in t"}!\\

  @{term "let x = snd p in t"} & \verb!@!\verb!{term "let x = snd p in t"}!\\

  @{term "let x = hd xs in t"} & \verb!@!\verb!{term "let x = hd xs in t"}!\\

  @{term "let x = tl xs in t"} & \verb!@!\verb!{term "let x = tl xs in t"}!\\

  @{term "let x = the y in t"} & \verb!@!\verb!{term "let x = the y in t"}!\\

  \end{tabular}

  \end{center}

*}

section "Proofs"

text {* Full proofs, even if written in beautiful Isar style, are

likely to be too long and detailed to be included in conference

papers, but some key lemmas might be of interest.

It is usually easiest to put them in figures like the one in Fig.\

\ref{fig:proof}. This was achieved with the \isakeyword{text\_raw} command:

*}

text_raw {*

  \begin{figure}

  \begin{center}\begin{minipage}{0.6\textwidth}  

  \isastyleminor\isamarkuptrue

*}

lemma True

proof -

  -- "pretty trivial"

  show True by force

qed

text_raw {*    

  \end{minipage}\end{center}

  \caption{Example proof in a figure.}\label{fig:proof}

  \end{figure}

*}

text {*

\begin{quote}

\small

\verb!text_raw {!\verb!*!\\

\verb!  \begin{figure}!\\

\verb!  \begin{center}\begin{minipage}{0.6\textwidth}!\\

\verb!  \isastyleminor\isamarkuptrue!\\

\verb!*!\verb!}!\\

\verb!lemma True!\\

\verb!proof -!\\

\verb!  -- "pretty trivial"!\\

\verb!  show True by force!\\

\verb!qed!\\

\verb!text_raw {!\verb!*!\\

\verb!  \end{minipage}\end{center}!\\

\verb!  \caption{Example proof in a figure.}\label{fig:proof}!\\

\verb!  \end{figure}!\\

\verb!*!\verb!}!

\end{quote}

Other theory text, e.g.\ definitions, can be put in figures, too.

*}

section {*Styles\label{sec:styles}*}

text {*

  The \verb!thm! antiquotation works nicely for single theorems, but

  sets of equations as used in definitions are more difficult to

  typeset nicely: people tend to prefer aligned @{text "="} signs.

  To deal with such cases where it is desirable to dive into the structure

  of terms and theorems, Isabelle offers antiquotations featuring

  ``styles'':

    \begin{quote}

    \verb!@!\verb!{thm (style) thm}!\\

    \verb!@!\verb!{prop (style) thm}!\\

    \verb!@!\verb!{term (style) term}!\\

    \verb!@!\verb!{term_type (style) term}!\\

    \verb!@!\verb!{typeof (style) term}!\\

    \end{quote}

  A ``style'' is a transformation of a term. There are predefined

  styles, namely \verb!lhs! and \verb!rhs!, \verb!prem! with one argument, and \verb!concl!.

  For example, 

  the output

  \begin{center}

  \begin{tabular}{l@ {~~@{text "="}~~}l}

  @{thm (lhs) foldl_Nil} & @{thm (rhs) foldl_Nil}\\

  @{thm (lhs) foldl_Cons} & @{thm (rhs) foldl_Cons}

  \end{tabular}

  \end{center}

  is produced by the following code:

  \begin{quote}

    \verb!\begin{center}!\\

    \verb!\begin{tabular}{l@ {~~!\verb!@!\verb!{text "="}~~}l}!\\

    \verb!@!\verb!{thm (lhs) foldl_Nil} & @!\verb!{thm (rhs) foldl_Nil}\\!\\

    \verb!@!\verb!{thm (lhs) foldl_Cons} & @!\verb!{thm (rhs) foldl_Cons}!\\

    \verb!\end{tabular}!\\

    \verb!\end{center}!

  \end{quote}

  Note the space between \verb!@! and \verb!{! in the tabular argument.

  It prevents Isabelle from interpreting \verb!@ {~~...~~}! 

  as an antiquotation. The styles \verb!lhs! and \verb!rhs!

  extract the left hand side (or right hand side respectively) from the

  conclusion of propositions consisting of a binary operator

  (e.~g.~@{text "="}, @{text "\<equiv>"}, @{text "<"}).

  Likewise, \verb!concl! may be used as a style to show just the

  conclusion of a proposition. For example, take \verb!hd_Cons_tl!:

  \begin{center}

    @{thm hd_Cons_tl}

  \end{center}

  To print just the conclusion,

  \begin{center}

    @{thm (concl) hd_Cons_tl}

  \end{center}

  type

  \begin{quote}

    \verb!\begin{center}!\\

    \verb!@!\verb!{thm (concl) hd_Cons_tl}!\\

    \verb!\end{center}!

  \end{quote}

  Beware that any options must be placed \emph{before}

  the style, as in this example.

  Further use cases can be found in \S\ref{sec:yourself}.

  If you are not afraid of ML, you may also define your own styles.

  Have a look at module @{ML_struct Term_Style}.

*}

(*<*)

end

(*>*)