\ifx\pdfoutput\undefined   % si on est pas en pdflatex
\documentclass[11pt,a4paper]{article}
\else
\documentclass[11pt,a4paper,pdftex]{article}
\fi
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{pslatex}
\usepackage{url}
\usepackage{verbatim}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{array}
\usepackage{fullpage}

\title{Translation from Coq V7 to V8}
\author{The Coq Development Team}

%% Macros etc.
\catcode`\_=13
\let\subscr=_
\def_{\ifmmode\sb\else\subscr\fi}

\def\NT#1{\langle\textit{#1}\rangle}
\def\NTL#1#2{\langle\textit{#1}\rangle_{#2}}
%\def\TERM#1{\textsf{\bf #1}}
\def\TERM#1{\texttt{#1}}
\newenvironment{transbox}
  {\begin{center}\tt\begin{tabular}{l|ll} \hfil\textrm{V7} & \hfil\textrm{V8} \\ \hline}
  {\end{tabular}\end{center}}
\def\TRANS#1#2
  {\begin{tabular}[t]{@{}l@{}}#1\end{tabular} &
   \begin{tabular}[t]{@{}l@{}}#2\end{tabular} \\}
\def\TRANSCOM#1#2#3
  {\begin{tabular}[t]{@{}l@{}}#1\end{tabular} &
   \begin{tabular}[t]{@{}l@{}}#2\end{tabular} & #3 \\}

%%
%%
%%
\begin{document}
\maketitle

\section{Introduction}

Coq version 8.0 is a major version and carries major changes: the
concrete syntax was redesigned almost from scratch, and many notions
of the libraries were renamed for uniformisation purposes. We felt
that these changes could discourage users with large theories from
switching to the new version.

The goal of this document is to introduce these changes on simple
examples (mainly the syntactic changes), and describe the automated
tools to help moving to V8.0. Essentially, it consists of a translator
that takes as input a Coq source file in old syntax and produces a
file in new syntax and adapted to the new standard library. The main
extra features of this translator is that it keeps comments, even
those within expressions\footnote{The position of those comment might
differ slightly since there is no exact matching of positions between
old and new syntax.}.

The document is organised as follows: first section describes the new
syntax on simple examples. It is very translation-oriented. This
should give users of older versions the flavour of the new syntax, and
allow them to make translation manually on small
examples. Section~\ref{Translation} explains how the translation
process can be automatised for the most part (the boring one: applying
similar changes over thousands of lines of code). We strongly advise
users to follow these indications, in order to avoid many potential
complications of the translation process.


\section{The new syntax on examples}

The goal of this section is to introduce to the new syntax of Coq on
simple examples, rather than just giving the new grammar. It is
strongly recommended to read first the definition of the new syntax
(in the reference manual), but this document should also be useful for
the eager user who wants to start with the new syntax quickly.

The toplevel has an option {\tt -translate} which allows
interactively translating commands. This toplevel translator accepts a
command, prints the translation on standard output (after a %
\verb+New syntax:+ balise), executes the command, and waits for another
command. The only requirements is that they should be syntactically
correct, but they do not have to be well-typed.

This interactive translator proved to be useful in two main
usages. First as a ``debugger'' of the translation. Before the
translation, it may help in spotting possible conflicts between the
new syntax and user notations. Or when the translation fails for some
reason, it makes it easy to find the exact reason why it failed and
make attempts in fixing the problem.

The second usage of the translator is when trying to make the first
proofs in new syntax. Well trained users will automatically think
their scripts in old syntax and might waste much time (and the
intuition of the proof) if they have to search the translation in a
document. Running a translator in the background will allow the user
to instantly have the answer.

The rest of this section is a description of all the aspects of the
syntax that changed and how they were translated. All the examples
below can be tested by entering the V7 commands in the toplevel
translator.


%%

\subsection{Changes in lexical conventions w.r.t. V7}

\subsubsection{Identifiers}

The lexical conventions changed: \TERM{_} is not a regular identifier
anymore. It is used in terms as a placeholder for subterms to be inferred
at type-checking, and in patterns as a non-binding variable.

Furthermore, only letters (Unicode letters), digits, single quotes and
_ are allowed after the first character.

\subsubsection{Quoted string}

Quoted strings are used typically to give a filename (which may not
be a regular identifier). As before they are written between double
quotes ("). Unlike for V7, there is no escape character: characters
are written normally except the double quote which is doubled.

\begin{transbox}
\TRANS{"abcd$\backslash\backslash$efg"}{"abcd$\backslash$efg"}
\TRANS{"abcd$\backslash$"efg"}{"abcd""efg"}
\end{transbox}


\subsection{Main changes in terms w.r.t. V7}


\subsubsection{Precedence of application}

In the new syntax, parentheses are not really part of the syntax of
application. The precedence of application (10) is tighter than all
prefix and infix notations. It makes it possible to remove parentheses
in many contexts.

\begin{transbox}
\TRANS{(A x)->(f x)=(g y)}{A x -> f x = g y}
\TRANS{(f [x]x)}{f (fun x => x)}
\end{transbox}


\subsubsection{Arithmetics and scopes}

The specialized notation for \TERM{Z} and \TERM{R} (introduced by
symbols \TERM{`} and \TERM{``}) have disappeared. They have been
replaced by the general notion of scope.

\begin{center}
\begin{tabular}{l|l|l}
type & scope name & delimiter \\
\hline
types & type_scope & \TERM{type} \\
\TERM{bool} & bool_scope & \\
\TERM{nat} & nat_scope & \TERM{nat} \\
\TERM{Z} & Z_scope & \TERM{Z} \\
\TERM{R} & R_scope & \TERM{R} \\
\TERM{positive} & positive_scope & \TERM{P}
\end{tabular}
\end{center}

In order to use notations of arithmetics on \TERM{Z}, its scope must
be opened with command \verb+Open Scope Z_scope.+ Another possibility
is using the scope change notation (\TERM{\%}). The latter notation is
to be used when notations of several scopes appear in the same
expression.

In examples below, scope changes are not needed if the appropriate scope
has been opened. Scope \verb|nat_scope| is opened in the initial state of Coq.
\begin{transbox}
\TRANSCOM{`0+x=x+0`}{0+x=x+0}{\textrm{Z_scope}}
\TRANSCOM{``0 + [if b then ``1`` else ``2``]``}{0 + if b then 1 else 2}{\textrm{R_scope}}
\TRANSCOM{(0)}{0}{\textrm{nat_scope}}
\end{transbox}

Below is a table that tells which notation is available in which
scope. The relative precedences and associativity of operators is the
same as in usual mathematics. See the reference manual for more
details. However, it is important to remember that unlike V7, the type
operators for product and sum are left-associative, in order not to
clash with arithmetic operators.

\begin{center}
\begin{tabular}{l|l}
scope & notations \\
\hline
nat_scope & \texttt{+ - * < <= > >=} \\
Z_scope & \texttt{+ - * / mod < <= > >= ?=} \\
R_scope & \texttt{+ - * / < <= > >=} \\
type_scope & \texttt{* +} \\
bool_scope & \texttt{\&\& || -} \\
list_scope & \texttt{:: ++}
\end{tabular}
\end{center}



\subsubsection{Notation for implicit arguments}

The explicitation of arguments is closer to the \emph{bindings}
notation in tactics. Argument positions follow the argument names of
the head constant. The example below assumes \verb+f+ is a function
with two implicit dependent arguments named \verb+x+ and \verb+y+.
\begin{transbox}
\TRANS{f 1!t1 2!t2 t3}{f (x:=t1) (y:=t2) t3}
\TRANS{!f t1 t2}{@f t1 t2}
\end{transbox}


\subsubsection{Inferred subterms}

Subterms that can be automatically inferred by the type-checker is now
written {\tt _}

\begin{transbox}
\TRANS{?}{_}
\end{transbox}

\subsubsection{Universal quantification}

The universal quantification and dependent product types are now
introduced by the \texttt{forall} keyword before the binders and a
comma after the binders.

The syntax of binders also changed significantly. A binder can simply be
a name when its type can be inferred. In other cases, the name and the type
of the variable are put between parentheses. When several consecutive
variables have the same type, they can be grouped. Finally, if all variables
have the same type, parentheses can be omitted.

\begin{transbox}
\TRANS{(x:A)B}{forall (x:~A), B ~~\textrm{or}~~ forall x:~A, B}
\TRANS{(x,y:nat)P}{forall (x y :~nat), P ~~\textrm{or}~~ forall x y :~nat, P}
\TRANS{(x,y:nat;z:A)P}{forall (x y :~nat) (z:A), P}
\TRANS{(x,y,z,t:?)P}{forall x y z t, P}
\TRANS{(x,y:nat;z:?)P}{forall (x y :~nat) z, P}
\end{transbox}

\subsubsection{Abstraction}

The notation for $\lambda$-abstraction follows that of universal
quantification. The binders are surrounded by keyword \texttt{fun}
and \verb+=>+.

\begin{transbox}
\TRANS{[x,y:nat; z](f a b c)}{fun (x y:nat) z => f a b c}
\end{transbox}


\subsubsection{Pattern-matching}

Beside the usage of the keyword pair \TERM{match}/\TERM{with} instead of
\TERM{Cases}/\TERM{of}, the main change is the notation for the type of
branches and return type. It is no longer written between \TERM{$<$ $>$} before
the \TERM{Cases} keyword, but interleaved with the destructured objects.

The idea is that for each destructured object, one may specify a
variable name (after the \TERM{as} keyword) to tell how the branches
types depend on this destructured objects (case of a dependent
elimination), and also how they depend on the value of the arguments
of the inductive type of the destructured objects (after the \TERM{in}
keyword). The type of branches is then given after the keyword
\TERM{return}, unless it can be inferred.

Moreover, when the destructured object is a variable, one may use this
variable in the return type.

\begin{transbox}
\TRANS{Cases n of\\~~ O => O \\| (S k) => (1) end}{match n with\\~~ 0 => 0 \\| S k => 1 end}
\TRANS{Cases m n of \\~~0 0 => t \\| ... end}{match m, n with \\~~0, 0 => t \\| ... end}
\TRANS{<[n:nat](P n)>Cases T of ... end}{match T as n return P n with ... end}
\TRANS{<[n:nat][p:(even n)]\~{}(odd n)>Cases p of\\~~ ... \\end}{match p in even n return \~{} odd n with\\~~ ...\\end}
\end{transbox}

The annotations of the special pattern-matching operators
(\TERM{if}/\TERM{then}/\TERM{else}) and \TERM{let()} also changed. The
only restriction is that the destructuring \TERM{let} does not allow
dependent case analysis.

\begin{transbox}
\TRANS{
 \begin{tabular}{@{}l}
 <[n:nat;x:(I n)](P n x)>if t then t1 \\
 else t2
 \end{tabular}}%
{\begin{tabular}{@{}l}
 if t as x in I n return P n x then t1 \\
 else t2
 \end{tabular}}
\TRANS{<[n:nat](P n)>let (p,q) = t1 in t2}%
{let (p,q) in I n return P n := t1 in t2}
\end{transbox}


\subsubsection{Fixpoints and cofixpoints}

An simpler syntax for non-mutual fixpoints is provided, making it very close
to the usual notation for non-recursive functions. The decreasing argument
is now indicated by an annotation between curly braces, regardless of the
binders grouping. The annotation can be omitted if the binders introduce only
one variable. The type of the result can be omitted if inferable.

\begin{transbox}
\TRANS{Fix plus\{plus [n:nat] : nat -> nat :=\\~~ [m]...\}}{fix plus (n m:nat) \{struct n\}: nat := ...}
\TRANS{Fix fact\{fact [n:nat]: nat :=\\
~~Cases n of\\~~~~ O => (1) \\~~| (S k) => (mult n (fact k)) end\}}{fix fact
  (n:nat) :=\\
~~match n with \\~~~~0 => 1 \\~~| (S k) => n * fact k end}
\end{transbox}

There is a syntactic sugar for single fixpoints (defining one
variable) associated to a local definition:

\begin{transbox}
\TRANS{let f := Fix f \{f [x:A] : T := M\} in\\(g (f y))}{let fix f (x:A) : T := M in\\g (f x)}
\end{transbox}

The same applies to cofixpoints, annotations are not allowed in that case.

\subsubsection{Notation for type cast}

\begin{transbox}
\TRANS{O :: nat}{0 : nat}
\end{transbox}

\subsection{Main changes in tactics w.r.t. V7}

The main change is that all tactic names are lowercase. This also holds for
Ltac keywords.

\subsubsection{Renaming of induction tactics}

\begin{transbox}
\TRANS{NewDestruct}{destruct}
\TRANS{NewInduction}{induction}
\TRANS{Induction}{simple induction}
\TRANS{Destruct}{simple destruct}
\end{transbox}

\subsubsection{Ltac}

Definitions of macros are introduced by \TERM{Ltac} instead of
\TERM{Tactic Definition}, \TERM{Meta Definition} or \TERM{Recursive
Definition}. They are considered recursive by default.

\begin{transbox}
\TRANS{Meta Definition my_tac t1 t2 := t1; t2.}%
{Ltac my_tac t1 t2 := t1; t2.}
\end{transbox}

Rules of a match command are not between square brackets anymore.

Context (understand a term with a placeholder) instantiation \TERM{inst}
became \TERM{context}. Syntax is unified with subterm matching.

\begin{transbox}
\TRANS{Match t With [C[x=y]] -> Inst C[y=x]}%
{match t with context C[x=y] => context C[y=x] end}
\end{transbox}

Arguments of macros use the term syntax. If a general Ltac expression
is to be passed, it must be prefixed with ``{\tt ltac :}''. In other
cases, when a \'{} was necessary, it is replaced by ``{\tt constr :}''

\begin{transbox}
\TRANS{my_tac '(S x)}{my_tac (S x)}
\TRANS{my_tac (Let x=tac In x)}{my_tac ltac:(let x:=tac in x)}
\TRANS{Let x = '[x](S (S x)) In Apply x}%
{let x := constr:(fun x => S (S x)) in apply x}
\end{transbox}

{\tt Match Context With} is now called {\tt match goal with}. Its
argument is an Ltac expression by default.


\subsubsection{Named arguments of theorems ({\em bindings})}

\begin{transbox}
\TRANS{Apply thm with x:=t 1:=u}{apply thm with (x:=t) (1:=u)}
\end{transbox}


\subsubsection{Occurrences}

To avoid ambiguity between a numeric literal and the optional
occurrence numbers of this term, the occurrence numbers are put after
the term itself and after keyword \TERM{as}.
\begin{transbox}
\TRANS{Pattern 1 2 (f x) 3 4 d y z}{pattern f x at 1 2, d at 3 4, y, z}
\end{transbox}


\subsubsection{{\tt LetTac} and {\tt Pose}}

Tactic {\tt LetTac} was renamed into {\tt set}, and tactic {\tt Pose}
was a particular case of {\tt LetTac} where the abbreviation is folded
in the conclusion\footnote{There is a tactic called {\tt pose} in V8,
but its behaviour is not to fold the abbreviation at all.}.

\begin{transbox}
\TRANS{LetTac x = t in H}{set (x := t) in H}
\TRANS{Pose x := t}{set (x := t)}
\end{transbox}

{\tt LetTac} could be followed by a specification (called a clause) of
the places where the abbreviation had to be folded (hypothese and/or
conclusion). Clauses are the syntactic notion to denote in which parts
of a goal a given transformation should occur. Its basic notation is
either \TERM{*} (meaning everywhere), or {\tt\textrm{\em hyps} |-
\textrm{\em concl}} where {\em hyps} is either \TERM{*} (to denote all
the hypotheses), or a comma-separated list of either hypothesis name,
or {\tt (value of $H$)} or {\tt (type of $H$)}. Moreover, occurrences
can be specified after every hypothesis after the {\TERM{at}}
keyword. {\em concl} is either empty or \TERM{*}, and can be followed
by occurrences.

\begin{transbox}
\TRANS{in Goal}{in |- *}
\TRANS{in H H1}{in H1, H2 |-}
\TRANS{in H H1 ...}{in * |-}
\TRANS{in H H1 Goal}{in H1, H2 |- *}
\TRANS{in H H1 H2 ... Goal}{in *}
\TRANS{in 1 2 H 3 4 H0 1 3 Goal}{in H at 1 2, H0 at 3 4 |- * at 1 3}
\end{transbox}

\subsection{Main changes in vernacular commands w.r.t. V7}


\subsubsection{Require}

The default behaviour of {\tt Require} is not to open the loaded
module.

\begin{transbox}
\TRANS{Require Arith}{Require Import Arith}
\end{transbox}

\subsubsection{Binders}

The binders of vernacular commands changed in the same way as those of
fixpoints. This also holds for parameters of inductive definitions.


\begin{transbox}
\TRANS{Definition x [a:A] : T := M}{Definition x (a:A) : T := M}
\TRANS{Inductive and [A,B:Prop]: Prop := \\~~conj : A->B->(and A B)}%
      {Inductive and (A B:Prop): Prop := \\~~conj : A -> B -> and A B}
\end{transbox}

\subsubsection{Hints}

Both {\tt Hints} and {\tt Hint} commands are beginning with {\tt Hint}.

Command {\tt HintDestruct} has disappeared.


The syntax of \emph{Extern} hints changed: the pattern and the tactic
to be applied are separated by a {\tt =>}.
\begin{transbox}
\TRANS{Hint name := Resolve (f ? x)}%
{Hint Resolve (f _ x)}
\TRANS{Hint name := Extern 4 (toto ?) Apply lemma}%
{Hint Extern 4 (toto _) => apply lemma}
\TRANS{Hints Resolve x y z}{Hint Resolve x y z}
\TRANS{Hints Resolve f : db1 db2}{Hint Resolve f : db1 db2}
\TRANS{Hints Immediate x y z}{Hint Immediate x y z}
\TRANS{Hints Unfold x y z}{Hint Unfold x y z}
%% \TRANS{\begin{tabular}{@{}l}
%%   HintDestruct Local Conclusion \\
%%   ~~name (f ? ?) 3 [Apply thm]
%%   \end{tabular}}%
%% {\begin{tabular}{@{}l}
%%  Hint Local Destuct name := \\
%%  ~~3 Conclusion (f _ _) => apply thm
%%  \end{tabular}}
\end{transbox}


\subsubsection{Implicit arguments}


{\tt Set Implicit Arguments} changed its meaning in V8: the default is
to turn implicit only the arguments that are {\em strictly} implicit
(or rigid), i.e. that remains inferable whatever the other arguments
are. For instance {\tt x} inferable from {\tt P x} is not strictly
inferable since it can disappears if {\tt P} is instantiated by a term
which erases {\tt x}.

\begin{transbox}
\TRANS{Set Implicit Arguments}%
{\begin{tabular}{l}
 Set Implicit Arguments. \\
 Unset Strict Implicits.
 \end{tabular}}
\end{transbox}

However, you may wish to adopt the new semantics of {\tt Set Implicit
Arguments} (for instance because you think that the choice of
arguments it sets implicit is more ``natural'' for you).


\subsection{Changes in standard library}

Many lemmas had their named changed to improve uniformity. The user
generally do not have to care since the translators performs the
renaming.

  Type {\tt entier} from fast_integer.v is renamed into {\tt N} by the
translator. As a consequence, user-defined objects of same name {\tt N}
are systematically qualified even tough it may not be necessary.  The
following table lists the main names with which the same problem
arises:
\begin{transbox}
\TRANS{IF}{IF_then_else}
\TRANS{ZERO}{Z0}
\TRANS{POS}{Zpos}
\TRANS{NEG}{Zneg}
\TRANS{SUPERIEUR}{Gt}
\TRANS{EGAL}{Eq}
\TRANS{INFERIEUR}{Lt}
\TRANS{add}{Pplus}
\TRANS{true_sub}{Pminus}
\TRANS{entier}{N}
\TRANS{Un_suivi_de}{Ndouble_plus_one}
\TRANS{Zero_suivi_de}{Ndouble}
\TRANS{Nul}{N0}
\TRANS{Pos}{Npos}
\end{transbox}


\subsubsection{Implicit arguments}

%% Hugo:
Main definitions of standard library have now implicit
arguments. These arguments are dropped in the translated files. This
can exceptionally be a source of incompatibilities which has to be
solved by hand (it typically happens for polymorphic functions applied
to {\tt nil} or {\tt None}).
%% preciser: avant ou apres trad ?

\subsubsection{Logic about {\tt Type}}

Many notations that applied to {\tt Set} have been extended to {\tt
Type}, so several definitions in {\tt Type} are superseded by them.

\begin{transbox}
\TRANS{x==y}{x=y}
\TRANS{(EXT x:Prop | Q)}{exists x:Prop, Q}
\TRANS{identityT}{identity}
\end{transbox}



%% Doc of the translator
\section{A guide to translation}
\label{Translation}

%%\subsection{Overview of the translation process}

Here is a short description of the tools involved in the translation process:
\begin{description}
\item{\tt coqc -translate}
is the automatic translator. It is a parser/pretty-printer. This means
that the translation is made by parsing every command using a parser
of old syntax, which is printed using the new syntax. Many efforts
were made to preserve as much as possible of the quality of the
presentation: it avoids expansion of syntax extensions, comments are
not discarded and placed at the same place.
\item{\tt translate-v8} (in the translation package) is a small
shell-script that will help translate developments that compile with a
Makefile with minimum requirements.
\end{description}

\subsection{Preparation to translation}

This step is very important because most of work shall be done before
translation. If a problem occurs during translation, it often means
that you will have to modify the original source and restart the
translation process. This also means that it is recommended not to
edit the output of the translator since it would be overwritten if
the translation has to be restarted.

\subsubsection{Compilation with {\tt coqc -v7}}

First of all, it is mandatory that files compile with the current
version of Coq (8.0) with option {\tt -v7}. Translation is a
complicated task that involves the full compilation of the
development. If your development was compiled with older versions,
first upgrade to Coq V8.0 with option {\tt -v7}. If you use a Makefile
similar to those produced by {\tt coq\_makefile}, you probably just
have to do

{\tt make OPT="-opt -v7"} ~~~or~~~ {\tt make OPT="-byte -v7"}

When the development compiles successfully, there are several changes
that might be necessary for the translation. Essentially, this is
about syntax extensions (see section below dedicated to porting syntax
extensions). If you do not use such features, then you are ready to
try and make the translation.

\subsection{Translation}

\subsubsection{The general case}

The preferred way is to use script {\tt translate-v8} if your development
is compiled by a Makefile with the following constraints:
\begin{itemize}
\item compilation is achieved by invoking make without specifying a target
\item options are passed to Coq with make variable COQFLAGS that
  includes variables OPT, COQLIBS, and OTHERFLAGS.
\end{itemize}
These constraints are met by the makefiles produced by {\tt coq\_makefile}

Otherwise, modify your build program so as to pass option {\tt
-translate} to program {\tt coqc}. The effect of this option is to
output the translated source of any {\tt .v} file in a file with
extension {\tt .v8} located in the same directory than the original
file.

\subsubsection{What may happen during the translation}

This section describes events that may happen during the
translation and measures to adopt.

These are the warnings that may arise during the translation, but they
generally do not require any modification for the user:
Warnings:
\begin{itemize}
\item {\tt Unable to detect if $id$ denotes a local definition}\\
This is due to a semantic change in clauses. In a command such as {\tt
simpl in H}, the old semantics were to perform simplification in the
type of {\tt H}, or in its body if it is defined. With the new
semantics, it is performed both in the type and the body (if any). It
might lead to incompatibilities

\item {\tt Forgetting obsolete module}\\
Some modules have disappeared in V8.0 (new syntax). The user does not
need to worry about it, since the translator deals with it.

\item {\tt Replacing obsolete module}\\
Same as before but with the module that were renamed. Here again, the
translator deals with it.
\end{itemize}

\subsection{Verification of the translation}

The shell-script {\tt translate-v8} also renames {\tt .v8} files into
{\tt .v} files (older {\tt .v} files are put in a subdirectory called
{\tt v7}) and tries to recompile them. To do so it invokes {\tt make}
without option (which should cause the compilation using {\tt coqc}
without particular option).

If compilation fails at this stage, you should refrain from repairing
errors manually on the new syntax, but rather modify the old syntax
script and restart the translation. We insist on that because the
problem encountered can show up in many instances (especially if the
problem comes from a syntactic extension), and fixing the original
sources (for instance the {\tt V8only} parts of notations) once will
solve all occurrences of the problem.

%%\subsubsection{Errors occurring after translation}
%%Equality in {\tt Z} or {\tt R}...

\subsection{Particular cases}

\subsubsection{Lexical conventions}

The definition of identifiers changed. Most of those changes are
handled by the translator. They include:
\begin{itemize}
\item {\tt \_} is not an identifier anymore: it is translated to {\tt
x\_}
\item avoid clash with new keywords by adding a trailing {\tt \_}
\end{itemize}

If the choices made by translation is not satisfactory
or in the following cases:
\begin{itemize}
\item use of latin letters
\item use of iso-latin characters in notations
\end{itemize}
users should change their development prior to translation.

\subsubsection{{\tt Case} and {\tt Match}}

These very low-level case analysis are no longer supported. The
translator tries hard to translate them into a user-friendly one, but
it might lack type information to do so\footnote{The translator tries
to typecheck terms before printing them, but it is not always possible
to determine the context in which terms appearing in tactics
live.}. If this happens, it is preferable to transform it manually
before translation.

\subsubsection{Syntax extensions with {\tt Grammar} and {\tt Syntax}}


{\tt Grammar} and {\tt Syntax} are no longer supported. They
should be replaced by an equivalent {\tt Notation} command and be
processed as described above. Before attempting translation, users
should verify that compilation with option {\tt -v7} succeeds.

In the cases where {\tt Grammar} and {\tt Syntax} cannot be emulated
by {\tt Notation}, users have to change manually they development as
they wish to avoid the use of {\tt Grammar}. If this is not done, the
translator will simply expand the notations and the output of the
translator will use the regular Coq syntax.

\subsubsection{Syntax extensions with {\tt Notation} and {\tt Infix}}

These commands do not necessarily need to be changed.

Some work will have to be done manually if the notation conflicts with
the new syntax (for instance, using keywords like {\tt fun} or {\tt
exists}, overloading of symbols of the old syntax, etc.) or if the
precedences are not right.

  Precedence levels are now from 0 to 200. In V8, the precedence and
associativity of an operator cannot be redefined. Typical level are
(refer to the chapter on notations in the Reference Manual for the
full list):

\begin{center}
\begin{tabular}{|cll|}
\hline
Notation & Precedence & Associativity \\
\hline
\verb!_ <-> _! & 95 & no \\
\verb!_ \/ _!  & 85 & right \\
\verb!_ /\ _!  & 80 & right \\
\verb!~ _!   & 75 & right \\
\verb!_ = _!, \verb!_ <> _!, \verb!_ < _!, \verb!_ > _!,
  \verb!_ <= _!, \verb!_ >= _!   & 70 & no \\
\verb!_ + _!, \verb!_ - _!   & 50 & left \\
\verb!_ * _!, \verb!_ / _!   & 40 & left \\
\verb!- _!  & 35 & right \\
\verb!_ ^ _!   & 30 & left \\
\hline
\end{tabular}
\end{center}


  By default, the translator keeps the associativity given in V7 while
the levels are mapped according to the following table:

\begin{center}
\begin{tabular}{l|l|l}
V7 level & mapped to & associativity \\
\hline
0 & 0 & no \\
1 & 20 & left \\
2 & 30 & right \\
3 & 40 & left \\
4 & 50 & left \\
5 & 70 & no \\
6 & 80 & right \\
7 & 85 & right \\
8 & 90 & right \\
9 & 95 & no \\
10 & 100 & left
\end{tabular}
\end{center}

If this is OK, just simply apply the translator.


\paragraph{Associativity conflict}

  Since the associativity of the levels obtained by translating a V7
level (as shown on table above) cannot be changed, you have to choose
another level with a compatible associativity.

  You can choose any level between 0 and 200, knowing that the
standard operators are already set at the levels shown on the list
above.

Assume you have a notation
\begin{verbatim}
Infix NONA 2 "=_S" my_setoid_eq.
\end{verbatim}
By default, the translator moves it to level 30 which is right
associative, hence a conflict with the expected no associativity.

To solve the problem, just add the "V8only" modifier to reset the
level and enforce the associativity as follows:
\begin{verbatim}
Infix NONA 2 "=_S" my_setoid_eq V8only (at level 70, no associativity).
\end{verbatim}
The translator now knows that it has to translate "=_S" at level 70
with no associativity.

Remark: 70 is the "natural" level for relations, hence the choice of 70
here, but any other level accepting a no-associativity would have been
OK.

Second example: assume you have a notation
\begin{verbatim}
Infix RIGHTA 1 "o" my_comp.
\end{verbatim}
By default, the translator moves it to level 20 which is left
associative, hence a conflict with the expected right associativity.

To solve the problem, just add the "V8only" modifier to reset the
level and enforce the associativity as follows:
\begin{verbatim}
Infix RIGHTA 1 "o" my_comp V8only (at level 20, right associativity).
\end{verbatim}
The translator now knows that it has to translate "o" at level 20
which has the correct "right associativity".

Remark: we assumed here that the user wants a strong precedence for
composition, in such a way, say, that "f o g + h" is parsed as
"(f o g) + h". To get "o" binding less than the arithmetical operators,
an appropriated level would have been close of 70, and below, e.g. 65.


\paragraph{Conflict: notation hides another notation}

Remark: use {\tt Print Grammar constr} in V8 to diagnose the overlap
and see the section on factorization in the chapter on notations of
the Reference Manual for hints on how to factorize.

Example:
\begin{verbatim}
Notation "{ x }" := (my_embedding x) (at level 1).
\end{verbatim}
overlaps in V8 with notation \verb#{ x : A & P }# at level 0 and with
x at level 99. The conflicts can be solved by left-factorizing the
notation as follows:
\begin{verbatim}
Notation "{ x }" := (my_embedding x) (at level 1)
  V8only (at level 0, x at level 99).
\end{verbatim}

\paragraph{Conflict: a notation conflicts with the V8 grammar}

Again, use the {\tt V8only} modifier to tell the translator to
automatically take in charge the new syntax.

Example:
\begin{verbatim}
Infix 3 "@" app.
\end{verbatim}
Since {\tt @} is used in the new syntax for deactivating the implicit
arguments, another symbol has to be used, e.g. {\tt @@}. This is done via
the {\tt V8only} option as follows:
\begin{verbatim}
Infix 3 "@" app V8only "@@" (at level 40, left associativity).
\end{verbatim}
or, alternatively by
\begin{verbatim}
Notation "x @ y" := (app x y) (at level 3, left associativity)
  V8only "x @@ y" (at level 40, left associativity).
\end{verbatim}

\paragraph{Conflict: my notation is already defined at another level
  (or with another associativity)}

In V8, the level and associativity of a given notation can no longer
be changed. Then, either you adopt the standard reserved levels and
associativity for this notation (as given on the list above) or you
change your notation.
\begin{itemize}
\item To change the notation, follow the directions in the previous
paragraph
\item To adopt the standard level, just use {\tt V8only} without any
argument.
\end{itemize}

Example:
\begin{verbatim}
Infix 6 "*" my_mult.
\end{verbatim}
is not accepted as such in V8. Write
\begin{verbatim}
Infix 6 "*" my_mult V8only.
\end{verbatim}
to tell the translator to use {\tt *} at the reserved level (i.e. 40
with left associativity). Even better, use interpretation scopes (look
at the Reference Manual).


\subsubsection{Strict implicit arguments}

In the case you want to adopt the new semantics of {\tt Set Implicit
 Arguments} (only setting rigid arguments as implicit), add the option
{\tt -strict-implicit} to the translator.

Warning: changing the number of implicit arguments can break the
notations.  Then use the {\tt V8only} modifier of {\tt Notation}.

\end{document}