[helm.git] / helm / papers / matita / matita.tex

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\begin{document}

\begin{opening}

 \title{The \MATITA{} Proof Assistant}

\author{Andrea \surname{Asperti} \email{asperti@cs.unibo.it}}
\author{Claudio \surname{Sacerdoti Coen} \email{sacerdot@cs.unibo.it}}
\author{Enrico \surname{Tassi} \email{tassi@cs.unibo.it}}
\author{Stefano \surname{Zacchiroli} \email{zacchiro@cs.unibo.it}}
\institute{Department of Computer Science, University of Bologna\\
 Mura Anteo Zamboni, 7 --- 40127 Bologna, ITALY}

\runningtitle{The Matita proof assistant}
\runningauthor{Asperti, Sacerdoti Coen, Tassi, Zacchiroli}

% \date{data}

\begin{motto}
``We are nearly bug-free'' -- \emph{CSC, Oct 2005}
\end{motto}

\begin{abstract}
 abstract qui
\end{abstract}

\keywords{Proof Assistant, Mathematical Knowledge Management, XML, Authoring,
Digital Libraries}

\end{opening}

\section{Introduction}
\label{sec:intro}
{\em Matita} is the proof assistant under development by the \HELM{} team
\cite{mkm-helm} at the University of Bologna, under the direction of 
Prof.~Asperti. 
The origin of the system goes back to 1999. At the time we were mostly 
interested to develop tools and techniques to enhance the accessibility
via web of formal libraries of mathematics. Due to its dimension, the
library of the \COQ{} proof assistant (of the order of 35'000 theorems) 
was choosed as a privileged test bench for our work, although experiments
have been also conducted with other systems, and notably with \NUPRL{}.
The work, mostly performed in the framework of the recently concluded 
European project IST-33562 \MOWGLI{}~\cite{pechino}, mainly consisted in the 
following teps:
\begin{itemize}
\item exporting the information from the internal representation of
 \COQ{} to a system and platform independent format. Since XML was at the 
time an emerging standard, we naturally adopted this technology, fostering
a content-based architecture for future system, where the documents
of the library were the the main components around which everything else 
has to be build;
\item developing indexing and searching techniques supporting semantic
 queries to the library; these efforts gave birth to our \WHELP{}
search engine, described in~\cite{whelp};
\item developing languages and tools for a high-quality notational 
rendering of mathematical information; in particular, we have been 
active in the MathML Working group since 1999, and developed inside
\HELM{} a MathML-compliant widget for the GTK graphical environment
which can be integrated in any application.
\end{itemize}
The exportation issue, extensively discussed in \cite{exportation-module},
has several major implications worth to be discussed. 

The first
point concerns the kind of content information to be exported. In a
proof assistant like \COQ{}, proofs are represented in at least three clearly
distinguishable formats: \emph{scripts} (i.e. sequences of commands issued by the
user to the system during an interactive session of proof), \emph{proof objects}
(which is the low-level representation of proofs in the form of
lambda-terms readable to and checked by kernel) and \emph{proof-trees} (which
is a kind of intermediate representation, vaguely inspired by a sequent
like notation, that inherits most of the defects but essentially
none of the advantages of the previous representations). 
Partially related to this problem, there is the
issue of the {\em granularity} of the library: scripts usually comprise
small developments with many definitions and theorems, while 
proof objects correspond to individual mathematical items. 

In our case, the choice of the content encoding was eventually dictated
by the methodological assumption of offering the information in a
stable and system-independent format. The language of scripts is too
oriented to \COQ, and it changes too rapidly to be of any interest
to third parties. On the other side, the language of proof objects 
merely depend on
the logical framework (the Calculus of Inductive Constructions, in
the case of \COQ), is grammatically simple, semantically clear and, 
especially, is very stable (as kernels of proof assistants 
often are). 
So the granularity of the library is at the level of individual 
objects, that also justifies from another point of view the need
for efficient searching techniques for retrieving individual 
logical items from the repository. 

The main (possibly only) problem with proof objects is that they are
difficult to read and do not directly correspond to what the user typed
in. An analogy frequently made in the proof assistant community is that of
comparing the vernacular language of scripts to a high level source language
and lambda terms to the assembly language they are compiled in. We do not
share this view and prefer to look at scripts as an imperative language, 
and to lambda terms as their denotational semantics; still, however,
denotational semantics is possibly more formal but surely not more readable 
than the imperative source.

For all the previous reasons, a huge amount of work inside \MOWGLI{} has
been devoted to automatic reconstruction of proofs in natural language
from lambda terms. Since lambda terms are in close connection 
with natural deduction 
(that is still the most natural logical language discovered so far)
the work is not hopeless as it may seem, especially if rendering
is combined, as in our case, with dynamic features supporting 
in-line expansions or contractions of subproofs. The final 
rendering is probably not entirely satisfactory (see \cite{ida} for a
discussion), but surely
readable (the actual quality largely depends by the way the lambda 
term is written). 

Summing up, we already disposed of the following tools/techniques:
\begin{itemize}
\item XML specifications for the Calculus of Inductive Constructions,
with tools for parsing and saving mathematical objects in such a format;
\item metadata specifications and tools for indexing and querying the
XML knowledge base;
\item a proof checker (i.e. the {\em kernel} of a proof assistant), 
 implemented to check that we exported form the \COQ{} library all the 
logically relevant content;
\item a sophisticated parser (used by the search engine), able to deal 
with potentially ambiguous and incomplete information, typical of the 
mathematical notation \cite{};
\item a {\em refiner}, i.e. a type inference system, based on complex 
existential variables, used by the disambiguating parser;
\item complex transformation algorithms for proof rendering in natural
language;
\item an innovative rendering widget, supporting high-quality bidimensional
rendering, and semantic selection, i.e. the possibility to select semantically
meaningful rendering expressions, and to past the respective content into
a different text area.
\NOTE{il widget non ha sel semantica}
\end{itemize}
Starting from all this, the further step of developing our own 
proof assistant was too
small and too tempting to be neglected. Essentially, we ``just'' had to
add an authoring interface, and a set of functionalities for the
overall management of the library, integrating everything into a
single system. \MATITA{} is the result of this effort. 

At first sight, \MATITA{} looks as (and partly is) a \COQ{} clone. This is
more the effect of the circumstances of its creation described 
above than the result of a deliberate design. In particular, we
(essentially) share the same foundational dialect of \COQ{} (the
Calculus of Inductive Constructions), the same implementative
language (\OCAML{}), and the same (script based) authoring philosophy.
However, as we shall see, the analogy essentially stops here. 

In a sense; we like to think of \MATITA{} as the way \COQ{} would 
look like if entirely rewritten from scratch: just to give an
idea, although \MATITA{} currently supports almost all functionalities of
\COQ{}, it links 60'000 lins of \OCAML{} code, against ... of \COQ{} (and
we are convinced that, starting from scratch again, we could furtherly
reduce our code in sensible way).\NOTE{righe \COQ{}}

\begin{itemize}
 \item scelta del sistema fondazionale
 \item sistema indipendente (da Coq)
  \begin{itemize}
   \item possibilit\`a di sperimentare (soluzioni architetturali, logiche,
    implementative, \dots)
   \item compatibilit\`a con sistemi legacy
  \end{itemize}
\end{itemize}

\section{\HELM{} library(??)}

\subsection{libreria tutta visibile}
\ASSIGNEDTO{csc}
\NOTE{assumo che si sia gia' parlato di approccio content-centrico}
Our commitment to the content-centric view of the architecture of the system
has important consequences on the user's experience and on the functionalities
of several components of \MATITA. In the content-centric view the library
of mathematical knowledge is an already existent and completely autonomous
entity that we are allowed to exploit and augment using \MATITA. Thus, in
principle, when the user starts to prove a new theorem she has complete
visibility of the library and she can refer to every definition and lemma,
also using the mathematical notation already developed. In a similar way,
every form of automation of the system must be able to analyze and possibly
exploit every notion in the library.

The benefits of this approach highly justify the non neglectable price to pay
in the development of several components. We analyse now a few of the causes
of this additional complexity.

\subsubsection{Ambiguity}
A rich mathematical library includes equivalent definitions and representations
of the same notion. Moreover, mathematical notation inside a rich library is
surely highly overloaded. As a consequence every mathematical expression the
user provides is highly ambiguous since all the definitions,
representations and special notations are available at once to the user.

The usual solution to the problem, as adopted for instance in Coq, is to
restrict the user's scope to just one interpretation for each definition,
representation or notation. In this way much of the ambiguity is removed,
burdening the user that must someway declare what is in scope and that must
use special syntax when she needs to refer to something not in scope.

Even with this approach ambiguity cannot be completely removed since implicit
coercions can be arbitrarily inserted by the system to ``change the type''
of subterms that do not have the expected type. Usually implicit coercions
are used to overcome the absence of subtyping that should mimic the subset
relation found in set theory. For instance, the expression
$\forall n \in nat. 2 * n * \pi \equiv_\pi 0$ is correct in set theory since
the set of natural numbers is a subset of that of real numbers; the
corresponding expression $\forall n:nat. 2*n*\pi \equiv_\pi 0$ is not well typed
and requires the automatic insertion of the coercion $real_of_nat: nat \to R$
either around both 2 and $n$ (to make both products be on real numbers) or
around the product $2*n$. The usual approach consists in either rejecting the
ambiguous term or arbitrarily choosing one of the interpretations. For instance,
Coq rejects the declaration of coercions that have alternatives
(i.e. already declared coercions with the same domain and codomain)
or that are obtained composing other coercions in order to
avoid making several terms highly ambiguous by choosing to insert any one of the
alternative coercions. Coq also arbitrarily chooses how to insert coercions in
terms to make them well typed when there is more than one possibility (as in
the previous example).

The approach we are following is radically different. It consists in dealing
with ambiguous expressions instead of avoiding them. As a last resource,
when the system is unable to disambiguate the input, the user is interactively
required to provide more information that is recorded to avoid asking the
same question again in subsequent processing of the same input.
More details on our approach can be found in \ref{sec:disambiguation}.

\subsubsection{Consistency}
A large mathematical library is likely to be logically inconsistent.
It may contain incompatible axioms or alternative conjectures and it may
even use definitions in incompatible ways. To clarify this last point,
consider two identical definitions of a set of elements of a given type
(or of a category of objects of a given type). Let us call the two definitions
$A-Set$ and $B-Set$ (or $A-Category$ and $B-Category$).
It is perfectly legitimate to either form the $A-Set$ of every $B-Set$
or the $B-Set$ of every $A-Set$ (the same for categories). This just corresponds
to assuming that a $B-Set$ (respectively an $A-Set$) is a small set, whereas
an $A-Set$ (respectively a $B-Set$) is a big set (possibly of small sets).
However, if one part of the library assumes $A-Set$s to be the small ones
and another part of the library assumes $B-Set$s to be the small ones, the
library as a whole will be logically inconsistent.

Logical inconsistency has never been a problem in the daily work of a
mathematician. The mathematician simply imposes himself a discipline to
restrict himself to consistent subsets of the mathematical knowledge.
However, in doing so he does not choose the subset in advance by forgetting
the rest of his knowledge. On the contrary he may proceed with a sort of
top-down strategy: he may always inspect or use part of his knowledge, but
when he actually does so he should check recursively that inconsistencies are
not exploited.

Contrarily to the mathematical practice, the usual tendency in the world of
assisted automation is that of building a logical environment (a consistent
subset of the library) in a bottom up way, checking the consistency of a
new axiom or theorem as soon as it is added to the environment. No lemma
or definition outside the environment can be used until it is added to the
library after every notion it depends on. Moreover, very often the logical
environment is the only part of the library that can be inspected,
that we can search lemmas in and that can be exploited by automatic tactics.

Moving one by one notions from the library to the environment is a costly
operation since it involves re-checking the correctness of the notion.
As a consequence mathematical notions are packages into theories that must
be added to the environment as a whole. However, the consistency problem is
only raised at the level of theories: theories must be imported in a bottom
up way and the system must check that no inconsistency arises.

The practice of limiting the scope on the library to the logical environment
is contrary to our commitment of being able to fully exploit as much as possible
of the library at any given time. To reconcile consistency and visibility
we have departed from the traditional implementation of an environment
allowing environments to be built on demand in a top-down way. The new
implementation is based on a modified meta-theory that changes the way
convertibility, type checking, unification and refinement judgements.
The modified meta-theory is fully described in \cite{libraryenvironments}.
Here we just remark how a strong commitment on the way the user interacts
with the library has lead to modifications to the logical core of the proof
assistant. This is evidence that breakthroughs in the user interfaces
and in the way the user interacts with the tools and with the library could
be achieved only by means of strong architectural changes.

\subsubsection{Accessibility}
A large library that is completely in scope needs effective indexing and
searching methods to make the user productive. Libraries of formal results
are particularly critical since they hold a large percentage of technical
lemmas that do not have a significative name and that must be retrieved
using advanced methods based on matching, unification, generalization and
instantiation.

The efficiency of searching inside the library becomes a critical operation
when automatic tactics exploit the library during the proof search. In this
scenario the tactics must retrieve a set of candidates for backward or
forward reasoning in a few milliseconds.

The choice of several proof assistants is to use ad-hoc data structures,
such as context trees, to index all the terms currently in scope. These
data structures are expecially designed to quickly retrieve terms up
to matching, instantiation and generalization. All these data structures
try to maximize sharing of identical subterms so that matching can be
reduced to a visit of the tree (or dag) that holds all the maximally shared
terms together.

Since the terms to be retrieved (or at least their initial prefix)
are stored (actually ``melted'') in the data structure, these data structures
must collect all the terms in a single location. In other words, adopting
such data structures means centralizing the library.

In the \MOWGLI{} project we have tried to follow an alternative approach
that consists in keeping the library fully distributed and indexing it
by means of spiders that collect metadata and store them in a database.
The challenge is to be able to collect only a smaller as possible number
of metadata that provide enough information to approximate the matching
operation. A matching operation is then performed in two steps. The first
step is a query to the remote search engine that stores the metadata in
order to detect a (hopefully small) complete set of candidates that could
match. Completeness means that no term that matches should be excluded from
the set of candiates. The second step consists in retrieving from the
distributed library all the candidates and attempt the actual matching.

In the last we years we have progressively improved this technique.
Our achievements can be found in \cite{query1,query2,query3}.

The technique and tools already developed have been integrated in \MATITA{},
that is able to contact a remote \WHELP{} search engine \cite{whelp} or that
can be directly linked to the code of the \WHELP. In either case the database
used to store the metadata can be local or remote.

Our current challenge consists in the exploitation of \WHELP{} inside of
\MATITA. In particular we are developing a set of tactics, for instance
based on paramodulation \cite{paramodulation}, that perform queries to \WHELP{}
to restrict the scope on the library to a set of interesting candidates,
greatly reducing the search space. Moreover, queries to \WHELP{} are performed
during parsing of user provided terms to disambiguate them.

In Sect.~\ref{sec:metadata} we describe the technique adopted in \MATITA.

\subsubsection{Library management}


\subsection{ricerca e indicizzazione}
\label{sec:metadata}
\ASSIGNEDTO{andrea}

\subsection{auto}
\ASSIGNEDTO{andrea}

\subsection{sostituzioni esplicite vs moduli}
\ASSIGNEDTO{csc}

\subsection{xml / gestione della libreria}
\ASSIGNEDTO{gares}


\section{User Interface (da cambiare)}

\subsection{assenza di proof tree / resa in linguaggio naturale}
\ASSIGNEDTO{andrea}

\subsection{Disambiguation}
\label{sec:disambiguation}
\ASSIGNEDTO{zack}

 \begin{table}
  \caption{\label{tab:termsyn} Concrete syntax of CIC terms: built-in
  notation\strut}
 \hrule
 \[
 \begin{array}{@{}rcll@{}}
   \NT{term} & ::= & & \mbox{\bf terms} \\
     &     & x & \mbox{(identifier)} \\
     &  |  & n & \mbox{(number)} \\
     &  |  & s & \mbox{(symbol)} \\
     &  |  & \mathrm{URI} & \mbox{(URI)} \\
     &  |  & \verb+_+ & \mbox{(implicit)}\TODO{sync} \\
     &  |  & \verb+?+n~[\verb+[+~\{\NT{subst}\}~\verb+]+] & \mbox{(meta)} \\
     &  |  & \verb+let+~\NT{ptname}~\verb+\def+~\NT{term}~\verb+in+~\NT{term} \\
     &  |  & \verb+let+~\NT{kind}~\NT{defs}~\verb+in+~\NT{term} \\
     &  |  & \NT{binder}~\{\NT{ptnames}\}^{+}~\verb+.+~\NT{term} \\
     &  |  & \NT{term}~\NT{term} & \mbox{(application)} \\
     &  |  & \verb+Prop+ \mid \verb+Set+ \mid \verb+Type+ \mid \verb+CProp+ & \mbox{(sort)} \\
     &  |  & \verb+match+~\NT{term}~ & \mbox{(pattern matching)} \\
     &     & ~ ~ [\verb+[+~\verb+in+~x~\verb+]+]
              ~ [\verb+[+~\verb+return+~\NT{term}~\verb+]+] \\
     &     & ~ ~ \verb+with [+~[\NT{rule}~\{\verb+|+~\NT{rule}\}]~\verb+]+ & \\
     &  |  & \verb+(+~\NT{term}~\verb+:+~\NT{term}~\verb+)+ & \mbox{(cast)} \\
     &  |  & \verb+(+~\NT{term}~\verb+)+ \\
   \NT{defs}  & ::= & & \mbox{\bf mutual definitions} \\
     &     & \NT{fun}~\{\verb+and+~\NT{fun}\} \\
   \NT{fun} & ::= & & \mbox{\bf functions} \\
     &     & \NT{arg}~\{\NT{ptnames}\}^{+}~[\verb+on+~x]~\verb+\def+~\NT{term} \\
   \NT{binder} & ::= & & \mbox{\bf binders} \\
     &     & \verb+\forall+ \mid \verb+\lambda+ \\
   \NT{arg} & ::= & & \mbox{\bf single argument} \\
     &     & \verb+_+ \mid x \\
   \NT{ptname} & ::= & & \mbox{\bf possibly typed name} \\
     &     & \NT{arg} \\
     &  |  & \verb+(+~\NT{arg}~\verb+:+~\NT{term}~\verb+)+ \\
   \NT{ptnames} & ::= & & \mbox{\bf bound variables} \\
     &     & \NT{arg} \\
     &  |  & \verb+(+~\NT{arg}~\{\verb+,+~\NT{arg}\}~[\verb+:+~\NT{term}]~\verb+)+ \\
   \NT{kind} & ::= & & \mbox{\bf induction kind} \\
     &     & \verb+rec+ \mid \verb+corec+ \\
   \NT{rule} & ::= & & \mbox{\bf rules} \\
     &     & x~\{\NT{ptname}\}~\verb+\Rightarrow+~\NT{term}
 \end{array}
 \]
 \hrule
 \end{table}
 

\subsubsection{Term input}

The primary form of user interaction employed by \MATITA{} is textual script
editing: the user modifies it and evaluate step by step its composing
\emph{statements}. Examples of statements are inductive type definitions,
theorem declarations, LCF-style tacticals, and macros (e.g. \texttt{Check} can
be used to ask the system to refine a given term and pretty print the result).
Since many statements refer to terms of the underlying calculus, \MATITA{} needs
a concrete syntax able to encode terms of the Calculus of Inductive
Constructions.

Two of the requirements in the design of such a syntax are apparently in
contrast:
\begin{enumerate}
 \item the syntax should be as close as possible to common mathematical practice
  and implement widespread mathematical notations;
 \item each term described by the syntax should be non-ambiguous meaning that it
  should exists a function which associates to it a CIC term.
\end{enumerate}

These two requirements are addressed in \MATITA{} by the mean of two mechanisms
which work together: \emph{term disambiguation} and \emph{extensible notation}.
Their interaction is visible in the architecture of the \MATITA{} input phase,
depicted in Fig.~\ref{fig:inputphase}. The architecture is articulated as a
pipline of three levels: the concrete syntax level (level 0) is the one the user
has to deal with when inserting CIC terms; the abstract syntax level (level 2)
is an internal representation which intuitively encodes mathematical formulae at
the content level~\cite{adams}\cite{mkm-structure}; the last level is that of
CIC terms.

\begin{figure}[ht]
 \begin{center}
  \includegraphics[width=0.9\textwidth]{input_phase}
  \caption{\MATITA{} input phase}
 \end{center}
 \label{fig:inputphase}
\end{figure}

Requirement (1) is addressed by a built-in concrete syntax for terms, described
in Tab.~\ref{tab:termsyn}, and the extensible notation mechanisms which offers a
way for extending available mathematical notations. Extensible notation, which
is also in charge of providing a parsing function mapping concrete syntax terms
to content level terms, is described in Sect.~\ref{sec:notation}.  Requirement
(2) is addressed by the conjunct action of that parsing function and
disambiguation which provides a function from content level terms to CIC terms.

\subsubsection{Sources of ambiguity}

The translation from content level terms to CIC terms is not straightforward
because some nodes of the content encoding admit more that one CIC encoding,
invalidating requirement (2).

\begin{example}
 \label{ex:disambiguation}

 Consider the term at the concrete syntax level \texttt{\TEXMACRO{forall} x. x +
 ln 1 = x} of Fig.~\ref{fig:inputphase}(a), it can be the type of a lemma the
 user may want to prove. Assuming that both \texttt{+} and \texttt{=} are parsed
 as infix operators, all the following questions are legitimate and must be
 answered before obtaining a CIC term from its content level encoding
 (Fig.~\ref{fig:inputphase}(b)):

 \begin{enumerate}

  \item Since \texttt{ln} is an unbound identifier, which CIC constants does it
   represent? Many different theorems in the library may share its (rather
   short) name \dots

  \item Which kind of number (\IN, \IR, \dots) the \texttt{1} literal stand for?
   Which encoding is used in CIC to represent it? E.g., assuming $1\in\IN$, is
   it an unary or a binary encoding?

  \item Which kind of equality the ``='' node represents? Is it Leibniz's
   polymorhpic equality? Is it a decidable equality over \IN, \IR, \dots?

 \end{enumerate}

\end{example}

In \MATITA, three \emph{sources of ambiguity} are admitted for content level
terms: unbound identifiers, literal numbers, and operators. Each instance of
ambiguity sources (ambiguous entity) occuring in a content level term is
associated to a \emph{disambiguation domain}. Intuitively a disambiguation
domain is a set of CIC terms which may be replaced for an ambiguous entity
during disambiguation. Each item of the domain is said to be an
\emph{interpretation} for the ambiguous entity.

\emph{Unbound identifiers} (question 1) are ambiguous entities since the
namespace of CIC objects is not flat and the same identifier may denote many
ofthem. For example the short name \texttt{plus\_assoc} in the \HELM{} library
is shared by three different theorems stating the associative property of
different additions.  This kind of ambiguity is avoidable if the user is willing
to use long names (in form of URIs in the \texttt{cic://} scheme) in the
concrete syntax, with the obvious drawbacks of obtaining long and unreadable
terms.

Given an unbound identifier, the corresponding disambiguation domain is computed
querying the library for all constants, inductive types, and inductive type
constructors having it as their short name (see the \LOCATE{} query in
Sect.~\ref{sec:metadata}).

\emph{Literal numbers} (question 2) are ambiguous entities as well since
different kinds of numbers can be encoded in CIC (\IN, \IR, \IZ, \dots) using
different encodings. Considering the restricted example of natural numbers we
can for instance encode them in CIC using inductive datatypes with a number of
constructor equal to the encoding base plus 1, obtaining one encoding for each
base.

For each possible way of mapping a literal number to a CIC term, \MATITA{} is
aware of a \emph{number intepretation function} which, when applied to the
natural number denoted by the literal\footnote{at the moment only literal
natural number are supported in the concrete syntax} returns a corresponding CIC
term. The disambiguation domain for a given literal number is built applying to
the literal all available number interpretation functions in turn.

Number interpretation functions can be defined in OCaml or directly using
\TODO{notazione per i numeri}.

\emph{Operators} (question 3) are intuitively head of applications, as such they
are always applied to a (possiblt empty) sequence of arguments. Their ambiguity
is a need since it is often the case that some notation is used in an overloaded
fashion to hide the use of different CIC constants which encodes similar
concepts. For example, in the standard library of \MATITA{} the infix \texttt{+}
notation is available building a binary \texttt{Op(+)} node, whose
disambiguation domain may refer to different constants like the addition over
natural numbers \URI{cic:/matita/nat/plus/plus.con} or that over real numbers of
the \COQ{} standard library \URI{cic:/Coq/Reals/Rdefinitions/Rplus.con}.

For each possible way of mapping an operator application to a CIC term,
\MATITA{} knows an \emph{operator interpretation function} which, when applied
to an operator and its arguments, returns a CIC term. The disambiguation domain
for a given operator is built applying to the operator and its arguments all
available operator interpretation functions in turn.

Operator interpretation functions could be added using the
\texttt{interpretation} statement. For example, among the first line of the
script \FILE{matita/library/logic/equality.ma} from the \MATITA{} standard
library we read:

\begin{grafite}
interpretation "leibnitz's equality"
 'eq x y =
   (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
\end{grafite}

Evaluating it in \MATITA{} will add an operator interpretation function for the
binary operator \texttt{eq} which expands to the CIC term on the right hand side
of the statement. That CIC term can be written using only built-in concrete
syntax, can contain no ambiguity source; still, it can refer to operator
arguments bound on the left hand side and can contain implicit terms (denoted
with \texttt{\_}) which will be expanded to fresh metavariables. The latter
feature is used in the example above for the first argument of Leibniz's
polymorhpic equality.

\subsubsection{Disambiguation algorithm}

A \emph{disambiguation algorithm} takes as input a content level term and return
a fully determined CIC term. The key observation on which a disambiguation
algorithm is based is that given a content level term with more than one sources
of ambiguity, not all possible combination of interpretation lead to a typable
CIC term. In the term of Ex.~\ref{ex:disambiguation} for instance the
interpretation of \texttt{ln} as a function from \IR to \IR and the
interpretation of \texttt{1} as the Peano number $1$ can't coexists. The notion
of ``can't coexists'' in the disambiguation of \MATITA{} is defined on top of
the \emph{refiner} for CIC terms described in~\cite{csc-phd}.

Briefly, a refiner is a function whose input is an \emph{incomplete CIC term}
$t_1$ --- i.e. a term where metavariables occur (Sect.~\ref{sec:metavariables}
--- and whose output is either:\NOTE{descrizione sommaria del refiner, pu\'o
essere spostata altrove}

\begin{enumerate}
 
 \item an incomplete CIC term $t_2$ where $t_2$ is a well-typed term obtained
  assigning a type to each metavariable in $t_1$ (in case of dependent types,
  instantiation of some of the metavariable occurring in $t_1$ may occur as
  well);

 \item $\epsilon$, meaning that no well-typed term could be obtained via
  assignment of type to metavariable in $t_1$ and their instantiation;

 \item $\bot$, meaning that the refiner is unable to decide whether of the two
  cases above apply (refinement is semi-decidable).

\end{enumerate}

On top of a CIC refiner \MATITA{} implement an efficient disambiguation
algorithm, which is outlined below. It takes as input a content level term $c$
and proceeds as follows:

\begin{enumerate}

 \item Create disambiguation domains $\{D_i | i\in\mathit{Dom}(c)\}$, where
  $\mathit{Dom}(c)$ is the set of ambiguity sources of $c$. Each $D_i$ is a set
  of CIC terms and can be built as described above.

 \item An \emph{interpretation} $\Phi$ for $c$ is a map associating an
  incomplete CIC term to each ambiguity source of $c$. Given $c$ and one of its
  interpretations an incomplete CIC term is fully determined replacing each
  ambiguity source of $c$ with its mapping in the interpretation and injecting
  the remaining structure of the content level in the CIC level (e.g. replacing
  the application of the content level with the application of the CIC level).
  This operation is informally called ``interpreting $c$ with $\Phi$''.
  
  Create an initial interpretation $\Phi_0 = \{\phi_i | \phi_i = \_,
  i\in\mathit{Dom}(c)\}$, which associates a fresh metavariable to each source
  of ambiguity of $c$. During this step, implicit terms are expanded to fresh
  metavariables as well.

 \item Refine the current incomplete CIC term (i.e.  the term obtained
  interpreting $t$ with $\Phi_i$).

  If the refinement succeeds or is undetermined the next interpretation
  $\Phi_{i+1}$ will be created \emph{making a choice}, that is replacing in the
  current interpretation one of the metavariable appearing in $\Phi_i$ with one
  of the possible choice from the corresponding disambiguation domain. The
  metavariable to be replaced is chosen following a preorder visit of the
  ambiguous term. Then, step 3 is attempted again with the new interpretation.
    
  If the refinement fails the current set of choices cannot lead to a well-typed
  term and backtracking of the current interpretation is attempted.
    
 \item Once an unambiguous correct interpretation is found (i.e. $\Phi_i$ does
  no longer contain any placeholder), backtracking is attempted anyway to find
  the other correct interpretations.

 \item Let $n$ be the number of interpretations who survived step 4. If $n=0$
  signal a type error. If $n=1$ we have found exactly one (incomplete) CIC term
  corresponding to the content level term $c$, returns it as output of the
  disambiguation phase. If $n>1$ we have found many different (incomplete) CIC
  terms which can correspond to the content level term, let the user choose one
  of the $n$ interpretations and returns the corresponding term.

\end{enumerate}

The efficiency of this algorithm resides in the fact that as soon as an
incomplete CIC term is not typable, no further instantiation of the
metavariables of the corresponding interpretation is attemped.
% For example, during the disambiguation of the user input
% \texttt{\TEXMACRO{forall} x. x*0 = 0}, an interpretation $\Phi_i$ is
% encountered which associates $?$ to the instance of \texttt{0} on the right,
% the real number $0$ to the instance of \texttt{0} on the left, and the
% multiplication over natural numbers (\texttt{mult} for short) to \texttt{*}.
% The refiner will fail, since \texttt{mult} require a natural argument, and no
% further instantiation of the placeholder will be tried.

Details of the disambiguation algorithm along with an analysis of its complexity
can be found in~\cite{disambiguation}, where a formulation without backtracking
(corresponding to the actual \MATITA{} implementation) is also presented.

\subsubsection{Disambiguation stages}

\subsection{notation}
\label{sec:notation}
\ASSIGNEDTO{zack}

Mathematical notation plays a fundamental role in mathematical practice: it
helps expressing in a concise symbolic fashion concepts of arbitrary complexity.
Its use in proof assistants like \MATITA{} is no exception. Formal mathematics
indeed often impose to encode mathematical concepts at a very high level of
details (e.g. Peano numbers, implicit arguments) having a restricted toolbox of
syntactic constructions in the calculus.

Consider for example one of the point reached while proving the distributivity
of times over minus on natural numbers included in the \MATITA{} standards
library. (Part of) the reached sequent can be seen in \MATITA{} both using the
notation for various arithmetical and relational operator or without using it.
The sequent rendered without using notation would be as follows:
\sequent{
\mathtt{H}: \mathtt{le} z y\\
\mathtt{Hcut}: \mathtt{eq} \mathtt{nat} (\mathtt{plus} (\mathtt{times} x (\mathtt{minus}
y z)) (\mathtt{times} x z))\\
(\mathtt{plus} (\mathtt{minus} (\mathtt{times} x y) (\mathtt{times} x z))
(\mathtt{times} x z))}{
\mathtt{eq} \mathtt{nat} (\mathtt{times} x (\mathtt{minus} y z)) (\mathtt{minus}
(\mathtt{times} x y) (\mathtt{times} x z))}
while the corresponding sequent rendered with notation enabled would be:
\sequent{
H: z\leq y\\
Hcut: x*(y-z)+x*z=x*y-x*z+x*z}{
x*(y-z)=x*y-x*z}


\subsection{mathml}
\ASSIGNEDTO{zack}

\subsection{selezione semantica, cut paste, hyperlink}
\ASSIGNEDTO{zack}

\subsection{pattern}
\ASSIGNEDTO{gares}\\
Patterns are the textual counterpart of the MathML widget graphical
selection.

Matita benefits of a graphical interface and a powerful MathML rendering
widget that allows the user to select pieces of the sequent he is working
on. While this is an extremely intuitive way for the user to
restrict the application of tactics, for example, to some subterms of the
conclusion or some hypothesis, the way this action is recorded to the text
script is not obvious.\\
In \MATITA{} this issue is addressed by patterns.

\subsubsection{Pattern syntax}
A pattern is composed of two terms: a $\NT{sequent\_path}$ and a
$\NT{wanted}$.
The former mocks-up a sequent, discharging unwanted subterms with $?$ and
selecting the interesting parts with the placeholder $\%$. 
The latter is a term that lives in the context of the placeholders.

The concrete syntax is reported in table \ref{tab:pathsyn}
\NOTE{uso nomi diversi dalla grammatica ma che hanno + senso}
\begin{table}
 \caption{\label{tab:pathsyn} Concrete syntax of \MATITA{} patterns.\strut}
\hrule
\[
\begin{array}{@{}rcll@{}}
  \NT{pattern} & 
    ::= & [~\verb+in match+~\NT{wanted}~]~[~\verb+in+~\NT{sequent\_path}~] & \\
  \NT{sequent\_path} & 
    ::= & \{~\NT{ident}~[~\verb+:+~\NT{multipath}~]~\}~
      [~\verb+\vdash+~\NT{multipath}~] & \\
  \NT{wanted} & ::= & \NT{term} & \\
  \NT{multipath} & ::= & \NT{term\_with\_placeholders} & \\
\end{array}
\]
\hrule
\end{table}

\subsubsection{How patterns work}
Patterns mimic the user's selection in two steps. The first one
selects roots (subterms) of the sequent, using the
$\NT{sequent\_path}$,  while the second 
one searches the $\NT{wanted}$ term starting from these roots. Both are
optional steps, and by convention the empty pattern selects the whole
conclusion.

\begin{description}
\item[Phase 1]
  concerns only the $[~\verb+in+~\NT{sequent\_path}~]$
  part of the syntax. $\NT{ident}$ is an hypothesis name and
  selects the assumption where the following optional $\NT{multipath}$
  will operate. \verb+\vdash+ can be considered the name for the goal.
  If the whole pattern is omitted, the whole goal will be selected.
  If one or more hypotheses names are given the selection is restricted to 
  these assumptions. If a $\NT{multipath}$ is omitted the whole
  assumption is selected. Remember that the user can be mostly
  unaware of this syntax, since the system is able to write down a 
  $\NT{sequent\_path}$ starting from a visual selection.
  \NOTE{Questo ancora non va in matita}

  A $\NT{multipath}$ is a CiC term in which a special constant $\%$
  is allowed.
  The roots of discharged subterms are marked with $?$, while $\%$
  is used to select roots. The default $\NT{multipath}$, the one that
  selects the whole term, is simply $\%$.
  Valid $\NT{multipath}$ are, for example, $(?~\%~?)$ or $\%~\verb+\to+~(\%~?)$
  that respectively select the first argument of an application or
  the source of an arrow and the head of the application that is
  found in the arrow target.

  The first phase selects not only terms (roots of subterms) but also 
  their context that will be eventually used in the second phase.

\item[Phase 2] 
  plays a role only if the $[~\verb+in match+~\NT{wanted}~]$
  part is specified. From the first phase we have some terms, that we
  will see as subterm roots, and their context. For each of these
  contexts the $\NT{wanted}$ term is disambiguated in it and the
  corresponding root is searched for a subterm $\alpha$-equivalent to
  $\NT{wanted}$. The result of this search is the selection the
  pattern represents.

\end{description}

\noindent
Since the first step is equipotent to the composition of the two
steps, the system uses it to represent each visual selection.
The second step is only meant for the
experienced user that writes patterns by hand, since it really
helps in writing concise patterns as we will see in the
following examples.

\subsubsection{Examples}
To explain how the first step works let's give an example. Consider
you want to prove the uniqueness of the identity element $0$ for natural
sum, and that you can relay on the previously demonstrated left
injectivity of the sum, that is $inj\_plus\_l:\forall x,y,z.x+y=z+y \to x =z$.
Typing
\begin{grafite}
theorem valid_name: \forall n,m. m + n = n \to m = O.
  intros (n m H).
\end{grafite}
\noindent
leads you to the following sequent 
\sequent{
n:nat\\
m:nat\\
H: m + n = n}{
m=O
}
\noindent
where you want to change the right part of the equivalence of the $H$
hypothesis with $O + n$ and then use $inj\_plus\_l$ to prove $m=O$.
\begin{grafite}
  change in H:(? ? ? %) with (O + n).
\end{grafite}
\noindent
This pattern, that is a simple instance of the $\NT{sequent\_path}$
grammar entry, acts on $H$ that has type (without notation) $(eq~nat~(m+n)~n)$
and discharges the head of the application and the first two arguments with a
$?$ and selects the last argument with $\%$. The syntax may seem uncomfortable,
but the user can simply select with the mouse the right part of the equivalence
and left to the system the burden of writing down in the script file the
corresponding pattern with $?$ and $\%$ in the right place (that is not
trivial, expecially where implicit arguments are hidden by the notation, like
the type $nat$ in this example).

Changing all the occurrences of $n$ in the hypothesis $H$ with $O+n$ 
works too and can be done, by the experienced user, writing directly
a simpler pattern that uses the second phase.
\begin{grafite}
  change in match n in H with (O + n).
\end{grafite}
\noindent
In this case the $\NT{sequent\_path}$ selects the whole $H$, while
the second phase searches the wanted $n$ inside it by
$\alpha$-equivalence. The resulting
equivalence will be $m+(O+n)=O+n$ since the second phase found two
occurrences of $n$ in $H$ and the tactic changed both.

Just for completeness the second pattern is equivalent to the
following one, that is less readable but uses only the first phase.
\begin{grafite}
  change in H:(? ? (? ? %) %) with (O + n).
\end{grafite}
\noindent

\subsubsection{Tactics supporting patterns}
In \MATITA{} all the tactics that can be restricted to subterm of the working
sequent accept the pattern syntax. In particular these tactics are: simplify,
change, fold, unfold, generalize, replace and rewrite.

\NOTE{attualmente rewrite e fold non supportano phase 2. per
supportarlo bisogna far loro trasformare il pattern phase1+phase2 
in un pattern phase1only come faccio nell'ultimo esempio. lo si fa
con una pattern\_of(select(pattern))}

\subsubsection{Comparison with Coq}
Coq has a two diffrent ways of restricting the application of tactis to
subterms of the sequent, both relaying on the same special syntax to identify
a term occurrence.

The first way is to use this special syntax to specify directly to the
tactic the occurrnces of a wanted term that should be affected, while
the second is to prepare the sequent with another tactic called
pattern and the apply the real tactic. Note that the choice is not
left to the user, since some tactics needs the sequent to be prepared
with pattern and do not accept directly this special syntax.

The base idea is that to identify a subterm of the sequent we can
write it and say that we want, for example, the third and the fifth
occurce of it (counting from left to right). In our previous example,
to change only the left part of the equivalence, the correct command
is
\begin{grafite}
  change n at 2 in H with (O + n)
\end{grafite} 
\noindent
meaning that in the hypothesis $H$ the $n$ we want to change is the
second we encounter proceeding from left toright.

The tactic pattern computes a
$\beta$-expansion of a part of the sequent with respect to some
occurrences of the given term. In the previous example the following
command
\begin{grafite}
  pattern n at 2 in H
\end{grafite}
\noindent
would have resulted in this sequent
\begin{grafite}
  n : nat
  m : nat
  H : (fun n0 : nat => m + n = n0) n
  ============================
   m = 0
\end{grafite}
\noindent
where $H$ is $\beta$-expanded over the second $n$
occurrence. This is a trick to make the unification algorithm ignore
the head of the application (since the unification is essentially
first-order) but normally operate on the arguments. 
This works for some tactics, like rewrite and replace,
but for example not for change and other tactics that do not relay on
unification. 

The idea behind this way of identifying subterms in not really far
from the idea behind patterns, but really fails in extending to
complex notation, since it relays on a mono-dimensional sequent representation.
Real math notation places arguments upside-down (like in indexed sums or
integrations) or even puts them inside a bidimensional matrix.  
In these cases using the mouse to select the wanted term is probably the 
only way to tell the system exactly what you want to do. 

One of the goals of \MATITA{} is to use modern publishing techiques, and
adopting a method for restricting tactics application domain that discourages 
using heavy math notation, would definitively be a bad choice.

\subsection{Tacticals}
\ASSIGNEDTO{gares}\\
There are mainly two kinds of languages used by proof assistants to recorder
proofs: tactic based and declarative. We will not investigate the philosophy
aroud the choice that many proof assistant made, \MATITA{} included, and we will not compare the two diffrent approaches. We will describe the common issues of the first one and how \MATITA{} tries to solve them.

First we must highlight the fact that proof scripts made using tactis are
particularly unreadable. This is not a big deal for the user while he is
constructing the proof, but is considerably a problem when he tries to reread
what he did or when he shows his work to someone else.

Another common issue for tactic based proof scripts is their mantenibility.
Huge libraries have been developed, and backward compatibility is a really time
consuming task. This problem is usually ameliorated with tacticals, that
help in structuring proofs and consequently their maintenance, but have a bad
counterpart in script readability.  Since tacticals are executed atomically,
the common practice of executing again a script to review all the proof steps
doesn't work at all. This issue in addition to the really poor feeling that a
list of tactics gives about the proof makes script rereading particularly hard.

\MATITA{} uses a language of tactics and tacticals, but adopts a peculiar
strategy to make this technique more user friendly without loosing in
mantenibility or expressivity.

\subsubsection{Tacticals overview}
Before describing the peculiarities of \MATITA{} tacticals we briefly introduce what tacticals are and where they can be useful.

Tacticals first appered in LCF(cita qualcosa) and can be seen as programming
constructs, like looping, branching, error recovery or sequential composition.
For example $tac_1~.~tac_2$ executes the first tactic and applies the second
only to the first goal opened by $tac_1$. Baranching can be used to specify a
diffrent tactic to apply to each new goal opened by another tactic, for example
$tac_1\verb+;[+~tac_{1.1}~\verb+|+~tac_{1.2}~\verb+|+~\cdots~|~tac_{1.n}~\verb+]+$
applies respectively $tac_{1.i}$ to the $i$-th goal opened by $tac_1$. Looping
can be used to iterate a tactic until it works: $\verb+repeat+~tac$ applies
$tac$ to the current goal, and again $tac$ to the eventually resulting goals
until all goal are closed or the tactic fails.

\begin{table}
 \caption{\label{tab:tacsyn} Concrete syntax of \MATITA{} tacticals.\strut}
\hrule
\[
\begin{array}{@{}rcll@{}}
  \NT{punctuation} & 
    ::= & \SEMICOLON \quad|\quad \DOT \quad|\quad \SHIFT \quad|\quad \BRANCH \quad|\quad \MERGE \quad|\quad \POS{\mathrm{NUMBER}~} & \\
  \NT{block\_kind} & 
    ::= & \verb+focus+ ~|~ \verb+try+ ~|~ \verb+solve+ ~|~ \verb+first+ ~|~ \verb+repeat+ ~|~ \verb+do+~\mathrm{NUMBER} & \\
  \NT{block\_delimiter} & 
    ::= & \verb+begin+ ~|~ \verb+end+ & \\
  \NT{tactical} & 
    ::= & \verb+skip+ ~|~ \NT{tactic} ~|~ \NT{block\_delimiter} ~|~ \NT{block\_kind} ~|~ \NT{punctuation} ~|~& \\
\end{array}
\]
\hrule
\end{table}

\MATITA{} tacticals syntax is reported in table \ref{tab:tacsyn}.
While one whould expect to find structured constructs like 
$\verb+do+~n~\NT{tactic}$ the syntax allows pieces of tacticals to be written.
This is essential for base idea behind matita tacticals: step-by-step execution.

\subsubsection{\MATITA{} Tinycals}
The low-level tacticals implementation of \MATITA{} allows a step-by-step execution of a tactical, that substantially means that a $\NT{block\_kind}$ is not executed as an atomic operation. This has two major benefits for the user, even being a so simple idea:
\begin{description}
\item[Proof structuring] 
  is much easyer. Consider for example a proof by induction, and imagine you are using classical tacticals. After applying the
  induction principle, with one step tacticals, you have to choose: structure
  the proof or not. If you decide for the former you have to branch with
  \verb+[+ and write tactics for all the cases separated by \verb+|+ and the
  close the tactical with
  \verb+]+. You can replace most of the cases by the identity tactic just to
  concentrate only on the first goal, but you will have to go one step back and
  one further every time you add something inside the tactical. And if you are
  boared of doing so, you will finish in giving up structuring the proof and
  write a plain list of tactics.\\
  With step-by-step tacticals you can apply the induction principle, and just
  open the branching tactical \verb+[+. Then you can interact with the system
  reaching a proof of the first case, without having to specify the whole
  branching tactical. When you have proved all the induction cases, you close
  the branching tactical with \verb+]+ and you are done with a structured proof.
\item[Rereading]
  is possible. Going on step by step shows exactly what is going on.
  Consider again a proof by induction, that starts applying the induction
  principle and suddenly baranches with a \verb+[+. This clearly separates all
  the induction cases, but if the square brackets content is executed in one
  single step you completely loose the possibility of rereading it. Again,
  executing step-by-step is the way you whould like to review the
  demonstration. Remember tha understandig the proof from the script is not
  easy, and only the execution of tactics (and the resulting transformed goal)
  gives you the feeling of what is goning on.
\end{description}



\subsection{named variable e disambiguazione lazy}
\ASSIGNEDTO{csc}

\subsection{metavariabili}
\label{sec:metavariables}
\ASSIGNEDTO{csc}

\begin{verbatim}

\end{verbatim}

\section{Drawbacks, missing, \dots}

\subsection{moduli}
\ASSIGNEDTO{}

\subsection{ltac}
\ASSIGNEDTO{}

\subsection{estrazione}
\ASSIGNEDTO{}

\subsection{localizzazione errori}
\ASSIGNEDTO{}

\acknowledgements
We would like to thank all the students that during the past
five years collaborated in the \HELM{} project and contributed to 
the development of Matita, and in particular
A.Griggio, F.Guidi, P. Di Lena, L.Padovani, I.Schena, M.Selmi, 
V.Tamburrelli.

\theendnotes

\bibliography{matita}


\end{document}