%%%% Includes
\usepackage{prova}
\pfshortnumbers{999}
\usepackage{color,graphicx}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{amsfonts}
%\mathfrak	gothic
%\mathcal	calligraphic
%\mathscr	calligraphic++
%\mathbb	blackboard
%\mathbf	bold 
%\usepackage{txfonts}
\usepackage{xypic}
\usepackage{pst-node}
\usepackage{verbatim}
\usepackage[dvips]{geometry}
\usepackage{color} %%% to use \textcolor{white}{lalalala}

%%%% Index
\usepackage{makeidx}
\makeindex
\newcommand{\ilabel}[1]{\label{#1}\index{#1}}
\newcommand{\xlabel}[1]{#1\ilabel{#1}}
\newcommand{\iemph}[1]{\emph{#1}\ilabel{#1}}
\newcommand{\fem}[1]{\emph{#1}}
%%%%

%%%% PDF
\RequirePackage{ifpdf} % running on pdfTeX?
\ifpdf
\usepackage[pdftex,bookmarks=true,
		   bookmarksnumbered=false,
		   bookmarksopen=true,
		   colorlinks=true,
		   linkcolor=webred] {hyperref}
\definecolor{webgreen}{rgb}{0, 0.5, 0} % less intense green
\definecolor{webblue}{rgb}{0, 0, 0.5} % less intense blue
\definecolor{webred}{rgb}{0.5, 0, 0}   % less intense red
\else
%\let\oldlabel\label
%\renewcommand{\label}[1]{ [#1] \oldlabel{#1}}
\newcommand{\href}[2]{ #1 }
\newcommand{\url}[1]{ #1 }
\fi
%%%%

%%%% Math commands
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{exthm}[thm]{Experimental Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ax}{Axiom}
\newtheorem{domanda}{Domanda}
\newtheorem{question}{Question}
\newtheorem{conje}[thm]{Conjecture}

\theoremstyle{definition}
\newtheorem{paradox}{Paradox}
\newtheorem{statm}{Statement}
\newtheorem{oss}{Observation}
\newtheorem{numberedproof}{Proof}
\newtheorem{algo}{Algorithm}
\newtheorem{defn}[thm]{Definition}
\newtheorem{notaz}[thm]{Notazione}
\newtheorem{notat}[thm]{Notation}
\newtheorem{exemp}[thm]{Example}
\newtheorem{data}{Verbatim}

\theoremstyle{remark}
\newtheorem{rem}{Remark}[section]
\newtheorem*{notation}{Notation}

\newcommand{\qq}{\qquad}
\newcommand{\Oo}[1]{\overline{#1}}
\def\ove#1{{\overline{#1}}}
\newcommand{\T}[1]{\textrm{#1}}

\newcommand{\I}{\mathcal{I}}
\newcommand{\V}{\mathcal{V}}
\newcommand{\C}{\mathcal{C}}
\newcommand{\B}{\mathcal{B}}
\newcommand{\A}{\mathcal{A}}
\newcommand{\TR}{{^t\!}}
\newcommand{\TRC}{{^\tau\!\!}}
\newcommand{\inte}[2]{\int_{[#1,#2]}}
\newcommand{\intd}[2]{\int_{#1}^{#2}}
\newcommand{\intc}[3]{(#1)\!\!\!\int_{#2}^{#3}}
\newcommand{\accu}[1]{D\left(#1\right)}
\newcommand{\fron}[1]{\operatorname{F}\left(#1\right)}
\newcommand{\den}[1]{{#1}^\circ}
\newcommand{\ext}{\operatorname{ext}}
\newcommand{\mis}{\operatorname{mis}}
\newcommand{\area}{\operatorname{area}}
\newcommand{\tg}{\tan}
\newcommand{\atan}{\oname{atan}}
\newcommand{\sen}{\sin}
\newcommand{\arsinh}{\operatorname{arsinh}}
\newcommand{\arsenh}{\operatorname{arsinh}}
\newcommand{\arcosh}{\operatorname{arcosh}}
\newcommand{\arcos}{\operatorname{arcos}}
\newcommand{\arcsinh}{\operatorname{arcsinh}}
\newcommand{\arcsenh}{\operatorname{arcsinh}}
\newcommand{\arccosh}{\operatorname{arccosh}}
\newcommand{\arctanh}{\operatorname{arctanh}}
\newcommand{\asse}[1]{\stackrel{\rightarrow}{#1}}
\newcommand{\eqpun}{\eqtxt{p.}}
\newcommand{\equni}{\eqtxt{u.}}
\newcommand{\plim}[3]{\lim_{n\to +\infty} #1 \stackrel{p.}{=} #2\;\T{in}\;#3}
\newcommand{\ulim}[3]{\lim_{n\to +\infty} #1 \stackrel{u.}{=} #2\;\T{in}\;#3}
\newcommand{\conv}[3]{#1 \stackrel{#2}{\longrightarrow } #3}
\newcommand{\pconv}[3]{#1 \stackrel{p.}{\longrightarrow } #2\;\T{in}\;#3}
\newcommand{\uconv}[3]{#1 \stackrel{u.}{\longrightarrow } #2\;\T{in}\;#3}
\newcommand{\tconv}[3]{#1 \stackrel{t.}{\longrightarrow } #2\;\T{in}\;#3}
\newcommand{\eqtxt}[1]{\stackrel{\T{#1}}{=}}
\newcommand{\serie}[1]{\sum_{#1}^{\infty}}
\newcommand{\serien}{\sum_{n=1}^{\infty}}
\newcommand{\limn}{\lim_{n\to +\infty}}
\newcommand{\pserie}[4]{\serie{#1}#2 \stackrel{p.}{=} #3\;\T{in}\;#4}
\newcommand{\userie}[4]{\serie{#1}#2 \stackrel{u.}{=} #3\;\T{in}\;#4}
\newcommand{\aserie}[4]{\serie{#1}#2 \stackrel{a.}{=} #3\;\T{in}\;#4}
\newcommand{\tserie}[4]{\serie{#1}#2 \stackrel{t.}{=} #3\;\T{in}\;#4}
\newcommand{\pserieconv}[4]{\serie{#1}#2 \stackrel{p.}{\longrightarrow } #3\;\T{in}\;#4}
\newcommand{\userieconv}[4]{\serie{#1}#2 \stackrel{u.}{\longrightarrow } #3\;\T{in}\;#4}
\newcommand{\aserieconv}[4]{\serie{#1}#2 \stackrel{a.}{\longrightarrow } #3\;\T{in}\;#4}
\newcommand{\tserieconv}[4]{\serie{#1}#2 \stackrel{t.}{\longrightarrow } #3\;\T{in}\;#4}
\newcommand{\ubrace}[2]{\underbrace{#1}_{#2}}
\def\uline#1{{\underline{#1}}}
\newcommand{\obrace}[2]{\overbrace{#1}^{#2}}
\newcommand{\deriv}[1]{D\left[#1\right]}
\newcommand{\derivn}[2]{D^{#1}\left[#2\right]}
\newcommand{\dclasseinf}[1]{C^{\infty}({#1})}
\newcommand{\sczero}[1]{C^{0}({#1})}
\newcommand{\sczn}[2]{C^{#1}({#2})}
\newcommand{\somma}[2]{\sum_{#1}^{#2}}
\newcommand{\sumrn}{\sum_{r=1}^{n}}
\newcommand{\sumin}{\sum_{i=1}^{n}}
\newcommand{\sumiN}{\sum_{i=1}^{N}}
\newcommand{\sumim}{\sum_{i=1}^{m}}
\newcommand{\sumik}{\sum_{i=1}^{k}}
\newcommand{\sumkn}{\sum_{k=1}^{n}}
\newcommand{\sumkm}{\sum_{k=1}^{m}}
\newcommand{\sumzin}{\sum_{i=0}^{n}}
\newcommand{\sumjn}{\sum_{j=1}^{n}}
\newcommand{\sumzjn}{\sum_{j=0}^{n}}
\newcommand{\sumjm}{\sum_{j=1}^{m}}
\newcommand{\sumip}{\sum_{i=1}^{p}}
\newcommand{\sumir}{\sum_{i=1}^{r}}
\newcommand{\sumis}{\sum_{i=1}^{s}}
\newcommand{\sumjr}{\sum_{j=1}^{r}}
\newcommand{\sumjs}{\sum_{j=1}^{s}}
\newcommand{\sumzkn}{\sum_{k=0}^{n}}
\newcommand{\prodin}{\prod_{i=1}^{n}}
\newcommand{\sumit}{\sum_{i=1}^{3}}
\newcommand{\xor}{\oplus}
\newcommand{\XOR}{\;\operatorname{XOR}\;}
\renewcommand{\succ}[1]{\left\{#1\right\}}
\renewcommand{\vec}[1]{\overrightarrow{#1}}
\def\vec#1{{\overrightarrow{#1}}}
\def\cev#1{{\vec{\vec{#1}}}}
\def\cev#1{{\overleftarrow{#1}}}
\newcommand{\vers}{\operatorname{vers}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\cod}{\operatorname{cod}}
\newcommand{\dom}{\operatorname{dom}}
\def\diag{{\operatorname{diag}}}
\def\dabs#1{{\Vert#1\Vert}}
\def\inv#1{{{#1}^{-1}}}
\def\gen#1{{\left\langle{#1}\right\rangle}}
\def\Aff{{\operatorname{Aff}}}
\def\id{{\operatorname{id}}}
\def\mcm{{\operatorname{mcm}}}
\def\MCD{{\operatorname{MCD}}}
\def\Hom{{\operatorname{Hom}}}
\def\div{{\Big/}}
\def\mid{\div}
\def\ndiv{{
\mid
\mkern-11mu
{{}_\backslash}
}}
\def\tleq{{\trianglelefteq}}
\def\treq{{\trianglerighteq}}
\def\immerso{{\hookrightarrow}}
\def\inniezione{{\hookrightarrow}}
\def\inject{{\hookrightarrow}}
\def\inniettato{{\hookrightarrow}}

\def\abs{{\operatorname{abs}}}
\newcommand{\diam}{\operatorname{diam}}
\def\definizione#1{{\;\;\stackrel{\T{def}}{#1}}\;\;}
\def\norm#1{{\Vert#1\Vert}}
\def\bnorm#1{{\BigVert#1\BigVert}}
\newcommand{\coord}[2]{[#1]_{#2}}
\def\coome#1{{\coord{#1}{\Omega}}}
\def\coorB#1{{\coord{#1}{B}}}
\def\dual#1{{{#1}^{\circ}}}
\def\nequiv{{\not\equiv}}
\def\eps{{\varepsilon}}

\def\floor#1{{\lfloor#1\rfloor}}

%% Defs
\def\*{{\times}}
\def\|{{\;\lor\;}}
\def\bigAND{{\;\bigwedge\;}}
\def\bigOR{{\;\bigvee\;}}
\def\&{{\;\land\;}}
\def\NOT{{\lnot}}
\def\True{{\oname{True}}}
\def\False{{\oname{False}}}
\def\Vero{{\oname{Vero}}}
\def\Falso{{\oname{Falso}}}
\def\-{{\setminus}}
\def\0{{\emptyset}}
\def\8{{\infty}}
\def\v{{\cup}}
\def\U{{\cup}}
\def\bigunion{{\bigcup}}
\def\biginter{{\bigcap}}
\def\^{{\cap}}
\def\<{{\;\Leftarrow\;}}
\def\>{{\;\Rightarrow\;}}
\def\={{\;\Leftrightarrow\;}}
\def\~{{\;\simeq\;}}
\def\({{\subseteq}}
\def\){{\supseteq}}
\def\'{{\;\;\;}}
\def\,{{,\;}}
%%%%

%%%% Misc
\newcommand{\sref}[1]{\T{passo \stepref{#1}}}
\newcommand{\spasso}[1]{\T{\stepref{#1}}}
\renewcommand{\see}[1]{\T{[\ref{#1},pg.\pageref{#1}]}}
\newcommand{\vedi}[1]{\T{vedi \see{#1}}}
\def\seeT#1{\T{\ref{#1}}}
\def\seeprop#1{\T{Proposizione \seeT{#1}}}
\def\seecor#1{\T{Corollario \seeT{#1}}}
\def\seelem#1{\T{Lemma \seeT{#1}}}
\def\seethm#1{\T{Teorema \seeT{#1}}}
\def\seep#1{\T{Prop \seeT{#1}}}
\def\seel#1{\T{Lem \seeT{#1}}}
\def\seec#1{\T{Cor \seeT{#1}}}
\def\seet#1{\T{Thm \seeT{#1}}}
%\newcommand{\vedi}[1]{\T{vedi \seeT{#1}}}

\newcommand{\pgra}[1]{\left\{#1\right\}}
\def\setformula#1{{\pgra{#1}}}
\newcommand{\pqua}[1]{\left[#1\right]}
\newcommand{\pton}[1]{\left(#1\right)}
\newcommand{\pass}[1]{\Big|#1\Big|}
\newcommand{\sist}[1]{{ \begin{cases} #1 \end{cases} }}
\newcommand{\eal}[1]{{\begin{align*} #1 \end{align*}}}
\def\linea{\rule{10cm}{.1pt}}
\def\lineapiccola{\rule{2cm}{.1pt}}
\def\nl{{\\&\qq }} % newline inside \eal{}
\def\NN{\\& } % newline inside \eal{}
\def\oname#1{\operatorname{#1}}
\def\ona#1{\operatorname{#1}}
\def\strut#1{\rule{0pt}{#1ex}}
\def\hstrut#1{\hline \strut#1}
\def\invisible#1{{\textcolor{white}{#1}}}

%%%%

%%%% Arrow
%%%% Something doesn't work (I'm beginning to hate LaTeX)
%\def\UP{{\ar[u]}}
%\def\DOWN{{\ar[d]}}
%\def\LEFT{{\ar[l]}}
%\def\RIGHT{{\ar[r]}}
%\def\UPeq{{\ar@{=}[u]}}
%\def\DOWNeq{{\ar@{=}[d]}}
%\def\LEFTeq{{\ar@{=}[l]}}
%\def\RIGHTeq{{\ar@{=}[r]}}

%%%% Proof
\renewenvironment{proof}{\begin{prova}\pf}{ \qed\end{prova}}

	%%%%%DEPRECATED%%%%%%%%%%%%55
\newenvironment{pnecessaria}{ \step{}{Dim. $\>$} \begin{prova}}{\end{prova}}
\newenvironment{psufficiente}{ \step{}{Dim. $\<$} \begin{prova}}{\end{prova}}
\newenvironment{pnecsuf}{ \step{}{Dim. $\=$} \begin{prova}}{\end{prova}}
	%%%%%DEPRECATED%%%%%%%%%%%%55

%\def\BEGINpnecessaria{\step{}{Dim. $\>$} \begin{prova}}
%\def\BEGINsufficiente{\step{}{Dim. $\<$} \begin{prova}}
%\def\BEGINpnecsuf{\step{}{Dim. $\=$} \begin{prova}}

\def\BEGINpnecessaria{\step{}{($\>$)} \begin{prova}}
\def\BEGINsufficiente{\step{}{($\<$)} \begin{prova}}
\def\BEGINpnecsuf{\step{}{($\<$)} \begin{prova}}
\def\END{\end{prova}}

%%%% Latex
\newcommand{\Comment}[1]{}

%Molte volte scrivo foonote al posto di footnote
\def\foonote#1{\footnote{#1}}