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\subsection{Types}

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\begin{definition}[Type syntax]\label{def:types} The set of types are the terms $t$ coinductively produced by the following grammar
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\[
\begin{array}{lrcl}
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\textbf{Types} & t & ::= & b\alt t\to t\alt t\times t\alt t\vee t \alt \neg t \alt \Empty 
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\end{array}
\]
and that satisfy the following conditions
\begin{itemize}
\item (regularity) the term has a finite number of different sub-terms;
\item (contractivity) every infinite branch of a type contains an infinite number of occurrences of the
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arrows or product type constructors.
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\end{itemize}
\end{definition}
We introduce the following abbreviations for types: $
    t_1 \land t_2 \eqdef \neg (\neg t_1 \vee \neg t_2)$, 
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    $t_ 1 \setminus t_2 \eqdef t_1 \wedge \neg t_2$, $\Any \eqdef \neg \Empty$.
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%
We refer to $ b $ and $ \to $ as \emph{type constructors}
and to $ \lor $, $ \land $, $ \lnot $, and $ \setminus $
as \emph{type connectives}.



\subsubsection{Subtyping}

\subsubsection{Operators for type constructors}

Let $t$ be a functional type (i.e., $t\leq\Empty\to\Any$) then

\begin{eqnarray}
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\dom t & = & \max \{ u \alt t\leq u\to \Any\} 
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\\
\apply t s & = &\min \{ u \alt t\leq s\to u\}
\\
\worra t s & = &\min\{u\leq \dom t\alt t\circ(\dom t\setminus u)\leq \neg s\}
\end{eqnarray}


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\subsubsection{Type schemes}

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Given a value $v$, the set of types $t$ such that $v \in t$ has no smallest element in general.
Indeed, the standard derivation rule for $\lambda$-abstraction values is:
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\begin{mathpar}
  \Infer[Abs]
      {t=(\wedge_{i\in I}\arrow {s_i} {t_i})\land (\wedge_{j\in J} \neg (\arrow {s'_j} {t'_j}))\\t\not\leq \Empty}
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      {\vdash \lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e : t}
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      {}
  \\
\end{mathpar}

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In the next parts, we will need an algorithmic type system.
In order for it to be complete in regards to lambda-abstractions, we'll need to use a
new syntactic category, called \textbf{type schemes}, that denotes a set of types.
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\[
\begin{array}{lrcl}
  \textbf{Type schemes} & \ts & ::= & t \alt \tsfun {\arrow t t ; \cdots ; \arrow t t} \alt \ts \tstimes \ts \alt \ts \tsor \ts \alt \tsempty
\end{array}
\]

\begin{definition}[Interpretation of type schemes]
  We define a function $\tsint {\_}$ that maps type schemes to set of types.

  \begin{align*}
    \begin{array}{lcl}
    \tsint t &=& \{s\alt t \leq s\}\\
    \tsint {\tsfunone {t_i} {s_i}_{i=1..n}} &=& \{s\alt
    \exists s_0 = \bigwedge_{i=1..n} \arrow {t_i} {s_i}
    \land \bigwedge_{j=1..m} \neg (\arrow {t_j'} {s_j'}).\ 
    \Empty \not\simeq s_0 \leq s \}\\
    \tsint {\ts_1 \tstimes \ts_2} &=& \{s\alt \exists t_1 \in \tsint {\ts_1}\ 
    \exists t_2 \in \tsint {\ts_2}.\ \pair {t_1} {t_2} \leq s\}\\
    \tsint {\ts_1 \tsor \ts_2} &=& \{s\alt \exists t_1 \in \tsint {\ts_1}\ 
    \exists t_2 \in \tsint {\ts_2}.\ {t_1} \vee {t_2} \leq s\}\\
    \tsint \tsempty &=& \varnothing
    \end{array}
  \end{align*}

Note that $\tsint \ts$ is closed under subsumption and intersection (straightforward induction).
\end{definition}

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\begin{definition}[Representative]
  We define a function $\tsrep {\_}$ that maps every non-empty type scheme into a type, \textit{representative} of the set of types denoted by the scheme.
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  \begin{align*}
    \begin{array}{lcl}
    \tsrep t &=& t\\
    \tsrep {\tsfunone {t_i} {s_i}_{i\in I}} &=& \bigwedge_{i\in I} \arrow {t_i} {s_i}\\
    \tsrep {\ts_1 \tstimes \ts_2} &=& \pair {\tsrep {\ts_1}} {\tsrep {\ts_2}}\\
    \tsrep {\ts_1 \tsor \ts_2} &=& \tsrep {\ts_1} \vee \tsrep {\ts_2}\\
    \tsrep \tsempty && \textit{undefined}
    \end{array}
  \end{align*}
\end{definition}
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Note that $\tsrep \ts \in \tsint \ts$.
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\begin{lemma}
  Let $\ts$ be a type scheme and $t$ a type. We can compute a type scheme, written $t \tsand \ts$, such that:
  \[\tsint {t \tsand \ts} = \{s \alt \exists t' \in \tsint \ts.\ t \land t' \leq s \}\]
\end{lemma}

\begin{lemma}
  Let $\ts$ be a type scheme and $t$ a type. We can decide the assertion $t \in \tsint \ts$,
  which we also write $\ts \leq t$.
\end{lemma}
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\subsection{Expressions}



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\[
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\begin{array}{lrcl}
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\textbf{Expressions} & e & ::= & c\alt x \alt \lambda^{\bigwedge \arrow t t}x.e \alt \ite k t e e \alt e e\\
\textbf{Conditions} & k & ::= & c\alt x \alt \lambda^{\bigwedge \arrow t t}x.e \alt k k\\
\textbf{Values} & v & ::= & c \alt \lambda^{\bigwedge \arrow t t}x.e\\
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\end{array}
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\]

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Let $e$ be an expression and $\varpi\in\{0,1\}^*$ a \emph{path}; we denote $\occ e\varpi$ the occurrence of $e$ reached by the path $\varpi$, that is
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\[
\begin{array}{r@{\downarrow}l@{\quad=\quad}l}
e&\epsilon & e\\
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e_0e_1& i.\varpi & \occ{e_i}\varpi\qquad i=0,1\\
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\end{array}
\]
undefined otherwise

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A type environment $\Gamma$ is a mapping from occurrences (i.e., expressions) to types, up to alpha-renaming (i.e., $\lambda^tx.x$ and $\lambda^ty.y$ are mapped to the same type, if any).
It is necessary to map alpha-equivalent expressions to the same type in order for our type system to be invariant by alpha-renaming.
We use $\Gamma_1,\Gamma_2$ for the type environment obtained by unioning the two type environments giving priority to $\Gamma_2$ (define formally).
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We suppose w.l.o.g that all variables abstracted in $\lambda$-abstractions are distincts (otherwise we can alpha-rename $\lambda$-abstractions).

\subsection{Declarative type system}

\begin{mathpar}
  \Infer[Occ]
      { }
      { \Gamma \vdash e: \Gamma(e) }
      { e\in\dom\Gamma }
  \qquad
  \Infer[Intersect]
      { \Gamma \vdash e:t_1\\\Gamma \vdash e:t_2 }
      { \Gamma \vdash e: t_1 \wedge t_2 }
      { }
  \qquad
  \Infer[Subs]
      { \Gamma \vdash e:t\\t\leq t' }
      { \Gamma \vdash e: t' }
      { }
  \qquad
  \\
  \Infer[EFQ]
  { }
  { \Gamma, (e:\Empty) \vdash e': t }
  { }
  \qquad
  \Infer[Const]
      { }
      {\Gamma\vdash c:\basic{c}}
      { }
  \qquad
  \Infer[App]
      {
        \Gamma \vdash e_1: \arrow {t_1}{t_2}\\
        \Gamma \vdash e_2: t_1
      }
      { \Gamma \vdash {e_1}{e_2}: t_2 }
      { }
      \\
  \Infer[Abs]
      {\Gamma,x:s_i\vdash e:t_i\\t\simeq \left(\bigwedge_{i\in I} \arrow {s_i} {t_i}\right)
      \land \left(\bigwedge_{j\in J} \neg (\arrow {s'_j} {t'_j})\right)\\t\not\simeq\Empty}
      {
      \Gamma\vdash\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:\textstyle t
      }
      { }
      \\
%      \Infer[If]
%            {\Gamma\vdash e:t_0\\
%            %t_0\not\leq \neg t \Rightarrow
%            \Gamma \cvdash + e t e_1:t'\\
%            %t_0\not\leq t \Rightarrow
%            \Gamma \cvdash - e t e_2:t'}
%            {\Gamma\vdash \ite {e} t {e_1}{e_2}: t'}
%            { }
    \Infer[If]
        {\Gamma\vdash e:t_0\\
        %t_0\not\leq \neg t \Rightarrow
        \Gamma \evdash + e t \Gamma^+ \\ \Gamma^+ \vdash e_1:t'\\
        %t_0\not\leq t \Rightarrow
        \Gamma \evdash - e t \Gamma^- \\ \Gamma^- \vdash e_2:t'}
        {\Gamma\vdash \ite {e} t {e_1}{e_2}: t'}
        { }
  \\
  \Infer[Proj]
  {\Gamma \vdash e:(t_1,t_2)}
  {\Gamma \vdash \pi_i e:t_i}
  { }
  \qquad
  \Infer[Pair]
  {\Gamma \vdash e_1:t_1 \and \Gamma \vdash e_2:t_2}
  {\Gamma \vdash (e_1,e_2):\pair {t_1} {t_2}}
  { }
\end{mathpar}

\begin{center} \line(1,0){300} \end{center}

\begin{mathpar}
%        \Infer[Base]
%            { \Gamma \vdash e':t' }
%            { \Gamma \cvdash p e t e':t' }
%            { }
%            \qquad
%        \Infer[Path]
%            { \pvdash \Gamma p e t \varpi:t_1 \\ \Gamma,(\occ e \varpi:t_1) \cvdash p e t e':t_2 }
%            { \Gamma \cvdash p e t e':t_2 }
%            { }
    \Infer[Base]
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      { }
      { \Gamma \evdash p e t \Gamma }
      { }
    \qquad
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    \Infer[Path]
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      { \pvdash {\Gamma'} p e t \varpi:t' \\ \Gamma \evdash p e t \Gamma' }
      { \Gamma \evdash p e t \Gamma',(\occ e \varpi:t') }
      { }
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\end{mathpar}

\begin{center} \line(1,0){300} \end{center}

\begin{mathpar}
    \Infer[PIntersect]
        { \pvdash \Gamma p e t \varpi:t_1 \\ \pvdash \Gamma p e t \varpi:t_2 }
        { \pvdash \Gamma p e t \varpi:t_1\land t_2 }
        { }
        \qquad
    \Infer[PSubs]
        { \pvdash \Gamma p e t \varpi:t_1 \\ t_1\leq t_2 }
        { \pvdash \Gamma p e t \varpi:t_2 }
        { }
        \\
    \Infer[PTypeof]
        { \Gamma \vdash \occ e \varpi:t' }
        { \pvdash \Gamma p e t \varpi:t' }
        { }
        \qquad
    \Infer[PEps+]
        { }
        { \pvdash \Gamma + e t \epsilon:t }
        { }
        \qquad
    \Infer[PEps-]
        { }
        { \pvdash \Gamma - e t \epsilon:\neg t }
        { }
        \\
    \Infer[PAppR]
        { \pvdash \Gamma p e t \varpi.0:\arrow{t_1}{t_2} \\ \pvdash \Gamma p e t \varpi:t_2' \\ t_2\land t_2' \simeq \Empty }
        { \pvdash \Gamma p e t \varpi.1:\neg t_1 }
        { }
        \\
    \Infer[PAppL]
        { \pvdash \Gamma p e t \varpi.1:t_1 \\ \pvdash \Gamma p e t \varpi:t_2 }
        { \pvdash \Gamma p e t \varpi.0:\neg (\arrow {t_1} {\neg t_2}) }
        { }
        \qquad
    \Infer[PPairL]
        { \pvdash \Gamma p e t \varpi:\pair{t_1}{t_2} }
        { \pvdash \Gamma p e t \varpi.l:t_1 }
        { }
        \\
    \Infer[PPairR]
        { \pvdash \Gamma p e t \varpi:\pair{t_1}{t_2} }
        { \pvdash \Gamma p e t \varpi.r:t_2 }
        { }
        \qquad
    \Infer[PFst]
        { \pvdash \Gamma p e t \varpi:t' }
        { \pvdash \Gamma p e t \varpi.f:\pair {t'} \Any }
        { }
        \qquad
    \Infer[PSnd]
        { \pvdash \Gamma p e t \varpi:t' }
        { \pvdash \Gamma p e t \varpi.s:\pair \Any {t'} }
        { }
        \qquad
\end{mathpar}

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\subsection{Algorithmic type system}
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\begin{mathpar}
  \Infer[EFQ]
  { }
  { \Gamma, (e:\Empty) \vdash e': \Empty }
  { \text{With priority over other rules} }
  \\
  \Infer[Occ]
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      { \Gamma\setminus\{e\} \vdash e : \ts }
      { \Gamma \vdash e: \Gamma(e) \tsand \ts }
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      { e\in\dom\Gamma}
  \qquad
  \Infer[Const]
      { }
      {\Gamma\vdash c:\basic{c}}
      {c\not\in\dom\Gamma}
   \\
  \Infer[Abs]
      {\Gamma,x:s_i\vdash e:\ts_i'\\ \ts_i'\leq t_i}
      {
      \Gamma\vdash\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e:\textstyle\tsfun {\arrow {s_i} {t_i}}_{i\in I}
      }
      {\lambda^{\wedge_{i\in I}\arrow {s_i} {t_i}}x.e\not\in\dom\Gamma}
      \\
  \Infer[App]
      {
        \Gamma \vdash e_1: \ts_1\\
        \Gamma \vdash e_2: \ts_2\\
        \ts_1 \leq \arrow \Empty \Any\\
        \ts_2 \leq \dom {\ts_1}
      }
      { \Gamma \vdash {e_1}{e_2}: \ts_1 \circ \ts_2 }
      { {e_1}{e_2}\not\in\dom\Gamma}
      \\
  \Infer[If]
        {\Gamma\vdash e:\ts_0\\
        %\makebox{$\begin{array}{l}
        %  \left\{
        %    \begin{array}{ll} %\Gamma,
        %    \Refine + {e,t} \Gamma \vdash e_1 : \ts_1 & \text{ if } \ts_0 \not\leq \neg t\\
        %    \ts_1 = \Empty & \text{ otherwise}
        %  \end{array}\right.\\
        %  \left\{
        %    \begin{array}{ll} %\Gamma,
        %    \Refine - {e,t} \Gamma \vdash e_2 : \ts_2 & \text{ if } \ts_0 \not\leq t\\
        %    \ts_2 = \Empty & \text{ otherwise}
        %  \end{array}\right.
        %\end{array}$}
        \Refine + {e,t} \Gamma \vdash e_1 : \ts_1\\
        \Refine - {e,t} \Gamma \vdash e_2 : \ts_2}
        {\Gamma\vdash \ite {e} t {e_1}{e_2}: \ts_1\tsor \ts_2}
        %{\ite {e} t {e_1}{e_2}\not\in\dom\Gamma}
        {\texttt{if}\dots\not\in\dom\Gamma}
  \\
  \Infer[Proj]
  {\Gamma \vdash e:\ts\and \ts\leq\pair\Any\Any}
  {\Gamma \vdash \pi_i e:\bpi_{\mathbf{i}}(\ts)}
  {\pi_i e\not\in\dom\Gamma}
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  \Infer[Pair]
  {\Gamma \vdash e_1:\ts_1 \and \Gamma \vdash e_2:\ts_2}
  {\Gamma \vdash (e_1,e_2):{\ts_1}\tstimes{\ts_2}}%\pair{t_1}{t_2}}
  {(e_1,e_2)\not\in\dom\Gamma}
\end{mathpar}
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\[
\begin{array}{lcl}
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  \typep+\epsilon{\Gamma,e,t} & = & t\\
  \typep-\epsilon{\Gamma,e,t} & = & \neg t\\
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  \typep{p}{\varpi.0}{\Gamma,e,t} & = & \neg(\arrow{\Gp p{\Gamma,e,t}{(\varpi.1)}}{\neg \Gp p {\Gamma,e,t} (\varpi)})\\
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  \typep{p}{\varpi.1}{\Gamma,e,t} & = & \worra{\tsrep {\tyof{\occ e{\varpi.0}}\Gamma}}{\Gp p {\Gamma,e,t} (\varpi)}\\
  \typep{p}{\varpi.l}{\Gamma,e,t} & = & \bpl{\Gp p {\Gamma,e,t} (\varpi)}\\
  \typep{p}{\varpi.r}{\Gamma,e,t} & = & \bpr{\Gp p {\Gamma,e,t} (\varpi)}\\
  \typep{p}{\varpi.f}{\Gamma,e,t} & = & \pair{\Gp p {\Gamma,e,t} (\varpi)}\Any\\
  \typep{p}{\varpi.s}{\Gamma,e,t} & = & \pair\Any{\Gp p {\Gamma,e,t} (\varpi)}\\ \\
  \Gp p {\Gamma,e,t} (\varpi) & = & \typep p \varpi {\Gamma,e,t} \land \tsrep {\tyof {\occ e \varpi} \Gamma}\\
  %\underbrace{\land \tyof {\occ e \varpi} {\Gamma\setminus\{\occ e \varpi\}}}_{\text{if $\occ e \varpi$ is not a variable}}}\\
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\end{array}
\]
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\begin{align*}
  &(\RefineStep p {e,t} (\Gamma))(e') = 
    \left\{\begin{array}{ll}
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      %\tyof {e'} \Gamma \tsand
      \bigwedge_{\{\varpi \alt \occ e \varpi \equiv e'\}}
      \Gp p {\Gamma,e,t} (\varpi) & \text{if } \exists \varpi.\ \occ e \varpi \equiv e' \\
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      \Gamma(e') & \text{otherwise, if } e' \in \dom \Gamma\\
      \text{undefined} & \text{otherwise}
    \end{array}\right.\\
  &\Refine p {e,t} \Gamma=\fixpoint_\Gamma (\RefineStep p {e,t})\qquad \text{(for the order $\leq$ on environments, defined in the proofs section)}
\end{align*}
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All the functions above are defined iff all their subexpressions are (e.g., $\occ e{\varpi.i}$ must be defined).
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The notation $\tyof{o}{\Gamma}$ denotes the type that can be deduced for the occurence $o$ under the type envirenment $\Gamma$.
That is, $\tyof{o}{\Gamma}=\ts$ if and only if $\Gamma\vdash o:\ts$ can be deduced by the typing rules.
\footnote{Note that the definition is well-founded.
This can be seen by analyzing the rule \Rule{If}: the definition of $\Refine + {e,t} \Gamma$ and  $\Refine - {e,t} \Gamma$ use
$\tyof{\occ e{\varpi}}\Gamma$, and this is defined for all $\varpi$ since the first premisses of \Rule{If} states that
$\Gamma\vdash e:\ts_0$ (and this is possible only if we were able to deduce under the hypothesis $\Gamma$ the type of every occurrence of $e$.)}


The reason for the definition of $\RefineStep p {e,t}$ is that the same subterm of $e$
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may occur several times in $e$ and therefore we can collect for every
occurrence a different type constraint. Since all the constraints must
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hold, then we take their intersection. For instance, if $f$ is a function
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of type $(s_1\times s_2\to t)\wedge( s_3\to\neg t)$ and we test
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whether $f(x,x)$ is of type $t$ or not, then the test succeed only if $(x,x)$ is of
type $s_1\times s_2$, that is, that $x$ has both type $s_1$ and type
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$s_2$ and thus their intersection $s_1{\wedge}s_2$.\\

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\subsection{Algorithms}
Define how to compute $t\circ s$ and $\worra t s$

If
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\[ t \simeq \bigvee_{i\in I}\left(\bigwedge_{p\in P_i}(s_p\to t_p)\bigwedge_{n\in N_i}\neg(s_n'\to t_n')\right) \]
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with \[\forall i\in I.\ \bigwedge_{p\in P_i}(s_p\to t_p)\bigwedge_{n\in N_i}\neg(s_n'\to t_n') \not\simeq \Empty\]
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then
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\begin{eqnarray*}
\dom{t}    & = & \bigwedge_{i\in I}\bigvee_{p\in P_i}s_p\\[4mm]
t\circ s   & = & \bigvee_{i\in I}\left(\bigvee_{\{Q\subsetneq P_i\alt s\not\leq\bigvee_{q\in Q}s_q\}}\left(\bigwedge_{p\in P_i\setminus Q}t_p\right)\right)\hspace*{1cm}\makebox[0cm][l]{(for $s\leq\dom{t}$)}\\[4mm]
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\worra t s & = & \dom t \wedge\bigvee_{i\in I}\left(\bigwedge_{\{P\subset P_i\alt s\leq \bigvee_{p \in P} \neg t_p\}}\left(\bigvee_{p \in P} \neg s_p\right) \right)
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\end{eqnarray*}
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\beppe{Explain, especially the last one}