### se

parent cafa32fd
 ... ... @@ -459,8 +459,7 @@ Let $M=\lambda x.(x \oplus \id)$. The following (incomplete) derivation can be b Note that $\multiset{M} \Redo \multiset{\two \id , \two \lambda x.\id}$; while $\der \id:\dist{\multiset{\dist{\At}} \arrow \At}$, it is necessary to have weakening in order to built a derivation proving $\der \lambda x. \id: \dist{\multiset{\dist{\At}} \arrow (\multiset{\dist{\Bt}} \arrow \Bt) }$. \end{example} \begin{enumerate} \item \antonio{ \antonio{ \begin{itemize} \item $[\lambda x. (x\oplus \id)]\Red[\frac 1 2 \lambda x. x, \frac 1 2 \lambda x. \id]$, \item $\der \lambda x.x:\dist{\multiset{\dist{\At}} \arrow \At}$, ... ... @@ -473,7 +472,7 @@ but this cannot be the case since the type of $\lambda x.(x\oplus \id)$ must ha $\dist{\two (\multiset{\dist{\At}} \arrow \At, \two (\multiset{\dist{\At}} \arrow \multiset{\dist{\Bt}} \arrow \Bt}$ (see the previous example) } \item \simona{ \simona{ Reading with attention the text of the subject expansion: \begin{itemize} \item $[\lambda x. (x\oplus \id)]\Red[\frac 1 2 \lambda x. x, \frac 1 2 \lambda x. \id]$, ... ... @@ -488,18 +487,18 @@ $\lambda x. (x \oplus I) : \two\dist{\multiset{}\arrow \dist{\multiset{\dist{\At We are NOT obliged to consider all together all the types of the reduct, but just a subset of them. In fact the rule$\oplus$allows to type just one component of a sum. In order to make it more clear we could modify the text as I will write after the old one. } \item \antonio{ \begin{itemize} \item$[\lambda x. (x\oplus \id)]\Red[\lambda x. x\oplus \lambda x. \id]$, \item$\der \lambda x. x\oplus \lambda x. \id:\dist{\frac 1 2 \multiset{\dist{\At}} \arrow \At, \frac 1 2 \multiset{}\arrow \dist{\multiset{\dist{\At}} \arrow \At}}$\end{itemize} Hence, by subject expansion (new formulation), it should exist a context$\Gamma$(greater than the empty context, hence whatsoever) such that$\Gamma \der \lambda x.(x\oplus \id):\dist{\frac 1 2 \multiset{\dist{\At}} \arrow \At, \frac 1 2 \multiset{}\arrow \dist{\multiset{\dist{\At}} \arrow \At}}$, but this cannot be the case since the type of$\lambda x.(x\oplus \id)$must have the shape$\dist{\two (\multiset{\dist{\At}} \arrow \At, \two (\multiset{\dist{\At}} \arrow \multiset{\dist{\Bt}} \arrow \Bt}$(see the previous example) } %\antonio{ %\begin{itemize} %\item$[\lambda x. (x\oplus \id)]\Red[\lambda x. x\oplus \lambda x. \id]$, %\item$\der \lambda x. x\oplus \lambda x. \id:\dist{\frac 1 2 \multiset{\dist{\At}} \arrow \At, \frac 1 2 \multiset{}\arrow \dist{\multiset{\dist{\At}} \arrow \At}}$%\end{itemize} %Hence, by subject expansion (new formulation), it should exist a context$\Gamma$(greater than the empty context, hence whatsoever) such that %$\Gamma \der \lambda x.(x\oplus \id):\dist{\frac 1 2 \multiset{\dist{\At}} \arrow \At, \frac 1 2 \multiset{}\arrow \dist{\multiset{\dist{\At}} \arrow \At}}$, %but this cannot be the case since the type of$\lambda x.(x\oplus \id)$must have the shape %$\dist{\two (\multiset{\dist{\At}} \arrow \At, \two (\multiset{\dist{\At}} \arrow \multiset{\dist{\Bt}} \arrow \Bt}$(see the previous example) %} \item \simona{ \simona{ Applying the subject expansion property it turns out : \begin{itemize} \item$[\lambda x. (x\oplus \id)]\Red[\two \lambda x. x, \two \lambda x. \id]$, ... ... @@ -529,7 +528,7 @@ In fact: and similarly for the other typing and the other term. The idea is that we can consider the multiset of the reduct collecting a submultiset of them (so, simplifying the property, one by one). } \end{enumerate} In fact the weakening rule is derivable, as the following property formalizes. \begin{property} \label{prop:weak} ... ... @@ -636,12 +635,12 @@ By induction on the length of the reduction, see the Appendix. \antonio{\begin{lemma}[Subject Expansion]\label{lem:subexp}$\multiset{M} \Red^* \multiset{p_iN_i \mid \iI}$,$k\in I$and$\Gamma \der N_k:\dist{q_j\At_j}_{\jJ}$imply$\forall l\in J\Delta \der M: p_kq_l\At_l$, for some$\Delta$,$\Gamma \leq \Delta$. \end{lemma}} %\antonio{\begin{lemma}[Subject Expansion]\label{lem:subexp} %$\multiset{M} \Red^* \multiset{p_iN_i \mid \iI}$,$k\in I$and$\Gamma \der % N_k:\dist{q_j\At_j}_{\jJ}$%imply$\forall l\in J\Delta \der % M: p_kq_l\At_l$, for some$\Delta$,$\Gamma \leq \Delta$. %\end{lemma}} In \cite{fscd} it has been proved that the system \Sis\ characterises$\hnf\$s: \begin{theorem}\label{th:char} ... ...
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