\env{\Gamma,e,t} (\varpi) & = &\constr\varpi{\Gamma,e,t}\land\tsrep{\tyof{\occ e \varpi}\Gamma}
\env{\Gamma,e,t} (\varpi) & = &\tsrep{\constr\varpi{\Gamma,e,t}\tsand\tyof{\occ e \varpi}\Gamma}
\end{eqnarray}
\begin{align*}
...
...
@@ -1053,11 +1053,11 @@
Now, let's prove the second property.
We perform a (nested) induction on $\varpi$.
Recall that $\env{\Gamma,e,t}(\varpi)=\constr\varpi{\Gamma,e,t}\land\tsrep{\tyof{\occ e \varpi}\Gamma}$.
We can easily derive $\pvdash\Gamma e t \varpi : \tsrep{\tyof{\occ e \varpi}\Gamma}$ by using the outer induction hypothesis
(by definition of $\tsrep{\_}$ we have $\forall\ts.\ \ts\leq\tsrep\ts$).
Recall that $\env{\Gamma,e,t}(\varpi)=\tsrep{\constr\varpi{\Gamma,e,t}\tsand\tyof{\occ e \varpi}\Gamma}$.
For any $t'$ such that $\tyof{\occ e \varpi}\Gamma\leq t'$, we can easily derive $\pvdash\Gamma e t \varpi : t'$ by using the outer induction hypothesis
(the first property that we have proved above) and the rule \Rule{PTypeof}.
We also have to derive $\pvdash\Gamma e t \varpi : \constr\varpi{\Gamma,e,t}$ (then it will be easy to conclude using the rule \Rule{PInter}).
Now we have to derive $\pvdash\Gamma e t \varpi : \constr\varpi{\Gamma,e,t}$ (then it will be easy to conclude using the rule \Rule{PInter}).
\begin{description}
\item[$\varpi=\epsilon$] We use the rule \Rule{PEps}.
\item[$\varpi=\varpi'.1$]
...
...
@@ -1174,6 +1174,7 @@
\begin{lemma}
\begin{align*}
&\forall t,\ts.\ \tsrep{t\tsand\ts}\leq t \land\tsrep{\ts}\\
&\forall t,\ts.\ t\tsand\ts\not\simeq\Empty\Rightarrow\tsrep{t\tsand\ts}\simeq t \land\tsrep{\ts}\\