\in {\cal P}^+(X)$, then $A \f B
    \in {\cal P}^+(X)$ and $B \f A \in {\cal P}^-(X)$.

-   If $A,B \in {\cal P}^+(X)$, then $A \et B, B \ou A \in {\cal
    P}^+(X)$.

-   If $A,B \in {\cal P}^-(X)$, then $A \et B, B \ou A \in {\cal
    P}^-(X)$.

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-   Let ${\cal F}=\{F_i \; / \; i \in I \}$ be a set of types in $\cal{T}$ (resp. in $\cal{T}'$). The congruence $\approx$ generated by ${\cal F}$ in $\cal{T}$ (resp. in $\cal{T}'$) is the least congruence such that $X_i \approx F_i$ for each $i \in I$.

-   We say that $\approx$ is good if, for each $X \in {\cal X}$, if $X \approx A$, then $A \in {\cal P}^+(X)$.

Strong normalization of ${\cal S}^{\m}_{\approx}$
-------------------------------------------------

Let $\approx$ be the congruence generated by a set ${\cal F}$ of types of $\cal{T}$.

\[mendler\] If $\approx$ is good, then the system ${\cal S}_{\approx}$ is strongly normalizing.

See [@Mend1] for the original proof and [@DaNo3] for an arithmetical one.

\[coding3\] If $\G \v_{{\cal S}^{\m}_{\approx}} M : A$, then $\G^{\diamond}
\v_{{\cal S}^c_{\approx}} M^{\diamond} : A$.

By induction on the typing $\G \v_{{\cal S}^{\m}_{\approx}} M : A$.

\[mendler\_mu\] If $\approx$ is good, then the 