Negative $N$ | $::=$ | $P \lol N \celse N \land N \celse \top \celse \dia P \celse a^- \celse \ups P$ |
Positive $P$ | $::=$ | $P \tensor P \celse P \lor P \celse 1 \celse 0 \celse \square N \celse a^+ \celse \dns N$ |
The expressions $p\trov f$ and $\star(f)$ are negative atoms. Every Language-A atom (regardless of polarity) is mapped to a Language-B positive atom. The expression $s \ge t$ is a positive atom. The expression $\forall s \ge t . N$ abbreviates $\forall s . (s \ge t) \imp N$. $$P^\bullet_t(\phi) = \forall\alpha \cn P^t . (\alpha\trov \phi)$$ $$N^\bullet_t(\alpha) = \forall\phi \cn N^t . (\alpha\trov \phi)$$
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Prenegative $M$ | $::=$ | $\dns N \celse a^+(t)\celse M \tensor M$ |
Negative $N$ | $::=$ | $\forall x\cn P.N \celse M\imp N \celse N \land N \celse \top \celse a^-(t)$ |
Positive $P$ | $::=$ | $\exists x \cn P.P \celse P \lor P \celse 0 \celse [t]\celse \eta x . M$ |
Contexts $\Gamma$ | $::=$ | $\cdot \celse \Gamma, M \celse \Gamma, x:P$ |
Conclusions $K$ | $::=$ | $t:P \celse N \celse M$ |
$P$ | $\forall x \cn P . N $ |
$\exists y \cn P_1 . P_2$ | $ \forall y \cn P_1 . \forall x \cn P_2.N$ |
$P_1 \lor P_2$ | $ \forall x \cn P_1 . N \land \forall x \cn P_2 . N$ |
$0$ | $ \top$ |
$1$ | $ \forall x . N$ |
$[t]$ | $ [t/x]N$ |
$\eta x \cn P . M$ | $ \forall x \cn P. (M\imp N)$ |
Prenegative $M$ | $::=$ | $\dns N \celse a^+(t)$ |
Negative $N$ | $::=$ | $\forall x .N \celse M\imp N \celse a^-(t) \celse N \land N \celse \top$ |
Contexts $\Gamma$ | $::=$ | $\cdot \celse \Gamma, M $ |