Provable formulas

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Important provable formulas are given by isomorphisms and by equivalences.

In many of the cases below the converse implication does not hold.

Distributivities

$A\plus (B\with C) \limp (A\plus B)\with (A\plus C)$

$A\tens (B\parr C) \limp (A\tens B)\parr C$

Factorizations

$(A\with B)\plus (A\with C) \limp A\with (B\plus C)$

Additive structure

$\begin{array}{rclcrclcrcl} A\with B &\limp& A &\quad& A\with B &\limp& B &\quad& A &\limp& \top\\ A &\limp& A\plus B &\quad& B &\limp& A\plus B &\quad& \zero &\limp& A \end{array}$

Exponential structure

Provable formulas involving exponential connectives only provide us with the lattice of exponential modalities.

$\begin{array}{rclcrcl} \oc A &\limp& A &\quad& A&\limp&\wn A\\ \oc A &\limp& 1 &\quad& \bot &\limp& \wn A \end{array}$

Monoidality of exponential

$\begin{array}{rcl} \wn(A\parr B) &\limp& \wn A\parr\wn B \\ \oc A\tens\oc B &\limp& \oc(A\tens B) \end{array}$