Provable formulas

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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

$\begin{array}{rcl} A\with B \longrightarrow A &\quad& A\with B \longrightarrow B\\ (C\limp A)\with(C\limp B) &\longrightarrow& C\limp(A\with B)\\ A &\longrightarrow& A\with A\\ A \longrightarrow A\plus B &\quad& B \longrightarrow A\plus B\\ (A\limp C)\with(B\limp C) &\longrightarrow& (A\plus B)\limp C\\ A\plus A &\longrightarrow& A\\ \end{array}$

Exponential structure

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

$\begin{array}{rclcrcl} \oc A &\longrightarrow& A &\quad& A&\longrightarrow&\wn A\\ \oc A &\longrightarrow& 1 &\quad& \bot &\longrightarrow& \wn A\\ \oc A &\longrightarrow& \oc A\tens\oc A &\quad& \wn A\parr\wn A &\longrightarrow& \wn A\\ \oc A &\longrightarrow& \oc\oc A &\quad& \wn\wn A &\longrightarrow& \wn A\\ \end{array}$

Monoidality of exponential

$\begin{array}{rclcrcl} \oc A\tens\oc B &\longrightarrow& \oc(A\tens B) &\quad& \one &\longrightarrow& \oc\one\\ \end{array}$