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

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

Exponential structure

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

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

Monoidality of exponential

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

Other provable formulas are given by isomorphisms.