Provable formulas

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(→‎Monoidality of exponential: Units removed: special case of equivalence for positive/negative formula)
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<math>
  
<math>
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\begin{array}{rcl}
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\begin{array}{rclcrclcrcl}
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A\with B \limp A &\quad& A\with B \limp B\\
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A\with B &\limp& A &\quad& A\with B &\limp& B &\quad& A &\limp& \top\\
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A \limp A\plus B &\quad& B \limp A\plus B\\
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A &\limp& A\plus B &\quad& B &\limp& A\plus B &\quad& \zero &\limp& A
 
\end{array}
  
\end{array}
 
</math>
  
</math>

Revision as of 19:06, 28 October 2013

This page is a stub and needs more content.

Important provable formulas are given by isomorphisms and by equivalences.

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

Contents

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}

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