Provable formulas

Important provable formulas are given by isomorphisms and by equivalences.

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

Distributivities

Standard distributivities

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

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

$\exists \xi . (A \with B) \limp (\exists \xi . A) \with (\exists \xi . B)$

Linear distributivities

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

$\exists \xi. (A \parr B) \limp A \parr \exists \xi.B \quad (\xi\notin A)$

$A \tens \forall \xi.B \limp \forall \xi. (A \tens B) \quad (\xi\notin A)$

Factorizations

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

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

$(\forall \xi . A) \plus (\forall \xi . B) \limp \forall \xi . (A \plus B)$

Additive structure

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

Quantifiers

$\begin{array}{rcll} A &\limp& \forall \xi.A &\quad (\xi\notin A) \\ \exists \xi.A &\limp& A &\quad (\xi\notin A) \end{array}$

$\begin{array}{rcl} \forall \xi_1.\forall \xi_2. A &\limp& \forall \xi. A[^\xi/_{\xi_1},^\xi/_{\xi_2}] \\ \exists \xi.A[^\xi/_{\xi_1},^\xi/_{\xi_2}] &\limp& \exists \xi_1. \exists \xi_2.A \end{array}$

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 exponentials

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