# Positive formula

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A ''positive formula'' is a formula <math>P</math> such that <math>P\limp\oc P</math> (thus a [[Wikipedia:F-coalgebra|coalgebra]] for the [[Wikipedia:Comonad|comonad]] <math>\oc</math>). As a consequence <math>P</math> and <math>\oc P</math> are [[Sequent calculus#Equivalences|equivalent]]. |
A ''positive formula'' is a formula <math>P</math> such that <math>P\limp\oc P</math> (thus a [[Wikipedia:F-coalgebra|coalgebra]] for the [[Wikipedia:Comonad|comonad]] <math>\oc</math>). As a consequence <math>P</math> and <math>\oc P</math> are [[Sequent calculus#Equivalences|equivalent]]. |
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+ | A formula <math>P</math> is positive if and only if <math>P\orth</math> is [[Negative formula|negative]]. |
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== Positive connectives == |
== Positive connectives == |
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== Generalized structural rules == |
== Generalized structural rules == |
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− | Positive formulas admit generalized left structural rules corresponding to a structure of <math>\tens</math>-comonoid: <math>P\limp P\tens P</math> and <math>P\limp\one</math>. The following rule is derivable: |
+ | Positive formulas admit generalized left structural rules corresponding to a structure of [[Wikipedia:Comonoid|<math>\tens</math>-comonoid]]: <math>P\limp P\tens P</math> and <math>P\limp\one</math>. The following rule is derivable: |

<math> |
<math> |
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{{Proof| |
{{Proof| |
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<math> |
<math> |
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− | \AxRule{\begin{array}{c}\\P_1\vdash\oc{P_1}\end{array}} |
+ | \AxRule{P_1\vdash\oc{P_1}} |

\AxRule{P_n\vdash\oc{P_n}} |
\AxRule{P_n\vdash\oc{P_n}} |
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\AxRule{\oc\Gamma,P_1,\dots,P_n\vdash A,\wn\Delta} |
\AxRule{\oc\Gamma,P_1,\dots,P_n\vdash A,\wn\Delta} |
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\LabelRule{\oc L} |
\LabelRule{\oc L} |
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− | \UnaRule{\oc\Gamma,P_1,\dots,\oc{P_n}\vdash A,\wn\Delta} |
+ | \UnaRule{\oc\Gamma,P_1,\dots,P_{n-1},\oc{P_n}\vdash A,\wn\Delta} |

− | \UnaRule{\vdots} |
+ | \VdotsRule{}{\oc\Gamma,P_1,\oc{P_2},\dots,\oc{P_n}\vdash A,\wn\Delta} |

\LabelRule{\oc L} |
\LabelRule{\oc L} |
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\UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash A,\wn\Delta} |
\UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash A,\wn\Delta} |
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\UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash \oc{A},\wn\Delta} |
\UnaRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_n}\vdash \oc{A},\wn\Delta} |
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\LabelRule{\rulename{cut}} |
\LabelRule{\rulename{cut}} |
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− | \BinRule{\begin{array}{c}\oc\Gamma,\oc{P_1},\dots,P_n\vdash \oc{A},\wn\Delta\\\vdots\end{array}} |
+ | \BinRule{\oc\Gamma,\oc{P_1},\dots,\oc{P_{n-1}},P_n\vdash \oc{A},\wn\Delta} |

+ | \VdotsRule{}{\oc\Gamma,\oc{P_1},P_2,\dots,P_n\vdash \oc{A},\wn\Delta} |
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\LabelRule{\rulename{cut}} |
\LabelRule{\rulename{cut}} |
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\BinRule{\oc\Gamma,P_1,\dots,P_n\vdash \oc{A},\wn\Delta} |
\BinRule{\oc\Gamma,P_1,\dots,P_n\vdash \oc{A},\wn\Delta} |

## Latest revision as of 19:49, 28 October 2013

A *positive formula* is a formula *P* such that (thus a coalgebra for the comonad ). As a consequence *P* and are equivalent.

A formula *P* is positive if and only if is negative.

## [edit] Positive connectives

A connective *c* of arity *n* is *positive* if for any positive formulas *P*_{1},...,*P*_{n}, is positive.

**Proposition** (Positive connectives)

, , , , and are positive connectives.

*Proof.*

More generally, is positive for any formula *A*.

The notion of positive connective is related with but different from the notion of asynchronous connective.

## [edit] Generalized structural rules

Positive formulas admit generalized left structural rules corresponding to a structure of -comonoid: and . The following rule is derivable:

*Proof.*

Positive formulas are also acceptable in the left-hand side context of the promotion rule. The following rule is derivable:

*Proof.*