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    <allpages gapfrom="Relational semantics" />
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      <page pageid="5" ns="0" title="Recommendations">
        <revisions>
          <rev xml:space="preserve">== User account ==

* Use &quot;FirstName LastName&quot; as Username.
* Use a valid e-mail address.

== Conventions ==

* Use English only, even for comments, discussions, ...
* Use &lt;tt&gt;&lt;nowiki&gt;== Header ==&lt;/nowiki&gt;&lt;/tt&gt; for higher-level headers, ''do not'' use only one equals sign on a side.

== Notations ==

For uniformity, use the [[notations|common notations]] when they are already defined, and add your notations to the [[notations|notations page]] when you introduce new ones.

== Definitions and theorems ==

For theorem-style environments, use the appropriate predefined [http://www.mediawiki.org/wiki/Templates templates]: [[Template:Definition|Definition]], [[Template:Theorem|Theorem]], [[Template:Proposition|Proposition]], [[Template:Lemma|Lemma]], [[Template:Corollary|Corollary]] (with an optional parameter &quot;title&quot;).
Additional templates are [[Template:Proof|Proof]] for proofs and [[Template:Remark|Remark]] for remarks.

{| border=&quot;1&quot; cellpadding=&quot;10&quot; cellspacing=&quot;1&quot;
|-
|&lt;pre&gt;
{{Definition|title=Concept|A new concept.}}

{{Theorem|This is a nice concept.}}

{{Proof|Left to the reader.}}

{{Remark|Really nice, isn't it?}}
&lt;/pre&gt;
|
{{Definition|title=Concept|A new concept.}}

{{Theorem|This is a nice concept.}}

{{Proof|Left to the reader.}}

{{Remark|Really nice, isn't it?}}
|-
|}

== Mathematical typesetting ==

=== Formulas ===

* use the macros defined in the [[LLWiki LaTeX Style]]

=== Proofs ===

The syntax for proofs is based on the [http://www.math.ucsd.edu/~sbuss/ResearchWeb/bussproofs/ bussproofs.sty] package, but available only through a sugar-coated specific to llwiki. Here is an example:
&lt;pre&gt;
\AxRule{}
\VdotsRule{s}{{}\vdash\Gamma,A\imp B}
\AxRule{}
\VdotsRule{t}{{}\vdash\Gamma, A}
\BinRule{{}\vdash \Gamma, B}
\LabelRule{\rulename{modus ponens}}
\DisplayProof
\quad
\longrightarrow_\beta
\quad
\AxRule{}
\VdotsRule{s[t/x]}{{}\vdash\Gamma, B}
\DisplayProof
&lt;/pre&gt;

&lt;math&gt;
\AxRule{}
\VdotsRule{s}{{}\vdash\Gamma,A\imp B}
\AxRule{}
\VdotsRule{t}{{}\vdash\Gamma, A}
\BinRule{{}\vdash \Gamma, B}
\LabelRule{\rulename{modus ponens}}
\DisplayProof
\quad
\longrightarrow_\beta
\quad
\AxRule{}
\VdotsRule{s[t/x]}{{}\vdash\Gamma, B}
\DisplayProof
&lt;/math&gt;

A list of all available macros for proofs is available on the page for the [[LLWiki_LaTeX_Style#Proof_trees|LLWiki LaTeX package]].

=== Set of rules ===

For a group of rules, separate rules on a line by &lt;nowiki&gt;\qquad&lt;/nowiki&gt; and separate lines by &lt;nowiki&gt;&lt;br /&gt;&lt;/nowiki&gt;.

&lt;pre&gt;
&lt;math&gt;
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_1}
\BinRule{\textit{Concl}}
\DisplayProof
\qquad
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_1}
\BinRule{\textit{Concl}}
\DisplayProof
&lt;/math&gt;

&lt;br /&gt;

&lt;math&gt;
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_3}
\BinRule{\textit{Concl}}
\DisplayProof
\qquad
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_4}
\BinRule{\textit{Concl}}
\DisplayProof
&lt;/math&gt;
&lt;/pre&gt;

&lt;math&gt;
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_1}
\BinRule{\textit{Concl}}
\DisplayProof
\qquad
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_2}
\BinRule{\textit{Concl}}
\DisplayProof
&lt;/math&gt;

&lt;br /&gt;

&lt;math&gt;
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_3}
\BinRule{\textit{Concl}}
\DisplayProof
\qquad
\AxRule{\textit{Hyp}_1}
\AxRule{\textit{Hyp}_2}
\LabelRule{\rulename{Rule}_4}
\BinRule{\textit{Concl}}
\DisplayProof
&lt;/math&gt;</rev>
        </revisions>
      </page>
      <page pageid="89" ns="0" title="Regular formula">
        <revisions>
          <rev xml:space="preserve">A ''regular formula'' is a formula &lt;math&gt;R&lt;/math&gt; such that &lt;math&gt;R\linequiv\wn\oc R&lt;/math&gt;.

A formula &lt;math&gt;L&lt;/math&gt; is ''co-regular'' if its dual &lt;math&gt;L\orth&lt;/math&gt; is regular, that is if &lt;math&gt;L\linequiv\oc\wn L&lt;/math&gt;.

== Alternative characterization ==

&lt;math&gt;R&lt;/math&gt; is regular if and only if it is [[Sequent calculus#Equivalences|equivalent]] to a formula of the shape &lt;math&gt;\wn P&lt;/math&gt; for some [[positive formula]] &lt;math&gt;P&lt;/math&gt;.

{{Proof|
If &lt;math&gt;R&lt;/math&gt; is regular then &lt;math&gt;R\linequiv\wn\oc R&lt;/math&gt; with &lt;math&gt;\oc R&lt;/math&gt; positive. If &lt;math&gt;R\linequiv\wn P&lt;/math&gt; with &lt;math&gt;P&lt;/math&gt; positive then &lt;math&gt;R&lt;/math&gt; is regular since &lt;math&gt;P\linequiv\oc P&lt;/math&gt;.
}}

== Regular connectives ==

A connective &lt;math&gt;c&lt;/math&gt; of arity &lt;math&gt;n&lt;/math&gt; is ''regular'' if for any regular formulas &lt;math&gt;R_1&lt;/math&gt;,...,&lt;math&gt;R_n&lt;/math&gt;, &lt;math&gt;c(R_1,\dots,R_n)&lt;/math&gt; is regular.

{{Proposition|title=Regular connectives|
&lt;math&gt;\parr&lt;/math&gt;, &lt;math&gt;\bot&lt;/math&gt; and &lt;math&gt;\wn\oc&lt;/math&gt; define regular connectives.
}}

{{Proof|
If &lt;math&gt;R&lt;/math&gt; and &lt;math&gt;S&lt;/math&gt; are regular, &lt;math&gt;R\parr S \linequiv \wn\oc R \parr \wn\oc S \linequiv \wn{(\oc R\plus\oc S)}&lt;/math&gt; thus it is regular since &lt;math&gt;\oc R\plus\oc S&lt;/math&gt; is positive.

&lt;math&gt;\bot\linequiv\wn\zero&lt;/math&gt; thus it is regular since &lt;math&gt;\zero&lt;/math&gt; is positive.

If &lt;math&gt;R&lt;/math&gt; is regular then &lt;math&gt;\wn\oc R&lt;/math&gt; is regular, since &lt;math&gt;\wn\oc\wn\oc R\linequiv \wn\oc R&lt;/math&gt;.
}}

More generally, &lt;math&gt;\wn\oc A&lt;/math&gt; is regular for any formula &lt;math&gt;A&lt;/math&gt;.</rev>
        </revisions>
      </page>
    </pages>
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