<?xml version="1.0" encoding="UTF-8"?>
<QEDEQ 
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    xsi:noNamespaceSchemaLocation="http://www.qedeq.org/0_04_01/xml/qedeq.xsd">
  <HEADER email="mime@qedeq.org">
    <SPECIFICATION name="predlogic" ruleVersion="1.00.00">
      <LOCATIONS>
        <LOCATION value="http://www.qedeq.org/0_04_01/sample"/>
      </LOCATIONS>
    </SPECIFICATION>
    <TITLE>
      <LATEX language="en">
         Mathematical Example Module
      </LATEX>
      <LATEX language="de">
         Mathematisches Beispielmodul
      </LATEX>
    </TITLE>
    <ABSTRACT>
      <LATEX language="en">
         This should contain predicate logic formulas.
      </LATEX>
      <LATEX language="de">
         Hier sollen prädikatenlogische Formeln stehen.
      </LATEX>
    </ABSTRACT>
    <AUTHORS>
      <AUTHOR email="michael@meyling.com">
        <NAME>
          <LATEX language="de">
            Michael Meyling
          </LATEX>
        </NAME>
      </AUTHOR>
    </AUTHORS>
    <IMPORTS>
      <IMPORT label="proplogic">
        <SPECIFICATION name="proplogic" ruleVersion="1.00.00">
          <LOCATIONS>
            <LOCATION value="."/>
            <LOCATION value="http://www.qedeq.org/0_04_01/sample"/>
          </LOCATIONS>
        </SPECIFICATION>
      </IMPORT>
    </IMPORTS>
  </HEADER>
  <CHAPTER>
    <TITLE>
      <LATEX language="en">
        Predicate Logic
      </LATEX>
      <LATEX language="de">
        Prädikatenlogik
      </LATEX>
    </TITLE>
    <INTRODUCTION>
      <LATEX language="en">
        <![CDATA[
          Here are some predicate logic propositions.
        ]]>
      </LATEX>
      <LATEX language="de">
        <![CDATA[
          Hier sind ein paar prädikatenlogische Formeln.
        ]]>
      </LATEX>
    </INTRODUCTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">Simple Formulas</LATEX>
        <LATEX language="de">Einfache Formeln</LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            These are the first ones.
          ]]>
        </LATEX>
        <LATEX language="de">
          <![CDATA[
            Das sind die ersten.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <NODE id="first:theorem" level="formal">
          <THEOREM>
            <FORMULA>
              <AND>
                <IMPL>
                  <FORALL>
                    <VAR id="x" />
                    <IMPL>
                      <PREDVAR id="\phi">
                        <VAR id="x" />
                      </PREDVAR>
                      <PREDVAR id="\psi">
                        <VAR id="x" />
                      </PREDVAR>
                    </IMPL>
                  </FORALL>
                  <IMPL>
                    <FORALL>
                      <VAR id="x" />
                      <PREDVAR id="\phi">
                        <VAR id="x" />
                      </PREDVAR>
                    </FORALL>
                    <FORALL>
                      <VAR id="x" />
                      <PREDVAR id="\psi">
                        <VAR id="x" />
                      </PREDVAR>
                    </FORALL>
                  </IMPL>
                </IMPL>
                

                <IMPL>
                  <FORALL>
                    <VAR id="x" />
                    <IMPL>
                      <PREDVAR id="\phi">
                        <VAR id="x" />
                      </PREDVAR>
                      <PREDVAR id="\psi">
                        <VAR id="x" />
                      </PREDVAR>
                    </IMPL>
                  </FORALL>
                  <IMPL>
                    <EXISTS>
                      <VAR id="x" />
                      <PREDVAR id="\phi">
                        <VAR id="x" />
                      </PREDVAR>
                    </EXISTS>
                    <EXISTS>
                      <VAR id="x" />
                      <PREDVAR id="\psi">
                        <VAR id="x" />
                      </PREDVAR>
                    </EXISTS>
                  </IMPL>
                </IMPL>

              </AND>
            </FORMULA>
          </THEOREM>
        </NODE>
        
        <NODE id="definition:identity">
          <NAME>
            <LATEX language="en">identity definition</LATEX>
            <LATEX language="de">Definition der Identität</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Identity\index{identity}\index{definition!of identity}</LATEX>
            <LATEX language="de">Identität\index{Identität}\index{Definition!der Identität}</LATEX>
          </TITLE>
          <PRECEDING>
            <LATEX language="en">
              <![CDATA[
                We define a predicate constant of arity two that shall stand for the identity of subjects.
              ]]>
            </LATEX>
            <LATEX language="de">
              <![CDATA[
                Es wird eine zweistellige Prädikatskonstante festgelegt, welche in der Interpretation die Identität von Subjekten ausdrücken soll.
              ]]>
            </LATEX>
          </PRECEDING>
          <DEFINITION_PREDICATE arguments="2" name="equal">
            <LATEXPATTERN>#1 \ =  \ #2</LATEXPATTERN>
            <VARLIST>
              <VAR id="x"/>
              <VAR id="y"/>
            </VARLIST>
          </DEFINITION_PREDICATE>
        </NODE>
        
      </SUBSECTIONS>
    </SECTION>
  </CHAPTER>
</QEDEQ>
