<?xml version="1.0" encoding="UTF-8"?>
<QEDEQ 
    xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
    xsi:noNamespaceSchemaLocation="http://www.qedeq.org/0_04_01/xml/qedeq.xsd">
  <HEADER email="mime@qedeq.org">
    <SPECIFICATION name="MCEC023" ruleVersion="1.00.00">
      <LOCATIONS>
        <LOCATION value="file:///."/>
      </LOCATIONS>
    </SPECIFICATION>
    <TITLE>
      <LATEX language="en">
         Identity Definition
      </LATEX>
    </TITLE>
    <ABSTRACT>
      <LATEX language="en">
         We define the identity constant and have some atoms and a proposition.
      </LATEX>
    </ABSTRACT>
    <AUTHORS>
      <AUTHOR email="michael@meyling.com">
        <NAME>
          <LATEX language="de">
            Michael Meyling
          </LATEX>
        </NAME>
      </AUTHOR>
    </AUTHORS>
  </HEADER>
  <CHAPTER>
    <TITLE>
      <LATEX language="en">
        The Only One
      </LATEX>
    </TITLE>
    <INTRODUCTION>
      <LATEX language="en">
        <![CDATA[
          This module is for testing purposes only.
        ]]>
      </LATEX>
    </INTRODUCTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">Identity</LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            We have a definition some axioms and a proposition.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <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>
        
        <NODE id="definition:notEqual">
          <NAME>
            <LATEX language="en">not identical definition</LATEX>
            <LATEX language="de">Definition der Verschiedenheit</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Not Identical\index{identical!is not}</LATEX>
            <LATEX language="de">Verschiedenheit\index{Verschiedenheit}\index{Definition!der Verschiedenheit}</LATEX>
          </TITLE>
          <PRECEDING>
            <LATEX language="en">
              <![CDATA[
                For convenience we also define the negation of the identity a predicate constant.
              ]]>
            </LATEX>
            <LATEX language="de">
              <![CDATA[
                Aus Bequemlichkeit definieren wir auch die Negation der Identitätskonstante.
              ]]>
            </LATEX>
          </PRECEDING>
          <DEFINITION_PREDICATE arguments="2" name="notEqual">
            <LATEXPATTERN>#1 \ \neq \ #2</LATEXPATTERN>
            <VARLIST>
              <VAR id="x"/>
              <VAR id="y"/>
            </VARLIST>
            <FORMULA>
              <NOT>
                <PREDCON ref="equal">
                  <VAR id="x"/>
                  <VAR id="y"/>
                </PREDCON>
               </NOT>
            </FORMULA>
          </DEFINITION_PREDICATE>
        </NODE>
        
        <NODE id="axiom:identityIsReflexive">
          <NAME>
            <LATEX language="en">reflexivity of identity</LATEX>
            <LATEX language="de">Reflexivität der Identität</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Reflexivity of Identity\index{reflexivity!of identity}\index{identy!reflexivity of}</LATEX>
            <LATEX language="de">Reflexivität der Identität\index{Reflexivität!der Identität}\index{Identität!Reflexivität der}</LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <PREDCON ref="equal">
                <VAR id="x" />
                <VAR id="x" />
              </PREDCON>
            </FORMULA>
          </AXIOM>
        </NODE>

        <NODE id="axiom:leibnizReplacement">
          <NAME>
            <LATEX language="en">Leibniz' replacement</LATEX>
            <LATEX language="de">Leibnizsche Ersetzbarkeit</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Leibniz' replacement\index{Leibniz' replacement}</LATEX>
            <LATEX language="de">Leibnizsche Ersetzbarkeit\index{Leibnizsche Ersetzbarkeit}</LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <PREDCON ref="equal">
                  <VAR id="x" />
                  <VAR id="y" />
                </PREDCON>
                <IMPL>
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                  <PREDVAR id="\phi">
                    <VAR id="y" />
                  </PREDVAR>
                </IMPL>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>

        <NODE id="axiom:symmetryOfIdentity">
          <NAME>
            <LATEX language="en">symmetry of identity</LATEX>
            <LATEX language="de">Symmetrie der Identität</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Symmetrie of identity\index{identity!symmetry of}</LATEX>
            <LATEX language="de">Symmetrie der Identität\index{Identität!Symmetrie der}</LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <PREDCON ref="equal">
                  <VAR id="x" />
                  <VAR id="y" />
                </PREDCON>
                <PREDCON ref="equal">
                  <VAR id="y" />
                  <VAR id="x" />
                </PREDCON>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>

        <NODE id="axiom:transitivityOfIdentity">
          <NAME>
            <LATEX language="en">transetivity of identity</LATEX>
            <LATEX language="de">Transitivität der Identität</LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">Transitivity of identity\index{identity!transetivity of}</LATEX>
            <LATEX language="de">Transitivität der Identität\index{Identität!Transitivität der}</LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <AND>
                  <PREDCON ref="equal">
                    <VAR id="x" />
                    <VAR id="y" />
                  </PREDCON>
                  <PREDCON ref="equal">
                    <VAR id="y" />
                    <VAR id="z" />
                  </PREDCON>
                </AND>
                <PREDCON ref="equal">
                  <VAR id="x" />
                  <VAR id="z" />
                </PREDCON>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        
        <NODE id="theorem:leibnizEquivalence" level="formal">
          <PRECEDING>
            <LATEX language="en">
              <![CDATA[
                We can reverse the second implication in the Leibniz replacement.
              ]]>
            </LATEX>
            <LATEX language="de">
              <![CDATA[
                Bei der Leibnizschen Ersetzbarkeit können wir die zweite Implikation umkehren.
              ]]>
            </LATEX>
          </PRECEDING>
          <THEOREM>
            <FORMULA>
              <IMPL>
                <PREDCON ref="equal">
                  <VAR id="x" />
                  <VAR id="y" />
                </PREDCON>
                <EQUI>
                  <PREDVAR id="\phi">
                    <VAR id="x" />
                  </PREDVAR>
                  <PREDVAR id="\phi">
                    <VAR id="y" />
                  </PREDVAR>
                </EQUI>
              </IMPL>
            </FORMULA>
          </THEOREM>
       </NODE>
      </SUBSECTIONS>
    </SECTION>
  </CHAPTER>
</QEDEQ>
