<?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_07/xml/qedeq.xsd">
  <HEADER email="mime@qedeq.org">
    <SPECIFICATION name="qedeq_sample4" ruleVersion="0.01.00">
      <LOCATIONS>
        <LOCATION value="http://www.qedeq.org/0_04_07/sample" />
      </LOCATIONS>
    </SPECIFICATION>
    <TITLE>
      <LATEX language="en">
        <![CDATA[
          Search for simple formal proofs.
        ]]>
      </LATEX>
    </TITLE>
    <ABSTRACT>
      <LATEX language="en">
        <![CDATA[
          Here we have some propositional calculus axioms and some propositions. Initially these propositions have no formal proofs. By using our proof finder we get such formal proofs.
        ]]>
      </LATEX>
    </ABSTRACT>
    <AUTHORS>
      <AUTHOR email="michael@meyling.com">
        <NAME>
          <LATEX language="en">
            <![CDATA[
              Michael Meyling
            ]]>
          </LATEX>
        </NAME>
      </AUTHOR>
    </AUTHORS>
  </HEADER>
  <CHAPTER>
    <TITLE>
      <LATEX language="en">
        <![CDATA[
          First
        ]]>
      </LATEX>
    </TITLE>
    <INTRODUCTION>
      <LATEX language="en">
        <![CDATA[
          After presenting some axioms we present some propositions.
        ]]>
      </LATEX>
    </INTRODUCTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">
          <![CDATA[
            Axioms
          ]]>
        </LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            Here we state our axioms.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <NODE id="axiom:disjunction_idempotence">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of disjunction idempotence
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Disjunction Idempotence
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <OR>
                  <PREDVAR id="A"/>
                  <PREDVAR id="A"/>
                </OR>
                <PREDVAR id="A"/>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:disjunction_weakening">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of weakening
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Axiom of Weakening
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <PREDVAR id="A"/>
                <OR>
                  <PREDVAR id="A"/>
                  <PREDVAR id="B"/>
                </OR>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:disjunction_commutative">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                commutativity of the disjunction
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Commutativity of the Disjunction
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <OR>
                  <PREDVAR id="A"/>
                  <PREDVAR id="B"/>
                </OR>
                <OR>
                  <PREDVAR id="B"/>
                  <PREDVAR id="A"/>
                </OR>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="axiom:disjunction_addition">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                axiom of disjunctive addition
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Disjunctive Addition
              ]]>
            </LATEX>
          </TITLE>
          <AXIOM>
            <FORMULA>
              <IMPL>
                <IMPL>
                  <PREDVAR id="A"/>
                  <PREDVAR id="B"/>
                </IMPL>
                <IMPL>
                  <IMPL>
                    <PREDVAR id="C"/>
                    <PREDVAR id="A"/>
                  </IMPL>
                  <IMPL>
                    <PREDVAR id="C"/>
                    <PREDVAR id="B"/>
                  </IMPL>
                </IMPL>
              </IMPL>
            </FORMULA>
          </AXIOM>
        </NODE>
        <NODE id="rule:addProvenFormula">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                add true formula
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Add true formula
              ]]>
            </LATEX>
          </TITLE>
          <RULE name="Add" version="0.01.00">
            <DESCRIPTION>
              <LATEX language="en">
                <![CDATA[
                  Addition of an axiom, definition or already proven formula. We have to reference to the location of a true formula.
                ]]>
              </LATEX>
            </DESCRIPTION>
          </RULE>
        </NODE>
        <NODE id="rule:modusPonens">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                modus ponens
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Modus Ponens\index{Modus Ponens}
              ]]>
            </LATEX>
            <LATEX language="de">
              <![CDATA[
                Abtrennung, Modus Ponens\index{Modus Ponens}\index{Abtrennungsregel}
              ]]>
            </LATEX>
          </TITLE>
          <RULE name="MP" version="0.01.00">
            <DESCRIPTION>
              <LATEX language="en">
                <![CDATA[
                  If both formulas $\alpha$ and $\alpha \rightarrow \beta$ are true, then we can conclude that $\beta$ is true as well.
                ]]>
              </LATEX>
            </DESCRIPTION>
          </RULE>
        </NODE>
        <NODE id="rule:substPred">
          <NAME>
            <LATEX language="en">
              <![CDATA[
                replace predicate variable
              ]]>
            </LATEX>
          </NAME>
          <TITLE>
            <LATEX language="en">
              <![CDATA[
                Replace Predicate Variable
              ]]>
            </LATEX>
          </TITLE>
          <RULE name="SubstPred" version="0.01.00">
            <DESCRIPTION>
              <LATEX language="en">
                <![CDATA[
                  Let $\alpha$ be a true formula that contains a predicate variable $p$ of arity $n$, let $x_1$, \ldots, $x_n$ be pairwise different subject variables and let $\beta(x_1, \ldots, x_n)$ be a formula where $x_1$, \ldots, $x_n$ are not bound. The formula $\beta(x_1, \ldots, x_n)$ must not contain all $x_1$, \ldots, $x_n$ as free subject variables. Furthermore it can also have other subject variables either free or bound.
                  
                  If the following conditions are fulfilled, then a replacement of all occurrences of $p(t_1, \ldots, t_n)$ each with appropriate terms $t_1$, \ldots, $t_n$ in $\alpha$ by $\beta(t_1, \ldots, t_n)$ results in another true formula.

                  \begin{itemize}

                  \item
                  the free variables of $\beta(x_1, \ldots, x_n)$ without $x_1$, \ldots, $x_n$ do not occur as bound variables in $\alpha$ 
                  
                  \item 
                  each occurrence of $p(t_1, \ldots, t_n)$ in $\alpha$ contains no bound variable of $\beta(x_1, \ldots, x_n)$ 

                  \item
                  the result of the substitution is a well-formed formula

                  \end{itemize}
                ]]>
              </LATEX>
            </DESCRIPTION>
          </RULE>
        </NODE>
      </SUBSECTIONS>
    </SECTION>
    <SECTION>
      <TITLE>
        <LATEX language="en">
          <![CDATA[
            One
          ]]>
        </LATEX>
      </TITLE>
      <INTRODUCTION>
        <LATEX language="en">
          <![CDATA[
            Here we start.
          ]]>
        </LATEX>
      </INTRODUCTION>
      <SUBSECTIONS>
        <NODE id="proposition:one" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <PREDVAR id="A"/>
                <PREDVAR id="A"/>
              </IMPL>
            </FORMULA>
          </THEOREM>
        </NODE>
        <NODE id="proposition:two" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <PREDVAR id="A"/>
                <OR>
                  <PREDVAR id="B"/>
                  <PREDVAR id="A"/>
                </OR>
              </IMPL>
            </FORMULA>
          </THEOREM>
        </NODE>
        <NODE id="proposition:three" level="formal">
          <THEOREM>
            <FORMULA>
              <IMPL>
                <OR>
                  <OR>
                    <PREDVAR id="A"/>
                    <PREDVAR id="B"/>
                  </OR>
                  <OR>
                    <PREDVAR id="C"/>
                    <PREDVAR id="D"/>
                  </OR>
                </OR>
                <OR>
                  <OR>
                    <PREDVAR id="C"/>
                    <PREDVAR id="D"/>
                  </OR>
                  <OR>
                    <PREDVAR id="A"/>
                    <PREDVAR id="B"/>
                  </OR>
                </OR>
              </IMPL>
            </FORMULA>
            <FORMAL_PROOF>
              <LINES>
                <L label="1">
                  <FORMULA>
                    <IMPL>
                      <OR>
                        <PREDVAR id="A"/>
                        <PREDVAR id="B"/>
                      </OR>
                      <OR>
                        <PREDVAR id="B"/>
                        <PREDVAR id="A"/>
                      </OR>
                    </IMPL>
                  </FORMULA>
                  <ADD ref="axiom:disjunction_commutative"/>
                </L>
                <L label="2">
                  <FORMULA>
                    <IMPL>
                      <OR>
                        <PREDVAR id="A"/>
                        <OR>
                          <PREDVAR id="C"/>
                          <PREDVAR id="D"/>
                        </OR>
                      </OR>
                      <OR>
                        <OR>
                          <PREDVAR id="C"/>
                          <PREDVAR id="D"/>
                        </OR>
                        <PREDVAR id="A"/>
                      </OR>
                    </IMPL>
                  </FORMULA>
                  <SUBST_PREDVAR ref="1">
                    <PREDVAR id="B"/>
                    <FORMULA>
                      <OR>
                        <PREDVAR id="C"/>
                        <PREDVAR id="D"/>
                      </OR>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
                <L label="3">
                  <FORMULA>
                    <IMPL>
                      <OR>
                        <OR>
                          <PREDVAR id="A"/>
                          <PREDVAR id="B"/>
                        </OR>
                        <OR>
                          <PREDVAR id="C"/>
                          <PREDVAR id="D"/>
                        </OR>
                      </OR>
                      <OR>
                        <OR>
                          <PREDVAR id="C"/>
                          <PREDVAR id="D"/>
                        </OR>
                        <OR>
                          <PREDVAR id="A"/>
                          <PREDVAR id="B"/>
                        </OR>
                      </OR>
                    </IMPL>
                  </FORMULA>
                  <SUBST_PREDVAR ref="2">
                    <PREDVAR id="A"/>
                    <FORMULA>
                      <OR>
                        <PREDVAR id="A"/>
                        <PREDVAR id="B"/>
                      </OR>
                    </FORMULA>
                  </SUBST_PREDVAR>
                </L>
              </LINES>
            </FORMAL_PROOF>
          </THEOREM>
        </NODE>
      </SUBSECTIONS>
    </SECTION>
  </CHAPTER>
</QEDEQ>
