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    <h2>Tarski Formalization Project Archives</h2>
         <p>The posted input files are now all mechanically generated from a master list of theorems.
    For more information about our methodology see the <a href="http://www.michaelbeeson.com/research/FormalTarski/index.php"> top page of this project</a>.
  </p>
     <p> Tarski works in n dimensions, and defines the concepts "plane" and "coplanar".  The plane 
    determined by line L and point p (not on L) is the set of points either on L, or on the same side of L as p,
    or on the opposite side of L from p.  
    </p>
    <p> To fix an oversight in Chapter 12 of SST, we needed to prove a version of the plane separation theorem
    that does not occur in SST; that is Lemma 9.13f described below.  We proved that lemma assuming the 
    dimension axiom for n=2.  One can formulate a similar lemma that should be valid without any dimension axiom,
    but it looks difficult to prove (having very many cases to deal with, each requiring about six constructed points).
    Since our aim was to work in two-dimensional geometry, this fix is sufficient.  We called this a "lemma" because it doesn't occur in SST, but with a proof length of 81 steps, it is really a difficult theorem.
    </p>
    <p>  SST does not use the dimension axiom in the first 12 chapters; we use it only in 
    Lemmas 9.13f and 9.13g,  which in turn are used only in Lemma 12.11A-3 and hence in Satz 12.11, the 
    verification of Hilbert's parallel axiom.  
    </p>
    <p> There are several other lemmas listed below with proof lengths of 7 to 22 steps.  These are lemmas
    that we needed to make other subsequent theorems from SST go through;  they became lemmas only when 
    we failed repeatedly to find certain Otter proofs without them.   Lemma 9.13b is particularly interesting,
    since it is one of the few cases where Otter found a really creative, unexpected proof. 
     </p>
     <p>The last two lemmas express the extensionality of <em>samesideline</em>, i.e. it does not matter which
     two points are used to specify the line. 
     </p>
    <table width="800" border="1">
      <colgroup>
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      <tr>
        <th  >Input File</th>
        <th  >Proof</th>
        <th  >Commentary</th>
    </tr>
        <tr>
        <td><a href="InputFiles/Lemma9.13a.in">Lemma 9.13a</a></td>
        <td><a href="Proofs/Lemma9.13a.prf">Lemma9.13a.prf  </a>
        <br> 16 steps  </td>
        <td>  if <em> p</em> and <em>q</em> are on the same side of   <em>Line</em>(<em>a,b</em>)</em>, then
        <em>p</em> and <em>q</em> are on the same side of <em>Line</em>(<em>b,a</em>)</em>
        </td>
      </tr>
      </tr>
        <tr>
        <td><a href="InputFiles/Lemma9.13b.in">Lemma 9.13b</a></td>
        <td><a href="Proofs/Lemma9.13b.prf">Lemma9.13b.prf  </a>
        <br> 10 steps  </td>
        <td>  if <em>u</em> and <em> w </em> both lie on <em>Line(a,b)</em>  then the midpoint of segment <em> uw</em> also lies on that line.  (Needed in Satz 10.10.)   </td>
      </tr>
        <tr>
        <td><a href="InputFiles/Lemma9.13c.in">Lemma 9.13c</a></td>
        <td><a href="Proofs/Lemma9.13c.prf">Lemma9.13c.prf </a>
        <br>  7 steps</td>
        <td>  if  <em>T(e,d,d1)</em>  and <em>p,q</em> are on the same side of <em>Line(e,d1)</em>, then  <em>p,q</em> are on the same side of <em>Line(e,d)</em>. SST doesn't need this because lines are sets.       </td>
      </tr>
        <tr>
        <td><a href="InputFiles/Lemma9.13d.in">Lemma 9.13d</a></td>
        <td><a href="Proofs/Lemma9.13d.prf">Lemma9.13d.prf </a>
        <br>  11 steps</td>
        <td>  if <em> p,q </em> are on the same side of <em> Line(a,b)</em> then <em> q,p </em> are on the same side of <em> Line(a,b)</em>.  This is "obvious",
              but the proof of Lemma 12.11A-2  would not go through without this lemma.
        </td>
      </tr>
         <tr>
        <td><a href="InputFiles/Lemma9.13e.in">Lemma 9.13e</a></td>
        <td><a href="Proofs/Lemma9.13e.prf">Lemma9.13e.prf </a>
        <br>  3 steps</td>
        <td> A trivial variant of Satz 9.8, but we seem to need it.</td>
      </tr>
       <tr>
        <td><a href="InputFiles/Lemma9.13g.in">Lemma 9.13g</a></td>
        <td><a href="Proofs/Lemma9.13g.prf">Lemma9.13g.prf</a>
        <br>26 steps </td>
        <td> Assuming the dimension axioms for dimension 2, any two perpendiculars
             to line L at point p are collinear. (Used to prove 9.13f.)
        </td>
      </tr>
        <tr>
        <td><a href="InputFiles/Lemma9.13f.in">Lemma 9.13f</a></td>
        <td><a href="Proofs/Lemma9.13f-case1.prf">not yet proved in one file. </a>
        
        <br> 65 steps  </td>
        <td> If <em> r</em> and <em>s</em> are not on line <em>Line(p,q)</em>, and not on opposite sides of  <em>L</em>,
         then they are 
             on the same side of <em>L</em>.  This relies on the dimension 2 axiom via 9.13g.
             <br>We could not get Otter to combine the two cases, so we present the two proofs separately. 
        </td>  
      </tr>
        <tr>
        <td><a href="InputFiles/Lemma9.13f-case1.in">Lemma 9.13f (case 1)</a></td>
        <td><a href="Proofs/Lemma9.13f-case1.prf">Lemma9.13f-case1.prf </a>
        <br> 65 steps  </td>
        <td> Let <em>e</em> be the foot of the perpendicular to <em>Line(p,q)</em> from <em>r</em>.  Case 1 is when <em>e=p</em>.
        </td>  
      </tr>
      <tr>
        <td><a href="InputFiles/Lemma9.13f-case2.in">Lemma 9.13f (case 2)</a></td>
        <td><a href="Proofs/Lemma9.13f-case2.prf">Lemma9.13f-case2.prf </a>
        <br> 65 steps  </td>
        <td> Case 2 is when <em>e</em> and <em>p</em> are not equal.
        </td>  
      </tr>
      
       <tr>
        <td><a href="InputFiles/ExtSameSide1.in">ExtSameSide1</a></td>
        <td><a href="Proofs/ExtSameSide1.prf">ExtCol.prf</a><br>19 steps</td>
        <td>If Col(a,b,c) and b,c are different from a, and p,q are on the same side of Line(a,b), then p,q are on the same side of Line(a,c).  </td>
      </tr> 
       <tr>
        <td><a href="InputFiles/ExtSameSide2.in">ExtSameSide2</a></td>
        <td><a href="Proofs/ExtSameSide1.prf">ExtCol.prf</a><br>16 steps</td>
        <td>If p,q are on the same side of Line(a,b), then p,q are on the same side of Line(b,a).  </td>
      </tr> 
    </table>
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