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    <h2>Tarski Formalization Project:  Results</h2>
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We used OTTER to find proofs of the theorems in Tarskian geometry in the first nine 
chapters of Szmielew's development in Part I of <em>Metamathematische Methoden in der Geometrie</em>.
 Those theorems 
include the four unsolved challenge problems from Quaife's 1992 book, and 
the verification of Hilbert's axioms.  Of the approximately 200 theorems, we proved
57% mechanically and without strategies (except for a good choice of settings), 
and by the use of the subformula strategy we increased that to 76%.  By using 
lemma adjunction and hint injection, which depend on starting with a book proof, 
we achieved a 100% success rate.

We have discussed the contents of the book, and the definition of "hard", "simple",
and "elementary" theorem (in terms of proof length), on <a href="index.php?include=AboutHardTheorems">this page</a>.
Please refer to that page for a discussion of the theorems mentioned by name in the table below.
 
Here is a table showing the hard theorems from the entire book,  with the 
number of steps in the proof(s) that we obtained.  A paired number like $(99,921)$ 
indicates  
a proof found mechanically using the subformula strategy, of length 99 and found in 921 seconds.
The last column gives the number of steps taken explicitly in the book proof. 
Not counting the lines with a question mark, we have 1900 Otter steps
corresponding to 465 book steps, giving a de Bruijn factor (the ratio of the two).
 
 
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  <tr>
     <th>Theorem</th>
     <th>Description</th>
     <th>Subformula Strategy</th>
     <th>Hints</th>
     <th>Book</th>
  </tr>
    <tr> <td>Satz 4.2  </td><td>  inner 5-segment theorem </td><td></td><td> 44  </td><td> 13 </td></tr>  
<tr> <td> Satz 5.1 </td><td> trichotomy </td><td></td><td>  127 </td><td>  70</td></tr> 
 <tr> <td>Satz 6.16b </td><td> transitivity of Col  </td><td></td><td> 46  </td><td>  1?</td></tr>
  <tr> <td>Satz 7.13 </td><td> refl. in pt. preserves congruence  </td><td> $(99, 921)$ </td><td> 72  </td><td> 28</td></tr>
<tr> <td> Satz 7.22a </td><td> Krippenlemma </td><td> $(108,3189)$</td><td> 96 </td><td> 27 </td></tr>
<tr> <td> Satz 7.25 </td><td> isosceles triangle has midpoint </td><td></td><td> 113 </td><td>34</td></tr>
<tr> <td> Satz 8.18 </td><td> Lotsatz (dropped perpendicular)  </td><td></td><td> 227 </td><td> 32</td></tr>
<tr> <td> Satz 8.21a </td><td> uniqueness of dropped perp   </td><td></td><td> 106</td><td> 5</td></tr>
<tr> <td>  Satz 8.22b </td><td> existence of midpoint   </td><td></td><td>201 </td><td> 42</td></tr>
<tr> <td>   Satz 8.24a </td><td>  midpoint   details      </td><td></td><td> 171</td><td> 42</td></tr>
<tr> <td>    Satz 8.24b </td><td>  midpoint details     </td><td></td><td> 163</td><td> 42</td></tr>
<tr> <td>  Satz 9.3 </td><td> opposite-side ray theorem   </td><td></td><td> 52 </td><td> 14</td></tr>
 <tr> <td> Satz 9.4b </td><td> $r=s$ case of 9.4       </td><td></td><td>44 </td><td>7</td></tr>
<tr> <td>   Satz 9.6 </td><td> inner Pasch implies outer Pasch  </td><td>$(98,186)$ </td><td>91</td><td> 27 </td></tr>
<tr> <td>    Satz 9.8 </td><td> opposite, same implies opposite  </td><td>$(63,1646)$ </td><td>48 </td><td>21</td></tr>
<tr> <td>    Satz 9.13 </td><td> transitivity of samesideline   </td><td></td><td> 71</td><td> 1?</td></tr>
<tr> <td>     Satz 10.2a </td><td> existence of reflection in line  </td><td></td><td> 50 </td><td> 23</td></tr>
 <tr> <td>    Satz 10.2b </td><td> uniqueness of reflection in line  </td><td></td><td> ??</td><td> 23</td></tr>
 <tr> <td>    Satz 10.10 </td><td>  refl. in line preserves congruence                    </td><td></td><td> 92 </td><td> 26</td></tr>
 <tr> <td>     Satz 10.12b </td><td>  right triangle theorem (Euclid 4)  </td><td></td><td> 100 </td><td>14</td></tr>
 <tr> <td>     Satz 10.16a </td><td> triangle construction  </td><td></td><td> 57 </td><td> 24?</td></tr>
 <tr> <td>     Satz 10.16b </td><td> triangle uniqueness   </td><td></td><td> 60</td><td>22</td></tr>
<tr> <td>      Satz 11.3a </td><td> angle congruence theorem  </td><td></td><td> 43</td><td> 18</td></tr>
 <tr> <td>     Satz 11.15a </td><td>  Hilbert's angle transport  </td><td></td><td> 77 </td><td> 1</td></tr>
</table>
 
 


 

 

 

 

 
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