Sindbad~EG File Manager

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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>
   <h3>Definition, congruence, and comparison of angles</h3>
 
    <p>Chapter 11 develops the theory of angles in Tarski's points-only theory.
       The fundamental theorem is Satz 11.4, which amounts to this:  suppose we know 
       that angles  abc and  ABC  are congruent because ab=AB,  ac=AC, and bc=BC.
       Now suppose  we move b,c,B, and C to other points on the same sides of the two angles,
       in such way that ab=AB and bc=bC still hold.  Then ac=AC must also still hold. 
       See Diagram 53 on p. 95 to
             understand this fundamental theorem about angle congruence. 
             </p>
             <p> Of course
       the formal expression of this involves more variables.  More importantly for Otter, there 
       are many possible cases for the orders of the points on the rays forming the angles.  These
       arguments by cases caused us a lot of trouble in finding Otter proofs.   The proofs exhibited
       here were found after many runs, in which the cases and subcases were proved in various combinations,
       and then the steps of the proofs taken as hints in attempts to eliminate the cases in favor
       of a tautology.  </p>
       
       <p> The book (SST) proves Satz 11.3 and Satz 11.4 together, in three steps, rather than
       separately as the theorems are stated. Our input files follow SST.  </p>
       
       <p>Hilbert's angle axioms follow from the theorems proved here.  Satz 11.15 is said by Szmielew to 
       be Hilbert's axiom III.4, but actually it is not literally, as it provides for constructing an angle 
       on the same side of a line L as a given point p, instead of on the opposite side.  That is a trivial 
       difference.  Satz 11.49 is the SAS congruence theorem, Hilbert's III.5. Nothing after Satz 11.4 
       is needed to prove either of these two Hilbert axioms, and strangely enough, these two axioms are 
       themselves not used in the rest of Chapter 11 or in Chapter 12. 
    </p>
    <p> The book says Satz 11.15 is an "easy consequence" of 10.15, and gives no proof at all; but 
    to prove it, we had to first discover and prove Satz 9.16 (which is not in the book and our proof 
    has 53 steps), and then Satz 11.15 has a 53 step proof.
    </p>
    
<?php
$descriptions["Satz11.3a"] = "left-to-right direction of Satz 11.3, i.e. (2) implies (3) p. 95";
$descriptions["Satz11.3d"] = "right-to-left direction of Satz 11.3";
$descriptions["Satz11.4b"] = "";
$descriptions["Satz11.4"] = "right-to-left direction of Satz 11.4, i.e. (4) implies (2) p. 95";
$descriptions["Satz11.4d"] = "left-to-right of Satz 11.4";
$descriptions["Satz11.6"] = "";
$descriptions["Satz11.7"] = "";
$descriptions["Satz11.8"] = "";
$descriptions["Satz11.9"] = "";
$descriptions["Satz11.15a"] = "Angle transport theorem (Hilbert's axiom III.4), existence part";
$descriptions["Satz11.15b"] = "Angle transport theorem (Hilbert's axiom III.4), uniqueness part";
$descriptions["Satz11.49a"] = "First half of SAS congruence theorem  (Hilbert's axiom III.5) <br>" .
              "(the opposite sides are congruent)";
$descriptions["Satz11.49b"] = "Second half of SAS congruence theorem (Hilbert's axiom III.5) <br>" .
               "(if $a \\neq c$ then the other angles are pairwise congruent)";
$Chapter = "11";
include("DisplayResults.php");

?>