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    <h2>Tarski Formalization Project: how TestChapter uses previous results</h2>
  
<p>  In repeatedly testing our code for the master list and the mechanical generation of 
input files, we needed a way to specify the way the input files are to be constructed
on a per-file basis.   To satisfy this need, the file <code>FinalResults.php</code> defines
 an array <code>FinalResults</code>.  This array is used by <code>TestChapter</code>
to generate the input files one chapter at a time.
</p>
<p> <code>FinalResults</code>  is an "associative array", meaning it is indexed not by 
integers but by the names of theorems.   Here is a part of its definition:

<code><pre>
    'Satz9.2' =>  10,    
	'Satz9.3a' =>  array(25, "subformula", 46),  // array(25, "hints"),  // 27 steps
	'Satz9.3' =>   array(50, "hints"),   // 52 steps 
	'SideReflect' => array(26, "subformula", 49), //  array(26, "hints"),  //52 steps  
	'Satz9.4a' => array(40, "hints"),  
	'Satz9.4a2' => array(35, "hints"),  // 35 steps
	'Satz9.4b' => array(42,"hints"),  // 48 steps
	'Satz9.4' =>  array(9, "max_weight = 9", "max_seconds = 450"),   // 350 seconds
	'Satz9.4c' => array(41, "hints"),   // 48 steps
	'Satz9.5' => array(30, "subformula",820), // array(12, "hints"),    // 38 steps
	'Satz9.6' => array(98, "subformula", 186), // array(91, "hints"),    // 111 steps
	'Satz9.8' => array(63, "subformula", 1646), //array(48, "hints"),    // 52 steps
	'Satz9.9' => array(6, "max_weight = 8", "max_seconds = 4000"),     
	'Satz9.13' => array(65, "hints"),   
	'Satz9.16' => array(14, "max_weight = 5"), // 7 seconds 
	'Satz9.19b' => array(8, "max_weight = 7"), //  11 seconds        
</pre></code>
The first line records the fact that Satz 9.2  can be proved without hints or diagram in 10 steps.
The second line says that Satz 9.3a was proved in 25 steps by the subformula strategey, and 46 seconds
is more than enough time to find the proof.  The third line says that Satz 9.3 was proved in 50 steps
using hints.   Usually when hints are used, 20 seconds is enough to find the proof; if not then a
value of <code>max_seconds</code> is also specified.  The entry for Satz 9.9 shows that it could be 
proved without hints or diagram, but finding that proof required setting a low <code>max_weight</code>
and waiting about an hour. (The time shown is a generous value for <code>max_seconds</code>, not the 
actual time required. Setting <code>max_seconds</code> too large is harmless, as the prover will 
terminate when it finds the proof.)

<code>TestChapter</code> ignores theorems that are listed as being proved by the subformula property.
Some theorems can be proved with hints,  or without hints using the subformula property.  We kept the 
resulting proofs and input files in different directories.   We used a separate program <code>TestSubformulaProperty</code>
to attempt to prove theorems by the subformula property.   Many of these proofs took more than an hour to find,
so having <code>TestChapter</code> attempt to regenerate them would have introduced unacceptable delays.  
<code>TestChapter</code> runs a whole chapter while you go to get a cup of coffee.


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