print qq{\nAn approach to the proof of the Distribution Rule for\n};
print qq{Summation and Union over the Natural Realm wrt MF(C):155 P.NJW\n};

# An approach to coding the proof for Distribution rule
# for Summation and Union over the Natural Realm
# re MF:C:155

#my($nu,$lhs,$rhs);
($kappa,$mu,$nu) = (3,4,5);


until( $kappa < $mu and $mu < $nu  ){ # skips this, already satisfied
 $mu *= $mu unless $mu > $nu;
 $kappa *= $kappa unless $kappa > $mu;
 print "\$kappa:$kappa \$mu:$mu \$nu:$nu\n";
}

print "\nEvaluating relation (P1) when \$kappa:$kappa < \$mu:$mu < \$nu:$nu\n";

$lhs =  $kappa + ( $mu < $nu ? $nu : $mu ); 
$rhs =  ( $kappa + $mu ) < ( $kappa + $nu ) ?  $kappa + $nu : $kappa + $mu ;
print "\$lhs:$lhs  = \$kappa:$kappa + \$nu:$nu\n";
print "\$rhs:$rhs  = \$kappa:$kappa + \$nu:$nu\n";
print "Evaluation of relation ", $lhs == $rhs ? 'succeeded' : 'failed';
print qq{\n};



until( $kappa < $nu and $nu < $mu  ){
 $mu *= $mu unless $mu > $nu;
 $nu *= $nu unless $nu > $kappa;
# print "\$kappa:$kappa \$nu:$nu \$mu:$mu\n";
}

print "\nEvaluating relation when (P2) \$kappa:$kappa < \$nu:$nu < \$mu:$mu\n";

$lhs =  $kappa + ( $mu < $nu ? $nu : $mu ); 
$rhs =  ( $kappa + $mu ) < ( $kappa + $nu ) ?  $kappa + $nu : $kappa + $mu ;
print "\$lhs:$lhs  = \$kappa:$kappa + \$mu:$mu\n";
print "\$rhs:$rhs  = \$kappa:$kappa + \$mu:$mu\n";
print "Evaluation of relation ", $lhs == $rhs ? 'succeeded' : 'failed';
print qq{\n};



until( $mu < $kappa and $kappa < $nu  ){
 $nu *= $nu unless $nu > $kappa;
 $kappa *= $kappa unless $kappa > $mu;
# print "\$mu:$mu \$kappa:$kappa \$nu:$nu\n";
}

print "\nEvaluating relation (P3) when \$mu:$mu < \$kappa:$kappa < \$nu:$nu\n";

$lhs =  $mu + ( $kappa < $nu ? $nu : $kappa ); 
$rhs =  ( $mu + $kappa ) < ( $mu + $nu ) ?  $mu + $nu : $mu + $kappa ;
print "\$lhs:$lhs  = \$mu:$mu + \$nu:$nu\n";
print "\$rhs:$rhs  = \$mu:$mu + \$nu:$nu\n";
print "Evaluation of relation ", $lhs == $rhs ? 'succeeded' : 'failed';
print qq{\n};



until( $mu < $nu and $nu < $kappa  ){
 $kappa *= $kappa unless $kappa > $nu;
 $nu *= $nu unless $nu > $mu;
# print "\$mu:$mu \$nu:$nu \$kappa:$kappa\n";
}

print "\nEvaluating relation (P4) when \$mu:$mu < \$nu:$nu < \$kappa:$kappa\n";

$lhs =  $mu + ( $kappa < $nu ? $nu : $kappa ); 
$rhs =  ( $mu + $kappa ) < ( $mu + $nu ) ?  $mu + $nu : $mu + $kappa ;
print "\$lhs:$lhs  = \$mu:$mu + \$nu:$nu\n";
print "\$rhs:$rhs  = \$mu:$mu + \$nu:$nu\n";
print "Evaluation of relation ", $lhs == $rhs ? 'succeeded' : 'failed';
print qq{\n};



until( $nu < $kappa and $kappa < $mu  ){
 $mu *= $mu unless $mu > $kappa;
 $kappa *= $kappa unless $kappa > $nu;
# print "\$nu:$nu \$kappa:$kappa \$mu:$mu\n";
}

print "\nEvaluating relation (P5) when \$nu:$nu < \$kappa:$kappa < \$mu:$mu\n";

$lhs =  $nu + ( $mu > $kappa ?  $mu : $kappa ); 
$rhs =  ( $nu + $mu ) > ( $nu + $kappa ) ?  $nu + $mu : $nu + $kappa ;
print "\$lhs:$lhs  = \$kappa:$kappa + \$mu:$mu\n";
print "\$rhs:$rhs  = \$kappa:$kappa + \$mu:$mu\n";
print "Evaluation of relation ", $lhs == $rhs ? 'succeeded' : 'failed'; 
print qq{\n};




until( $nu < $mu and $mu < $kappa  ){
 $kappa *= $kappa unless $kappa > $mu;
 $mu *= $mu unless $mu > $nu;
# print "\$nu:$nu \$mu:$mu \$kappa:$kappa\n";
}

print "\nEvaluating relation (P6) when \$nu:$nu < \$mu:$mu < \$kappa:$kappa\n";

$lhs =  $nu + ( $mu > $kappa ?  $mu : $kappa ); 
$rhs =  ( $nu + $mu ) > ( $nu + $kappa ) ?  $nu + $mu : $nu + $kappa ;
print "\$lhs:$lhs  = \$kappa:$kappa + \$mu:$mu\n";
print "\$rhs:$rhs  = \$kappa:$kappa + \$mu:$mu\n";
print "Evaluation of relation ", $lhs == $rhs ? 'succeeded' : 'failed'; 
print qq{\n\n};