# 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);
if( $kappa < $mu and $mu < $nu ){
  print "Evaluating relation when \$kappa:$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 > $mu and $mu > $nu  ){
 $mu *= $mu unless $mu > $nu;
 $kappa *= $kappa unless $kappa > $mu;
# print "\$kappa:$kappa \$mu:$mu \$nu:$nu\n";

}

#print "\$kappa:$kappa > \$mu:$mu > \$nu:$mu"

#now we have interchanged the values not the operators 

if( $nu < $mu and $mu < $kappa ){
  print "Evaluating relation when \$nu:$nu < \$mu:$mu < \$kappa:$kappa\n";
  $lhs =  $nu + ( $mu > $kappa ?  $mu : $kappa ); 
#  $rhs =  ( $nu > $mu ? $nu : $mu ) + ( $nu < $kappa ? $nu : $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};

}
__END__
Evaluating relation when $kappa:3 < $mu:4 < $nu:5
$lhs:8  = $kappa:3 + $nu:5
$rhs:8  = $kappa:3 + $nu:5
Evaluation of relation succeeded
Evaluating relation when $nu:5 < $mu:16 < $kappa:81
$lhs:86  = $kappa:81 + $mu:16
$rhs:86  = $kappa:81 + $mu:16
Evaluation of relation succeeded