/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES


The dependency pairs transformation was applied.

Remains to prove termination of the 29-rule system
{ (r1,true) (a,false) -> (a,true) (a,false) (a,false) (r1,false) ,
  (r1,true) (a,false) -> (a,true) (a,false) (r1,false) ,
  (r1,true) (a,false) -> (a,true) (r1,false) ,
  (r1,true) (a,false) -> (r1,true) ,
  (r2,true) (a,false) -> (a,true) (a,false) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (r2,false) ,
  (r2,true) (a,false) -> (r2,true) ,
  (a,true) (l1,false) -> (a,true) (a,false) (a,false) ,
  (a,true) (l1,false) -> (a,true) (a,false) ,
  (a,true) (l1,false) -> (a,true) ,
  (a,true) (a,false) (l2,false) -> (a,true) (a,false) ,
  (a,true) (a,false) (l2,false) -> (a,true) ,
  (r1,true) (b,false) -> (b,true) ,
  (r2,true) (b,false) -> (a,true) (b,false) ,
  (r2,true) (b,false) -> (b,true) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l1,false) -> (r2,true) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (b,true) (l2,false) -> (r1,true) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }




The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

(r1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 27-rule system
{ (r1,true) (a,false) -> (a,true) (a,false) (a,false) (r1,false) ,
  (r1,true) (a,false) -> (a,true) (a,false) (r1,false) ,
  (r1,true) (a,false) -> (a,true) (r1,false) ,
  (r1,true) (a,false) -> (r1,true) ,
  (r2,true) (a,false) -> (a,true) (a,false) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (r2,false) ,
  (r2,true) (a,false) -> (r2,true) ,
  (a,true) (l1,false) -> (a,true) (a,false) (a,false) ,
  (a,true) (l1,false) -> (a,true) (a,false) ,
  (a,true) (l1,false) -> (a,true) ,
  (a,true) (a,false) (l2,false) -> (a,true) (a,false) ,
  (a,true) (a,false) (l2,false) -> (a,true) ,
  (r2,true) (b,false) -> (a,true) (b,false) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l1,false) -> (r2,true) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (b,true) (l2,false) -> (r1,true) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

(r1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

Remains to prove termination of the 25-rule system
{ (r1,true) (a,false) -> (a,true) (a,false) (a,false) (r1,false) ,
  (r1,true) (a,false) -> (a,true) (a,false) (r1,false) ,
  (r1,true) (a,false) -> (a,true) (r1,false) ,
  (r1,true) (a,false) -> (r1,true) ,
  (r2,true) (a,false) -> (a,true) (a,false) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (r2,false) ,
  (r2,true) (a,false) -> (r2,true) ,
  (a,true) (l1,false) -> (a,true) (a,false) (a,false) ,
  (a,true) (l1,false) -> (a,true) (a,false) ,
  (a,true) (l1,false) -> (a,true) ,
  (a,true) (a,false) (l2,false) -> (a,true) (a,false) ,
  (a,true) (a,false) (l2,false) -> (a,true) ,
  (r2,true) (b,false) -> (a,true) (b,false) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

(r1,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 22-rule system
{ (r1,true) (a,false) -> (r1,true) ,
  (r2,true) (a,false) -> (a,true) (a,false) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (a,false) (r2,false) ,
  (r2,true) (a,false) -> (a,true) (r2,false) ,
  (r2,true) (a,false) -> (r2,true) ,
  (a,true) (l1,false) -> (a,true) (a,false) (a,false) ,
  (a,true) (l1,false) -> (a,true) (a,false) ,
  (a,true) (l1,false) -> (a,true) ,
  (a,true) (a,false) (l2,false) -> (a,true) (a,false) ,
  (a,true) (a,false) (l2,false) -> (a,true) ,
  (r2,true) (b,false) -> (a,true) (b,false) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

(r1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(r2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(l2,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 18-rule system
{ (r1,true) (a,false) -> (r1,true) ,
  (r2,true) (a,false) -> (r2,true) ,
  (a,true) (l1,false) -> (a,true) (a,false) (a,false) ,
  (a,true) (l1,false) -> (a,true) (a,false) ,
  (a,true) (l1,false) -> (a,true) ,
  (a,true) (a,false) (l2,false) -> (a,true) (a,false) ,
  (a,true) (a,false) (l2,false) -> (a,true) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 2:

(r1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(r2,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r2,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(l1,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(l2,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 13-rule system
{ (r1,true) (a,false) -> (r1,true) ,
  (r2,true) (a,false) -> (r2,true) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 3:

(r1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(a,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(a,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 3 |
 \     /
(r2,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r2,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 3 |
 \     /
(l1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(l2,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 12-rule system
{ (r1,true) (a,false) -> (r1,true) ,
  (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 3:

(r1,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(a,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(a,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 3 |
 \     /
(r2,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r2,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 3 |
 \     /
(l1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(l2,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 11-rule system
{ (b,true) (l1,false) -> (b,true) (r2,false) ,
  (b,true) (l2,false) -> (b,true) (r1,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 6:

(r1,true) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(a,false) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 1 0 0 |
| 0 0 1 0 0 0 |
| 0 0 0 0 0 1 |
| 0 0 0 0 1 0 |
 \           /
(a,true) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(r1,false) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 0 1 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(r2,true) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(r2,false) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 0 1 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(l1,false) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 1 0 |
| 0 0 0 0 0 1 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(l2,false) is interpreted by
 /           \
| 1 0 0 0 1 0 |
| 0 1 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(b,false) is interpreted by
 /           \
| 1 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /
(b,true) is interpreted by
 /           \
| 1 0 1 0 0 0 |
| 0 1 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
 \           /

Remains to prove termination of the 10-rule system
{ (b,true) (l1,false) -> (b,true) (r2,false) ,
  (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 3:

(r1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(a,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(a,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
(r2,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r2,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(l1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
(l2,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 9-rule system
{ (r1,false) (a,false) ->= (a,false) (a,false) (a,false) (r1,false) ,
  (r2,false) (a,false) ->= (a,false) (a,false) (a,false) (r2,false) ,
  (a,false) (l1,false) ->= (l1,false) (a,false) (a,false) (a,false) ,
  (a,false) (a,false) (l2,false) ->= (l2,false) (a,false) (a,false) ,
  (r1,false) (b,false) ->= (l1,false) (b,false) ,
  (r2,false) (b,false) ->= (l2,false) (a,false) (b,false) ,
  (b,false) (l1,false) ->= (b,false) (r2,false) ,
  (b,false) (l2,false) ->= (b,false) (r1,false) ,
  (a,false) (a,false) ->= }


The system is trivially terminating.