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


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


YES


The system was reversed.

Remains to prove termination of the 15-rule system
{ 0 r0 -> r0 0 ,
  1 r0 -> r0 1 ,
  m r0 -> r0 m ,
  0 r1 -> r1 0 ,
  1 r1 -> r1 1 ,
  m r1 -> r1 m ,
  b r0 -> b 0 qr ,
  b r1 -> b 1 qr ,
  qr 0 -> 0 qr ,
  qr 1 -> 1 qr ,
  qr m -> m ql ,
  ql 0 -> 0 ql ,
  ql 1 -> 1 ql ,
  0 ql b -> r0 b 0 ,
  1 ql b -> r1 b 1 }


The dependency pairs transformation was applied.

Remains to prove termination of the 41-rule system
{ (0,true) (r0,false) -> (0,true) ,
  (1,true) (r0,false) -> (1,true) ,
  (m,true) (r0,false) -> (m,true) ,
  (0,true) (r1,false) -> (0,true) ,
  (1,true) (r1,false) -> (1,true) ,
  (m,true) (r1,false) -> (m,true) ,
  (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r0,false) -> (0,true) (qr,false) ,
  (b,true) (r0,false) -> (qr,true) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (b,true) (r1,false) -> (1,true) (qr,false) ,
  (b,true) (r1,false) -> (qr,true) ,
  (qr,true) (0,false) -> (0,true) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (1,true) (qr,false) ,
  (qr,true) (1,false) -> (qr,true) ,
  (qr,true) (m,false) -> (m,true) (ql,false) ,
  (qr,true) (m,false) -> (ql,true) ,
  (ql,true) (0,false) -> (0,true) (ql,false) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (1,true) (ql,false) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,true) (ql,false) (b,false) -> (b,true) (0,false) ,
  (0,true) (ql,false) (b,false) -> (0,true) ,
  (1,true) (ql,false) (b,false) -> (b,true) (1,false) ,
  (1,true) (ql,false) (b,false) -> (1,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }




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

(0,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(ql,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 39-rule system
{ (0,true) (r0,false) -> (0,true) ,
  (1,true) (r0,false) -> (1,true) ,
  (m,true) (r0,false) -> (m,true) ,
  (0,true) (r1,false) -> (0,true) ,
  (1,true) (r1,false) -> (1,true) ,
  (m,true) (r1,false) -> (m,true) ,
  (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r0,false) -> (0,true) (qr,false) ,
  (b,true) (r0,false) -> (qr,true) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (b,true) (r1,false) -> (1,true) (qr,false) ,
  (b,true) (r1,false) -> (qr,true) ,
  (qr,true) (0,false) -> (0,true) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (1,true) (qr,false) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (0,false) -> (0,true) (ql,false) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (1,true) (ql,false) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,true) (ql,false) (b,false) -> (b,true) (0,false) ,
  (0,true) (ql,false) (b,false) -> (0,true) ,
  (1,true) (ql,false) (b,false) -> (b,true) (1,false) ,
  (1,true) (ql,false) (b,false) -> (1,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 37-rule system
{ (0,true) (r0,false) -> (0,true) ,
  (1,true) (r0,false) -> (1,true) ,
  (m,true) (r0,false) -> (m,true) ,
  (0,true) (r1,false) -> (0,true) ,
  (1,true) (r1,false) -> (1,true) ,
  (m,true) (r1,false) -> (m,true) ,
  (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r0,false) -> (0,true) (qr,false) ,
  (b,true) (r0,false) -> (qr,true) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (b,true) (r1,false) -> (1,true) (qr,false) ,
  (b,true) (r1,false) -> (qr,true) ,
  (qr,true) (0,false) -> (0,true) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (1,true) (qr,false) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,true) (ql,false) (b,false) -> (b,true) (0,false) ,
  (0,true) (ql,false) (b,false) -> (0,true) ,
  (1,true) (ql,false) (b,false) -> (b,true) (1,false) ,
  (1,true) (ql,false) (b,false) -> (1,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

Remains to prove termination of the 33-rule system
{ (0,true) (r0,false) -> (0,true) ,
  (1,true) (r0,false) -> (1,true) ,
  (m,true) (r0,false) -> (m,true) ,
  (0,true) (r1,false) -> (0,true) ,
  (1,true) (r1,false) -> (1,true) ,
  (m,true) (r1,false) -> (m,true) ,
  (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r0,false) -> (0,true) (qr,false) ,
  (b,true) (r0,false) -> (qr,true) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (b,true) (r1,false) -> (1,true) (qr,false) ,
  (b,true) (r1,false) -> (qr,true) ,
  (qr,true) (0,false) -> (0,true) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (1,true) (qr,false) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 29-rule system
{ (0,true) (r0,false) -> (0,true) ,
  (1,true) (r0,false) -> (1,true) ,
  (m,true) (r0,false) -> (m,true) ,
  (0,true) (r1,false) -> (0,true) ,
  (1,true) (r1,false) -> (1,true) ,
  (m,true) (r1,false) -> (m,true) ,
  (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (qr,true) (0,false) -> (0,true) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (1,true) (qr,false) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 27-rule system
{ (0,true) (r0,false) -> (0,true) ,
  (1,true) (r0,false) -> (1,true) ,
  (m,true) (r0,false) -> (m,true) ,
  (0,true) (r1,false) -> (0,true) ,
  (1,true) (r1,false) -> (1,true) ,
  (m,true) (r1,false) -> (m,true) ,
  (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r0,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(1,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(r1,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(0,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(qr,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(qr,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(1,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(m,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(ql,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(ql,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 21-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (0,false) -> (ql,true) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 20-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (qr,true) ,
  (ql,true) (1,false) -> (ql,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 19-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (qr,true) (0,false) -> (qr,true) ,
  (qr,true) (1,false) -> (qr,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,true) 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 |
 \     /

Remains to prove termination of the 18-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (qr,true) (1,false) -> (qr,true) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,true) 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 |
 \     /

Remains to prove termination of the 17-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (m,false) (r1,false) ->= (r1,false) (m,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 1 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,true) 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 |
 \     /

Remains to prove termination of the 16-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (b,true) (r1,false) -> (b,true) (1,false) (qr,false) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
(ql,true) 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 |
 \     /

Remains to prove termination of the 15-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (b,false) (r1,false) ->= (b,false) (1,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
(ql,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 14-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (m,false) (r0,false) ->= (r0,false) (m,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(ql,true) 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 |
 \     /

Remains to prove termination of the 13-rule system
{ (b,true) (r0,false) -> (b,true) (0,false) (qr,false) ,
  (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


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

(0,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(r0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
(1,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(m,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 0 |
 \     /
(b,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(0,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(qr,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(qr,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(1,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(m,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(ql,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 0 |
 \     /
(ql,true) 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 |
 \     /

Remains to prove termination of the 12-rule system
{ (0,false) (r0,false) ->= (r0,false) (0,false) ,
  (1,false) (r0,false) ->= (r0,false) (1,false) ,
  (0,false) (r1,false) ->= (r1,false) (0,false) ,
  (1,false) (r1,false) ->= (r1,false) (1,false) ,
  (b,false) (r0,false) ->= (b,false) (0,false) (qr,false) ,
  (qr,false) (0,false) ->= (0,false) (qr,false) ,
  (qr,false) (1,false) ->= (1,false) (qr,false) ,
  (qr,false) (m,false) ->= (m,false) (ql,false) ,
  (ql,false) (0,false) ->= (0,false) (ql,false) ,
  (ql,false) (1,false) ->= (1,false) (ql,false) ,
  (0,false) (ql,false) (b,false) ->= (r0,false) (b,false) (0,false) ,
  (1,false) (ql,false) (b,false) ->= (r1,false) (b,false) (1,false) }


The system is trivially terminating.