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


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


YES




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

a is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
b is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
c is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
d is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
e is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
f is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 8-rule system
{ a b -> b c a ,
  b c -> c b b ,
  a c -> c a b ,
  a a -> a d d d ,
  d a -> d d c ,
  a d d c -> a a a d ,
  e e f f -> f f f e e ,
  b d -> d d }


The system was reversed.

Remains to prove termination of the 8-rule system
{ b a -> a c b ,
  c b -> b b c ,
  c a -> b a c ,
  a a -> d d d a ,
  a d -> c d d ,
  c d d a -> d a a a ,
  f f e e -> e e f f f ,
  d b -> d d }


The dependency pairs transformation was applied.

Remains to prove termination of the 33-rule system
{ (b,true) (a,false) -> (a,true) (c,false) (b,false) ,
  (b,true) (a,false) -> (c,true) (b,false) ,
  (b,true) (a,false) -> (b,true) ,
  (c,true) (b,false) -> (b,true) (b,false) (c,false) ,
  (c,true) (b,false) -> (b,true) (c,false) ,
  (c,true) (b,false) -> (c,true) ,
  (c,true) (a,false) -> (b,true) (a,false) (c,false) ,
  (c,true) (a,false) -> (a,true) (c,false) ,
  (c,true) (a,false) -> (c,true) ,
  (a,true) (a,false) -> (d,true) (d,false) (d,false) (a,false) ,
  (a,true) (a,false) -> (d,true) (d,false) (a,false) ,
  (a,true) (a,false) -> (d,true) (a,false) ,
  (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (a,true) (d,false) -> (d,true) (d,false) ,
  (a,true) (d,false) -> (d,true) ,
  (c,true) (d,false) (d,false) (a,false) -> (d,true) (a,false) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (f,true) (f,false) (e,false) (e,false) -> (f,true) (f,false) (f,false) ,
  (f,true) (f,false) (e,false) (e,false) -> (f,true) (f,false) ,
  (f,true) (f,false) (e,false) (e,false) -> (f,true) ,
  (d,true) (b,false) -> (d,true) (d,false) ,
  (d,true) (b,false) -> (d,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }




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

(b,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 |
 \   /
(c,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(c,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(d,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(d,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(f,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(f,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(e,false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

Remains to prove termination of the 30-rule system
{ (b,true) (a,false) -> (a,true) (c,false) (b,false) ,
  (b,true) (a,false) -> (c,true) (b,false) ,
  (b,true) (a,false) -> (b,true) ,
  (c,true) (b,false) -> (b,true) (b,false) (c,false) ,
  (c,true) (b,false) -> (b,true) (c,false) ,
  (c,true) (b,false) -> (c,true) ,
  (c,true) (a,false) -> (b,true) (a,false) (c,false) ,
  (c,true) (a,false) -> (a,true) (c,false) ,
  (c,true) (a,false) -> (c,true) ,
  (a,true) (a,false) -> (d,true) (d,false) (d,false) (a,false) ,
  (a,true) (a,false) -> (d,true) (d,false) (a,false) ,
  (a,true) (a,false) -> (d,true) (a,false) ,
  (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (a,true) (d,false) -> (d,true) (d,false) ,
  (a,true) (d,false) -> (d,true) ,
  (c,true) (d,false) (d,false) (a,false) -> (d,true) (a,false) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (d,true) (b,false) -> (d,true) (d,false) ,
  (d,true) (b,false) -> (d,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


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

(b,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(c,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(c,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(d,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(d,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(f,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(e,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 24-rule system
{ (b,true) (a,false) -> (a,true) (c,false) (b,false) ,
  (b,true) (a,false) -> (c,true) (b,false) ,
  (b,true) (a,false) -> (b,true) ,
  (c,true) (b,false) -> (b,true) (b,false) (c,false) ,
  (c,true) (b,false) -> (b,true) (c,false) ,
  (c,true) (b,false) -> (c,true) ,
  (c,true) (a,false) -> (b,true) (a,false) (c,false) ,
  (c,true) (a,false) -> (a,true) (c,false) ,
  (c,true) (a,false) -> (c,true) ,
  (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (d,true) (b,false) -> (d,true) (d,false) ,
  (d,true) (b,false) -> (d,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


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

(b,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 0 |
 \     /
(a,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(c,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 2 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(c,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(e,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 22-rule system
{ (b,true) (a,false) -> (a,true) (c,false) (b,false) ,
  (b,true) (a,false) -> (c,true) (b,false) ,
  (b,true) (a,false) -> (b,true) ,
  (c,true) (b,false) -> (b,true) (b,false) (c,false) ,
  (c,true) (b,false) -> (b,true) (c,false) ,
  (c,true) (b,false) -> (c,true) ,
  (c,true) (a,false) -> (b,true) (a,false) (c,false) ,
  (c,true) (a,false) -> (a,true) (c,false) ,
  (c,true) (a,false) -> (c,true) ,
  (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


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

(b,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 |
 \     /
(c,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(c,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(e,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 17-rule system
{ (c,true) (b,false) -> (b,true) (b,false) (c,false) ,
  (c,true) (b,false) -> (b,true) (c,false) ,
  (c,true) (b,false) -> (c,true) ,
  (c,true) (a,false) -> (b,true) (a,false) (c,false) ,
  (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


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

(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(c,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(c,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(d,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(f,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(e,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 14-rule system
{ (c,true) (b,false) -> (c,true) ,
  (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


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

(b,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 0 |
 \     /
(a,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(c,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 2 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 1 1 |
 \     /
(c,true) is interpreted by
 /     \
| 1 0 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(e,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 13-rule system
{ (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (f,false) (f,false) (e,false) (e,false) ->= (e,false) (e,false) (f,false) (f,false) (f,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

(b,true) is interpreted by
 /         \
| 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 |
 \         /
(a,false) is interpreted by
 /         \
| 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 |
 \         /
(a,true) is interpreted by
 /         \
| 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 |
 \         /
(c,false) is interpreted by
 /         \
| 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 |
 \         /
(b,false) is interpreted by
 /         \
| 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 |
 \         /
(c,true) is interpreted by
 /         \
| 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 |
 \         /
(d,true) is interpreted by
 /         \
| 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 |
 \         /
(d,false) is interpreted by
 /         \
| 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 |
 \         /
(f,true) is interpreted by
 /         \
| 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 |
 \         /
(f,false) is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
| 0 0 1 0 0 |
| 0 0 1 0 0 |
 \         /
(e,false) is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 2 |
| 0 1 1 1 0 |
 \         /

Remains to prove termination of the 12-rule system
{ (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) ,
  (d,false) (b,false) ->= (d,false) (d,false) }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 4:

(b,true) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(a,false) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 1 0 0 |
 \       /
(a,true) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(c,false) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 2 1 |
| 0 1 0 0 |
 \       /
(b,false) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 1 1 0 |
| 0 0 0 1 |
 \       /
(c,true) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(d,true) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(d,false) is interpreted by
 /       \
| 1 0 1 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(f,true) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(f,false) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /
(e,false) is interpreted by
 /       \
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
 \       /

Remains to prove termination of the 11-rule system
{ (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) (a,false) ,
  (c,true) (d,false) (d,false) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) }


The system was filtered by the following matrix interpretation
of type E_J with J = {1,...,2} and dimension 5:

(b,true) is interpreted by
 /         \
| 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 |
 \         /
(a,false) is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
| 0 0 0 1 0 |
| 0 1 0 1 0 |
 \         /
(a,true) is interpreted by
 /         \
| 1 0 0 1 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
(c,false) is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 1 0 0 |
| 0 0 1 0 0 |
| 0 0 0 0 1 |
 \         /
(b,false) is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 1 0 0 0 |
 \         /
(c,true) is interpreted by
 /         \
| 1 0 1 0 0 |
| 0 1 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
| 0 0 0 0 0 |
 \         /
(d,true) is interpreted by
 /         \
| 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 |
 \         /
(d,false) is interpreted by
 /         \
| 1 0 0 0 0 |
| 0 1 0 0 0 |
| 0 0 0 1 0 |
| 0 0 0 0 1 |
| 0 0 0 0 0 |
 \         /
(f,true) is interpreted by
 /         \
| 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 |
 \         /
(f,false) is interpreted by
 /         \
| 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 |
 \         /
(e,false) is interpreted by
 /         \
| 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 |
 \         /

Remains to prove termination of the 8-rule system
{ (a,true) (a,false) -> (a,true) ,
  (a,true) (d,false) -> (c,true) (d,false) (d,false) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,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:

(b,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(a,true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
(c,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(b,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(c,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(d,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(d,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(f,true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(f,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
(e,false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 7-rule system
{ (a,true) (a,false) -> (a,true) ,
  (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,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:

(b,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 1 |
| 0 1 0 |
| 0 0 0 |
 \     /
(c,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(b,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 1 |
 \     /
(c,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(d,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,true) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(f,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /
(e,false) is interpreted by
 /     \
| 1 0 0 |
| 0 1 0 |
| 0 0 0 |
 \     /

Remains to prove termination of the 6-rule system
{ (b,false) (a,false) ->= (a,false) (c,false) (b,false) ,
  (c,false) (b,false) ->= (b,false) (b,false) (c,false) ,
  (c,false) (a,false) ->= (b,false) (a,false) (c,false) ,
  (a,false) (a,false) ->= (d,false) (d,false) (d,false) (a,false) ,
  (a,false) (d,false) ->= (c,false) (d,false) (d,false) ,
  (c,false) (d,false) (d,false) (a,false) ->= (d,false) (a,false) (a,false) (a,false) }


The system is trivially terminating.