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


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


YES


Applying context closure of depth 1 in the following form: System R over Sigma
maps to { fold(xly) -> fold(xry) | l -> r in R, x,y in Sigma } over Sigma^2,
where fold(a_1,...,a_n) = (a_1,a_2)...(a_{n-1}a_{n})

Remains to prove termination of the 64-rule system
{ [A, A] [A, b] [b, A] -> [A, b] [b, a] [a, B] [B, A] [A, A] ,
  [A, B] [B, a] [a, A] -> [A, a] [a, b] [b, A] [A, B] [B, A] ,
  [A, A] [A, a] [a, A] -> [A, A] ,
  [A, B] [B, b] [b, A] -> [A, A] ,
  [A, A] [A, b] [b, b] -> [A, b] [b, a] [a, B] [B, A] [A, b] ,
  [A, B] [B, a] [a, b] -> [A, a] [a, b] [b, A] [A, B] [B, b] ,
  [A, A] [A, a] [a, b] -> [A, b] ,
  [A, B] [B, b] [b, b] -> [A, b] ,
  [A, A] [A, b] [b, a] -> [A, b] [b, a] [a, B] [B, A] [A, a] ,
  [A, B] [B, a] [a, a] -> [A, a] [a, b] [b, A] [A, B] [B, a] ,
  [A, A] [A, a] [a, a] -> [A, a] ,
  [A, B] [B, b] [b, a] -> [A, a] ,
  [A, A] [A, b] [b, B] -> [A, b] [b, a] [a, B] [B, A] [A, B] ,
  [A, B] [B, a] [a, B] -> [A, a] [a, b] [b, A] [A, B] [B, B] ,
  [A, A] [A, a] [a, B] -> [A, B] ,
  [A, B] [B, b] [b, B] -> [A, B] ,
  [b, A] [A, b] [b, A] -> [b, b] [b, a] [a, B] [B, A] [A, A] ,
  [b, B] [B, a] [a, A] -> [b, a] [a, b] [b, A] [A, B] [B, A] ,
  [b, A] [A, a] [a, A] -> [b, A] ,
  [b, B] [B, b] [b, A] -> [b, A] ,
  [b, A] [A, b] [b, b] -> [b, b] [b, a] [a, B] [B, A] [A, b] ,
  [b, B] [B, a] [a, b] -> [b, a] [a, b] [b, A] [A, B] [B, b] ,
  [b, A] [A, a] [a, b] -> [b, b] ,
  [b, B] [B, b] [b, b] -> [b, b] ,
  [b, A] [A, b] [b, a] -> [b, b] [b, a] [a, B] [B, A] [A, a] ,
  [b, B] [B, a] [a, a] -> [b, a] [a, b] [b, A] [A, B] [B, a] ,
  [b, A] [A, a] [a, a] -> [b, a] ,
  [b, B] [B, b] [b, a] -> [b, a] ,
  [b, A] [A, b] [b, B] -> [b, b] [b, a] [a, B] [B, A] [A, B] ,
  [b, B] [B, a] [a, B] -> [b, a] [a, b] [b, A] [A, B] [B, B] ,
  [b, A] [A, a] [a, B] -> [b, B] ,
  [b, B] [B, b] [b, B] -> [b, B] ,
  [a, A] [A, b] [b, A] -> [a, b] [b, a] [a, B] [B, A] [A, A] ,
  [a, B] [B, a] [a, A] -> [a, a] [a, b] [b, A] [A, B] [B, A] ,
  [a, A] [A, a] [a, A] -> [a, A] ,
  [a, B] [B, b] [b, A] -> [a, A] ,
  [a, A] [A, b] [b, b] -> [a, b] [b, a] [a, B] [B, A] [A, b] ,
  [a, B] [B, a] [a, b] -> [a, a] [a, b] [b, A] [A, B] [B, b] ,
  [a, A] [A, a] [a, b] -> [a, b] ,
  [a, B] [B, b] [b, b] -> [a, b] ,
  [a, A] [A, b] [b, a] -> [a, b] [b, a] [a, B] [B, A] [A, a] ,
  [a, B] [B, a] [a, a] -> [a, a] [a, b] [b, A] [A, B] [B, a] ,
  [a, A] [A, a] [a, a] -> [a, a] ,
  [a, B] [B, b] [b, a] -> [a, a] ,
  [a, A] [A, b] [b, B] -> [a, b] [b, a] [a, B] [B, A] [A, B] ,
  [a, B] [B, a] [a, B] -> [a, a] [a, b] [b, A] [A, B] [B, B] ,
  [a, A] [A, a] [a, B] -> [a, B] ,
  [a, B] [B, b] [b, B] -> [a, B] ,
  [B, A] [A, b] [b, A] -> [B, b] [b, a] [a, B] [B, A] [A, A] ,
  [B, B] [B, a] [a, A] -> [B, a] [a, b] [b, A] [A, B] [B, A] ,
  [B, A] [A, a] [a, A] -> [B, A] ,
  [B, B] [B, b] [b, A] -> [B, A] ,
  [B, A] [A, b] [b, b] -> [B, b] [b, a] [a, B] [B, A] [A, b] ,
  [B, B] [B, a] [a, b] -> [B, a] [a, b] [b, A] [A, B] [B, b] ,
  [B, A] [A, a] [a, b] -> [B, b] ,
  [B, B] [B, b] [b, b] -> [B, b] ,
  [B, A] [A, b] [b, a] -> [B, b] [b, a] [a, B] [B, A] [A, a] ,
  [B, B] [B, a] [a, a] -> [B, a] [a, b] [b, A] [A, B] [B, a] ,
  [B, A] [A, a] [a, a] -> [B, a] ,
  [B, B] [B, b] [b, a] -> [B, a] ,
  [B, A] [A, b] [b, B] -> [B, b] [b, a] [a, B] [B, A] [A, B] ,
  [B, B] [B, a] [a, B] -> [B, a] [a, b] [b, A] [A, B] [B, B] ,
  [B, A] [A, a] [a, B] -> [B, B] ,
  [B, B] [B, b] [b, B] -> [B, B] }



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

[A, A] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[A, b] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[b, A] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[b, a] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[a, B] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[B, A] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[A, B] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[B, a] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[a, A] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[A, a] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[a, b] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[B, b] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[b, b] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[a, a] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[b, B] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[B, B] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

Remains to prove termination of the 41-rule system
{ [A, A] [A, b] [b, A] -> [A, b] [b, a] [a, B] [B, A] [A, A] ,
  [A, B] [B, b] [b, A] -> [A, A] ,
  [A, A] [A, b] [b, b] -> [A, b] [b, a] [a, B] [B, A] [A, b] ,
  [A, B] [B, a] [a, b] -> [A, a] [a, b] [b, A] [A, B] [B, b] ,
  [A, A] [A, a] [a, b] -> [A, b] ,
  [A, B] [B, b] [b, b] -> [A, b] ,
  [A, A] [A, b] [b, a] -> [A, b] [b, a] [a, B] [B, A] [A, a] ,
  [A, B] [B, a] [a, a] -> [A, a] [a, b] [b, A] [A, B] [B, a] ,
  [A, B] [B, b] [b, a] -> [A, a] ,
  [A, A] [A, b] [b, B] -> [A, b] [b, a] [a, B] [B, A] [A, B] ,
  [A, B] [B, a] [a, B] -> [A, a] [a, b] [b, A] [A, B] [B, B] ,
  [A, B] [B, b] [b, B] -> [A, B] ,
  [b, A] [A, b] [b, A] -> [b, b] [b, a] [a, B] [B, A] [A, A] ,
  [b, B] [B, b] [b, A] -> [b, A] ,
  [b, A] [A, b] [b, b] -> [b, b] [b, a] [a, B] [B, A] [A, b] ,
  [b, B] [B, a] [a, b] -> [b, a] [a, b] [b, A] [A, B] [B, b] ,
  [b, A] [A, a] [a, b] -> [b, b] ,
  [b, B] [B, b] [b, b] -> [b, b] ,
  [b, A] [A, b] [b, a] -> [b, b] [b, a] [a, B] [B, A] [A, a] ,
  [b, B] [B, a] [a, a] -> [b, a] [a, b] [b, A] [A, B] [B, a] ,
  [b, B] [B, b] [b, a] -> [b, a] ,
  [b, A] [A, b] [b, B] -> [b, b] [b, a] [a, B] [B, A] [A, B] ,
  [b, B] [B, a] [a, B] -> [b, a] [a, b] [b, A] [A, B] [B, B] ,
  [b, B] [B, b] [b, B] -> [b, B] ,
  [a, B] [B, a] [a, b] -> [a, a] [a, b] [b, A] [A, B] [B, b] ,
  [a, B] [B, a] [a, a] -> [a, a] [a, b] [b, A] [A, B] [B, a] ,
  [a, B] [B, b] [b, a] -> [a, a] ,
  [a, B] [B, a] [a, B] -> [a, a] [a, b] [b, A] [A, B] [B, B] ,
  [a, B] [B, b] [b, B] -> [a, B] ,
  [B, A] [A, b] [b, A] -> [B, b] [b, a] [a, B] [B, A] [A, A] ,
  [B, A] [A, b] [b, b] -> [B, b] [b, a] [a, B] [B, A] [A, b] ,
  [B, B] [B, a] [a, b] -> [B, a] [a, b] [b, A] [A, B] [B, b] ,
  [B, A] [A, a] [a, b] -> [B, b] ,
  [B, A] [A, b] [b, a] -> [B, b] [b, a] [a, B] [B, A] [A, a] ,
  [B, B] [B, a] [a, a] -> [B, a] [a, b] [b, A] [A, B] [B, a] ,
  [B, A] [A, a] [a, a] -> [B, a] ,
  [B, B] [B, b] [b, a] -> [B, a] ,
  [B, A] [A, b] [b, B] -> [B, b] [b, a] [a, B] [B, A] [A, B] ,
  [B, B] [B, a] [a, B] -> [B, a] [a, b] [b, A] [A, B] [B, B] ,
  [B, A] [A, a] [a, B] -> [B, B] ,
  [B, B] [B, b] [b, B] -> [B, B] }


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

[A, A] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[A, b] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[b, A] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[b, a] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[a, B] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[B, A] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[A, B] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[B, a] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[a, A] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[A, a] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[a, b] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[B, b] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[b, b] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[a, a] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
[b, B] is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
[B, B] is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 28-rule system
{ [A, A] [A, b] [b, A] -> [A, b] [b, a] [a, B] [B, A] [A, A] ,
  [A, B] [B, b] [b, A] -> [A, A] ,
  [A, A] [A, b] [b, b] -> [A, b] [b, a] [a, B] [B, A] [A, b] ,
  [A, B] [B, a] [a, b] -> [A, a] [a, b] [b, A] [A, B] [B, b] ,
  [A, A] [A, a] [a, b] -> [A, b] ,
  [A, B] [B, b] [b, b] -> [A, b] ,
  [A, A] [A, b] [b, a] -> [A, b] [b, a] [a, B] [B, A] [A, a] ,
  [A, B] [B, a] [a, a] -> [A, a] [a, b] [b, A] [A, B] [B, a] ,
  [A, B] [B, b] [b, a] -> [A, a] ,
  [A, B] [B, a] [a, B] -> [A, a] [a, b] [b, A] [A, B] [B, B] ,
  [b, A] [A, b] [b, A] -> [b, b] [b, a] [a, B] [B, A] [A, A] ,
  [b, A] [A, b] [b, b] -> [b, b] [b, a] [a, B] [B, A] [A, b] ,
  [b, A] [A, a] [a, b] -> [b, b] ,
  [b, A] [A, b] [b, a] -> [b, b] [b, a] [a, B] [B, A] [A, a] ,
  [a, B] [B, a] [a, b] -> [a, a] [a, b] [b, A] [A, B] [B, b] ,
  [a, B] [B, a] [a, a] -> [a, a] [a, b] [b, A] [A, B] [B, a] ,
  [a, B] [B, b] [b, a] -> [a, a] ,
  [a, B] [B, a] [a, B] -> [a, a] [a, b] [b, A] [A, B] [B, B] ,
  [B, A] [A, b] [b, A] -> [B, b] [b, a] [a, B] [B, A] [A, A] ,
  [B, A] [A, b] [b, b] -> [B, b] [b, a] [a, B] [B, A] [A, b] ,
  [B, B] [B, a] [a, b] -> [B, a] [a, b] [b, A] [A, B] [B, b] ,
  [B, A] [A, a] [a, b] -> [B, b] ,
  [B, A] [A, b] [b, a] -> [B, b] [b, a] [a, B] [B, A] [A, a] ,
  [B, B] [B, a] [a, a] -> [B, a] [a, b] [b, A] [A, B] [B, a] ,
  [B, A] [A, a] [a, a] -> [B, a] ,
  [B, B] [B, b] [b, a] -> [B, a] ,
  [B, B] [B, a] [a, B] -> [B, a] [a, b] [b, A] [A, B] [B, B] ,
  [B, A] [A, a] [a, B] -> [B, B] }


The dependency pairs transformation was applied.

Remains to prove termination of the 72-rule system
{ ([A, A],true) ([A, b],false) ([b, A],false) -> ([a, B],true) ([B, A],false) ([A, A],false) ,
  ([A, A],true) ([A, b],false) ([b, A],false) -> ([B, A],true) ([A, A],false) ,
  ([A, A],true) ([A, b],false) ([b, A],false) -> ([A, A],true) ,
  ([A, B],true) ([B, b],false) ([b, A],false) -> ([A, A],true) ,
  ([A, A],true) ([A, b],false) ([b, b],false) -> ([a, B],true) ([B, A],false) ([A, b],false) ,
  ([A, A],true) ([A, b],false) ([b, b],false) -> ([B, A],true) ([A, b],false) ,
  ([A, B],true) ([B, a],false) ([a, b],false) -> ([b, A],true) ([A, B],false) ([B, b],false) ,
  ([A, B],true) ([B, a],false) ([a, b],false) -> ([A, B],true) ([B, b],false) ,
  ([A, A],true) ([A, b],false) ([b, a],false) -> ([a, B],true) ([B, A],false) ([A, a],false) ,
  ([A, A],true) ([A, b],false) ([b, a],false) -> ([B, A],true) ([A, a],false) ,
  ([A, B],true) ([B, a],false) ([a, a],false) -> ([b, A],true) ([A, B],false) ([B, a],false) ,
  ([A, B],true) ([B, a],false) ([a, a],false) -> ([A, B],true) ([B, a],false) ,
  ([A, B],true) ([B, a],false) ([a, B],false) -> ([b, A],true) ([A, B],false) ([B, B],false) ,
  ([A, B],true) ([B, a],false) ([a, B],false) -> ([A, B],true) ([B, B],false) ,
  ([A, B],true) ([B, a],false) ([a, B],false) -> ([B, B],true) ,
  ([b, A],true) ([A, b],false) ([b, A],false) -> ([a, B],true) ([B, A],false) ([A, A],false) ,
  ([b, A],true) ([A, b],false) ([b, A],false) -> ([B, A],true) ([A, A],false) ,
  ([b, A],true) ([A, b],false) ([b, A],false) -> ([A, A],true) ,
  ([b, A],true) ([A, b],false) ([b, b],false) -> ([a, B],true) ([B, A],false) ([A, b],false) ,
  ([b, A],true) ([A, b],false) ([b, b],false) -> ([B, A],true) ([A, b],false) ,
  ([b, A],true) ([A, b],false) ([b, a],false) -> ([a, B],true) ([B, A],false) ([A, a],false) ,
  ([b, A],true) ([A, b],false) ([b, a],false) -> ([B, A],true) ([A, a],false) ,
  ([a, B],true) ([B, a],false) ([a, b],false) -> ([b, A],true) ([A, B],false) ([B, b],false) ,
  ([a, B],true) ([B, a],false) ([a, b],false) -> ([A, B],true) ([B, b],false) ,
  ([a, B],true) ([B, a],false) ([a, a],false) -> ([b, A],true) ([A, B],false) ([B, a],false) ,
  ([a, B],true) ([B, a],false) ([a, a],false) -> ([A, B],true) ([B, a],false) ,
  ([a, B],true) ([B, a],false) ([a, B],false) -> ([b, A],true) ([A, B],false) ([B, B],false) ,
  ([a, B],true) ([B, a],false) ([a, B],false) -> ([A, B],true) ([B, B],false) ,
  ([a, B],true) ([B, a],false) ([a, B],false) -> ([B, B],true) ,
  ([B, A],true) ([A, b],false) ([b, A],false) -> ([a, B],true) ([B, A],false) ([A, A],false) ,
  ([B, A],true) ([A, b],false) ([b, A],false) -> ([B, A],true) ([A, A],false) ,
  ([B, A],true) ([A, b],false) ([b, A],false) -> ([A, A],true) ,
  ([B, A],true) ([A, b],false) ([b, b],false) -> ([a, B],true) ([B, A],false) ([A, b],false) ,
  ([B, A],true) ([A, b],false) ([b, b],false) -> ([B, A],true) ([A, b],false) ,
  ([B, B],true) ([B, a],false) ([a, b],false) -> ([b, A],true) ([A, B],false) ([B, b],false) ,
  ([B, B],true) ([B, a],false) ([a, b],false) -> ([A, B],true) ([B, b],false) ,
  ([B, A],true) ([A, b],false) ([b, a],false) -> ([a, B],true) ([B, A],false) ([A, a],false) ,
  ([B, A],true) ([A, b],false) ([b, a],false) -> ([B, A],true) ([A, a],false) ,
  ([B, B],true) ([B, a],false) ([a, a],false) -> ([b, A],true) ([A, B],false) ([B, a],false) ,
  ([B, B],true) ([B, a],false) ([a, a],false) -> ([A, B],true) ([B, a],false) ,
  ([B, B],true) ([B, a],false) ([a, B],false) -> ([b, A],true) ([A, B],false) ([B, B],false) ,
  ([B, B],true) ([B, a],false) ([a, B],false) -> ([A, B],true) ([B, B],false) ,
  ([B, B],true) ([B, a],false) ([a, B],false) -> ([B, B],true) ,
  ([B, A],true) ([A, a],false) ([a, B],false) -> ([B, B],true) ,
  ([A, A],false) ([A, b],false) ([b, A],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([A, B],false) ([B, b],false) ([b, A],false) ->= ([A, A],false) ,
  ([A, A],false) ([A, b],false) ([b, b],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([A, B],false) ([B, a],false) ([a, b],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([A, A],false) ([A, a],false) ([a, b],false) ->= ([A, b],false) ,
  ([A, B],false) ([B, b],false) ([b, b],false) ->= ([A, b],false) ,
  ([A, A],false) ([A, b],false) ([b, a],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, a],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([A, B],false) ([B, b],false) ([b, a],false) ->= ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, B],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([b, A],false) ([A, b],false) ([b, A],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([b, A],false) ([A, b],false) ([b, b],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([b, A],false) ([A, a],false) ([a, b],false) ->= ([b, b],false) ,
  ([b, A],false) ([A, b],false) ([b, a],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([a, B],false) ([B, a],false) ([a, b],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([a, B],false) ([B, a],false) ([a, a],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([a, B],false) ([B, b],false) ([b, a],false) ->= ([a, a],false) ,
  ([a, B],false) ([B, a],false) ([a, B],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, b],false) ([b, A],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([B, A],false) ([A, b],false) ([b, b],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([B, B],false) ([B, a],false) ([a, b],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([B, A],false) ([A, a],false) ([a, b],false) ->= ([B, b],false) ,
  ([B, A],false) ([A, b],false) ([b, a],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([B, B],false) ([B, a],false) ([a, a],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([B, A],false) ([A, a],false) ([a, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, b],false) ([b, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, a],false) ([a, B],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, a],false) ([a, B],false) ->= ([B, B],false) }




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

([A, A],true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([A, b],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([b, A],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([a, B],true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([B, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, A],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([B, A],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, B],true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([B, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, b],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([B, a],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([a, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, A],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, a],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([a, B],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([B, B],false) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([B, B],true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /

Remains to prove termination of the 35-rule system
{ ([b, A],true) ([A, b],false) ([b, A],false) -> ([a, B],true) ([B, A],false) ([A, A],false) ,
  ([b, A],true) ([A, b],false) ([b, b],false) -> ([a, B],true) ([B, A],false) ([A, b],false) ,
  ([b, A],true) ([A, b],false) ([b, a],false) -> ([a, B],true) ([B, A],false) ([A, a],false) ,
  ([B, A],true) ([A, b],false) ([b, A],false) -> ([a, B],true) ([B, A],false) ([A, A],false) ,
  ([B, A],true) ([A, b],false) ([b, b],false) -> ([a, B],true) ([B, A],false) ([A, b],false) ,
  ([B, A],true) ([A, b],false) ([b, a],false) -> ([a, B],true) ([B, A],false) ([A, a],false) ,
  ([B, A],true) ([A, a],false) ([a, B],false) -> ([B, B],true) ,
  ([A, A],false) ([A, b],false) ([b, A],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([A, B],false) ([B, b],false) ([b, A],false) ->= ([A, A],false) ,
  ([A, A],false) ([A, b],false) ([b, b],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([A, B],false) ([B, a],false) ([a, b],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([A, A],false) ([A, a],false) ([a, b],false) ->= ([A, b],false) ,
  ([A, B],false) ([B, b],false) ([b, b],false) ->= ([A, b],false) ,
  ([A, A],false) ([A, b],false) ([b, a],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, a],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([A, B],false) ([B, b],false) ([b, a],false) ->= ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, B],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([b, A],false) ([A, b],false) ([b, A],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([b, A],false) ([A, b],false) ([b, b],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([b, A],false) ([A, a],false) ([a, b],false) ->= ([b, b],false) ,
  ([b, A],false) ([A, b],false) ([b, a],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([a, B],false) ([B, a],false) ([a, b],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([a, B],false) ([B, a],false) ([a, a],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([a, B],false) ([B, b],false) ([b, a],false) ->= ([a, a],false) ,
  ([a, B],false) ([B, a],false) ([a, B],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, b],false) ([b, A],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([B, A],false) ([A, b],false) ([b, b],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([B, B],false) ([B, a],false) ([a, b],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([B, A],false) ([A, a],false) ([a, b],false) ->= ([B, b],false) ,
  ([B, A],false) ([A, b],false) ([b, a],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([B, B],false) ([B, a],false) ([a, a],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([B, A],false) ([A, a],false) ([a, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, b],false) ([b, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, a],false) ([a, B],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, a],false) ([a, B],false) ->= ([B, B],false) }


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

([A, A],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, B],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, A],true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([A, B],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, A],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, B],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 31-rule system
{ ([b, A],true) ([A, b],false) ([b, A],false) -> ([a, B],true) ([B, A],false) ([A, A],false) ,
  ([b, A],true) ([A, b],false) ([b, b],false) -> ([a, B],true) ([B, A],false) ([A, b],false) ,
  ([b, A],true) ([A, b],false) ([b, a],false) -> ([a, B],true) ([B, A],false) ([A, a],false) ,
  ([A, A],false) ([A, b],false) ([b, A],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([A, B],false) ([B, b],false) ([b, A],false) ->= ([A, A],false) ,
  ([A, A],false) ([A, b],false) ([b, b],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([A, B],false) ([B, a],false) ([a, b],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([A, A],false) ([A, a],false) ([a, b],false) ->= ([A, b],false) ,
  ([A, B],false) ([B, b],false) ([b, b],false) ->= ([A, b],false) ,
  ([A, A],false) ([A, b],false) ([b, a],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, a],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([A, B],false) ([B, b],false) ([b, a],false) ->= ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, B],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([b, A],false) ([A, b],false) ([b, A],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([b, A],false) ([A, b],false) ([b, b],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([b, A],false) ([A, a],false) ([a, b],false) ->= ([b, b],false) ,
  ([b, A],false) ([A, b],false) ([b, a],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([a, B],false) ([B, a],false) ([a, b],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([a, B],false) ([B, a],false) ([a, a],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([a, B],false) ([B, b],false) ([b, a],false) ->= ([a, a],false) ,
  ([a, B],false) ([B, a],false) ([a, B],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, b],false) ([b, A],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([B, A],false) ([A, b],false) ([b, b],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([B, B],false) ([B, a],false) ([a, b],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([B, A],false) ([A, a],false) ([a, b],false) ->= ([B, b],false) ,
  ([B, A],false) ([A, b],false) ([b, a],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([B, B],false) ([B, a],false) ([a, a],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([B, A],false) ([A, a],false) ([a, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, b],false) ([b, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, a],false) ([a, B],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, a],false) ([a, B],false) ->= ([B, B],false) }


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

([A, A],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, B],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, A],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, A],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, B],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, b],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, A],true) is interpreted by
 /   \
| 1 1 |
| 0 1 |
 \   /
([A, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([b, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([A, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, a],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([a, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, B],false) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /
([B, B],true) is interpreted by
 /   \
| 1 0 |
| 0 1 |
 \   /

Remains to prove termination of the 28-rule system
{ ([A, A],false) ([A, b],false) ([b, A],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([A, B],false) ([B, b],false) ([b, A],false) ->= ([A, A],false) ,
  ([A, A],false) ([A, b],false) ([b, b],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([A, B],false) ([B, a],false) ([a, b],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([A, A],false) ([A, a],false) ([a, b],false) ->= ([A, b],false) ,
  ([A, B],false) ([B, b],false) ([b, b],false) ->= ([A, b],false) ,
  ([A, A],false) ([A, b],false) ([b, a],false) ->= ([A, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, a],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([A, B],false) ([B, b],false) ([b, a],false) ->= ([A, a],false) ,
  ([A, B],false) ([B, a],false) ([a, B],false) ->= ([A, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([b, A],false) ([A, b],false) ([b, A],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([b, A],false) ([A, b],false) ([b, b],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([b, A],false) ([A, a],false) ([a, b],false) ->= ([b, b],false) ,
  ([b, A],false) ([A, b],false) ([b, a],false) ->= ([b, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([a, B],false) ([B, a],false) ([a, b],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([a, B],false) ([B, a],false) ([a, a],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([a, B],false) ([B, b],false) ([b, a],false) ->= ([a, a],false) ,
  ([a, B],false) ([B, a],false) ([a, B],false) ->= ([a, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, b],false) ([b, A],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, A],false) ,
  ([B, A],false) ([A, b],false) ([b, b],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, b],false) ,
  ([B, B],false) ([B, a],false) ([a, b],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, b],false) ,
  ([B, A],false) ([A, a],false) ([a, b],false) ->= ([B, b],false) ,
  ([B, A],false) ([A, b],false) ([b, a],false) ->= ([B, b],false) ([b, a],false) ([a, B],false) ([B, A],false) ([A, a],false) ,
  ([B, B],false) ([B, a],false) ([a, a],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, a],false) ,
  ([B, A],false) ([A, a],false) ([a, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, b],false) ([b, a],false) ->= ([B, a],false) ,
  ([B, B],false) ([B, a],false) ([a, B],false) ->= ([B, a],false) ([a, b],false) ([b, A],false) ([A, B],false) ([B, B],false) ,
  ([B, A],false) ([A, a],false) ([a, B],false) ->= ([B, B],false) }


The system is trivially terminating.