Term Rewriting System R:
[x]
a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

A(a(x)) -> B(b(x))
A(a(x)) -> B(x)
B(b(a(x))) -> A(b(b(x)))
B(b(a(x))) -> B(b(x))
B(b(a(x))) -> B(x)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

B(b(a(x))) -> B(x)
B(b(a(x))) -> B(b(x))
A(a(x)) -> B(x)
B(b(a(x))) -> A(b(b(x)))
A(a(x)) -> B(b(x))


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A(a(x)) -> B(b(x))
one new Dependency Pair is created:

A(a(b(a(x'')))) -> B(a(b(b(x''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Narrowing Transformation


Dependency Pairs:

A(a(b(a(x'')))) -> B(a(b(b(x''))))
B(b(a(x))) -> B(b(x))
A(a(x)) -> B(x)
B(b(a(x))) -> A(b(b(x)))
B(b(a(x))) -> B(x)


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

B(b(a(x))) -> A(b(b(x)))
one new Dependency Pair is created:

B(b(a(b(a(x''))))) -> A(b(a(b(b(x'')))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 3
                 ↳Narrowing Transformation


Dependency Pairs:

A(a(x)) -> B(x)
B(b(a(b(a(x''))))) -> A(b(a(b(b(x'')))))
B(b(a(x))) -> B(x)
B(b(a(x))) -> B(b(x))
A(a(b(a(x'')))) -> B(a(b(b(x''))))


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

B(b(a(x))) -> B(b(x))
one new Dependency Pair is created:

B(b(a(b(a(x''))))) -> B(a(b(b(x''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 4
                 ↳Narrowing Transformation


Dependency Pairs:

B(b(a(b(a(x''))))) -> B(a(b(b(x''))))
A(a(b(a(x'')))) -> B(a(b(b(x''))))
B(b(a(b(a(x''))))) -> A(b(a(b(b(x'')))))
B(b(a(x))) -> B(x)
A(a(x)) -> B(x)


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

A(a(b(a(x'')))) -> B(a(b(b(x''))))
one new Dependency Pair is created:

A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 5
                 ↳Narrowing Transformation


Dependency Pairs:

A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))
A(a(x)) -> B(x)
B(b(a(b(a(x''))))) -> A(b(a(b(b(x'')))))
B(b(a(x))) -> B(x)
B(b(a(b(a(x''))))) -> B(a(b(b(x''))))


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

B(b(a(b(a(x''))))) -> A(b(a(b(b(x'')))))
one new Dependency Pair is created:

B(b(a(b(a(b(a(x'))))))) -> A(b(a(b(a(b(b(x')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 6
                 ↳Narrowing Transformation


Dependency Pairs:

A(a(x)) -> B(x)
B(b(a(b(a(b(a(x'))))))) -> A(b(a(b(a(b(b(x')))))))
B(b(a(b(a(x''))))) -> B(a(b(b(x''))))
B(b(a(x))) -> B(x)
A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Narrowing SCC transformation can be performed.
As a result of transforming the rule

B(b(a(b(a(x''))))) -> B(a(b(b(x''))))
one new Dependency Pair is created:

B(b(a(b(a(b(a(x'))))))) -> B(a(b(a(b(b(x'))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 7
                 ↳Forward Instantiation Transformation


Dependency Pairs:

B(b(a(b(a(b(a(x'))))))) -> B(a(b(a(b(b(x'))))))
A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))
B(b(a(b(a(b(a(x'))))))) -> A(b(a(b(a(b(b(x')))))))
B(b(a(x))) -> B(x)
A(a(x)) -> B(x)


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

A(a(x)) -> B(x)
two new Dependency Pairs are created:

A(a(b(a(x'')))) -> B(b(a(x'')))
A(a(b(a(b(a(b(a(x''')))))))) -> B(b(a(b(a(b(a(x''')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Forward Instantiation Transformation


Dependency Pairs:

A(a(b(a(b(a(b(a(x''')))))))) -> B(b(a(b(a(b(a(x''')))))))
A(a(b(a(x'')))) -> B(b(a(x'')))
A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))
B(b(a(b(a(b(a(x'))))))) -> A(b(a(b(a(b(b(x')))))))
B(b(a(x))) -> B(x)
B(b(a(b(a(b(a(x'))))))) -> B(a(b(a(b(b(x'))))))


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




On this DP problem, a Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

B(b(a(x))) -> B(x)
two new Dependency Pairs are created:

B(b(a(b(a(x''))))) -> B(b(a(x'')))
B(b(a(b(a(b(a(b(a(x'''))))))))) -> B(b(a(b(a(b(a(x''')))))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Polynomial Ordering


Dependency Pairs:

A(a(b(a(x'')))) -> B(b(a(x'')))
B(b(a(b(a(b(a(b(a(x'''))))))))) -> B(b(a(b(a(b(a(x''')))))))
B(b(a(b(a(x''))))) -> B(b(a(x'')))
B(b(a(b(a(b(a(x'))))))) -> B(a(b(a(b(b(x'))))))
A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))
B(b(a(b(a(b(a(x'))))))) -> A(b(a(b(a(b(b(x')))))))
A(a(b(a(b(a(b(a(x''')))))))) -> B(b(a(b(a(b(a(x''')))))))


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




The following dependency pairs can be strictly oriented:

A(a(b(a(x'')))) -> B(b(a(x'')))
B(b(a(b(a(b(a(b(a(x'''))))))))) -> B(b(a(b(a(b(a(x''')))))))
B(b(a(b(a(x''))))) -> B(b(a(x'')))
B(b(a(b(a(b(a(x'))))))) -> B(a(b(a(b(b(x'))))))
A(a(b(a(b(a(x')))))) -> B(a(b(a(b(b(x'))))))
B(b(a(b(a(b(a(x'))))))) -> A(b(a(b(a(b(b(x')))))))
A(a(b(a(b(a(b(a(x''')))))))) -> B(b(a(b(a(b(a(x''')))))))


Additionally, the following usable rules for innermost w.r.t. to the implicit AFS can be oriented:

b(b(a(x))) -> a(b(b(x)))
a(a(x)) -> b(b(x))


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(B(x1))=  x1  
  POL(b(x1))=  x1  
  POL(a(x1))=  1 + x1  
  POL(A(x1))=  x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Dependency Graph


Dependency Pair:


Rules:


a(a(x)) -> b(b(x))
b(b(a(x))) -> a(b(b(x)))


Strategy:

innermost




Using the Dependency Graph resulted in no new DP problems.

Innermost Termination of R successfully shown.
Duration:
0:00 minutes