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

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)))





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)))
two new Dependency Pairs are created:

B(b(a(a(x'')))) -> A(a(b(b(x''))))
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
             ↳Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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'')))))
two new Dependency Pairs are created:

B(b(a(b(a(a(x')))))) -> A(b(a(a(b(b(x'))))))
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 3
                 ↳Polynomial Ordering


Dependency Pairs:

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


Rules:


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





The following dependency pairs can be strictly oriented:

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


Additionally, the following usable rules 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 4
                 ↳Dependency Graph


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.

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