Term Rewriting System R:
[X]
c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

C(b(a(X))) -> A(a(b(b(c(c(X))))))
C(b(a(X))) -> A(b(b(c(c(X)))))
C(b(a(X))) -> B(b(c(c(X))))
C(b(a(X))) -> B(c(c(X)))
C(b(a(X))) -> C(c(X))
C(b(a(X))) -> C(X)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

C(b(a(X))) -> C(X)
C(b(a(X))) -> C(c(X))


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(X))) -> C(c(X))
two new Dependency Pairs are created:

C(b(a(b(a(X''))))) -> C(a(a(b(b(c(c(X'')))))))
C(b(a(X''))) -> C(e)

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

C(b(a(b(a(X''))))) -> C(a(a(b(b(c(c(X'')))))))
C(b(a(X))) -> C(X)


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(X''))))) -> C(a(a(b(b(c(c(X'')))))))
seven new Dependency Pairs are created:

C(b(a(b(a(X'''))))) -> C(e)
C(b(a(b(a(X'''))))) -> C(a(e))
C(b(a(b(a(X'''))))) -> C(a(a(e)))
C(b(a(b(a(X'''))))) -> C(a(a(b(e))))
C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))
C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))
C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))
C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))
C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))
C(b(a(b(a(X'''))))) -> C(a(a(b(e))))
C(b(a(b(a(X'''))))) -> C(a(a(e)))
C(b(a(b(a(X'''))))) -> C(a(e))
C(b(a(X))) -> C(X)


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(X'''))))) -> C(a(e))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))
C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))
C(b(a(b(a(X'''))))) -> C(a(a(b(e))))
C(b(a(b(a(X'''))))) -> C(a(a(e)))
C(b(a(X))) -> C(X)
C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(X'''))))) -> C(a(a(e)))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))
C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))
C(b(a(b(a(X'''))))) -> C(a(a(b(e))))
C(b(a(X))) -> C(X)
C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(X'''))))) -> C(a(a(b(e))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))
C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))
C(b(a(X))) -> C(X)
C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(X'''))))) -> C(a(a(b(b(e)))))
no new Dependency Pairs are created.
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:

C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))
C(b(a(X))) -> C(X)
C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(b(a(X'))))))) -> C(a(a(b(b(c(a(a(b(b(c(c(X'))))))))))))
no new Dependency Pairs are created.
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:

C(b(a(X))) -> C(X)
C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(b(a(X'''))))) -> C(a(a(b(b(c(e))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Forward Instantiation Transformation


Dependency Pair:

C(b(a(X))) -> C(X)


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





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

C(b(a(X))) -> C(X)
one new Dependency Pair is created:

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 10
                 ↳Polynomial Ordering


Dependency Pair:

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


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





The following dependency pair can be strictly oriented:

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


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

b(X) -> e


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(C(x1))=  1 + x1  
  POL(e)=  0  
  POL(b(x1))=  x1  
  POL(a(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 11
                 ↳Dependency Graph


Dependency Pair:


Rules:


c(b(a(X))) -> a(a(b(b(c(c(X))))))
c(X) -> e
a(X) -> e
b(X) -> e





Using the Dependency Graph resulted in no new DP problems.

Termination of R successfully shown.
Duration:
0:01 minutes