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