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`
`         ↳Forward Instantiation 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 Forward Instantiation SCC transformation can be performed.
As a result of transforming the rule

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

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polynomial Ordering`

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

The following dependency pair can be strictly oriented:

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

There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳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:
0:00 minutes