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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Forward Instantiation Transformation`

Dependency Pair:

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

Rule:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

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

Dependency Pair:

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

Rule:

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

Strategy:

innermost

The following dependency pair can be strictly oriented:

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

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

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

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

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳FwdInst`
`           →DP Problem 2`
`             ↳Polo`
`             ...`
`               →DP Problem 3`
`                 ↳Dependency Graph`

Dependency Pair:

Rule:

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

Strategy:

innermost

Using the Dependency Graph resulted in no new DP problems.

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