Term Rewriting System R:
[x]
a(f, a(f, a(g, a(g, x)))) -> a(g, a(g, a(g, a(f, a(f, a(f, x))))))

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

A(f, a(f, a(g, a(g, x)))) -> A(g, a(g, a(g, a(f, a(f, a(f, x))))))
A(f, a(f, a(g, a(g, x)))) -> A(g, a(g, a(f, a(f, a(f, x)))))
A(f, a(f, a(g, a(g, x)))) -> A(g, a(f, a(f, a(f, x))))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, a(f, x)))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) -> A(f, x)

Furthermore, R contains one SCC.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`

Dependency Pairs:

A(f, a(f, a(g, a(g, x)))) -> A(f, x)
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, a(f, x)))

Rule:

a(f, a(f, a(g, a(g, x)))) -> a(g, a(g, a(g, a(f, a(f, a(f, x))))))

Strategy:

innermost

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

A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, a(f, x)))
two new Dependency Pairs are created:

A(f, a(f, a(g, a(g, a(g, a(g, x'')))))) -> A(f, a(g, a(g, a(g, a(f, a(f, a(f, x'')))))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, a(g, a(f, a(f, a(f, x''))))))))

The transformation is resulting in one new DP problem:

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

Dependency Pairs:

A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, a(g, a(f, a(f, a(f, x''))))))))
A(f, a(f, a(g, a(g, a(g, a(g, x'')))))) -> A(f, a(g, a(g, a(g, a(f, a(f, a(f, x'')))))))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, x)))) -> A(f, x)

Rule:

a(f, a(f, a(g, a(g, x)))) -> a(g, a(g, a(g, a(f, a(f, a(f, x))))))

Strategy:

innermost

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

A(f, a(f, a(g, a(g, x)))) -> A(f, x)
three new Dependency Pairs are created:

A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, x''))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, a(g, a(g, x''''))))))))) -> A(f, a(f, a(g, a(g, a(g, a(g, x''''))))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, a(f, a(g, a(g, x'''')))))))))) -> A(f, a(f, a(g, a(g, a(f, a(g, a(g, x'''')))))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 3`
`                 ↳Polynomial Ordering`

Dependency Pairs:

A(f, a(f, a(g, a(g, a(f, a(g, a(g, a(f, a(g, a(g, x'''')))))))))) -> A(f, a(f, a(g, a(g, a(f, a(g, a(g, x'''')))))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, a(g, a(g, x''''))))))))) -> A(f, a(f, a(g, a(g, a(g, a(g, x''''))))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, x''))))
A(f, a(f, a(g, a(g, a(g, a(g, x'')))))) -> A(f, a(g, a(g, a(g, a(f, a(f, a(f, x'')))))))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, a(g, a(f, a(f, a(f, x''))))))))

Rule:

a(f, a(f, a(g, a(g, x)))) -> a(g, a(g, a(g, a(f, a(f, a(f, x))))))

Strategy:

innermost

The following dependency pair can be strictly oriented:

A(f, a(f, a(g, a(g, a(g, a(g, x'')))))) -> A(f, a(g, a(g, a(g, a(f, a(f, a(f, x'')))))))

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

a(f, a(f, a(g, a(g, x)))) -> a(g, a(g, a(g, a(f, a(f, a(f, x))))))

Used ordering: Polynomial ordering with Polynomial interpretation:
 POL(g) =  0 POL(a(x1, x2)) =  x1 POL(f) =  1 POL(A(x1, x2)) =  x2

resulting in one new DP problem.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 2`
`             ↳FwdInst`
`             ...`
`               →DP Problem 4`
`                 ↳Remaining Obligation(s)`

The following remains to be proven:
Dependency Pairs:

A(f, a(f, a(g, a(g, a(f, a(g, a(g, a(f, a(g, a(g, x'''')))))))))) -> A(f, a(f, a(g, a(g, a(f, a(g, a(g, x'''')))))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, a(g, a(g, x''''))))))))) -> A(f, a(f, a(g, a(g, a(g, a(g, x''''))))))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, x''))))
A(f, a(f, a(g, a(g, x)))) -> A(f, a(f, x))
A(f, a(f, a(g, a(g, a(f, a(g, a(g, x''))))))) -> A(f, a(f, a(g, a(g, a(g, a(f, a(f, a(f, x''))))))))

Rule:

a(f, a(f, a(g, a(g, x)))) -> a(g, a(g, a(g, a(f, a(f, a(f, x))))))

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:00 minutes