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

Innermost Termination of R to be shown.

`   R`
`     ↳Dependency Pair Analysis`

R contains the following Dependency Pairs:

F(a, f(b, x)) -> F(a, f(a, f(a, x)))
F(a, f(b, x)) -> F(a, f(a, x))
F(a, f(b, x)) -> F(a, x)
F(b, f(a, x)) -> F(b, f(b, f(b, x)))
F(b, f(a, x)) -> F(b, f(b, x))
F(b, f(a, x)) -> F(b, x)

Furthermore, R contains two SCCs.

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Narrowing Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pairs:

F(a, f(b, x)) -> F(a, x)
F(a, f(b, x)) -> F(a, f(a, x))
F(a, f(b, x)) -> F(a, f(a, f(a, x)))

Rules:

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

Strategy:

innermost

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

F(a, f(b, x)) -> F(a, f(a, f(a, x)))
one new Dependency Pair is created:

F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, f(a, x'')))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Narrowing Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pairs:

F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, f(a, x'')))))
F(a, f(b, x)) -> F(a, f(a, x))
F(a, f(b, x)) -> F(a, x)

Rules:

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

Strategy:

innermost

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

F(a, f(b, x)) -> F(a, f(a, x))
one new Dependency Pair is created:

F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, x''))))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Nar`
`             ...`
`               →DP Problem 4`
`                 ↳Forward Instantiation Transformation`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pairs:

F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, x''))))
F(a, f(b, x)) -> F(a, x)
F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, f(a, x'')))))

Rules:

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

Strategy:

innermost

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

F(a, f(b, x)) -> F(a, x)
two new Dependency Pairs are created:

F(a, f(b, f(b, x''))) -> F(a, f(b, x''))
F(a, f(b, f(b, f(b, x'''')))) -> F(a, f(b, f(b, x'''')))

The transformation is resulting in one new DP problem:

`   R`
`     ↳DPs`
`       →DP Problem 1`
`         ↳Nar`
`           →DP Problem 3`
`             ↳Nar`
`             ...`
`               →DP Problem 5`
`                 ↳Polynomial Ordering`
`       →DP Problem 2`
`         ↳Remaining`

Dependency Pairs:

F(a, f(b, f(b, f(b, x'''')))) -> F(a, f(b, f(b, x'''')))
F(a, f(b, f(b, x''))) -> F(a, f(b, x''))
F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, f(a, x'')))))
F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, x''))))

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, f(a, x'')))))
F(a, f(b, f(b, x''))) -> F(a, f(a, f(a, f(a, x''))))

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

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

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

resulting in one new DP problem.

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

The following remains to be proven:
• Dependency Pairs:

F(a, f(b, f(b, f(b, x'''')))) -> F(a, f(b, f(b, x'''')))
F(a, f(b, f(b, x''))) -> F(a, f(b, x''))

Rules:

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

Strategy:

innermost

• Dependency Pairs:

F(b, f(a, x)) -> F(b, x)
F(b, f(a, x)) -> F(b, f(b, x))
F(b, f(a, x)) -> F(b, f(b, f(b, x)))

Rules:

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

Strategy:

innermost

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

The following remains to be proven:
• Dependency Pairs:

F(a, f(b, f(b, f(b, x'''')))) -> F(a, f(b, f(b, x'''')))
F(a, f(b, f(b, x''))) -> F(a, f(b, x''))

Rules:

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

Strategy:

innermost

• Dependency Pairs:

F(b, f(a, x)) -> F(b, x)
F(b, f(a, x)) -> F(b, f(b, x))
F(b, f(a, x)) -> F(b, f(b, f(b, x)))

Rules:

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

Strategy:

innermost

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