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

Innermost Termination of R to be shown.

R
Dependency Pair Analysis

R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC.

R
DPs
→DP Problem 1
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 3
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 4
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 5
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 6
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

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:

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

Rules:

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

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:

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

Rules:

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

Strategy:

innermost

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

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

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

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 Pairs:

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

Rules:

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

Strategy:

innermost

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

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

F(c, f(a, f(a, x''))) -> F(c, f(a, x''))
F(c, f(a, f(a, f(a, f(a, x'''))))) -> F(c, f(a, f(a, f(a, 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 Pairs:

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

Rules:

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

Strategy:

innermost

The following dependency pairs can be strictly oriented:

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

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

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

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

resulting in one new DP problem.

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 11
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 12
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 13
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 14
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 15
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 16
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 17
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 18
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 19
Narrowing Transformation

Dependency Pairs:

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

Rules:

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

Strategy:

innermost

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

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

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

The transformation is resulting in one new DP problem:

R
DPs
→DP Problem 1
Nar
→DP Problem 2
Nar
...
→DP Problem 20
Remaining Obligation(s)

The following remains to be proven:
Dependency Pairs:

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

Rules:

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

Strategy:

innermost

Innermost Termination of R could not be shown.
Duration:
0:01 minutes