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

Termination of R to be shown.



   R
Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains two SCCs.


   R
DPs
       →DP Problem 1
Polynomial Ordering
       →DP Problem 2
Nar


Dependency Pairs:

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


Rules:


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





The following dependency pairs can be strictly oriented:

F(a, f(c, x)) -> F(a, x)
F(a, f(b, x)) -> F(a, 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)=  0  
  POL(b)=  0  
  POL(a)=  0  
  POL(F(x1, x2))=  x2  
  POL(f(x1, x2))=  1 + x2  

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
           →DP Problem 3
Dependency Graph
       →DP Problem 2
Nar


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



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


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, x'')) -> F(d, f(b, f(c, x'')))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 5
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(b, x''))) -> F(d, f(b, f(a, x'')))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 6
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 7
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(b, x'''))) -> F(d, f(b, f(b, f(c, x'''))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 8
Polynomial Ordering


Dependency Pairs:

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


Rules:


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





The following dependency pairs can be strictly oriented:

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


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

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


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

resulting in one new DP problem.



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 9
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, x'''))) -> F(d, f(c, f(b, f(c, x'''))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 10
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, x')))) -> F(d, f(c, f(b, f(a, x'))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 11
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

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

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 12
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, x'')))) -> F(d, f(c, f(b, f(b, f(c, x'')))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 13
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(b, x''))))) -> F(d, f(c, f(b, f(b, f(a, x'')))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 14
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(c, x''))))) -> F(d, f(c, f(b, f(c, f(a, x'')))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 15
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(c, f(b, x''))))) -> F(d, f(c, f(c, f(b, f(a, x'')))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 16
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(b, x'''))))) -> F(d, f(c, f(b, f(b, f(b, f(c, x'''))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 17
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(b, f(b, x')))))) -> F(d, f(c, f(b, f(b, f(b, f(a, x'))))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 18
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(b, f(c, x')))))) -> F(d, f(c, f(b, f(b, f(c, f(a, x'))))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 19
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(c, x'''))))) -> F(d, f(c, f(b, f(c, f(b, f(c, x'''))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 20
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(c, f(b, x')))))) -> F(d, f(c, f(b, f(c, f(b, f(a, x'))))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 21
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(c, f(c, x')))))) -> F(d, f(c, f(b, f(c, f(c, f(a, x'))))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 22
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(c, f(b, f(b, x')))))) -> F(d, f(c, f(c, f(b, f(b, f(a, x'))))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 23
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(c, f(b, f(c, x')))))) -> F(d, f(c, f(c, f(b, f(c, f(a, x'))))))
three new Dependency Pairs are created:

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

The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 24
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(b, f(b, x'')))))) -> F(d, f(c, f(b, f(b, f(b, f(b, f(c, x'')))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 25
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(b, f(c, x'')))))) -> F(d, f(c, f(b, f(b, f(c, f(b, f(c, x'')))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 26
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(c, f(b, x'')))))) -> F(d, f(c, f(b, f(c, f(b, f(b, f(c, x'')))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 27
Narrowing Transformation


Dependency Pairs:

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


Rules:


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





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

F(d, f(c, f(c, f(b, f(c, f(c, x'')))))) -> F(d, f(c, f(b, f(c, f(c, f(b, f(c, x'')))))))
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem:



   R
DPs
       →DP Problem 1
Polo
       →DP Problem 2
Nar
           →DP Problem 4
Nar
             ...
               →DP Problem 28
Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rules:


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




Termination of R could not be shown.
Duration:
0:09 minutes