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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rule:


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





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

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

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

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 Pair:

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


Rule:


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





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

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

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

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


Rule:


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





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

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

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

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


Rule:


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





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

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

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

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 8
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Nar
           →DP Problem 2
             ↳Nar
             ...
               →DP Problem 9
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rule:


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




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