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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pair:

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rule:


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





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

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

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


Rule:


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





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

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

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


Rule:


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





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

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

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


Rule:


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





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

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

F(a, f(f(a, f(f(a, f(f(a, f(f(a, x'''), a)), a)), a)), a)) -> F(a, f(f(f(a, f(a, f(f(a, f(a, x''')), a))), a), a))
F(a, f(f(a, f(f(a, f(f(a, f(f(a, x'''), a)), a)), a)), a)) -> F(a, f(f(a, f(f(f(a, f(a, f(a, x'''))), a), a)), a))
F(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, x'), a)), a)), a)), a)), a)) -> F(a, f(f(a, f(f(a, f(f(a, f(f(a, f(a, 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 6
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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





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

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

F(a, f(f(a, f(f(a, f(f(a, f(f(a, x''), a)), a)), a)), a)) -> F(a, f(f(f(a, f(f(a, f(a, f(a, x''))), a)), a), a))
F(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, x''), a)), a)), a)), a)), a)) -> F(a, f(f(f(a, f(a, f(f(a, f(f(a, f(a, 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 7
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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





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

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

F(a, f(f(a, f(f(a, f(f(a, f(f(a, x''''), a)), a)), a)), a)) -> F(a, f(f(f(a, f(f(a, f(a, f(a, x''''))), a)), a), a))
F(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, x'), a)), a)), a)), a)), a)) -> F(a, f(f(f(a, f(a, f(f(a, f(f(a, f(a, 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 8
                 ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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





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

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

F(a, f(f(a, f(f(a, f(f(a, f(f(a, f(f(a, x'), a)), a)), a)), a)), a)) -> F(a, f(f(a, f(f(f(a, f(a, f(f(a, f(a, 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 9
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pairs:

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


Rule:


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




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