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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Narrowing Transformation


Dependency Pairs:

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


Rule:


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





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pairs:

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


Rule:


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





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

F(f(a, a), a) -> F(f(f(f(a, f(a, a)), f(a, a)), a), a)
no new Dependency Pairs are created.
The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rule:


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





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

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

F(f(a, a), a) -> F(f(a, f(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 Pair:

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


Rule:


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





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

F(f(a, a), a) -> F(f(a, f(a, a)), a)
no new Dependency Pairs are created.
The transformation is resulting in no new DP problems.


Termination of R successfully shown.
Duration:
0:00 minutes