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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(s(x), y, y) -> F(y, x, s(x))

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Instantiation Transformation


Dependency Pair:

F(s(x), y, y) -> F(y, x, s(x))


Rule:


f(s(x), y, y) -> f(y, x, s(x))





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

F(s(x), y, y) -> F(y, x, s(x))
one new Dependency Pair is created:

F(s(x0), s(x'''), s(x''')) -> F(s(x'''), x0, s(x0))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳Forward Instantiation Transformation


Dependency Pair:

F(s(x0), s(x'''), s(x''')) -> F(s(x'''), x0, s(x0))


Rule:


f(s(x), y, y) -> f(y, x, s(x))





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

F(s(x0), s(x'''), s(x''')) -> F(s(x'''), x0, s(x0))
one new Dependency Pair is created:

F(s(s(x'''''')), s(x'''0), s(x'''0)) -> F(s(x'''0), s(x''''''), s(s(x'''''')))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 3
                 ↳Instantiation Transformation


Dependency Pair:

F(s(s(x'''''')), s(x'''0), s(x'''0)) -> F(s(x'''0), s(x''''''), s(s(x'''''')))


Rule:


f(s(x), y, y) -> f(y, x, s(x))





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

F(s(s(x'''''')), s(x'''0), s(x'''0)) -> F(s(x'''0), s(x''''''), s(s(x'''''')))
one new Dependency Pair is created:

F(s(s(x''''''0)), s(s(x''''''''')), s(s(x'''''''''))) -> F(s(s(x''''''''')), s(x''''''0), s(s(x''''''0)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Forward Instantiation Transformation


Dependency Pair:

F(s(s(x''''''0)), s(s(x''''''''')), s(s(x'''''''''))) -> F(s(s(x''''''''')), s(x''''''0), s(s(x''''''0)))


Rule:


f(s(x), y, y) -> f(y, x, s(x))





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

F(s(s(x''''''0)), s(s(x''''''''')), s(s(x'''''''''))) -> F(s(s(x''''''''')), s(x''''''0), s(s(x''''''0)))
one new Dependency Pair is created:

F(s(s(s(x''''''''''''))), s(s(x'''''''''0)), s(s(x'''''''''0))) -> F(s(s(x'''''''''0)), s(s(x'''''''''''')), s(s(s(x''''''''''''))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Instantiation Transformation


Dependency Pair:

F(s(s(s(x''''''''''''))), s(s(x'''''''''0)), s(s(x'''''''''0))) -> F(s(s(x'''''''''0)), s(s(x'''''''''''')), s(s(s(x''''''''''''))))


Rule:


f(s(x), y, y) -> f(y, x, s(x))





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

F(s(s(s(x''''''''''''))), s(s(x'''''''''0)), s(s(x'''''''''0))) -> F(s(s(x'''''''''0)), s(s(x'''''''''''')), s(s(s(x''''''''''''))))
one new Dependency Pair is created:

F(s(s(s(x''''''''''''0))), s(s(s(x'''''''''''''''))), s(s(s(x''''''''''''''')))) -> F(s(s(s(x'''''''''''''''))), s(s(x''''''''''''0)), s(s(s(x''''''''''''0))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 6
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(s(s(s(x''''''''''''0))), s(s(s(x'''''''''''''''))), s(s(s(x''''''''''''''')))) -> F(s(s(s(x'''''''''''''''))), s(s(x''''''''''''0)), s(s(s(x''''''''''''0))))


Rule:


f(s(x), y, y) -> f(y, x, s(x))




Termination of R could not be shown.
Duration:
0:00 minutes