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
                 ↳Argument Filtering and Ordering


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))





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the AFS that need to be oriented.
Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(F(x1, x2))=  1 + x1 + x2  

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2, x3) -> F(x1, x2)
s(x1) -> s(x1)


   R
     ↳DPs
       →DP Problem 1
         ↳Inst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Dependency Graph


Dependency Pair:


Rule:


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





Using the Dependency Graph resulted in no new DP problems.

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