Term Rewriting System R:
[y, x]
f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

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

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Forward Instantiation Transformation


Dependency Pairs:

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


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)





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

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



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


Dependency Pair:

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


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)





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

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

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

The transformation is resulting in one new DP problem: 



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


Dependency Pair:

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


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 4
                 ↳Polynomial Ordering


Dependency Pair:

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


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)





The following dependency pair can be strictly oriented:

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


There are no usable rules w.r.t. to the implicit AFS that need to be oriented.

Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(s(x1))=  1 + x1  
  POL(F(x1, x2))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳FwdInst
           →DP Problem 2
             ↳FwdInst
             ...
               →DP Problem 5
                 ↳Dependency Graph


Dependency Pair:


Rules:


f(0, y) -> 0
f(s(x), y) -> f(f(x, y), y)





Using the Dependency Graph resulted in no new DP problems.

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