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
Argument Filtering and Ordering


Dependency Pair:

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


The following rules can be oriented:

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


Used ordering: Polynomial ordering with Polynomial interpretation:
  POL(0)=  0  
  POL(s(x1))=  1 + x1  
  POL(F(x1, x2))=  x1 + x2  

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


   R
DPs
       →DP Problem 1
FwdInst
           →DP Problem 2
AFS
             ...
               →DP Problem 3
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