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

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(x), s(0), y) -> F(y, y, g(x))
G(s(x)) -> G(x)

Furthermore, R contains two SCCs. 


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


Dependency Pair:

G(s(x)) -> G(x)


Rules:


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





The following dependency pair can be strictly oriented:

G(s(x)) -> G(x)


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(G(x1))=  x1  
  POL(s(x1))=  1 + x1  

resulting in one new DP problem. 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
           →DP Problem 3
             ↳Dependency Graph
       →DP Problem 2
         ↳FwdInst


Dependency Pair:


Rules:


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





Using the Dependency Graph resulted in no new DP problems.


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


Dependency Pair:

F(g(x), s(0), y) -> F(y, y, g(x))


Rules:


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





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

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

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

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Polo
       →DP Problem 2
         ↳FwdInst
           →DP Problem 4
             ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

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


Rules:


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




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