Term Rewriting System R:
[X]
f(X, g(X)) -> f(1, g(X))
g(1) -> g(0)

Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(X, g(X)) -> F(1, g(X))
G(1) -> G(0)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Modular Removal of Rules


Dependency Pair:

F(X, g(X)) -> F(1, g(X))


Rules:


f(X, g(X)) -> f(1, g(X))
g(1) -> g(0)





We have the following set of usable rules:

g(1) -> g(0)
To remove rules and DPs from this DP problem we used the following monotonic and CE-compatible order: Polynomial ordering.
Polynomial interpretation:
  POL(0)=  0  
  POL(g(x1))=  x1  
  POL(1)=  0  
  POL(F(x1, x2))=  x1 + x2  

We have the following set D of usable symbols: {0, g, 1, F}
No Dependency Pairs can be deleted.
1 non usable rules have been deleted.

The result of this processor delivers one new DP problem.



   R
     ↳DPs
       →DP Problem 1
         ↳MRR
           →DP Problem 2
             ↳Non Termination


Dependency Pair:

F(X, g(X)) -> F(1, g(X))


Rule:


g(1) -> g(0)





Found an infinite P-chain over R:
P =

F(X, g(X)) -> F(1, g(X))

R =

g(1) -> g(0)

s = F(1, g(1))
evaluates to t =F(1, g(1))

Thus, s starts an infinite chain.

Non-Termination of R could be shown.
Duration:
0:00 minutes